Abstract

In a related paper (Bodin and Gass, 2003), we described the basic concepts that we believe must be covered when teaching the Analytic Hierarchy Process (AHP) to MBA students and outlined six exercises that can be used as in-class examples 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 covered are: (a) the AHP fundamental pairwise comparison scale, (b) inconsistency and sensitivity analysis, (c) ratio scales, (d) the ratings model, (e) the team approach for solving an AHP problem, (f) AHP and resource allocation, and (g) if class 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 of the AHP are given in Saaty, 1980 and Saaty, 1994.

Since the AHP has the proven ability to resolve (or assist in resolving) a wide class 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 decision analysis problem, an MBA graduate must have the background and experience to 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) with relevant 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 the University of Maryland, the AHP module was covered in about 2-2.5 weeks. In this module, we used the software package, Expert Choice. The trial version of Expert Choice can be downloaded for free from the website. Other software packages 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 by itself for carrying out the AHP analysis and computations. It must be noted, however, that the ratings version of the AHP in Expert Choice forms a table called a ratings spreadsheet (called 'spreadsheet' in section 3 of this paper) for determining the weight for each alternative. The weight of the alternative in this spreadsheet is a measure of how close that alternative is to the perfect alternative (weight = 1).

In Section 2 of this paper, six varied exercises that we found useful in the classroom for conveying the essentials of an AHP analysis and the features of the Expert Choice software are presented. As in Bodin and Gass, 2003, these exercises are outlined as follow:

• EX1 contains a simple direct comparison model for the purchase of a new automobile. Variants of this example have appeared in numerous publications including Saaty, 1990. The criteria and the alternatives are specified.
• EX2 and EX3 are problems involving the integration of the ratings model version 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 set of criteria and alternative tax strategies (over and above the tax strategies 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 a set 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 literature or on the Internet are described in Bodin and Gass, 2003. As noted in Bodin and Gass, 2003:

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 implementation by the Expert Choice software. The hierarchy is easy to build and the instructor can demonstrate the replication command that simplifies the building of the hierarchy. 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. The student can readily see that there is no one best alternative, as none of the automobiles is best across all criteria (as indicated by the asterisks).

The problem has both quantitative and qualitative data. The price data can be used directly in the EC comparison matrix by the data entry mode, but the data entry 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 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 also be used indirectly by asking the usual pairwise comparison question, e.g., "Is Avalon 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 equivalent verbal 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 utility evaluation on the dollars - the dollar spent for the cheaper auto has a greater utility 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 faculty member first illustrate the data entry mode and then illustrating the 1-9 pairwise comparison 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 the individual auto's MPG number divided by the sum of all the MPG numbers. For prestige and comfort, the student must make pairwise comparisons that respect the 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 small team of students. They illustrate the use of the AHP ratings model to determine weights for competing projects, with the weights then used in a 0-1 optimization problem to select a subset of the projects subject to a budget constraint.

BMGT Industries has an internal Advanced Technology Project Committee (ATP) responsible for selecting new projects for funding. The selection is made from those projects suggested by its division managers. The selection cycle is now upon 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 not continue the status quo. It has instructed its division managers (Research and Development, Manufacturing, Marketing, Logistics, Finance, Human Resources) to come up with a set of new projects that addresses the future of each division 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 new products the R&D group expects to develop over the next two years. The staffs have determined the two year R&D costs and initial production costs for each robotic project. Further, with help from the Marketing Division, the staffs have also estimated the return, represented by net present value (NPV), of each robotic product, assuming that the product comes to market in the next five years. Although the ATP Committee is impressed by the excellence of the eleven projects and would like to fund them all, there is not enough money to do so.

Faced with this problem, the ATP Committee has asked BMGT's new MBA employee to devise a way to select a subset of the competing projects to undertake 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 code names 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. The following 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 ability to 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 2.1.

Table 2.1 Data for EX2

Project	Cost	NPV	Prob. Success	Market Share
P1	$30,000	$425,000	0.50	Good
P2	$40,000	$380,000	0.75	Low
P3	$65,000	$400,000	0.25	High
P4	$95,000	$250,000	1.00	Good
P5	$100,000	$900,000	0.25	Good
P6	$125,000	$800,000	0.75	Fair
P7	$145,000	$1,000,000	0.50	Fair
P8	$165,000	$750,000	0.50	High
P9	$170,000	$800,000	0.75	Good
P10	$185,000	$950,000	0.50	Fair
P11	$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 the weight 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 best subset of projects.

