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**The Analytic Hierarchy Process**

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**The analytic hierarchy process (AHP), developed by Thomas L**

The analytic hierarchy process (AHP), developed by Thomas L. Saaty is designed to solve complex problems involving multiple criteria. The process requires the decision maker to provide judgments about the relative importance of each of the criteria and then to specify a preference for each decision alternative on each criterion. The output of the AHP is a prioritized ranking indicating the overall preference for each of the decision alternatives. In order to introduce the AHP, we consider the problem faced by Dave Payne. Dave is planning to purchase a new car. Alter a preliminary analysis of the makes and models available, Dave has narrowed the list of decision alternatives to three cars, which we will refer to as car A, car B, and car C. A summary of the information Dave has collected about the cars has been provided.

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Car A Car B Car C Price $13,100 $11,200 $9,500 MPG 18 23 29 Interior Deluxe Above average Standard Body 4-door midsize 2-door sport 2-door compact Radio AM/FM, tape AM/FM AM Engine 6-cylinder 4-cylinder turbo 4-cylinder Based on the information in the table - as well as his own personal feelings resulting from driving each car—Dave decided that there were several criteria that he needed to consider in making the purchase decision. After some thought, he selected purchase price , miles per gallon ( MPG ), comfort, and style as the four criteria to be considered. Quantitative data regarding the purchase price and MPG criteria are provided directly in the table.

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**However, measures of comfort and style cannot be specified so easily**

However, measures of comfort and style cannot be specified so easily. Dave will need to consider factors such as car interior, type of radio, ease of entry and exit, seat-adjustment features etc., in order to determine the comfort level for each car. The style criterion will need to be measured in terms of Dave’s subjective evaluation of each car. Even when we deal with a criterion as easily measured as purchase price, however, subjectivity becomes an issue whenever a particular decision maker indicates his or her personal preferences. For instance, car A costs $3600 more than car C; this difference might represent a great deal of money to one person, but not very much money to another person. Thus, whether car A is considered extremely more expensive than car C or only moderately more expensive than car C is a subjective judgment that will depend primarily on the financial status of the person making the comparison. AHP’s advantage is that it can handle situations in which the subjective judgments of individuals constitute an important part of the decision process

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**Developing the Hierarchy**

The first step in the AHP is to develop a graphical representation of the problem in terms of the overall goal, the criteria, and the decision alternatives. Such a graph depicts the hierarchy for the problem. The following figure shows the hierarchy for the car-selection problem.

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Note that the first level of the hierarchy shows that the overall goal is to select the best car. At the second level, we see that the four criteria (purchase price, MPG, comfort, and style) will contribute to the achievement of the overall goal. Finally, at the third level we see that each decision alternative (car A, car B, and ear C) can contribute to each criterion in a unique way. The approach AHP takes is to have the decision maker specify his or her judgments about the relative importance of each criterion in terms of its contribution to the achievement of the overall goal. At the next level, the AHP asks the decision maker to indicate a preference or priority for each decision alternative in terms of how it contributes to each criterion. For example, in the car-selection problem, Dave will need to specify his judgment about the relative importance of each of the four criteria. He will also need to indicate his preference for each of the three cars relative to each criterion. Given information on relative importance and preferences, a mathematical process is used to synthesize the information and provide priority ranking of the three cars in terms of their overall preference.

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**ESTABLISHING PRIORITIES USING THE AHP**

In this section we will show how the AHP utilizes pair wise comparisons to establish priority measures for both the criteria and the decision alternatives. The sets of priorities that need to be determined in the car-selection problem are as follows: The priorities of the four criteria in terms of the overall goal The priorities of the three cars in terms of the purchase-price criterion The priorities of the three cars iii terms of the MPG criterion The priorities of the three cars in terms of the comfort criterion The priorities of the three cars in terms of the style criterion In the following discussion we will demonstrate how to establish priorities for the three cars in terms of the comfort criterion. The other sets of priorities can be determined in a similar fashion.

