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Introduction to Management Science 8th Edition by Bernard W. Taylor III Chapter 14 Multicriteria Decision Making Chapter 14 - 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 Chapter 14 - Multicriteria Decision Making 2 Overview Study of problems with several criteria, multiple criteria, instead of a single objective when making a decision. Two techniques discussed: goal programming, and the analytical hierarchy process. 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. Chapter 14 - Multicriteria Decision Making 3 Goal Programming Model Formulation (1 of 2) Beaver Creek Pottery Company Example: Maximize Z = $40x1 + 50x2 subject to: 1x1 + 2x2 40 hours of labor 4x2 + 3x2 120 pounds of clay x1, x2 0 Where: x1 = number of bowls produced x2 = number of mugs produced Chapter 14 - Multicriteria Decision Making 4 Goal Programming Model Formulation (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 14 - 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 goals in the order of the goal priorities. Chapter 14 - Multicriteria Decision Making 6 Goal Programming Goal Constraints and Objective Function (1 of 2) Labor goals constraint (1, less than 40 hours labor; 4, minimum overtime): Minimize P1d1-, P4d1+ Add profit goal constraint (2, achieve profit of $1,600): Minimize P1d1-, P2d2-, P4d1+ Add material goal constraint (3, avoid keeping more than 120 pounds of clay on hand): Minimize P1d1-, P2d2-, P3d3+, P4d1+ Chapter 14 - Multicriteria Decision Making 7 Goal Programming Goal Constraints and Objective Function (2 of 2) Complete Goal Programming Model: 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 Chapter 14 - Multicriteria Decision Making 8 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 14 - Multicriteria Decision Making 9 Goal Programming Alternative Forms of Goal Constraints (2 of 2) Complete Model with 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 14 - Multicriteria Decision Making 10 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 14.1 Goal Constraints Chapter 14 - Multicriteria Decision Making 11 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 14.2 The First-Priority Goal: Minimize Chapter 14 - Multicriteria Decision Making 12 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 14.3 The Second-Priority Goal: Minimize Chapter 14 - Multicriteria Decision Making 13 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 14.4 The Third-Priority Goal: Minimize Chapter 14 - Multicriteria Decision Making 14 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 14.5 The Fourth-Priority Goal: Minimize Chapter 14 - Multicriteria Decision Making 15 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 x1 = 15 bowls x2 = 20 mugs d1- = 15 hours Chapter 14 - Multicriteria Decision Making 16 Goal Programming Computer Solution Using QM for Windows (1 of 3) 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 Exhibit 14.1 Chapter 14 - Multicriteria Decision Making 17 Goal Programming Computer Solution Using QM for Windows (2 of 3) Exhibit 14.2 Chapter 14 - Multicriteria Decision Making 18 Goal Programming Computer Solution Using QM for Windows (3 of 3) Exhibit 14.3 Chapter 14 - Multicriteria Decision Making 19 Goal Programming Computer Solution Using Excel (1 of 3) Exhibit 14.4 Chapter 14 - Multicriteria Decision Making 20 Goal Programming Computer Solution Using Excel (2 of 3) Exhibit 14.5 Chapter 14 - Multicriteria Decision Making 21 Goal Programming Computer Solution Using Excel (3 of 3) Exhibit 14.6 Chapter 14 - Multicriteria Decision Making 22 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 14 - Multicriteria Decision Making 23 Goal Programming Solution for Altered Problem Using Excel (2 of 6) Exhibit 14.7 Chapter 14 - Multicriteria Decision Making 24 Goal Programming Solution for Altered Problem Using Excel (3 of 6) Exhibit 14.8 Chapter 14 - Multicriteria Decision Making 25 Goal Programming Solution for Altered Problem Using Excel (4 of 6) Exhibit 14.9 Chapter 14 - Multicriteria Decision Making 26 Goal Programming Solution for Altered Problem Using Excel (5 of 6) Exhibit 14.10 Chapter 14 - Multicriteria Decision Making 27 Goal Programming Solution for Altered Problem Using Excel (6 of 6) Exhibit 14.11 Chapter 14 - Multicriteria Decision Making 28 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 14 - Multicriteria Decision Making 29 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 14 - Multicriteria Decision Making 30 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 14 - Multicriteria Decision Making 31 Analytical Hierarchy Process General Mathematical Process Mathematically determine preferences for each site for each criteria. Mathematically determine preferences for criteria (rank order of importance). Combine these two sets of preferences to mathematically derive a score for each site. Select the site with the highest score. Chapter 14 - Multicriteria Decision Making 32 Analytical Hierarchy Process Pair-wise Comparisons In a pair-wise 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 14 - Multicriteria Decision Making 33 Analytical Hierarchy Process Pair-wise Comparisons (2 of 2) Table 14.1 Preference Scale for Pair-wise Comparisons Chapter 14 - Multicriteria