BODIN & GASS - Exercises for Teaching the Analytic Hierarchy Process.pdf

8/10/2019 BODIN & GASS - Exercises for Teaching the Analytic Hierarchy Process.pdf

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Exercises for Teaching the Analytic HierarchyProcess

Lawrence Bodin and Saul I. GassRobert H. Smith School of Business

University of Maryland

College Park, MD 20742

[email protected] [email protected]

Abstract

In a related paper (Bodin and Gass, 2003), we described the basic concepts

that webelieve must becovered when teaching the Analytic Hierarchy Process(AHP) to MBA students and outlined six exercises that can be used as in-classexamples or homework problems. In this paper, we present the details of these

exercises and an example of an AHP analysis.

1. INTRODUCTION

When teaching the AHP to MBA students, the key points that should be coveredare: (a) the AHP fundamental pairwise comparison scale, (b) inconsistency and

sensitivity analysis, (c) ratio scales, (d) the ratings model, (e) the teamapproach for solving an AHP problem, (f) AHP and resource allocation, and (g) ifclass time is available, the notion of rank reversal although rank reversal is not

essential to a basic understanding of the AHP (Bodin and Gass, 2003). Details ofthe AHP are given in Saaty, 1980and Saaty, 1994.

Since the AHP has the proven ability to resolve (or assist in resolving) a wideclass of important decision problems, we believe that AHP must be part of the

common-knowledge base of an MBA. When faced with a multi-criteria decisionanalysis problem, an MBA graduate must have the background and experienceto ask the right questions of their staff and/or fellow workers and understand

how the AHP can be used to resolve multi-criteria decision analysis problems.The AHP is a decision-aid that can provide the decision maker (DM) withrelevant information to assist the DM in choosing the "best" alternative or to

rank a set of alternatives.

In the quantitative MBA class (Decision Analysis and Models) taught at theUniversity of Maryland, the AHP module was covered in about 2-2.5 weeks. Inthis module, we used the software package, Expert Choice. The trial version of

Expert Choice can be downloaded for free from the website . Other softwarepackages that contain an implementation of the AHP are HIPRE and Criterium.

We have not used HIPRE and Criterium and, hence, cannot comment on them.As an aid to the reader, the appendix describes the introductory operations

research and quantitative methods textbooks that discuss the AHP.

Given the ease of use of the Expert Choice software, we see no pedagogical

advantage in implementing the AHP in a spreadsheet program such as Excel byitself for carrying out the AHP analysis and computations. It must be noted,

Volume 4, Number 2, January 2004
mailto:[email protected]:[email protected]://www.expertchoice.com/http://www.expertchoice.com/http://-/?-http://-/?-http://-/?-http://-/?-mailto:[email protected]:[email protected]


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however, that the ratings version of the AHP in Expert Choice forms a tablecalled a ratings spreadsheet (called 'spreadsheet' in section 3 of this paper) fordetermining the weight for each alternative. The weight of the alternative in

this spreadsheet is a measure of how close that alternative is to the perfectalternative (weight = 1).

In Section 2 of this paper, six varied exercises that we found useful in theclassroom for conveying the essentials of an AHP analysis and the features of

the Expert Choice software are presented. As in Bodin and Gass, 2003, theseexercises are outlined as follow:

EX1 contains a simple direct comparison model for the purchase of a newautomobile. Variants of this example have appeared in numerouspublications including Saaty, 1990. The criteria and the alternatives arespecified.

EX2 and EX3 are problems involving the integration of the ratings modelversion of the AHP with a resource allocation problem.EX4 contains an analysis of alternative income tax structures. The criteria

to be used are not explicitly specified. The student must determine a setof criteria and alternative tax strategies (over and above the taxstrategies specified in the example). This problem works well for teams of

three to five students.EX5 is a problem of determining the best long distance telephone service.The student or team must collect data on each of these services

(generally from the Internet), determine a set of criteria, and develop aset of alternatives for the associated ratings model.

EX6 contains the analysis of the relative size of five geometric figures.EX6 is designed to validate the use of the 1-9 pairwise comparison scale.This validation example should be presented soon after AHP fundamentals

and examples of the AHP are discussed. The problem is due to Saaty,1994.

The availability of additional AHP examples that have appeared in the literatureor on the Internet are described in Bodin and Gass, 2003. As noted in Bodin and

Gass, 2003:

Our experience has shown that the AHP is a winning topic for MBA

students. The MBA students like the AHP, they easily learn how touse the AHP and, in many cases, they get very enthusiastic aboutthe AHP. We often have to "rein-in" the students because they getso excited about the material. AHP should be a required topic for any

introductory MBA course in decision making.

2. EXERCISES IN USING THE AHP

In this section, six exercises (called EX1-EX6) that can be used in class

problems or as homework problems on the AHP are presented.

2.1 EX1: Choosing the Best Automobile

This basic example illustrates the key aspects of the AHP and its implementat ion

by the Expert Choice software. The hierarchy is easy to build and the instructorcan demonstrate the replication command that simplifies the building of thehierarchy. The overall goal of the example is to choose the best automobile with

respect to the four criteria. Figure 2.1 gives the data for the problem. Thestudent can readily see that there is no one best alternative, as none of theautomobiles is best across all criteria (as indicated by the asterisks).

