Lecture note on introduction to industrial engineering

ĐSTANBUL KÜLTÜR UNIVERSITYFACULTY OF ENGINEERING AND ARCHITECTURE

Department of Industrial Engineering

IE 250 Introduction to Industrial Engineering

Prof. Tülin AKTĐN

Spring 2011

1. INTRODUCTION TO BASIC CONCEPTS

1.1. Definition of Industrial Engineering

Industrial Engineering (IE) is concerned with the design, improvement

and installation of integrated systems of people, materials,

information, equipment and energy. It draws upon specialized

knowledge and skill in the mathematical, physical and social sciences

together with the principles and methods of engineering analysis and

design to specify, predict and evaluate the results to be obtained from

such systems.

INDUSTRIAL ENGINEERING

“5M” of Industrial Engineering

Manpower

Material

Method

Machine

Money

1.2. History of Industrial Engineering

The origins of industrial engineering can be traced back to many different

sources. Fredrick Winslow Taylor is most often considered as the father of

industrial engineering even though all his ideas where not original. Some of

the preceding influences may have been Adam Smith, Thomas Malthus,

David Ricardo and John Stuart Mill. All of their works provided classical

liberal explanations for the successes and limitations of the Industrial

Revolution.

Another major contributor to the field was Charles W. Babbage, a

mathematics professor. One of his major contributions to the field was his

book On the Economy of Machinery and Manufacturers in 1832. In this

book he discusses many different topics dealing with manufacturing, a few

of which will be extremely familiar to an IE. Babbage discusses the idea of

the learning curve, the division of task and how learning is affected, and

the effect of learning on the generation of waste.

In the late nineteenth century more developments where being made

that would lead to the formalization of industrial engineering. Henry R.

Towne stressed the economic aspect of an engineer's job. Towne belonged

to the American Society of Mechanical Engineers (ASME) as did many other

early American pioneers in this new field. The IE handbook says the, "ASME

was the breeding ground for industrial engineering. Towne along with

Fredrick A. Halsey worked on developing and presenting wage incentive

plans to the ASME. It was out of these meetings that the Halsey plan of

wage payment developed. The purpose was to increase the productivity of

workers without negatively affecting the cost of production. The plan

suggested that some of the gains be shared with the employees. This is

one early example of one profit sharing plan.

Henry L. Gantt belonged to the ASME and presented papers to the ASME

on topics such as cost, selection of workers, training, good incentive plans,

and scheduling of work. He is the originator of the Gantt chart, currently

the most popular chart used in scheduling of work.

What would Industrial Engineering be without mentioning Fredrick

Winslow Taylor? Taylor is probably the best known of the pioneers in

industrial engineering. His work, like others, covered topics such as the

organization of work by management, worker selection, training, and

additional compensation for those individuals that could meet the standard

as developed by the company through his methods.

The Gilbreths are accredited with the development of time and motion

studies. Frank Bunker Gilbreth and his wife Dr. Lillian M. Gilbreth worked on

understanding fatigue, skill development, motion studies, as well as time

studies. Lillian Gilbreth had a Ph.D. in psychology which helped in

understanding the many people issues. One of the most significant things

the Gilbrethss did was to classify the basic human motions into seventeen

types, some effective and some non-effective. They labeled the table of

classification therbligs. Effective therbligs are useful in accomplishing work

and non-effective therbligs are not. Gilbreth concluded that the time to

complete an effective therblig can be shortened but will be very hard to

eliminate. On the other hand non-effective therbligs should be completely

eliminated if possible.

1.3. “Systems Approach” in Industrial Engineering

Some basic definitions

System: A set of components which are related by some form of

interaction, and which act together to achieve some objective or

purpose.

Components: The individual parts, or elements, that collectively

make up a system.

Relationships: The cause-effect dependencies between components.

Objective or Purpose: The desired state or outcome which the

system is attempting to achieve.

An example of a system:

System: The air-conditioning system in a home.

Objective: To heat or to cool the house, depending on the need.

Components: The house (walls, ceiling, floors, furniture, etc.), the heat pump, the thermostat, the air within the system, and the electricity that drives the system.

An example of a system (continued):

Relationships:

(1) The air temperature depends on:(a) Heat transfer through the walls, ceiling, floor

and windows of the house.(b) Heat input or output due to heat pump action.