2.2.2 Further Analysis

After being presented with the solution, the R&D director says that he does not believe 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 the knapsack problem, where the expected value is defined as NPV*P(success of project). A second knapsack problem is solved and analyzed. If the two solutions are different, the student should make a recommendation as to which subset of projects BMGT Industries should select, and why. Some discussion in class 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 supplies services to business and government. It has a fairly active research program directed towards improving the company's internal operations. BDW is now going through its planning cycle to determine which internal information system projects suggested by its managers it should fund. Out of the 30 projects that were originally proposed, BDW's Software Development Board has selected 11 projects 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 of programmer hours to complete, and it is clear that there are not enough programmer hours available to do all 11 projects. The Board needs some way of selecting 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 the projects in terms of the following three criteria:

• Improving accuracy in its clerical operations.
• Improving general information processing efficiency.
• Promoting organizational learning.

A further concern deals with the cost of each project and the number of programmer hours each project uses. For each of the projects, the project managers, working with the Board, have determined the following characteristics for each of the projects that are code-named P1, P2, ..., P11.

• The impact of each project with respect to its ability to improve accuracy evaluated in terms of High, Above Average or Good.
• The impact of each project with respect to improving efficiency evaluated in terms of Excellent, Very Good, Good or Fair.
• The impact of each project with respect to promoting organizational learning evaluated 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 the projects is given in Table 2.2.

BDW has a budget of $500,000 and 7,500 programmer hours to allocate to the eleven internal projects. The student is to rank the eleven projects and determine the associated weights using the EC ratings mode, and then select the "best" subset of the eleven projects that does not exceed the total budget and available programmer hours by solving a two-constraint 0-1 maximizing optimization (knapsack) problem.

Table 2.2 BDW Information Systems Project Information -- Planning Cycle FY 2000 (Confidential)

PROJECTS	ACCURACY	EFFICIENCY	LEARNING	COST ($000)	HOURS (00)
P1	HIGH	VERY GOOD	YES	80	10
P2	ABOVE AV	EXCELLENT	SO-SO	55	9
P3	HIGH	FAIR	MAYBE	90	11
P4	GOOD	EXCELLENT	YES	100	15
P5	GOOD	GOOD	YES	40	8
P6	ABOVE AV	FAIR	YES	60	7
P7	HIGH	FAIR	MAYBE	85	6
P8	ABOVE AV	EXCELLENT	MAYBE	110	13
P9	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 of the projects is selected for implementation. The selected projects is presented to the Board. The Board approved the analysis and voted to accept the recommendations.

2.3.2 The Addition of a 12^th Project

After the presentation, the President of BDW calls the student's Boss and asks the 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 believes that 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 are selected. 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 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 should compare 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	LEARNING	COST ($000)	HOURS (00)
P12	GOOD	FAIR	SO-SO	30	10

2.4 EX4: Simplifying the Income Tax Structure in the United States

2.4.1 Background

In the 1996 Republican presidential primaries, some discussion centered on flat tax proposals, limitations on deductions, etc. Steve Forbes, a Republican presidential candidate, made the flat tax a cornerstone of his platform for winning 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 too complex.
2. Any tax structure has to be "affordable" in that it cannot adversely increase the deficit that the current tax structure generates. For the purpose 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 the AHP 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 by group number). In addition to the current flat tax proposals, one can consider a progressive tax rate structure or a regressive tax rate structure. In each tax rate proposal, certain deductions are allowed and other deductions are not allowed.

Table 2.4 Income Distribution and Types of Deductions for Analyzing Various Tax Proposals

Group Number	No. of Households (millions)	Average Income (x1000)	Class 1 Deductions (x1000)	Class 2 Deductions (x1000)	Class 3 Deductions (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 been subtracted from gross income.
b.	Class 1 Deductions:	Interest on Home, Property Taxes, State and Local Taxes.
c.	Class 2 Deductions:	Investment Deductions, Tax Shelters, etc.
d.	Class 3 Deductions:	All Other Deductions such as medical, contributions, office, miscellaneous, basic business 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 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 and analyze the proposal(s) along with the 4 given tax proposals.

2.4.3 Determining the Revenue Generated by Proposal 1

Group 1: 20 million people. Net income is $14,000. Total tax generated is 20 x 14 x .15 = $42 billion.

Group 2: 30 million people. Net income is $35,000. Total tax generated is 30 x 35 x .15 = $157.5 billion.

Group 3: 10 million people. Net income is $65,000. Total tax generated is 10 x (35 x .15 + 30 x .25) = $127.5 billion.

