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 inclass
examples or homework problems. In this paper, we present the
details of these exercises and an example of an AHP analysis.
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 commonknowledge base of an MBA. When faced with a multicriteria 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 multicriteria decision analysis problems. The AHP is a decisionaid 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 22.5 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 19 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:
Our experience has shown that the AHP is a winning topic for MBA
students. The MBA students like the AHP, they easily learn how to
use the AHP and, in many cases, they get very enthusiastic about
the AHP. We often have to "reinin" the students because they
get so excited about the material. AHP should be a required topic
for any introductory MBA course in decision making.

In this section, six exercises (called EX1EX6) that can be used in class problems or as homework problems on the AHP are presented.
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).
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 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 19 scale (or equivalent verbal scale) and the student has to decide how the $15,000 compares to the $18,000. Using the 19 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 19 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 19 scale does a very good job in capturing the preferences (e.g., High/Low = 7, High/Good = 5, High/Fair = 3, and so on).
EX2 and EX3 are takehome 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 01 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 
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 01 optimization (knapsack) problem that selects the best subset of projects.
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.
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 codenamed 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 SoSo.
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 twoconstraint 01 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 
SOSO 
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 
SOSO 
80 
12 
P11 
HIGH 
FAIR 
YES 
115 
14 
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.
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
daughterinlaw. Also, the President's daughterinlaw 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 P1P11 are selected. P12 has a low cost, but uses
a lot of programmer hours (it uses lowlevel programmers who
are at the lowend 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 
SOSO 
30 
10 
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:
 The American public believes that the existing tax structure is too complex.
 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. 
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. 
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.
a. 
The student should develop 13 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. 
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. 18005973028. 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 telephoneno internet!! 1 877shop880. Is there a special code to dial before using? Prepay a specified amount?
 Net2phone: 4.9c/call in US and Canada. 18004388735. 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 connectionno 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? 18777676569.

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 easytouse, 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 dialup service for long phone calls.
 A calling card service that gives an inexpensive but convenient way to make long distance calls away from home.
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 offpeak.
 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 offpeak.
 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 offpeak.
 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 offpeak.
 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.
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 810 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. 
The student should write a 510 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.
The following exercise demonstrates that the weights generated
by the AHP, using subjective judgments and the 19 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 twolevel 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
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 decisionmaker in the model and what
are the personal beliefs of the decisionmaker. As an
illustration, in this example, the pairwise comparisons
(as well as the criteria and subcriteria) can differ,
depending upon whether the decisionmaker 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. 
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 3.4Figure
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 (AmenitiesComfort) + .006 (AmenitiesPrestige) +.002 (AmenitiesStyle) 
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
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.
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 2229 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 criterionsubcriterion, AmenitiesPrestige
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 criterionsubcriterion, AmenitiesComfort, 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 criterionsubcriterion, AmenitiesStyle, 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.
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.
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 Email 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 multicriteria decisionmaking 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 decisionaiding 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.
Bodin, L. and S.
Gass (2003), "On Teaching the Analytic Hierarchy Process,"
Computers and Operations Research, Vol. 30, No. 10,
pp. 14871498.
Saaty, T. L. (1980),
The Analytic Hierarchy Process, McGrawHill, 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.
