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Introduction to Management Science 9th Edition by Bernard W. Taylor III Chapter 9 Multicriteria Decision Making © 2007 Pearson Education Chapter 9 - Multicriteria Decision Making 1 Chapter Topics Goal Programming Graphical Interpretation of Goal Programming Computer Solution of Goal Programming Problems with QM for Windows and Excel The Analytical Hierarchy Process Scoring Models Chapter 9 - Multicriteria Decision Making 2 Overview Study of problems with several criteria, multiple criteria, instead of a single objective when making a decision. Three techniques discussed: goal programming, the analytical hierarchy process and scoring models. Goal programming is a variation of linear programming considering more than one objective (goals) in the objective function. The analytical hierarchy process develops a score for each decision alternative based on comparisons of each under different criteria reflecting the decision makers preferences. Scoring models are based on a relatively simple weighted scoring technique. Chapter 9 - Multicriteria Decision Making 3 Goal Programming Example Problem Data (1 of 2) Beaver Creek Pottery Company Example: Maximize Z = $40x1 + 50x2 subject to: 1x1 + 2x2 40 hours of labor 4x1 + 3x2 120 pounds of clay x1, x2 0 Where: x1 = number of bowls produced x2 = number of mugs produced Chapter 9 - Multicriteria Decision Making 4 Goal Programming Example Problem Data (2 of 2) Adding objectives (goals) in order of importance, the company: Does not want to use fewer than 40 hours of labor per day. Would like to achieve a satisfactory profit level of $1,600 per day. Prefers not to keep more than 120 pounds of clay on hand each day. Would like to minimize the amount of overtime. Chapter 9 - Multicriteria Decision Making 5 Goal Programming Goal Constraint Requirements All goal constraints are equalities that include deviational variables d- and d+. A positive deviational variable (d+) is the amount by which a goal level is exceeded. A negative deviation variable (d-) is the amount by which a goal level is underachieved. At least one or both deviational variables in a goal constraint must equal zero. The objective function in a goal programming model seeks to minimize the deviation from the respective goals in the order of the goal priorities. Chapter 9 - Multicriteria Decision Making 6 Goal Programming Model Formulation Goal Constraints (1 of 3) Labor goal: x1 + 2x2 + d1- - d1+ = 40 (hours/day) Profit goal: 40x1 + 50 x2 + d2 - - d2 + = 1,600 ($/day) Material goal: 4x1 + 3x2 + d3 - - d3 + = 120 Chapter 9 - Multicriteria Decision Making (lbs of clay/day) 7 Goal Programming Model Formulation Objective Function (2 of 3) Labor goals constraint (priority 1 - less than 40 hours labor; priority 4 - minimum overtime): Minimize P1d1-, P4d1+ Add profit goal constraint (priority 2 - achieve profit of $1,600): Minimize P1d1-, P2d2-, P4d1+ Add material goal constraint (priority 3 - avoid keeping more than 120 pounds of clay on hand): Minimize P1d1-, P2d2-, P3d3+, P4d1+ Chapter 9 - Multicriteria Decision Making 8 Goal Programming Model Formulation Complete Model (3 of 3) Complete Goal Programming Model: Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to: x1 + 2x2 + d1- - d1+ = 40 (labor) 40x1 + 50 x2 + d2 - - d2 + = 1,600 (profit) 4x1 + 3x2 + d3 - - d3 + = 120 (clay) x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0 Chapter 9 - Multicriteria Decision Making 9 Goal Programming Alternative Forms of Goal Constraints (1 of 2) Changing fourth-priority goal “limits overtime to 10 hours” instead of minimizing overtime: d1- + d4 - - d4+ = 10 minimize P1d1 -, P2d2 -, P3d3 +, P4d4 + Addition of a