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**Pair-wise Comparisons**

Pair-wise comparisons are fundamental building blocks of the AHP. In establishing the priorities for the three cars in terms of comfort, we will ask Dave to state a preference for the comfort of the cars when the cars are considered two at a time (pair wise). That is Dave will be asked to compare the comfort of car A to car B, car A to car C, and car B to car C in three separate comparisons. The AHP employs an underlying scale with values from 1 to 9 to rate the relative preferences for two items. Researchers and experience have confirmed the 9-unit scale as a reasonable basis for discriminating between the preferences for two items.

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**Verbal Judgment of Preference**

Numerical Rating Extremely preferred 9 Very strongly to extremely 8 Very strongly preferred 7 Strongly to very strongly 6 Strongly preferred 5 Moderately to strongly 4 Moderately preferred 3 Equally to moderately 2 Equally preferred 1

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In the car-selection example, suppose that Dave has compared the comforts of car A with those of car B and is convinced that car A is more comfortable. Dave is then asked to state his preference for the comfort of car A compared to that of car B using one of the verbal descriptions shown in earlier table. If he believes that car A is moderately preferred to car B, a value of 3 is utilized in the AHP; if he believes that car A is strongly preferred, a value of 5 is utilized; if he believes that car A is very strongly preferred, a value of 7 is utilized; if he believes that car A is extremely preferred, a value of 9 is utilized. Values of 2, 4, 6, and 8 are the intermediate values for the scale. A value of 1 is reserved for the case where the two items are judged to be equally preferred.

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Suppose that when asked his preference between cars A and B with respect to the comfort criterion, Dave states that car A is between equally and moderately more preferred than car B; the numerical measure that reflects this judgment is 2. Dave is then asked to provide his preference between car A and car C. Suppose in this case he states that car A is very strongly to extremely more preferred than car C; this corresponds to a numerical rating of 8. Finally, Dave is asked to state his preference for car B compared to car C. Suppose in this case he indicates that car B is strongly to very strongly preferred to car C; the AHP would assign a numerical rating of 6

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**The Pairwise Comparison Matrix**

In order to develop the priorities for the three cars in terms of the comfort criterion, we need to develop a matrix of the pairwise comparison ratings. Since three cars are being considered, the pairwise comparison matrix will consist of three rows and three columns. As shown below: Comfort Car A Car B Car C 2 8 6 Note: In the pairwise comparison matrix, the value in row i and column j is the measure of preference of the car in row i when compared to the car in column j.

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We see that the value in the matrix that corresponds to comparing car A with car B is 2, the value that corresponds to comparing car A with cur C’ is 8 and the value that corresponds to comparing car B with car C is 6. In order to determine the remaining entries in the pairwise comparison matrix, first note that when we compare any car against itself , the judgment must be that they are equally preferred. Hence, the AHP assigns a 1 to all elements on the diagonal of the pairwise comparison matrix. Given these entries, all that remains is to determine the rating for car B compared to car A, car C compared to car A, and car C compared to car B. Obviously, we could follow the same procedure and ask Dave to provide his preferences for these pairwise comparisons. However, since we already know that Dave has rated his preference for car A compared to car B as 2, there is no need for him to make another pairwise comparison with these two cars.

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In fact, we will conclude that the preference rating for car B when compared to car A is simply the reciprocal of the preference rating for car A when compared to car B: 1/2. Using this logic, the AHP obtains the preference rating of car B compared to car A by computing the reciprocal of the rating of car A compared to car B. Using this inverse, or reciprocal, relationship, we find that the rating of car C compared to car A is ¼ and the rating of car C compared to car B is ¼. Using these numerical values of preference, the complete pairwise comparison matrix for the comfort criterion is shown in completed table.

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Comfort Car A Car B Car C 1 2 8 1/2 6 1/8 1/6

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Synthesis Once the matrix of pairwise comparisons has been developed, we can calculate what is called the priority of each of the elements being compared. For example, we would now like to use the pairwise comparison information to estimate the relative priority for each of the cars in terms of the comfort criterion. This part of the AHP is referred to as synthesization. The exact mathematical procedure required to perform this synthesization involves the computation of eigenvalues and eigenvectors and is beyond the scope of this text. However, the following three-step procedure provides a good approximation of the synthesized priorities.

