Deterministic Models in OR Joseph Khamalah, Ph.D. K327:02

Deterministic Models in OR

Joseph Khamalah, Ph.D.

K327:02

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Agenda: Week of August 26, 2010

Introduction– Syllabus

– Student Info Sheet

– Exams

– Cases/Assignments

– Peer Evaluation

Group Formation

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Grade Weights

Within Area Total

a) INDIVIDUAL WORK (55-75%)…………...……60%Exams (2@ 20 – 35%) ………40%

Assignments (10 – 20%) …….20%

b) GROUP WORK (25-45%)……………….…..40%

Exams (2@ 10 – 20%)…..…...20%

Cases (4@ 01 – 03) …………..12%

Project (01 – 10%) …....……...08%

Grand Total…………...………..………... 100%

Deterministic Models in Operations Research

Introduction – Chapter 1

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Learning Objectives

Develop a general understanding of the management science / operations research approach to decision making.

Realize that quantitative applications begin with a problem situation. Obtain a brief introduction to quantitative techniques and their

frequency of use in practice. Understand that managerial problem situations have both quantitative

and qualitative considerations that are important in the decision making process.

Learn about models in terms of what they are and why they are useful (the emphasis is on mathematical models).

Identify the step-by-step procedure that is used in most quantitative approaches to decision making.

Learn about basic models of cost, revenue, and profit and be able to compute the breakeven point.

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Introduction to Operations Research

Operations Research/Management Science

– Winston: “a scientific approach to decision making, which seeks to determine how best to design and operate a system, usually under conditions requiring the allocation of scarce resources.”

– Kimball & Morse: “a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control.”

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Introduction to OR cont’

Provides rational basis for decision making– Solves the type of complex problems that turn up in

the modern business environment

– Builds mathematical and computer models of organizational systems composed of people, machines, and procedures

– Uses analytical and numerical techniques to make predictions and decisions based on these models

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Introduction to OR cont’

Draws upon

– engineering, management, mathematics

Closely related to the "decision sciences"

– applied mathematics, computer science, economics, industrial engineering and systems engineering

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Body of Knowledge

The body of knowledge involving quantitative approaches to decision making is referred to as – Management Science

– Operations Research

– Decision Science

– Operations Management

– Industrial Engineering

– Industrial Management

– Systems Management

– Systems Science

It has its early roots in World War II and is flourishing in business and industry with the aid of computers.

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What is OR? Cont’

Origins in the 1940's with the management of the Cold War armed forces. Currently used in a very broad range of organizations and systems.

The linear programming model for solving OR problems was developed in the 1940's. The first computer implementation to solve linear programs, known as the simplex method was developed in 1947 by George Dantzig.

The Nobel Prize in Economics was awarded in 1975 to L.V. Kantorovich and T.C. Koopmans for their work in linear programming. Linear programming was also used in portfolio management by H.M. Marcowitz, who won the Nobel Prize in 1990.

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Features of OR

Emphasis on:– large, complex operations– mathematical models– computer implementation

Extensive use in:– manufacturing– transportation– entertainment– construction/development– communication– computer/database systems– economics/investing– armed forces– biology/genomics

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Examples of OR Problems

resource allocation scheduling routing inventory management system design and construction production pricing forecasting capital budgeting customer service

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The role of the OR Analyst

1. Formulate the Problem: Determine the nature and constraints of the problem, and the goal(s) of the client for the problem. Determine a model class appropriate for the problem.

2. Observe the system: Collect the facts and data necessary for precisely specifying and solving the model.

3. Formulate a mathematical model needed for the problem solution and identify the software tools used to solve this mathematical model.

4. Verify the Model: Check the results of the model on known situations to see if it gives reasonable and predictable answers in these cases.

5. Determine a solution to the new case according to the model developed above.

6. Present the solution in a form which the client can use.7. Assist in the implementation of the solution

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7 Steps of Problem Solving(First 5 steps are the process of decision making)

– Define the problem.

– Identify the set of alternative solutions.

– Determine the criteria for evaluating alternatives.

– Evaluate the alternatives.

– Choose an alternative (make a decision).

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

– Implement the chosen alternative.

– Evaluate the results.