(2) The thermostat action depends on:(a) Air temperature.(b) Thermostat setting.

(3) The heat pump status depends on:(a) Thermostat action.(b) Availability of electricity.

Other examples of systems

• production system of a factory,

• information system of a business firm,

• computer system of an airlines company,

• circulatory system of the human body,

• nervous system of the human body, etc.

System classifications

• Natural vs. Man-Made Systems

Natural systems ⇒ exist as a result of processes occurring in the natural world.

e.g. a river.

Man-made systems ⇒ owe their origin to human activity.

e.g. a bridge built to cross over a river.

System classifications (continued)

• Static vs. Dynamic Systems

Static systems ⇒ have structure, but no associated activity.

e.g. a bridge crossing a river.

Dynamic systems ⇒ involve time-varying behaviour.

e.g. the Turkish economy.


• Physical vs. Abstract Systems

Physical systems ⇒ involve physically existing components.

e.g. a factory (since it involves machines, buildings, people, and so on).

Abstract systems ⇒ involve symbols representing the system components.

e.g. an architect’s drawing of a factory (consists of lines, shading, and dimensioning).


• Open vs. Closed Systems

Open systems ⇒ interact with their environment, allowing materials (matter), information, and energy to cross their boundaries.

Closed systems ⇒ operate with very little interchange with its environment.

“Systems approach” attempts to resolve the conflicts of interest

among the components of the system in a way that is best for the

system as a whole.

1.4. Definition of Operations Research

Operations Research (OR) is a scientific approach to decision making

and modeling of deterministic and probabilistic systems that originate

from real life. These applications, which occur in government,

business, engineering, economics, and the natural and social sciences,

are largely characterized by the need to allocate limited resources.

The approach attempts to find the best, or optimal solution to the

problem under consideration.

The definitions of IE and OR indicate that they have common features.

However, the primary difference is that, OR has a higher level of

theoretical and mathematical orientation, providing a major portion of

the science base of IE.

Many industrial engineers work in the area of OR, as do

mathematicians, statisticians, physicists, sociologists, and others.

OR incorporates both scientific and artistic features:

Provides mathematical techniques and algorithms ⇒⇒⇒⇒ science

Modeling and interpretation of the model results require creativity and

personal competence ⇒⇒⇒⇒ art

Some application areas of Operations Research

• Military (origin of OR - the urgent need to allocate scarce resources to the various military operations and to the activities within each operation in an effective manner during World War II)

• Aircraft and missile • Communication

• Electronics • Computer

• Food • Transportation

• Metallurgy • Financial institutions

• Mining • Health and medicine

• Paper

• Petroleum

Some of the problems that are solved using Operations Research techniques

• Linear programming

- assignment of personnel

- blending of materials

- distribution and transportation

- investment portfolios

Some of the problems that are solved using Operations Research techniques (continued)

• Dynamic programming

- planning advertising expenditures

- distributing sales effort


• Queueing theory

- traffic congestion

- air traffic scheduling

- production scheduling

- hospital operation


• Simulation

- simulation of the passage of traffic across a junction with time-sequenced traffic lights to determine the best time sequences

- simulation of the Turkish economy to predict the effect of economic policy decisions

- simulation of large-scale distribution and inventory control systems to improve the design of these systems

Some of the problems that are solved using OperationsResearch techniques (continued)

• Simulation

- simulation of the overall operation of an entire business firmto evaluate broad changes in the policies and operation of the firm, and also to provide a business game for trainingexecutives

- simulation of the operation of a developed river basin to determine the best configuration of dams, power plants, and irrigation works that would provide the desired level of flood control and water resource development

2. OPTIMIZATION

2.1. Basic Definitions

Optimization is finding the best solution of a problem by maximizing or

minimizing a specific function called the objective function, which

depends on a finite number of decision variables, whose values are

restricted to satisfy a number of constraints.

In mathematical terms, the problem becomes:

Optimize (i.e., maximize or minimize) z = f(x1, x2, …, xn) (Objective function)

subject to:g1(x1, x2, …, xn) b1g2(x1, x2, …, xn) ≥ b2 (Constraints)

. = .

. ≤ .gm(x1, x2, …, xn) bm

• The problem stated above involves “n” decision variables, and “m”

constraints.

