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1 1 Slide Chapter 10 Multicriteria Decision Making n A Scoring Model for Job Selection n Spreadsheet Solution of the Job Selection Scoring Model n The Analytic Hierarchy Process n Establishing Priorities Using AHP n Using AHP to Develop an Overall Priority Ranking

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2 2 Slide Multicriteria Decision Problems n In this chapter we consider decision problems involving two or more criteria. n These criteria are often in conflict as the decision maker attempts to identify the best decision alternative. n A trade-off among the multiple criteria is often necessary. n Management science techniques can be used to: quantify the importance of each criterion,quantify the importance of each criterion, evaluate how each decision alternative contributes to each criterion, andevaluate how each decision alternative contributes to each criterion, and identify the decision alternative that comes closest to meeting the multiple criteria.identify the decision alternative that comes closest to meeting the multiple criteria.

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3 3 Slide A Scoring Model for Job Selection n A graduating college student with a double major in Finance and Accounting has received the following three job offers: financial analyst for an investment firm in Chicagofinancial analyst for an investment firm in Chicago accountant for a manufacturing firm in Denveraccountant for a manufacturing firm in Denver auditor for a CPA firm in Houstonauditor for a CPA firm in Houston

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4 4 Slide A Scoring Model for Job Selection n The student made the following comments: “The financial analyst position provides the best opportunity for my long-run career advancement.”“The financial analyst position provides the best opportunity for my long-run career advancement.” “I would prefer living in Denver rather than in Chicago or Houston.”“I would prefer living in Denver rather than in Chicago or Houston.” “I like the management style and philosophy at the Houston CPA firm the best.”“I like the management style and philosophy at the Houston CPA firm the best.” n Clearly, this is a multicriteria decision problem.

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5 5 Slide A Scoring Model for Job Selection n Considering only the long-run career advancement criterion, the financial analyst position in Chicago is the best decision alternative. n Considering only the location criterion, the accountant position in Denver is the best decision alternative. n Considering only the style criterion, the auditor position in Houston is the best alternative.

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6 6 Slide A Scoring Model for Job Selection n Steps Required to Develop a Scoring Model Step 1: List the decision-making criteria.Step 1: List the decision-making criteria. Step 2: Assign a weight to each criterion.Step 2: Assign a weight to each criterion. Step 3: Rate how well each decision alternative satisfies each criterion.Step 3: Rate how well each decision alternative satisfies each criterion. Step 4: Compute the score for each decision alternative.Step 4: Compute the score for each decision alternative. Step 5: Order the decision alternatives from highest score to lowest score. The alternative with the highest score is the recommended alternative.Step 5: Order the decision alternatives from highest score to lowest score. The alternative with the highest score is the recommended alternative.

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7 7 Slide A Scoring Model for Job Selection n Mathematical Model S j = w i r ij S j = w i r ijiwhere: r ij = rating for criterion i and decision alternative j S j = score for decision alternative j

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8 8 Slide A Scoring Model for Job Selection n Step 1: List the criteria (important factors). Career advancementCareer advancement LocationLocation ManagementManagement SalarySalary PrestigePrestige Job SecurityJob Security Enjoyable workEnjoyable work

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9 9 Slide A Scoring Model for Job Selection n Five-Point Scale Chosen for Step 2 Importance Weight Importance Weight Very unimportant1 Very unimportant1 Somewhat unimportant2 Somewhat unimportant2 Average importance3 Average importance3 Somewhat important4 Somewhat important4 Very important5 Very important5

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10 Slide A Scoring Model for Job Selection n Step 2: Assign a weight to each criterion. Criterion Importance Weight Career advancementVery important5 LocationAverage importance3 ManagementSomewhat important4 SalaryAverage importance3 PrestigeSomewhat unimportant2 Job securitySomewhat important4 Enjoyable workVery important5

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11 Slide A Scoring Model for Job Selection n Nine-Point Scale Chosen for Step 3 Level of Satisfaction Rating Level of Satisfaction Rating Extremely low1 Extremely low1 Very low2 Very low2 Low3 Low3 Slightly low4 Slightly low4 Average5 Average5 Slightly high6 Slightly high6 High7 High7 Very high8 Very high8 Extremely high9 Extremely high9

