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 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

Model Formulation (1 of 2) 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

Model Formulation (2 of 2) 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

Goal Constraint Requirements 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 Constraints and Objective Function (3 of 3) 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

Alternative Forms of Goal Constraints (1 of 2) 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

Alternative Forms of Goal Constraints (2 of 2) 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

Graphical Interpretation (1 of 6) 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

The First-Priority Goal: Minimize 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

The Second-Priority Goal: Minimize 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

The Third-Priority Goal: Minimize 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

The Fourth-Priority Goal: Minimize 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

Graphical Interpretation (6 of 6) 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

Computer Solution Using QM for Windows (1 of 3) 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

Computer Solution Using QM for Windows (2 of 3) Goal Programming Computer Solution Using QM for Windows (2 of 3) Exhibit 9.2 Chapter 9 - Multicriteria Decision Making

Computer Solution Using QM for Windows (3 of 3) Goal Programming Computer Solution Using QM for Windows (3 of 3) Exhibit 9.3 Chapter 9 - Multicriteria Decision Making

Computer Solution Using Excel (1 of 3) Goal Programming Computer Solution Using Excel (1 of 3) Exhibit 9.4 Chapter 9 - Multicriteria Decision Making

Computer Solution Using Excel (2 of 3) Goal Programming Computer Solution Using Excel (2 of 3) Exhibit 9.5 Chapter 9 - Multicriteria Decision Making

Computer Solution Using Excel (3 of 3) Goal Programming Computer Solution Using Excel (3 of 3) Exhibit 9.6 Chapter 9 - Multicriteria Decision Making

Solution for Alternate Problem Using Excel (1 of 6) 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

Solution for Alternate Problem Using Excel (2 of 6) Goal Programming Solution for Alternate Problem Using Excel (2 of 6) Exhibit 9.7 Chapter 9 - Multicriteria Decision Making

Solution for Alternate Problem Using Excel (3 of 6) Goal Programming Solution for Alternate Problem Using Excel (3 of 6) Exhibit 9.8 Chapter 9 - Multicriteria Decision Making

Solution for Alternate Problem Using Excel (4 of 6) Goal Programming Solution for Alternate Problem Using Excel (4 of 6) Exhibit 9.9 Chapter 9 - Multicriteria Decision Making

Solution for Alternate Problem Using Excel (5 of 6) Goal Programming Solution for Alternate Problem Using Excel (5 of 6) Exhibit 9.10 Chapter 9 - Multicriteria Decision Making

Solution for Alternate Problem Using Excel (6 of 6) Goal Programming Solution for Alternate Problem Using Excel (6 of 6) Exhibit 9.11 Chapter 9 - Multicriteria Decision Making

Excel Spreadsheets (1 of 4) Goal Programming Excel Spreadsheets (1 of 4) Exhibit 9.12 Chapter 9 - Multicriteria Decision Making

Excel Spreadsheets (2 of 4) Goal Programming Excel Spreadsheets (2 of 4) Exhibit 9.13 Chapter 9 - Multicriteria Decision Making

Excel Spreadsheets (3 of 4) Goal Programming Excel Spreadsheets (3 of 4) Exhibit 9.14 Chapter 9 - Multicriteria Decision Making

Excel Spreadsheets (4 of 4) 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