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**Procedure for Synthesizing Judgments**

Step 1: Sum the values in each column of the pairwise comparison matrix. Step 2: Divide each element in the pairwise comparison matrix by its column total; the resulting matrix is referred to as the normalized pairwise column. Step 3: Compute the average of the elements in each row of the normalized matrix; these averages provide an estimate of the relative priorities of the elements being compared. To see how the synthesization process works for our example problem, we carry out the procedure using the pairwise comparison matrix shown in table.

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**Comfort Car A Car B Car C 1 2 8 1/2 6 1/8 1/6 Column totals 13/8 19/6**

Step 1: Sum the values in each column. Comfort Car A Car B Car C 1 2 8 1/2 6 1/8 1/6 Column totals 13/8 19/6 15

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**Comfort Car A Car B Car C 8/13 12/19 8/15 4/13 6/19 6/15 1/13 1/19**

Step 2: Divide each element of the matrix by its column total. Comfort Car A Car B Car C 8/13 12/19 8/15 4/13 6/19 6/15 1/13 1/19 1/15 Note that all columns in the normalized pairwise comparison matrix now have a sum of 1.

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**Step 3: Average the elements in each row**

Step 3: Average the elements in each row. (The values in the normalized pairwise comparison matrix have been converted to decimal form.) Comfort Car A Car B Car C 0.615 0.632 0.533 0.593 0.508 0.316 0.400 0.341 0.677 0.053 0.067 0.066 Total 1.000 This synthesis provides the relative priorities for the three cars with respect to the comfort criterion. Thus, we see that, considering comfort, the must preferred car is car A (with a priority of 0.593). Car B (with a priority of 0.341) is second, followed by car C (with a priority of 0.066).

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Consistency A key step in the AHP is the establishment of priorities through the use of the pairwise comparison procedure. An important consideration in terms of the quality of the ultimate decision relates to the consistency of judgments that the decision maker demonstrated during the series of pairwise comparisons. For example, consider a situation involving the comparison of three job offers with respect to the salary criterion. Suppose that the following pairwise comparison matrix was developed. Salary Job 1 Job 2 Job 3 1 2 8 1/2 3 1/8 1/3

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The interpretation of the preference scores is that the preference for job 1 is twice the preference for job 2, and the preference for job 2 is three limes the preference for job 3. Using these two pieces of information, we would logically concIude that the preference for job 1 should be 2 x 3 = 6 times the preference for job 3. The fact that the pairwise comparison matrix showed a preference of instead of 6 indicates that some lack of consistency exists in the pairwise comparisons. However, it has to be realized that perfect consistency is very difficult to achieve and that some lack of consistency is expected to exist in almost any set of pairwise comparisons. To handle the consistency question, the AHP provides a method for measuring the degree of consistency among the pairwise judgments provided by the decision maker, If the degree of consistency is acceptable, the decision process can continue. However, if the degree of consistency is unacceptable, the decision maker should reconsider and possibly revise the pairwise comparison judgments before proceeding with the analysis.

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The AHP provides a measure of the consistency of pairwise comparison judgments by computing a consistency ratio. This ratio is designed in such a way that values of the ratio exceeding 0.10 are indicative of inconsistent judgments; in such cases the decision maker would probably want to reconsider and revise the original values in the pairwise comparison matrix. Values of the consistently ratio of 0.10 or less are considered to indicate a reasonable level of consistency in the pairwise comparisons. Although the exact mathematical computation of the consistency ratio is beyond the scope of this text, an approximation of the ratio can be obtained. We will illustrate this computational procedure for the car-selection problem by considering Dave’s pairwise comparison for the comfort criterion.