Alternatives Price Miles/Gallon (MPG) Prestige Comfort

Avalon $15,000* 26 Low Good

Babylon $18,000 28* Fair Fair

Carryon $24,000 20 High* High*

Figure 2.1 Data for Automobile Purchase Example (* denotes best alternative)

The problem has both quantitative and qualitative data. The price data can beused directly in the EC comparison matrix by the data entry mode, but the dataentry has to be inverted (invert button) in that a low price is better than a

higher price (EC considers a higher number as being better than a lower number
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unless told otherwise). Note that the prices are of the same order of magnitude- we are not comparing a cheap Ford Falcon to a Jaguar. Comparing items of

the same "Order of Magnitude" is an axiom of the AHP. The price data can alsobe used indirectly by asking the usual pairwise comparison question, e.g., "IsAvalon preferred to Babylon with respect to price and how more is it preferred?"

Here the preference needs to be established using the 1-9 scale (or equivalentverbal scale) and the student has to decide how the $15,000 compares to the

$18,000. Using the 1-9 scale for the dollar figures tends to build a utilityevaluation on the dollars - the dollar spent for the cheaper auto has a greaterutility than a dollar spent on a more expensive auto. The data entry mode

treats all dollars as having the same utility. We suggest that the facultymember first illustrate the data entry mode and then illustrating the 1-9 pairwisecomparison mode. The final rankings will probably stay the same but the weights

assigned to the different elements will probably be different.

The MPG numbers are direct data entry; the weights obtained are just theindividual auto's MPG number divided by the sum of all the MPG numbers. For

prestige and comfort, the student must make pairwise comparisons that respectthe individual criterion transitivity relationship (High>Good>Fair>Low). The 1-9

scale does a very good job in capturing the preferences (e.g., High/Low = 7,High/Good = 5, High/Fair = 3, and so on).

2.2 EX2: Ratings/Resource Allocation-Case 1

EX2 and EX3 are take-home examples to be done by each student or a smallteam of students. They illustrate the use of the AHP ratings model to determineweights for competing projects, with the weights then used in a 0-1

optimization problem to select a subset of the projects subject to a budgetconstraint.

BMGT Industries has an internal Advanced Technology Project Committee (ATP)

responsible for selecting new projects for funding. The selection is made fromthose projects suggested by its division managers. The selection cycle is nowupon us. The ATP Committee feels that the time is right for it to restructure and

redirect its various R&D projects. BMGT wants to ensure that its divisions do notcontinue the status quo. It has instructed its division managers (Research andDevelopment, Manufacturing, Marketing, Logistics, Finance, Human Resources)

to come up with a set of new projects that addresses the future of eachdivision and BMGT.

The R&D and Manufacturing managers have joined forces and have agreed on

eleven new robotic manufacturing projects to go along with the other newproducts the R&D group expects to develop over the next two years. The staffshave determined the two year R&D costs and initial production costs for each

robotic project. Further, with help from the Marketing Division, the staffs havealso estimated the return, represented by net present value (NPV), of eachrobotic product, assuming that the product comes to market in the next five

years. Although the ATP Committee is impressed by the excellence of theeleven projects and would like to fund them all, there is not enough money todo so.

Faced with this problem, the ATP Committee has asked BMGT's new MBAemployee to devise a way to select a subset of the competing projects toundertake and fund. Each student (or team) assumes the role of the new hire.The student must sell the AHP methodology to the Committee and to the R&D

and Manufacturing managers.

The eleven competing robotic R&D/Manufacturing projects are given codenames P1 to P11. Each project is associated with a single new product that

could be developed by R&D, with a prototype to be built by Manufacturing. Thefollowing is known for each project:

a. The projected two year R&D and initial manufacturing cost.b. The estimated five year NPV.

c. The R&D division's estimate of the probability of success of making the

new product.d. The marketing division's qualitative estimate of the new product's abilityto capture a 35% market share.

The ATP Committee has allocated a budget of $400,000 to the eleven projects.The problem is to select the most beneficial subset of the eleven projects that

does not exceed the total budget. The data for this example are given in Table


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2.1.

Table 2.1 Data for EX2

Project Cost NPV Prob. Success Market Share

P1 $30,000 $425,000 0.50 GoodP2 $40,000 $380,000 0.75 LowP3 $65,000 $400,000 0.25 High

P4 $95,000 $250,000 1.00 GoodP5 $100,000 $900,000 0.25 GoodP6 $125,000 $800,000 0.75 Fair

P7 $145,000 $1,000,000 0.50 FairP8 $165,000 $750,000 0.50 High

P9 $170,000 $800,000 0.75 GoodP10 $185,000 $950,000 0.50 FairP11 $200,000 $850,000 0.75 High

2.2.1 Basic Analysis

The ratings model intensity levels are given in Figures 2.2 - 2.4.

NPV Intensity Levels

$900,000+ Excellent

$800,000 to $899,999 Very Good

$500,000 to $799,999 Good$250,000 to $499,000 Fair

Figure 2.2 Intensity Levels for NPV

Probability of Success Intensity Levels

1.00 Sure thing

0.50 Go for it

0.25 A bit chancy

Figure 2.3 Intensity Levels for Probability of Success

Market Share Intensity Levels

High High

Good Good

Fair Fair

Low Low

Figure 2.4 Intensity Levels For Market Share

The analysis of this problem is carried out in two steps:

Step 1: Using the Ratings mode, rank the eleven projects and determine theweight for each project.

