İSTANBUL KÜLTÜR UNIVERSITY FACULTY OF ENGINEERING Department of Industrial Engineering

İSTANBUL KÜLTÜR UNIVERSITYFACULTY OF ENGINEERING

Department of Industrial Engineering

IE2401 Introduction to Industrial Engineering

Prof. Tülin AKTİN

Spring 2012

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 ENGINEERIN

G

“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 Gilbreths 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


• Simulation

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

- 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) b1

g2(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’s variable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapetto’s variable 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 beef contains 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 25 TL, 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?

Factory

Toy productions mix

A 1,2,3

B 2,3

C 1,4

D 3,4

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.

Workstation

Product

1 2 3 4

1 5 7 4 10

2 612

8 15

313

14

9 17Workstation

Product

1 2 3 4

110

8 6 9

218

2015

17

315

1613

17

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).

???

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.

FactorMaximum

PointCandidate

ACandidate

BNearness to market 300 150 250

Availability of power 450 300 400

Availability of raw materials

500 400 325

Climate 150 100 90

Housing 175 125 100

Community services and attitude

250 150 100

Transportation flexibility 400 275 400

Labor and wages 350 200 275

Laws and taxation 275 100 225

Total 2850 1800 * 2165 *

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.

Candidate A Candidate B Candidate C Candidate D

FactorWeight

(1)Point (2)

Result (1x2)

Point

Result

Point

Result

Point

Result

Total area (m2)

1.00 10 10.00 8 8.00 7 7.00 8 8.00

Soil conditions

0.80 8 6.40 9 7.20 7 5.60 8 6.40

Distance to settlement centers

0.60 7 4.20 8 4.80 6 3.60 8 4.80

Proximity to transportation facilities

0.75 8 6.00 3 2.25 8 6.00 3 2.25

Suitability of weather conditions

0.85 6 5.10 7 5.95 3 2.55 7 5.95

Passenger potential

0.95 6 5.70 3 2.85 9 8.55 6 5.70

Height of the buildings in the surrounding

0.45 7 3.15 9 4.05 3 1.35 9 4.05

Cost of land 0.95 5 4.75 9 8.55 5 4.75 10 9.50

Suitability to the natural environment

0.65 9 5.85 10 6.50 5 3.25 3 1.95

Proximity to strategic regions

0.35 10 3.50 8 2.80 6 2.10 1 0.35

Total * 54.65 * 52.95 44.75 48.95

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:

EquipmentProductlayout

Grouplayout

Processlayout

Saw 2 1 1

Lathe 2 2 2

Mill 3 2 2

Drill 4 2 2

Weld 1 1 1

Grind 2 1 1

Paint 2 2 1

Total req’s 16 11 10

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.

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

crea

m s

ales

(n

o o

f b

oxe

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:

ttbatx

dependentvariable

independentvariable

where:

ba ,

t: unknown parameters

: random error component

This model has the following assumptions:

0)( tE 2)( tV

jiforCov ji ,0),(

),0(~ 2 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

ttbatx ˆˆˆ

T

tSSE

1) tperiod of residual( 2

T

ttba

tx

T

ttx

txSSE

1)ˆˆ(

1)ˆ( 22

0)ˆˆ(2ˆ 1

T

tt tbax

a

SSE

0)ˆˆ(2ˆ

1

T

tt ttbax

b

SSE

As a result, the least-squares normal equations are

obtained as follows:

T

tt

T

t

T

t

xtba111

ˆ)1(ˆ

T

tt

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

tt

T

tt

)(ˆ)1(

6

)1(

12ˆ11

2Tbx

TTxt

TTb

T

tt

T

tt

Hence, the forecast equation can be written as:

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.

Week

Number of cars sold

1 10

2 12

3 15

4 18

5 20

y = 2,6x + 7,2

5

10

15

20

25

0 1 2 3 4 5 6

Week

Nu

mb

er o

f c

ars

so

ld

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

22

)(

)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

Month

Number of newly married couples

Refrigerator sales

(x 103 TL)

1 100 461

2 110 473

3 96 450

4 114 472

5 120 481

6 160 538

7 150 540

8 124 517

9 93 449

10 88 452

11 104 454

12 116 495


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

Ref

riger

ator

sal

es

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

Patients who havedied from lung cancer

(x 103)

The value of tobaccoproduction in U.S.A

(x 1012 $)

26.7 1319

27.4 1326

28.4 1387

28.7 1406

28.6 1390

27.5 1354

26.1 1293

24.7 1228

İSTANBUL KÜLTÜR UNIVERSITY FACULTY OF ENGINEERING Department of Industrial Engineering

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Transcript of İSTANBUL KÜLTÜR UNIVERSITY FACULTY OF ENGINEERING Department of Industrial Engineering