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**Estimating the Consistency Ratio**

Step 1: Multiply each value in the first column of the pairwise comparison matrix by the relative priority of the first item considered; multiply each value in the second column of the matrix by the relative priority of the second item considered; multiply each value in the third column of the matrix by the relative priority of the third item considered. Sum the values across the rows to obtain a vector of values labeled weighted sum. This computation for the car-selection example is: Weighted Sum Vector

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Step 2: Divide the elements of the vector of weighted sums obtained in 1 by the corresponding priority value. For the car-selection example, we obtain:

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Step 3: Compute the average of the values computed in step 2; this average is denoted by λmax. For the car-selection example, we obtain Step 4: Compute the consistency index (CI), which is defined us follows: Where n= the number of items being compared. For the car-selection example with n = 3, we obtain

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**Step 5: Compute the consistency ratio (CR), which is defined as follows**

where RI, the random index, is the consistency index of a randomly generated pairwise comparison matrix, It can be shown that RI depends on the number of elements being compared and takes on the following values: n 3 4 5 6 7 8 RI 0.58 0.9 1.12 1.24 1.32 1.41 Thus, for our car- example with n = 3 and RI = 0.5 we obtain the following consistency ratio

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**Other Pairwise Comparisons for the Car-Selection Example**

In continuing with the AHP analysis of the car-selection problem, we need to use the pairwise comparison procedure to determine the priorities of the three cars in terms of the purchase price, MPG, and style criteria. This requires that Dave express pairwise comparison preferences for the cars, considering each of these criteria one at a time. Dave’s preferences are summarized in the pairwise comparison matrices shown. Price Car A Car B Car C 1 1/3 1/4 3 1/2 4 2 MPG Car A Car B Car C 1 1/4 1/6 4 1/3 6 3

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Style Car A Car B Car C 1 1/3 4 3 7 1/7 The interpretation of the numerical values in the earlier tables is the same as the interrelation of the preference values we observed for the comfort criterion. For example, consider the comparison of car A and car B in terms of the purchase price criterion. Car B ($11,200) is considered more preferable than car A ($13,100). In fact, the pairwise comparison matrix shows Dave’s preference for car B is three times greater than his preference for car A in terms of purchase price. Similarly, car A is only ¼ as preferred as car B. Recall that the pairwise comparison matrix is set up to show the preference of the item in row i when compared to the item in column j

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Following the same synthesis procedure that we used for the comfort criterion, the priority vectors for these criteria can be computed. The result of this synthesis is shown below. Price MPG Style In interpreting these priorities we see that car C is the most preferable in terms of purchase price (0.557) and miles per gallon (0.639). Car B is the most preferable in terms of style (0.655). No car is the most preferred with respect to all criteria. Thus, before a final decision can be made, we must assess the relative importance of the criteria.

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In addition to the pairwise comparisons for the decision alternatives, we must use the same pairwise comparison procedure to set priorities for all four criteria in terms of the importance of each in contributing toward the overall goal of selecting time best car. To develop this final pairwise comparison matrix, Dave would have to specify how important he thought each criterion was compared to each of the other criteria. In order to do this, six pairwise judgments have to be made: purchase price compared to MPG; purchase compared to comfort; purchase price compared to style; MPG compared to comfort; MPG compared to style; and comfort compared to style. For example, in the pairwise comparison of the purchase price and MPG criteria, Dave indicated that purchase price was moderately more important than MPG. Using the AHP 9-point numerical rating scale, a value of 3 was recorded to show the higher importance of the purchase-price criterion.

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The summary of the pairwise comparison matrix preferences for the four criteria is shown in table below. Criterion Price MPG Comfort Style 1 3 2 1/3 4

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The synthesization process described earlier in this section can now be used to convert the pairwise comparison information into the priorities for the four criteria. The results obtained are as follows Criteria Priorities Price 0.398 MPG 0.085 Comfort 0.218 Style 0.299 We see that the purchase price (0.398) has been identified as the highest-priority or most important criterion in the car-selection decision. Style (0.299) and comfort (0.218) rank next in importance. MPG (0.085) is a relatively unimportant criterion in terms of the overall goal of selecting the best car.

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**Using The AHP To Develop An Overall Priority Ranking**

A matrix that summarizes the priorities for each car in terms of each criterion is given below. This matrix is referred to as the priority matrix. Price MPG Comfort Style Car A 0.123 0.087 0.593 0.265 Car B 0.320 0.274 0.341 0.655 Car C 0.557 0.639 0.066 0.080 The overall priority for each decision alternative is obtained by summing the product of the criterion priority times the priority of the decision alternative with respect to that criterion. Recall that the criterion priorities were found to be for purchase price, for MPG, for comfort, and for style.