Problem Solving and Decision Making

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Quantitative Analysis and Decision Making

Potential Reasons for a Quantitative Analysis Approach to Decision Making– The problem is complex.

– The problem is very important.

– The problem is new.

– The problem is repetitive.

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Quantitative Analysis

Quantitative Analysis Process– Model Development

– Data Preparation

– Model Solution

– Report Generation

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Model Development

Models are representations of real objects or situations Three forms of models are:

– Iconic models - physical replicas (scalar representations) of real objects;

– Analog models - physical in form, but do not physically resemble the object being modeled; and

– Mathematical models - represent real world problems through a system of mathematical formulae and expressions based on key assumptions, estimates, or statistical analyses.

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Advantages of Models

Generally, experimenting with models (compared to experimenting with the real situation):– requires less time

– is less expensive

– involves less risk

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Mathematical Models

Cost/benefit considerations must be made in selecting an appropriate mathematical model.

Frequently a less complicated (and perhaps less precise) model is more appropriate than a more complex and accurate one due to cost and ease of solution considerations.

For every complex problem, there is a simple solution,

which is neat andwrong!!

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Mathematical Models, cont’

Relate decision variables (controllable inputs) with fixed or variable parameters (uncontrollable inputs)

Frequently seek to maximize or minimize some objective function subject to constraints

Are said to be stochastic if any of the uncontrollable inputs is subject to variation, otherwise are deterministic

Generally, stochastic models are more difficult to analyze. The values of the decision variables that provide the

mathematically-best output are referred to as the optimal solution for the model.

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Transforming Model Inputs into Output

Uncontrollable Inputs(Environmental Factors)

Uncontrollable Inputs(Environmental Factors)

ControllableInputs

(DecisionVariables)

ControllableInputs

(DecisionVariables)

Output(ProjectedResults)

Output(ProjectedResults)

MathematicalModel

MathematicalModel

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Example: Project Scheduling

Consider the construction of the Phase III student housing complex. The project consists of hundreds of activities

involving excavating, framing,

wiring, plastering, painting, land-

scaping, and more. Some of the

activities must be done sequentially

and others can be done at the same

time. Also, some of the activities

can be completed faster than normal

by purchasing additional resources (workers, equipment, etc.).

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Example: Project Scheduling, cont’

Question: What is the best schedule for the activities and for which activities should additional resources be purchased? How could management science be used to solve this problem?

*****Management science can provide a structured, quantitative approach for determining the minimum project completion time based on the activities' normal times and then based on the activities' expedited (reduced) times.

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Question:

What would be the uncontrollable inputs?

***** –Normal and expedited activity completion times–Activity expediting costs–Funds available for expediting–Precedence relationships of the activities

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Question:What would be the:

1. objective function of the mathematical model?2. the decision variables? 3. the constraints?

*****1. Objective function: minimize project completion time2. Decision variables: which activities to expedite and by

how much, and when to start each activity3. Constraints: do not violate any activity precedence

relationships and do not expedite in excess of the funds available.

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Question:

Is the model deterministic or stochastic?

*****

Stochastic. Activity completion times, both normal and expedited, are uncertain and subject to variation. Activity expediting costs are uncertain. The number of activities and their precedence relationships might change before the project is completed due to a project design change.

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Question:

Suggest assumptions that could be made to simplify the model.

*****

Make the model deterministic by assuming normal and expedited activity times are known with certainty and are constant. The same assumption might be made about the other stochastic, uncontrollable inputs.

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Data Preparation

Data preparation is not a trivial step, due to the time required and the possibility of data collection errors.

A model with 50 decision variables and 25 constraints could have over 1300 data elements!

Often, a fairly large data base is needed. Information systems specialists might be needed.

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Model Solution

The analyst attempts to identify the alternative (the set of decision variable values) that provides the “best” output for the model.

The “best” output is the optimal solution. If the alternative does not satisfy all of the model

constraints, it is rejected as being infeasible, regardless of the objective function value.

If the alternative satisfies all of the model constraints, it is feasible and a candidate for the “best” solution.

