Decision analysis - Swinburne · 2016. 11. 21. · Decision Analysis E. Shayan Decision analysis is a philosophy and a methodology for answering questions. It provides a logical framework

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DECISION ANALYSIS

Monetary Decisions Decisions under uncertainty Multi Criteria Decisions Mathematical Models in Decision Making

MM481 Decision Analysis,

School of Mechanical and Manufacturing Engineering

\ R S

Ebrahim Shayan

Industrial Engineering I Management Group

IRIS

Swinburne University of Technology

1996

Author's notes:

The set of material intends to cover the syllabus of the subject mainly with the view of assisting students, as there is no available text book covering all these issues. It will be complemented by additional hand outs which were not included due to lack of time. Some particular pieces missing are the table of contents, the match of some of the references to the text, interest tables and probability distributions.

Students may find errors of different kinds in the text. It is expected and very much

appreciated to receive their comments on errors and any other suggestion towards extending and improving the quality of the material presented for the next round.

E. Shayan

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«;z.S¢..0\{:, Decision Analysis

Chapter 1

E. Shayan

Single Criterion Decisions, on monetary value

Decision can be defined as "An irrevocable allocation of resources". At its simplest terms, the resource may be time but usually capital, opportunities and alike are involved. Due to the scarcity of the resources, it pays to make 11good11 decisions although it does not guarantee the 11 good 11 outcomes due to factors which are out of our control.

1.1 Decision making process:

Everyone is concerned with making decisions of one sort or another. Some decisions have greater dimensions and impact than others, forcing us to be more careful due to the fear of repercussions involved. However any decision made goes through some

steps.

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*

*

*

*

*

*

*

*

*

Recognition of the problem Understanding the problem Observation, data collection consultation with experts Formal definition of the problem - criteria Formulation, model building, experimentation Generating solutions (alternatives), meeting the basic criteria Selecting the most suitable alternative Pilot run - testing Implementation and Training Follow-up, Feed back, modifications

Recognition and understanding of the problem:

We are used to start looking at a problem which is either well documented or has been indicated to us by others. In reality this is not the way it happens. You would be able to only see the symptoms (if lucky) in weird forms not necessarily directly indicative of the cause. Obviously attacking the symptom is not a permanent solution as the problem will recur unless the causes are removed. This is what has to be recognised and understood.

Use of a problem analysis diagram assists in determination of the logical relationship of possible causes (usually more than one candidate)

Data collection:

To determine the actual cause, creating the problem, appropriate data must be collected. You may need to run tests, formulate and set up special systems, investigate previous errors, . . . . . Use of graphs/charts is recommended. Data should discover the deterministic and probabilistic characteristics of the problem. When the problem involves people blaming others for all the mistakes will be wrong most of the time. Make the problem the common enemy not the people who can help with the solution.

Cash Flows 1 Chapter 1

Decision Analysis E. Shayan

Formal definition of the problem:

It is constructive and to the benefit of all parties involved to come up with a written statement of the problem (Decision) at hand. Definition should include what is to be achieved, criteria to examine solutions and the restrictions applied to the environment in which the question is addressed.

Develop solutions:

Successful identification of the cause usually leads to a solution. However in cases finding solution is not so obvious. The best solution is usually the most permanent one which is initially costly but returns greater over longer terms. It is highly recommended to develop several (3) solutions which may include the existing one. In the political environments it is a protection to the decision maker and the company to have alternative solutions studied as a precaution for the failure of the initial proposal.

Selection of the best solution:

Some people call this step "The decision making". No "good" decision can be made without prior knowledge about its underlying mechanisms.

Depending on the nature of the problem, choice of the best solution can be by itself a larger problem. It is worth mentioning that a decision is made when the decision maker has to do trade-off between at least two actions which is in a sense "irrevocable allocation of resources".

Unless a real problem is discussed in a real environment this definition and idea does not ensemble its significance. Most students do not appreciate the meaning of the definition because they can not relate to the problem at hand and the difference between two different courses of action. In a company this might mean existence or non-existence in the market place. The irony is that in most companies decision makers are not even aware of how to make a "good" choice.

There are several techniques and methodologies available which will be covered in this subject to assist in making choices scientifically.

Implementation and follow up:

Putting the best choice into practice is a major development. It is necessary that people who were involved in the analysis stage and decision making continue to work in implementation stage until a satisfactory level is achieved. In the meantime a new group of people are being trained during the implementation who can take over when the initial group is relieved of this task. This ensures that the experience gained in early stages is utilised productively.

One important issue is the fact that every solution is only suitable for the conditions assumed which are subject to change over time. This mandates the constant monitoring of the condition and modification of the solution to match it. Drastic changes call for new solutions. That is why Decision making is a Process.



Decision analysis is a philosophy and a methodology for answering questions. It

provides a logical framework for decision-making based on what you know, what you can do, and what you prefer.

Most people in most circumstances, you simply decide in a particular moment based on prior experience, intuition and advice from others, etc. However, some people have a hard time making decisions in even the mundane situations and in major decisions, we all have trouble. Furthermore, even if we know with certainty what our decision is to be, we must still convince others ( eg bosses, business partners, colleagues, spouse, parents, etc. ) that we know what we are doing.

Decision making is the most important and most difficult function managers perform The difference between a good decision and a bad decision can mean the difference between success or failure of the company itself The problems faced by management can involve multiple criteria and alternative choice - complex decisions.

In such cases, intuition rarely suffices- the answer "just because I wanted to" never worked as a teenager when we confronted our parents, and surely won't work with our bosses! Thus, for most decisions we either approach the problem from a 'holistic point of view in which we simply choose the best decision, or 'somehow' break the decision down into components in order to a) better understand the problem we are faced with and/or b) communicate with someone else as to why a particular course of action was chosen.

One might argue the whole process of decision making is so unstructured and so distorted that it is no use trying to be precise. One is then tempted to go further and conclude that "participant satisfaction" is the main objective of decision making Were this the case, then multi-criteria decision making would be a simple matter of using mathematical terminology, to improve numbers that please people. But different sets of arbitrary numbers are likely to result, producing different decisions and we are Tight back where we started.

One set of numbers pleases a group of people who might be equally pleased with another set of numbers that contradicts the recommendations of the first set. This is mere number crunching. If a decision support theory is to be trustworthy there must be uniqueness in the representation of judgements and in the scale derived from these judgements. The examples used in the Analytical Hierarchy Process illustrate the point.

Everyday individuals and organisations are faced with making decisions. Some of these decisions are made front habit. Occasionally an important decision has to made that can have immediate or long ten-n effects on the future of the person or the organisation concerned ie in the presence of uncertainty in the surrounding environment.

If all the parameters in the decision model are known with certainty (or are assumed to fixed and known for the time of the decision), then the models are deterministic Examples of these decision models are linear programming and integer programming models which are used to determine the best course of action that will lead to maximum profit or minimum cost. These models are single criterion models. The goal programming models are also deterministic models with known parameters and



resource availabilities, but there is more than one criterion or objective to be maximised or minimised. Such models are called Multiple Objective or Multi criteria

Models.

Uncertainty makes decision making complicated, it raises both methodological and philosophical questions. How should choices be evaluated when the consequences are uncertain? When do we have enough information to make a decision? How much should we pay for information that can reduce our uncertainty ?

Decision analysis can be based on the principles of probability theory and a set of standard conceptions derived from utility theory. The principles of decision analyst have been applied to a made range of commercial and public policy problems.

Unlike the simple textbook examples, many realistic decision analyses problems are large and complex. They have many uncertain variables and complete relationships among these variables. The existing decision analysis tools are not completely addressing the complexities.

Attempts are made to highlight realistic problems or 'real life' case studies. Some of the decision analysis tools have also been presented and outcomes generated via decision making methodologies. Examples include multi-criteria decisions solved by analytical hierarchy processes (AHP).

1.2 Time value of money

The fact that a borrower of money is prepared to pay the principal amount plus an agreed amount depending on the length of use implies acceptance of time value of money. Therefore two money figures equal in magnitude but at different times are not equal in value. The extra amount paid, usually called interest, is the cost of using the money, similar to rent as cost of using a property.

In this context any engineering project is just a set of different money figures representing either expenses incurred or income generated at different times. It is important to realise the source of these figures. This is explained in Appendix I , II.

To formalise we will use a cash flow diagram which abstracts the time horizon and the transactions.

0 1 2 Now

Cash Flows

3

$9

$5 *1000

One perio

4 Chapter 1


number 1, ... , n show the end of each period respectively. It is assumed for simplicity that all transactions during a period are lumped to the end of that period unless otherwise specified.

Given that magnitude is not representing that real value, due to time effect, the natural question is how to sum up the figures in a meaningful way. To do this we should come up with conversion process between Figures F.P.A.

Interest is usually measured by interest rate ,i, the percentage of the principal taken as cost of money per period. The period is usually a year but could be month, week, ...

Compound interest:

Commonly, interest is charged on the principal, the sum of money owed including the unpaid interest.

If P is the sum of money borrowed at a given time k, then one period later the borrower has to provide P + iP = P (1 + i) to pay off the total loan that is to say Pat time k is equivalent to P(1 + i) at time k + 1. Applying the same principle successively, shows that Pat time zero is equal to P(l + i)n, n periods later.

F = P (1 + i)n

( 1 ) P = F (1 + i)-n

Notice i is given per period of the same length used in the above formulation. Other cases will be considered later.

Exercise 1.2.1

If you deposit $200 in a bank account paying 1% interest compounded per month, how much do you expect after a full year.

F = 200 * (1+.01/2 •

Future and Present Values of Annuities

An annuity has the general form of a series of equal cash flows made or received over regular and equal intervals of time. There are different classifications of annuities depending on the timing of payments and compounding periods. Only the ordinary annuity is considered here where equal payments of $A are made or received at the end of each compounding period

$A $A $A

I I I 0 1

Cash Flows

2 3

$A $A

I I n-1 n

5 Chapter 1


Other main types of annuity are annuity due, when equal payments are made at the beginning of the time periods, and deferred annuities where payment commences at some future time.

Considering the cash flow model above, every $A can be considered as a P for a future value at period n. Therefore (1) can be applied as follows:

F = A(l + i)n-1 + A(l + i)n-2 ............ +A

By multiplying this expression by (1 + i) and obtaining

F(l + i) = A(l + i)n + A(l + i)n-l ......... + A(l + i)

and subtracting (a difference equation approach) a simpler expression is obtained

F =A [(1 + i)n-1] / i

The present value of this series of equal payments can be similarly found from the expression below using a difference equation approach

P = Al(l+ i) + AI(I+ i)2 + ... + AI (1 + i )n

which reduces to

P=A[(l +i)n-111 [i (l +i)n]

(2)

(3)

The formulae (2) may be used to calculate the compounded amount of a series of regular equal payments made or received while formulae (3) may be used to estimate the present value of such a series. Notice that (3) can be directly derived from (2) using (1).

Expressions for annuities due and deferred annuities may be likewise derived.

Sinking Funds

If formula (2) is solved for A,

A = F { i I [ ( 1 + i )n - 1]} (4)

the amount of a regular instalment that would have to be made to compound at an interest rate i, to a sum, F in the future may be calculated. For large scale investments and in equipment replacement planning such sinking funds are often established to provide for expenditure at a later date ie. after n time periods.

Solving formula (3) for A giving

A= P i(l + i)n I [(I + i)n - 1] (5)



allows an estimate of the equal payments that might be withdrawn at the end of successive periods provided by the present amount P. At the end of the last period when the final amount A was withdrawn the balance in the fund would be zero. This is sometimes referred to as capital recovery from a sinking fund.

Gradual payments with growth

It is sometimes desired to make a series of regular payments at the end of successive periods that are increasing (or decreasing) in a linear fashion.

$A A+G A+2G A+ (n -1 )G

0 1 2 3 n

If a payment, made or received is adjusted by a constant amount G after the first payment, an expression for its present value may be derived.

(6)

If payments are adjusted in a compound fashion so that they are increased by a constant proportion, starting with the first payment

A(1 + g)1 A(1 + g)2 A(1 + g)3 A(1 + g)n

0 1 2 3 n

the expression for the present value

p = A(1 +g) I (1 + i) + A(1 +g)2 I (1 +i)2 + A (1 + g)n I (1 + i)n

may be expressed as

using formula (3), P = A [(1 + i *)n- 1] I i* (1 + i*)n

where i* = [(1 +g) I (1 + i)] -1

Alternatively a difference equation approach with multiplication of the equation by (1 + g)/(1 + i) may be used to derive this same result.

(7)



Solving for Interest Rates and Number of Payments

In many cases it is necessary to solve the preceding formulae for the interest or discount rate. an iterative procedure must be used because there is not algebraic method of finding an exact solution and linear interpolation is not any applicable except in occasional circumstances. As with estimating interest rates often upper and lower bounds are found by trial and error and linear interpolation used if a high degree of accuracy is not required.

Effective rate:

What is the effect of compounding faster? If an interest rate of i per year is compounded monthly at a rate of i/12 what would the difference be if any?

In general what is the relation between a rate i per year and a rate of i/m applied m times per year ?

F = P (1 + i/m)m

=P {1+[(1+i/m)m-1]} =P{1+r}

In comparison with F = P(1 + i), r = (1 + i)m-1 is the effective rate

Exercise 1.2.2: 12% compounded 6 monthly at 6% has an effect of

(1 + .1212)2 = 12.36% per year ¥

Short compounding periods

If i% per year is compounded m time per year ie. (weekly, daily, hourly at rate i/m, then Future (F) value of P, after n years compounded at rate of i/m, m times per year is

F = P (1 + i/m)m.n

Substituting i / m = 1 I k in (8) yields,

F = P [(1+1 I k)k]ni

If m is increased to a large number (theoretically infinity), then for a fixed i, k also tends to infinity. Therefore

or

Exercise 1.2.3

Lim F = P [ Lim (1 + 1 I k)k ]ni

k�oo k�oo

F = P exp (ni), n = 0,1, ... (9)

What is the value of an investment which pays $1000imonth forever, at i = 12%

Hint: Find the limiting relationship between P and A first

(8)



In any of the formulas derived above there are four parameters including i, n and two of P, A, F, g. Usually one of these factors is unknown and must be calculated.

It is possible to collect the terms for i, n in any of the relations so that there is always one of the (P, A, F, g) being expressed in terms of the other multiplied by that term. For example in F = P [(1 + i)n] the term is [(1 + i)nl For any fixed (i, n) this term is fixed. Therefore it can be tabulated. A standard notation simplifies the term.

When P is given and F is to be calculated for a given i, n, the term to multiply by P is read "F given P at i, n" which is symbolised as (F/P, i, n). Similarly if A is given and P is desired then we need (P/ A, i, n).

Care must be taken that these terms can only be applied to situations identical to the cash flow model for which the formulation was derived. For example, annual payments must start at period one and ends at period n of the respective cash flow for (3) to be applicable. Any variation would violate the conditions and must be treated with care.

Table 1.1 Summary of Discrete Compounding Interest Factors

Find Given Factor Symbol Name

p F 1/(l+it (PIP, i%, n) Single-payment present worth factor

F p (1 + i) n (F IP, i%, n) Single-payment compound amoun factor

p A [(1 + i)n -1]/ i(l+ i) (I'/ A, i%, n) Uniform series present worth factor

A p i(l +i) n / [ (1 + i) n - 1 ] (NP, i%, n) Uniform series capital recovery factor

F A [{I +i) n - 1] / i (F/A, i%, n) Uniform-n series compound amount factor

A F i / [{I +i) n - 1] (NF, i %,n) Uniform series sinking fund factor

Key: P = present worth, F = future worth, A = uniform series amount, i = interest rate per period, n= number of interest periods.

Tables in Appendix 3 are examples of tabulated factors. It is an interesting exercise to prepare a model (preferably spreadsheet) to reproduce such tables for all factors in troduced in tables 1.1 , 1.2.

Note that the formulae apply to ordinary annuities where payments are made or received at the end of each time period.

Table 1.2

Find Given

p

Cash Flows

Summary of Discrete Compounding Arithmetic Gradient .Series Interest Factors

Factor Symbol Name

(P/G,i%,n) PW of gradient sries

9 Chapter 1


G p {[(1 + i )n - 1] l i(l + i )n- n l(1 + i )n}li (G/ P,i%,n) Gradient series of PW

F G {[(1 + i )n -1] I i- n}l i (FIG, i%,n) Future worth of Gradient

G F {[(1 + i )n -1] I i- n}i (G/F,i%,n) Gradient series of future

A G 1 1i- n/[(1 + i )n -1] (NG, i %, n) Gradient to uniform

G A 11{ 11i- n/[(1 + i )n- 1]} (G IA,i %,n) Uniform to gradient

Key. P =present worth, F =future worth, G= gradient series amount, A= uniform series amount.

Exercise 1.2.4

Money was invested in I 00 shares of mutual funds 5 years ago at $20 .I 0 per share. A year later an additional 20 shares were bought at $14.50. The dividends for the first years were $400, $300 and $200 respectively. $100 was paid in dividends for the last 2 years. Since the price of the shares were steadily declining, all 120 shares held were sold at the end of the fifth year at $14 .I 7. Could one have done better by placing the money in the bank at 4.5% interest ? what return was earned on mutual fund investment?

Solution:

Dividends:

Year I $400 5th Year 2 $300 4th Year 3 $200 3rd Year 4 $100 2nd Year 5 $100 1st

Normal Gradient

0 1 2 3 4 5 + I 2 3 4 5

l l l l �

FV= 100 x $20.10 (F/P,i%, n) + 20 x $14.50 (F/P, i%, n) + [400(F/A,i%,n)IOO(F/G,i%,n)] + 100 + 120 x $14.17

Take i = 5%, Using interest tables given in Appendix gives:

FV= -2010 ( 1.276)- 290 ( 1 .216) + [400 ( 4.310 )- 100 ( 6.20 )] + 100 + 1700.40

= -13 . This means that the rate of return i is very close to five, in fact less than 5%. If the man put his money in the bank he would have received 4.5% interest and the return would have been: Total dividends: $1100 Total money after 5 years from shares: 120 x 14.17 = $1700.40 Total money = $1100 + $1700.40 = $2800.40 Money made from bank (interest rate 4.5% ) 100 X $20.10 ( 1.308 ) + 20 X $14.50 ( 1.239 ) = $2986.25

CashF1ows 10 Chapter 1

'I I


Therefore, if the money was put in the bank it would have received a further

( $2986.25-$2800.40 ) = $184.85 from the investment. ¥

Exercise 1.2.5 An investment of$15,000 followed by annual deposits of$500 value yields 15% interest over a 20 year period. A second investment of$15,000 value yield 20% for 15 years. Determine which plan will require the least uniform annual cash flow?

Solution:

Investment 1 : A=P(P/A, 15%, 20) + 500 = 15,000 * .15973 + 500 = 2896.06 Alternatively

F = $15,000 ( F/ P, 15%,20) + $500 ( F/ A, 15%, 20) = $15,000 ( 16.367) + $500 ( 102.44) = $296,727, thus

A = 296,727 ( 0.00976 )= $2896.06

Investment 2: A=P(A/P, 20%, 15) = 15,000* .21386 = $3208 Alternatively

F = $15,000 ( F/P, 20%, 15)= $15,000 ( 14.407 )= $231,105

Or A = $231105 ( 0.013 88 ) = $3208.227

From the two annuity payments calculated, the best choice would be to sell

investment 1, as it has a smaller annuity or least uniform annual cash flow. ¥

These methods are practical, easy to use and are relatively fast ( ie. short time period ) especially for those individuals looking to invest, whether- it be in share in stock market, bank or any other investment portfolio. These methods allow you to determine future worth, present worth, required interest rates, number of years for a investment, annuity payments etc. All these 'particulars' enhance the individual' chance for making a better decision based on the calculated values they are able to deduce. The examples illustrate a common problem when it comes to investments which one to choose!!! Deciding on investment types is covered later in the Section.

CASE 1.1 Dragon Company

Dragon Ltd. is currently making a product with machinery bought five years ago an new machinery that has 'just come onto the market. Data for these two pieces of equipment follows:

Cost Life Depreciation Current book value & tax basis Estimated salvage value now Estimated salvage value at end of life Cash operating costs Marginal tax rate, 40% After tax discount rate 9%

Cash Flows

Existing ·

$800,000 ( 5 yr. ago) 20 years SL/20 yr.

$600,000 $600,000

0 $ 50,000

11

Proposed $875,000

15 years SL/ 5 yr.

$ 35,000

Chapter 1


Therefore the net cash flows would be: 1. Net investment of $875,000 minus $600,000 net salvage value on the existing

equipment. 2. Annual operating savings of$50,000 minus $35,000 in cash operating costs. Th

present value ( net of taxes) is ($15,000 x 0.6 x 8.0607) = $72,546. 3. The difference between $40,000 a year straight line depreciation for 15 years

compared with depreciation over 5 years for the replacement. Old asset LS tax savings ($40,000 x 0.4 x 8.0607) = $128,971 New Asset LS tax savings ($175,000 x 0.4 x 3.8897) = $272,279 Additional savings $143,308 = ($272,279 -$129,971)

The new equipment does not have net present-value savings to the company an should not be acquired. The old machine was worth $800,000 five years ago and has a current-it book value of$600,000. Assuming that the machine is worthless todaytherefore the firm would receive nothing if it were to sell the asset. The books would be adjusted to show a loss of$600,000 and the only cash flow would be the taxes

saved owing to the loss ($240,000 = $600,000 x 0.4).

The approach outlined dealt with incremental cash flow. It would give the same result as if looking at separate cash flows. From the results shown below we determine the present value of future flows associated with each. The one with the lower present value is better.

Keep the existing equipment

Cash operating costs [ $50,000 x ( 1 -0.40)] Less: Tax shield SL depreciation ($40,000 x 0.40) Total

Multiply by (Present Value annuity factor: 15 yr., 9% =) Total present value

Buy new equipment

$ 30,000 -$ 16.000 $ 14,000

8.0607 $112,850

Present value cash operating costs [ $35,000 x ( 1 -0.40) x 8.0607 1 Less: PV of SL depreciation

$169,275 $ 272.279 -$103,004 Total present value

Purchase price - new equipment Net cash from sale - old equipment

Total present value

$875,000 ($600,000) $275,000

$171,996

The difference calculated between the two alternatives is $59,146 ie. ($171,996

$112,850). The idea is to choose the alternative with the lowest present value cost¥

An article titled, 'Cost justification of the New Ohio Edison Company's Energy Management System's discusses systematic methods for analysing alternative strategies that the Ohio Edison (OE) Company used during a feasibility study of it Energy Management System ( EMS). 'Me Ohio Edison Company believes:

"A key element in the requirements study of an EMS is determining the value of no only the present system, but also of several feasible alternatives in terms of present worth dollars. The ten-n value represents the difference between a system's benefit



(or the loss of those benefits) and its cost. When the values of various alternatives a compared over time with that of the existing system, the most cost effective point for system replacement can be accurately defined along with the best alterative.

This article is a good representation of a company's strategy for decision making o replacing equipment. It sets a good example for other companies to follow.

1.3 Cost Considerations in Replacement Decision Analysis

It has already been suggested that equipment replacement is a necessary function of any manufacturing business. The drive behind plant replacement analysis, as with any aspect of business, is the search for a greater profit. So, since we are concerned with money, the cost considerations in replacement analysis are paramount to the overall decision making process.

Listed below are some of the cost factors which might be considered initially.

1. Is the cost of keeping the present equipment too high. 2. Will the cost of changing or re-modelling it for new work be too great. 3. Will spoiled work be reduced by the greater accuracy of the new equipment. 4. Will greater output, or faster rate of production be obtained. 4. Will one new machine do the work of two or more existing machines of the same

kind. 6. Can machine operatives be substituted for skilled craftsman, thus

lowering labour costs by the change. 7. If several machines are to be replaced, can one operative tend two or more of the

new machines. 8. Will the maintenance cost of the new equipment be less than that of the old? 9. Will the new equipment save manufacturing space. 10. Will it be conducive to better work and higher output by the worker. 11. Will it smooth out the production curve. 12. Will it provide the basis for a better service to the customer. 13. Can the current product being made be later made on another machine. 14. How soon must the machine pay for itself to justify its purchase,

especially if products change ? 14. How many years of effective service may be expected from the machine ? 16. How would the costs of operating the new equipment be charged to the product 17. Are funds available for the purchase of the equipment or can the

investment be specially financed ?

When going about replacement decision analysis, we consider the current year operating with our current 'old' equipment. The decision to be made is whether to keep using the 'old' machinery next year or replace it with new equipment. As such when considering costs, we must also take into account the further deterioration o the current equipment between now and the new year. Some subjective nominal sum may be decided upon.

There are also some other additional costs which should be taken into account. These may include,


Decision Analysis

(a) The displacement of employees if the new process is introduced. (b) Any -new hazards introduced by the new equipment.

E. Shayan

(c) Loss of customer confidence if old equipment continues to produce good at decreasing quality levels.

(d) Again subjective nominal sums can be chosen, they should be added as expenses to each of the equipment choices.

Below is a check list suggesting the cost factors which may enter into next year operational comparison:

1. Direct labour saving-a. Allowable speeds or feeds of the machines may differ. b. Location of controls and indicating devices may change the operating time . C. Automatic features may provide multiple operation. d. Differences may exist in set-up time. e. Handling time may be reduced through automatic loading and unloading. f. The effect of the machines on the labour requirements of operations which

precede and follow them may vary. 2. Supervisory and administrative costs. A reduction in direct labour cost may

reduce the necessary supervisory time. 3. Maintenance costs

a. The old machine may be in such a state of disrepair that its maintenance costs far exceed that of the proposed machine.

b. The different preventative maintenance costs between the two machine should be considered.

C. The risk of down time, and its effect on overall production. 4. Cost of supplies.

a. Each machine may require different cutting tools, ie. superior tools if faster cutting speeds are achievable.

b. May require different jigs and fixtures. C. May give varying replacement times for cutting tools, due perhaps to

faster cutting speeds or deeper depths of cut. d. May require different cutting fluids.

4. Miscellaneous cost items. a. Power requirements ( gas, electricity, or compressed air ). b. Floor space. C. Taxes. d. Insurance.

6. Cost and quality. a. Improved quality may reduce scrap and hence reduce the cost per component. b. Improvements in market reputation.

1.3.4 Actual Cost Analysis

Since we are comparing two alternatives, when performing a decision analysis stud we should be looking at those costs that will actually change upon replacement Those items that should be in the actual analysis are:


Decision Analysis

OLD EQUIPMENT

Do Consider

1. Operators costs 2. Repair and maintenance cost 3. Down time cost 4. Salvage value

4. Costs involved in producing fewer or inferior parts

6. Rebuilding or reconditioning costs

NEW EQUIPMENT

Do Consider

I . Initial cost 2. Interest charged on money invested 3. Salvage value at end of useful life 4. Cost advantages of improvements 4. Labour savings 6. Primary service life 7. Any effect of a change in the firm's

tax status

E. Shayan

Do Not Conside

1. Original cost 2. Past $ on repair or maintenance 3. Unrealistic book value

Do Not Consider 1. Any savings not clearly recognisable 2. Arbitrary burden or overhead charges

Performing the analysis using these criterion should be able to tell us which decision value put the company in a more profitable position.

There are numerous factors that affect the decision making process in this example such as : Is there a need for the new equipment, or with the current equipment suffice? Can the current equipment be repaired ? What will be advantageous for the company in both the long term and the short ten-n ? Is purchasing new equipment necessary, or will leasing benefit the company more ? These are just some of the questions that should be asked when considering the replacement of equipment' All of these problems can be solved systematically following different methodologies so as to come to the best decision to benefit the company.

Exercise 1.3.1 : Replace or Purchase Problem

A company is considering replacing old machinery that has a written down and scrap value of$20,000 with a new machine costing $100,000. The old machine in fully depreciated (zero residual value) can be sold for $50,000 and has running cost of

$40,000 p.a. The new machine will save $19,000 p.a. in operating costs and be depreciated at a rate of 10% p.a. (straight line basis) and will have a salvage value of

$10,000 after 5 years. If the old machine will last 5 more years and the company has a 48% marginal tax rate and requires an 8% return on investment, which is the optimal decision using NPV approach.

