Anderson Sweeney Williams QUANTITATIVE METHODS FOR BUSINESS 8

Anderson Sweeney Williams QUANTITATIVE METHODS FOR BUSINESS 8 e Slides Prepared by JOHN LOUCKS © 2001 South-Western College Publishing/Thomson Learning Slide 1

Chapter 18 Multicriteria Decision Problems n n n Goal Programming: Formulation and Graphical Solution Scoring Models The Analytic Hierarchy Process Establishing Priorities Using AHP to Develop an Overall Priority Ranking Slide 2

Goal Programming n n Goal programming may be used to solve linear programs with multiple objectives, with each objective viewed as a "goal". In goal programming, di+ and di- , deviation variables, are the amounts a targeted goal i is overachieved or underachieved, respectively. The goals themselves are added to the constraint set with di+ and di- acting as the surplus and slack variables. One approach to goal programming is to satisfy goals in a priority sequence. Second-priority goals are pursued without reducing the first-priority goals, etc. Slide 3

Goal Programming n n For each priority level, the objective function is to minimize the (weighted) sum of the goal deviations. Previous "optimal" achievements of goals are added to the constraint set so that they are not degraded while trying to achieve lesser priority goals. Slide 4

Goal Programming Approach n n Step 1: Decide the priority level of each goal. Step 2: Decide the weight on each goal. If a priority level has more than one goal, for each goal i decide the weight, wi , to be placed on the deviation(s), di+ and/or di-, from the goal. Step 3: Set up the initial linear program. Min w 1 d 1+ + w 2 d 2 s. t. Functional Constraints, and Goal Constraints Step 4: Solve this linear program. If there is a lower priority level, go to step 5. Otherwise, a final optimal solution has been reached. Slide 5

Goal Programming Approach n Step 5: Set up the new linear program. Consider the next-lower priority level goals and formulate a new objective function based on these goals. Add a constraint requiring the achievement of the nexthigher priority level goals to be maintained. The new linear program might be: Min w 3 d 3+ + w 4 d 4 s. t. Functional Constraints, Goal Constraints, and w 1 d 1+ + w 2 d 2 - = k Go to step 4. (Repeat steps 4 and 5 until all priority levels have been examined. ) Slide 6

Example: Conceptual Products is a computer company that produces the CP 400 and the CP 500 computers. The computers use different mother boards produced in abundant supply by the company, but use the same cases and disk drives. The CP 400 models use two floppy disk drives and no zip disk drives whereas the CP 500 models use one floppy disk drive and one zip disk drive. Slide 7

Example: Conceptual Products The disk drives and cases are bought from vendors. There are 1000 floppy disk drives, 500 zip disk drives, and 600 cases available to Conceptual Products on a weekly basis. It takes one hour to manufacture a CP 400 and its profit is $200 and it takes one and one-half hours to manufacture a CP 500 and its profit is $500. Slide 8

Example: Conceptual Products The company has four goals which are given below: Priority 1: Meet a state contract of 200 CP 400 machines weekly. (Goal 1) Priority 2: Make at least 500 total computers weekly. (Goal 2) Priority 3: Make at least $250, 000 weekly. (Goal 3) Priority 4: Use no more than 400 man-hours per week. (Goal 4) Slide 9

Example: Conceptual Products Variables x 1 = number of CP 400 computers produced weekly x 2 = number of CP 500 computers produced weekly di- = amount the right hand side of goal i is deficient di+ = amount the right hand side of goal i is exceeded n Functional Constraints Availability of floppy disk drives: 2 x 1 + x 2 < 1000 Availability of zip disk drives: x 2 < 500 Availability of cases: x 1 + x 2 < 600 n Slide 10

Example: Conceptual Products n Goals (1) 200 CP 400 computers weekly: x 1 + d 1 - - d 1+ = 200 (2) 500 total computers weekly: x 1 + x 2 + d 2 - - d 2+ = 500 (3) $250(in thousands) profit: . 2 x 1 +. 5 x 2 + d 3 - - d 3+ = 250 (4) 400 total man-hours weekly: x 1 + 1. 5 x 2 + d 4 - - d 4+ = 400 Non-negativity: x 1, x 2, di-, di+ > 0 for all i Slide 11