fifth-priority goal- “important to achieve the goal for mugs”: x1 + d5 - = 30 bowls x2 + d6 - = 20 mugs minimize P1d1 -, P2d2 -, P3d3 +, P4d4 +, 4P5d5 - + 5P5d6 - Chapter 9 - Multicriteria Decision Making 10 Goal Programming Alternative Forms of Goal Constraints (2 of 2) Complete Model with Added New Goals: Minimize P1d1-, P2d2-, P3d3+, P4d4+, 4P5d5- + 5P5d6subject to: x1 + 2x2 + d1- - d1+ = 40 40x1 + 50x2 + d2- - d2+ = 1,600 4x1 + 3x2 + d3- - d3+ = 120 d1+ + d4- - d4+ = 10 x1 + d5- = 30 x2 + d6- = 20 x1, x2, d1-, d1+, d2-, d2+, d3-, d3+, d4-, d4+, d5-, d6- 0 Chapter 9 - Multicriteria Decision Making 11 Goal Programming Graphical Interpretation (1 of 6) Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to: x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0 Figure 9.1 Goal Constraints Chapter 9 - Multicriteria Decision Making 12 Goal Programming Graphical Interpretation (2 of 6) Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to: x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0 Figure 9.2 The First-Priority Goal: Minimize Chapter 9 - Multicriteria Decision Making 13 Goal Programming Graphical Interpretation (3 of 6) Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to: x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0 Figure 9.3 The Second-Priority Goal: Minimize Chapter 9 - Multicriteria Decision Making 14 Goal Programming Graphical Interpretation (4 of 6) Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to: x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0 Figure 9.4 The Third-Priority Goal: Minimize Chapter 9 - Multicriteria Decision Making 15 Goal Programming Graphical Interpretation (5 of 6) Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to: x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0 Figure 9.5 The Fourth-Priority Goal: Minimize Chapter 9 - Multicriteria Decision Making 16 Goal Programming Graphical Interpretation (6 of 6) Goal programming solutions do not always achieve all goals and they are not “optimal”, they achieve the best or most satisfactory solution possible. Minimize P1d1-, P2d2-, P3d3+, P4d1+ subject to: x1 + 2x2 + d1- - d1+ = 40 40x1 + 50 x2 + d2 - - d2 + = 1,600 4x1 + 3x2 + d3 - - d3 + = 120 x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0 Solution: x1 = 15 bowls x2 = 20 mugs d1- = 15 hours Chapter 9 - Multicriteria Decision Making 17 Goal Programming Computer Solution Using Excel (1 of 3) Exhibit 9.4 Chapter 9 - Multicriteria Decision Making 18 Goal Programming Computer Solution Using Excel (2 of 3) Exhibit 9.5 Chapter 9 - Multicriteria Decision Making 19 Goal Programming Computer Solution Using Excel (3 of 3) Exhibit 9.6 Chapter 9 - Multicriteria Decision Making 20 Goal Programming Solution for Altered Problem Using Excel (1 of 6) Minimize P1d1-, P2d2-, P3d3+, P4d4+, 4P5d5- + 5P5d6subject to: x1 + 2x2 + d1- - d1+ = 40 40x1 + 50x2 + d2- - d2+ = 1,600 4x1 + 3x2 + d3- - d3+ = 120 d1+ + d4- - d4+ = 10 x1 + d5- = 30 x2 + d6- = 20 x1, x2, d1-, d1+, d2-, d2+, d3-, d3+, d4-, d4+, d5-, d6- 0 Chapter 9 - Multicriteria Decision Making 21 Goal Programming Solution for Altered Problem Using Excel (2 of 6) Exhibit 9.7 Chapter 9 - Multicriteria Decision Making 22 Goal Programming Solution for Altered Problem Using Excel (3 of 6) Exhibit 9.8 Chapter 9 - Multicriteria Decision Making 23 Goal Programming Solution for Altered Problem Using Excel (4 of 6) Exhibit 9.9 Chapter 9 - Multicriteria Decision Making 24 Goal Programming Solution for Altered Problem Using Excel (5 of 6) Exhibit 9.10 Chapter 9 - Multicriteria Decision Making 25 Goal Programming Solution for Altered Problem Using Excel (6 of 6) Exhibit 9.11 Chapter 9 - Multicriteria Decision Making 26 Analytical Hierarchy Process