Solution: (Students).

The following case study examines the feasibility of replacing existing semi-trailer trucks with new B-do-utes. Many of the above factors and criteria are shown in the example.


Decision Analysis

Case 1.2 Australian Petroleum (AP)

E. Shayan

Background: Recent road transport milestones in the Australian road transport history are generally acknowledged to be in two major related areas; primarily the changes i regulations affecting economics of scale and as a flow-on opportunity for industry t develop new vehicle types.

Australian Petroleum has been quick to seize on technological developments and t pioneer new areas which lower distribution costs and provide a competitive marketing edge. The drive for increased efficiency in fuel distribution has and will continue to benefit the vehicle industry, because in the search for low tare weight an maximum payload, the vehicle suppliers are able to extract the best in technical an manufacturing excellence from component manufacturers.

Australian Petroleum conducted feasibility studies and determined that B-Double provided an opportunity to improve the productivity and cost reductions in the distribution of liquid fuels. Most predicted benefits were attributed to obtaining large payloads towed with a single prime mover operating over long distances at

maximum vehicle utilisation. However, the question which needed to be answered was at what distances did the B-Doubles become more economically justifiable (break even point) than the single prime movers. 'Me decision-making process which follows outlines all the processes and assumptions which were taken in order t produce a result. It will supply alternatives before stating the final result.

Feasibility Of A B-Double Versus A Semi Trailer

To justify the purchase of a B-Double, Australian Petroleum has broken down the costs associated with the operation of these units. Graphs were used to predict the savings required (forecasting) to justify a B-Double for any location. The associate costs were calculated using the following equations: -

Fixed Costs F = ( v + r + I) where : F = Annual fixed vehicle operating costs

v = Vehicle insurance r = Vehicle registration I = Depreciation of lease charges

Variable Costs I. Personnel D = (w +I + s + a)

where: D = Annual driver costs w =wages I = Driver Insurance s = Superannuation a = Accident compensation

II. Other Costs V= ( m+ f + t) where: V = Variable costs per kilometre

m = Maintenance

Cash Flows

f = Annual fuel consumption t = Tyres

16 Chapter 1

Decision Analysis

Total Annual Vehicle Operating Operating Cost OC= (F+ D + V)

E. Shayan

Assumptions : The following assumption have been listed for the benefit of this report, they include:

The cost of a B-Double being approximately $80,000 higher than that of a standard semi-trailer. A minimum rate or return ( DCF - Discounted Cash Flow ) of 20% on investment is required to JustifY expenditure

B-Double utilisation is expected to be I 1 0 hours per week for SO weeks per year Trip time for a B-Double has been calculated by adding an additional 30 minutes for loading and unloading to the times for a semi-trailer. A further allowance of S to 10 minutes, (both cases were looked at,) for delays in traffic.

Trip Distance and DCF Overleaf two graphs indicate the operation of a B-Double running on average 11 hours per week for SO weeks per year. For various round trip distances the economy of purchasing a B-Double in preference to a semi-trailer can be read off the graph at 20% DCF being achieved at a minimum round trip distance of 98 Km if a 10 minute allowance is used and 64 Km if a S minute allowance is used. Note that there is significant difference between the two graphs for short trips and the relative DCF.

For a typical example with a B-Double doing country work of, say ISO Km round trip the DCF can be read off as being 30% (38% with aS minute allowance, which would easily justifY the additional expenditure of $80,000 over the cost of a semitrailer.

Therefore, in summary, it can be stated that B-Doubles become more economically attractive as the distance of the trip becomes larger. The decision-making process outlined the distance at which a break even point (point at which capital expenditure equals the accepted return on investment) can be reached, given 20% DCF, this distance being in the range of 6S to 90 kilometre depending on the allowances given as stated above.

In other words, the results show that it would not be economically viable to operate with B-Double units if the distance was only SO kilometres (for example driving around town)- it would be best to use semi-trailers in this case. However, if the trial distance was to say, exceed 100 kilometres as for many country deliveries, then the results suggest it would be best to operate with the B-Double units.

B -DOUBLE ECONOMICS DCF by Distance SO 1 =Not Economical

2 =Marginal 40 3 = Economical

30

20

10

Cash Flows

S Min Difference

3

17 Chapter 1

Decision Analysis

0

70

60

50

40

50 100 ISO

FIGURE 1 RETURN DISTANCE- KILOMETRES

B -DOUBLE ECONOMICS DCF by Distance 1: Not Economic 2: Marginal 3: Economic

E. Shayan

200

Note-. Travel Time 15° o

Difference of 10 minu 30

20

10 1

0 50

1.4 Risk Attitudes of people

2

100 150 200

So far we have only considered the mathematics of cash flows. Every person can utilise them to get the same results. But often different individuals come up with different decisions having the same information. It is appropriate to consider this issue in decision making. Next section is a brief explanation of idea.

Risk Aversion Feeling of dis-utility caused by uncertainty. Usually (not always) people shy away from risk and uncertainty.

Risk A verters : Most people and companies exhibit an aversion to risk taking. Risk is regarded "bad". People take risk only if they at the same time expect to gain sufficient return, usually much more than their commitment. This behaviour can be explained by the following graph. a shows the risk. Every curve shows a region of money values against risk on which the person has an indifferent attitude.

B

t> .._ __ _,__....__....-...� ___ .....__ ________ risk Cash Flows 18 Chapter 1


a1 a2 a3 a4 a

Point A shows that for E2 dollars the person will be willing to take a risk at level cr2. Point A is preferred to D (A>D) because it demonstrates the same risk for better value. Point D and 0 Show the statuesque, A > B Less risk Same Value. Low/High degree of risk aversion is shown by the slope of the curves.

Other types of behaviour can be explained as well. For example, some have a risk preference attitude. They will take more risks for the same $ value.

$ $

r-------------12

11 t r--------------- 10

Gamblers Athletes

In order to quantifY the issue, lets, define a measure utilising risk and dollar value.

Cv =Coefficient ofVariation Criterion= cr/EMV inRisk/$

RULE: Choose alternative with lowest Cv

Certainty - equivalent criterion: of a decision is the sum of money with certainty for which the DM is indifferent between the money and the alternative.

RULE: Choose the alternative with highest certainty - equivalent

If you have a choice between a lottery ticket that can win $10,000 with a chance of 1 in million and $100 cash, which one would you prefer? How much offered makes you undecided.

1.5 EXERCISES:

1. If$10,000 is placed in a savings account that pays 12% p.a. compounded quarterly what will be the balance after 5 years. What is the effective interest rate per annum?

2. Suppose a regular income of $8,000 per year was required by someone contemplating retirement, to be paid in equal instalments at the end of each year for 20 years. How much would need to be invested if the invested balance earned 12%?

3. What monthly payment has to be made to pay of a housing loan of$20,000 at an interest rate of 12.5% per annum, where the interest is calculated monthly, in 5 years?

4. What is the effective rate of interest on a hire purchase loan where equipment costing $20,000 can be acquired over two years with monthly payments of$990. Cash Flows 19 Chapter 1


4. A company can construct a factory building for $800,000 or it can lease an equivalent building for $75,000 a year for 25 years with the option of purchasing the building for $100,000 at the end of the 25-year period. The company can earn 14% per year before taxes on its invest capital. Compare the present values of the alternatives and determine what the company should do.

6. A machine can be purchased that will produce earnings of$1500 per year. The machine costs $9,000 and will have a salvage value at the end of its useful life of

$2000 regardless of the number of years that it is used. If 12% can be earned on alternative investments, how many years must the machine last to make it equally attractive with alternative investments?

7. In Australia two methods of calculating depreciation for taxation purposes are allowed; the straight line method and the diminishing value method.

In the straight line Method an asset costing $P initially depreciated over n years and written off at the end of this time with zero written down value, is depreciated by a constant amount P/n each year.

(i) Show that the present value of depreciation allowances is

=P [1-11(1 +i)n] I i n

where i is the discount rate.

In the diminishing value method under Australian tax law an asset may be depreciated at one and a half times the straight line rate and is calculated as a proportion of the written down value.

(ii) Show that the present value of the depreciation allowances if the equipment is scrapped after k years

P = Lm=1 [3P(1-3n/2)m-1] I [2n(1 + i)m]

Note that these results may be useful when the taxation effect of depreciation is also considered in determining equipment replacement policy.

9. In hire purchase contracts a flat rate of interest is calculated over the period of the loan on the whole of the original principal and no account is taken of reductions in the amount owed due to payments.

Thus the interest payable p.a. = P x R x n

IfR is the flat rate and n is the number of repayments required to pay off the loan P the effective interest rate is r = 2n R/(n + 1) which is approximately double the flat rate.

Alternatively if C is the total cost of the loan (original amount borrowed plus interest or payment amount) by the number of years n the total number of repayments, P the original loan amount and m the number of payments per annum required, the effective rate is r = 2m C/p(n + 1 ). Try to derive the above expressions from first principles.


Decision Analysis

1.6 CAPITAL INVESTMENT APPRAISAL METHODS

E. Shayan

Typical investment problems involve expansion of operations, replacement of equipment or modernisation of facilities, evaluation of leasing versus purchase opportunities, changes to financing methods and simple selection of profitable alternatives.

Despite the importance of capital investment decisions to individual firms' profit and growth, futures, debate about appropriate appraisal methods and interpretation of the results has continued in recent years. In practice it appears that crude methods, such as the payback period approach, are still widely used without complete appreciation of their shortcomings.

The discounting techniques such as the net present value method and the internal rate of return method when applied are similarly, frequently misused. The emphasis on using a single criteria to evaluate the worth of a complex investment project has perhaps been misplaced. The use of a single measure to compare and rank projects in order of desirability, generally assumes that the objective of the firm is to achieve a maximum rate of growth. However, this simplistic perspective ignores wider realities such as budget constraints, short versus long term corporate goals, the nature of the particular enterprise and diversification opportunities and many other practical considerations. Consideration of such factors should be the first priority and less stress placed on simple measures of investment's attractiveness.

In this section the mathematics of the main appraisal techniques are examined and their interpretation considered. The factors that must be input in the required calculations are also outlined.

1.6.1 Accounting Rate of Return

Essentially, the accounting rate of return is the earings from a project, usually after deducting both depreciation and taxes and is usually e>cpressed as a percentage of the investment outlay. It is compared with a required rate of return o-r cut off rate t determine a project's acceptability. If the accounting rate of return is greater than the required rate of return, the project is acceptable; if it is less than the required rate of return, the project is unacceptable.

There are many ways to calculate rate of return or return on investment. One of the most popular methods is:

Ra = average earings I initial investment x 100 %

Exercise 1.6.1.1: Assume a company is considering replacing equipment which cost $10,000 and generates returns in year 1 ( $2,000 ) year 2 ( $3,000 ) and year 3 ($4,000).The accounting rate of return may be calculated as follows, ie. based on the

initial investment. [(2000 + 3000 + 4000) I 3] I 10000 = 30%¥

1.6.2 Payback Period

The payback period is the length of time that it takes for the stream of net cash inflow



received from a project to equal the initial outlay on the project. For example, the payback period of a machine which costs $3,000 and has net cash inflows of $1,000 per annum is three years. A project with a payback period shorter than the maximum acceptable period will be accepted, while a project with a payback period greater the the maximum acceptable period will be rejected. In practice, most payback periods are

from 2 to 5 years.

Calculation of the payback period takes into account only the net cash inflows up t the point where they equal the capital outlay. The calculation of the payback period completely ignores any net cash inflows after that point. As a result, the payback method of evaluation discriminates against projects with long gestation periods an large cash flows late in their lives. The major weakness of the payback method is it failure to take into account the magnitude and timing of all the project's cash inflow and outflows.

1.6.3 Discounted Cash Flow Techniques

In making a capital investment cash and possibly other resources are committed in the expectation of receiving larger amounts of cash and other benefits at some time in the future. To apply discounting techniques such as net present value and internal rate of return, all costs and benefits must be expressed in monetary terms, even if they are fairly intangible, hence the use of the word "cash".

The concept of "discounting" relates to the concept money has a time value measured by the difference between the relative preference for cash available now for consumption and cash available sometime in the future. The timing and magnitude of future cash flows to be received from an investment are taken account of by discounting future values to present values.

1.6.3.1 Net Present Value

The net present value (NPV) of an investment is the difference between the present value of the cash inflows and the cash outflows ("net"). A discount rate used to convert further values to present values and may be the firms required rate of return or its opportunity cost of acquitting funds to finance the project. The criterion should measure the increase in the value of the firm from undertaking the investment.

Mathematically: NPV = Algebraic Sum of all Cash Flow Figures discounted to present time

=l:n{(Revenue- Cost)k /(1 + i)k }, where i =interest rate per period

k = o n =life of project

Exercise 1.6.3.1 : :A machine is leased for a four year period. The terms of the lease are as follows. Initial payment $2,000 followed by $2,400 a year later continuing at a diminishing rate of $3 00/year until the end of the contract.



The machine is expected to return an average of $3, 000/year starting from the first year. Company is considering an interest rate of 12% for this investment. what is the NPV of this investment?

period inflow outflow

0 1 2 3 4 3000 3000 3000 3000

2000 2400 2100 1800 1500

In order to develop a better understanding of the process lets work out this example in detail using formula developed earlier.

It is not difficult to realise that the given cash flow is equivalent to the sum of the following cash flows, using the same interest rate $ figures are in $1 000.

0 1 2

3 4 1 2

3 4

I $2 0

$L 6.41 J

31 31 31 31 6 9

31 I 0 1 2

3 4

0 1 2

3 4

Clearly we are capable of handling the first three cases using previous notation. the 4th cash flow is a gradual increment which is directly converted to an F value at time 4 by F4 = 300 (F/g, 2%, 4). Notice that although the first $300 occur at time 2 we use n = 4 in the formula, compatible with the model.

Now F4 can be directly converted toP. The NPV of the whole cash flow therefore is NPV = -2000 + 3000 (PIA ,12%, 4)- 2400 (P/A,12%, 4) + 300 (F/g, 12%, 4) * (P/F, 12%, 4)

=?"

1.6.3.2 Internal Rate of Return (IRR)

Any investment is to generate a return of some sort. Here we only consider money figures. The more the return the better the investment scheme. How could this be measured? The internal rate of return is the rate of interest which results from the recovery of the investment outlay plus a return on future cash flow during the life of the project.

If you deposit 100 dollars in a bank account and receive $112 at the end of the year you would say that the bank has paid you 12% interest or the investment has generated 12% interest. In fact the $112 received a year later has the same value of your initial deposit, ie. the NPV of the system is zero. With the same analogy, when investment is made in a project, it is expected to generate some income in the time horizon under consideration. This is due to certain internal mechanism of interest generation. However, the total value of the system will be zero again.



Therefore, internal rate of return can be defined as an interest rate which makes the NPV of a cash flow equal to zero. In order to calculate the IRR, we have to formulate NPV for an11nknown i. Set it equal to zero and solve for i. i.e.

NPV== Ln { (Rev�ll�e - Cost )k /(1 + i)k } =0 ' solve for i.

This equation is non linear and usually difficult to solve analytically. However, a trial and error approach yields a narrow interval in which interpolation can be used to get a practically satisfactory result.

Exercise 1.6.3.2.1: Management is considering the investment of updating the machine (an investment of$9,000 ) that would]! return net cash inflows of$5,090,

$4,500 and $4,00 at the end of years 1, 2 and 3. Assuming that the required rate of return is 15% should the equipment be replaced?

To answer this question, the project's internal rate of ret-urn may be calculated a follows:

NPV= -9,000/( 1 + r ) + 4,500/(1 + r)2 + 4,000/(1 + r)3 = 0.

By trial and error, r = 25%. As the project's internal rate of return of 25% exceeds the required rate of 15%, the

project is acceptable¥

1.7 Cost/Benefit Ratio:

In public sector financial planning a benefit cost ratio is often used to assess the acceptability of a project. Like the pay-back period, it is not meant to be a measure of profitability and may be refined by discounting costs and benefits. The simple decision rule is that if the ratio is less than unity the project is acceptable. The sensitivity of the ratio of changes in the enumerator (costs) and nominator (benefits) need to be carefully tested when this ratio method is used, particularly as it is used in situations where often relatively intangible benefits must be valued in monetary terms? (Give Example)

1.8 Incremental Analysis

In situations where financial decisions based on comparisons between two mutually exclusive projects are made or where the level of investment in a divisible (a proportion of the project may be undertaken) project is of interest, an incremental approach may be useful. This approach also allows the difference in capital outlays to be taken some account of.

Exercise 1.8.1 Projects A and B are mutually exclusive where B can be considered as a package of A and B-A.

Projects Initial Capital Annual Cash Inflows IRR A -$60,000 +$14,310 20% B -$158,000 $32,690 15% B-A -$98,000 $18,380 13.4%

An incremental return of 13.4% is the return on additional expenditure above what is

provided by project A¥ Cash Flows 24 Chapter 1


Exercise 1.8.2 A retail company is extending credit to its customers for the first time in its history. a study of competitors experience shows the expected net profits under various plans. What choice would you make? Minimum desired rate of return is 14.0%.

Plans A

Total Investment. 10,000 Annual net income. 500 Rate of return. 4. 0% Increment on preceding plan. Increment net Income. Rate on increment

B

30,000 4,200

14.0%

20,000 3,7000

18.5%

If we had $100,000 to invest in each plan Plan A B Total funds 100,000 100,000 Available 10,000 30,000 Amount. Return on Investment. Return on funds left at 14.0%. Total Return. Return on

$100;000.

500

12,600

13,100 13.1%

4,200

9,800

14,000 14,0%

c

c

50,000 7,800

14.6%

20,000 3,6000 18.0%

100,000 50,000

7,800

7,000

14,800 14.8%

D

D

80,000 12,120 14.2%

30,000 4,320

14.4%

100,000 80,000

12,120

2,800

14,920 14.92%

- if we chose E the extra $20,000 investment would earn only 11% return.

1.9 Exercises

1. A machine is leased for a 6 year period as follows: $2,000 now, $2400 a year later and $300 less each year after.

E

100,000 14,320

4.3%

20,000 2,200

11.0%

E

100,000 100,000

14,320

0

14,320 14.32%

A similar machine could be leased for $2200 a year, paid at the end of each year. If the interest rate is 8% which one is more economical?

2. A bank advertised the following yearly loan scheme: Scheme 1 :"We charge 7% only. i.e. for $10,000 you will reply 1/12th of$10,700 each month, 12 times." There is also scheme 2 claiming 7% with the repayment of 1/12th of $10,000 each month for 12 months but deducts $700 from the principal before payment. Which of them do you prefer? What is the real interest rate involved in each scheme?

What is the correct monthly repayment for 7% rate?


Decision Analysis

3. A new 20 year Bond of$10,000 offers 7% payable semi-annually

E. Shayan

(i.e. $350 each 6 months for 20 years). What should be the current price of the Bond, if one wants to consider the Bond an investment with a 9% rate of return?

4. An investor bought a building for $99,500 to rent. It cost him $9,500 to renovate the house for more rent. The first year's rent was $15,000, and for the following 5 years he could get $18,000/year. At the end of the sixth year he decided to sell the house, so he signed only another one year lease for $1 7, 000. He then sold the house at the end of that year for $220,000. The maintenance figures for this period were (5, 4.5, 4.7, 4.5, 3.6, 4.3, 4.1) thousand dollars each year respectively, assumed to occur at the end of each year.

( i ) What was the rate of return for this investment? If the rate of 12% is common in such investments, what is the payback period?

1.10 Spreadsheet Implementation of NPV and IRR

Programming the cash flow relations in any high level computer language is a simple job. Most of companies use spreadsheets (Lotus 123, Excel, ... ) to manipulate their financial data. Forming cash flows in such environments is made easy by the facilities and utilities available in the spreadsheets. For example if the cash flow is given in a range named 11CASH11 and interest rate is stored in a cell called 11RATE11 in Lotus 123 then NPV and IRR of such cash flow is easily calculated in respective cells with formula entries as: @ NPV(CASH, RATE) , @ IRR (GUESS, CASH) where GUESS is a rough estimate ofiRR.

Example:

C31: @NPV(0.01 *C22,C24 .. C30)

A B c

interest rate = 1 0 period cash flow

0 -20 1 5 2 5 3 5 4 5 5 5 6 5

D

21 22 23 24 25 26 27 28 29 30 31

32

net present value

IRR

1.614821

0.129780.

READY

E F G H

1.11 Factors to be considered in Investment Appraisal using NPV or IRR

The consideration of the following factors in applying NPV or IRR is necessary.

1.11.1 Opportunity cost



of capital in conjunction with financing structure of the firm and nature of capital market operation. This aspect which primarily relates to the NPV approach is considered later.

1.11.2 Inflation

Until the present it has been assumed that where cash flows are estimated in current value and not nominal (inflated terms) terms and the discounted factor or IRR is in zero inflation terms. In cases where it is necessary to take account of inflation explicitly, the discount rate should be expressed as [(1 +i) (1 +f) -1 ] where f is the inflation rate. This is sometimes called the money discount rate. This particularly necessary where it is expected there are differential rates of change in purchasing power (ie. inflation) in cash inflows and

outflows.

The current cash inflow R in 11money11 terms may therefore be re-expressed as (1+f)Rand PV= (l+f)R/[(1+ i)(1+f)-1]

The effect of inflation is not considered further here, but a fuller treatment is given in Merrett and Sykes (1973) and Carsberg and Hope (1976).

1.11.3 Taxation

The impact of taxation is important in two main regards. (a) Company tax on profits means cash flows should be adjusted to measure after tax profits. Thus a cash inflow of $1000 before tax is $540 after tax if a company pays a rate of 46 cents in the dollar. It may be necessary to further consider the effect of taxation on shareholders if the project involves dividend payment or capital gains but this is not considered here.

(b) Depreciation - Australian taxation allows that depreciation (a non cash charge) may be claimed as a cost thereby reducing tax payable. However, if say, equipment is sold for an amount above its written-down value, tax must be paid at the normal rate on the difference. Depreciation may be calculated using the straight line or the diminishing value method. One switch between methods is allowable by the Taxation Department and the optimal timing of this switch is discussed in NPV evaluation by Everett J.E. and Davision A. G. (1978). Generally, practice is to initially use the diminishing value method then switch to the straight line method after a few years. Problems in valuation of assets may arise in considering residual values.

1.11.4 Life of Investment/Planning Horizon

In comparison of projects with different lives an attempt should sometimes be made to consider reinvestment or replacement opportunities over the same time horizon. This enables more direct comparison if the discrepancy between project lives is substantial.

If planning decision involve optimal life determination, consideration must be given to valuation of assets and depreciation as in 3. (Sometimes bail-out opportunities are also of interest). Equivalent annual costs (as an annuity



series of costs flows) of replacement alternatives are often used in these situations where projects have unequal lives.

Chambers (1977) proposed an extension ofiRR ofPV calculations where emphasis on whether or not this investment's assets are "revocable" or "locked in" and this during the projects life is taken into account using a DCEF (discounted cash equivalent flow) approach. This has been criticised and commented on by Ma and Scott (1980),

Peasnell (1981 ).

CASE 1.3 Black Kettle's Company Cash Flow Analysis

The level of inflation and interest rates are of vital importance when a company i contemplating expanding their manufacturing operations due to a greater demand for their product.

In the case of the Black Kettle Company, their "Black Kettle" products have found niche market and demand for them is growing. The time has come for the company to expand. But before the final decision is made, an in depth analysis of what the inflation and interest rate will do in the next five years is required. This needs to made so that the company can see if they can still finance their expansions if inflation was to rise sharply and interest rates were raised in order to maintain inflation. Below are the Balance Sheets and Income Statements for the Black Kettle Company.

BLACK KETTLE COMPANY Balance Sheet December 31/1993 December 31/1994

Assets

Current Assets: Cash and marketable securities Accounts receivable - trade

Total current assets Fixed assets - net Total assets$

Liabilities and Net Worth

Current liabilities : Accounts payable Accrued taxes Other accrued liabilities

Total current liabilities First mortgage bonds, 9.5% due in 1995 Total liabilities Common stock ( 5,000 shares) Retained earnings Total liabilities and net worth

Cash Flows

$ 20,000 32,000 29,000 81,000 88,000

160,000

$ 25,000 3,000

20,000 48,000

$ 55,000 103,000

103,000 26,000

$ 179,000

28

$ 36,000 34,000 30,000 100,000 90,000

190,000

$ 33,000 6,000

26,000 65,000

$ 55,000 200,000

120,000 30,000

$ 200,000

Chapter 1

Decision Analysis

BLACK KETTLE COMPANY Income Statement

December 31/1993

Net Sales $ 350,000 $

Cost of goods sold 200.000 Gross profit 150,000

Expenses: Selling general, and

administrative expenses 41,000

Interest expense 5.000

Total expenses 46,000

Net profit before taxes 104,000

Income taxes (federal and state) 7.000

Net profit after taxes s 97,000 $

December 31/1994

410,000

240.000

170,000

55,000

5,000

60,000

120,000

10,000

100,000

E. Shayan

If inflation and interest rates were to use, a number of other factors are likely to b influenced, these include future wages and future initials costs.

Presently inflation is set at 2.5% with an interest rate of 10.95% and the Black Kettle Company is looking to invest $20,000 in their expansion. From economic indicators, i appears that inflation may jump to 6. 5% in the next five years. In order to calculate the interest rate that is required for the company to make a profit in the expansion the following equation needs to be utilised:

IRR= i+ f + i*f= (i+ l)(f+ I)- 1

for i (interest rate)= 0.1095

f (inflation rate)= 0.065

IRR (interest profit rate)= 18.16%

In order for the company to make a profit on the investment, the interest rate would need to be above 18.16% if the corresponding inflation rate was to rise to 6.5%. Fro this small calculation it can be seen that it is really not quite the right time to invest a it is highly improbable that the interest rate will reach 18.16%. If the company was t go ahead with the investment then it would actually be losing money in the Ion ten-n. However, if inflation was to remain constant at 2.5% for the next five years then : i = 0.1095 and f-= 0.025, IRR = 13.72%

The corresponding interest rate would only need to be 13.72%. This rate of interest i rather more attainable than the previous rate so it is more likely the company would choose to invest. However, as mentioned previously, the down side is that it i highly unlikely that the inflation rate is going to stay constant at 2.5%. R ealistically there will be slight fluctuations in its value. If the present interest rate was slightly lower say, 9.85% and inflation was still to increase to 6.5%, then, i = 0.0895 and f-= 0.065, IRR = 16.03%

The corresponding interest rate would only -need to be 16.03%. 'this is still very hi and the company would not be likely to invest as the interest rate would probably no be attained. If, in fact, the inflation rate did not rise, then i = 0.0895 and f= 0.025

IRR = 11.67%. The corresponding interest rate would only need to be 11.67%.



The rate of interest is possibly attainable, therefore the company is more likely to make an investment. As can be seen from the previously shown calculations it is a straight forward process to calculate the interest rate required in order to raise a profit from an investment. However, the difficulty is in being able to predict with some accuracy what the inflation and interest rates are likely to do in the future. If the prediction of these values is incorrect and the investment has already been made then the

repercussions could be damaging 'I

1.12 Comparison ofNPV and IRR Criteria

A common textbook assertion in the debate about the technical merits of IRR and NPV is that conflict between them arises because of implicit assumptions concerning the reinvestment rate from the cash inflows.

Typically an example is given to illustrate this conflict. Suppose two investment alternatives each cost $1000 initially and the first yields $11 00 after one year and the second $1166 in two years time. If the discount rate is 5% the NPV of the first is $48 and the second is $58, and the corresponding IRR's are 10% and 7.5%. However, it can be pointed out that if the $1100 received from the first project is reinvested at 5%

for one year, then $1210 is received at the end of second year. It is then concluded that the NPV does not include consideration of reinvestment while IRR does and so the question of which measure is preferable hinges on this.

NPV

0 Fischer rate identical NPV

Discount rate

IRR2

The general rule advanced, illustrated by the figure above is that when reinvestment opportunities are available with a return in excess of the Fischer rate then the IRR provides the preferred measure. This is sometimes further illustrated by estimating the net terminal values. (ie. future values) of the alternatives as done in the above example ($1210 vs $1166).