Step 2:Using the project weights determined in Step 1, formulate and solve a

budget constrained 0-1 optimization (knapsack) problem that selects the bestsubset of projects.

2.2.2 Further Analysis

After being presented with the solution, the R&D director says that he does notbelieve in the AHP weights, but does believe in expected value. He now wants

to use the expected value of a project in the objective function of theknapsack problem, where the expected value is defined as NPV*P(success ofproject). A second knapsack problem is solved and analyzed. If the two

solutions are different, the student should make a recommendation as to whichsubset of projects BMGT Industries should select, and why. Some discussion inclass on the accuracy of the probabilities of success and the need for

sensitivity studies would be of value.

2.3 EX3. Ratings/Resource Allocation - Case 2

BMGT DecisionWare Inc. (BDW) is a software consulting company that suppliesservices to business and government. It has a fairly active research programdirected towards improving the company's internal operations. BDW is now going

through its planning cycle to determine which internal information system


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projects suggested by its managers it should fund. Out of the 30 projects thatwere originally proposed, BDW's Software Development Board has selected 11projects that it feels are meritorious candidates for funding. Of course, there is

not enough money to do all 11! Also, each project requires an estimated level ofprogrammer hours to complete, and it is clear that there are not enoughprogrammer hours available to do all 11 projects. The Board needs some way ofselecting a subset of the 11 that would be of most value to the company.

The student is the analyst in this case. The Board wants to evaluate theprojects in terms of the following three criteria:

Improving accuracyin its clerical operations.Improving general information processing efficiency.Promoting organizational learning.

A further concern deals with the cost of each project and the number ofprogrammer hours each project uses. For each of the projects, the project

managers, working with the Board, have determined the followingcharacteristics for each of the projects that are code-named P1, P2, ..., P11.

The impact of each project with respect to its ability to improve

accuracyevaluated in terms of High, Above Average or Good.The impact of each project with respect to improving efficiencyevaluated in terms of Excellent, Very Good, Good or Fair.

The impact of each project with respect to promoting organizationallearningevaluated in terms of Yes, Maybe or So-So.

The project managers have also estimated the cost of each project and the

number of programmer hours required. A summary of the information on theprojects is given in Table 2.2.

BDW has a budget of $500,000 and 7,500 programmer hours to allocate to theeleven internal projects. The student is to rank the eleven projects and

determine the associated weights using the EC ratings mode, and then selectthe "best" subset of the eleven projects that does not exceed the total budgetand available programmer hours by solving a two-constraint 0-1 maximizing

optimization (knapsack) problem.

Table 2.2 BDW Information Systems Project Information -- Planning Cycle FY2000 (Confidential)

PROJECTS ACCURACY EFFICIENCY LEARNINGCOST

($000)HOURS

(00)

P1 HIGH VERY GOOD YES 80 10P2 ABOVE AV EXCELLENT SO-SO 55 9P3 HIGH FAIR MAYBE 90 11

P4 GOOD EXCELLENT YES 100 15P5 GOOD GOOD YES 40 8P6 ABOVE AV FAIR YES 60 7

P7 HIGH FAIR MAYBE 85 6P8 ABOVE AV EXCELLENT MAYBE 110 13P9 GOOD VERY GOOD YES 45 5

P10 ABOVE AV EXCELLENT SO-SO 80 12

P11 HIGH FAIR YES 115 14

2.3.1 Approach

The analysis of these 11 projects is carried out using the AHP and a subset ofthe projects is selected for implementation. The selected projects is presented

to the Board. The Board approved the analysis and voted to accept therecommendations.

2.3.2 The Addition of a 12thProject

After the presentation, the President of BDW calls the student's Boss and asksthe Boss to consider a twelfth project P12. P12 was proposed as one of the

original 30, but did not meet the initial cut. The manager who would run P12 is

the President's daughter-in-law. Also, the President's daughter-in-law believesthat there is an excess of programmer hours and she is concerned that some

programmers will have to be fired if only a subset of projects P1-P11 areselected. P12 has a low cost, but uses a lot of programmer hours (it uses low-level programmers who are at the low-end of the pay scale). The Boss wants

the student to furnish some ammunition to shoot P12 down, as the Boss does


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not think much of the project. The information on P12 is the given in Table 2.3.The analysis is now repeated with the twelve projects. The student shouldcompare both solutions and make a recommendation to the Boss. Question:Should the Boss shoot down P12?

Table 2.3 .Information for Project P12

PROJECTS ACCURACY EFFICIENCY LEARNINGCOST

($000)

HOURS

(00)P12 GOOD FAIR SO-SO 30 10

2.4 EX4: Simplifying the Income Tax Structure in the UnitedStates

2.4.1 Background

In the 1996 Republican presidential primaries, some discussion centered on flat

tax proposals, limitations on deductions, etc. Steve Forbes, a Republicanpresidential candidate, made the flat tax a cornerstone of his platform forwinning the Republican nomination (he failed to get the nomination). The

Forbes's campaign led to the following conclusions:

1. The American public believes that the existing tax structure is toocomplex.

2. Any tax structure has to be "affordable" in that it cannot adversely

increase the deficit that the current tax structure generates. For thepurpose of this example, assume that the deficit under the current

tax structure is $100 billion.

With this as background, the following example allows the student to use theAHP to analyze the costs and benefits of different types of tax proposals.