• The objective may be to maximize a function (such as profit,

expected return, or efficiency) or to minimize a function (such as

cost, time, or distance).

• The decision variables are controlled or determined by the

decision-maker.

• Each of the “m” constraint relationships involves one of the three

signs ≥≥≥≥, =, ≤≤≤≤

• Every problem will have certain limits or constraints within which

the solution must be found. These constraints are:

- the physical laws (which indicate the way that physical

quantities behave and interact)

- the rules of society (e.g., government regulations regarding

environmental pollution, public health and safety)

- the availability of resources (e.g., limits on materials, energy,

water, money, manpower and information)

An example of an optimization problem:

A small manufacturing firm that produces one item is interested in

determining the optimal amount of the product. The objective of the

firm is to maximize the profit.

First of all, the decision variable of the problem has to be specified.

Here,

x = the number of units produced and sold

is the decision variable of the problem.

In order to determine the profit, the revenue and the total cost need to be considered.

Revenue is generated by selling the product at a particular price:

revenue = price * items sold, or

r = p x

Total cost, on the other hand, has two components:

Fixed costs (costs of being in business) - must be met even if the firm does not produce a single item (such as rent, license fees, etc.).

Variable costs (costs of doing business) – are influenced by the number of units produced (such as labor costs, raw material costs, etc.).

total cost = fixed costs + variable costs

total cost = fixed costs + (variable costs per unit) * (number of units produced and sold)

total cost = f + c x

Thus,

profit = revenue – total cost

profit = p x – f – c x

The problem formulation becomes:

maximize z = p x – f – c x

subject to:

x ≤ C (capacity limitation on the number of units produced)

x ≥ D (demand should be met)

x ≥ 0 (non-negativity constraint)

Models

Optimization Models Heuristic Models

Deterministic Stochastic

(values are known with certainty)

(values are not known with certainty)

2.2. Some Linear Programming Models

A linear programming (LP) model seeks to optimize a linear objective

function subject to a set of linear constraints.

One method to solve LP problems is the Graphical Solution

Procedure.

The procedure consists of two steps:

1. Determination of the feasible solution space.

2. Determination of the optimum solution from among all the feasible

points in the solution space.

This procedure is not convenient when more than three variables are

involved.

Example 1:

Giapetto’s Woodcarving, Inc., manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto’svariable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapetto’svariable labor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor. A train requires 1 hour of finishing labor and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw material, but only 100 finishing hours and 80 carpentry hours. Demand for trains is unlimited, but at most 40 soldiers are bought each week.

Giapetto wants to maximize weekly profit.

Formulate and solve the above problem using the Graphical Solution Procedure.

Example 2:

Hızlı Auto manufactures luxury cars and trucks. The company believes that its most likely customers are high-income women (HIW) and men (HIM). To reach these groups, Hızlı Auto has embarked on an ambitious TV advertising campaign and has decided to purchase 1-minute commercial spots on two types of programs: comedy shows and football games. Each comedy commercial is seen by 7 million HIW and 2 million HIM. Each football commercial is seen by 2 million HIW and 12 million HIM. A 1-minute comedy ad costs 50,000 TL, and a 1-minute football ad costs 100,000 TL. Hızlı Auto would like the commercials to be seen by at least 28 million HIW and 24 million HIM.

Hızlı Auto wants to meet its advertising requirements at minimum cost.


Example 3:

A company owns two different mines that produce an ore which, after being crushed, is graded into three classes: high-, medium-, and low-grade. Each grade of ore has a certain demand. The company has contracted to provide a smelting plant with 12 tons of high-grade, 8 tons of medium-grade, and 24 tons of low-grade ore per week. Operating costs are $200 per day for mine 1, and $160 per day for mine 2. The two mines have different capacities. Mine 1 produces 6, 2, and 4 tons per day of high-, medium-, and low-grade ores, respectively. Mine 2, on the other hand, produces 2, 2, and 12 tons per day of the three ores.

How many days per week should each mine be operated to satisfy the orders and minimize operating costs?