However, this explanation is somewhat superficial and a more subtle explanation is warranted. Dorfman (1981) summarises some of the technical work in this area and points out the following aspects:



since the NPV calculation does not consider reinvestment rates, these must be explicitly considered if relevant in a comparison of alternatives. Thus a reinvestment policy that allows the cash flows from 11daughter" projects (and their daughters add infinitives) to be estimated must be used. While this is daunting, to ignore these assumes that after the initial investment project no opportunities are more attractive than the market rate of interest (discount rate) arise.

that in situations where the firm's objective is to maximise rate of growth, the preferred measure is internal rate of return providing similar investment opportunities continue to open up in the future. On the other hand when the objective is to maximise the value of its distributions, then the net present value approach is more likely to be appropriate.

Merritt and Sykes (1973) earlier noted this misconception, noting the basic definite of IRR and its information content. That is that the rate of return on capital outstanding in a project during every year of its life, and this no more assumes reinvestment that the statement "that a bank is receiving 6% on an overdraft" implies the bank is reinvesting repayments of the overdraft at 6%.

The rate of return carried on reinvestment has no relevance, per se, but is only relevant if alternative investment opportunities are available in the future which are dependent on the project under consideration.

Keane (1981) argues that "neither the IRR nor the NPV method contains any implicit assumptions about the reinvestment rates available for the intermediate cash flows" by reference to the basic definitions of the measures. He points out that the IRR simply "represents the project's wealth increment". He further argues, correctly, that even if capital is rationed the assertion that IRR assumes reinvestment at the internal rate of return, is not validated and that in both cases all reinvestment's ("dependent projects") should be explicitly considered. These arguments are not reproduced here.

In applying either the IRR or NPV criteria, therefore the objectives of the firm are relevant as are reinvestment opportunities. In practical application, relevant reinvestment opportunities should be considered if capital is rationed or if there projects are dependent in both methods. If capital is not rationed, opportunities are irrelevant.

On the question of whether IRR or NPv is to be preferred, the controversy between, and amongst, academics and business analysts will continue but the basic definitions and informational content of each should be noted. The final decision on the appropriateness of each depends as outlined on the firms availability of capital and

policy.

Baker (1981) provides an excellent summary outline of the implications of various corporate objectives on aspects of financial policy including capital structure and dividend policy as well as capital investment.


Decision Analysis

1.13 Cash flow simulation

E. Shayan

A major assumption made implicitly in construction of any cash flow is the certainty of the magnitude of the cash occurrence in the future. In the dynamic market environment such assumption is doomed to be incorrect in most of the cases.

In fact what is really done is to evaluate one of infinite possible realisations of the scenario. In reality every figure in the model is a snap shot of a random variable. Perhaps it is a specific measure such as mean or median. Accordingly the calculated

NPV or IRR is also a single point from a range of possible values. Thus the analysis can be quite misleading. For example a positive value for NPV indicates that the investment has potential's and a IRR >current interest rate suggests a good investment opportunity. However these are based on the figures used. It is possible that under different circumstances the result is quite the opposite? How can we get a good picture of all the possibilities?

One way of treating the above problems is by simulation. The experts dealing with these cases have vast amount of information. Their knowledge can be utilised to estimate a probability distribution for every suspicious figure. For example maintenance engineer would be able to explain the possible outcome of a repair expenditure in a reasonable accuracy. Such consultations inevitably results in better understanding and improved information, communication, etc. anyway.

The idea is to associate a probability distribution to every figure which may have a different value in the future. Then randomly generate a number from the given distribution. Use these generated figures in the cash flow model to calculate NPV and/or IRR. By repeating this process sufficient number of times, a set of values are obtained for NPV 1IRR which can be analysed to deduce their distribution. This information is far more enlightening than a single figure. In fact the single figure is only one of the possibilities.

Several questions can be examined from this result. What is the chance ofNPV<O, Pr { a<NPV <b}, Pr {IRR <R}. Even though the single figure NPV is positive, if there is a high chance ofNPV< 0 the decision maker may choose not to consider the alternative.

1.13. 1 Introduction to Simulation

By definition, to simulate is to get close to, to copy, mimic, imitate. With the increasing readiness and complications of change in the manager's environment, the ease with which off-the-job training can bridge the gap with the reality of managing is going to be even more evident with managers today. How often do managers return from off-the-job courses with ambitious action plans only to be thwarted by the 'transfer of learning' problems. This is frequently because the learning isn't close enough to the manager's reality [ 46].

It is not possible to claim any remedies for 'bridging this gap'. It requires an increase of initiatives by trainers. I'll offer one - and one which works: High Reality Business Simulations. The work of researchers such as Mintzenberg, (1) Stewart, (2) and

Kotter



(3) serves to substantiate that managers are not naturally methodical and reflective planners.

Many simulations are designed which are highly sophisticated, increasingly computer based and require great details and in-depth analysis for a basic understanding. This can often result in managers and company teams finding for themselves an objective of teasing out the particular marketing model on which the simulation is based.

Such simulations can leave you feeling rather uncertain about forging links with the day to day realities of managing. Indeed, it is common for such simulations to either neglect or give too low a priority to people's interactions and skills which are the day to day 'elements' of managing. Whilst discrete learning can be developed to focus learning outputs on particular management principles, e.g. planning and control, such learning may not be internalised by managers because of a lack of context in which to relate specific learning points. The simulation has been deliberately designed to provide this essential contextual underpinning so that part learning becomes holistic learning [ 43].

1.13 .2 Advantages and Benefits:

A better understanding of how systems operate and respond to various factors which enhance engineering capability to improve performance and respond to present problems associated with your industry.

This advantage is that it increases the ability to try out various alternatives which may result in better designs, reduced investment and risks and reduced operating costs.

Often the various systems interact on the plant floor in a manner which was not detected during the system's design. These interactions can affect the overall systems performance ,leading to delays and/or extra costs. These extra costs and delays can be minimised by using simulation to reveal these interactions and predict overall system performances thus resulting in fewer surprises!

Another advantage is that it provides the ability to impose a single chance on the system and observe its effect on performance without the influences by other unrelated changes ( i.e. a controlled environment for analysis ).

Simulation is an excellent training tool useful in a variety of decision-making applications.

1. 13.3 Limitations

Simulation is an assistant to (not a substitute for) engineering judgement.

Simulation does not provide automatic optimisation, it provides only the result of "what if' questions.

Simulation results are as accurate as input data. Therefore it requires that the use has a thorough understanding of the subject. Given inaccurate or incomplete data simulation will not produce accurate or useful results. Simulation cannot be used to



describe characteristics that have not been explicitly modelled.

Managers' preferences for particular leaning styles can be used and highlighted

within the simulation. Advantage is that feedback and results are entirely dependent on actions and decisions taken by managers, all of which are visible and can be clearly understood.

This deliberately introduces into the simulation a level of simplicity to ensure that learning outputs are achieved. Managers are not left wondering why it was that well thought out decisions on numbers and types of packages, projected investments, changes in the market, etc. have not yielded the desired outcome. This can be the situation with some poorer computer simulations which ignore the people interactive element and frequently focus too much on the marketing model of the simulation.

1.13.4 Simulation Example Using Portfolio Analysis:

Simulation is a technique that systematically repeats the application of a rule or formula to a given set of data. For this problem we will be dealing with a special kind of simulation, Monte Carlo simulation. To use Monte Carlo simulation, all we need to do is specify the probability distributions, either discrete or continuous and the decision rule or formula that should be applied to the selected values of the variables.

Monte Carlo simulation is named after the famous casino because of the procedure of selecting variables by chance (randomly). For example, let us assume that we are trying to forecast the piece of a share of a stock for one year from now, its probability and that this estimate must be based on forecasts of sales, profit margins, net income, the number of shares outstanding, and piece earnings ratio. Now that the variables have been enumerated, it is necessary to specify the probability distribution for each variable, as can be seen in figure 1.13 .1.1 below.

Figure 13.1 Probability Distribution Using Monte Carlo Simulation

We can see that there is a 30 percent probability, in the analyst's judgement, that the next year's sales will be $10 million.

After the probabilities are specified, the next. step is to set up the formulas that are to be used. In this example, the formulas are as follows:


Decision Analysis

Earnings per share= [Sales x Margin %] I [No. of shares outstanding]. Earnings per share x PIE= Price per share.

E. Shayan

The computer program that has been written by or for the analyst, will then randomly select numbers and match them against the corresponding values in the figure above. One value of each variable will be selected.

The expenses have been replaced with approximate net income profit margins. Simulation can be used whenever decision tree analysis can be applied; however simulation is much more practical when there are many alternatives. In fact, if sufficient alternative values are specified, we would represent the data as a continuous distribution.

For example the computer will select four numbers randomly and match each of the four against the distributions of sales, profit margin, shares outstanding (in this cas all values will be matched against 1,000,000 shares since it is the only possible case),

and PIE ratio. Assume that the random values selected are $10 million, 33-1 I 3 %, 1 million and 20 respectively. Then these values can be substituted into the two equations above. This would lead to first, [($10 million x 33-113%) I 1 million]= $3.33 earnings per share. Then earnings per share (BPS) of $3.33 times the PIE of20 yields a piece of $66.60. This process is called the first iteration.

The process is then repeated as many times as the analyst desires. It is generally repeated at least several hundred times, which only takes a few seconds of computer time. Thus many prices of returns will have been generated. Next probability measures can be calculated based on the distribution of these prices and returns.

At this point, the analyst has a calculated distribution of prices with a mean and a standard deviation. This information alone is extremely helpful, for these two statistics are necessary inputs for modem portfolio analysis. However, this is not the only information the simulation provides to the analyst. The distribution itself is very important. It can be portrayed graphically in a histogram (a bar graph) when the distribution is discrete, in a continuous curve when the distribution is continuous, or in tabular form as a frequency distribution.

This allows the analyst to see directly by inspection in which ranges the most likely outcomes will be. Note that if the outcomes are normally distributed (which is very likely), techniques of statistical interference can be used in conjunction with the mean and standard deviation.

This example can be made considerably more sophisticated and more accurate with the addition of more decision variables such as input costs, variable sales prices, etc. More alternative values of these variables can be hypothesised. These present more of a practical implementation problem than a conceptual one.

The model as it has been set up here assumes that the various distributions are entirely independent. This assumption is made because we randomly select values from the various distributions without regard to the other values that have been an will be selected from other distributions. For example, we draw a value from the



profit margin distribution without regard to the value we have selected from the sales distribution.

This can be somewhat unrealistic because we might select a very low sales figure and a very high profit margin. This event is certainly unlikely to occur in reality since

low sales imply under-utilisation of facilities, which in tum implies low profit margins- One way of overcoming this potential problem is to 11constrain11 the distributions so that the selection of certain values from one distribution is conditional upon the value of a previously selected value from another distribution.

For example, if low sales values have been randomly selected from a sale distributions, only a low profit margin can be selected from the profit margin

distribution. This is accomplished in the simulation program.

Another way of introducing more economic rationale into a simulation model is to generate possible values of key variables in the program itself via regression analysis That is, generate values from a behavioural model that has been developed from past experience and then select values randomly from this distribution of outcomes outcomes that make economic sense.

For example, experience may have taught us that obtainable margins are dependent on certain variables. These obtainable margins are generated from the regression that

utilises the 11proven11 explanatory variables and then we select from this internal (internal to the simulation) distribution. The Monte Carlo simulation provides much of the information the analyst requires in terms of price forecasts and overcomes the objections to other techniques.

1.13.5 Summary

A simulation package provides opportunities for managers to respond to problemsolving and decision-making in situations of ambiguity with real time pressures. They also have opportunities to practice a whole range of people skills. They are frequently re-evaluating their experience and learning from them. Importantly, they are starting to use new technology to obtain competitive advantage which its use offers. Overall, such a high reality simulation package provides opportunities to think and act. It link things together so that conceptual models become real working tools, i.e. having put concepts into practice managers feel more comfortable to do so in the working situation, which is of course where the vast majority of real development takes place. Also most importantly they are starting to use new technology interactively.

1.13.6 Spread Sheet Implementation of cash flow simulation - Example on

PROPS

Load PROPS onto Lotus 123 (tutorial is given in lab) and implement the following example according to the instructions.

Exercise 1.13.1: Suppose you invest $20,000 in a new machine and expect to realise a cash flow of $5,000 in each of the next 6 years with an interest rate of 10% per annum. What is the net present value (NPV) and internal rate of return (IRR)



generated by the machine? What happens if the cash flow is not $5,000 in each year, but is a random variable?

To analyse this problem, first access the Frob/Simulation worksheet. Then set up the financial analysis model in the spreadsheet in some clear area, say [A21 .. C32]. Enter the interest rate in, say cell C22, and the cash flows for each year in [C24 .. C30]. To compute the net present value and internal rate of return, you can use the spreadsheet functions @NPV and @IRR. In this sheet, in cell C31 put the formula @NPV(.01 *C22,C24 .. C30) and in C32 the formula @IRR(.01 *C22,C24 .. C30). Placing year number and titles in appropriate cells, you have the following worksheet:

C31: @NPV(0.01 *C22,C24 .. C30) READY

A B c D E F G H 21 22 interest rate = 10 23 period cash flow 24 0 -20 25 1 5 26 2 5 27 3 5 28 4 5 29 5 5 30 6 5 31 net present value 1.614821 32 IRR 0.129780

Now, what happens if the cash flows are random? Assume, for example, that the cash flow in each period is normally distributed with a mean of 5 (thousand) and a standard deviation of 1 (thousand), and observations are independent from period to period. Begin by defining this distribution in a convenient space in the worksheet as follows. Move the cursor to a cell with a clear area below and right, say to A34 in this case. Call up the menu with [Alt]-M and select Distributions, Continuous, and Normal in sequence, and provide the name (CASH), the mean (5), the standard deviation (1) and answere other questions when requested. Before calling the simulation model, move the cursor to a cell with a clear space below and right (say, A42). Call up the menu with [Alt]-M and select Simulation. PROPS prompts and Analyst Name, Simulation Name, Number oflnput Variables, and Number of Output Variables. Respond with 6 input variables (one for each year's cash flow), and 2 output variables. PROPS then creates the following tables and returns to READY mode.

A B c D E F G

43 PROPS -- Spreadsheet Simulation

44

45 Analyst Name IE/Mgroup

46 Simulation Namefinancial

47

A B c D E F G H

48 Input Name oflnput

49 Variable Random Cell

50 Number VariableLocation

51 1


Decision Analysis

52

53

54

55

56

57

58

60

61

62

63

64

65

66 67

68

69

70

71

72 73

74

75

76

77

78

79

80

81

82

2 3

4

5

6

7

8

A B

Output c

Name

D E

Output F G H

Variable of Cell ----- frequency distribution ----Number Output Location record start stop interval

1

2

Total number of simulated points?

Simulation Sample:

No. 0

n n

0

0

1

Note: The spreadsheet formula (for simulation) and the pdfs

for the (input) random variables must all be defined

before attempting to 'Execute' the simulation.

Fill in the above tables, then hit Alt-M for the Menu.

0

0 1 1

E. Shayan

Fill this table to create the linkages to the spreadsheet being simulated. Each input will use the same random variable distribution CASH, with input 1 being related to cell C25, input 2 to cell C26, and so on. (That is, to enter these, type c25 in cell CSO, c26 in C51, and so on.) Name the output variables NPV and IRR respectively, giving cell locations C31 and C32. Initialise the number of points to be simulated to, say, 10. Then the spreadsheet appears:

47

48 49 50

51 52 53

54

55

56

57

58

A

A

B c D

Input Name Input VariableofRandom Cell Number VariableLocation

1

2 3

4

5

6

B

Name

cash

cash

cash

cash cash

cash

c

Output

c25

c26

c27

c 28 c 29

c 30

E G G

D E F

60

61

62 Output Variable of Cell ------frequency distribution ------

Cash Flows 38

H

G H

Chapter 1

Decision Analysis

63 64 65 66 67 70 71

Number Output Locationrecord? start stop interval

1 npv c31 n 0 2 irr c32 n 0

Total number of simulate4 points? 10

To carry out the simulation, press [Alt]-M and you will see the menu:

47

F47: Execute Continue Distributions Quit Begin Simulating A B C D E F G

E. Shayan

0 0

MENU

H

1 1

Select Execute by pressing [Enter], and PROPS will set up the simulation and run 10

independent simulations of the spreadsheet. The output in the worksheet appears:

A B c D E F G

Total number of simulated points? 10

Simulation Sample: No. in 1 in 2 in 3 in 4 in 5 in 6 10 6 6 7 4 4 6

69 70 71 72 73 74 75 76

---------------------------------------------------------------------------------------------

77 Output # 1: npv 78 79 mean 2.534913 80 std dev 1.837398 81 82 83 Ouput #2: irr 84 85 mean 0.147274 86 std dev 0.033594

H

out_ 1 3.886372

This initial run of 10 sample points is useful to ensure that the output is correct and all random variables are defined properly. Having done so, to obtain some reasonable statistics on the simulation, change the number of points to a larger value, say 200,

press [Alt]-M, and select Continue from the menu. PROPS will then continue the simulation until it has collected 200 independent simulations ofthe spreadsheet model. The output now appears:

69 70 71 72 73 74

A B c

Total number of simulated points?

Simulation Sample: No. in 1 in 2 200 7 6

D E F G

200

in 3 in 4 in 5 in 6 7 6 6 5

7 5 ---------------------------------------------------------------------------------------------

77 Output # 1: npv 78 79 mean 1.736485

Cash Flows 39

H

out 1 6.570450

Chapter 1


80 std dev 1.731317

83 Output #2: irr

84

85 mean 0.131554

86 std dev 0.031712

You may wish to repeat the simulation and create a frequency distribution for each of the output variables. To do so, change the default value in the record? column from n to y for the output variables and define the range for each. Here, let NPV run from -3

to 5 with an interval size of 1, and IRR from . 09 to .17 in steps of 0. 0 1. Your screen should appear:

A B c D E F G H 60

61 Output Name Output 62 Variable of Cell ------frequency distribution ------63 Number Output Locationrecord? start stop interval 64 ---------- --------65 1 npv c31 y -3 5 1

66 2 irr c32 y 0.09 .017 0.01

67

To repeat the simulation, press [Alt]-M and select Execute. The simulation will be initialised and then run, giving the output:

A B c D E F G H I J

73 Simulation Sample:

74 No. in 1 in 2 in_3 in 4 in 5 in 6 out 1 out_2

75 200 4 3 5 5 5 6 0.20109 0.096521

76 ----------------------------------------------------------------------------------------------

77

78 Output #1: npv

79 frequency distribution

80 mean 1.491139 x -3 -2 -1 0 1 2 3

81 std dev 1.624478 freq. 0 8 14 39 38 43 39

82

83 84 Output #2: irr

85 frequency distribution

86 mean 0.127244 x 0.09 0.1 0.11 0.12 0.13

87 std dev 0.029939 freq. 26 24 20 23 26

K

4

12

The reported frequency distribution is the number of occurrences for the output variable in the range [ x-(interval/2), x +(interval/2). Occurrences outside this range are assigned to the lowest or highest range, as appropriate. The mean and standard deviation are computed from the actual occurrence data, and not the frequency distribution.

Samples for the random variable inputs are generated independently for each input variable, using the pdf values in the distribution as the relative likelihood of each possible random variable value. In simulating, PROPS normalises these relative values to create a corresponding probability distribution (summing to one).


L

5 7


Notice that the initial value calculated for NPV was $1,614 which is not far from the mean value ofNPV of$1,491. However now we have a distribution of the random variable NPV which spreads between -$3,000 and $5,000.

Pr{NPV � 0} = (39 + 14 + 8)/200 = 61/200 = 30.5% Pr{NPV � $1,614} > (61 + 38) I 200 = 49.%

This information is totally new to the decision maker who is used to rely on single NPV figures. There was no evidence of about 50% chance of having NPV < 1,614 before. It is an interesting exercise to plot the NPV, IRR using graphic capabilities of

Lotus 123 \f

In cases where accuracy is of more concern regarding the assumed distributions, sensitivity analysis can be performed by changing parameters or types of the distributions to fully examine their impact on the final result.

Note that you can construct the spreadsheet model or enter the distributions either before or after you select simulation from the menu. if you should happen to specify a distribution in the simulation which is not created, on executing the simulation PROPS will stop and inform you of this. You can then press [Alt]-M, select Distributions, and generate the required distribution. Then execution can proceed.

PROPS also allows you to add or delete variables from the model you have constructed. For example, suppose in the simulation above you with to have a record of the cash flow in period 1. To do so, add a new output variable 3 to the output variable table, name it, and give its cell location, C25 here. The table will then appear:

60

61

62 63

64

65

66 67

A

Output Variable of

Number Output

--------- --------

1

2 3

B

Name

npv irr

cash

c D E F

Output Cell ------frequency distribution ------Locationrecord? start ---------- -------- -----

c31

c32 c25

stop

y y

y

interval

-3 5

0.09 017 2 8

G H

1

0.01 1

You can now re-run the simulation by pressing [Alt]-M and selecting Execute. The output will now appear:

A B c D E F G H I J

77 Output #:npv Sum(x) 32.477 78 frequency distribution 79 mean 1.6238 X -3 -2 -1 0 1 2 3

80 std dev 1.6170 freq. 0 0 2 4 4 4 3

81 82 83 Output #2: irr Sum(x) 2.5826 84 frequency distribution 85 mean 0.1291 X 0.09 0.1 0.11 0.12 0.13 0.14 0.15 86 std dev0.0288 freq. 2 4 2 2 2 1 3 87



88

89

90

91

92

Output #3: cash

mean 4.85 x std dev 1.1079 freq.

frequency distribution 2 3

0 3

4

4

Sum(x) 97

5 6

7 5

7

1

8

0

In the output, a simulation sample table is included. This table displays the number of the current simulation run and the current input and output values for each repetition. You can observe how these values change as the simulation proceeds. The final values shown in this table are for the (n+ 1 )st sample point, where n is the number of simulation runs specified. This table is particularly useful in ensuring that the simulation is working correctly, in that each time you press the [Calc] key (F9 in Lotus 1-2-3), new random selections for each input will be shown, with the corresponding output variable values.

If you wish to interrupt the simulation when it is executing, press [Ctr][Break], followed by [Esc]. You could then, for example, save the worksheet and carry on with some other tasks. When you wish to continue with the simulation, you can retrieve the worksheet and restart the simulation from the break point pressing [Alt]-M and selecting Continue.

When finished with a simulation, press [Alt]-M and select Quit. This deletes all internal references to the input distributions and deletes any range name definitions created during the simulation.

1.14 Exercises

A company is considering replacing its power generator system. According to proposal A, a new power generator could be purchased at a price of $60,000. This system will cost $4,000 annually to maintain and operate. In return it will reduce the company's fuel cost by $12,000 a year. The useful life of this system is 10 years.

Proposal B is a modular type of generator which can be expanded according to the needs of the company. The initial module which has to be paid for at the beginning of the project costs $30,000. Two more units would be added to this system, one at the end of the fourth year and the second one at the end of the seventh year. Each unit will cost $20,000. The maintenance and operating costs for this system are $2,000 for the first four years, $3,000 for the next three years and $4,000 for the last three years. The savings in fuel cost for this project will be $8,000 for the first four years, $12,000 for the next three years and $15,000 for the last three years. The salvage values at the end of 10 years for proposals A and B are $10,000 and $15,000, respectively.

You are consulted to assist in choosing the best alternative. It is stated that 15% interest rate is acceptable by the company.

a) Describe your actions towards making such a decision.

Suppose in your investigations you have noticed that some of the figures given are subject to variations. In particular the maintenance and operating costs could be anywhere between $1,000 to $3,500 for the first four years ofProject B due to learning curve effects. These cost figures for the remaining years of Project B will



have normal distributions with mean value as figures given and a standard deviation of $800, $1,000 for the next three years and last three years respectively. On the other hand, fuel cost reduction in Project A is not well known. The best estimate puts it as an exponential distribution with mean $12,000 while the maintenance cost (A) follows

a normal distribution with mean $3,500 and standard deviations of $500, each year¥

1.15 Financial Management

A typical firm consists of 3 subsystems, production marketing and finance. In this section the nature of the finance function and its effect on the objectives of the firm is examined. These objectives may range from maximising profit or return on investment to satisfying to simple survival.

The two basic decision areas are financing and investment decisions. Investment decisions involve allocating resources for project and assets which will provide returns in the future. Financing decisions involve the manner of operations from shareholders and in the capital market. Dividend decisions are sometimes considered as a decision area separate from financing. The results of these decisions is reflected in the financial structure for the firm.

The financing of and investment areas and their inter-relationships a finance manager must examine are:

(1) Working capital management. The way in which current assets and liabilities are managed. This includes short term consideration of financing for production and inventory levels and creditors and debtors positions. (2) Capital investment in projects and assets. (3) Medium and long term financing from financial institutions and other elements of the capital market. Return on investment, dividend policy and riskiness of operations are important, imparting factors on the cost of capital to the firm. (4) Valuation and capital structure. The attitude of investors to balancing the firm depends on the risk versus return and is related to the financial structure of the firm.

1.16 Summary Advantages and Disadvantages of Appraisal Methods

Advantages

• NPV preferred by economists as more properly emphasising the theory of the firms'

basic goal to increase in worth or value. taking into account timing of each flow and effect of time value of money (as

does IRR).

• IRR

more direct and useful measure of profitability when riskiness of investment needs to be taken account of since risk is measured in the same terms (quantity and time dimension).

more easily understood by businessmen than N.P.V.



• Payback

easy to understand and where projects are risky and a weak cash flow position exists, may be appropriate. appropriate if company goals are best described as satisfying and survival. avoids need to estimate future cash flow accurately and investment life.

Disadvantages

• NPV

may be difficult to determine an appropriate discount rate ie. cost of capital. more difficult for businessmen to interpret. doesn't indicate magnitude of investment required and emphasise cash flow situation. The same criticism applies to IRR.

life of investment may not be at all clearly defined making it difficult to calculate NPV. This sometimes important factor also causes similar problems with IRR calculation. a recent general criticism of discounting techniques by Hayes and Garvin (1982) suggests that it results in companies being unwilling to make long term investments, because of biases in misapplying the techniques.

• IRR more difficult than NPV to use and evaluate ranking and selection of projects involves aspects such as divisibility, mutual exclusiveness and capital rationing. gives a measure of return relative to the outlay required but ignores the absolute size of the return. cannot be used (without modification) if only cash outflows of two alternatives are being compared.

• Payback

over-emphasises liquidity and ignores profitability, economic life, earnings after payback period etc.

NPV and IRR methods have been used in Engineering Management decisions in the past, in particular, where similar machines were compared economically. In the new manufacturing environments such as CIM and FMS areas, care must be taken while using these methods as allocation of costs or profits are much more difficult if not impossible due to variety of products being produced on the same machine.

1.17 Financial Ratios

Traditionally analysis of a companies position by examining its financial statements is know as fundamental analysis. It may assist potential investors, both shareholders and creditors to gauge the position and performance of the firm to some extent. It may also be of some use to management in monitoring the current assets position.



The ratios however, provided limited information and comparisons with similar

companies may be misleading unless special circumstances are considered. Financial ratio analysis is less popular than it used to be but may have some limited value.

The performance measures in particular do not take account of risk and other competitive factors. Financial analysts therefore, who consider related hypotheses on capital market efficiency and portfolio theory operative, believe share prices (company value) to be only influenced by future events and discount the use of historical data and measures.

Performance Ratios:

Net Earnings (net profit after tax) Return on investment (ROI) Total Assets return on all capital invested.

Net Earnings Share Price

Net Earnings Net worth (Owners equity)

Net Earnings Net Sales

Cash, Securities Receivable Current Liabilities

Cost of Goods sold

Inventory

Financial Structure ratios

Long term debt Equity of Shareholders

Total Current assets Total Current Liabilities

Total Liabilities Total Assets

Cash Flows

Yield measures attractiveness of company company to investors at perceived risk level.

Return on equity indicates return on owners capital.