Table 2.4 contains some very simple data (fictitious) for analyzing various tax

proposals. The population is stratified into 5 population groups (indicated bygroup number). In addition to the current flat tax proposals, one can consider aprogressive tax rate structure or a regressive tax rate structure. In each tax

rate proposal, certain deductions are allowed and other deductions are notallowed.

Table 2.4 Income Distribution and Types of Deductions for Analyzing Various TaxProposals

Group

Number

No. ofHouseholds

(millions)

AverageIncome

(x1000)

Class 1Deductions

(x1000)

Class 2Deductions

(x1000)

Class 3Deductions

(x1000)

1 20 20 2 2 2

1 30 50 5 5 5

3 10 100 10 10 15

4 5 200 25 50 25

5 2 500 75 150 150

Notes on Table 2.4

a. Average Income: After allowances for all dependents have beensubtracted from gross income.

b. Class 1 Deductions: Interest on Home, Property Taxes, State andLocal Taxes.

c. Class 2 Deductions: Investment Deductions, Tax Shelters, etc.

d. Class 3 Deductions: All Other Deductions such as medical,contributions, office, miscellaneous, basicbusiness deductions, home etc.

2.4.2 Tax Proposals

The following tax proposals have been suggested for this analysis.

Proposal 1:Emulation of the Existing Tax Code

All three classes of deductions are allowed.

Tax Structure:

15% of net income up to $35K/year

25% of net income from $35K to $80K/year


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20 million people.

Net income is $14,000.Total tax generated is 20 x 14 x .15 = $42 billion.

Group 1:

30 million people.Net income is $35,000.

Total tax generated is 30 x 35 x .15 = $157.5 billion.10 million people.Net income is $65,000.

Total tax generated is 10 x (35 x .15 + 30 x .25) = $127.5 billion.

Group 3:

5 million people.Net income is $100,000.

Total tax generated is 5 x (35 x .15 + 45 x .25 + 20*.35) = $117.5billion.

Group 4:

2 million people.

Net income is $125,000.Total tax generated is 2 x (35 x .15 + 45 x .25 + 45 * .35) = $64.5billion.

Group 5:

Group 2:

35% of net income over $80K/year

Proposal 2:Flat Tax Proposal 1

Class 1 deductions only are allowed

Tax Structure: 13% of net income.

Proposal 3:Flat Tax Proposal 2

Class 1 deductions only are allowed

Tax Structure: 15% of net income.

Proposal 4:Progressive Tax Proposal

Class 1 deductions only are allowed.

Tax Structure:

10% of net income up to $50K/year.

20% of net income over $50K/year.

Proposal 5:, etc: Your Proposal(s)

The student must make between 1 and 3 additional tax proposals andanalyze the proposal(s) along with the 4 given tax proposals.

2.4.3 Determining the Revenue Generated by Proposal 1

Thus, the total taxes generated under Proposal 1 is $509 billion, the sum of thetaxes generated by the five groups. This proposal generates a deficit of $100billion since the total budget for the government is $609 billion. Assume that

$609 billion is the budget under any of the proposals.

2.4.4 The Assignment

a. The student should develop 1-3 tax proposals. Call these tax

proposal(s), Proposal 5 Proposal 6, etc.b. Using the above numbers, the student should determine the total

taxes generated under each proposal. The student should thencompute the deficit or surplus under each of the tax proposals.

c. The student should then use the AHP to rank the proposalsaccording the student's goals, objectives and prejudices. Either theAHP ratings approach or the direct comparisons of alternatives


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should be used or the student can carry out the analysis both ways.

The student should carefully write up the solution found, describethe assumptions, goals, objectives, prejudices etc. A diskette shouldbe included with the writeup for evaluation purposes.

2.5 EX5 Selecting a New Long Distance Telephone Service

2.5.1 Introduction

One of the most confusing issues that confront many people is what is the most

appropriate long distance service (or services) to employ. The question to beanswered by the student is to determine the most appropriate long distance

service for an individual (where the individual is assumed to be the student). InFigure 2.5, we list several long distance carriers that existed in the Fall, 2000.Some have Internet addresses attached; missing addresses have to be to be

determined by the faculty member or the student.

1. ATT: $4.95 or $5.95/month, 10c/minute part of the time, 5c/minuteat other times, no special code to dial, occasional specials such as

1 free hour/month, etc. Calling card exists but is more expensive.2. MCI: $4.95 or $5.95/month, 10c/minute part of the time, 5c/minute

at other times, no special code to dial, occasional specials such as

1 free hour/month, etc. Calling card exists but is more expensive.3. SPRINT: $4.95 or $5.95/month, 10c/minute part of the time,

5c/minute at other times, no special code to dial, occasionalspecials such as 1 free hour/month, etc. Calling card exists but ismore expensive.

4. IDT Global Call: . 6.9c/minute in US and Canada. 99c/month

monthly service charge. 800 access number from remote site. Donot know if you need 800 access code from home phone. 1-800-597-3028. Prepay a specified amount?

5. SHOPSS.COM: . $9.95/month fixed fee. No additional charges. Is

this too good to be true? Is this site an Internet only site? Ad saysthat it is a high quality ordinary telephone to ordinary telephone-nointernet!! 1- 877-shop-880. Is there a special code to dial before

using? Prepay a specified amount?