Example 4:

A pie shop that specializes in plain and fruit pies makes delicious pies and sells them at reasonable prices, so that it can sell all the pies it makes in a day. Every dozen plain pies nets a 1.5 TL profit, and requires 12 kg. of flour, 50 eggs, and 5 kg. of sugar (and no fruit mixture). Every dozen fruit pies nets a 2.5 TL profit, and uses 10 kg. of flour, 40 eggs, 10 kg. of sugar, and 15 kg. of fruit mixture.

On a given day, the bakers at the pie shop found that they had 150 kg. of flour, 500 eggs, 90 kg. of sugar, and 120 kg. of fruit mixture with which to make pies.

Find the optimal production schedule of pies for the day.


Example 5:

A company produces two products: Model A and Model B. A single unit of Model A requires 2.4 minutes of punch press time and 5 minutes of assembly time, and yields a profit of 8 TL per unit. A single unit of Model B requires 3 minutes of punch press time and 2.5 minutes of welding time, and yields a profit of 7 TL per unit.

If the punch press department has 1200 minutes available per week, the welding department 600 minutes, and the assembly department 1500 minutes per week, what is the product mix (quantity of each to be produced) that maximizes profit?


Example 6:

The Village Butcher Shop traditionally makes its meat loaf from a combination of lean ground beef and ground lamb. The ground beefcontains 80 percent meat and 20 percent fat, and costs the shop 8 TL per kilogram; the ground lamb contains 68 percent meat and 32 percent fat, and costs 6 TL per kilogram.

How much of each kind of meat should the shop use in each kilogram of meat loaf if it wants to minimize its cost and to keep the fat content of the meet loaf to no more than 25 percent?


Example 7:

A furniture maker has 6 units of wood and 28 hours of free time, in which he will make decorative screens. Two models have sold well in the past, so he will restrict himself to those two. He estimates that model I requires 2 units of wood and 7 hours of time, while model II requires 1 unit of wood and 8 hours of time. The prices of the models are 120 TL and 80 TL, respectively.

How many screens of each model should the furniture maker assemble if he wishes to maximize his sales revenue?


Example 8:

Four factories are engaged in the production of four types of toys. The following table lists the toys that can be produced by each factory. The unit profits of toys 1, 2, 3, and 4 are; 50 TL, 40 TL, 55 TL, and 25TL, respectively.

All toys require approximately the same per-unit labor and material. The daily capacities of the four factories are 250, 180, 300, and 100 toys, respectively. The daily demands for the four toys are 200, 150, 350, and 100 units, respectively.

Formulate the above problem. Can you solve it using the Graphical Solution Procedure?

3,4D

1,4C

2,3B

1,2,3A

Toy productions mixFactory

Example 9: A company makes three products and has available four workstations. The production time (in minutes) per unit produced varies from workstation to workstation (due to different manning levels) as shown below:

Similarly, the profit (£) contribution per unit varies from workstation to workstation as below:

If one week, there are 35 working hours available at each workstation, how much of each product should be produced given that we need at least 100 units of product 1, 150 units of product 2, and 100 units of product 3? Formulate this problem as an LP.

17914133

1581262

104751

4321Product

Workstation

171316153

171520182

968101

4321Product

Workstation

3. FACILITIES LOCATION AND LAYOUT

Facility: Something (plant, office, warehouse, etc.) built or established

to serve a purpose.

Facilities management: A location decision for that facility, and the

composition or internal layout of the facility once located

(⇒ facility location + facility layout).

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3.1. Facilities Location

Facilities location is the determination of which of several possible

locations should be operated in order to maximize or minimize some

objective function, such as profit, cost, distance or time.

Examples:

• locate a new warehouse relative to production facilities and customers

• locate an emergency service (police station, fire station, blood bank, etc.)

• locate branch offices for banks

• locate supply centers for construction projects

Figure 2

Figure 1

Steps in a facility location decision:

1. Define the location objectives and associated variables.

2. Identify the relevant decision criteria.

Quantitative - economic

Qualitative - less tangible

3. Relate the objectives to the criteria in the form of a model, or

models (such as break-even, linear programming, qualitative factor

analysis, point rating).

4. Generate necessary data and use the models to evaluate the

alternative locations.

5. Select the location that best satisfies the criteria.

Example 1: Locating a new plant using point rating

Steps of the method:

1. Identify the factors.

2. Assign a point rating to each factor (this is the maximum point that

can be achieved by an ideal location).