Decision Making 34 Analytical Hierarchy Process Pair-wise Comparison Matrix A pair-wise comparison matrix summarizes the pair-wise comparisons for a criteria. Customer Market A B C 1 3 2 1/3 1 1/5 1/2 5 1 Site A B C Income Level A B C 1 6 1/3 1/6 1 1/9 3 9 1 Chapter 14 - Multicriteria Decision Making Infrastructure 1 1/3 1 3 1 7 1 1/7 1 Transportation 1 1/3 1/2 3 1 2 1/4 4 1 35 Analytical Hierarchy Process Developing Preferences Within Criteria (1 of 3) In synthetization, decision alternatives are prioritized with each criterion and then normalized: Site A B C Site A B C Customer Market A B C 1 3 2 1/3 1 1/5 1/2 5 1 11/6 9 16/5 Customer Market A B C 6/11 3/9 5/8 2/11 1/9 1/16 3/11 5/9 5/16 Chapter 14 - Multicriteria Decision Making 36 Analytical Hierarchy Process Developing Preferences Within Criteria (2 of 3) Table 14.2 The Normalized Matrix with Row Averages Chapter 14 - Multicriteria Decision Making 37 Analytical Hierarchy Process Developing Preferences Within Criteria (3 of 3) Table 14.3 Criteria Preference Matrix Chapter 14 - Multicriteria Decision Making 38 Analytical Hierarchy Process Ranking the Criteria (1 of 2) Pair-wise Comparison Matrix: Criteria Market Income Infrastructure Transportation Market 1 5 1/3 1/4 Income Infrastructure Transportation 1/5 3 4 1 9 7 1/9 1 2 1/7 1/2 1 Table 14.4 Normalized Matrix for Criteria with Row Averages Chapter 14 - Multicriteria Decision Making 39 Analytical Hierarchy Process Ranking the Criteria (2 of 2) Preference Vector: Market Income Infrastructure Transportation Chapter 14 - Multicriteria Decision Making 0.1993 0.6535 0.0860 0.0612 40 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 14 - Multicriteria Decision Making Score 0.5314 0.3091 0.1595 1.0000 41 Analytical Hierarchy Process Summary of Mathematical Steps Develop a pair-wise comparison matrix for each decision alternative for each criteria. Synthetization Sum the values of each column of the pair-wise 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 pair-wise 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 14 - Multicriteria Decision Making 42 Goal Programming Excel Spreadsheets (1 of 4) Exhibit 14.12 Chapter 14 - Multicriteria Decision Making 43 Goal Programming Excel Spreadsheets (2 of 4) Exhibit 14.13 Chapter 14 - Multicriteria Decision Making 44 Goal Programming Excel Spreadsheets (3 of 4) Exhibit 14.14 Chapter 14 - Multicriteria Decision Making 45 Goal Programming Excel Spreadsheets (4 of 4) Exhibit 14.15 Chapter 14 - Multicriteria Decision Making 46 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 criteria 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 14 - Multicriteria Decision Making 47 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 Mall 3 90 65 79 80 50 Mall 4 60 90 85 90 70 S1 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80) = 62.75 S2 = (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30) = 73.50 S3 = (.30)(90) + (.25)(65) + (.25)(79) + (.10)(80) + (.10)(50) = 76.00 S4 = (.30)(60) + (.25)(90) + (.25)(85) + (.10)(90) + (.10)(70) = 77.75 Mall 4 preferred because of highest score, followed by malls 3, 2, 1. Chapter 14 - Multicriteria Decision Making 48 Scoring Model Excel Solution Exhibit 14.16 Chapter 14 - Multicriteria Decision Making 49 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 14 - Multicriteria Decision Making 50 Goal Programming Example Problem Solution (1 of 2) Step 1: Model Formulation: Minimize P1d1-, P2d2-, P3d3subject to: 80x1 + 40x2 + d1- - d1+ = 3,000 interviews 50x1 + 70x2 + d2- - d2 + = $2,500 budget 80x1 + d3- - d3 + = 1,000 telephone interviews where: x1 = number of telephone interviews x2 = number of personal interviews Chapter 14 - Multicriteria Decision Making 51 Goal Programming Example Problem Solution (2 of 2) Step 2: QM for Windows Solution: Chapter 14 - Multicriteria Decision Making 52 Analytical Hierarchy Process Example Problem Problem Statement Purchasing decision, three model alternatives, three decision criteria. Pair-wise 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 Chapter 14 - Multicriteria Decision Making Gears 3 1 1/2 Weight 5 2 1 53 Analytical Hierarchy Process Example Problem Problem Solution (1 of 4) Step 1: Develop normalized matrices and preference vectors for all the pair-wise comparison matrices for criteria. Price Bike X Y Z Bike X Y Z X 0.6667 0.2222 0.1111 Y 0.6667 0.2222 0.1111 Z 0.6667 0.2222 0.1111 Row Averages 0.6667 0.2222 0.1111 1.0000 X 0.0909 0.2727 0.6364 Gear Action Y 0.0625 0.1875 0.7500 Z 0.1026 0.1795 0.7179 Row Averages 0.0853 0.2132 0.7014 1.0000 Chapter 14 - Multicriteria Decision Making 54 Analytical Hierarchy Process Example Problem Problem Solution (2 of 4) Step 1 continued: Develop normalized matrices and preference vectors for all the pair-wise 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 Chapter 14 - Multicriteria Decision Making 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 55 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 Chapter 14 - Multicriteria Decision Making Weight 0.6250 0.2500 0.1250 Row Averages 0.6479 0.2299 0.1222 1.0000 0.6479 0.2299 0.1222 56 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 14 - Multicriteria Decision Making 57 Chapter 14 - Multicriteria Decision Making 58