Example: Conceptual Products n Objective Functions Priority 1: Minimize the amount the state contract is not met: Min d 1 Priority 2: Minimize the number under 500 computers produced weekly: Min d 2 Priority 3: Minimize the amount under $250, 000 earned weekly: Min d 3 Priority 4: Minimize the man-hours over 400 used weekly: Min d 4+ Slide 12

Example: Conceptual Products n Formulation Summary Min P 1(d 1 -) + P 2(d 2 -) + P 3(d 3 -) + P 4(d 4+) s. t. 1000 600 200 500 250 2 x 1 +x 2 x 1 < < 500 < +d 1 - -d 1+ +x 2 . 2 x 1+. 5 x 2 = +d 2 - -d 2+ = +d 3 - -d 3+ = - + Slide 13

Example: Conceptual Products n Graphical Solution, Iteration 1 To solve graphically, first graph the functional constraints. Then graph the first goal: x 1 = 200. Note on the next slide that there is a set of points that exceed x 1 = 200 (where d 1 - = 0). Slide 14

Example: Conceptual Products n Functional Constraints and Goal 1 Graphed x 2 1000 2 x 1 + x 2 < 1000 800 Goal 1: x 1 > 200 x 2 < 500 x 1 + x 2 < 600 400 Points Satisfying Goal 1 200 400 600 800 1000 1200 x 1 Slide 15

Example: Conceptual Products n Graphical Solution, Iteration 2 Now add Goal 1 as x 1 > 200 and graph Goal 2: x 1 + x 2 = 500. Note on the next slide that there is still a set of points satisfying the first goal that also satisfies this second goal (where d 2 - = 0). Slide 16

Example: Conceptual Products n Goal 1 (Constraint) and Goal 2 Graphed x 2 1000 2 x 1 + x 2 < 1000 800 Goal 1: x 1 > 200 x 2 < 500 x 1 + x 2 < 600 400 Points Satisfying Both Goals 1 and 2 200 Goal 2: x 1 + x 2 > 500 200 400 600 800 1000 1200 x 1 Slide 17

Example: Conceptual Products n Graphical Solution, Iteration 3 Now add Goal 2 as x 1 + x 2 > 500 and Goal 3: . 2 x 1 +. 5 x 2 = 250. Note on the next slide that no points satisfy the previous functional constraints and goals and satisfy this constraint. Thus, to Min d 3 -, this minimum value is achieved when we Max. 2 x 1 +. 5 x 2. Note that this occurs at x 1 = 200 and x 2 = 400, so that. 2 x 1 +. 5 x 2 = 240 or d 3 - = 10. Slide 18

Example: Conceptual Products n Goal 2 (Constraint) and Goal 3 Graphed x 2 1000 2 x 1 + x 2 < 1000 800 Goal 1: x 1 > 200 x 2 < 500 x 1 + x 2 < 600 (200, 400) 400 Points Satisfying Both Goals 1 and 2 200 Goal 2: x 1 + x 2 > 500 200 400 600 800 1000 Goal 3: . 2 x 1 +. 5 x 2 = 250 x 1 1200 Slide 19

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 Chicago • accountant for a manufacturing firm in Denver • auditor for a CPA firm in Houston Slide 20

A Scoring Model for Job Selection n n The student made the following comments: • “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 like the management style and philosophy at the Houston CPA firm the best. ” Clearly, this is a multicriteria decision problem. Slide 21

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. Considering only the location criterion • the accountant position in Denver is the best decision alternative. Considering only the style criterion • the auditor position in Houston is the best alternative. Slide 22

A Scoring Model for Job Selection n Steps Required to Develop a Scoring Model • Step 1: List the decision-making criteria. • Step 2: Assign a weight to each criterion. • Step 3: Rate how well each decision alternative satisfies each criterion. • 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. Slide 23