Measures profitability of sales.

Acid Test Ratio measures ability of company to quickly meet obligations by using the highly liquid assets cash, marketable securities and receivable (debtor).

Measures turnover of inventory.

Measures extent to which the company is financed by borrowed capital.

Current ratio measures short term solvency; ability to meet its obligations.

Average ratio measures companies reliance on debt finance.

45 Chapter 1


Appendix 1. I

Cash flows

Data from income statements and balance sheets will be used for such purposes as performance measurement and budgeting. The year- end balance sheet shows the cumulative effects of the past operating, investing, and financing decisions- the firm's

remaining assets, remaining unpaid borrowings, equity capital raised, and results of past operations. The income statement shows some of the current effects of these past decisions. The third formal statement commonly used in financial accounting is the statement of cash flows1. The cash flow statement is needed because neither of the other statements directly addresses the firm's current objectives in these three important areas and the balancing of the cash flows associated with them. Like the others, the cash flow statement is after-the-fact (or historical). But it provides the interested reader with further information with which to assess managerial decisions and performance and the

firm's prospects for future profit, payments to financing sources, and growth. The formal cash flows statement has three basic parts, which classify the firm's cash inflows and out flows as related to operating, investing, or financing activities. Operating2 flows: inflows include cash received from sales or products, performance of services and perhaps for interest and dividends on investments.

Outflows include cash paid for purchases of inventory, services of employees and others, taxes, and other expense items, including interest.

1 cash flows means cash movements resulting from transactions with parties external to the company (or economic entity). entity means any legal, administrative, of fiduciary arrangement, organisational structure or other party (including a person) having the capacity to deploy scare resources in order to achieve objectives.

2 operating activities means those activities which relate to the provisions of goods and services. Examples of cash flows from operating activities include payments to suppliers and employees for goods and services; and receipts in respect of the provision of goods and services



Investing1 flows: inflows include receipts from sales of long-lived operating assets, such as property, plan, equipment, and patents, from repayments on loans made to others, and from sales of investments in securities of other entities. Cash out lays to acquire these same types of assets, or to make loans to others, are examples of investing outflows. Financing2 flows: inflows include cash received fro long term or short-term borrowing and from issuing stock. Dividends, purchases of treasury stock, and

repayments of borrowings are examples of financing outflows. The formal cash flow statement is based on the combination of cash3 and those cash equivalents4 .

Reporting operating cash flows:

Income statement is prepared using the accrual basis of accounting and does not , either specifically or exclusively, reflect cash transactions relating to operation, thus

desired information about cash flows relating to operating activities is not available directly from an income statement.

To meet the need for information about the cash flows relating to operating activities, two reporting approaches have achieved general acceptance. The first approach, called the direct method, reports operating cash inflows and outflows separately. A reporting of operating cash flows under this approach would resemble a cash budget showing receipts and disbursements for operating activities. The second approach, called the indirect method, reports only the net cash flow from operations, but it does so by presenting the reported net income and then adjusting the amount for the effects of

1 Investing activities means those activities which relate to the acquisition and disposal of non-current assets, including property, plant and equipment and other productive assets, and investments, such as securities, not falling within the definition of cash.Examples of cash flows from investing activities include payments to acquire property, plant and equipment, and proceeds from the sale of such assets, payments to acquire equity instruments of other companies, and proceeds from the sale of such instruments; and other equity contributions, for example acquisition of an ownership interest in a partnership.

2 Financial activities means those activities which relate to changing the size and composition of the financial structure of the entity, including equity, and borrowings not falling within the definition of cash. Examples of cash flow from financing activities includes proceeds from issuing equity instruments and outlays to buy back such instruments, proceeds from short-term or long-term borrowing and repayments of borrowings, and payments of dividents.

·

3 Cash mens cash on hand and cash equivalents. Cash on hand means notes and coins held, and deposits held at call with a bank or financial institution.

4 Cash equivalents means highly liquid investments which are readily convertible to cash on hand at the investor's option and which a company or an economic entity uses in its cash management function on a day-to-day basis; and borrowings which are integral to the cash management function and which are not subject to a term facility.



noncash items that affected net income. Thus, the indirect approach neither reports inflows and outflows separately nor describes the major outflows. Rather, it reconciles (explains the difference between) reported income and the net cash flow from operations. ( The calculation of operating cash flows using this method is often called a reconciliation of net income and cash flow from operations.) This approach is the more widely used, perhaps because it gives the reader an explanation of the differences between reported income and net cash flows from operations. Because of the popularity of the indirect method, we use it in the sample format and illustration to be presented shortly.

For simplicity the following format is used to show a skeleton outline of a formal statement of cash flow1 . The sample outline uses the indirect method of presenting cash flow from operating activities and reports important noncash activities in a separate schedule.

Example Company, statement of cash flows for 199X

Net cash flow from operating activities:

N<:<t income Adjustments for noncash expenses, revenues, losses, and gains included in income

Net cash flow from (for) operating activities Cash flows from investing activities:

Net cash provided (used) by investing activities

Cash flows from financing activities:

Net cash provided (used) by financing activities Net increase (decrease) in cash

Schedule of Noncash investing and financing Activities:

XXX

XXX

(xxx) XXX

XXX

XXX

(xxx) XXX

XXX

XXX

(xxx) XXX

XXX

XXX

XXX

XXX

XXX

Note the descriptions of the net cash flow in each of the three major sections of the statement. Depending on the transactions during the year, the net flow in any of the sections could be an inflow or an outflow and the description would be chosen accordingly.

1 Note: Significant activities not involving cash flows must also be reported in the formal cash flow statement. For example, suppose that a firm issued a 10- year note payable in a transaction to acquire machinery. Acquiring machinery is obviously an investment, issuing the note is a financing activity. Alternatively, suppose the firm settles some of its long -term debt by issuing common stock. Both the issuance of stock and the reduction of debt affect the financing of the firm. Cash Flows 48 Chapter 1


Appendix 1. II Accounting Terminology

is the process of identifying, measuring and communicating economic information to permit informed judgments and decisions by users of the information.

Accounting is an information-gathering and communication system maintained for the purposes of making decisions about the use of economic resources, for enabling effective control over the utilisation of those resources, and for evaluating the results of economic activity. It may be briefly described as a financial information system to facilitate the effective use of resources.

Bookkeeping covers the procedural aspects of accounting work, and embraces the recording function. Bookkeeping procedures are governed by the end products - the accounting reports-which in turn are determined by the informational needs of users. Such procedures must be understood in order that the accounting reports can be interpreted correctly.

Management accounting is directed towards the provision of accounting data, including that on activities within the entity and for segments of the entity's activities, for use by management. It embraces cost accounting, budgeting, and use of accounting data for decision-making, control, and evaluation purposes.

Financial accounting, on the other hand, is directed towards the overall measurement of the financial results of operations of the firm-for example, cash surplus, income, and financial position; to reporting such data to management and outside users; and to internal administration, such as ensuring that debts are paid or collected on time.

Financial accounting is concerned with aggregate data for the firm as a whole or major segments of it, whereas management accounting as conventionally defined is concerned primarily with desegregated data about the detailed internal operations of the enterprise.

Journals:

Journal are the normal books of originating entry in the handwritten accounting system. Journals are merely lists of transactions in chronological order, prepared in a specified format. The journal is a day-book in the nature of a memorandum record or diary, used as the basis for the subsequent entries in the ledger accounts.

Two main types of journals are used- a general journal, and a series of specialised journals. Specialised journals are used where there are large numbers of transactions of the same type. All transactions and events not recorded in specialised journals are recorded in the general journal.

The transaction or event is analysed according to the accounts affected and its twofold aspects, and a short description (the narration) of the transaction is given, together with any pertinent information concerning it.

Specialised journals are used to record the everyday transactions of the firm. Rather than each transaction being accorded a separate double entry in the general journal, all



transactions of the same type are merely listed in the relevant specialised journal, and the totals are posted to the main ledger accounts. This system is more economical to

operate; and, when the specialised journal includes a series of classification columns for analyses of the items, much more separate information can be provided. Typically, specialised journals are kept for cash receipts, cash payments, credit purchases, credit sales, payroll and so on.

Accounts and the Ledger:

Accounts are merely chronological lists of transactions relating to the same item and which are prepared on a double-entry basis.

The complete collection of accounts is called the ledger.

The twofold aspect of every transaction is recorded in each account as a debit aspect and a credit aspect, and a column is provided for each. A third column is provided for the balance of the account.

At the end of the accounting period, a list of the balances remaining in each account is made, and, if for every debit entry there is a credit entry, the sum of the debit and credit balances must be equal. This list is called a trial balance. Although the trial balance provides a valuable check on the ledger, it is not a conclusive test of accuracy since it will not indicate that some transactions have been omitted altogether, that some transactions have been recorded in the wrong accounts, or that incorrect amounts have been recorded. It provides nothing more than a check on the equality of debit and credit entries.

Recording the Twofold Aspect of Transactions:

All inward flows are debited in the appropriate account, and all outward flows are credited in the related account. Since each inward flow comes from a particular source and the related outflow necessarily goes to that source, or conversely each outflow from the firm goes to a particular person and the related inflow comes from that source, the rules for debit and credit may be restated simply as: debit the account into which the item flows (where to?), and credit the account from which it flows (where from?). The credit account can be referred to in general term as the source account, and the debit account as the inflow account. These two rules for debit and credit are completely general and they cover all credit transactions plus all internal transfers of values (for example, "balance day adjustment") or of resources (for example, inventory issues) within the firm where no transactions are involved.

Ledgers:

Along with the use of general and specialised journals, the ledger is divided into the main general ledger and specialised subsidiary ledgers. The general ledger comprises all the main accounts and a complete double-entry record must be maintained in it. The trial balanced are prepared from the general ledger. Subsidiary ledgers comprise supplementary accounts which record, in detail, items included in the general ledger. The purpose of maintaining subsidiary ledgers is to permit the accumulation of extra detail on some aspects of the firm's operations without unduly expanding the main



ledger and to enable the physical separation of work by the data processing staff. It is

important to recognise that entries in subsidiary ledgers are in addition to the double entry in the subsidiary ledgers. Rather, these ledgers are essentially just additional memoranda records. Accounts in the general ledger which are supported by additional data in specialised ledgers are known as control account.

The documents of transactions and events (sales invoices, purchase orders and goods received reports, creditors' invoices, electricity bills, cash receipts, and so on) are collated and sorted. These are known as the source documents or evidence records. From these documents, the transactions and events are then recorded in either specialised journals (e.g., everyday operating transactions) or the general journal (e.g., non-regular transactions, balance day adjustments, closing and reversing entries).

Job Cost System:

A job cost system is used where the products are made as individual job lots or in batches. Many different products may be made simultaneously or in succession. This is typical of small scale manufacturing, building, printing and many service industries such as car maintenance. Frequently jobs are done in response to customers orders and specifications. Job costing is also used in many professional services areas such as medical, legal, accountancy, consulting engineering and architecture, as a basis for assessing fees charged to clients.

Each individual job is a cost unit. A detailed cost record, called a job cost card or sheet, is kept for each job, and the costs incurred on it are recorded and accumulated on the cost card. It is a subsidiary ledger account of the Works-in-Process Control account while the job is incomplete, and to the Finished Goods Inventory Control account upon completion and prior to delivery. Normally, the job cost card shows the costs of direct materials issued to the job, direct labor cost charged to it, and the share of manufacturing overhead expense charged to it.

Standard Cost System: A standard cost system involves the use of predetermined product costs. In the system, product costs are predetermined at what they ought to be, and finished goods inventories are valued at this cost. The costing process also involves the measurement of actual costs, and the differences between the actual cost and standard cost are highlighted as cost variances in reports to management so that they can investigate the causes of the variances and take remedial action if warranted.

Net working capital is current assets less current liabilities.



Statement of Manufacturing Costs for year 19xx

(at standard cost)

Direct materials used direct labor charged Factory overhead expense applied

Total cost of completed production

XXX

XXX

$xxx

Income Statement for year 19xx

Sales Less Cost of Sales

Standard gross profit

Less Variances Material price Material use Labor rate Labor use Factory overhead use

Actual gross profit

Cash Flows 52

XXX

XXX

XXX

XXX

XXX

XXX

XXX

XXX

XXX

$xxx

Chapter 1

Decision Analysis

Balance Sheet for year 19xx Assets

Current assets: Cash Accounts receivable Inventories Total current assets

Fixed assets Plant and equipment Less Accumulated depreciation Fixed assets - net

Total assets

Current liabilities Notes payable (current) Accounts payable Total current liabilities

Long term liabilities Long-term debt Total liabilities

Owner's equity Common Stock Retained earnings

XXX

XXX

XXX

XXX

XXX

XXX

XXX

XXX

XXX

XXX

Total liabilities and owners equity

Income Statements

XXX

XXX

XXX

!!! XXX

(Profit & Loss Accounts for year 19xx)

Sales Less cost of goods sold

Gross profit

Less Expenses: Selling, general &

administrative expenses

Interest expenses Total expenses Net profit before tax Less tax(SO%)

Net profit after taxes

Cash Flows 53

XXX

XXX

XXX

!!!

XXX

XXX

XXX

XXX

E. Shayan

Chapter 1

Decision Analysis

Cash Flows

Jim (Pi F. i%, N) =lim 1 v = 0

�-· .v-"' (1 + i)"

Jim (F!P. i%, N) =lim (1 + i)"' = x .v-x .Y-�

. ·pf ·ot. N)- I" ( 1 + i).v-1 - 1 hm � ,-\, 1 ,o, - tm .(1 ") - -:--· N-• I +1 l

"( .. 1 ).v lim (A!P. i%, N) =lim (; 1 �t 1 = i ,_, N-"' +I -

0 0 (1 + i)"'- 1 lim (FlA. 1%, N) = hm . = :10 .v-x N-ao l

lim (AI F. i%. N) = lim (1 .),v 1 = 0 ...... _""' .v-OQ + l -

. . . 1 [ ( 1 + i)"'- 1 N ] 1 hm (PIG. z%. N) = hm -:- .(1 " )N - (1 . ).v =

"72 .V-• N-» I I + l + I I

. . . _. (!((l+i)"v-1_ N )]-t=iz l�m (GIP. z%, N)- hm . .(1 ") N (1 .).v .''i-• N-• I I +I +I

1 [ ( 1 + :) 'V -1 ] lim(F/G.i%,N)=lim -:- '. -N =x .V-x N-!IC l l

. . . . [1 "(l+i).V-1 )]-I hm (G/F. 1%, N) = hm -:- ( · . - N = 0 .V-x .V-x l • I

lim (A/G. i%. N) = lim [�- �'" ] = � .v-x .v-• I (1+1. -1 I

l�m (G/A, i%, N) = lim [�- ( '\v 1 ]-' = i ,\-x N-• I 1 + I

I

Similarly. as iCC goes to zero, the limits of compound interest factors are lim (PI F. i%, N) = 1 i-U

lim (FIP, i%, N) = 1 i-0

lim (PIA, i%, N) = N ;-o

lim (AlP, i%, N) = _Nl •-0

54

E. Shayan

Chapter 1

Decision Analysis

INTnREST llACTORS FOit 12% .INTEitf:Sl'* ...

Sing I�: Payment �

Compound • , Present-amount ·worth

factor factor

To find F To find P Given P Given 'F

F/P, i, n P/F, i; n

1.120 0.8929

1.254 0.7972

1.405 0.7118

1.574 0.6355

1.762 0.5674

1.974 0.5066

2.211 0.4524 2.476 0.4039

2.773 0.3606

3.106 0.3220

3.479 0.2875

3.896 0.2567

4.364 0.2292

4.887 0.2046

5.474 0.1827

6.130 0.1631

6.866 0.1457

7.690 0.1300

8.613 0.1161

9.646 0.10i7

10.804 0.0926

12.100 0.0827

13.552 0.0738

15.179 0.0659

17.000 0.0588

19.040 0.0525

21.325 0.0469

23.884 0.0419

26.750 0.0374

29.960 0.1)334

33.555 0.0298

37.582 0.0266

42.092 0.0238

.C7.143 0.0212

52.800 0.0189

93.051 0.0108

163.9BS 0.0061

289.002 0.0035

Cash Flows

•

Compound· amount

factor

To find F

Given A

F/A, i, n

1.000

2.120

3.374

4.779 6.353

8.115

10.089

12.300

14.776

17.549

20.655

24.133

28.029

32.393

37.280

42.753

48.884

55.750

63.440

72.052

81.699

92.503 104.603

118.155

133.334

150.334

169.374

190.699

214.583

241.333

271.293

304.848

342.429

384.521

431.664

767.091

1358.230 2400.018

;

Equal Payment Series

Sinking-fund

factor

To find A

Given F A/F,.i, tt

1.0000

0.4717 0.2964

0.2092 0.1574

. 0.1232

0.0991

0.0813 0.0677 0.0570

0.0484 0.0414 0.0357

0.0309

0.0268

0.0234

0.0205

O.o179

0.0158

0.013 9

0.0123

0.0108

0.0096

0.0085 0.0075

0.0067

0.0059

0.0053 0.0047

0.0042

0.0037

0.0033

0.0029

0.0026

0.0023

0.0013

0.0007

0.0004.

55

Present-worth factor

To find P Given A P/A, i, tt

0.8929

1.6901

2.4018

'3.0374 3.6048

4.1114 4.5638

4.9676· 5.3283 5.6502

5.9377 6.1944

6.4236

6.6282

6.8109

6.9740 7.1196

7.2497 7.3658 7.4695

7.5620

7.6447

7.7184 7.7843 7.8431

7.8957

7.9426 7.9844

8.0218

8.0552.

8.0850

a·. 1116

8.1354

8.1566 8.1755

8.2438

8.2825

8:3045, .. ... , ... ....

E. Shayan

Uniform Capital- gradient-recovery series

factor factor

To find A To find A

Given P GivcnG A/P, i, n A/G, i, 11

1.1200 0.0000

0.5917 0.4717

0.4164 0.9246

0.3292 1.3589

0.2774 1.7746

0.2432 2.1721

0.2191 2.5515

0.2013 2.9132

0.1877 3.2574

0.1770 3.5847

0.1684 3.8953

0.1614 4.1897

0.1557 4.4683

0.1509 4.7317

0.1468 4.9803

0.1434 5.2-147

0.1405 5.4353

0.1379 5.6427

0.1358 5.8375

0.1339 6.0202

0.1323 6.1913

0.1308 6.3514

0.1296 6.5010

0.1285 6.6407

0.1275 6.7708

0.1267 6.8921

0.125 9 7.0049

0.1253 7.1098

0.1247 7.2071

0.1242 7.2974

0 1237 7.381 1

0.1233 7.4586

0.1229 7.5303

0.1226 7.5965

0.1223 7.6577

0.1213 7.8988

0.1207 8.0572

. 0.12�.4 ·. 8.1597 . . . ..

Chapter 1


Chapter 2

DECISION MAKING UNDER UNCERTAINTY

We have all had to make important decisions where we were uncertain about factor that were relevant to the decisions. In this section we will examine situations in which decisions are made in an uncertain environment.

The following model encompasses several aspects of making a decision in the absence of certainty. The decision make-r first chooses an action from a set A= { al, a2, a3 .... an }of available actions. Then the state of the world is observed. Let pj be the probability that the state of the world is observed to be sj from the set S = { sl, s2, s3, ... } . If action ai is chosen and the state of the world is sj, the decision maker receives a reward rij. We refer to this method as the "state of the world decision-making model".

The following example looks at a generalised situation, however, it still depicts the fundamentals of decision-making in an uncertain environment. The scenario utilises inventory control with the objective of buying just enough (in other words as much as you can sell). This question is a valid one in practice where almost all manufacturing industries strive to purchase as much inventory as required from the demand for products set by the consumer.

CASE2.1 Pie Producer

Sallyanne Gillespie(SG) produces and sells Pies . Each day SG must determine how many Pies to produce. SG 's cost is 45c per Pie which sells for 60 each. Unsold Pies at the end of the day are worthless. SG knows that each day she can sell anywhere between 41 and 50 Pies in the space of one hour in the lunch time. With the possibility of selling 41 or 50 Pies being equally likely. We will show how this problem - case study fits into the "state of the world model".

Solution: In this example, the members of S = {41,42,43,44,45,46,46,47,48,49,50} are the possible values of the daily demand for Pies. We are given that P41 = P42 =

P43= P44 = P45 = P46 = P47 = P48 = P49 = PSO = 1/10. SG must choose an action ( the number ofPies to produce each day) from A= ( 41,42,43,44,45,46,47,48,49,50 }.

If SG produces i Pies and j Pies are demanded, min (i, j ) Pies are sold earning a net profit of rij.

where rij = 60i- 45i =lSi ( i<j) 60j - 45i ( i >j)

Values of rij are given in table 2.1

Uncertainties 56

Chapter 2


Table 2.1.1 Rewards for SG Pies example Papers papers Demanded (S.c) Ordered

41 42 43 44 45 46 47 41 6.15 6.15 6.15 6.15 6.15 6.15 6.15 42 4.70 6.30 6.30 6.30 6.30 6.30 6.30 43 4.40 4.85 6.45 6.45 6.45 6.45 6.45 44 4.25 4.70 6.15 6.60 6.60 6.60 6.60 45 4.95 4.40 4.85 6.30 6.75 6.75 6.75 46 4.65 4.10 4.55 6.00 6.45 6.90 6.90 47 4.35 4.80 4.25 4.70 6.15 6.60 7.05 48 4.05 4.50 4.05 4.50 4.95 6.40 6.85 49 3.75 4.20 4.65 4.10 4.55 6.00 6.45 50 3.45 3.90 4-35 4.80 4.25 4.70 6.15

2.1.2 The Maximum Criterion

For each action, determine the worst outcome (smallest reward). criterion chooses the action with the best "worst" outcome.

48 49 6.15 6.15 6.30 6.30 6.45 6.45 6.60 6.60 6.75 6.75 6.90 6.90 7.05 7.05 7.20 7.20 6.90 7.35 6.60 7.05

For example we obtain the results in Table 2.1.2. Thus the maximin criterion recommends ordering 41 units. This ensures that SG will, no matter "what

50 6.15 6.30 6.45 6.60 6.75 6.90 7.05 7,20 7.35 7.50

the state of the world", earn a profit of$6.15c. The maximin criterion is concerned with making the worst possible outcome as pleasant as possible. Unfortunately choosing a decision to mitigate the worst case may prevent the decision-maker fro taking advantage of good fortune. For example, if Sallyanne follows the maxmin criterion, she will never make less than $6.15c, but she all never make more than

$6.15c.

Table 2.1.2 Computation of Maximin Decision for Example

papers Worst State Re,ard inaorst State Ordered of the world of the world

41 42 43 44 45 46 47 48 49 50

41,42,43,44,45,46,47,48,49,50 41 41 41 41 41 41 41 41 41

2.1.3 The Maximax Criterion

$6.15 $4.70 $4.40 $4.25 $4.95 $4.65 $4.35 $4.05

$3.75 $3.45

For each action determine the best outcome (largest reward). The maximax criterion chooses the action with the "best" outcome. For example we obtain the results from

the Table 2.1. 3. Thus the maximax criterion would recommend ordering 50 units. In

Uncertainties 57

Chapter 2


the best state (when 50 units are demanded) this yields a profits of $7.50c. Of course, making a decision according to the maximax criterion leaves Sallyanne open to the disastrous possibility that only 4 Pies shall be demanded, in which case she loses money because left-over Pies are not refundable.

Table 2.1.3 Comp-Litation ofMaximax Decision for Pie Vendor Pies State Yielding Best outcome

Best Outcome $ 41 41,42,43,44,45,46,47,48,49,50 6.15 42 4 2,4 3,4 4,4 5,4 6,4 7,4 8,4 9,5 0 6.30 43 4 3,4 4,4 5,4 6,4 7,4 8,4 9,5 0 6.45 44 44,45,46,47,48,49,50 6.60 45 45,46,47,48,49,50 6.75 46 46,47,48,49,50 6.90 47 47,48,49,50 7.05 48 48,49,50 7.20 49 49,50 7.35 50 5 0 7.50

2.1.4 Minimax Regret

The minimax regret criterion uses the concept of opportunity cost to arive at a decision. For each possible state of world sj , findan action i*G) that maximises rij. That is, i*G) is the best possible action to choose if the state of the world is actually sj. Then for any action ai and state sj, the opportunity loss or regret for ai in sj is ri+(j)J - rij. For example, if j = 42 units demanded, the best decision is to order i*(42) = 42 units, yielding a profit of r = 42(60)- 42(45) = $6.3 Suppose we chose to order i =41 units. Since r = 41(60)- 41(45) = $6.14. Therefor the loss or regret for i= 41 and j = 42 is $6.30 - $6.15 = $0.14. If we order 41 units in hindsight we realise that by making the optimal choice in this case ordering 42 units we would have done $0.15 better than we did by ordering 41.Table shows the opportunity cost or regret matrix. The minimax regret criterion chooses an action by applying the minimax criterion the regret matrix. In other wordsit attempts to avoid disappointment over what might have been. From the regret matrix in Table 2.1 we obtain the minimax regret decision in Table 2.1.4. Thus the minimax criterion recommends ordering 44 units.

Table 2-1-4 Regret Table for PiesVendor Example Units Units Demanded

41 42 43 44 45 46 47 48 49 50 41 0 .15 .30 .45 .60 .75 .90 1.05 1.20 1.35 42 .45 0 .15 .30 .45 .60 .75 .90 1.05 1.20 43 .75 .45 0 .15 .30 .45 .60 .75 .90 1.05 44 .90 .40 .30 0 .15 .30 .45 .60 .75 .90 45 1.20 .90 .60 .30 0 .15 .30 .45 .60 .75 46 1.50 1.20 .90 .60 .15 0 .15 .30 .45 .60 47 1.80 1.50 .20 .90 .45 .30 0 .15 .30 .45 48 2.10 1.80 1.40 1.10 .65 .50 .20 0 .15 .30 49 2.40 2.10 1.80 1.50 1.05 .90 .60 .30 0 .15 50 2.70 2.40 2.10 1.80 1.35 1.20 .90 .60 .30 0

Uncertainties 58

Chapter 2


Table 2.1.5 Computation ofMininiax Regret Decision for Pies Vendor Example Papers ordered Maximum Regret

41 1.35 42 1.20 43 1.05 44 0.90 45 1.20 46 1.50 47 1.80 48 2.10 49 2.40 50 2.70

2.1.5 The Expected Value Criterion

The expected value criterion chooses the action that yields the largest expected reward For this example the expected value criterion would recommend ordering 44 units This can be seen in Table 2.1.6 as 44 units receives the highest reward of $6.33

Table 2.1.6 Computation ofExpected Value Decision for News Vendor Example Units Workings Expected Reward 41 1 I 10 (10 X 6.15) $6.15 42 1/ 10 (9 X 6.30 + 4.70)= $6.24 43 1/ 10 (8 X 6.45 + 4.4 + 4.85) $6.29 44 1/10 ( 7 X 6.60 + 4.25 + 4.70 + 6.15 )= $6.3 3 45 1/10 ( 6 X 6.75 + 4.95 + 4.40 + 4.85 + 6.30)= $6.30 46 1/10 ( 5 X 6.90 + 4.65 + 4.10 + 4.55 + 6.00 + 6.45)= $6.22 47 1/10 ( 4 X 7.01 + 4.35 + 4.80 + 4.25 + 4.70 + 6.15 + 6.60)= $6.10 48 1/10 ( 3 X 7.20 + 4.05 + 4.50 + 4.05 + 4.50 + 4.95 + 6.40 + 6.85)= $4.99 49 1/10 ( 2 X 7.35 + 3.75 + 4.20 + 4.65 + 4.10 + 4.55 + 6.00 + 6.45 + 6.90)=$4.73 50 1/10 (3.45 + 3.90 + 4.35 + 4.80 + 4.25 + 4.70 + 6.15 + 6.60 + 7.05 + 7.5)= $4.48

2.2 RISK ASSESSMENT & RETURN THROUGH MARKETING STUDIES

2.2.1 Introduction To Risk Assessment And Return

There is one major rule that normally describes the relationship between risk and return, which is : " The higher the risk, the greater the return." This means that the investor must be able to decide when the return outweighs the risk and go ahead and invest his money!