Overview AHP is a method for ranking several decision alternatives and selecting the best one when the decision maker has multiple objectives, or criteria, on which to base the decision. The decision maker makes a decision based on how the alternatives compare according to several criteria. The decision maker will select the alternative that best meets his or her decision criteria. AHP is a process for developing a numerical score to rank each decision alternative based on how well the alternative meets the decision maker’s criteria. Chapter 9 - Multicriteria Decision Making 27 Analytical Hierarchy Process Example Problem Statement Southcorp Development Company shopping mall site selection. Three potential sites: Atlanta Birmingham Charlotte. Criteria for site comparisons: Customer market base. Income level Infrastructure Chapter 9 - Multicriteria Decision Making 28 Analytical Hierarchy Process Hierarchy Structure Top of the hierarchy: the objective (select the best site). Second level: how the four criteria contribute to the objective. Third level: how each of the three alternatives contributes to each of the four criteria. Chapter 9 - Multicriteria Decision Making 29 Analytical Hierarchy Process General Mathematical Process Mathematically determine preferences for sites with respect to each criterion. Mathematically determine preferences for criteria (rank order of importance). Combine these two sets of preferences to mathematically derive a composite score for each site. Select the site with the highest score. Chapter 9 - Multicriteria Decision Making 30 Analytical Hierarchy Process Pairwise Comparisons (1 of 2) In a pairwise comparison, two alternatives are compared according to a criterion and one is preferred. A preference scale assigns numerical values to different levels of performance. Chapter 9 - Multicriteria Decision Making 31 Analytical Hierarchy Process Pairwise Comparisons (2 of 2) Table 9.1 Preference Scale for Pairwise Comparisons Chapter 9 - Multicriteria Decision Making 32 Analytical Hierarchy Process Pairwise Comparison Matrix A pairwise comparison matrix summarizes the pairwise comparisons for a criteria. Customer Market A B 1 3 1/3 1 1/2 5 Site A B C Income Level A B C 1 6 1/3 1/6 1 1/9 3 9 1 Infrastructure 1 1/3 1 3 1 7 1 1/7 1 Chapter 9 - Multicriteria Decision Making C 2 1/5 1 Transportation 1 1/3 1/2 3 1 2 1/4 4 1 33 Analytical Hierarchy Process Developing Preferences Within Criteria (1 of 3) In synthesization, decision alternatives are prioritized with each criterion and then normalized: Customer Market Site A B C A 1 3 2 B 1/3 1 1/5 C 1/2 5 1 11/6 9 16/5 Site A B C A 6/11 2/11 3/11 Customer Market B C 3/9 5/8 1/9 1/16 5/9 5/16 Chapter 9 - Multicriteria Decision Making 34 Analytical Hierarchy Process Developing Preferences Within Criteria (2 of 3) The row average values represent the preference vector Table 9.2 The Normalized Matrix with Row Averages Chapter 9 - Multicriteria Decision Making 35 Analytical Hierarchy Process Developing Preferences Within Criteria (3 of 3) Preference vectors for other criteria are computed similarly, resulting in the preference matrix Table 9.3 Criteria Preference Matrix Chapter 9 - Multicriteria Decision Making 36 Analytical Hierarchy Process Ranking the Criteria (1 of 2) Pairwise Comparison Matrix: Criteria Market Income Infrastructure Transportation Market 1 1/5 3 4 Income 5 1 9 7 Infrastructure 1/3 1/9 1 2 Transportation 1/4 1/7 1/2 1 Table 9.4 Normalized Matrix for Criteria with Row Averages Chapter 9 - Multicriteria Decision Making 37 Analytical Hierarchy Process Ranking the Criteria (2 of 2) Preference Vector for Criteria: Market Income Infrastructure Transportation 0.1993 0.6535 0.0860 