6. Net2phone: 4.9c/call in US and Canada. 1-800-438-8735. Addclaims no activation charge, no connection charges, no minimum calllength and you keep your existing phone line. Is this a high quality

ordinary telephone to ordinary telephone connection-no Internet??99c/month service charge. Prepay a specified amount?

7. Net2phone: No Internet address given. 1c/call in the United States.

Appears to be call from a PC to an ordinary phone. Minimumpurchase of $5.95. Prepay a specified amount? 1-877-767-6569.

Figure 2.5 Long Distance Carriers - Fall 2000

A faculty member using this example should update the list in Figure 2.5 and

should add in cell phone options.

The problem is to apply the ratings version of the AHP to determine the bestlong distance service plan for the individual carrying out the analysis. The planthat the student puts together must satisfy the following needs:

The plan must provide for long distance service from the individual's hometo anywhere in the United States.The plan must provide reasonable calling card service.

The plan must have reasonable expected cost.The service under the plan must be easy-to-use, have high quality

service, good technical support, etc. (The student can figure out whatthe etc. means.)

The long distance service plan can be a combination of two or more services.For example, a plan might consist of the following:

ATT for long distance services in the home.

A cheap dial-up service for long phone calls.A calling card service that gives an inexpensive but convenient way tomake long distance calls away from home.

2.5.2 Telephone Usage Information
http://www.net2phonedirect.com/http://www.shopss.com/http://www.global-call.com/


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In building this model, the student will need information on the demand usagefor the current telephone service. Use the following usage data for this study:

Case 1:

20 long distance phone calls in a month.

These long distance phone calls required 200 minutes in total.There were 4 long distance calls over 20 minutes.50% of the calls and 50% of the minutes used were during the peak

period and the remainder of the calls took place in the off-peak.Probability of Case 1 occurring is .1

One calling card call of 10 minutes in duration.

Case 2:


These long distance phone calls required 500 minutes in total.There were 10 long distance calls over 20 minutes.

50% of the calls and 50% of the minutes were during the peak period andthe remainder of the calls took place in the off-peak.Probability of Case 2 occurring is .3.

One calling card call of 10 minutes in duration.One additional calling card call of 5 minutes in duration.

Case 3:


These long distance phone calls required 900 minutes in total.There were 18 long distance calls over 20 minutes.50% of the calls and 50% of the minutes were during the peak period and

the remainder of the calls took place in the off-peak.Probability of Case 3 occurring is .4.Two calling card calls - each 10 minutes in duration.

One additional calling card call of 5 minutes in duration.

Case 4:

120 long distance phone calls in a month.These long distance phone calls required 1500 minutes in total.

There were 35 long distance calls over 20 minutes.50% of the calls and 50% of the minutes were during the peak period andthe remainder of the calls took place in the off-peak.

Probability of Case 4 occurring is .2.Three calling card calls - each of 10 minutes in duration.Two additional calling card calls - each of 5 minutes in duration.

2.5.3 Analysis

a. Collect data from each of the long distance carriers. In this study,any long distance carrier, including those given in Figure 2.5, is acandidate. The number of alternatives to be considered can be

limited to between 8-10 to ease the hand computations. At leastone alternative must be a combined strategy of 2 or more ofthe carriers. One of the alternatives (or part of an alternative)

can be a wireless service that allows roaming as part of thepackage.

b. Compute the expected cost of each alternative that c reated using

the telephone usage information given above.c. Carry out the analysis of the alternatives using the AHP ratings

model.

2.5.4 Report

The student should write a 5-10 page report describing the results of theanalysis. this report should contain a one page executive summary describingthe results, a couple of pages describing the model and the remainder of the

report describing the analysis. A careful description of the telephone data thatwas collected should also be included.

2.6 EX6: Geometric Validation Exercise

The following exercise demonstrates that the weights generated by the AHP,using subjective judgments and the 1-9 scale, can yield a close approximation


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of true known values. Five geometric figures are displayed in Figure 2.6. Wewant to estimate the following ratios:

Weight Figure i = (Area of Figure i)/(Total Area of the Five Figures),i= A, B, C, D, and E.

To accomplish this, the simple AHP two-level hierarchy is first developed, Figure2.7. Then, using the pairwise comparison mode, the data of the comparisonmatrix shown in Figure 2.8 are entered. Synthesizing the data finds the AHP

area ratio weight vector. The results should compare very well to the true arearatio weights given below.

A = .471B = .050

C = .234D = .149E = .096

In most cases, the estimates determined by the AHP differ by no more than 5%from the true values. The problem can be done by each student or as a class"team" analysis that uses majority vote in determining the comparison values for

the matrix (see related discussion of this problem in Sect ion 2.1).

Figure 2.6 Geometric Validation Figures, Saaty 1994

enlarge

Figure 2.7 AHP Hierarchy for the Geometric Validation Problem

enlarge
http://archive.ite.journal.informs.org/Vol4No2/BodinGass/images/figure2_lg.gifhttp://archive.ite.journal.informs.org/Vol4No2/BodinGass/images/figure1_lg.jpghttp://-/?-


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Figure 2.8Geometric Validation Problem Pairwise Comparison Matrix

enlarge

3. Example

This example is a variant of example EX1 that was described in Section 2.1. Thedata for this example can be found in Figure 2.9.