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**Thus, the computation of the overall priority for car A is as follows:**

Overall car A priority = 0.398(0.123) (0.087) (0.593) (0.265) = 0.265 Repeating this calculation for cars B and C provides their overall priorities as follows: Overall car B priority = 0.398(0.320) (0.274) (0.341) (0.655) = 0.421 Overall car C priority = 0.398(0.557) (0.639) (0.066) (0.080) = 0.314

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**Ranking these priority values, we have the following AHP ranking of the decision alternatives:**

Car B 0.421 Car C 0.314 Car A 0.265 Total 1.000 These results provide a basis for Dave to make a decision regarding the purchase of a car. Based on the AHP priorities, Dave should select car B. If Dave believes that the judgments that he has made regarding the importance of the criteria and his preferences for the cars in terms of the criteria are valid, then the AHP priorities show that car B is the preferred car.

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**USING EXPERT CHOICE TO IMPLEMENT THE AHP**

Expert Choice (EC), a software package marketed by Decision Support Software, provides a user-friendly procedure for implementing the AHP on a microcomputer. We now provide an introduction to this software package by showing how it cart be used to compute the priorities for the car-selection problem. Expert Choice enables the user to simply construct a graphical representation of the hierarchy. For example, to create the hierarchy for the car-selection example, the user selects the option to develop a new application; what appears on the computer’s monitor is a request to define the overall goal. After the user defines the overall goal, a rectangular box, or node, appears on the screen, with the goal description written directly above it.

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The user selects the EDIT command and then the INSERT option; another rectangular box or node appears below the goal node, and the user now types the name of a criterion, which will be entered inside the box. This process continues until all four criterion nodes have been specified. The figure given shows the partial hierarchy appearing on the computer screen after the four criteria have been specified.

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In the figure we see that in addition to the names of each criterion , the criterion nodes also contain the decimal value of This value represents the initial weight, or priority, given to each criterion at the start of the EC session. The user can now continue the process of using the EDIT command with the INSERT option to define the decision alternative nodes associated with each of the criterion nodes. In the following figure we show the result of defining the decision alternative nodes for the price criterion; now that since there are three alternatives, the initial priorities are set at

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**A similar set of decision alternatives is then identified for each of the other three criteria.**

Once the user has developed the complete hierarchy for the problem, he or she can focus on any particular part of time hierarchy through time use of the REDRAW command.

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In fact, to show the detail displayed in the earlier figure, all we did was to point to the price node (using the arrows on the keyboard’s numeric key pad) and then type R for redraw. Our intent here is not to attempt to show you how to use EC but merely to let you develop some appreciation for the ease with which the analysis can be performed using this software package. Now that the hierarchy has been input to EC, we are ready to begin developing the pairwise comparisons needed to establish priorities for the decision alternatives. In order to illustrate the type of approach used, we moved back to the goal node with EC and then selected time COMPARE command by typing C. After selecting the option to make comparisons based on the importance of the decision criteria, the EC system begins to go through the pairwise comparison analysis.

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One portion of this analysis, which shows the approach used by EC to establish the comparative importance between the purchase price and MPG criteria, is shown in following figure. Note that this figure indicates to time EC system that price is moderately more important than MPG. This process continues until all the entries in the pairwise comparison matrix for criteria have been developed. The synthesization process is then performed to compute the priorities for the criteria. The process of entering pairwise preferences for the cars relative to each of the criteria was then performed in a similar manner. The overall decision was then arrived at by entering the command S which is an abbreviation for synthesizing; this command is used only when we have entered all the data for the pairwise comparison matrices and want to obtain an overall prioritization of the decision alternatives.

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**The priorities that were obtained after synthesization**

The following figure shows the results obtained. Note that the results indicate that the final priority for car B, the most preferable, is The EC system is a very helpful software package in performing the multiple-criteria decision analysis of the AHP. In addition to providing the overall priorities for the decision alternatives, EC has the capability of doing “what if” types of analyses, where the decision maker can begin to learn how the overall priorities for the decision alternative are affected by changes in the preference input data.

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