Group 4: 5 million people. Net income is $100,000. Total tax generated is 5 x (35 x .15 + 45 x .25 + 20*.35) = $117.5 billion.

Group 5: 2 million people. Net income is $125,000. Total tax generated is 2 x (35 x .15 + 45 x .25 + 45 * .35) = $64.5 billion.

Thus, the total taxes generated under Proposal 1 is $509 billion, the sum of the taxes generated by the five groups. This proposal generates a deficit of $100 billion 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 then compute the deficit or surplus under each of the tax proposals.
c.	The student should then use the AHP to rank the proposals according the student's goals, objectives and prejudices. Either the AHP ratings approach or the direct comparisons of alternatives should be used or the student can carry out the analysis both ways. The student should carefully write up the solution found, describe the assumptions, goals, objectives, prejudices etc. A diskette should be 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 be answered by the student is to determine the most appropriate long distance service for an individual (where the individual is assumed to be the student). In Figure 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.

ATT: $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. 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. SPRINT: $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. IDT Global Call: . 6.9c/minute in US and Canada. 99c/month monthly service charge. 800 access number from remote site. Do not know if you need 800 access code from home phone. 1-800-597-3028. Prepay a specified amount? 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 says that it is a high quality ordinary telephone to ordinary telephone-no internet!! 1- 877-shop-880. Is there a special code to dial before using? Prepay a specified amount? Net2phone: 4.9c/call in US and Canada. 1-800-438-8735. Add claims no activation charge, no connection charges, no minimum call length 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? Net2phone: No Internet address given. 1c/call in the United States. Appears to be call from a PC to an ordinary phone. Minimum purchase 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 best long distance service plan for the individual carrying out the analysis. The plan that the student puts together must satisfy the following needs:

• The plan must provide for long distance service from the individual's home to 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 what the 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 to make long distance calls away from home.

2.5.2 Telephone Usage Information

In building this model, the student will need information on the demand usage for 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:

• 40 long distance phone calls in a month.
• 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 and the 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:

• 70 long distance phone calls in a month.
• 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 and the 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 a candidate. The number of alternatives to be considered can be limited to between 8-10 to ease the hand computations. At least one alternative must be a combined strategy of 2 or more of the carriers. One of the alternatives (or part of an alternative) can be a wireless service that allows roaming as part of the package.
b.	Compute the expected cost of each alternative that created 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 the analysis. this report should contain a one page executive summary describing the results, a couple of pages describing the model and the remainder of the report describing the analysis. A careful description of the telephone data that was 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 of true known values. Five geometric figures are displayed in Figure 2.6. We want 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, Figure 2.7. Then, using the pairwise comparison mode, the data of the comparison matrix 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 area ratio weights given below.

• A = .471
• B = .050
• C = .234
• D = .149
• E = .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 Section 2.1).

Figure 2.6 Geometric Validation Figures, Saaty 1994
enlarge

Figure 2.7 AHP Hierarchy for the Geometric Validation Problem
enlarge

Figure 2.8 Geometric Validation Problem Pairwise Comparison Matrix
enlarge

3. Example

This example is a variant of example EX1 that was described in Section 2.1. The data 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 and Carryon. 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 whereas Amenities 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) in order to evaluate the three alternatives. The student is very concerned about capital expense, demands comfort and wants a reasonably prestigious car. The student is not very concerned about the car's styling and operating cost. The student converts these personal beliefs into pairwise comparisons. The AHP uses 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 the decision-maker in the model and what are the personal beliefs of the decision-maker. As an illustration, in this example, the pairwise comparisons (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 who is 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. The pairwise 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 3.3 and the pairwise comparisons for the alternatives - Avalon, Babylon and Carryon - under the appropriate criteria and subcriteria are given in Figure 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 car and the Carryon is the lowest rated car.