Risk basically refers to the "volatility of returns in various sorts of investments" Which means the possibility of receiving no return or a lower than expected return o investment.

Uncertainties 59

Chapter 2


The share market offers one of the greatest challenges for a decision maker in that i has all of the ingredients described above. 'Me major factor being 'risks'. However the idea of investing is to receive a higher return than would be made if the money were placed in a bank and interest received.

Case 2.2: 'Truck Stop Analysis

The Director of Truck Stop Analysis (TSA), Dr. Martin Montgomery, undertook feasibility study to predict future outcomes so as to examine whether or not the implementation of a new (latest technology) truck analysis testing plant Transportation Aust. in North Ringwood would be viable before purchasing the equipment from Canada.

Economic Feasibility

The economic feasibility of Truck Analysis Testing Facility ( TATF ) at Transportation Aust. has been re-evaluated based on:

1. Acquisition of quotes for the capital investment for the purchase, shipping and delivery of the VL T equipment.

2. Estimation of the capital investment for the installation of the equipment. 3. Estimatiot of the annual operating cost of the TATF only (excludes NAB) 4. Estimation of the five year rental lease agreement for the site. 4. The market survey interviews with 20 potential customers representing large

segment of the major suppliers of Prime Movers and Trailers, and Fleet Operators.

Findings were as follows:

(A) CAPITAL INVESTMENT($000's)

- Purchase of VL T Equipment . 7 5 - Installation of the TATF ( TSA cost only) excludingTransportation Aust. 133 - Total capital investment by TSA 208 -Total capital investment by TSA and Transportation Aust. 258

(B) OPERATING EXPENSES FOR TATF ONLY

1. Salaries and Wages -Dr. Martin Montgomery (50%) - Engineer (50%) - Operator (base salary plus 20% overtime) - Superannuation - Workcare

Total Salaries and Wages

2. Employee Expenses ( based on 1993 I 94 Plan ) - Travel I Hire -Accommodation - Taxi's, Couriers and parking

Uncertainties 60

Chapter 2

90 18 36 4.3 2 150

5 3 3

Decision Analysis

-Petrol (80%)

- Car allowance (50%)

- Entertainment

-All others (50%) Total employee Expenses

3. Office Supplies

-Stationary and Postage (75%) - Food and beverages -Others

Total Office Supplies

4. Utilities

-Phone and Fax (50%) -Power and Water -Rates

Total Utilities

4. Lease Equipment

6. Plant -Plant Rental - Insurance estimate -Security - Maintenance repair

Total Plant

7. Miscellaneous - Accounting and Legal Fees - Bank Charges - Advertising and Promotion - Company Tax

Total Miscellaneous Total operating Cost Per Annum (p.a.) Add 5% contingency

Total Cost

(C) Income

Income from TATF is generated from two types of services namely: 1. axle alignment and brake testing on a frequent or once-off basis, or

E. Shayan

1.0

4

15 4 6 40

3.0 2.0

6. 0

5 8 1.2

14.2

4. 4

24 4 2 2

32

2 3 3 1 9

257 13

270

2. Special testing for load axle weighing and brake timing between the Prime Mover

and the Trailer.

The marketing survey conducted on 20 potential customers revealed that the maximum charge the customers can bear is in the order of $300 per test for the service carried out in item 1 from above, and $500 per test for item 2 above. It is estimate that 80% of customers will request services for item 1, and 20% for item 2. On that basis the average income will be $340.

(D) Economic Analysis

Uncertainties 61

Chapter 2


The following economic analyses were carried out to test the sensitivity of the income or the number of tests per annum that need to be generated to yield a good return o investment.

CASE CAPITAL($) SAVING ($) ga. DCF% 1 208,000 50,000 -4 2 208,000 60,000 1 3 208,000 70,000 6 4 208,000 98,000 20 5 258,000 70,000 0 6 258,000 80,000 3 7 258,000 100,000 11 8 258,000 126,000 20

Where DCF means discounted cash flow analysis. For further details on the calculations of the above data, refer to figures 2.2.1 & 2.2.2

DCF%

20

15

10

5

0 40 50 60 70 80 SAVINGS -$000's

90 100

$98

110

Figure 2.2.1 Maximum DCF% at $208,000 Capital Investment

This economic analysis indicates that an annual saving of$126,000 needs to generated above the annual operating cost of $270,000 which is based on an average income per test of $340 equating to a total of 1165 tests per annum to yield a 20 minimum based on capital investment of$258,000.

Based on a capital investment of$208,000 and a saving of$98,000,1080 tests per annum has to be generated to yield a return on investment of 20%.

DCF%

25

20

15

Unc rtainties Chapter 2

Decision Analysis

10

5

0 60 70 80 90

SAVINGS - $000's

100 110 120 130

Figure 2.2.2 Maximum DCF % at $258,000 Capital Investment

(F) Marketing Survey

E. Shayan

Estimated maximum no. of Road Based Tests p.a. based on following n-iarketin survey.

VOVLO STEVENSON ML

TNT MACK SHELL

BUS ASSOCIATION

0 0 0 0 0

0 0

COOTES 40*

MOBIL 50 BP 50 AMPOL 30 CAL TEX (LINFOX) 25 AUSTRALIA POST 40

KENWORTH 30 MERCEDES 30

MAXICUBE 20 E.A.ROCKE 30

HOCKNEY 25 FREIGHTER 25

INTERNATIONAL 15

TOTAL TESTS p.a. APPROXIMATELY 400

* The whole fleet tested once only and no guarantee of a repeat business. The marketing survey highlighted the following points worthy of mention.

1. Seven out of the twenty customers surveyed ( 35% ) we'll not use the truck testing facility.

2. One out of the twenty customers (5%) surveyed will initially test all his fleet. 3. The remaining 60% will only use the truck testing facility on a needs basis

- infrequent, depending on the operating problems, a new design, or a perceived problem by the driver.

4. The unknown factor to determine is the percentage of the total market the 20 customers represent. Therefore to test the sensitivity of the number of tests per annum available to Truck Stop Analysis from the work sampling carried out on 20 customers, the folioing assumptions are made in terms of % of the market the survey represents

% OfMarket Potential Number Of Tests Max. 50%

Uncertainties 63

Max. Full

50%

Chapter 2

Decision Analysis

25 50

75

400

400 400

200 200 200

1600

800

533

800

400

266

The truck testing facility needs to generate at least 1 000 tests per annum t make it economically attractive. 'Merefore the marketing survey ( 20 customers) has to equate to 3 0% of the total market potential customers in Melbourne.

E. Shayan

4. 20 customers selected for the marketing survey represent more than 50% of the potential market in Melbourne for the following three reasons.

a. All the major Prime Movers and Trailer manufacturers were surveyed. b. All the Oil Company Fleet operators were selected. c. A selected number of the large fleet operators were also included.

6. The remaining potential fleet operators that were not included in the survey are unlikely to generate the required amount of tests to be completed to make the truck analysis project a viable one.

7. Some of the operating problems identified by the customers surveyed are:

Conclusions:

Unfortunately, the marketing survey did not support the required number of tests to make this project an economically viable proposition. The market environment remains uncertain, it can not be predicted. The surveys showed little support by the customers to use the TATF on an on-going basis. 'Me only issue that will give it an incentive for the transport operators to utilise the TATF and make it an economically attractive proposition, is forced legislation by the Government of Victoria. Even if this situation eventuates, the major customers all no doubt investigate the economics of installing their own testing facility, rather than test outside and incur downtime penalty in travel and a service fee.

Note also that the above economic analysis did not take into account the penalty associated in the cancellation of the 5 year lease if TATF terminated the operation less than the 5 year stated period. For example, if the operation ceased after a year the TATF would have had to pay Transportation Aust. a penalty of$100,000.

Considering the high cost of investment, the lack of support from the market place and the risks associated with the rental of the property from Transportation Aust. it was recommended that the project not proceed any further.

Case 2.3 Coles Myer Versus Bt Global

The second example in this section deals with the stock market. It examines the opportunities available to an investor, and how an 'uncertain' market can, With the

help of research data, become more attractive to the investor/decision-maker. In

Uncertainties 64

Chapter 2


other words reducing the risks of the 'uncertain' environment by simple understanding or gaining as much knowledge of it prior to any commitments being made.

An investor has $5000 dollars and wishes to invest the money where the return of

investment is the highest. However, the investor does not want to take too great a risk so intends to invest his $5000 for a term of five years. The investor finds that the best interest rate currently being offered for $5000 is from the Bank Of Melbourne for a term deposit of five years duration at 9.25%. The investor decides that t determine the rate of return from the stock market, past history of share fluctuation needs to be looked at, a trend found and future fluctuations predicted.

The investor decides to employ a financial adviser to select possible shares and to the actual purchasing of the shares, as the investor has no experience of the stock market. A prospectus for industrial shares, as these are longer tern, low risk shares is obtained. Coles Myer and Bt Global Asset Management Limited are the companies selected from the advice given ( othe company's shares may be recommended, but for the sake of this example only tw companies will be looked at). 'Me rate of return on investment from the last five years for both companies is studied.

1. An investment of$5000 in Coles Myer from 1988 to 1992 would have return $6290.32 that is a total of $11290.32 this is an interest rate greater than 9.25% as the total return would only have been $7781.75 if invested in the bank. However, the period of 1988 to 1992 was a time where the bank interest rates were decreasing. This usually results in stock market prices increasing. However currently (1995), bank interest rates are increasing indicating that the stock market prices will reduce. Cycles need to be looked at so that the investor can make an estimate of how interest rates, inflation and the economy are going to affect the investment. The investor must also consider forecasted budgets by the government as this inva@ably influences the market.

2. An investment of $5000 in the'Bt Global Assets from the same time span as above

would have returned $10454.54.

The investor must make a decision by looking at the graphs from both companies past performances. He determines that the general trend of both graphs is upwards (see figures 2.2.3 & 2.2.4). The Coles Myer graph shows an overall trend which indicates that the share prices are on the rise. The Bt Global Asset Managemen Limited graph shows that share prices before 1993 were fairly steady. After 1993 the share prices seem to be increasing at a good rate and show a trend for increasing more rapidly than the Coles Myer shares.

With the help of the adviser (another decision-maker), the investor determines-nines that greater return will be achieved from the stock market with these shares than from any bank for the next five years.

The investor decides that he all divide up his $5000 so that he can invest in both the companies. The reasoning behind dividing up his shares is as follows: If over 50 shares in Coles Myer are bought, additional benefits and discounts are offered to the shareholder. Each time they make a purchase at a Coles Myer store the sales price i

Uncertainties 65

Chapter 2


reduced. This is an additional saving. Since the investor does purchase goods fro these stores at least 500 shares of Coles Myer will be purchased.

The Melbourne 'Age' of 29 I 03 I 95 lists the Coles Myer share price at $4.55, therefor $2275 worth of shares need to be invested. Since the Coles Myer past history ha shown a steadier share price increase over a number of years (see figures 2.2.3 & 2.2.4 than the Bt Global Asset Management Limited shares, a greater proportion of the $5000 will be invested with Coles Myer. Since an even-hundred number of share must be purchased, 700 Coles Myer shares will be purchased. The remaining fund will purchase an even number ofBt Global Assets Management Limited.

The investor in the next months should study the stock market to determine-nine the time to buy i.e. when the prices are low - this will be an essential part of the decision making process. As these shares normally do not fluctuate markedly over a period of a few months and are on a general upward trend, the time for buying is now as the prices of the shares are on the increase. If these were short ten-n shares the fluctuations of each day of trade will have to be studied and plotted very carefully.

For the next five years the investor needs to monitor -regularly the shares he has purchased and the All Ordinaries Index. He should keep charts of monthly average so that he can study trends and cycles (good decision-making practice). Any additional company information such as prospectuses should be obtained and studied each year Other information such as rumours of the chosen companies making loses should b studied and the investment term reviewed

However, the source of such rumours needs to be reliable, keeping in mind that rumours may be just that! Although the before-mentioned studies of the stock market should be carried out for the five year period, the stock chosen and the investment term are a good combination for security and high returns and are recommended aspects of good practice in the investment decision-making process.

Share Price

5000 "

i� r'f """ 4000

II "\. _,... L 3000

2000 ( I ...A.. I

fV 1/

1000

1986 1987 1988 1989 1990 1991 1992 1993

Figure 2.2.3 Coles Myer Share Prices Over Recent Years.

The comments and statements made above do not mean that if you follow the step and watch for the signs, you will succeed in making money on your investment! It is obvious that there are no guarantees when it comes to the stock market, however, if you broaden your understanding and knowledge of the subject and carefully look for the signs by monitoring your stocks, you should be in a better position to make the best decision when it is required of you.

Uncertainties 66

Chapter 2

Share Price

1500

1400

1300

1200

1100

1000

900

800

700

Uncertainties

---

1991

J

1 ............... 1

�"-._ - ......... I .......... I"""" �ll

1992 1993

Bt Qobal Asset Management Ltd

Share Prices Over Recent Years.

Figure 2.2.4

67

11

� ._1 �

Chapter 2


Uncertainties 68

Chapter 2


Uncertainties 69

Chapter 2


Chapter 3 DECISION TREES

Decision tree is a graphical representation of a sequence of decisions taken as state of the nature changes along time. The idea is to put together all the possibilities in advance in order to make a decision on which steps to take in order to achieve better results. When the future dirction is decided, it is a decision point and when the future dirction is determined as a result of a random phenomenon it is called a chance event on which the decision maker has no control. However by contemplation on these events in advance, corrective actions may be taken to avoid worst cases.

3.1.1 Formulating a Decision Tree

The decision tree is formed, sequentially, front the left to the right, as decision point and chance points, both known as nodes, are envisaged in time. Decision nodes are indicated by squares and chance nodes by circles. Connecting the nodes are branches.

Branches emanating to the right of a decision node represents the course of action chosen at that decision, and a branch emanating from an event node represents the event that can occur at that point. Costs or returns can accompany any of the decisions, along the decision branch. At every decision node the decision-maker must select one branch.

At the ends of the branches all costs and rewards are totalled to give the outcome of that path. The analysis then starts from this point on the Tight and works backward

towards the goal [ 46].

3.2.1. Decision Tree Analysis - Portfolio Example

A decision tree contains within its branches all possible outcomes at a given stage of the decision-making process. Therefore, when one adds the probabilities of the end points of the branches, the sum will be one, much as the probability of tossing head or tails on the flip of a coin is one.

Figure 3 .1 .1 is an example of the application of decision-tree analysis. Assume that there are 1 million shares of stock outstanding. There are three key variables which subjectively determined probabilities that have been highlighted by the analyst base on his experience : sales ( S ), expenses (E), and Price Earings ratio (PIE). As seen the partially completed tree, sales can be $12 million, $11 million, and $10 million with probabilities of0.2,0.5, and 0. 3 respectively. Expenses can be either $4 million or $8 million, with probabilities of0.4 and 0.6 respectively; and finally, the price earning ratio can be 30, 20, or 10 with probabilities of0.2, 0.6 , 0.2 respectively. Even in the simplified example, the inherent advantages and disadvantages of this technique can be seen [23].

Decision Trees 70 Chapter 3

Decision Analysis

PRICE �-------_. $240(0.16)

P/E 10(02)

Figure3.1.1 PARTIAL DECISION TREE.

E. Shayan

We see that a complete set of possible outcomes (prices), together with the probabilities of occurrence, is generated from the analysis. It enables the analyst to set up a frequency distribution of prices in the future. With current price and

projected dividend, the analyst can calculate the frequency distribution of expected returns.

From the above figure the maximum piece of a share of this stock obtainable in one year is $240 which can be seen in figure 3 .1.1 at the end of the top branch. The probability of this is about 2 percent. We also see that the minimum possible price will be $20, with a probability of about 4 percent. Two intermediate prices that are considerably more likely to occur are also given.

Tracing the branches leading to a piece of $140 and a probability of 12 percent, we se that sales of$11 million and expenses of$4 million are projected. Since we are summing I million shares outstanding earnings per share ( EPS ), would be 7 =[($11m

- $4 m) I $1m]. The price earnings in this sen'es of branches is 20. Tis leads to a place of $140 20 x $7). To obtain the likelihood of this event, we must ultiply together the probabilities associated with these levels of sales, expenses, and PIE. From the tree we see that these are 0.5, 0.4, 0.6. When multiplied together, they come to 0.12 or 12 percent. The other branches can be completed by redoing this process.

When all other sequences of events (branches) are completed and the ptices have bee converted to Holding Peiiod Yield (HFY), where HPY equals:

HPY = [(Pl- PO)+ D1] I PO

where: P1 =price one year front now of one share of stock. Po = current price of one share of stock. Dl =dividends received on one share of stock one year from now.

The analyst can calculate a measure of central tendency, such as the mean, and measure of dispersion about the mean, such as the standard deviation. With all wor completed, the analyst and investor are in a much better position to make a



Informed judgement about the metits of this stock purchase, since they have measures of profitable returns and risks, rather than only a poirit of estimate of return.

Analysis has also cited additional advantages:

1. Better investment decisions should result from a procedure that demands more intensive analysis. Breaking down investment uncertainties into manageable parts and analysing them individually should lead toimproved decisions when the parts are systematically combined.

2. Communication of investment ideas should be improved by the ability to express degrees of uncertainty more precisely, and by proading those taking investment action a deeper understanding of the thought processes behind the final recommendation.

In summary, we see that decision tree analysis is particularly useful when the number of sequential decisions (sales, expenses and PI E, for example) is limited, a manageable number of alternatives outcomes are possible, and the analyst can assess the associate probabilities.

On the surface, decision-tree analysis seems to) have done away xm'th most of our objections to other techniques of company analysis. Unfortunately, we should no celebrate to soon. In real world situations, there would be many possible alternatives and many steps before we arrived at a final solution. Under these circumstances, the number of calculations and the plotting of a decision tree would become extensive an quite an arduous task. Therefore we need another technique that does not require u to specify all the possible branches in our tree. The use of a computer would also help Such a technique, would be known as Simulation and is illustrated 1-n greater detail in section 4 of the report.

3.1.3 Probability Relationships With Decision Trees

The following examples illustrate the use of probability distributions in decision problems. The examples are set up in such a way as to take the reader through the example thoroughly and at the same time emphasising the main concepts o probability distributions. The examples will contain discussions regarding decision problems encountered [38],[25].

The first example looks at a furniture manufacturer ( Pinewood Designs)

Case 3.1.1 Pinewood Designs manufactures furniture products.

The manager, Jim Pot, has a decision to make on what type of trlick to purchase for use in the company operations. The truck purchased must be able to pick up raw material, make deliveries and transport sample products to exhibitions. The manager does not wish to tie up a lot of money in a large truck if demand is slow in the next year. However, if demand is high, the company's capacity to meet the demand will be lessened if a small truck is all they have to perform the necessary functions with. There are three choice of trucks available to Jim to purchase. A small truck, A standard size truck, A large flatted truck.


Decision Analysis

It is expected that the next year sales will fall into one of four categories:

Level (1)

Level (2) Level (3) Level (4)

$0 - $20,000 (low) $20,000- $40,000 $40,000 - $60,000 $60,000 + (high)

E. Shayan

Considering the price of the trucks, the sales levels and the cost associated with hitting of other trucks in case of a wrong decision, jim and his marketing manager have the

following pay off table: (Pay off is the reward obtained when a particular decision is made and a certain event o state of nature occurs. The table containing this information for all alternatives an events is called a Pay off table).

Sales Levels ( Events )

Type of Truck I (low) 2 3 4 (high) Action

Impoll 20 10 15 25

Standard 15 25 12 20

Large Hatbed -20 -5 30 40

Pay off table for Pinewood Design in $000 profit for next year

A payoff table can always be converted to a loss table. The elements of a loss table represent the 'regret' or 'opportunity losses', for not choosing the alterative corresponding to the highest pay off. Calculation of loss table for Pinewood Design is described below:

Start with column 1. The highest entry in that column is 20, so we subtract each entry in column I from 20 to obtain the corresponding loss. The three losses are calculated to be 20-20 = 0, 20-15 = 5 and 20- (-20) = 40. Thus column I entries become 0.5 and 40. Note the meaning of these losses: If event 1 occurs and if we had purchased the import, we would have chosen the best action possible give that event 1 occurs - that is, our regret or opportunity loss would be 0.

If we had purchased the standard truck, the payoff would be $15,000, which is $5,000 less than the best possible payoff, our regret or opportunity loss then would be $5,000. Similarly, the opportunity loss for purchasing the flatbed truck would be $40,000.

Repeat the above conversion procedure for columns 2,3 and 4. The resulting loss table is shown for Pinewood Design in $1,000.

Import Sales levels

Standard 1 2 3 4

Flatbed 0 15 15 15

5 0 18 20 40 30 0 0



Maximin (Pessimist's Decision Model)

The decision-maker has a pessimistic view about the states of nature or perhaps feel because of economic insecurity, high losses must be avoided at the risk of possible

loosing high profits. Therefore, lean towards using the decision model known as The Maximin model. The major concept behind this model is the avoidance of high or

unacceptable losses.

To implement this model, we determine the lowest outcome for each alternative (strategy), and then select the strategy having the highest of these lowest outcome and therefore maximising the minimum outcomes (this procedure was also used in

the news vendor example- see section 2.2.1 of the report). The procedure may be described -using the Pinewood example as follows :

Identify the minimum pay off for each decision alternative. The minimum pay off if the imp ott is purchased is I 0. The minimum pay off if the standard

truck is purchased is 12. The minimum pay offif the flatbed is purchased is -20.

Select the decision alternative with the largest minimum pay off The maximum of the above pay off is 12, corresponding to the decision to purchase the standard truck.

Thus, using the maximin rule, Pinewood should decide to buy a standard size pick -up truck.

Maximum ( Optimist's Decision Model)

The decision-maker here chooses to view the environment as being friendly and she is optimistic about the outcome rather than pessimistic. Under such an assumption the decision-maker determines the highest pay off for each alternative and the chooses the

maximum of these.

The maximax decision model can be explained as follows using the Pinewood

example:

Identify the maximum pay off for each decision alternative. The maximum pay off if the import is purchased is 24. The maximum pay off if the standard is purchased is 24.

The maximum pay off if the flatbed is purchased is 40.

Select the decision alternative with the largest pay off The maximum of these pay off is 40, corresponding to the decision to purchase the flatbed. Thus, using the maximax rule, Pinewood should purchase the flatbed truck.

Note that by doing so, Pinewood could realise a first year profit of$40,000, but if sales are in range 1, there is the risk of losing $20,000.

Minimax Regret Model

In the minimax model the decision-maker tries to minimise the maximum opportunity loss or regret that might be experienced. The model's described in the following steps :



IdentifY the maximum possible loss for each decision alternative. The maximum loss if the import is purchased is 14. The maximum loss if the standard truck is purchased is 20 and if the flatbed is purchased is 40.

Select the decision alternative having the smallest loss. The minimum of these losses is 15, corresponding to the purchase of the import truck, thus, using the minimax rule,

Pinewood should decide to purchase the import truck .

Probabilistic Decision Models

When the decision-maker has access to prior data, or knowledge based on a past experience, he/she can assign probabilities to various events, it is then possible to use a

probabilistic decision model called Bayes Model.

The Bayes Criterion selects the alternative which has the maximum expected pay off if a loss table is used, the Bayes Model selects the alternative with expected loss.

The Bayes Decision Model ( maximising expected pay off) is implemented as follows:

For each decision alternative, compute the expected pay off This is done by weighting each pay off in the row corresponding to the decision alternative by the probability of the corresponding event and then summing these terms.

Select the decision alternative having the maximum expected pay off. This decision is called a Bayes Decision. Ties are broken arbitrarily. Notationally, we shall let R denote pay off(reward) and L denote loss. Also the expected pay offif we choose action A will be written ER(A).

Let Pj = probability of event j occurring where L Pj = 1 Oij = pay off of alternative i given the event j has occured. Ai, i = 1,2 ...... m are alternative or strategies to choose from.

Then A*= Max L Pj Oij is the Bayes strategy.

Suppose Jim, the manager ofPinewood Design (previous example), can assign The following probabilities to the levels of sales for the next year:

Level Probability Low 1 0.20 Low 2 0.35 Low 3 0.30 High 4 0.15

The Bayes decision using the maximum expected pay off would be as follows:

Expected return (ER) for each truck would be ER (import )= 0.20(20) + 0.35(10) + 0.30(15) + 0.15(25) = 14.75 ER (standard)= 0.20(15) + 0.35(25) + 0.30(12) + 0.15(20) = 18.35 ER (flatbed )= 0.20(-20) + 0.35(-5) + 0.30(30) + 0.15(40) = 9.25



The strategy that minimises the expected pay off is al which is buying the standard

truck.

Bayes decision using the loss table:

EL EL (import ) EL ( standard ) EL ( flatbed )

= represents expected loss = 0.20(0) + 0.35(15) + 0.30(15) + 0.15(15) = 0.20(5) + 0.35( 0 ) + 0.30(18) + 0.15(20)

= 0.20(40) + 0.35(30) + 0.30 (0) + 0.15 (0)

= 12

= 9 .4 = 18.5

Therefore the strategy that minimised the expected loss is the second strategy that 1 buying the standard truck, which is the same as the Bayes strategy using maximum

expected pay off.

Case 3.1.2 Adabama Flechic - Bidding on Supply of Instruments for American Paper Company-

Alabama Electric manufactures and installs instruments throughout the Paper Industry in Amelica. Alabama Flectn'c has decided to bid on supplying American Paper Company with 100 of its most recent type of instruments. It estimates it can build them for $4,000 each; the President of Alabama Electric has decided to offer the

100 instruments for $500,000. Alabama Electric's only real competitor is Southern Electric, a company with the same reputation for reliability and quality as Alabam Electric. For this reason, the president of Alabama Electric is sure the order will be given to the lower bidder. If the bids were tied, Southern Electtic would receive the order because of their long-term standing relationship with American Paper Company.

Alabama Electric is having a meeting to discuss allocating resources to filling the order should its bid be accepted. In the back of everybody's mind is the possibility of losing the bid. During discussions about the bid, one person mentioned that he feared winning bid of $500,000 might still lose money for Alabama Electric.

The discussion also revealed another three areas of concern, these were the optimal level of Alabama's bid, the uncertain size of Southern Electrics bid, and the uncertain cost of production if Alabama won the bid. The marketing and manufacturing departments (production) initially raised the concerns. Assignments for further stud

were made accordingly.

Probability Assessments

The Alabama staff agreed on the definitions of uncertainties. Cost was a well define measure that excluded depreciation, allocated costs and truly fixed costs. Southern ElectTics bid was interpreted slightly in tens of the 'deliverable' called for in American Paper Company request for bids.

After some work, the Alabama Electric's staff concluded by giving an estimate o Southern Electrics bid, it stated that it could be less than $500,000, between $500,000 and $700,000, or greater than $700,000. Similarly Alabama Electric's cost could turn



out to be less than $300,000, between $300,000 and $500,000 or greater than

$500,000.

The Alabama Staff used their judgements to assign probabilities to each possible outcome. They assigned a 35% chance the Southern Electric bid will be less that $500,000, a 50% chance the bid would be between $500,000 and $700,000 and a 15

chance the bid will be greater than $700,000. Similarly, the-re is a 25% chance Alabama's cost will be less than $300,000, a 50% chance it all be between $300,000

and $500,000, and a 25% chance it will be greater than $500,000.

To simplify the problem further, the staff chose a single number to represent the ranges on Southern's bid and Alabama's bid. Southern's representative bids are $400,000, $600,000 and $800,000. Similarly, possible Alabama costs were assumed to be $200,000, $400,000 and $600,000.