0.0612 Chapter 9 - Multicriteria Decision Making 38 Analytical Hierarchy Process Developing an Overall Ranking Overall Score: Site A score = .1993(.5012) + .6535(.2819) + .0860(.1790) + .0612(.1561) = .3091 Site B score = .1993(.1185) + .6535(.0598) + .0860(.6850) + .0612(.6196) = .1595 Site C score = .1993(.3803) + .6535(.6583) + .0860(.1360) + .0612(.2243) = .5314 Overall Ranking: Site Charlotte Atlanta Birmingham Chapter 9 - Multicriteria Decision Making Score 0.5314 0.3091 0.1595 1.0000 39 Analytical Hierarchy Process Summary of Mathematical Steps Develop a pairwise comparison matrix for each decision alternative for each criteria. Synthesization Sum the values of each column of the pairwise comparison matrices. Divide each value in each column by the corresponding column sum. Average the values in each row of the normalized matrices. Combine the vectors of preferences for each criterion. Develop a pairwise comparison matrix for the criteria. Compute the normalized matrix. Develop the preference vector. Compute an overall score for each decision alternative Rank the decision alternatives. Chapter 9 - Multicriteria Decision Making 40 Analytical Hierarchy Process: Consistency (1 of 3) Consistency Index (CI): Check for consistency and validity of multiple pairwise comparisons Example: Southcorp’s consistency in the pairwise comparisons of the 4 site selection criteria Step 1: Multiply the pairwise comparison matrix of the 4 criteria by its preference vector Market Income Infrastruc. Transp. Criteria Market 1 1/5 3 4 0.1993 Income 5 1 9 7 X 0.6535 Infrastructure 1/3 1/9 1 2 0.0860 Transportation 1/4 1/7 1/2 1 0.0612 (1)(.1993)+(1/5)(.6535)+(3)(.0860)+(4)(.0612) = 0.8328 (5)(.1993)+(1)(.6535)+(9)(.0860)+(7)(.0612) = 2.8524 (1/3)(.1993)+(1/9)(.6535)+(1)(.0860)+(2)(.0612) = 0.3474 (1/4)(.1993)+(1/7)(.6535)+(1/2)(.0860)+(1)(.0612) = 0.2473 Chapter 9 - Multicriteria Decision Making 41 Analytical Hierarchy Process: Consistency (2 of 3) Step 2: Divide each value by the corresponding weight from the preference vector and compute the average 0.8328/0.1993 = 4.1786 2.8524/0.6535 = 4.3648 0.3474/0.0860 = 4.0401 0.2473/0.0612 = 4.0422 16.257 Average = 16.257/4 = 4.1564 Step 3: Calculate the Consistency Index (CI) CI = (Average – n)/(n-1), where n is no. of items compared CI = (4.1564-4)/(4-1) = 0.0521 (CI = 0 indicates perfect consistency) Chapter 9 - Multicriteria Decision Making 42 Analytical Hierarchy Process: Consistency (3 of 3) Step 4: Compute the Ratio CI/RI where RI is a random index value obtained from Table 9.5 Table 9.5 Random Index Values for n Items Being Compared CI/RI = 0.0521/0.90 = 0.0580 Note: Degree of consistency is satisfactory if CI/RI < 0.10 Chapter 9 - Multicriteria Decision Making 43 Analytical Hierarchy Process Excel Spreadsheets (1 of 4) Exhibit 9.12 Chapter 9 - Multicriteria Decision Making 44 Analytical Hierarchy Process Excel Spreadsheets (2 of 4) Exhibit 9.13 Chapter 9 - Multicriteria Decision Making 45 Analytical Hierarchy Process Excel Spreadsheets (3 of 4) Exhibit 9.14 Chapter 9 - Multicriteria Decision Making 46 Analytical Hierarchy Process Excel Spreadsheets (4 of 4) Exhibit 9.15 Chapter 9 - Multicriteria Decision Making 47 Scoring Model Overview Each decision alternative graded in terms of how well it satisfies the criterion according to following formula: Si = gijwj where: wj = a weight between 0 and 1.00 assigned to criterion j; 1.00 important, 0 unimportant; sum of total weights equals one. gij = a grade between 0 and 100 indicating how well alternative i satisfies criteria j; 100 indicates high satisfaction, 0 low satisfaction. Chapter 9 - Multicriteria Decision Making 48 Scoring Model Example Problem Mall selection