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Model Solution, cont’

One solution approach is trial-and-error.– Might not provide the best solution

– Inefficient (numerous calculations required)

Special solution procedures have been developed for specific mathematical models.– Some small models/problems can be solved by hand

calculations

– Most practical applications require using a computer

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Computer Software

OR would not have developed as a field without the computer. Currently there are many software packages each of which

handles a particular class of models. Examples of decision models for which there is extensive software available:– linear/integer programming models– network & routing models– decision tree models– simulation models– queuing models– investment models

A variety of these s/w packages are available on IPFW’s network– Spreadsheet packages such as Microsoft Excel– The Management Scientist, developed by the textbook authors– DS - Decision Science– POM– HOM

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Quantitative Methods in Practice

Linear Programming Integer Linear

Programming PERT/CPM Inventory models Waiting Line Models Simulation

Decision Analysis Goal Programming Analytic Hierarchy

Process Forecasting Markov-Process

Models

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Model Testing and Validation

Often, goodness/accuracy of a model cannot be assessed until solutions are generated.

Small test problems having known, or at least expected, solutions can be used for model testing and validation.

If the model generates expected solutions, use the model on the full-scale problem.

If inaccuracies or potential shortcomings inherent in the model are identified, take corrective action such as:

– Collection of more-accurate input data

– Modification of the model

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Report Generation

A managerial report, based on the results of the model, should be prepared.

The report should be easily understood by the decision maker.

The report should include:– the recommended decision

– other pertinent information about the results (for example, how sensitive the model solution is to the assumptions and data used in the model)

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Implementation and Follow-Up

Successful implementation of model results is of critical importance.

Secure as much user involvement as possible throughout the modeling process.

Continue to monitor the contribution of the model. It might be necessary to refine or expand the model.

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Example: Austin Auto Auction

An auctioneer has developed a simple mathematical model for deciding the starting bid s/he will require when auctioning a used automobile.

Essentially, s/he sets the starting bid at seventy (70) percent of what s/he predicts the final winning bid will (or should) be. S/he predicts the winning bid by starting with the car's original selling price and making two deductions, one based on the car's age and the other based on the car's mileage.

The age deduction is $800 per year and the mileage deduction is $0.025 per mile.

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Example: Austin Auto Auction, cont’

Question: Develop the mathematical model that will give the starting bid (B) for a car in terms of the car's original price (P), current age (A) and mileage (M).

*****

The expected winning bid can be expressed as:

P - 800(A) - 0.025(M)

The entire model is:

B = 0.7(expected winning bid)

B = 0.7(P - 800(A) - 0.025(M))

B = 0.7(P) - 560(A) - 0.0175(M) or

B = 0.7P - 560A - 0.0175M

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Question:

Suppose a four-year old car with 60,000 miles on the odometer is up for auction. If its original price was $12,500, what starting bid should the auctioneer require?

*****

B = 0.7(12,500) - 560(4) - 0.0175(60,000)

= 8,750.00 - 2,240.00 – 1,050.00

= $5,460.00

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Question:

The model is based on what assumptions?

*****

The model assumes that the only factors influencing the value of a used car are the original price, age, and mileage (not condition, rarity, or other factors).Also, it is assumed that age and mileage devalue a car in a linear manner and without limit. (Note, the starting bid for a very old car might be negative!)

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Example: Iron Works, Inc.

Iron Works, Inc. manufactures two

products made from steel. It has just received

this month's allocation of b pounds of steel.

It takes a1 pounds of steel to make a unit of product 1

and a2 pounds of steel to make a unit of product 2.

Let x1 and x2 denote this month's production level of

product 1 and product 2, respectively. Denote by p1 and

p2 the unit profits for products 1 and 2, respectively.

Iron Works has a contract calling for at least m units of

product 1 this month. The firm's facilities are such that at

most u units of product 2 may be produced monthly.

