Introduction to Management Science 8 th Edition by
Introduction to Management Science 8 th Edition by Bernard W. Taylor III Chapter 9 Multicriteria Decision Making 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 Overview Study of problems with several criteria, multiple criteria, instead of a single 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 2
Goal Programming Model Formulation (1 of 2) Beaver Creek Pottery Company Example: Maximize Z = $40 x 1 + 50 x 2 subject to: 1 x 1 + 2 x 2 40 hours of labor 4 x 2 + 3 x 2 120 pounds of clay x 1, x 2 0 Where: x 1 = number of bowls produced x 2 = number of mugs produced Chapter 9 - Multicriteria Decision Making 3
Goal Programming Model Formulation (2 of 2) Adding objectives (goals) in order of importance (i. e. priorities), 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 4
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 5
Goal Programming: Goal Constraints (1 of 3) x 1 + 2 x 2 = 40 - d 1 - + d 1+ 40 x 1 + 50 x 2 = 1, 600 - d 2 - + d 2+ 4 x 1 + 3 x 2 = 120 - d 3 - + d 3+ x 1 , x 2 , d 1 -, d 1 + , d 2 -, d 2 + , d 3 -, d 3 + 0 Chapter 9 - Multicriteria Decision Making 6
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 P 1 d 1 -, P 4 d 1+ Add profit goal constraint (2, achieve profit of $1, 600): Minimize P 1 d 1 -, P 2 d 2 -, P 4 d 1+ Add material goal constraint (3, avoid keeping more than 120 pounds of clay on hand): Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3+, P 4 d 1+ Chapter 9 - Multicriteria Decision Making 7
Goal Programming Goal Constraints and Objective Function (3 of 3) Complete Goal Programming Model: Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3+, P 4 d 1+ subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2 + = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3 + = 120 x 1 , x 2 , d 1 -, d 1 + , d 2 -, d 2 + , d 3 -, d 3 + 0 Chapter 9 - 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: d 1 - + d 4 - - d 4+ = 10 minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 4 + Addition of a fifth-priority goal- due to limited warehouse space, cannot produce more than 30 bowls and 20 mugs daily. x 1 + d 5 - = 30 bowls x 2 + d 6 - = 20 mugs minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 -, P 4 d 4 -, 4 P 5 d 5 -, 5 P 5 d 6 Chapter 9 - Multicriteria Decision Making 9
Goal Programming Alternative Forms of Goal Constraints (2 of 2) Complete Model with New Goals: Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 -, P 4 d 4 -, 4 P 5 d 5 -, 5 P 5 d 6 subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2+ = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3+ = 120 d 1+ + d 4 - - d 4+ = 10 x 1 + d 5 - = 30 x 2 + d 6 - = 20 x 1, x 2, d 1 -, d 1+, d 2 -, d 2+, d 3 -, d 3+, d 4 -, d 4+, d 5 -, d 6 - 0 Chapter 9 - Multicriteria Decision Making 10
Goal Programming Graphical Interpretation (1 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3+, P 4 d 1+ subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2 + = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 + 0 Figure 9. 1 Goal Constraints Chapter 9 - Multicriteria Decision Making 11
Goal Programming Graphical Interpretation (2 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3+, P 4 d 1+ subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2 + = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 + 0 Figure 9. 2 The First-Priority Goal: Minimize Chapter 9 - Multicriteria Decision Making 12
Goal Programming Graphical Interpretation (3 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3+, P 4 d 1+ subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2 + = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 + 0 Figure 9. 3 The Second-Priority Goal: Minimize Chapter 9 - Multicriteria Decision Making 13
Goal Programming Graphical Interpretation (4 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3+, P 4 d 1+ subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2 + = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 + 0 Figure 9. 4 The Third-Priority Goal: Minimize Chapter 9 - Multicriteria Decision Making 14
Goal Programming Graphical Interpretation (5 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3+, P 4 d 1+ subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2 + = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 + 0 Figure 9. 5 The Fourth-Priority Goal: Minimize Chapter 9 - 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 P 1 d 1 -, P 2 d 2 -, P 3 d 3+, P 4 d 1+ subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2 + = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3 + = 120 x 1 , x 2 , d 1 -, d 1 + , d 2 -, d 2 + , d 3 -, d 3 + 0 x 1 = 15 bowls x 2 = 20 mugs d 1 - = 15 hours Chapter 9 - Multicriteria Decision Making 16
Goal Programming Computer Solution Using QM for Windows (1 of 3) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3+, P 4 d 1+ subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2 + = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3 + = 120 x 1 , x 2 , d 1 -, d 1 + , d 2 -, d 2 + , d 3 -, d 3 + 0 Exhibit 9. 1 Chapter 9 - Multicriteria Decision Making 17
Goal Programming Computer Solution Using QM for Windows (2 of 3) Exhibit 9. 2 Chapter 9 - Multicriteria Decision Making 18
Goal Programming Computer Solution Using QM for Windows (3 of 3) Exhibit 9. 3 Chapter 9 - Multicriteria Decision Making 19
Goal Programming Computer Solution Using Excel (1 of 3) Exhibit 9. 4 Chapter 9 - Multicriteria Decision Making 20
Goal Programming Computer Solution Using Excel (2 of 3) Exhibit 9. 5 Chapter 9 - Multicriteria Decision Making 21
Goal Programming Computer Solution Using Excel (3 of 3) Exhibit 9. 6 Chapter 9 - Multicriteria Decision Making 22
Goal Programming Solution for Alternate Problem Using Excel (1 of 6) Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 -, P 4 d 4 -, 4 P 5 d 5 -, 5 P 5 d 6 subject to: x 1 + 2 x 2 + d 1 - - d 1+ = 40 40 x 1 + 50 x 2 + d 2 - - d 2+ = 1, 600 4 x 1 + 3 x 2 + d 3 - - d 3+ = 120 d 1+ + d 4 - - d 4+ = 10 x 1 + d 5 - = 30 x 2 + d 6 - = 20 x 1, x 2, d 1 -, d 1+, d 2 -, d 2+, d 3 -, d 3+, d 4 -, d 4+, d 5 -, d 6 - 0 Chapter 9 - Multicriteria Decision Making 23
Goal Programming Solution for Alternate Problem Using Excel (2 of 6) Exhibit 9. 7 Chapter 9 - Multicriteria Decision Making 24
Goal Programming Solution for Alternate Problem Using Excel (3 of 6) Exhibit 9. 8 Chapter 9 - Multicriteria Decision Making 25
Goal Programming Solution for Alternate Problem Using Excel (4 of 6) Exhibit 9. 9 Chapter 9 - Multicriteria Decision Making 26
Goal Programming Solution for Alternate Problem Using Excel (5 of 6) Exhibit 9. 10 Chapter 9 - Multicriteria Decision Making 27
Goal Programming Solution for Alternate Problem Using Excel (6 of 6) Exhibit 9. 11 Chapter 9 - Multicriteria Decision Making 28
Goal Programming Excel Spreadsheets (1 of 4) Exhibit 9. 12 Chapter 9 - Multicriteria Decision Making 29
Goal Programming Excel Spreadsheets (2 of 4) Exhibit 9. 13 Chapter 9 - Multicriteria Decision Making 30
Goal Programming Excel Spreadsheets (3 of 4) Exhibit 9. 14 Chapter 9 - Multicriteria Decision Making 31
Goal Programming Excel Spreadsheets (4 of 4) Exhibit 9. 15 Chapter 9 - Multicriteria Decision Making 32
Goal Programming Example 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 33
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