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## Chapter 9 Multicriteria Decision Making

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**Introduction to Management Science**8th Edition by Bernard W. Taylor III Chapter 9 Multicriteria Decision Making Chapter 9 - Multicriteria Decision Making**Chapter Topics**• Goal Programming • Graphical Interpretation of Goal Programming • Computer Solution of Goal Programming Problems with QM for Windows and Excel • Overview • Study of problems with several criteria, multiple criteria, instead of asingle objective when making a decision. • Goal programming is a variation of linear programming considering more than one objective (goals) in the objective function. Chapter 9 - Multicriteria Decision Making**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 9 - Multicriteria Decision Making**Goal Programming**Model Formulation (2 of 2) • Adding objectives (goals) in order of importance (i.e. priorities), thecompany: • 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**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 9 - Multicriteria Decision Making**Goal Programming: Goal Constraints (1 of 3)**x1 + 2x2 = 40 - d1- + d1+ 40x1 + 50 x2 = 1,600 - d2- + d2+ 4x1 + 3x2 = 120 - d3- + d3+ x1, x2, d1 -, d1 +, d2 -, d2 +, d3 -, d3 + 0 Chapter 9 - Multicriteria Decision Making**Goal Programming: Objective Function (2 of 3)**• Let Pi= Priority i, where i = 1, 2, 3, and 4. • 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 9 - Multicriteria Decision Making**Goal Programming**Goal Constraints and Objective Function (3 of 3) 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 9 - Multicriteria Decision Making**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- due to limited warehouse space, cannot produce more than 30 bowls and 20 mugs daily. • x1 + d5 - = 30 bowls • x2 + d6 - = 20 mugs • minimize P1d1 -, P2d2 -, P3d3 -, P4d4 -, 4P5d5 -, 5P5d6 - Chapter 9 - Multicriteria Decision Making**Goal Programming**Alternative Forms of Goal Constraints (2 of 2) Complete Model with New Goals: Minimize P1d1-, P2d2-, P3d3-, P4d4-, 4P5d5-, 5P5d6- subject 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**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**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**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**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**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**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 9 - Multicriteria Decision Making**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 9.1 Chapter 9 - Multicriteria Decision Making**Goal Programming**Computer Solution Using QM for Windows (2 of 3) Exhibit 9.2 Chapter 9 - Multicriteria Decision Making**Goal Programming**Computer Solution Using QM for Windows (3 of 3) Exhibit 9.3 Chapter 9 - Multicriteria Decision Making**Goal Programming**Computer Solution Using Excel (1 of 3) Exhibit 9.4 Chapter 9 - Multicriteria Decision Making**Goal Programming**Computer Solution Using Excel (2 of 3) Exhibit 9.5 Chapter 9 - Multicriteria Decision Making**Goal Programming**Computer Solution Using Excel (3 of 3) Exhibit 9.6 Chapter 9 - Multicriteria Decision Making**Goal Programming**Solution for Alternate Problem Using Excel (1 of 6) Minimize P1d1-, P2d2-, P3d3-, P4d4-, 4P5d5-, 5P5d6- subject 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**Goal Programming**Solution for Alternate Problem Using Excel (2 of 6) Exhibit 9.7 Chapter 9 - Multicriteria Decision Making**Goal Programming**Solution for Alternate Problem Using Excel (3 of 6) Exhibit 9.8 Chapter 9 - Multicriteria Decision Making**Goal Programming**Solution for Alternate Problem Using Excel (4 of 6) Exhibit 9.9 Chapter 9 - Multicriteria Decision Making**Goal Programming**Solution for Alternate Problem Using Excel (5 of 6) Exhibit 9.10 Chapter 9 - Multicriteria Decision Making**Goal Programming**Solution for Alternate Problem Using Excel (6 of 6) Exhibit 9.11 Chapter 9 - Multicriteria Decision Making**Goal Programming**Excel Spreadsheets (1 of 4) Exhibit 9.12 Chapter 9 - Multicriteria Decision Making**Goal Programming**Excel Spreadsheets (2 of 4) Exhibit 9.13 Chapter 9 - Multicriteria Decision Making**Goal Programming**Excel Spreadsheets (3 of 4) Exhibit 9.14 Chapter 9 - Multicriteria Decision Making**Goal Programming**Excel Spreadsheets (4 of 4) Exhibit 9.15 Chapter 9 - Multicriteria Decision Making**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