ĐSTANBUL KÜLTÜR UNIVERSITY FACULTY 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 an d
inst instal alla lati tion on
of
int integ egra rate ted d
syst system emss
of
peo peopl ple, e,
mat mater eria ials ls,,
information, equipment and energy. It draws upon specialized knowledge and skill in the mathematical, physical and social sciences together with the principles and methods methods of engineering engineering analysis and design to specify, specify, predict predict and evaluate evaluate the the results to be obtained from such systems.
1. INTRODUCTION TO BASIC CONCEPTS 1.1. Definition of Industrial Engineering Industrial Engineering (IE) is concerned with the design, improvement
and an d
inst instal alla lati tion on
of
int integ egra rate ted d
syst system emss
of
peo peopl ple, e,
mat mater eria ials ls,,
information, equipment and energy. It draws upon specialized knowledge and skill in the mathematical, physical and social sciences together with the principles and methods methods of engineering engineering analysis and design to specify, specify, predict predict and evaluate evaluate the 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.
System classifications (continued) • 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).
System classifications (continued) • 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
Some of the problems that are solved using Operations Research techniques (continued) • Queueing theory - traffic congestion
- air traffic scheduling
- production scheduling
- hospital operation
Some of the problems that are solved using Operations Research techniques (continued) • 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 Operations Research Resear ch tech techniqu niques es (con (continue tinued) d) • Simulation - simu simula lati tion on of the the ove overa rallll oper operat atio ion n of of an an ent entir ire e busi busine ness ss firm firm to evaluate evaluate broad broad change changess in the policies policies and operatio operation n of the firm, firm, and and also also to pro provid vide e a busin busines esss game game for train trainin ing g executives
- simulatio simulation n of the the operat operation ion of of a develo developed ped river river basin basin to determ determine ine the best best con config figur urati ation on of dams dams,, power power plant plants, s, and irrigatio irrigation n works works that would would provid provide e the the desired desired level level of flood control control and water water resource resource develo developme pment nt
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, function, which depends on a finite number of decision variables, variables, whose values are restricted to satisfy a number of constraints of constraints.. In mathematical terms, the problem becomes: Optimize (i.e., maximize or minimize) z = f(x 1 , x 2 , …, x n 1 2 n )
(Objective function) function)
subject to: g 1 , x 2 , …, x n 1(x 1 1 2 n ) g 2 , x 2 , …, x n 2(x 1 1 2 n ) . . g m , x 2 , …, x n m (x 1 1 2 n )
≥
= ≤
b 1 1 b 2 2 . . b m m
(Constraints Constraints))
•
The The proble blem sta stated ted above invo involv lves es “n ” decisi decision on vari variab ables les,, and 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 1minute 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 1minute 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. Formulate and solve the above problem using the Graphical Solution Procedure.
Example 3: A company owns two different mines that produce an ore which, after being crushed, is graded into three classes: high-, medium-, and lowgrade. 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? Formulate and solve the above problem using the Graphical Solution Procedure.
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. Formulate and solve the above problem using the Graphical Solution Procedure.
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? Formulate and solve the above problem using the Graphical Solution Procedure.
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? Formulate and solve the above problem using the Graphical Solution Procedure.
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? Formulate and solve the above problem using the Graphical Solution Procedure.
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.
Factory Toy productions mix A
1,2,3
B
2,3
C
1,4
D
3,4
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?
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:
Workstation 1 2 3 4
Product 5 7 1 6 12 2 13 14 3
4
10
8
15
9
17
Similarly, the profit (£) contribution per unit varies from workstation to workstation as below: Workstation
Product 1 2 3 4 10 8 6 9 1 18 20 15 17 2 15 16 13 17 3 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.
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 Figure 1
• locate an emergency service (police station, fire station, blood bank, etc.) Figure 2
• locate branch offices for banks • locate supply centers for construction projects
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.
Factor
Candidate Candidate Maximum Point A B
Nearness 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
2850
1800
* 2165 *
Total
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 Factor
Candidate B
Candidate C
Candidate D
Weight (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 Figure 3
• layout of the various departments within an office or building 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
construction industry (building a house, dam, bridge, etc.).
industry,
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
interdepartmental flow.
characterized
by
high
degrees
of
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:
Equipment
Product layout
Group layout
Process layout
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. 140
?
d l 120 o s s 100 t c u d 80 o r p f 60 o r e b 40 m u 20 N 0 1
2
3
4
5
6
Month
7
8
9
10
Some forecasting examples •
Manufacturing firms forecast demand for their products in order to have the necessary manpower and raw materials to support production.