Criteria PurchasePrice MPG --- Amenities ---

Subcriteria Prestige Comfort Style

Avalon $18,000 30 Low Fair Fair

Babylon $28,000 26 Very Good Excellent

Carryon $35,000 18 OK Excellent Good

Figure 2.9 Data for Automobile Purchase Example

In this example, a student wishes to purchase an automobile and has reduced

his search to the following three alternatives called the Avalon, Babylon andCarryon. The student plans to use the AHP to help him make his decision. The

student's criteria are capital cost (represented by Purchase Price in Figure 2.9),operating cost (represented by Miles/Gallon (MPG) in Figure 2.9) and Amenities.Purchase Price and MPG can be considered quantitative criteria whereasAmenities can be considered a qualitative criterion. The student's subcriteria

under Amenities are Prestige, Comfort and Style.

The student has established the following considerations (or personal beliefs) inorder to evaluate the three alternatives. The student is very concerned aboutcapital expense, demands comfort and wants a reasonably prestigious car. The

student is not very concerned about the car's styling and operating cost. Thestudent converts these personal beliefs into pairwise comparisons. The AHPuses these pairwise comparisons to generate a weight for each alternative so

that the alternatives can be ranked.

Note: The faculty member must force the student to state who is thedecision-maker in the model and what are the personal beliefs of the

decision-maker. As an illustration, in this example, the pairwisecomparisons (as well as the criteria and subcriteria) can differ,depending upon whether the decision-maker is (i) a student, (ii)a

person who is established and has a high income or (iii) a person whois retired and living on a fixed income.

3.1 Direct Comparison Model of the AHP

The AHP tree for the direct comparison model is given in Figure 3.1. Thepairwise comparisons for the criteria under the Goal node is given in Figure 3.2.The pairwise comparisons for the subcriteria under the criterion, Amenities, is

given in Figure 3.3and the pairwise comparisons for the alternatives - Avalon,Babylon and Carryon - under the appropriate criteria and subcriteria are given inFigure 3.4-Figure 3.8. In the AHP synthesis for this problem given in Figure 3.9,

the Avalon is the highest rated car, the Babylon is the next highest rated carand the Carryon is the lowest rated car.
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Weights .54 .297 .163

Inconsistency Measure = .01

Figure 3.5 Pairwise Comparisons of Alternatives from the Criterion, MPG, and theWeights Determined by the AHP Tree

Avalon Babylon Carryon

Avalon 1 1/6 1/3

Babylon 6 1 4

Carryon 3 1/4 1

Weights .091 .691 .218


Figure 3.6 Pairwise Comparisons of Alternatives from the Subcriterion, Prestige,and the Weights Determined by the AHP Tree


Avalon 1 1/5.5 1/8

Babylon 5.5 1 1/3

Carryon 8 3 1

Weights .064 .271 .657


Figure 3.7 Pairwise Comparisons of Alternatives from the Subcriterion, Comfort,and the Weights Determined by the AHP Tree


Avalon 1 1/7 1/4

Babylon 7 1 3.5

Carryon 4 1/3.5 1

Weights .077 .679 .271


Figure 3.8 Pairwise Comparisons of Alternatives from the Subcriterion, Style and

the Weights Determined by the AHP Tree

The summary portion of Figure 3.9gives the overall weight for each alternativeas determined by the AHP. The detailed portion of Figure 3.9gives a breakout

of the weights as a function of the criteria and subcriteria. The weight of .481

for the Avalon (with the criteria/subcriteria noted in parentheses) is computedas follows:

.481 =.415 (Purchase Price) + .046 (MPG)+ .011 (Amenities-Comfort)

+ .006 (Amenities-Prestige) +.002 (Amenities-Style)

Similar computations can be carried out for the Babylon and the Carryon.

a. Summary:

Avalon = .481 Babylon = .315 Carryon = .204


b. Detailed Analysis

Purchase Price=.644


Amenities = .271

Comfort = .172

Avalon = .011 Babylon = ..048 Carryon = .113

Prestige = .070


Style = .028


MPG = .085


Figure 3.9 Analysis of the Results for the Direct Comparison Model

3.2 The AHP Ratings Model

Although the Avalon is the highest rated car in the direct comparison modelpresented in 3.1, the student wants to ensure that he has made the

appropriate decision. To accomplish this, he employs the AHP ratings model.
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$30,000). The pairwise comparisons for the intensities under the criterion,Purchase Price, are given in Figure 3.11.The intensities for the criterion, MPG, are inexpensive (29mpg),

reasonable 22-29 mpg), and expensive (< 22 mpg). The pairwisecomparisons for the intensities under the criterion, MPG, are given inFigure 3.12.

The intensities for the subcriterion, Comfort, under the criterion,Amenities are very prestigeous, OK prestigeous, and low prestigeous.Thepairwise comparisons for the intensities under the criterion-subcriterion,

Amenities-Prestige are given in Figure 3.13.

The intensities for the subcriterion, Comfort under the criterion, Amenitiesare excellent comfort, good comfort and fair comfort. The pairwise

comparisons for the intensities under the c riterion-subcriterion, Amenities-Comfort, are given in Figure 3.14.The intensities for the subcriterion, Style under the criterion, Amenities

are excellent styling, good styling and fair styling. The pairwisecomparisons for the intensities under the c riterion-subcriterion, Amenities-Style, are given in Figure 3.15.