A Scoring Model for Job Selection n Mathematical Model Sj = S wi rij i where: rij = rating for criterion i and decision alternative j Sj = score for decision alternative j Slide 24

A Scoring Model for Job Selection n Step 1: List the criteria (important factors). • Career advancement • Location • Management • Salary • Prestige • Job Security • Enjoyable work Slide 25

A Scoring Model for Job Selection n Five-Point Scale Chosen for Step 2 Importance Very unimportant Somewhat unimportant Average importance Somewhat important Very important Weight 1 2 3 4 5 Slide 26

A Scoring Model for Job Selection n Step 2: Assign a weight to each criterion. Criterion Importance Weight Career advancement Very important 5 Location Average importance 3 Management Somewhat important 4 Salary Average importance 3 Prestige Somewhat unimportant 2 Job security Somewhat important 4 Enjoyable work Very important 5 Slide 27

A Scoring Model for Job Selection n Nine-Point Scale Chosen for Step 3 Level of Satisfaction Rating Extremely low 1 Very low 2 Low 3 Slightly low 4 Average 5 Slightly high 6 High 7 Very high 8 Extremely high 9 Slide 28

A Scoring Model for Job Selection Step 3: Rate how well each decision alternative satisfies each criterion. Decision Alternative Analyst Accountant Auditor Criterion Chicago Denver Houston Career advancement 8 6 4 Location 3 8 7 Management 5 6 9 Salary 6 7 5 Prestige 7 5 4 Job security 4 7 6 Enjoyable work 8 6 5 n Slide 29

A Scoring Model for Job Selection n Step 4: Compute the score for each decision alternative. Decision Alternative 1 - Analyst in Chicago Criterion Weight (wi ) Rating (ri 1) w i ri 1 Career advancement 5 x 8 = 40 Location 3 3 9 Management 4 5 20 Salary 3 6 18 Prestige 2 7 14 Job security 4 4 16 Enjoyable work 5 8 40 Score 157 Slide 30

A Scoring Model for Job Selection n Step 4: Compute the score for each decision alternative. Sj = S wi rij i 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 Slide 31

A Scoring Model for Job Selection Step 4: Compute the score for each decision alternative. Decision Alternative Analyst Accountant Auditor Criterion Chicago Denver Houston Career advancement 40 30 20 Location 9 24 21 Management 20 24 36 Salary 18 21 15 Prestige 14 10 8 Job security 16 28 24 Enjoyable work 40 30 25 Score 157 167 149 n Slide 32

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. • 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. Slide 33

A Scoring Model for Job Selection n Partial Spreadsheet Showing Steps 1 - 3 Slide 34

A Scoring Model for Job Selection n Partial Spreadsheet Showing Formulas for Step 4 Slide 35

A Scoring Model for Job Selection n Partial Spreadsheet Showing Results of Step 4 Slide 36

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. Slide 37

Analytic Hierarchy Process n Step 1: List the Overall Goal, Criteria, and Decision Alternatives ------- For each criterion, perform steps 2 through 5 ------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 Slide 38

Analytic Hierarchy Process n Step 2: Pairwise Comparison Matrix (continued) Compared to the second alternative, the first alternative is: extremely preferred very strongly preferred moderately preferred equally preferred Numerical rating 9 7 5 3 1 Slide 39

Analytic Hierarchy Process n Step 2: Pairwise Comparison Matrix (continued) 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. Slide 40

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. Slide 41

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. Slide 42

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). Slide 43

Determining the Consistency Ratio n n 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. Step 3: Determine the average, max, of the results of step 2. Slide 44

Determining the Consistency Ratio n n n Step 4: Compute the consistency index, CI, of the n alternatives by: CI = ( max - n)/(n - 1). Step 5: Determine the random index, RI, as follows: Number of Random Alternative (n) Index (RI) 3 0. 58 6 1. 24 4 0. 90 7 1. 32 5 1. 12 8 1. 41 Step 6: Compute the consistency ratio: CR = CR/RI. Slide 45