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12 Slide A Scoring Model for Job Selection n Step 3: Rate how well each decision alternative satisfies each criterion. Decision Alternative Decision Alternative Analyst Accountant Auditor Analyst Accountant Auditor Criterion Chicago Denver Houston Criterion Chicago Denver Houston Career advancement864 Location387 Management569 Salary675 Prestige754 Job security476 Enjoyable work865

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13 Slide A Scoring Model for Job Selection n Step 4: Compute the score for each decision alternative. Decision Alternative 1 - Analyst in Chicago Decision Alternative 1 - Analyst in Chicago Criterion Weight ( w i ) Rating ( r i 1 ) w i r i 1 Criterion Weight ( w i ) Rating ( r i 1 ) w i r i 1 Career advancement 5 x 8 =40 Location 3 3 9 Management 4 520 Salary 3 618 Prestige 2 714 Job security 4 416 Enjoyable work 5 840 Score 157

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14 Slide A Scoring Model for Job Selection n Step 4: Compute the score for each decision alternative. S j = w i r ij S j = w i r ijI S 1 = 5(8)+3(3)+4(5)+3(6)+2(7)+4(4)+5(8) = 157 S 2 = 5(6)+3(8)+4(6)+3(7)+2(5)+4(7)+5(6) = 167 S 3 = 5(4)+3(7)+4(9)+3(5)+2(4)+4(6)+5(5) = 149

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15 Slide A Scoring Model for Job Selection n Step 4: Compute the score for each decision alternative. Decision Alternative Decision Alternative Analyst Accountant Auditor Analyst Accountant Auditor Criterion Chicago Denver Houston Criterion Chicago Denver Houston Career advancement403020 Location 92421 Management202436 Salary182115 Prestige1410 8 Job security162824 Enjoyable work403025 Score 157 167 149 Score 157 167 149

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16 Slide A Scoring Model for Job Selection n Step 5: Order the decision alternatives from highest score to lowest score. The alternative with the highest score is the recommended alternative. The accountant position in Denver has the highest score and is the recommended decision alternative.The accountant position in Denver has the highest score and is the recommended decision alternative. Note that the analyst position in Chicago ranks first in 4 of 7 criteria compared to only 2 of 7 for the accountant position in Denver.Note that the analyst position in Chicago ranks first in 4 of 7 criteria compared to only 2 of 7 for the accountant position in Denver. But when the weights of the criteria are considered, the Denver position is superior to the Chicago job.But when the weights of the criteria are considered, the Denver position is superior to the Chicago job.

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17 Slide A Scoring Model for Job Selection n Partial Spreadsheet Showing Steps 1 - 3

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18 Slide A Scoring Model for Job Selection n Partial Spreadsheet Showing Formulas for Step 4

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19 Slide A Scoring Model for Job Selection n Partial Spreadsheet Showing Results of Step 4

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20 Slide Analytic Hierarchy Process The Analytic Hierarchy Process (AHP), is a procedure designed to quantify managerial judgments of the relative importance of each of several conflicting criteria used in the decision making process. The Analytic Hierarchy Process (AHP), is a procedure designed to quantify managerial judgments of the relative importance of each of several conflicting criteria used in the decision making process.

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21 Slide Analytic Hierarchy Process n Step 1: List the Overall Goal, Criteria, and Decision Alternatives n Step 2: Develop a Pairwise Comparison Matrix Rate the relative importance between each pair of decision alternatives. The matrix lists the alternatives horizontally and vertically and has the numerical ratings comparing the horizontal (first) alternative with the vertical (second) alternative. Ratings are given as follows:... continued... continued For each criterion, perform steps 2 through 5

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22 Slide Analytic Hierarchy Process n Step 2: Pairwise Comparison Matrix (continued) Compared to the second alternative, the first alternative is: Numerical rating extremely preferred 9 extremely preferred 9 very strongly preferred 7 very strongly preferred 7 strongly preferred 5 strongly preferred 5 moderately preferred 3 moderately preferred 3 equally preferred 1 equally preferred 1 Intermediate numeric ratings of 8, 6, 4, 2 can be assigned. A reciprocal rating (i.e. 1/9, 1/8, etc.) is assigned when the second alternative is preferred to the first. The value of 1 is always assigned when comparing an alternative with itself.