Figure 3.1 AHP Tree for Direct Comparison Model for Purchasing an Automobile

	Purchase Price	MPG	Amenities
Purchase Price	1	6	3
MPG	1/6	1	1/4
Amenities	1/3	4	1
Weights	.644	.085	.271
Inconsistency Measure = .05

Figure 3.2 Pairwise Comparisons of Criteria from the Goal Node and Weights Determined by the AHP

	Prestige	Comfort	Style
Prestige	1	1/3	3
Comfort	3	1	5
Style	1/3	1/3	1
Weights	.258	.637	.105
Inconsistency Measure =   .04

Figure 3.3 Pairwise Comparisons of the Subcriteria from the Criterion, Amenities, and the Weights Determined by the AHP Tree

	Avalon	Babylon	Carryon
Avalon	1	3	6
Babylon	1/3	1	4
Carryon	1/6	1/4	1
Weights	.644	.271	.085
Inconsistency Measure =  .05

Figure 3.4 Pairwise Comparisons of Alternatives from the Criterion, Purchase Price, and the Weights Determined by the AHP Tree

	Avalon	Babylon	Carryon
Avalon	1	2	3
Babylon	1/2	1	2
Carryon	1/3	1/2	1
Weights	.54	.297	.163
Inconsistency Measure =  .01

Figure 3.5 Pairwise Comparisons of Alternatives from the Criterion, MPG, and the Weights 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
Inconsistency Measure =  .05

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

	Avalon	Babylon	Carryon
Avalon	1	1/5.5	1/8
Babylon	5.5	1	1/3
Carryon	8	3	1
Weights	.064	.271	.657
Inconsistency Measure =  .06

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

	Avalon	Babylon	Carryon
Avalon	1	1/7	1/4
Babylon	7	1	3.5
Carryon	4	1/3.5	1
Weights	.077	.679	.271
Inconsistency Measure =  .05

Figure 3.8 Pairwise Comparisons of Alternatives from the Subcriterion, Style and the Weights Determined by the AHP Tree

The summary portion of Figure 3.9 gives the overall weight for each alternative as determined by the AHP. The detailed portion of Figure 3.9 gives 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 computed as 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
Inconsistency Measure = .05
b. Detailed Analysis
Purchase Price= .644
	Avalon = .415	Babylon = .174	Carryon = .055
Amenities = .271
Comfort = .172
	Avalon = .011	Babylon = ..048	Carryon = .113
Prestige = .070
	Avalon = .006	Babylon = .048	Carryon = .015
Style = .028
	Avalon = .002	Babylon = .025	Carryon = .014
MPG = .085
	Avalon = .046	Babylon = .025	Carryon = .014

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 model presented in 3.1, the student wants to ensure that he has made the appropriate decision. To accomplish this, he employs the AHP ratings model. Generally, the ratings model is used when there is a large number of alternatives. The ratings model gives a Score to each alternative and the alternative with the highest Score is the highest rated alternative.

One of the authors has used the Score for an alternative in the ratings model to give a measure of how close that alternative is to the perfect alternative (Score = 1). This interpretation of the Score is described in this section and used in Section 3.3. It is possible for an alternative to rate the best in the direct comparison analysis but have a Score so low when employing the ratings model that the user might decide to not implement that alternative (or at least cast some doubt on using that alternative).

The Score of an alternative can differ as a function of the degree of stratification used in setting up the model. To illustrate what we mean by degree of stratification, let us examine the process of computing the Grade Point Average (GPA) for students at a university. One stratification for the grades and points earned for the grade is the following: A(4 points), B(3 points), C(2 points), D(1 point) and F(0 points). A finer stratification is A (4 points), A-(3.67 points), B+(3.33 points), B(3 points), B-(2.67 points), etc. A student's GPA under the first stratification need not be the same as the student's GPA under the finer stratification. The perfect GPA for a student under both stratifications is a GPA of 4 and the Score can differ somewhat by the stratification (or intensities) used.

3.3 Ratings Version of the AHP

The AHP tree for the ratings model is given in Figure 3.10. The pairwise comparisons for the criteria under the Goal node is given in Figure 3.2 and the pairwise comparisons for the subcriteria under the criterion, Amenities, is given in Figure 3.3. We have constructed this example so that that the pairwise comparisons for the criteria out of the Goal node and the subcriteria under the criterion, Amenities, are the same, regarding of whether the user employs the direct comparison model or the ratings model.

Figure 3.10 AHP Tree for the Ratings Model for Purchasing an Automobile

The intensities (denoted in italics below) are defined as follows:

• The intensities for the criterion, Purchase Price, are reasonable ($15,000 to $22,000), expensive ($22,000 to $30,000) and very expensive (> $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 pairwise comparisons for the intensities under the criterion, MPG, are given in Figure 3.12.
• The intensities for the subcriterion, Comfort, under the criterion, Amenities are very prestigeous, OK prestigeous, and low prestigeous. The pairwise 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, Amenities are excellent comfort, good comfort and fair comfort. The pairwise comparisons for the intensities under the criterion-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 pairwise comparisons for the intensities under the criterion-subcriterion, Amenities-Style, are given in Figure 3.15.