3. Evaluate each candidate according to these factors.

4. Select the candidate with the highest score as the location of the

new facility.

* 2165 *18002850Total

225100275Laws and taxation

275200350Labor and wages

400275400Transportation flexibility

100150250Community services andattitude

100125175Housing

90100150Climate

325400500Availability of raw materials

400300450Availability of power

250150300Nearness to market

Candidate B

Candidate A

Maximum PointFactor

Example 2: Locating a new airport using a weighted

method

• Ten critical factors are identified for this problem.

• The weight of each factor is selected from a range of [0,1],

where; 0 is the lowest weight, and 1 is the highest.

• The point that will be assigned to each candidate airport

location is selected from a range of [0,10], where; 0 is the

lowest point, and 10 is the highest.

48.9544.7552.95* 54.65 *Total

0.3512.1062.8083.50100.35Proximity to strategic regions

1.9533.2556.50105.8590.65Suitability to the natural environment

9.50104.7558.5594.7550.95Cost of land

4.0591.3534.0593.1570.45Height of the buildings in the surrounding

5.7068.5592.8535.7060.95Passenger potential

5.9572.5535.9575.1060.85Suitability of weather conditions

2.2536.0082.2536.0080.75Proximity to transportation facilities

4.8083.6064.8084.2070.60Distance to settlement centers

6.4085.6077.2096.4080.80Soil conditions

8.0087.0078.00810.00101.00Total area (m2)

ResultPointResultPointResultPointResult (1x2)Point (2)Weight (1)Factor

Candidate DCandidate CCandidate BCandidate A

3.2. Facilities Layout

Facilities layout is the joint determination of the locations, sizes and

configurations of multiple activities within a facility.

Examples:

• layout of the manufacturing cells, workstations, etc. within a plant

• layout of the various departments within an office or building

Figure 3

Figure 4

Steps in a layout design process:

1. Formulating the layout design problem.

2. Analyzing the design problem.

3. Searching for alternative layout designs.

4. Evaluating the layout design alternatives.

5. Selecting the preferred design.

6. Specifying the layout design to be installed.

Some of the objectives of the plant layout process

• Minimize investment in equipment.

• Minimize overall production time.

• Utilize existing space most effectively.

• Provide for employee convenience, safety, and comfort.

• Maintain flexibility of arrangement and operation.

• Minimize material handling cost.

• Minimize variation in types of material handling equipment.

• Facilitate the manufacturing process.

• Facilitate the organizational structure.

Types of layout

There exist four general layout categories:

1. Fixed layout / static product layout

2. Product layout / production-line layout

3. Process layout / functional layout

4. Group layout / group technology layout / cellular layout

1. Fixed layout / static product layout

• It is used when the product is too large or cumbersome (massive)

to move through the various processing steps.

• Rather than taking the product to the processes, the processes are

brought to the product.

1. Fixed layout / static product layout (continued)

• Some examples: shipbuilding industry, aircraft industry,

construction industry (building a house, dam, bridge, etc.).

2. Product layout / production-line layout

• It results when processes are located according to the processing

sequence for the product. Material flows directly from a

workstation to the adjacent workstation.

• Product layouts are employed when one or a few standardized

products with high-volume are produced.

2. Product layout / production-line layout (continued)

• Some examples: a car washing line, the final assembly line in the

automotive industry.

3. Process layout / functional layout

• In a process layout, all machines involved in performing a

particular process are grouped together. Hence, it consists of a

collection of processing departments or cells.

• Process layouts are used when there exist many low-volume,

dissimilar products.

• Process layout is characterized by high degrees of

interdepartmental flow.

3. Process layout / functional layout (continued)

• Some examples: auto repair workshops, the different clinics (x-ray,

cardiology, surgery, neurology, etc.) in a hospital.

4. Group layout / group technology layout / cellular layout

• It is used when production volumes for individual products are not

sufficient to justify product layouts. But by grouping products into

logical product families, a product layout can be justified for the

family.

• The group layout typically has a high degree of intradepartmental

flow; it is a compromise (middle term) between the product layout

and the process layout.

• It possesses both the efficiency of the product layout and the

flexibility of the process layout.