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23 Slide Analytic Hierarchy Process n Step 3: Develop a Normalized Matrix Divide each number in a column of the pairwise comparison matrix by its column sum. n Step 4: Develop the Priority Vector Average each row of the normalized matrix. These row averages form the priority vector of alternative preferences with respect to the particular criterion. The values in this vector sum to 1.

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24 Slide Analytic Hierarchy Process n Step 5: Calculate a Consistency Ratio The consistency of the subjective input in the pairwise comparison matrix can be measured by calculating a consistency ratio. A consistency ratio of less than.1 is good. For ratios which are greater than.1, the subjective input should be re-evaluated. n Step 6: Develop a Priority Matrix After steps 2 through 5 has been performed for all criteria, the results of step 4 are summarized in a priority matrix by listing the decision alternatives horizontally and the criteria vertically. The column entries are the priority vectors for each criterion.

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25 Slide Analytic Hierarchy Process n Step 7: Develop a Criteria Pairwise Development Matrix This is done in the same manner as that used to construct alternative pairwise comparison matrices by using subjective ratings (step 2). Similarly, normalize the matrix (step 3) and develop a criteria priority vector (step 4). n Step 8: Develop an Overall Priority Vector Multiply the criteria priority vector (from step 7) by the priority matrix (from step 6).

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26 Slide Determining the Consistency Ratio n Step 1: For each row of the pairwise comparison matrix, determine a weighted sum by summing the multiples of the entries by the priority of its corresponding (column) alternative. Step 2: For each row, divide its weighted sum by the priority of its corresponding (row) alternative. n Step 3: Determine the average, max, of the results of step 2.

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27 Slide Determining the Consistency Ratio n Step 4: Compute the consistency index, CI, of the n alternatives by: CI = ( max - n )/( n - 1). n Step 5: Determine the random index, RI, as follows: Number of RandomNumber of Random Number of RandomNumber of Random Alternative ( n ) Index (RI) Alternative ( n ) Index (RI) Alternative ( n ) Index (RI) Alternative ( n ) Index (RI) 3 0.58 6 1.24 3 0.58 6 1.24 4 0.90 7 1.32 4 0.90 7 1.32 5 1.12 8 1.41 5 1.12 8 1.41 n Step 6: Determine the consistency ratio, CR, as follows: CR = CR/RI. CR = CR/RI.

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28 Slide Example: Gill Glass Designer Gill Glass must decide which of three manufacturers will develop his "signature" toothbrushes. Three factors seem important to Gill: (1) his costs; (2) reliability of the product; and, (3) delivery time of the orders. The three manufacturers are Cornell Industries, Brush Pik, and Picobuy. Cornell Industries will sell toothbrushes to Gill Glass for $100 per gross, Brush Pik for $80 per gross, and Picobuy for $144 per gross. Gill has decided that in terms of price, Brush Pik is moderately preferred to Cornell and very strongly preferred to Picobuy. In turn Cornell is strongly to very strongly preferred to Picobuy.

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29 Slide Example: Gill Glass n Hierarchy for the Manufacturer Selection Problem Select the Best Toothbrush Manufacturer CostReliabilityDelivery Time Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Overall Goal Criteria Decision Alternatives

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30 Slide Example: Gill Glass n Forming the Pairwise Comparison Matrix For Cost Since Brush Pik is moderately preferred to Cornell, Cornell's entry in the Brush Pik row is 3 and Brush Pik's entry in the Cornell row is 1/3.Since Brush Pik is moderately preferred to Cornell, Cornell's entry in the Brush Pik row is 3 and Brush Pik's entry in the Cornell row is 1/3. Since Brush Pik is very strongly preferred to Picobuy, Picobuy's entry in the Brush Pik row is 7 and Brush Pik's entry in the Picobuy row is 1/7.Since Brush Pik is very strongly preferred to Picobuy, Picobuy's entry in the Brush Pik row is 7 and Brush Pik's entry in the Picobuy row is 1/7. Since Cornell is strongly to very strongly preferred to Picobuy, Picobuy's entry in the Cornell row is 6 and Cornell's entry in the Picobuy row is 1/6.Since Cornell is strongly to very strongly preferred to Picobuy, Picobuy's entry in the Cornell row is 6 and Cornell's entry in the Picobuy row is 1/6.