For Alabama Electric, the certainties of Southern Electric's bid and Alabama's costs are independent, i.e., probabilistically they are considered independent. Before we look further into this question let us review and summarise the major issues of concern. The three areas of concern are:

1. Optimal level of Alabama Electric bid 2. Southern Electric's bid - what is it ( $ ) 3. Production costs associated with the bid ( Alabama Electric )

The major issue is profit - that is the 'bottom line' (production costs), how much profit is required. what if Southern ElectTics bid is less than $500,000- does Alabam lose? What if Alabamacouldleam of Southern E]ectTicsbid-how much would the help in answering profit margins required? The information for costs would help to identify the profit. These are just some of the questions that are required to be asked and wherever possible answered. Below are the representations for both bids in the 'decision tree' format. The values are then computed and shown in figure 3 .1.2

0.35 fL....::O�.S0=----�0.15

---------

< 500 500-700 > 700

Alabama Southern Alabama

Electric Bid Electric Bid Electric Bid

< 300 300-500 > 500

End point values ($ lXXI) aoo 500 zoo no bid tOO 0 0 0

-tOO 0 0 0

-aoo 0 0 0

tOO 300 0 0

-tOO too 0 0

-SOO -too 0 0

tOO 300 500 0

-tOO tOO 300 0

-300 -too too o

Fig 3.12 Decision Tree with end point values



A summary of the expected value (EV) can be seen in figure 3 .1.3. This figure -uses cumulative probability approach, where the probabilities are summed up to one, and the values plotted over the ranges. By doing this, it is shows that the distribute values range from being negative in value (as with the 700 bid) to significantly positive. The difference with this type of representation is that it shows all the possible values not just theexpected value. Figure 3 .1. 4 shows a discrete probability distribution in the form of a histogram. Examining this histogram you notice that the expected value for the 700 bid equals 45 and the 500 bid equals 65, these values falling in the 0.35 and 0.33 section of the histogram respectively.

Cumulative Probability

1. Bid 700 .8 Inferior . 6 .4 .2

.

00

E�V=45 :

: : Dominant EV=65

0�---L--�----------------------�

-200 -100 0 100 200 300 400 500 *($000)

Figure 3 .1. 3 The Entire Probability distribution and single point

Probability

0.5

0.4

0.3

0.2

0.1

0.0

0.35 0.33

0.16 0.16

-150 -50 50 150 250 350

Net Contributions to Profit ( $ 000)

Figure 3.1.4 A "Discrete" Probability Distribution Plotted as a Histogram.

The above figures have shown the distribution of values and their associated probability. Two bids have been illustrated in figure 3.1.3, from these it can be sho that the 'inferior' alternatives never cross 'below' or to the 'light' of the 'dominant alternatives. If the alternatives in the distribution do 'cross', the outcome the depends on the decision-maker's risk attitude! The next step is to list each unique profit outcome and its associated probability. The probabilities listed are the join probabilities for each endpoint given a bid of $500,000 from Alabama Electiic.


Decision Analysis

Tree Endpoints

Profit ($) Probability

0 0.0875 0 0.1750 0 0.0875

300,000 0.1250 100,000 0.2500

-100,000 0.1250 300,000 0.0375 100,000 0.0750

-100,000 0.0375 1.000

Probability Distributions

Profit($) Probabilty Cum. Prob.

-100,000 0

100,000 300,000

0.1625 0.3500 0.3250 0.1625

0.1625 0.5125 0.8375 1.0000

E. Shayan

The Value Function : Alabama Electric decided that profit was the prime value in this case, as defined by the following equation. Given the currently contemplated bid of $500,000, the value associated with each possible Southern Electlic bid an Alabama's cost can be easily computed.

Profit Bid - Cost 0

If Bid < Southern Electric's Bid Otherwise

Bid Albama' s Bid for 100 Instruments cost Alabama's Production Costs

After all considerations have been given to the various bids, is was shown that the bid of$500,000 by Alabama Electric would have been the dominant bid in comparison t the other bids of $300,000 and $700,000. Also noted that by using probability distributions, the user can see that the expected value is only just one outcome ( that is the most likely), it does not represent all the alternatives. In this case, negative profits, in other words losses, have been shown in these distributions, although the probability rating is low, there is evidence that they do exist. In more ways than on this information can be useful to the decision-maker who is analysing the problem is that the analyst receives the whole picture of likely events.

3.1.4 Bayes'Theorem

If the decision-maker has access to son-ie extra information which we call sample information ( this information might col-ne from niarket research, collection of data testing, etc.), the decision-maker can revise his/her prior information on the set of probabilities for events.

The original probabilities of the various events are called prior probabilities. These exist prior to the use of sample information. Once the sample information is obtained, the prior probabilities and the sample information are used to determine revised probabilities, called posterior probabilities. Posterior probabilities a calculated after the sample information is obtained. To calculate posterior probabilities, a formula called Bayes theorem is used [38].


Decision Analysis

Bayes Theorem can be written as:

P(A I B) =probability of event A given that event B has occurred. = P(B I A) * P(A) I P(B)

Let's define the notation required: Ai = strategy ( or alternative ) is available to decision-maker Ej = event j ( state of nature ) p(Ej) = Pj = is the prior probability of event Ej

Let Ti represent the sample outcome or a given prediction by a survey, P(Ti) = predictive probability

E. Shayan

The likelihood probabilities are given by the sample information and represented a P(Ti I Ej) = given the outcome Ei, the indication was Ti

If we have the prior probabilities P(Ej) and the likelihood probabilities from sample information, then the posterior probabilities or revised probabilities can be calculate

using Bayes theorem as follows:

P(Ej I Tj) = P(Ti I Ej) P(Ej) I P(Ti) = P(Ej , Ti) I P(Ti)

where P(Ej, Ti) is the joint probability of Ej and Ti.

The predictive probabilities P(Ti) can be calculated as P(Ti) = :L P(Ej , Ti)

Having calculated the revised probabilities, they would be used in the decision tree t calculate the appropriate decision, based on the expected profit or expected cost. The above procedure is described in example below, : The Pinewood Furniture Company

(revisited)

The PW Co. acquires the services of a market research consulting firm, Intelligent Research Inc. run by Dr. Eugene Bright. Dr. E. Bright conducts market research and gives one of two indications:

Tl =A favourable indication of market for Pinewood products T2 = An unfavourable indication of market for Pinewood products

Tl and T2 are referred to as sample outcomes. Smart supplies the following likelihood probabilities as an indication of the accuracy of his market research.

P(Ti I Ej) Ej =the sales level, j = 1,2,3,4 Ti = indication, i = 1,2

El (low) E2 E3 Favourable market 0.05 0.03 0.7 Indication T1 Unfavourable 0.95 0.7 0.3 indication T2

Decision Trees 79

E4 (high) 0.9

0.1

Chapter 3


That is P(Ti I Ej) = 0.05 means that the outcome is El, that is, the sales in level 1 (low there is 0.05 probability of a favourable indication of the market.

The calculation ofP(Ti,Ej) and the revised probabilities are shown in the following table: The event probabilities are

P(Ej) = 0.2, 0.35, 0.3, 0.15 for j = 1,2,3,4 respectively

Values ofP(Ti,Ej) are given below El E2 E3 E4

T1 0.2 X 0.05 = 0.01 T2 0.2 X 0.95 = 0.19

0.35 X 0.3 = 0.105 0.3 X 0.7 = 0.21 0.15 X 0.9 = 0.135 0.35 X 0.7 = 0.245 0.3 X 0.3 = 0.09 0.1 5 X 0.1 = 0.01 5

P(Tl) = 0.01 + 0.105 + 0.21 + 0.135 = 0.46 P(T2) = 0.19 + 0.245 + 0.09 + 0.015 = 0.54

The revised probabilities are calculated as:

P(Ej I Ti) = P(Ti,Ej) I P(Ti)

E1 P(Ei I Ti)

E2 E3 E4

T1 T2

0. 01/0 .46=0. 022 0.1910.56=0.352

0.10510.46=0.228 0.2110.46=0.457 0.135 I 0.46=0.293 0.245 I 0.56=0.454 0.0910.56=0.1 67 0.015 I 0.56= 0.028

Comparing the above revised probabilities you can see that probability of sales at level 1 has decreased from 0.2 to 0.022 and the probability of level 3 has increased from 0.3 to 0.457. The probabilities should sum up to 1.0. Due to round off errors the above probabilities should sum up to 1. 001.

The above information can be summarised in the following Pinewood Decision Tree Representation.

111122 _. 20

Take the

marlcet

�

23.85

ll22ll_.lo 0.457-.15 11293 _.25

.,c�.......,'+OE-= � � 0.457 _.12 11293 _.20

0022 _. 20 11221_.5 O.CS7 _.30 11293 _.co

0.352_.20 o.uc-.1o 0.167-.15 D.028 _. 25

0.352-.15 o.cs. _. 25

'-J�==��; � �� 0.352 _. ·20 o.csc -..5 0.167 _. 30 DJl28 -.co

Figure 3 .1. 5 Pinewood Tree Representation


Decision Analysis

3.1.5 Expected Value of Sample I nformation ( EVSI)

E. Shayan

We have already considered expected values throughout this section, however, this is by no means the only type of outcome associated with decision trees or, in fact, The ideal solution. In this section we will examine the value of sample information with respect to the above example (Pinewood Furniture Company), and take into consideration the value placed upon improved probabilities, and if so, explore The different outcomes of results. The example below will emphasise these points.

Consider the Pinewood Tree Representation ( see the illustration given in figure 3 .1. for additional information ), if we were to change (improve) just one of the probabilities in the distribution - what would be the outcome ?

Overall Outcome 23.85

Don't take the market research 16.00

Flatbed

0.15__. 20 0.15---. 10 o . .so-. 15 U20_. 25

Standard 0.15 __. 15 �0.15 --. 25 --- "' 0.50 __. 12

19.25 0.20 __. 20

0.15__. -20 0.15__. -5 o.5o_. 30 020_. 40

Figure 3 .1. 6 Expected Value of Sample Information Results in 'No• change in Outcome.

From the above representation you can see, that as a result of the change 1 probability values, the value for not taking the market research has increased from 18.35 to 19.24. However, it did not change the overall decision and therefore additional payment should be given. It can be shown that improving the probabilities can actually enhance/change the overall outcome. The next example all illustrate this, again considering the Pinewood Tree Representation.

From this change in the sample information probabilities, it can be shown that the value for not taking the market research has been improved from 18.3 5 to 24.00. The change all actually affect the entire decision outcome. The expected gain from using sample information is 25 - 23.85 ($ 000) = $1150, more than the expected pay without that information. Therefore the cost of obtaining such information should not exceed $1150.

3.1.6 Expected Value of Perfect Information ( EVPI)

Having the probability of events and the pay off or loss table, it is possible to calculate the value of having access to a perfect source of information which could tell The decision-maker exactly what event is going to occur. The expected value of such information is called the expected value of perfect information ( EVPI ). The generated procedure for calculating the EVPI is:



For each known event identify the best possible pay off, given that it is k-novvn for certain the outcome will occur. Weight each of these pay off by the probability of The respective events and s-um these products. This sums equals the expected mean value

of perfect information ( EMVPI ). It can be written as follows.

EMVPI L (best pay off given event j ) x Pj

where the sum is taken over all events, j

Let EVPI denote the expected value of perfect information. Then EVPI = EMVPI -(the expected pay off using Bayes decision )= (expected loss using the Bayes decision EL( A*)

Using the example used throughout this section of the report, we shall illustrate to use of the value of perfect information. Consider the -pinewood Furniture example.

Given that the probabilities for the sales level ( 0. 2, 0.35, 0.3 and 0.15 ), the expected money value= EMVPI = 0. 2(20) + 0.35(25) + 0.3(30) + 0.15(40) = 27.75

From the above equation we see that the EVPI = 27.75 - 18.35 = 9. 4 ($ 000). Therefore can be stated that the value of perfect information is worth $9,400 in this case.

3.2 IMPLEMENTATION OF DECISION TREES

3.2-1 Uses of decision trees

A new form of sensitivity analysis using a decision tree is presented. The analysis allows decision-makers to conduct "what if questioning of multiple future conditions to simultaneously relax assumptions about these conditions. A range of values for the future conditions under which various alternatives are preferred is identified.

Using this approach, a manager is able to explore various combinations of favourable and -unfavourable outcomes for key future conditions. This, in tum, allows the manager to explore the amount of risk inherent in adopting an alternative.

An opportunity to clarify what is known about important future conditions, such as demand, often emerges in a decision process. Such information is both costly and

imperfect, but it improves the manager's prospects of selecting the best course of action.

Systematically representing such a decision with a decision tree has many benefits They include the ability to clarify options, specify necessary information, manage complex relationships, communicate, manage conflict, uncover and deal with uncertainty, carry out sequential decisions, learn about the value of expert advice and help decision-makers increase their understanding thereby improving the 'decision making' capacity of the individual involved.

Decision Trees 8 2 Chapter 3


3.2.2 Using A Decision Tree

A decision tree provides a flow diagram, much like a road map. There are frequent

forks in the road, depicting significant chance events that can occur. At each fork

these events are described so that they will be mutually exclusive and exhaustive. As a result the probabilities sum to one. This is clearly indicated in the Fashion Accessories case study in section 3 .2.4 of the report. The tree depicts the chronological order of a possible set of acts that are governed by chance.

The tree•s branch points distinguish between chance and choice forks. A choice for is designated as a square and a chance fork as a circle. The branches of a tree terminate with a payoff or an expected value. The branches can be traced to specify a series managerial acts that make up an alternative.

A decision tree can incorporate a variety of acts and consequences, including not

acting or choosing to continue to use a current system or procedure. The tree provides graphic representation listing crucial events that make up a decision.

Five steps are required to construct a decision tree;

1. Identify alternatives, acts that can influence alternatives and plausible chance

events. 2. Map out the logical flow of the tree specifying choice and chance nodes and

connecting them by specifying hybrid alternatives or outcomes. 3. Determine payoffs or utilities for the ends of the tree•s branches.

4. Assign probabilities to all chance nodes. 4. Select a decision rule to value each alternative in the tree,

A basic tree is shown in figure 3 .2.1 . A decision tree has three components: nodes branches and evaluations. Nodes describe choice and outcomes. Branches depict consequences that flow from an alternative. The branches of a tree terminate with evaluations.

Decision Trees

Choice Node

Alternative Branches

Option C

( Current Practice )

Outcome Node

Consequence Branches•

• Consequences are typically measured in terms of demand

Figure 3 .2.1 The Decision Tree Framework

83

Evaluation Node

Payoff1

Payoff3

Payoff 4

Payoff6

Current

Performance

Chapter 3


The decision-maker begins with a choice node and lists the alternative course of action. Next, outcomes are listed which describe the consequences that stem from the adoption of each alternative.

The current practice - option C, describes current system performance which serves a a 'benchmark' for other options. The current system can also be included to see how it would fare if demand or some other consequence factor were to change. Each consequence branch creates a payoff valued by a particular set of criteria. For multiple criteria one of two options can be used.

A tree can be built for each relevant criterion called for to value alternatives, or the effects of the criteria can be merged into a single tree. For simplicity, in this section of the report we will only consider single criteria examples.

3.2.3 Using Sensitivity Analysis To Make A Decision

Conclusions are drawn from a decision tree depending on the precision of likelihood estimates for future conditions. These estimates can be treated as assumptions to progressively relaxed. "What if' questions are posed to find out what must assumed about the estimates to change a decision. Sensitivity analysis can be used to examine demand estimates from an optimistic and pessimistic point of view.

Many decisions require assumptions about several future conditions. A different type of analysis is required if none of these future conditions can be treated in expected value terms to be explored one at a time. Treating the likelihoods of all future conditions as unknowns allows the decision-makers to penetrate a tough decision characterised by uncertain or unreliable information about several important conditions.

Case 3.2A Fashion Accessories-

To demonstrate this analysis, assume a large women's department store called Fashion Accessories is considering opening a new branch. The cost of leasing building, buying fixtures and other fixed costs is $1 million. Experience shows that

such funds, once committed, are not recoverable. Leases can not be broken and fixtures have little salvage value. The standard practice of Fashion accessories is to offer a large selection of goods and engage in advertising to stimulate sales when the store first opens.

In the past, Fashion Accessories has used two advertising firm that offer different advertising plans. Experience and marketing studies suggest that advertising improves the prospect of strong demand.

Payoffs in this example are expressed as the present worth of operating profits, determined by discounting an income stream of revenues under each alternative, less the costs needed to operate a store at the volume produced by advertising. Data that summarises the present worth of operating profits produced by the two advertising


Decision Analysis

plans for the store option are shown below:

Level of Advertising Worth)

Major

Minor

Advertising Cost

$ 100,000 (firm I)

$ 50,000 (fin-n 2)

Demand

Large Small

Large Small

Prospects

P=0.6 p = 0.4 P=0.2

p=0.8

E. Shayan

Profits (Present

$ 10,000,000 -$ 1,000,000

$5,000,000 -$ 500,000

Fashion Accessories has a speciality competitor called Unlimited. Unlimited has an 'uncanny' ability to identify a selection of clothing that sells better than anything Fashion accessories has to offer. The competitor does not compete with all of Fashion Accessories items, but in areas that it does, Unlimited always manages to capture what it wants of the market. The president of Fashion Accessories has become paranoid about the competition from Unlimited.

For one in every four previous store openings, Unlimited opened at the same time as Fashion Accessories. From these situations, historical data has been obtained and evaluated that Unlimited took a volume valued at $3 million in present worth terms from Fashion Accessories. However, it does not matter what steps Fashion Accessories took to counter Unlimited's entry into the market. To reflect the impact of competition when it occurs, $3 million must be subtracted from the profits that result from the two advertising programs.

Fashion Accessories does not wish to enter any market that Unlimited has selected. However, the time needed to set up a store makes it impossible to back out of the market after Fashion Accessories finds out Unlimited has decide to enter. Fashion Accessories has no way to determine whether unlimited has decided to enter before Fashion Accessories must make it's own commitments. To make matters worse, Unlimited seems to have inside information on Fashion Accessories plans.

Fashion Accessories is competing with several stores (other than it's major competitor Unlimited) at all of its current locations. It can choose to concentrate advertising of present stores, which creates a 'no store' alternative. However, in this case, advertising would have a smaller impact because of competition from Unlimited and others. The payoffs for the 'no store' alternative are also expressed as the present worth of increases in operating profit that may occur when increased advertising is used for a store that has been in operation for several years. The payoffs are as follows:

Level of Advertising Advertising Cost

Major $ 100,000 (firm 1 )

Minor $ 50,000 (firm 2)

Decision Trees 85

Demand

Large Small Large Small

Prospects

p =0.6 P=0,4 P=0.2 P=0.8

Profits PV

$750,000 $ 300,000 $500,000 $ 0

Chapter 3

Decision Analysis

The decision is summarised in figure 3 .2.2

Store (-$1M)

Operating Profit $10,000,000

��------�.��00,000

-$1�00,000

-$4,000,000

-$500,000 __ __,...

,_,_ ___ -llo..... -$3,500,000

Small ( P=0.4)

,......___�l.a=rg�e (�P:_: =0�2:!,_ )

___ .,..... $500,000 "-"'-------___, .... $ 0

Small ( P=O.B)

Figure 3.3.2 Decision Tree for Store Purchase Decision

Estimation ofPayoffs.

E. Shayan

The decision tree in figure 3.2.2 is collapsed according to the following format:

Payoff= -Store Cost - Advertising Cost + Large Demand Prospects (No competition profit+ competition profit)+ Small Demand Prospects (No competition profit+ competition profit.

Using the expected value decision rule with the best estimates of demand prospects (large demand= 0.6, small demand= 0.4, competition= 0.25, no competition= 0.7 and profits (present worth of operating profit), the decision-maker finds the following payoffs for the options:

Store and Major Advertising = -$1,000,000- $100,000 + 0.6 [ 0.75( $10,000,000 ) + 0.25 ( $7,000,000 )] + 0,4[ 0.75( -$1,000,000 ) + 0.25( -$4,000,000 )] = $3,750,000

Store and Minor Advertising = -$I,200,000 No Store and Major Advertising = 0- $100,000 + 0.6( $750,000) + 0.4( $300,000)

= $470,000 No Store and Minor Advertising = $50,000

The expected values suggest that opening a store with a major advertising package is the best alternative. However, Fashion Accessories' president worries that if the strategy leaks have not been plugged, the chance of competition from Unlimited may increase. The president also believes that the historical impact of advertising may have been eroded because local markets have been flooded with advertising recently which seems to have diluted its impact.

Treating the prospects of competition as unknowns: Each of the four options can be



subjected to a sensitivity analysis that treats the likelihood of both future conditions as unknowns. The option calling for a store and major advertising, with the likelihoods

expressed as unknown, is shown below;

Target Profit (P) = -$1,100,000 + P[p($10,000,000) + (f-p)($7,000,000)] +

(1-P)[p(-$1,000,000) + (1-p)(-$4,000,000)] Simplifying the equation in terms of the unknowns yields:

Target Profit = -5,100,000 + ll,OOO,OOOP + 3,000,000p

P represents the prospects of a large demand and p the likelihood that Fashion Accessories can avoid competition from unlimited. Solving for P yields:

P = [ (5, 100,000 +Target Profit ) - 3,000,000p ] I 11,000,000

To carry out sensitivity analysis with three unknowns and one equation, the value of P is determined by selecting feasible values for target profit and all possible values for p, the chance of avoiding competition from Unlimited. The maximum profit in present

worth terms) that the store can produce is $8.9 million. The $10 million maximum operating profit is less than $1 million in store costs and $100,000 in advertising .

For the purpose of exploration, the decision-maker sets profit at zero, beginning with the smallest target value of any interest. Setting the target profit at zero reduces the equation as shown;

P = ( 5,100,000- 3,000,000p ) I 11,000,000

Now let p, the prospect of avoiding competition, take the extreme values of zero an one. Letting p = 0, or making competition a certainty, yields;

P = ( 5,100,000- o) I 11,000,000 = 0.4636

Letting p = 1, making no competition a certainty, yields;

P = ( 5,100,000- 3,000,000 ) I 11,000,000 = 0.191

Considering target profit in $1 million increments and solving for extreme values of p produces the following values for P, the prospect of a large demand:

Target Profit 0 $1,000,000 $2,000,000 $3,000,000 $4,000,000 $5,000,000 $6,000,000 $7,000,000 $8,000,000

Decision Trees

Values ofP When P=O P=l

0.4636 0.191 0.545 0.282 0.645 0.372 0.736 0.464 0.820 0.554 0.920 0.645

0.736 0.827 0.918

87 Chapter 3


A family of straight lines results, which can be plotted on a grid for the unknown P (prospects of large demand) and p (prospects of avoiding competition) shown in figure 3.2.3

Figure 3.2.3 is subjected to "what if' questioning by the decision-maker. To conduct a sensitivity analysis, Fashion Accessory's president notes that the previous estimates of p and P were 0.75 and 0.6, respectively. Entering these into figure 3.2.3 produces a expected value for profit of between $3 and $4 million.

A more conservative estimate would be to assume a 50/50 chance of avoiding competition and having a large demand. Using the graph and entering p =0.5 and P=0.5, the decision-maker discovers that profit for this contingency would be about $ 2 million.

Using this kind of logic, the decision-maker can carry out various kinds of surveys For instance, the prospect of a large demand can be clarified by experts who may contend that it falls between 0.4 and 0.6. Drawing vertical lines at these points on Figure 3 .2.4 allows the decision-maker to explore the implications of making a range of supporting assumptions about the competitor. To make these assumptions clear the payoff lines outside the expected demand region are shown as dashed lines (see figure 3.2.4). Payoff possibilities outside the 0.4 to 0.6 region are ruled out.

".

Prospects of No Profit

Competitors (p) $0 $1M $2M $3M $4M $5M $6M $1M $8M 1.0 ....----.--...---.--....-----.--.----.:---,---,

0.9

0.8

0.7

0.6

0.5

0.4

03

0.2

0.1

0

0 0.1 02 03 0.4 05 0.6 0.7 0.8 0.9 1.0

Prospects of Large Demand (P)

Figure 3.2.3 Sensitivity Analysis for Future Conditions and Expected Values.


Decision Analysis Prospects of No Competitors (p)

Profit

tD �--�r=�=r��;r��

0.9

0.8

0.7

0.6

0.5

0.4

0.3 Loss Region

0.2

0.1

0

0 0.1 02 0.3 0.4 05 0.6 0.7 0.8 0.9 1.0

Prospects of Large Demand (P)

Figure 3.2.4 Sensitivity Analysis for Ranges ofExpectation.

E. Shayan

To conduct sensitivity analysis, various assumptions about competition are entertained. Competition must be quite likely (80 percent) to push profit to zero. A 50150 competition assessment results in a minimum profit of $1 million and maximum profit of$4 million. If the competitor's past history is used as a guide (75% chance of no competition), a zero profit only results when the prospect of a large demand falls below 3 0%, well below the demand thought by experts to be attainable. Very conservative one in five estimate of no competition produces a profit range of zero to

$2 million.

The prospects of competition can be given a similar treatment. A conservative range of values between 0.20 and 0. 75 for no competition is bracketed by the top and bottom horizontal lines in figure 3.2.4. Prospects of a high demand above 0.4 will avoid losses and can produce profits of up to $ 8 million.

The assessment shows that Fashion Accessories' president does not need to be oncemed with Unlimited's plans. Unlimited does not take enough of Fashion Accessories revenue to make its store strategically unprofitable, even when it makes very conservative estimates of future demand.

The best option identified by an expected value analysis is usually selected for multiple

ensitivity analysis. However, the same type of analysis can be applied to all options. In this example, the options identified by "store - minor advertising" and "use of existing stores with each advertising package" can also be subjected to sensitivity analysis. If the computations produce different choices by treating the projects associated with all future conditions as unknowns. The analysis is used to prune the options, as well as to reassure the decision-maker about how payoffs can change as the result of chance events.

3.3 DECISION PROGRAMMING LANGUAGE (DPL)

DPL is a synthesis of many of the underlying concepts of the influence diagrams and tree based approaches. It introduces several new representational and computational concepts that are not contained in existing approaches.



An essential feature ofDPL is that it is a programming environment designed for decision analysis problems. Problems are specified in an English-like language created

from the vocabulary commonly used in decision analysis. The source code is then compiled to produce an internal representation of both an 'influence diagram' and 'decision tree'. The figure 3.3.1 shows how both the influence diagram and the decision tree combines to become part of the decision programming language system.

Influence Diagram \cision Pro��ming Language/ De aston.._

Chance.._ Value ...

DecideAbouL Gamble On._

Pay ....

Figure 3.3.1 Representation oflnfluence Diagram and Decision Tree

The DPL source code comprises two sections. The first is a data definition section which contains the specifications of events, values and theie relationships. The second section provides a specification of the decision sequence, which is a formal ordering of the decisions and state of information. These two sections correspond roughly to influence diagrams and decision tress respectively, (see figure 3.3.2 ).

However, the clear separation of these two sections allows DPL to use different forms for representing each part of the problem and a different computational 'engine' for each part. This allows DPL to take the advantage of some of the virtues of each approach. Influence diagrams are best at precisely describing probabilistic relationships and trees are best at clearly describing the decision sequence.

3.3.1 comparison of DPL against other techniques

Decision analysis is a systematic technique for making decisions based on ..... .

What you can do What you believe


Decision Analysis

What you can do

Introduce Product

What you prefer

Profits

Figure 3.3.2 Availability Of Options

The Decision Analysis Process

Confusion

What you believe

Profits

3.3.3 The Decision Analysis Process

E

E. Shayan

I

-9� Light

Two "Standard" approaches in decision programming languages have been developed for building and solving decision analysis models.

DEOSION TREES INFLUENCE DIAGRAMS

Figure 3. 3. 4 Illustration of Decision trees and Influence Diagrams



Case "Wildcatter Oil Drilling"

The Wildcatter is prospecting for oil and has several decisions to make, some of these include the following:

1. That they may drill for oil and face uncertain consequences. That they may strike oil or a dry hole. And if they drill they will face uncertain drilling costs.

2. Or, conduct either a core sample or seismic sounding test to help make the drilling decision. The core sample is conclusive but more expensive. Once the test is conducted the drilling decision must be made.