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Example: Iron Works, Inc., cont’

Mathematical Model– The total monthly profit =

(profit per unit of product 1)

x (monthly production of product 1)

+ (profit per unit of product 2)


= p1x1 + p2x2

We want to maximize total monthly profit:

Max p1x1 + p2x2

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Mathematical Model (continued)– The total amount of steel used during monthly production

equals: (steel required per unit of product 1)

x (monthly production of product 1) + (steel required per unit of product 2)


= a1x1 + a2x2

This quantity must be less than or equal to the allocated b pounds of steel:

a1x1 + a2x2 < b

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Mathematical Model (continued)– The monthly production level of product 1 must be

greater than or equal to m :

x1 > m

– The monthly production level of product 2 must be less than or equal to u :

x2 < u

– However, the production level for product 2 cannot be negative:

x2 > 0

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Mathematical Model Summary

Max p1x1 + p2x2

s.t. a1x1 + a2x2 < b

x1 > m

x2 < u

x2 > 0

ObjectiveObjectiveFunctionFunction

““Subject to”Subject to”

ConstraintConstraintss

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Question:

Suppose b = 2000, a1 = 2, a2 = 3, m = 60, u = 720, p1 = 100, p2 = 200. Rewrite the model with these specific values for the uncontrollable inputs.

*****

Substituting, the model is:

Max 100x1 + 200x2

s.t. 2x1 + 3x2 < 2000

x1 > 60

x2 < 720

x2 > 0

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Question:The optimal solution to the current model is x1 = 60 and x2 = 626 2/3. If the product were engines, explain why this is not a true optimal solution for the "real-life" problem.

*****One cannot produce and sell 2/3 of an engine. Thus the problem is further restricted by the fact that both x1 and x2 must be integers. They could remain fractions if it is assumed these fractions are work in progress to be completed the next month.

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Uncontrollable InputsUncontrollable InputsUncontrollable InputsUncontrollable Inputs

$100 profit per unit Prod. 1$200 profit per unit Prod. 22 lbs. steel per unit Prod. 13 lbs. Steel per unit Prod. 2

2000 lbs. steel allocated60 units minimum Prod. 1

720 units maximum Prod. 20 units minimum Prod. 2

$100 profit per unit Prod. 1$200 profit per unit Prod. 22 lbs. steel per unit Prod. 13 lbs. Steel per unit Prod. 2

2000 lbs. steel allocated60 units minimum Prod. 1

720 units maximum Prod. 20 units minimum Prod. 2

60 units Prod. 1626.67 units Prod. 2

60 units Prod. 1626.67 units Prod. 2

Controllable InputsControllable InputsControllable InputsControllable Inputs

Profit = $131,333.33Steel Used = 2000

Profit = $131,333.33Steel Used = 2000

OutputOutputOutputOutput

Mathematical ModelMathematical ModelMathematical ModelMathematical Model

Max 100(60) + 200(626.67)s.t. 2(60) + 3(626.67) < 2000 60 > 60 626.67 < 720 626.67 > 0

Max 100(60) + 200(626.67)s.t. 2(60) + 3(626.67) < 2000 60 > 60 626.67 < 720 626.67 > 0

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Example: Ponderosa Development Corp.

Ponderosa Development Corporation (PDC) is a small real estate developer that builds only one style house. The selling price of the house is $115,000.

Land for each house costs $55,000 and lumber, supplies, and other materials run another $28,000 per

house. Total labor costs are approximately $20,000 per house.

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Ponderosa leases office space for $2,000 per month. The cost of supplies, utilities, and leased equipment runs another $3,000 per month.

The one salesperson of PDC is paid a commission

of $2,000 on the sale of each house. PDC has seven

permanent office employees whose monthly salaries

are given on the next slide.

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Employee Monthly Salary

President $10,000

VP, Development 6,000

VP, Marketing 4,500

Project Manager 5,500

Controller 4,000

Office Manager 3,000

Receptionist 2,000

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Question:

Identify all costs and denote the marginal cost and marginal revenue for each house.

*****–The monthly salaries total $35,000 and monthly office lease and supply costs total another $5,000. This $40,000 is a monthly fixed cost.–The total cost of land, material, labor, and sales commission per house, $105,000, is the marginal cost for a house.–The selling price of $115,000 is the marginal revenue per house.

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Question:

Write the monthly cost function c (x), revenue function r (x), and profit function p (x).