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31 Slide Example: Gill Glass n Pairwise Comparison Matrix for Cost Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Cornell 1 1/3 6 Cornell 1 1/3 6 Brush Pik 3 1 7 Picobuy 1/6 1/7 1

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32 Slide Example: Gill Glass n Normalized Matrix for Cost Divide each entry in the pairwise comparison matrix by its corresponding column sum. For example, for Cornell the column sum = 1 + 3 + 1/6 = 25/6. This gives: Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Cornell 6/25 7/31 6/14 Cornell 6/25 7/31 6/14 Brush Pik 18/25 21/31 7/14 Picobuy 1/25 3/31 1/14

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33 Slide Example: Gill Glass n Priority Vector For Cost The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cornell: ( 6/25 + 7/31 + 6/14)/3 =.298 Cornell: ( 6/25 + 7/31 + 6/14)/3 =.298 Brush Pik: (18/25 + 21/31 + 7/14)/3 =.632 Brush Pik: (18/25 + 21/31 + 7/14)/3 =.632 Picobuy: ( 1/25 + 3/31 + 1/14)/3 =.069 Picobuy: ( 1/25 + 3/31 + 1/14)/3 =.069

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34 Slide Example: Gill Glass n Checking Consistency Multiply each column of the pairwise comparison matrix by its priority:Multiply each column of the pairwise comparison matrix by its priority: 1 1/3 6.923 1 1/3 6.923.298 3 +.632 1 +.069 7 = 2.009.298 3 +.632 1 +.069 7 = 2.009 1/6 1/7 1.209 1/6 1/7 1.209 Divide these number by their priorities to get:Divide these number by their priorities to get:.923/.298 = 3.097.923/.298 = 3.097 2.009/.632 = 3.179 2.009/.632 = 3.179.209/.069 = 3.029.209/.069 = 3.029

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35 Slide Example: Gill Glass n Checking Consistency Average the above results to get max.Average the above results to get max. max = (3.097 + 3.179 + 3.029)/3 = 3.102 max = (3.097 + 3.179 + 3.029)/3 = 3.102 Compute the consistence index, CI, for two terms by:Compute the consistence index, CI, for two terms by: CI = ( max - n )/( n - 1) = (3.102 - 3)/2 =.051 CI = ( max - n )/( n - 1) = (3.102 - 3)/2 =.051 Compute the consistency ratio, CR, by CI/RI, where RI =.58 for 3 factors:Compute the consistency ratio, CR, by CI/RI, where RI =.58 for 3 factors: CR = CI/RI =.051/.58 =.088 CR = CI/RI =.051/.58 =.088 Since the consistency ratio, CR, is less than.10, this is well within the acceptable range for consistency.

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36 Slide Example: Gill Glass Gill Glass has determined that for reliability, Cornell is very strongly preferable to Brush Pik and equally preferable to Picobuy. Also, Picobuy is strongly preferable to Brush Pik.

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37 Slide Example: Gill Glass n Pairwise Comparison Matrix for Reliability Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Cornell 1 7 2 Cornell 1 7 2 Brush Pik 1/7 1 5 Picobuy 1/2 1/5 1

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38 Slide Example: Gill Glass n Normalized Matrix for Reliability Divide each entry in the pairwise comparison matrix by its corresponding column sum. For example, for Cornell the column sum = 1 + 1/7 + 1/2 = 23/14. This gives: Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Cornell 14/23 35/41 2/8 Cornell 14/23 35/41 2/8 Brush Pik 2/23 5/41 5/8 Picobuy 7/23 1/41 1/8

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39 Slide Example: Gill Glass n Priority Vector For Reliability The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cornell: (14/23 + 35/41 + 2/8)/3 =.571 Cornell: (14/23 + 35/41 + 2/8)/3 =.571 Brush Pik: ( 2/23 + 5/41 + 5/8)/3 =.278 Brush Pik: ( 2/23 + 5/41 + 5/8)/3 =.278 Picobuy: ( 7/23 + 1/41 + 1/8)/3 =.151 Picobuy: ( 7/23 + 1/41 + 1/8)/3 =.151 n Checking Consistency Gill Glass’ responses to reliability could be checked for consistency in the same manner as was cost.