with four alternatives and five criteria: Grades for Alternative (0 to 100) Weight Decision Criteria (0 to 1.00) School proximity 0.30 Median income 0.25 Vehicular traffic 0.25 Mall quality, size 0.10 Other shopping 0.10 Mall 1 40 75 60 90 80 Mall 2 60 80 90 100 30 S1 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80) S2 = (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30) S3 = (.30)(90) + (.25)(65) + (.25)(79) + (.10)(80) + (.10)(50) S4 = (.30)(60) + (.25)(90) + (.25)(85) + (.10)(90) + (.10)(70) Mall 3 90 65 79 80 50 = = = = Mall 4 60 90 85 90 70 62.75 73.50 76.00 77.75 Mall 4 preferred because of highest score, followed by malls 3, 2, 1. Chapter 9 - Multicriteria Decision Making 49 Scoring Model Excel Solution Exhibit 9.16 Chapter 9 - Multicriteria Decision Making 50 Goal Programming Example Problem Problem Statement Public relations firm survey interviewer staffing requirements determination. One person can conduct 80 telephone interviews or 40 personal interviews per day. $50/ day for telephone interviewer; $70 for personal interviewer. Goals (in priority order): At least 3,000 total interviews. Interviewer conducts only one type of interview each day. Maintain daily budget of $2,500. At least 1,000 interviews should be by telephone. Formulate a goal programming model to determine number of interviewers to hire in order to satisfy the goals, and then solve the problem. Chapter 9 - Multicriteria Decision Making 51 Analytical Hierarchy Process Example Problem Problem Statement Purchasing decision, three model alternatives, three decision criteria. Pairwise comparison matrices: Bike X X 1 Y 1/3 Z 1/6 Price Y 3 1 1/2 Z 6 2 1 Bike X Y Z Gear Action X Y Z 1 1/3 1/7 3 1 1/4 7 4 1 Weight/Durability Bike X Y Z X 1 3 1 Y 1/3 1 1/2 Z 1 2 1 Prioritized decision criteria: Criteria Price Gears Weight Price 1 1/3 1/5 Gears 3 1 1/2 Chapter 9 - Multicriteria Decision Making Weight 5 2 1 52 Analytical Hierarchy Process Example Problem Problem Solution (1 of 4) Step 1: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Price Bike X Y Z Bike X Y Z X 0.6667 0.2222 0.1111 X 0.0909 0.2727 0.6364 Y 0.6667 0.2222 0.1111 Gear Action Y 0.0625 0.1875 0.7500 Z 0.6667 0.2222 0.1111 Z 0.1026 0.1795 0.7179 Chapter 9 - Multicriteria Decision Making Row Averages 0.6667 0.2222 0.1111 1.0000 Row Averages 0.0853 0.2132 0.7014 1.0000 53 Analytical Hierarchy Process Example Problem Problem Solution (2 of 4) Step 1 continued: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Bike X Y Z Weight/Durability X Y Z 0.4286 0.5000 0.4000 0.1429 0.1667 0.2000 0.4286 0.3333 0.4000 Bike X Y Z Price 0.6667 0.2222 0.1111 Criteria Gears 0.0853 0.2132 0.7014 Row Averages 0.4429 0.1698 0.3873 1.0000 Weight 0.4429 0.1698 0.3873 Chapter 9 - Multicriteria Decision Making 54 Analytical Hierarchy Process Example Problem Problem Solution (3 of 4) Step 2: Rank the criteria. Criteria Price Gears Weight Price 0.6522 0.2174 0.1304 Gears 0.6667 0.2222 0.1111 Price Gears Weight Weight 0.6250 0.2500 0.1250 Row Averages 0.6479 0.2299 0.1222 1.0000 0.6479 0.2299 0.1222 Chapter 9 - Multicriteria Decision Making 55 Analytical Hierarchy Process Example Problem Problem Solution (4 of 4) Step 3: Develop an overall ranking. Bike X Bike Y Bike Z 0.6667 0.0853 0.4429 0.6479 0.2222 0.2132 0.1698 0.2299 0.1111 0.7014 0.3837 0.1222 Bike X score = .6667(.6479) + .0853(.2299) + .4429(.1222) = .5057 Bike Y score = .2222(.6479) + .2132(.2299) + .1698(.1222) = .2138 Bike Z score = .1111(.6479) + .7014(.2299) + .3873(.1222) = .2806 Overall ranking of bikes: X first followed by Z and Y (sum of scores equal 1.0000). Chapter 9 - Multicriteria Decision Making 56 End of chapter Chapter 9 - Multicriteria Decision Making 57