*****

c (x) = variable cost + fixed cost = 105,000x + 40,000

r (x) = 115,000x

p (x) = r (x) - c (x) = 10,000x - 40,000

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Question:

What is the breakeven point for monthly sales

of the houses? *****

r (x ) = c (x )

115,000x = 105,000x + 40,000

Solving for x gives:

115,000x - 105,000x = 40,000

10,000x = 40,000

Therefore, x = 40,000/10,000 = 4.

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Question:

What is the monthly profit if 12 houses per

month are built and sold?

*****

Simply plug 12 into the profit formula and solve

p (12) = 10,000(12) - 40,000 = $80,000 monthly profit

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Graph of Break-Even Analysis

00

200200

400400

600600

800800

10001000

12001200

00 11 22 33 44 55 66 77 88 99 1010Number of Houses Sold (x)

Th

ousa

nds

of

Dolla

rs

Break-Even Point = 4 Houses

Total Cost = 40,000 + 105,000x

Total Revenue = 115,000x

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More Examples of Optimization Problems:A Product Mix Problem

Woody's Furniture Company makes chairs, tables, and desks. Chairs are made entirely out of pine, and use 8 linear feet of pine per chair. Tables and desks are made of pine and mahogany, tables using 12 linear feet of pine and 15 linear feet of mahogany, and desks use 16 linear feet of pine and 20 linear feet of mahogany. Chairs require 3 hours of labor to produce, tables 6 hours, and desks 9 hours. Chairs provide $35 profit, tables $60 profit, and desks $75 profit.

Woody has 120 linear feet of pine and 60 linear feet of mahogany delivered each day, and has a work force of 6 carpenters, each of whom puts in an 8-hour day. How can Woody make the best use of these resources?

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Tabular description of Woody’s problem

Amount of resource/profit per Resource AvailableChair Table Desk

Pine 8 12 16 120

Mahogany 0 15 20 60

Man-hours 3 6 9 48

Profit 35 60 75

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A Diet Problem

The minimum daily requirements of protein, carbohydrates, and iron in the diet are 500, 400, and 75 respectively, and the cost of beef, potatoes, and spinach are $0.35, $0.20, and $0.15 per ounce, respectively.

How should the dietician plan her menu?

Food

Nutrient

Protein Carbs Iron

Beef 20 10 5

Potatoes 10 30 6

Spinach 6 7 20

A dietician wishes to plan a meal using ground beef, potatoes, and spinach which satisfies minimum daily requirements of protein, carbohydrates, and iron. The nutritional makeup of each ounce of foodstuff (in the appropriate nutritional units) is given below.

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Tabular description of the diet problem

Amount of resource/cost per ounce of Demand

Beef Potatoes Spinach

Protein 20 10 6 500

Carbs 10 30 7 400

Iron 5 6 20 75

Cost $0.35 $0.20 $0.15

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An Assignment Problem

The OR IDEAS Consulting Company has six crack OR analysts that it wants to assign to six projects it has contracted. The specialized skills of each analyst has been carefully evaluated with respect to each job to determine the amount of profit the company can expect to earn if the analyst is assigned to that particular job. The results are given in the table below:

Project

1 2 3 4 5 6

Anderson 10500 9000 8000 10000 12000 11500

Carlucci 12000 8000 7000 11000 11500 11000

Nataraja 10000 9000 6000 9000 12000 10500

Chin 12000 11000 9000 9000 10000 11000

Yohana 9500 7000 8000 8000 9000 7000

Yamanaka 12500 12000 10000 10000 11000 14000

♦ How should OR IDEAS put its analysts to best use?

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A Transportation Problem

After a week of renting cars that travel all across the country, Avis finds that it has a shortage of cars in Los Angeles, San Diego, Pittsburgh, New York, Atlanta, and Chicago, while it has a surplus of cars in San Francisco, Denver, Miami, and Houston. There is a certain cost of shipping cars from each supply point to each demand point, as given in the chart in the following table:

Outlet Surplus

LA SD Pgh NY Atl Chi

SF 5 4 19 16 17 13 6

Denver 4 7 9 7 8 4 250

Miami 7 8 17 10 9 12 20

Houston 5 7 10 8 12 13 13

Shortage 4 16 10 9 7 16

How can Avis return the cars to the correct locations?

Documents

Deterministic Models in OR Joseph Khamalah, Ph.D. K327:02