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40 Slide Example: Gill Glass Gill Glass has determined that for delivery time, Cornell is equally preferable to Picobuy. Both Cornell and Picobuy are very strongly to extremely preferable to Brush Pik.

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41 Slide Example: Gill Glass n Pairwise Comparison Matrix for Delivery Time Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Cornell 1 8 1 Cornell 1 8 1 Brush Pik 1/8 1 1/8 Picobuy 1 8 1

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42 Slide Example: Gill Glass n Normalized Matrix for Delivery Time Divide each entry in the pairwise comparison matrix by its corresponding column sum. Cornell Brush Pik Picobuy Cornell Brush Pik Picobuy Cornell 8/17 8/17 8/17 Cornell 8/17 8/17 8/17 Brush Pik 1/17 1/17 1/17 Picobuy 8/17 8/17 8/17

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43 Slide Example: Gill Glass n Priority Vector For Delivery Time The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cornell: (8/17 + 8/17 + 8/17)/3 =.471 Cornell: (8/17 + 8/17 + 8/17)/3 =.471 Brush Pik: (1/17 + 1/17 + 1/17)/3 =.059 Brush Pik: (1/17 + 1/17 + 1/17)/3 =.059 Picobuy: (8/17 + 8/17 + 8/17)/3 =.471 Picobuy: (8/17 + 8/17 + 8/17)/3 =.471 n Checking Consistency Gill Glass’ responses to delivery time could be checked for consistency in the same manner as was cost.

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44 Slide Example: Gill Glass The accounting department has determined that in terms of criteria, cost is extremely preferable to delivery time and very strongly preferable to reliability, and that reliability is very strongly preferable to delivery time.

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45 Slide Example: Gill Glass n Pairwise Comparison Matrix for Criteria Cost Reliability Delivery Cost Reliability Delivery Cost 1 7 9 Cost 1 7 9 Reliability 1/7 1 7 Delivery 1/9 1/7 1

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46 Slide Example: Gill Glass n Normalized Matrix for Criteria Divide each entry in the pairwise comparison matrix by its corresponding column sum. Cost Reliability Delivery Cost Reliability Delivery Cost 63/79 49/57 9/17 Reliability 9/79 7/57 7/17 Delivery 7/79 1/57 1/17

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47 Slide Example: Gill Glass n Priority Vector For Criteria The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cost: (63/79 + 49/57 + 9/17)/3 =.729 Cost: (63/79 + 49/57 + 9/17)/3 =.729 Reliability: ( 9/79 + 7/57 + 7/17)/3 =.216 Reliability: ( 9/79 + 7/57 + 7/17)/3 =.216 Delivery: ( 7/79 + 1/57 + 1/17)/3 =.055 Delivery: ( 7/79 + 1/57 + 1/17)/3 =.055

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48 Slide Example: Gill Glass n Overall Priority Vector The overall priorities are determined by multiplying the priority vector of the criteria by the priorities for each decision alternative for each objective. Priority Vector Priority Vector for Criteria [.729.216.055 ] for Criteria [.729.216.055 ] Cost Reliability Delivery Cost Reliability Delivery Cornell.298.571.471 Brush Pik.632.278.059 Picobuy.069.151.471

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49 Slide Example: Gill Glass n Overall Priority Vector (continued) Thus, the overall priority vector is: Cornell:(.729)(.298) + (.216)(.571) + (.055)(.471) =.366 Brush Pik: (.729)(.632) + (.216)(.278) + (.055)(.059) =.524 Picobuy: (.729)(.069) + (.216)(.151) + (.055)(.471) =.109 Brush Pik appears to be the overall recommendation.

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50 Slide Example: Gill Glass n Spreadsheet Layout Pairwise Compar. Normalized Wgt. Consist. Matrices Matrices Priority Sum Meas. Matrices Matrices Priority Sum Meas. Prior. Summary Overall Prior. Prior. Summary Overall Prior. Criteria (3x3) Cost (3x3) Reliability (3x3) Deliv. Time (3x3) Criteria (3x3) (3x3) Cost (3x3) Reliability (3x3) Deliv. Time (3x3) (3x1) Consist. Ratio (3x1)

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