4. Group layout / group technology layout / cellular layout(continued)

Equipment statistics related with the figures:

101116Total req’s

122Paint

112Grind

111Weld

224Drill

223Mill

222Lathe

112Saw

Process

layout

Group

layout

Product

layoutEquipment

4. FORECASTING SYSTEMS

4.1. Introduction

Forecasting is the process of analyzing the past data of a time-

dependent variable and predicting its future values by the help of a

qualitative or quantitative method.

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10

Month

Nu

mb

er

of

pro

du

cts

so

ld ?

Some forecasting examples

• Manufacturing firms forecast demand for their products in order to

have the necessary manpower and raw materials to support

production.

• Companies specializing in service operations forecast customer

arrival patterns in an effort to maintain adequate staffing to serve

customer needs.

• Security analysts forecast company revenues, profits, and debt

ratios, as well as general trends in financial markets, in order to

make investment recommendations.

Why is forecasting important?

Proper forecasting ⇒ better use of capacity,

⇒ reduced inventory costs,

⇒ lower overall personnel costs,

⇒ increased customer satisfaction.

Poor forecasting ⇒ decreased profitability,

⇒ collapse of the firm.

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4.2. Forecasting Methods

Forecasts should be sufficiently accurate and flexible to plan for future

activities, and this subject highly depends on the forecasting technique

that will be employed.

In selecting the appropriate forecasting method, the period (span) of

the forecasting decisions plays an important role.

Long-range forecasts require techniques with long-term horizons

(� 1-5 years).

Used for:

- facility location,

- capacity expansion,

- technology selection,

- new product decisions,...

Medium-range forecasts necessitate techniques having medium-term

horizons (� 3 months-1 year).

Used for:

- production and inventory control,

- labor level decisions,

- allocation of financial resources,...

Short-range forecasts can rely more on recent history (� 1-2 weeks).

Used for:

- scheduling,

- inventory replenishment,

- lot-sizing decisions,...

Table 1 summarizes some of the most commonly used forecasting

methods together with their effective time horizons and relative

application costs.

Table 1

An example of selecting the proper forecasting techniques during the

life cycle of a product is given in Figure 5.

Figure 5

4.3. Time Series

Time series is a set of observations of a variable over time (in other

words, a past history of data values). Often, it is available, and can

be helpful in developing the forecast.

0

5

10

15

20

Ice

cre

am

sa

les

(no

of

bo

xe

s)

Month

Series1 5 7 6 8 10 15 17 20 11 8

1 2 3 4 5 6 7 8 9 10

Components of a time series

A time series is comprised of one or more of the following four

components:

1. trend (a continuous long-term directional movement, indicating

growth or decline, in the data).

0

1000

2000

3000

4000

5000

6000

7000

0 1 2 3 4 5 6 7 8 9 10

Figure 6a – Time Series with Linear Trend

0

200

400

600

800

1000

1200

0 2 4 6 8 10 12 14

Figure 6b – Time Series with Linear Nonlinear Trend

Components of a time series (continued)

2. seasonal variation (a decrease or increase in the data during

certain time intervals, due to calendar or climatic changes. May contain

yearly, monthly or weekly cycles).

Figure 6c – Time Series with Trend and Seasonality

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14 16 18 20

Components of a time series (continued)

3. cyclical variation (a temporary upturn or downturn that seems to

follow no observable pattern. Usually results from changes in economic

conditions such as inflation, stagnation).

4. random effects (occasional and unpredictable effects due to

chance and unusual occurrences. They are the residual after the trend,

seasonal, and cyclical variations are removed).

Steps in the time series forecasting process

1. Collect historic data, graph the data versus time to aid in

hypothesizing a form for the time series model, and verify this

hypothesis statistically.

2. Select an appropriate forecasting technique for the time series

model and determine the values of its parameters.

3. Prepare a forecast using the selected forecasting technique.

4. Validate the model by calculating the forecast errors.

4.4. Regression Methods

Consider the following simple linear model:

ttba

tx ε+⋅+=

dependentvariable

independentvariable

where:

ba ,

tε

: unknown parameters

: random error component

This model has the following assumptions:

0)( =tE ε

2)( εσε =tV

jiforCov ji ≠= ,0),( εε

),0(~ 2

εσε Nt

⇒ the errors are uncorrelated random variables

Now, let us assume that there are T periods of data available

(x1,...,xT). The unknown parameters a & b will be estimated such

that, the sum of squares of the residuals is minimized.