ReasonablyExpensive

ExpensiveVery

Expensive

ReasonablyExpensive

1 3 6

Expensive 1/3 1 4

Very Expensive 1/6 1/4 1

Weights .0644 .271 .085


Figure 3.11 Pairwise Comparisons of the Intensities from the Criterion, PurchasePrice, and the Weights Determined by the AHP Tree

Inexpensive Reasonable Expensive

Inexpensive 1 2 3

Reasonable 1/2 1 2

Expensive 1/3 1/2 1

Weights .54 .297 .163

Inconsistency Measure = .01Figure 3.12 Pairwise Comparisons of the Intensities from the Criterion, MPG, and

the Weights Determined by the AHP Tree

High Prestige OK Prestige Low Prestige

High Prestige 1 4 6

OK Prestige 1/4 1 3

Low Prestige 1/6 1/3 1

Weights .091 .691 .218


Figure 3.13Pairwise Comparisons of the Intensities from the Subcriterion,Prestige, and the Weights Determined by the AHP Tree

ExcellentComfort

Good Comfort Fair Comfort

ExcellentComfort

1 5.5 8

Good Comfort 1/5.5 1 3

Fair Comfort 1/8 1/3 1

Weights .752 .174 .074


Figure 3.14 Pairwise Comparisons of the Intensities from the Subcriterion,Comfort, and the Weights Determined by the AHP Tree

Excellent Styling Good Styling Fair Styling

Excellent Styling 1 4 7

Good Styling 1/4 1 3.5

Fair Styling 1/7 1/3.5 1

Weights .700 .221 .079


Figure 3.15 Pairwise Comparisons of the Intensities from the Subcriterion, Style
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and the Weights Determined by the AHP Tree

For the quantitative criteria, Purchase Price and MPG, we establish intervals foreach of the intensities so that any alternative falling in the same interval forthis criterion gets the same value for that criterion or subcriterion in the

spreadsheet. In the example, we used three intensities for each criterion orsubcriterion needing intensities. It is generally advised to use more intensities ateach level of the tree (generally around 5) to give a finer stratification of the

criterion or subcriterion in the spreadsheet.

3.4 AnalysisThe spreadsheet analysis of the three alternatives is given in Figure 3.16 andthe computation of the elements in the spreadsheet is given in Figure 3.17.From this analysis, the Avalon has the highest Score, the Babylon has the

second highest Score and the Carryon has the lowest Score. Thus, the Avalonis the highest rated car regardless of whether the direct comparison model or

the ratings model is employed.

--- Amenities ---

Alternative ScorePurchase

PriceMPG PrestigeComfort Style

Avalon 0.759 0.644 0.085 .010 .017 .003

Babylon 0.456 .271 .046 0.07 .040 .029

Carryon 0.314 .085 .026 .022 .172 .009

Figure 3.16 Ratings Spreadsheet

Purchase Price(.644)

Reasonably Expensive(.644) Weight = .644*.644/.644 = .644

Expensive(.271) Weight = .644*.271/.644 = .271

Very Expensive(.085) Weight = .644*.085/.644 = .084

MPG(.085)

Inexpensive(.54) Weight = .085*.54/.54 = .085

Reasonable(.297) Weight = .085*.297/.54 = .046

Expensive (.163) Weight = .085*.163/.54 = .026

Amenities(.271)

Prestige(.258)

High Prestige(.691) Weight = .271*.258*.691/.691 = .07

OK Prestige (.218)Weight = .271*.258*.218/.691 =.022

Low Prestige (.091) Weight = .271*.258*.091/.691 = .01

Comfort(.637)

Excellent Comfort(.752)Weight = .271*.637*.752/.752 =.174

Good Comfort (.174) Weight = .271*.637*.172/.752 = .04

Fair Comfort (.074)Weight = .271*.637*.074/.752 =.017

Style(.105)

Excellent Styling(.7) Weight = .271*.105*.7/.7 = .029 Good Styling(.221) Weight = .271*.105*.221/.7 = .009

Fair Styling(.079) Weight = .271*.105*.079/.7 = .003

Figure 3.17 Computation of the Weights of the Intensities in the Spreadsheet inFigure 3.16

The student now has to decide if Avalon's Score of .756 is high enough towarrant purchasing the car. In other words, the student has to decide if

Avalon's Score of .756 is close enough to the perfect Score of 1 under thestudent's beliefs and prejudices that the student can make the decision to

purchase the car. If the student does not find the Score of the Avalon highenough to warrant purchasing the car, then the student might decide toexamine other alternatives and repeat the above analysis.

Further analysis shows that most of the Score for the Avalon comes from the

criteria, Purchase Price and MPG, and that the Avalon has only fair amenities. Ifthe student reruns the model and makes Amenities the most important criterion,

then our tests indicate that the Carryon most likely will become the preferredautomobile. However, the Scores for the alternatives when Amenities is madethe most important criterion are generally quite low (under .6 in most cases).
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Appendix

AHP Resources

prepared by

Thaddeus SimUniversity of Iowa

[email protected]

A. Textbooks

We inspected a total of 38 introductory management science, operationsresearch, and quantitative methods/techniques/analysis textbooks for AHP

content. For some textbooks we checked multiple editions. 15 of these 38books (about 40%) had some AHP content. The content varied from a fewparagraphs to a full chapter. The number of pages set aside for AHP varied from

3 to 19. It is fair to say that the AHP coverage in introductory textbooks hasincreased over time. A good example if the Quantitative Analysis for

Management text by Render and Stair. The second edition of this text(published in 1985) had no AHP coverage. In contrast, the fifth edition (1994)had 6 pages on AHP, and the seventh edition (1999) had 19 pages. The tablebelow lists the 38 textbooks considered and the AHP coverage (in the number of

pages) in the order of decreasing AHP-coverage.