	Reasonably Expensive	Expensive	Very Expensive
Reasonably Expensive	1	3	6
Expensive	1/3	1	4
Very Expensive	1/6	1/4	1
Weights	.0644	.271	.085
Inconsistency Measure =  .05

Figure 3.11 Pairwise Comparisons of the Intensities from the Criterion, Purchase Price, 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 =  .01

Figure 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
Inconsistency Measure = .05

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

	Excellent Comfort	Good Comfort	Fair Comfort
Excellent Comfort	1	5.5	8
Good Comfort	1/5.5	1	3
Fair Comfort	1/8	1/3	1
Weights	.752	.174	.074
Inconsistency Measure = .06

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
Inconsistency Measure = .05

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

For the quantitative criteria, Purchase Price and MPG, we establish intervals for each of the intensities so that any alternative falling in the same interval for this criterion gets the same value for that criterion or subcriterion in the spreadsheet. In the example, we used three intensities for each criterion or subcriterion needing intensities. It is generally advised to use more intensities at each level of the tree (generally around 5) to give a finer stratification of the criterion or subcriterion in the spreadsheet.

3.4 Analysis

The spreadsheet analysis of the three alternatives is given in Figure 3.16 and the 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 Avalon is the highest rated car regardless of whether the direct comparison model or the ratings model is employed.

				---   Amenities  ---
Alternative	Score	Purchase Price	MPG	Prestige	Comfort	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 in Figure 3.16

The student now has to decide if Avalon's Score of .756 is high enough to warrant 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 the student'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 high enough to warrant purchasing the car, then the student might decide to examine 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. If the student reruns the model and makes Amenities the most important criterion, then our tests indicate that the Carryon most likely will become the preferred automobile. However, the Scores for the alternatives when Amenities is made the most important criterion are generally quite low (under .6 in most cases). This analysis is not presented in this paper. When Amenities is made the most important criterion, our interpretation may be that no alternative scores high enough so that the decision to purchase one of these cars based solely on the AHP analysis can be made.

4. DISCUSSION

We have presented some exercises that we have found useful in teaching the AHP and a detailed example that illustrates the AHP direct pairwise comparison approach and the AHP rating model. Based on Bodin and Gass, 2003, we have received over 30 E-mail requests for copies of the exercises given in Section 2 of this paper. Many requests were from Asia (China, India and Japan) and several were from Europe. Most requests came from persons teaching either quantitative methods or multi-criteria decision-making courses. Several inquiries were from practitioners who were using the AHP on a project and/or were interested in learning more about AHP.

The AHP is a powerful decision-aiding tool. Those who teach the AHP must ensure that the student understands how to use the AHP and how to interpret the results correctly. We trust that the material in this paper is of value in accomplishing this end.

References

Bodin, L. and S. Gass (2003), "On Teaching the Analytic Hierarchy Process," Computers and Operations Research, Vol. 30, No. 10, pp. 1487-1498.

Saaty, T. L. (1980), The Analytic Hierarchy Process, McGraw-Hill, New York.

Saaty, T. L. (1990), "Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in a Complex World," RWS Publications, Pittsburgh, PA.

Saaty, T. L. (1994), "Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process," RWS Publications, Pittsburgh.

To reference this paper, please use:  Bodin, L. and S. Gass (2004), "Exercises for Teaching the Analytic Hierarchy Process," INFORMS Transactions on Education, Vol. 4, No 2,  http://ite.pubs.informs.org/Vol4No2/BodinGass/

A. Textbooks

We inspected a total of 38 introductory management science, operations research, and quantitative methods/techniques/analysis textbooks for AHP content. For some textbooks we checked multiple editions. 15 of these 38 books (about 40%) had some AHP content. The content varied from a few paragraphs 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 has increased 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 table below lists the 38 textbooks considered and the AHP coverage (in the number of pages) in the order of decreasing AHP-coverage.

B. Software

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, of these software packages use AHP. Reduced trial or educational versions of these packages are available for downloading from company web sites. The table below lists the relevant software names and companies. OR/MS Today has published 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	Criterium DecisionPlus	www.infoharvest.com
Arlington Software Corporation	ERGO	www.arlingsoft.com
Expert Choice	Expert Choice	www.expertchoice.com
Helsinki University of Technology	HIPRE 3+	www.sal.hut.fi/Downloadables/hipre3.html
Krysalis Ltd.	OnBalance	www.krysalis.co.uk
Catalyze Ltd.	Hiview	www.catalyze.co.uk