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. Slide 46

Example: Gill Glass n Hierarchy for the Manufacturer Selection Problem Overall Goal Select the Best Toothbrush Manufacturer Criteria Cost Cornell Decision Brush Pik Alternatives Picobuy Reliability Delivery Time Cornell Brush Pik Picobuy Slide 47

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 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. Slide 48

Example: Gill Glass n Pairwise Comparison Matrix for Cost Cornell Brush Pik Picobuy Cornell 1 Brush Pik Picobuy 1/3 3 1/6 6 1 1/7 7 1 Slide 49

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 6/25 7/31 6/14 Brush Pik 18/25 21/31 Picobuy 1/25 3/31 7/14 1/14 Slide 50

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 Brush Pik: (18/25 + 21/31 + 7/14)/3 =. 632 Picobuy: ( 1/25 + 3/31 + 1/14)/3 =. 069 Slide 51

Example: Gill Glass n Checking Consistency • Multiply each column of the pairwise comparison matrix by its priority: 1 1/3 6. 923. 298 3 +. 632 1 +. 069 7 = 2. 009 1/6 1/7 1. 209 • Divide these number by their priorities to get: . 923/. 298 = 3. 097 2. 009/. 632 = 3. 179. 209/. 069 = 3. 029 Slide 52

Example: Gill Glass n Checking Consistency • Average the above results to get max = (3. 097 + 3. 179 + 3. 029)/3 = 3. 102 • Compute the consistence index, CI, for two terms by: CI = ( max - n)/(n - 1) = (3. 102 - 3)/2 =. 051 • Compute the consistency ratio, CR, by CI/RI, where RI =. 58 for 3 factors: CR = CI/RI =. 051/. 58 =. 088 Since the consistency ratio, CR, is less than. 10, this is well within the acceptable range for consistency. Slide 53

Example: 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. Slide 54

Example: Gill Glass n Pairwise Comparison Matrix for Reliability Cornell Brush Pik Picobuy Cornell 1 Brush Pik Picobuy 7 1/2 2 1 1/5 5 1 Slide 55

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 14/23 Brush Pik Picobuy 35/41 2/23 7/23 2/8 5/41 1/41 5/8 1/8 Slide 56

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 Brush Pik: ( 2/23 + 5/41 + 5/8)/3 Picobuy: ( 7/23 + 1/41 + 1/8)/3 n = = = . 571. 278. 151 Checking Consistency Gill Glass’ responses to reliability could be checked for consistency in the same manner as was cost. Slide 57

Example: 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. Slide 58

Example: Gill Glass n Pairwise Comparison Matrix for Delivery Time Cornell Brush Pik Picobuy Cornell 1 Brush Pik Picobuy 8 1/8 1 1 1 8 1/8 1 Slide 59

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 8/17 Brush Pik Picobuy 8/17 1/17 8/17 Slide 60

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)/3 Brush Pik: (1/17 + 1/17)/3 Picobuy: (8/17 + 8/17)/3 n = = = . 471. 059. 471 Checking Consistency Gill Glass’ responses to delivery time could be checked for consistency in the same manner as was cost. Slide 61

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. Slide 62

Example: Gill Glass n Pairwise Comparison Matrix for Criteria Cost Reliability Delivery 1 1/7 1/9 7 1 1/7 9 7 1 Slide 63

Example: Gill Glass n Normalized Matrix for Criteria Divide each entry in the pairwise comparison matrix by its corresponding column sum. Cost Reliability Delivery 63/79 9/79 7/79 49/57 7/57 1/57 9/17 7/17 1/17 Slide 64

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 Reliability: ( 9/79 + 7/57 + 7/17)/3 =. 216 Delivery: ( 7/79 + 1/57 + 1/17)/3 =. 055 Slide 65

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 for Criteria [. 729. 216. 055 ] Cost Reliability Delivery Cornell. 298. 571. 471 Brush Pik. 632. 278. 059 Picobuy. 069. 151. 471 Slide 66

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. Slide 67

The End of Chapter 18 Slide 68