A decision tree for the oil Wildcatter problem would look like the following;

Initial Action Test Drilling Drill ing Amount Results Decision Cost Oil

Drill

s E

c::( No No Test

Drill

� E

c(No Drill

:: <C

c:(No

E <=-

0 Drill

� E

c(No Drill

c:(No � <C

5 E

Figure 3.3.5 Decision Tree ofWildcatter problem

Decision trees have important advantages in that they;

1. Provide intuitive framework for presenting decision structure, data and results 2. Handle problem ??? asymmetries easily 3. Make solution algorithm transparent


Decision Analysis

However, decision trees also leave important drawbacks in that;

1. Size grows exponentially with increased complexity

E. Shayan

2. Two versions often required: one for assessment and one for computation 3. Probabilistic relationships not clear from tree structure

Let us now look at the other "standard" approach which has been developed for building and solving decision models.

An influence diagram for the oil Wildcatter problem can be seen below

Figure 3.3 .6 Influence Diagram for Oil Wildcatter Problem

Influence diagrams have important advantages in that they: 1. Display probabilistic relationships clearly and compactly 2 . Show where coalescence can speed computation

However, influence diagrams also have important drawbacks in that they have: 1 . Limited representation of decision structure 2. No natural structure for display of data or results 3. Poor handling of asymmetries

The result is existing decision analysis tools can have some troublesome limitations such as: 1. Small Problems 2. Slow Performance 3. Simple Models 4. Incomplete Outputs 4. Cumbersome Inputs

Decision Programming Language ( DPL) was designed to overcome these limitations. We have already looked at some background information into DPL in section 3.3. Now we will try to illustrate via the use of a flow sheet how DPL works.


Decision Analysis

Gra hical Interfaces

Q� Influence Dia am

Decision Tree

ABC

Decision

Programming

Language

Complete Outputs

Risk Profile

DPL GO, NO GO

Engine Optimal Policy

Sensitivity

Figure 3.3.7 Operations OfDecision Programming Language (DPL)

E. Shayan

What makes Decision Programming Languages different from other techniques. The main factors can be summarised as they combine the use of advanced concepts "intelligent" algorithms and state of the art computer implementation programs to evaluate and decide upon making principles and approaches etc.

3.4 SOFTWARE APPLICATIONS -DECISION SUPPORT TECHNIQUES

The following section has been introduced for completeness. This section does not contain a working example, however it shows the application and uses of the

DEXTRA package.

The introduction of decision support techniques into a process whose participants are not familiar with the decision analysis approach is frequently difficult. As the level of seniority increases, the manager at that level naturally feels more and more strongly that the main factor which secured his appointment to a senior position is his perceived ability to cope with policy decisions of ever increasing scope and importance.

To come face to face with a decision analyst who claims to offer decision support through the use of micro based software is, therefore, understandably disconcerting. To some managers, acceptance of such support may seem to strike at the very foundations of their managerial ability. The decision support systems must be introduced with a proper awareness of this hazard.

The individual making decisions on his behalf generally has no such inhibitions and can quickly accustom himself to working with the micro. Initially, the interaction needs to be three cornered, entailing three way communication between the decision maker, the decision analyst and the micro. If the software is 'user friendly' the decision analyst should ideally withdraw to leave the decision-maker interacting alone with the



micro, i.e. in other words - the individual can 'figure out for himself backed up with the opportunity for consultation by a decision analyst from time to time.

The relationship between the decision analyst and a corporate decision maker or decision-making group is necessarily more formal and the analyst generally needs to act as an interface between the decision-maker and the software. In some environments, it may be advisable for the analyst to disguise the fact that the software being used to support the analytical technique is being run on a micro - since managers who accept only with reluctance that software supported techniques can be beneficial in the making of 'big' decisions, may be immediately disenchanted if they see their deliberations being processed on a 'small' computer.

3.4.1 Operation And Applications Using the Package DEXTRA

An example of such a software package which relates to this topic is DEXTRA - A decision tree analysis package.

DEXTRA is a proprietary package developed by P ACTEL for use on the 'Apple' under CP/M, to analyse decision trees of up to 180 links.

An important policy decision in the development ofDEXTRA was concerned with the use of graphics: should we go for the graphical representation of the tree on the screen, or should we concentrate on providing the numerical back up through DEXTRA leaving the user with the chart and therefore requiring modification to his tree on paper? It became clear that the former course would lead to too much graphical programming. Accordingly the latter idea was adopted. The process of drawing up and modifying a decision tree on paper is in itself a good learning experience and a good basis for a thorough understanding - it acquaints oneself with the underlying structure of the problem. Formulating a decision tree has already be described in section 3 .1. 5 of this report - it can be used as a reference if required.

DEXTRA is menu driven - it includes rapid reaction facilities for moving up and down branches, as well as hopping from limb to limb to see what data is attached to each part of the tree. The value held at each node may be a composite of up to four components, weighted to represent the utilities of different aspects affecting the initial and subsequent decisions. Previously this has been shown in the Pinewood Tree Representation as four branches emanating from one point.

Each node can be related to a time scale and discounted cash flow calculations can be applied to display the appropriate net cash flow folded back to any branch of the tree displayed on the screen. Different utility components can be discounted at different rates if required.

Utility functions can be used instead of probabilities. However, it should be noted that utility functions are totally dependent on the user - in terms of the form which they represent. Utility functions, are not as common as other forms mentioned like expected values or probability distributions.



In many analyses, the user's understanding of the problem results in a tree design showing repeated clusters of branches, identical in structure, and often in the associated probabilities and outcomes as well. These clusters are tedious and time consuming to set up and require the user to ensure at all times that a changed value in one part of the tree is then calculated throughout the existing branches in which it relates. DEXTRA handles this using a 'cloning' effect which enables tree links to be tagged together, both on set up and when values are altered. A change in value on one branch can, if so required, be implemented on all other branches which are cloned with it - a powerful aid in ensuring the integrity of the data being input and analysed.

DEXTRA's application is, of course, to that category of decisions concerned with the selection of a best course of action having regard to the outcome of events an subsequent decisions downstream. Decision tree analysis is not a technique that would normally expect a decision-maker or group to be familiar with detail and the use of the package reflects this view: the decision analyst needs to act as the interface between the package and the decision-maker in most circumstances.

DEXTRA's present capacity of 180 branches could clearly be increased with the addition of a facility to view complete trees as sub trees, and their then to combine them - this facility is currently under consideration for development. In practice though, 180 branches can accommodate trees of some power and complexity.

Very large trees concerned with complex decisions (such arise in defence, oil field development, or the construction of tidal flow barrages) are better analysed using more comprehensive software products on larger machines. An example of such machine would be the 6 P AK suite developed by Decisions and Designs in the USA

As I mentioned in the introduction to the section of the report, I am not addressing issues relating to case study or an example. The above describes the applications and the uses of DEXTRA within the context of decision trees.










Decision Analysis

Chapter 4.0 MULTI CRITERIA DECISIONS

4.1 Introduction to Analytical Hierarchy Process ( AHP)

E. Shayan

This section of the report serves as an introduction to Analytical Hierarchy Process -multi-criteria decision-making approach in which factors are arranged in a hierarchical structure. The principles and the philosophy of the theory are summarised giving general background information of the type of measurement utilised, its properties

and applications [ 42].

AHP is an intuitive and relatively easy method for formulating and analysing decisions. There are three major concepts behind the AHP ; these are analytical, hierarchy & process. We will briefly describe the philosophy of these three

components.

Analytical - Simply put, the AHP uses numbers. Mathematics are used understand others and/or describe your choice to others. In a sense, all methods which seek to describe a decision are analytical since they must use mathematical

logical reasoning.

Hierarchy - The analytical hierarchy process structures a decision problem in levels which correspond to one's understanding of the situation: goals, criteria, sub-criteria

and alternatives. It breaks down complex problems into a Simple hierarchical construction by questioning each element. The main objective (goal) takes place in the root of the hierarchy, the sub objectives (criteria) are structured the second level according to their contributions to the goal (main objective), a criteria I sub-criteria and alternatives take place in the subsequent levels.

Goal Criteria

Sub Criteria Alternatives

Process - As most know, decisions which are truly important cannot be made in a

single meeting one cannot expect the analytical hierarchy process to counteract this asic

human tendency. People need time to think about a decision, gather new information, negotiate if it is a group decision, etc. Thus any real decision problem involves a process of learning, debating and revising one's priorities. Saaty states that the AHP is meant to be used to aid and hopefully shotten this decision process through the insights

which this method can generate- it never did and neve will, replace the overall decision making process. The AHP points to where more information is needed and where major points of disagreement lie, etc. The AHP is meant to aid and not to destroy the natural process of decision-making .

Multi Criteria Decisions 101 Chapter 4


4.2 How To Structure A Decision Problem Using AHP

Perhaps the most creative task in making a decision is to choose the factors that are important for that decision. In the AHP we arrange these factors, once selected, in hierarchical structure descending from an overall goal to criteria, sub-criteria and alternatives in successive levels.

To a person unfamiliar with the subject there may be some concern about what to include and where to include it. When constructing hierarchies one must include enough relevant detail to:

1 . Represent the problem as thoroughly as possible, but not so thoroughly as to lose sensitivity to change in elements;

2. Consider the environment surrounding the problem;

3. Identify the issues or attributes that contribute to the solution; and 4. Identify the participants associated with the problem.

Arranging the goals, attributes, issues and stakeholders in a hierarchy serves two

purposes. It provides an overall view of the complex relationships inherent in the situation and helps the decision-maker assess whether the issues in each level are of the same order of magnitude so he can compare such similar elements accurately.

To obtain a meaningful answer, you must not compare sets of objects outside their class, for example 'a small one bedroom apartment with a luxurious million dollar mansion' and have any hope of a meaningful answer. A fundamental scale is use when comparing objects in the same class. It consists of verbal judgements ranging from equal to extreme (equal, moderately more, strongly more, very strongly more extremely more) corresponding to the verbal judgements are the numerical judgements ( 1,3,5,7 &

9) and comprises between these values.

A hierarchy does not need to be complete, that is, an element in a given level does not have to function as an attribute (or criteria ) for all the elements in the level below. A hierarchy is not a traditional decision tree. Each level may represent different slice of the problem. One level may represent social factors another political factors (these can be seen readily in the following two worked examples) to be evaluated. Further, a decision maker can insert or eliminate levels and elements as necessary to classify the task and set priorities or to sharpen the focus on one or more parts in the system. Elements that have global characteristics can be represented at higher levels in the hierarchy, others that specifically characterise the problem at hand can be developed in greater depth.

The task of setting priorities requires that the criteria, the properties or features of the alternatives being compared and the alternatives themselves, are gradually layered the hierarchy so that it is meaningful to compare them amongst themselves in relation to the elements of the next higher level.

Finally, after judgements have been made on the impact of all elements and computed for the hierarchy as a whole, the less important elements can be dropped from further



consideration because of their relatively small impact on the overall objective. The

priorities can be re-computed throughout, either with or without changing the remaining judgements.

The AHP also tests the consistency of judgements. In decision-making problems it is important to know how good our consistency is, because we may not want the decision to be based on other judgements that have such a low consistency that they appear to

be random. On the other hand, perfect consistency is hard to live up to Therefore a compromise has to be met.

4.3 Pairwise Comparisons As Ratios

When we Measure something with respect to a property, we use some known scale for that purpose. How do you derive relative scales using judgements or data from standard scale and how do you perform the subsequent arithmetic operations on such scales avoiding useless number crunching?

Judgements are given in the form of paired comparisons. One of the uses of a hierarchy is that it allows the user to focus judgements separately on each of several properties essential for making a sound decision. The most effective way to concentrate judgements is to take a pair of elements and compare them on a single property without concern for other properties or other elements. This is why pairwise comparisons in combination with the hierarchical structure is so useful in deriving measurement. We also note that sometime comparisons are made on the basis of standards established in memory through experience or training.

Hierarchical analysis is a technique used to estimate the weights of the possible outcomes. 'Me highest level in the hierarchy identifies the general objective for the decision or choosing the best alternative. The following levels consist of decision attributes which contribute to related goals or criteria. The number of levels in the

hierarchy varies depending on the complexity of the problem and the amount of detail needed to reach a satisfactory solution.

After structuring the hierarchy to represent the real life situation, the next step is to evaluate the elements in the hierarchy. To perform this process, one has to find the impact of each element in a level on elements in the next higher level by comparison method .

The AHP is used with two types of comparisons, relative and absolute:

(i) Absolute comparison (scoring) which is performed with respect to a set of

standards. (ii) Relative comparison (scaling) which is carried out in pairs and is also called

pairwise comparison.

The AHP uses two types of measurement, relative and absolute. In both, paired comparisons are performed to derive priorities for criteria with respect to the goal. In relative measurement, paired comparisons are performed throughout the hierarch including the alternatives in the lowest level of the hierarchy with respect to the criteria



above. In absolute measurement, paired comparisons are performed through the hierarchy with the exceptions of the alternatives themselves. The level just above the alternatives consists of intensities or grades which are refinements of the criteria or subcriteria governing the alternatives.

AHP utilises pairwise comparison to establish priority measures. Pairwise eases the measurement and helps achieve more accurate considerations. It is very useful on large problems as the mind can only deal with 7+/-2 simultaneous considerations as quoted by Miller.

4.4 Example Choosing The Best House To Buy

A family of average income is considering buying a house; the family identified eight criteria which they thought they should look for in a house. The three categories identified were: economic, geographic and physical. Although one may have begun by examining the relative importance of these clusters, the family felt that they wanted to prioritise the relative importance of all the criteria without working with clusters. The problem was to decide which of the three candidate houses to choose. The first step is the structuring of the problem as a hierarchy.

In the first (or top) level is the overall goal of'Satisfaction with House'. In the second level are the eight criteria which contribute to the goal and the third (or bottom) level are the three candidate houses which are to be evaluated from in terms of the criteria in the second level. The definitions of the criteria and the hierarchy is shown in figure 4. 1. 1

Figure 4. 1. 1 Decomposition of the Problem into a Hierarchy

The criteria important to the individual family were:

(1) Size of house: storage space, size of rooms, number of rooms, total area of house.


\


(2) Location to bus lines: convenient, close bus service. (3) Neighbourhood: Little traffic, secure, nice and friendly, low taxes, good

condition of neighbourhood. (4) Age of house: Not too old. (5) Yard space: Includes front, back and side; space from neighbours. (6) Modem facilities: Dishwasher, garbage disposal, air conditioning alarm

system and other fixtures in the house. (7) General condition: Repairs needed, walls, carpet, drapes, cleanliness, wiring. (8) Financing available: Vendor financing, Lending Institution financing,

other personal arrangements

Table 4.1.2 The Fundamental Scale

-

Intensity of Importance Definition Explanation on an absolute scale

1

3

5

7

9

Equal Importance

Moderate importance of one over another

Essential or strong importance

Very strong importance

Extreme importance

Two activities contribute equally to the objective

Experience and judgement strongly favour one activity over another

Experience and judgement strongly favour one activity over another

An activity is strongly favoured and its dominance demonstrated in practice

The evidence favouring one activity over another is of the highest possible order of affirmation

2,4,6,8 Intermediate values between When compromise is needed two adjacent judgements

The second step is the elicitation of pairwise comparison judgements. Arrange the elements in the second level into a matrix and extract judgement from the people who

have the problem about the relative importance of the elements with respect to the overall goal and satisfaction with the house.

The scale to use these judgement is given in Table 4 .1. 2. The question to ask when comparing two criteria are of the following kind: of the two criteria being compared,



which is considered more important by the family buying the house with respect to the overall goal of family satisfaction with the house?

The matrix of pairwise comparisons of the criteria given by the home buyers in this case is shown in table 4 .1. 3, along with the resulting vector of priorities. The vector of priorities is the principal ??? eigenvector of the matrix. It gives the relative priority of

the criteria measured on a ratio scale. In this case, financing has the highest priority with 33% of the influence. For the consistency index (CI), adopt the value (???maxn) I (n-1) If the ratio (called the consistency ratio (CR ) ) of CI to that from the random matrices is significantly small, consistency is defined as the rating of quality during pairwise comparisons. The AHP provides a measure of the consistency of pairwise comparison judgements by computing an inconsistency ratio. This ratio is designed such that values of more than 0.1 are not acceptable. In such cases the decision-maker should revise the judgements to reduce inconsistency to an acceptable level <0. 1.

Table 4.1.3 Pairwise Comparison Matrix Level 1 1 2 3 4 5 6 7 8 Ptiority Vector

<) 'l "\_ '

1 1 5 3 7 6 6 1/3 1/4 0.173 2 1/2 1 1/3 5 3 3 1/2 1/7 0.540 3 1/3 3 1 6 3 4 6 1/2 0.188 4 1/7 1/2 1/6 1 1/3 1/4 1/7 1/8 0.018 5 1/7 1/3 1/3 3 - 1 1/2 1/5 1/7 0.031 6 1/7 1/3 1/4 4 2 1 1/2 1/7 0.036 7 3 5 1/6 7 5 5 1 1/2 0.167 8 4 7 5 8 6 6 2 1 0.333

???max= 9.669, CI = 0.238, CR=0.169

In Table 4.1.3 instead of naming the criteria, we use the number previously associate with each. Next, move to the pairwise comparisons of the elements in the lowest level. The elements to be compared are the houses with respect to how much better one is in comparison to the other whilst satisfYing each criterion in level 2. There will be eight 3 (3 matrices of judgements since there are eight elements in level 2 and 3 houses to be pairwise compared for each element. Again, the matrices contain the judgements of the family involved. To understand the judgements, a brief description of the house follows.

House A Is the largest. It is located in a neighbourhood with little traffic and low taxes. It's yard space is comparably larger than houses B and C. However, the general condition is not very good and it needs cleaning and painting. Financing is unsatisfactory because it would have to be bank financed at high interest, House B. Is a little smaller than House A, it is not close to a bus route and the neighbourhood gives the feeling of insecurity because of traffic conditions. The yard space is fairly small and



lacks the basic modem facilities. On the other hand, the general condition is very good. Also the mortgage is obtainable which means the financing is good with a rather low interest rate.

House C- Is very small and has few modem facilities, but is in good condition wit pretty carpets and drapes. Although the neighbourhood has high taxes it seem secure. The yard space is bigger than that of house B, but is not comparable to house A's spacious surroundings.

The matrices of comparisons of the houses with respect to the criteria and their local priorities are given in 4.1.4.

Table 4.1.4. Comparison Matrices and Local priorities

Size of House A B c Priotity Yard A B c Priority

Vector Space Vector

A 1 6 8 0.754 A 1 5 4 0.674

B 1/6 1 4 0.181 B 1/5 1/3 0.101

c 1/8 1/4 0.065 c 1/4 3 0226

£max= 3.136, Cl = 0.068, CR = 0.117 £max= 3.ll86, Cl = 0.0 43, CR = 0.074

Transport. A B c Priority Modem A B c Priority

Vector Facilities Vector

A 1 7 1/5 0.233 A 8 6 0.747

B 1/7 1 1/8 0.005 B 1/8 1 1/5 0.060

c 5 8 0.713 c 1/6 5 0.193

£max= 3247, Cl = 0.124, CR = 0213 £max = 3.197, Cl = O.D99, CR = 0.170

Neighbourhood A B c Priority General A B c

Priority

Vector Condition Vector

A 1 8 6 0.745 A 1/2 1/2 0.200

B 1/8 1/4 0.065 B 2 0.400

c 1/6 4 0.181 c 2 0.400

£max= 3.130, Cl = 0.068, CR = 0.117 £max= 3.000, Cl = 0.000, CR = 0.000

Age of House A B c Priority Financing A B c Priority

Vector Bank Vector

A 0.333 A 1/7 1/5 0.072

B 0.333 B 7 1 3 0. 650

c 0.333 c 5 1/3 1 0.278

£max= 3.000, Cl = 0.000, CR = 0.000 £max = 3.065, Cl = O.D32, CR = O.D56



The third step is to establish the composite or global priorities of the houses. We lay out the local priorities of the houses with respect to each criterion in a matrix and

multiply each column of vectors by the priority of the corresponding criterion and add across each row which results in the desired vector of the houses seen in Table 4.1.4.

House A, which is the least desirable with respect to financing (the highest priority criterion), contrary to expectation, had the highest priority. It was the house that was selected and ultimately bought by the family.

Table 4.1.5 Local and Global Priorities

1 2 3 4 5 6 7 8 (0.173) (0.054) (0.188) (0.018) (0.031) (0.036) (0.167) (0,333)

A 0.754 0.233 0.754 0.333 0.674 0.747 0.200 0.072 B 0.181 0.055 0.065 0.333 0.101 0.060 0.400 0.650 c 0.065 0.713 0.181 0.333 0.226 0.193 0.400 0.278

4.2 AHP A FLEXIBLE MODEL FOR DECISION MAKING

0.396 0.341 0.263

Basic observations of human nature, analytical thinking and measurement have led to the development of a useful model for solving problems quantitatively. The analytical hierarchy process is a flexible model that allows individuals or groups to shape ideas and define problems by making their own assumptions and deriving the desired solution from them. It also enables people to test the sensitivity of the solution or the outcome to changes in information. It is designed to accommodate

our human nature rather than force us into a mode of thinking that may interfere with our better judgement. The AHP is a powerful process for tackling complex political and socioeconomic problems.

The AHP incorporates judgements and personal values in a logical way. It depends on imagination, experience and knowledge to structure the hierarchy of a problem and its logic, intuition and experience to provide judgements. Once accepted an followed, the AHP shows the user how to connect elements of one part of a problem

with those of another to obtain the combined outcome. It is a process for identifying understanding and assessing the system as a whole. In this manner it is flexible.

To define a complex problem and develop sound judgements, the AHP must be progressively repeated or re-iterated over time; one can hardly expect instant solutions to complicated problems. The AHP is flexible enough to allow revision - decision-makers can both expand the elements of a problem hierarchy and change

their judgements.

Another feature of the AHP is that it provides a framework for group participation in decision-making or problem-solving. We have seen that ideas and judgements can be questioned and strengthened or weakened by evidence that other people present.


Decision Analysis

4.3 SITE SELECTION USING AHP

E. Shayan

The next example considers a real life case study of the actual events that took place during the site selection process at BTR Kennon, and shows how the 'though

processes' could have been adapted to AHP to evaluate the selection.

4.3.1 Example Using Site Selection At Nylux

Nylux, a manufacturer of carpet goods for the Automotive Industry, is planning to

expand and rent an existing factory to manufacture and store carpets. A team of company managers must decide on the appropriate location for the factory taking

many different criteria into account. This real life example reports on the decision process that the project team employed to model the site selection problem.

Outline Of The Project

In May 1993, a project team was selected and was charged with the task of proposing

site selection plan for the existing factory. The project team reported directly to the Managing Director and consisted of the following members.

Group A - Consisted of 3 managers with experience in manufacturing technology and production.

Group B - Consisted of 2 experts in design and process engineering Group C - Consisted of 3 leaders in factory management that will be in charge

of the existing factory once it is complete.

The general manager of the company headed the project team.

The objectives of the project team were (I) to clearly define the requirements of the factory site - what was required and suitable for the perceived operation, (2) to select the appropriate site to achieve this and (3), to report their results to the managing director, upon which a decision would be made by the managing director within three months.

The first step in the decision-making process required the project team to discuss the overall plan of the new factory and gather information relevant to the project. For example, the team needed to decide upon (1) how much area of the factory was required for production and how much was required for excess storage, (2) the different types of layouts available to fit an existing factory, (3) a suitable infrastructure for the processes and ( 4), a desirable working environment at the factory.

After the potential sites were identified, the project team screened this initial set on

four key decision factors and eliminated several sites. These sites were dropped from further consideration since they lacked (1) proximity close to other major factories from the same Organisation, (2) building structure/open areas and high roofs for



greater storage and access, (3) size of building-total area of building and surrounding and ( 4), access to site - main roads due to large transport vehicles required.

To screen the remaining candidates, the project team gathered more data about the sites and modified a checklist of decision factors that had been used in a previous site selection decision. A partial list of the team factors and the team's evaluation of the three sites are shown in Table 4. 3 .1. From this table we can see that the three decision factors - technical, economic and social - are broken down into important items an each item is further decomposed into sub-items.

The next step is to judge - rate each site by evaluating each criteria within that site selection. The ratings which will be used in this example have already been shown in

Table 4.1.2

This approach -helped the project team narrow down the candidate list to three sites: Site A (factory size 1400 sq.m, good facilities, suitable interior and access, rent

$moderate - high). Site B (factory size 1100 sq.m , good facilities, requires work on interior, rent

$moderate ). Site C (factory size 1250 sq.m, excellent facilities, suitable interior and access,

rent$ moderate - high ).

The final step in the decision-making process called for the project team to compare sites A, B and C in detail to select the best site and then to make recommendations to the Managing Director, however, the team found it difficult to make a selection based only on the checklist approach. The three sites exhibited the same score on many of the sub-items and the team had no way of weighting the factors, items, sub-items, listed in Table 4. 3.1. For example, the attitudes and judgements of three groups represented on the project team widely differed on the importance of the technical, economic and social factors. As a result, it became very difficult for the team to reach a consensus on the best site for the new factory.

At this step in the process, a group of three managers, (one from groups A,B,C) was organised and charged with the task of ranking the three candidates. If they had decided to apply the Analytical Hierarchy Process to help model their decision problem, they potentially could provide a rational way of ranking sites when conflicts exist in weighting key decision factors



Table 4.3 .1 Checklist For Site Selection

Eactoe2 lliml Sub-il�m Sites A B c

Situation of Site Base G FE

Area E E E

Environment F F F

Approach Roads G F G

Technical Water Supply Amount E E E

Quality E E E

Site Interior Open GGE

High Ceiling E E E

Large Access Doors EGG

Size of Building Optimal Size F G E

Initial Investment Real Estate GF F

Facilities Cost E E E

Electrical, Power & Water Supply GGG

Energy Cost Water E E E

Electricity GGG

Economic Material F F F

Tax Tax F F F

Exemption F F F

Labour Recruitment GGG

Wage Rate GGG

Quality E E E

Union E E E

Legal Regulation Factory Location Act E E E

Building Stds. Act E E E

Environmental Pollution E E E

Rre I Safety Act F F F

Social Regional Characteristics Living Expenses E F E

Shift System E E E

Rate of Absenteeism F F F

Productivity F F F

Key : E = Excellent, G = Good, F = Fair.

Multi Criteria Decisions Ill Chapter 4

Decision Analysis

4.3.2. The Decision Hierarchy Used in Site Selection

E. Shayan

To construct the decision hierarchy for the site selection problem, the three manager excluded the sub-items listed in the table above for which the three sites A, B, C have

the same rating. For example, the sites received identical evaluations on the two tax sub-items (i.e. A,B,C each received the same 'fair' rating on tax and exemption). therefore, the tax item is not included in the hierarchy. Many other items were found to be excluded in this way.

To generate the pairwise comparison matrices at each level, the group considered three different scenarios. (1) adequate building size with suitable interior is rated the highest (case 1). (2) proximity of building in regard to other existing operations in the area and (3) costs involved in getting the site 'up to a standard', i.e. ma-king sure the legal regulations are approved.

Pairwise comparisons can be generated using the data obtained. However, since I have already illustrated this approach using mathematical analysis in the previous example, I have chosen to describe the outcomes of events and in doing so, explain the determining factors, which affected the decision-making process.

The site chosen was C. The managers agreed that there is no use buying a site that is too large, even if there is a chance of a further expansion in the future as you are paying for it now and the likelihood of that expansion is not yet known. They also concluded that they could not wait for facilities to be upgraded as time is becoming another important criteria.

Therefore when the comparisons were completed it showed these results in the ratings. The other major factor was the costs associated with the factory. The cost associated with the three sites did not differ too much, instead they differed considerably in what they offered. Site C obviously stood out as being the best choice more so than any of the other options. The ratings given in the pairwise comparisons would have again illustrated this point significantly.

4.4 AHP AND EXPERT CHOICE

It allows the decision-maker to visually portray a complex problem in the form of hierarchy and to use either verbal or numerical judgement to compare criteria and alternatives in a pairwise fashion, i.e. one to another. This has been previously illustrated throughout section 4. of the report. Common sense tells us that every decision we make is based on some criteria and involves at least two choices.