Author Title Edition Publisher Year Pages

Render, StairQuantitative Analysis forManagement

7Prentice

Hall1999 19

Anderson,Sweeney,

Williams

An Introduction to

Management Science:Quantitative Approaches toDecision Making

6 West 1991 16

Anderson,Sweeney,Williams

Quantitative Methods forBusiness

7 West 1997 15

Winston,

AlbrightPractical Management Science 2 Duxbury 2000 14

TaylorIntroduction to ManagementScience

6Prentice

Hall1999 12

WinstonOperations Research:

Applications and Algorithms3 Duxbury 1993 12

Camm, EvansManagement Science:Modeling, Analysis and

Interpretation

South-Western

1995 11

Hanna Introduction to ManagementScience: MasteringQuantitative Analysis

South-Western

1995 9

Render, StairIntroduction to ManagementScience

Allyn &Bacon

1992 9

ClaussApplied Management Science

and Spreadsheet Modeling Duxbury 1996 8

Ragsdale

Spreadsheet Modeling andDecision Analysis: A PracticalIntroduction to Management

Science

Course 1995 8

Eppen, Gould,Schmidt,

Moore,Weatherford

Introductory Management

Science 5

Prentice

Hall 1998 7

Render, StairQuantitative Analysis for

Management5

Allyn &

Bacon1994 6

Forginonne Quantitative Management Dryden 1989 3
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Moore,Weatherford,Eppen, Gould,

Schmidt

Decision Modeling withMicrosoft Excel

6Prentice

Hall2001 CD ROM

Austin,Ghandforoush

Management Science forDecision Makers West 1993 0

Baker, Kropp

Management Science: An

Introduction to the Use ofDecision Models

Wiley 1985 0

BellManagementScience/Operations Research:A Strategic Approach

South-Western

1999 0

BroshQuantitative Techniques for

Managerial Decision Making Reston 1985 0

Burton,Chandler,Holzer

Quantitative Approaches to

Business Decision Making

Harper &

Row1986 0

Cook, RussellIntroduction to ManagementScience

4Prentice

Hall1989 0

Dennis, Dennis Management Science West 1991 0

Eppen, Gould,

Schmidt

Introductory Management

Science

4Prentice

Hall

1984 0

Evans,Anderson,

Sweeney,Williams

Applied Production &

Operations Management3 West 1990 0

Groebner,Shannon

Introduction to ManagementScience

Dellen 1991 0

HesseManagerial SpreadsheetModeling and Analysis

Irwin 1997 0

Hillier, Hillier,

Lieberman

Introduction to ManagementScience: A Modeling and Case

Studies Approach withSpreadsheets

Irwin

McGraw-Hill

2000 0

KnowlesManagement Science: Building

and Using Models Irwin 1989 0

Lawrence,Pasternack

Applied Management Science:A Computer-Integrated

Approach for Decision Making

2 Wiley 2001 0

LeeIntroduction to ManagementScience

2 Dryden 1987 0

Levin, Rubin,

Stinson

Quantitative Approaches to

Management6

McGraw

Hill1986 0

Markland,

Sweigart

Quantitative Methods:Applicat ions to ManagerialDecision Making

Wiley 1987 0

Mathur, Solow Management Science: The Artof Decision Making

PrenticeHall

1994 0

PlaneManagement Science: A

Spreadsheet Approach

Boyd &

Fraser1996 0

Render, StairQuantitative Analysis forManagement

2Allyn &Bacon

1985 0

TaylorIntroduction to Management

Science4

Allyn &

Bacon1993 0

Thierauf,

Klekamp, Ruwe

Management Science: AModel Formulation Approachwith Computer Applications

Merrill 1985 0

Vazsonyi/SpirerQuantitative Analysis for

Business

Prentice

Hall1984 0

B. Software


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We found six organizations that offer a multiobjective decision making software.While the method used is not always explicit, we believe that most, if not all, ofthese software packages use AHP. Reduced trial or educational versions of

these packages are available for downloading from company web sites. Thetable below lists the relevant software names and companies. OR/MS Today haspublished a decision analysis software survey that may be of interest to

readers:

http://www.lionhrtpub.com/orms/surveys/das/das.html

Company Software Webpage

InfoHarvest CriteriumDecisionPlus

www.infoharvest.com

ArlingtonSoftware

Corporation

ERGO www.arlingsoft.com

Expert ChoiceExpertChoice

www.expertchoice.com

Helsinki

University ofTechnology

HIPRE 3+ www.sal.hut.fi/Downloadables/hipre3.html

Krysalis Ltd. OnBalance www.krysalis.co.uk

Catalyze Ltd. Hiview www.catalyze.co.uk
http://www.catalyze.co.uk/http://www.krysalis.co.uk/http://www.sal.hut.fi/Downloadables/hipre3.htmlhttp://www.expertchoice.com/http://www.arlingsoft.com/http://www.infoharvest.com/http://www.lionhrtpub.com/orms/surveys/das/das.html


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Documents

BODIN & GASS - Exercises for Teaching the Analytic Hierarchy Process.pdf