The estimated values of the parameters are shown as .)ˆ&ˆ( ba

ttba

tx ε+⋅+= ˆˆˆ

∑=

=T

t

SSE

1

) tperiod of residual( 2

∑=

⋅−−∑=

=−=T

t

tbat

xT

tt

xt

xSSE

1

)ˆˆ(

1

)ˆ( 22

0)ˆˆ(2ˆ 1

=⋅−−−=∂

∂∑

=

T

t

t tbaxa

SSE

0)ˆˆ(2ˆ

1

=⋅⋅−−−=∂

∂∑

=

T

t

t ttbaxb

SSE

As a result, the least-squares normal equations are obtained

as follows:

∑∑∑===

=+T

t

t

T

t

T

t

xtba111

ˆ)1(ˆ

∑∑∑===

⋅=+T

t

t

T

t

T

t

xttbta11

2

1

ˆˆ

Recall the following closed forms:

6

)12)(1(&

2

)1(

1

2

1

++=

+= ∑∑

==

TTTt

TTt

T

t

T

t

Then the least-squares normal equations become:

)(ˆ)1(

6

)1(

)12(2ˆ

11

TaxtTT

xTT

Ta

T

t

t

T

t

t ≡⋅−

−−

+= ∑∑

==

)(ˆ)1(

6

)1(

12ˆ

112

TbxTT

xtTT

bT

t

t

T

t

t ≡−

−⋅−

= ∑∑==

Hence, the forecast equation can be written as:

[ ]ττ ++=+ TTbTaxT )(ˆ)(ˆˆ

Example 1:

The following table displays the weekly sales of a car. Estimate

the sales for weeks 6 and 10 using the linear trend model.

205

184

153

122

101

Number of cars soldWeek

y = 2,6x + 7,2

5

10

15

20

25

0 1 2 3 4 5 6

Week

Nu

mb

er

of

ca

rs s

old

The answer of Example 1 using Microsoft Excel:

Coefficient of determination (r2)

How much of the total deviation in xt (dependent variable) is

explained by t or the trend line?

⇒ calculate the coefficient of determination!

∑∑

−

−==

2

2

2

)(

)ˆ(

total

explained

tt

tt

xx

xxrCoefficient of determination

10 2 ≤≤r

Correlation coefficient (r)

Displays the relative importance of the relationship between xt

and t.

Sign of r ⇒ direction of the relationship

r ⇒ strength of the relationship

Correlation coefficient

( ) ( )∑∑∑∑∑ ∑∑

−⋅−

⋅−⋅==

2222

2

)()( tt

tt

xxTttT

xtxtTrr

11 ≤≤− r

Interpretation of the correlation coefficient

(a) Perfect positive correlation:

(b) Positive correlation:

(c) No correlation:

(d) Perfect negative correlation:

(e) Negative correlation:

Example 2:

It is assumed that the monthly refrigerator sales in a city is

directly proportional to the number of newly married couples in

that month. The data is given below.

a) Can the closed form equations of and be used in

estimating the future values?

b) Determine and .

c) Determine and interpret r and r2.

)(ˆ Ta

)(ˆ Ta

)(ˆ Tb

)(ˆ Tb

49511612

45410411

4528810

449939

5171248

5401507

5381606

4811205

4721144

450963

4731102

4611001

Refrigerator sales (x 103 TL)

Number of newly married couples

Month


y = 1,4515x + 315,52

R2 = 0,907

440

460

480

500

520

540

85 95 105 115 125 135 145 155 165

Number of newly married couples

Re

frig

era

tor

sa

les

Example 3:

A magazine has conducted a survey on the number of patients

who have died from lung cancer, together with the tobacco

production in U.S.A. The result of this survey is presented

below.

Using regression analysis:

a) Determine and .

b) Determine and interpret r and r2.

)(ˆ Ta )(ˆ Tb

122824.7

129326.1

135427.5

139028.6

140628.7

138728.4

132627.4

131926.7

The value of tobaccoproduction in U.S.A

(x 1012 $)

Patients who havedied from lung cancer

(x 103)


Documents

Lecture note on introduction to industrial engineering