In fact, the great majority of decisions we make are simple and our decisions are intuitive - we rank the alternatives mentally and choose the one that satisfies the criteria we have established. But as the decisions become more complex we have more difficulty handling all the competing criteria and alternatives mentally, and as a result, are more likely to make the wrong decision which could lead to monetary, social, political or personal loss [ 46].



Any mathematical-based process involves time and energy to perform calculation. Fortunately, a software package known as 'Expert Choice', developed by Decision Support Software Inc., eliminates all the manual labour associated with AHP and makes structuring the hierarchy quick and simple. As stated in the 'Expert Choice documentation, 'An 'Expert Choice' solution reflects the expertise of the decisionmaker, not the computer.' 'Expert Choice' presents screens structured according to the AHP methodology [12].

After the decision-maker identifies the criteria and alternatives in response to the 'Expert' system prompts, the system performs all mathematical calculations required to produce a ranking of alternatives based on the criteria and judgement inputs.

Once the process is complete, the decision-maker can perform 'what if exercises to test the impact of changes in his or her judgement concerning the relative weights of the criteria or alternatives. A new alternative or additional criterion can quickly be assimilated into the model and their impacts determined literally at the touch of button.

'Expert Choice' does not do away with the customary requirement to perform a technical evaluation through cost/price analysis of each proposal. What it does is to help organise the process, convert subjective judgements to a form suitable for mathematical synthesis, perform the calculations and, perhaps most importantly provide a clear, logical, concise, medium for communicating the recommends course of action to others.

The first thing 'Expert Choice' does is to ask the decision maker what his or her overall goal is. Then it asks the decision-maker to state the criteria and sub-criteria on which the decision will be based. Finally, the alternatives being considered are added. After the hierarchy is established, the decision-maker must then compare each alternative to other alternatives with respect to each sub-criteria. Following that, each of the sub-criteria is compared to the others with respect to its 'parent' criterion. Finally, each criterion is compared, one at a time, to the other criteria with respect to the goal.

'Expert Choice' asks whether the factor being considered is either more important, more preferable or more likely than the factor it is compared to. This process forces the user to formally structure his thoughts, not only about the overall goal, but about every factor bearing on the decision,

Source selections are well suited to the application of 'Expert Choice'. Derivations of the precise numerical numbers is done by 'Expert Choice', based on the answers give to questions asked by the system.

Once the decision-maker tells 'Expert Choice' what the primary criteria are, the sub criteria (if any) must be specified.

By simply moving the screen cursor to the next node representing the next primary criterion and requesting 'Expert Choice' to re-draw the hierarchy, all the sub-criteria can easily be specified under the appropriate node. Once all the sub-criteria are added



the actual alternatives being considered in the decision are specified.

Once the complete hierarchy has been drawn, the process of determining the relative importance or preference of the alternatives, sub-criteria and criteria begins. 'Expert Choice' asks the decision-maker to state his judgement concerning each of the comparisons in the hierarchy.

All of the above procedure for 'Expert Choice' can be seen in the example given in Appendix 6. In this appendix, the problem outlines actual screen representations and the 'Expert Choice' command line to a problem involving "determining the best way to raise capital".

Some of the problems that can be resolved using 'Expert Choice' [381 are: Corporate Decision-Making

1. Strategic Planning 2. Acquisitions and Mergers 3. Marketing and Advertising 4. Research and Development 4. Employee Evaluations 6. Product Life Cycle 7. Proposal Planning 8. Public Relations

Small Business Decision-Making 1. Strategic Planning 2. New Products I life Cycle 3. Financial Planning 4. Marketing and Advertising 4. Bid I No Bid on Request for Proposal 6. Sue I Don't Sue or Settle I Don't Settle 7. Purchasing 8. Hiring I Firing

National Policy Decision-Making 1 . Strategic Planning 2. Nuclear Arms Limitation Agreement 3. Budget Allocation 4. National Crisis 4. Support/ Oppose Legislation 6. Proposal Planning 7. Intentional Trade Policy 8. Military Decisions

Public Administration Decision- Making (Federal I State I Local) 1. Resource and Budget Allocations 2. Policy and Legal Decisions 3. Evaluating RFP's 4. Tactical Planning


Decision Analysis

4. Tax Policy 6. Reduction in Forces 7. Public Relations

Personal Decision-Making 1. Career Planning 2. Educational Planning 3. Job Selection Change 4. Geographical (Re)Location 4. Marital Decisions 6. Financial Investments 7. Voting 8. Acquisitions

Summary

E. Shayan

What can 'Expert Choice' do? 'Expert Choice' has the capability to deal with any type of complex problem within every department or division in an Organisation, such a finance, marketing, advertising, purchasing, personnel, engineering and production.

'Expert Choice' can establish a forum for group decision, manage complexity, apply knowledge, intuition, derive priorities and rank alternatives, measure consistency of judgements. It is able to recommend decisions or justify the rationale for a decision.



Chapter 5.0 AUTOMATING DECISION ANALYSIS

5.1 Introduction To Automating Decision Analysis [10]

One of the most difficult problems faced by managers today is how to make good decisions under uncertainty. Uncertainty makes decision-making complicated, 1 raises both methodological and philosophical questions. How should choices be evaluated when the consequences are uncertain? When do we have enough information to make a decision? How much should we pay for information that can reduce our uncertainty ?

Decision analysis is a philosophy and a methodology for answering questions like these. It provides a logical framework for decision-making based on what you know, what you can do and what you prefer.

Decision analysis is based on the principles of probability theory and a set of standard conceptions derived from utility theory, The principles of decision analysis have been applied to a wide range of commercial and public policy problems.

Despite the promise of decision analysis, there are a number of practical limitations to its use. One important limitation is due to the weaknesses of existing decision analysis tools. Unlike the simple textbook examples, many realistic decision analysis

problems are large and complex. They have many uncertain variables and complex relationships among these variables.

Since computational burden tends to increase exponentially with respect to the number of model variables, decision analysis models frequently require troublesome compromises between realism and practicality. Too often, the result is a model that fits the limitations of the modelling tools rather than the understanding and needs of the decision-maker.

The next section reviews the major features of a typical decision analysis problem. hypothetical problem will be introduced to illustrate each step of the decision analysis process.

5.2 An Dlustrative Decision Analysis Problem

Decision analysis problems typically share two major features. They have a large number of inter-related and uncertain variables and a process that links each decision with what is known about the variables at the time each decision is made.

Consider the following sequential decision-making problem, although the problem is hypothetical.

An electric utility is re-starting a nuclear power plant that has been out of operation for routine maintenance and repairs. The utility managers, however, are concerned about the reliability of several major systems of the plant. The design and construction materials are similar to many other plants that have experienced major

Automating Decision Analysis 116 Chapter 5


system failures.

Many of these failures have been associated with a phenomenon known as intergranular stress corrosion cracking ( IGSCC). Although the causes ofiGSCC are still uncertain, it is believed to result from a physical and chemical interaction of pressurised cooling water with construction materials used in piping systems.

The utility has a number of alternatives. One choice is simply to re-start the plant a planned, however, if any major system fails during re-start, other systems could also suffer major damage. The resulting repairs and system downtime would be very costly. The utility is also concerned that major failures during re-start will cause it to lose credibility with the Nuclear Regulatory Commission and the public.

Alliteratively, the utility could delay the start-up and conduct a 'reliability program'. The cost of the reliability program is thought to be high, although the exact amount is uncertain. The effectiveness of the reliability program is more certain since it would replace all piping and systems affected by IGSCC. The plant could be re-started with very low chance of failure due to IGSCC. However, since IGSCC is not the on] possible cause of system failure, the utility still could not be certain that no system failures will occur.

Because the reliability program is so expensive and the re-start option is so risky, plan engineers have presented another option. Before choosing between re-start or the reliability program, the utility could conduct one or both of two tests to estimate the level ofiGSCC in the plant. One test is inexpensive but is considered somewhat unreliable.

It involves a limited sample of easily accessible systems for evidence ofiGSCC. The second test is much more reliable, but much more expensive. It involves an extensive inspection of internal systems for IGSCC. In either case, the level ofiGSCC discovered in the plant will not perfectly foretell whether any of the systems will fail but can be used to estimate, by influence, the likelihood of system failure.

What should the utility do: proceed with re-starting the plant, conduct the reliability program or conduct one or both of the tests for IGSCC and then decide? Should the utility attempt to resolve other uncertainties, such as the cost of the reliability program? If so, how much should it be willing to invest ?

Which course of action will minimise expected costs? The next section shows how the hypothetical problem is addressed in each step of the decision analysis process The basic inputs to a decision analysis model are developed for the illustrative problem.

5.3 The Decision Analysis Process

The decision analysis process is typically viewed as an interactive series of steps. for this example the major steps have been outlined below. It has been characterised into five main areas:



1. Problem structuring. 2. Deterministic modelling. 3. Probabilistic modelling. 4. Problem analysis. 4. Communication of results.

5.3.1 Step 1. Problem Structuring

The objective of the problem structuring is to develop a clear statement of the decision- maker's alternatives, values and uncertainties. This step is often simply a process for clarifying 'fuzzy' ideas about the nature of the problem and getting a clear statement of the major issues. Alternatives for the utility:

In the case of the hypothetical problem, ·the decision-maker's alternatives can be described as a dynamic decision process, summarised as follows. The utility must decide between re-starting the plant or conducting a reliability program. Alliteratively the utility could first conduct either one of the tests. If the tests are completed the utility has new information about the chance of system failure based on the results of the test.

The utility must then decide whether to conduct the second test, re-start the plant or buy the reliability program. If the second test is conducted, the utility repeats the base decision. Again, the utility has new information from the second test about the level ofiGSCC and thus the chance of system failure.

If testing is to be conducted, which text should come first? The decision structure must allow for either sequence and the optimal sequence is determined as part of the analysis. 'Me unreliable and inexpensive test would examine several easily accessible external systems. The more reliable test would inspect internal systems. Consequently, the decision sequence for testing should consider the simple test first and the extensive test second - just to see the results from the simple test first.

Values:

The utility is concerned primarily about the economic costs of its choices. These include direct costs of failures as well as indirect costs of delay. The utility is concerned about the loss of credibility with the Nuclear Regulatory Commission and the public if major system failures occur during re-start.

The major uncertainties identified are as follows : 1. The level ofiGSCC in the plant. 2. The number of major system failures in re-start. 3 . The cost of the reliability program. 4. The outcomes of each test, and 4. 'Me cost of each test.

5.3.2 Step 2. Deterministic Modelling



The deterministic phase of the decision analysis process has three major objectives. For this example only one is relevant and therefore only one shall be discussed. The one which will be discussed is to develop a quantitative deterministic model of the structural relationships among decisions and state variables and is referred to as an 'outcome model'.

The outcome model:

The outcome model is essentially a calculator for computing the level of each type of impact for each alternative (test, re-start, etc.) and for every state of uncertain variable. In the example, the utility is concerned about losing credibility as well as the direct economic impact of its choices. A simple outcome model is used. For example, total costs are equal to test costs plus the cost of the reliability program and the cost of any failures. For simplicity, assume that all outcomes can be measured as cost impacts and are already reflected in the cost impacts of other variables.

5.3.3 Step 3. Probabilistic Modelling And Data Gathering

The purpose of this phase is two-fold: (1) to develop a model of the probabilistic relationships (i.e. dependencies and independence) among uncertain variables and (2), to develop the probabilistic inputs from each uncertain variable. Probabilistic independence in practical terms means simply that for any two variables A and B, knowing the value of A does not change the probability distribution ofB (and vis versa). Probabilistic dependence implies that the preceding is not true.

The treatment of probabilistic relationships and uncertain quantities are critical features of decision analysis models. Assessment of expert judgement is the traditional process by which all information (statistical) is incorporated into the model. A formal and highly structured interview process is used to help minimise the well known effects of judgmental biases when assessing expert judgement 5 [24].

Several variables in the sample problem are independent on the level ofiGSCC. The outcomes of both tests can be expressed as conditioned by the actual level ofiGSCC in the plant's systems. The likelihood of system failures can also be expressed as conditioned by the level ofiGSCC and whether the utility conducts the reliability program. If the utility does not conduct the reliability program, then system failures are uncertain and of relatively high probability. If the utility conducts the reliability program on the other hand, then system failures are still uncertain, but since the

reliability program removes all IGSCC, the probability of failure is much lower.

The cost of testing and the cost of the reliability program are both independent of the level ofiGSCC. Test costs are also thought to be relatively certain and relatively low, since the utility is familiar \with the testing procedures required. The cost of the

reliability program on the other hand, is viewed as uncertain and possibly very high.

5.3.4 Step 4. Problem Analysis

In the problem analysis phase, components of the model are linked together and a



formal evaluation of the problem is conducted. The 'standard' outputs of a decision analysis include the following.

Optimal policy - where the policy is the fundamental result of a decision analysis. It

indicates the best choice for each decision, what the utility should do first, conduct tests, what it should do given test outcomes.

Risk profiles. The optimal policy is the 'what' result of the analysis; risk profiles are one of the 'why' outputs. Risk profiles are generated for the optimal policy and for other policies that may be of interest to the decision-maker. Risk profiles help the decision-maker understand the policy results and to identify issues that may not have been treated.

Sensitivity problem analysis steps usually include additional deterministic sensitivity analysis on remaining deterministic values to ensure confidence in judgement that they are relatively certain. The problem analysis step also includes a probabilistic sensitivity analysis which tests results for sensitivity to the probability distributions for uncertain variables.

5.3.5 Step 4. Communication Of Results

The objective of the last step is to promote the analyst's and ultimately the decision maker's understanding of the model results. The model must be able to show 'why' an answer is 'right'.





Chapter 6.0 MATHEMATICAL MODELS FOR DECISION

ANALYSIS

6.1 LINEAR PROGRAMMING

Linear Programming is a tool for solving optimisation problems. In 1947, George Dantzig developed an efficient method, the simplex algorithm, for solving linear programming problems. Since the development of the simplex algorithm, linear programming has been used to solve optimisation problems in industries as diverse as banking, education, forestry, petroleum and trucking [ 461.

What is linear programming? In this section we will introduce linear programming and define some important terms that are used to describe linear pr�gramming problems.

A linear programming problem (LP) consists of three parts:

1. A linear function (the objective function) of decision variables (say X1, X2, .... , Xn) that are to be maximised or minimised.

2. A set of constraints (each of which must be linear equality or linear inequality) that restricts the values that may be assumed by the decision variable.

3. The sign restrictions which specify for each decision variable Xj either (1) variable must Xj must be non negative- Xj > 0; or (2) variable Xj may be positive, zero, or negative- Xj is unrestricted in sign (urs).

The coefficient of the variable in the objective function is the variables objective function coefficient. The coefficient of a variable in a constraint is a technological coefficient.

A point is simply a specification of the values of each decision variable. The feasible region of a linear programme consists of all points satisfying the linear programmes constraints and sign restrictions. Any point in the feasible region that has the largest Z value of all points in the feasible region (for max problem) is an optimal solution to the linear programme. A linear programme may have no optimal solution, one optimal solution or an infinite -number of optimal solutions.

A constraint in a linear programme is binding at an optimal solution if the constraint holds with equality when the values of the variables in the optimal solution are substituted into the constraint.

Graphical solution of linear programming problems:

The feasible region for any linear programme is a convex set. If a linear programme has an optimal solution, there is an extreme (or corner) point of the feasible region that is an optimal solution to the linear programme.

Mathematical approaches 122 Chapter 6


We may graphically solve a linear programme (max problem) with two decision

variables as follows:

Step 1: Graph the feasible region. Step 2: Draw an isoprofit line.

Step 3: Move parallel to the isoprofit line in the direction of increasing z.

The last point in the feasible region that contacts an isoprofit line is an optimal solution to the linear programme.

Linear programming - four different types of solutions- later on in this section we will be examining one of these in greater detail.

When you solve a linear programme, one of the following four cases will occur:

Case 1: The linear programme has a unique solution. Case 2: The linear programme has more than one (actually all infinite

number of) optimal solutions. This is the case of alternative optimal solutions. Graphically we recognise this case when the isoprofit line hits an entire line segment before leaving the feasible regiOn.

Case 3: The linear programme is not feasible (it has no feasible solution). This means that the feasible region contains no points.

Case 4: The linear programme is unbounded. This means (in a max problem) that there are points in the feasible region with arbitrarily large z values. Graphically we recognise this case by the fact that when we move parallel to an isoprofit line in the direction of increasing z, we never lose contact with the linear programmes feasible region.

Formulating linear programmes

The most important step in formulating linear programmes is to correctly determine the decision variables. In any constraints the terms must have the same units. For example, one term cannot have "kilograms of raw materials" while another term has the units "tonnes of raw material".

The following example of linear programming can be utilised to solve workplace scheduling problems. This 1make believe1 problem displays the fundamentals of linear programming and how these principles are used to schedule work.

6.1.1 Example Using linear Programming In Relation To Work Scheduling

CSL is a chain of computer service stores. The number of hours of skilled repair time that CSL requires during the -next five months is as follows:

Month 1 (January) : Month 2 (February):

Mathematical approaches

6000 hours 7000 hours

123 Chapter 6

( .

Decision Analysis

Month 3 (March): Month 4 (April): Month 5 (May):

8000 hours 9500 hours 1 1000 hours

E. Shayan

At the beginning of January, 50 skilled technicians worked for CSL. Each skilled technician can work up to 160 hours per month. In order to meet future demands new technicians must be trained. It takes one month to train a new technician. During the month of training a trainee must be supervised for 50 hours by an experienced technician. Each experienced technician is paid $2000 a month (even if 160 hours of work is not completed per month). During the month of training, a

trainee is paid $1000 per month. At the end of each month, 5% of CSL's experienced technicians quit to join Plum Computers. This question requires you to formulate a linear programme whose solution will enable CSL to minimise the labour cost

incurred in meeting the service requirements for the next five months.

Solution

CSL must determine the number of technicians that should be trained during month t ( t = 1,2,3,4,5). Thus, we define

Xt =number of technicians trained during month t ( t = 1,2,3,4,5 )

CSL wants to minimise total labour cost during the next five months. Note that Total Labour Cost = cost of paying trainees + cost of paying experienced technicians.

To express the cost of paying experienced technicians, we need to define, for t = 1,2,3,4,4. Yt =number of experienced technicians at the beginning of the month t.

Then Total Labour Cost = (1000 X1 + 1000 X2 + 1000 X3 + 1000 X4 + 1000 X5) +

(2000 Y1 + 2000 Y2 + 2000 Y3 + 2000 Y 4 + 2000 Y5) Thus, CSL's objective function is min z = 1000 X1 + 1000 X2 + 1000 X3 + 1000 X4 + 1000 X5 + 2000 Y1 +

2000 Y2 + 2000 Y3 + 2000 Y4 + 2000 Y5)

What constraints does CSL face? Note that we are given Y1 = 50, and that for t=1,2,3,4,5, CSL must ensure that Number of available technician hours during month t ( number of technician hours required during month t.

Since each trainee requires 50 hours of experienced technician time and each skilled technician is available for 160 hours per month,

number of available technician hours during month t = 160 Y1-50 X1

Now, yields the following five constraints:

160 Y1 -50 X1 ( 6000 ( Month 1 constraint ) 160 Y2 -50 X2 ( 7000 ( Month 2 constraint ) 160 Y3 -50 X3 ( 8000 ( Month 3 constraint ) 160 Y4-50 X4 ( 9500 ( Month 4 constraint )

Mathematical approaches 1 2 4 Chapter 6


160 Y5- 50 X5 ( 11000 (Month 5 constraint )

We need constraints that relate variables from different periods. In the CSL problem, it is important to realise that the number of skilled technicians available at the beginning of any month is determined by the number of skilled technicians available during the previous month and the number of technicians trained during the previous month:

Experienced technicians available at the beginning of month t =

experienced technicians available at the beginning of month ( t - 1 ) + technicians trained during month (t-1)

experienced technicians who quit during month ( t - 1 )

For example, for February, yields

Y2 = Y1 + X1- 0.05 Y1, or Y2 = 0.95 Y1 + X1

Similarly for March, April, May, respectively, yields

Y3 = 0.95 Y2 + X2

Y 4 = 0.95 Y3 + X3 Y5 =0.95 Y4+X4

Adding the sign restrictions, Xt ( 0 and Yt ( 0 ( t = 1,2,3,4,5 ), we obtain the following linear programme:

min z = 1000 X1 + 1000 X2 + 1000 X3 + 1000 X4 + 1000 X5 + 2000 Y1 + 2000 Y2 + 2000 Y3 + 2000 Y4 + 2000 Y4.

s.t 160 Y1- 50 X1 .GE. 6000 160 Y2- 50 X2 .GE. 7000 160 Y3- 50 X3 .GE. 8000 160 Y4- 50 X4 .GE. 9500 160 Y5 - 50 X5 .GE. 11000

Y1 =50 0.95 Y1 +X1 =Y2 0. 95 Y2 + X2 = Y3 0. 95 Y3 + X3 = Y 4 0.95 Y4+X4 =Y5

The optimal solution is z = 593,777; X1 = 0; X2 = 8.45; X3= 11.45; X4= 9.52; X5= 0; Y1= 50; Y2= 46.5; Y3= 53.58; Y4= 62.34; and Y5= 68.74.

In reality the Xj's must be integers, so this solution is difficult to interpret. Of course, we could obtain a feasible integer solution by rounding X2 up to 9, X3 up to 12, and X4 up to 10, but there is no guarantee that this is the optimal solution.

6.2 INTEGER PROGRAMMING:

Simply stated, an integer programming problem (IP) is a linear programming problem in which some or all of the variables are required to be non-negative integers.



A non-linear integer programming problem is an optimisation problem in which either the objective function or the left hand side of some of the constraints are nonlinear functions and some or all of the variables must be integers. Such problems are beyond the scope of this report.

An integer programming problem in which all variables are required to be integers is called a pure integer programming problem, an example of this type of problem will be examined later on in this section. An integer programming problem in which only some of the variables are required to be integers is called a mixed integer programming problem [ 46].

6.2.1 Example Using Integer Programming For Investment Decisions In Stockco.

Stockco is considering four investments. -Investment 1 will yield a net present value (NPV) of $16, 000; Investment 2, an NPV of $22, 000; Investment 3, an NPV of $12,000

and Investment 4, an NPV of $8,000. Each investment requires a certain cash outflow at the present time: Investment 1, $5,000; Investment 2, $7,000; Investment 3, $4,000; and Investment 4, $3,000. At present, $14,000 is available for investment.

formulate an IP whose solution will tell Stockco how to maximise the NPV obtained from Investments 1 to 4.

Solution :

As in linear programming formulations, we begin by defining a variable for each decision that Stockco must make. This leads us to define a 0-1 variable;

Xj ( j = 1 ,2,3, 4 ) = { 1 - if investment j is made and 0 - if otherwise}.

For example, X2 = 1 if investment 2 is made, and X2 = 0 if investment 2 is not.

The NPV obtained by Stockco (in thousands of dollars ) is:

Total NPV obtained by Stockco = 16 XI+ 22 X2 + I2 X3 + 8 X4

To see this, note that ifXj =I, the equation above includes the NPV of investment j, and ifX1 = 0, equation above does not include the NPV oflnvestment j. This means that whatever the combination of investments is undertaken, the equation above gives the NPV of that combination of projects. For example, if Stockco invests in Investments 1 and 4, then the NPV of $16,000 + $8,000 = $24,000 is obtained, since this combination of investments correspond to X1 = X4 = 1, X2 = X3 = 0. The equation above indicates that the NPV for this investment combination is 16(1) + 22(0)

+ I2(0) 8(I)=$24(thousand). This reason implies that Stockcds objective function is max z = I6 XI + 22 X2 + I2 X3 + 8 X4

Stockco faces the constraint that at most $I4,000 can be invested. By the same reasoning used to develop the first equation above, we can show that

Mathematical approaches I26 Chapter 6


Total amount invested (in thousands of dollars)= 5 XI+ 7 X2 + 4 X3 + 3 X4

For example, if XI= 0, X2 = X3 = X4 =I, then Stockco makes investments 2,3 and 4. In this case, Stockco must invest 7 + 4 + 3 = $I4 (thousand). Substituting these values into the equation above yields 5(0) + 7(1) + 4(1) + 3(1) = $I4 (thousand). Since at most $I4,000 can be invested, XI, X2, X3 and X4 must satisfy

5 XI+ 7 X2 + 4 X3 + 3 X4 ( I4

Combining the last two equations with the constraints Xj = 0 or I ( j = I,2,3,4 ) yields the following 0-I integer programming problem

max z = I6 XI+ 22 X2 + I2 X3 + 8 X4

s.t 5 XI + 7 X2 + 4 X3 +3 X4 Xj = 0 or I 0 = I,2,3,4 )

The total NPV obtained by Stockco shows that the optimal solution to the above three equations is XI = 0, X2 = X3 = X4 =I, z = $42,000. Hence, Stockco should invest in Investments 2,3 and 4 and not invest in Investment 1. Since Investment I yields a higher NPV per dollar invested than any of the other Investments (Investment I yields $3.20 per dollar invested, Investment 2 yields $3.I4 per dollar invested, Investment 3 yields $3 per dollar invested and Investment 4 yields $2.67 per dollar invested. It may surprise you that Investment I is not undertaken.

To see why the optimal solution to the above equations does not involve making the 'best' investment, note that any investment that uses Investment I cannot use more than $12,000. This means that using Investment 1 forces Stockco to forgo investing $2,000. On the other hand, the optimal investment combination uses all the $I4, 000 of the investment budget. This enables the optimal combination to obtain a higher NPV than any other combination that includes Investment I. If fractional investments

were allowed, the optimal solution to the above equations would change dramatically to XI= X2 = 1, X3 = 0.50, X4 = 0, z = $44,000, and Investment 1 would be used.

This simple example shows that the choice of modelling a capital budgeting problem as a linear programming or as an integer programming problem can significantly affect the optimal solution to the problem.

6.3 Overview of Deterministic and Probabilistic Models

Examples of decision models can either be deterministic or probabilistic. Section 5. 3 .2 and 5.3.3 have also covered deterministic and probabilistic models in their literature survey. If all the parameters in the decision model are known with certainty (or are assumed to be fixed and known for the time of the decision), then the models are deterministic. Examples of these decision models are linear programming and integer programming models which are used to determine the best course of action that will lead to maximum profit or minimum cost. These models are single criterion models. The goal programming models are also deterministic models with known parameters and resource availabilities, but there is more than one criterion or objective to be maximised or minimised. Such models are called Multiple Objective



or multi-criteria Models. The last two models were mentioned for completeness only and will not be discussed further due to the length of this section of the report.

CONCLUSION

The objective was to examine systematic techniques and applications of decisionmaking processes, analyse them and draw conclusions. This objective has been hopefully fulfilled to clearly enable a lay person with little or no understanding of the processes involved, as well as those with a background in decision-analysis applications, processes and techniques to gain from the materials and examples.

Extensive coverage of the literature survey is presented. The format simplifies the understanding of the processes by taking the reader step-by-step through the analysis of each problem, systematically offering a sequence of easily understood applications which lead to an eventual conclusion thereby deciding the issue(s) under review.

The broad spectrum of examples both hypothetical (original thought) and 'real life' case studies, cover not only the manufacturing sector but also current 'every-day' issues likely to arise in any decision-making process in civilian life. Common sense tells us that being in possession of all/almost all the facts will enable a worthwhile decision to be made; however, when these facts are not available for review it is of inestimable value to examine through the decision analysis processes presented in this or similar reports, ways in which one may reach a successful conclusion.

In many of the examples presented in this report, the rather lengthy step-by-step process of analysing each decision is a vital prerequisite to obtaining the 'best' outcome. For those companies who are in possession of the software packages or decision-analysis 'tools' (knowledge of the applications and processes) the process and time required to obtain the desired results is significantly reduced.

It is recommended that before a decision is made, the analyst examines all the possible alternatives, obtains all the relevant data and evaluates each step in thedecision-making process before drawing a conclusion. It makes good sense to reduce the risks and uncertainties involved in any decision-making process prior to reaching a conclusion. However, even after taking all the necessary precautions there are no guarantees of successful outcomes.





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