•
Comp Compan anie iess spe speci cial aliz izin ing g in in ser servi vice ce oper operat atio ions ns fore foreca cast st customer arrival patterns in an effort to maintain adequate staffing to serve customer needs.
•
Security analysts ffo orecast company revenues, revenues, profits profits,, and debt ratios,, as well as general trends in financial markets, ratios 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.
20 s ) e l s 15 a e s x o b 10 m a f e r o c o 5 e n ( c I 0
Month Series1
1
2
3
4
5
6
7
8
9
10
5
7
6
8
10
15
17
20
11
8
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). 7000
1200
6000
1000
5000
800
4000
600 3000
400
2000
200
1000
0
0 0
1
2
3
4
5
6
7
8
9
10
Figure 6a – Time Series with Linear Trend
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).
3000
2500 2000
1500
1000 500
0 0
2
4
6
8
10
12
14
16
18
20
Figure 6c – Time Series with Trend and Seasonality
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:
x = a + b ⋅ t + ε t t dependent variable
independent variable
where:
a,b
: unknown parameters
ε t
: random error component
This model has the following assumptions:
E ( ε t ) = 0
V (ε t ) =
2 σ ε
ε t
~ N (0 ,
2 σ ε
Cov(ε i , ε j ) = 0 , for i ≠ j ⇒
the errors are uncorrelated random variables
)
Now, let us assume that there are T periods of data available (x 1 ,...,x The unknown parameters a & b will be estimated such T). that, the sum of squares of the residuals is minimized. The estimated values of the parameters are shown as ( aˆ & bˆ) .
xˆ = aˆ + bˆ ⋅ t + ε t t T
SSE = ∑ ( residual of period t ) t = 1
2
T
T
SSE = ∑ ( x − xˆ ) = ∑ ( x − aˆ − bˆ ⋅ t ) 2 t t t t = 1 t = 1
∂SSE ∂aˆ
∂ SSE ∂ bˆ
2
T
= −2
∑
( xt − aˆ − bˆ ⋅ t ) = 0
t =1
T
= −2
∑ ( x t =1
t
− aˆ − bˆ ⋅ t ) ⋅ t = 0
As a result, the least-squares normal equations are obtained as follows:
T
aˆ
∑
(1) + bˆ
aˆ
∑ t =1
t =1
t + bˆ
T
∑ t = ∑ x
t =1
T
T
T
t
t =1
T
∑ t = ∑ t ⋅ x 2
t =1
t =1
t
Recall the following closed forms: T
∑ t = t =1
T
T (T + 1)
∑ t
2
&
2
=
T (T + 1)(2T + 1)
6
t =1
Then the least-squares normal equations become:
aˆ =
2(2T + 1)
T
x ∑ T (T − 1)
t
t =1
bˆ =
12 2
T (T
−
t =1
T
t ⋅ x ∑ T (T − 1)
T
∑ − 1)
6
t ⋅ xt −
t
≡ aˆ (T )
t =1
6
T
∑ T (T − 1) t =1
xt ≡ bˆ(T )
Hence, the forecast equation can be written as:
xˆT + = aˆ (T ) + bˆ(T )[T + τ ] τ
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
The answer of Example 1 using Microsoft Excel:
25 d l o 20 s s r a c f 15 o r e b m10 u N
y = 2,6x + 7,2
5 0
1
2
3
Week
4
5
6
Coefficient of determination (r 2 ) How much of the total deviation in x t (dependent variable) is explained by t or the trend line?
⇒ calculate the coefficient of determination!
Coefficient of determination
2
r =
explained
0 ≤ r 2 ≤1
total
∑ = ∑ ( x − x ) ( xˆt − xt ) t
t
2 2
Correlation coefficient (r ) Displays the relative importance of the relationship between x t and t . Sign of r ⇒ direction of the relationship
r ⇒ strength of the relationship
Correlation coefficient
2
r = r =
−1≤
r
≤1
∑ t ⋅ x − ∑ t ⋅ ∑ x (T ∑ t − (∑ t ) )⋅ (T ∑ x − (∑ x ) ) T
2
t
2
t
2 t
2
t
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 aˆ (T ) and bˆ(T ) be used in estimating the future values? b) Determine aˆ (T ) and bˆ(T ) . c) Determine and interpret r and r 2 .
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
The answer of Example 2 using Microsoft Excel:
540 s e 520 l a s r o500 t a r e g480 i r f e R460
y = 1,4515x + 315,52 R2 = 0,907
440 85
95
105
115
125
135
145
Number of new ly married couples
155
165
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 aˆ (T ) and bˆ(T ) . b) Determine and interpret r and r 2 .
Patients who have died from lung cancer (x 103)
The value of tobacco production 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
