Multicriteria Decision Making

Transcription

1 Multicriteria Decision Making Chapter 9 91

2 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 92

3 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. 93

4 Example Problem Data (1 of 2) Beaver Creek Pottery Company Example: Maximize Z = $40x 1 50x 2 subject to: 1x 1 2x 2 40 hours of labor 4x 1 3x pounds of clay x 1, x 2 0 Where: x 1 = number of bowls produced x 2 = number of mugs produced 94

5 Example Problem Data (2 of 2) Adding objectives (goals) in order of importance, the company: 1. Does not want to use fewer than 40 hours of labor per day. 2. Would like to achieve a satisfactory profit level of $1,600 per day. 3. Prefers not to keep more than 120 pounds of clay on hand each day. 4. Would like to minimize the amount of overtime. 95

6 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 seeks to minimize the deviation from the respective goals in the order of the goal priorities. 96

7 Model Formulation Goal Constraints (1 of 3) Labor goal: x 1 2x 2 d 1 d 1 = 40 (hours/day) Profit goal: 40x 1 50 x 2 d 2 d 2 = 1,600 ($/day) Material goal: 4x 1 3x 2 d 3 d 3 = 120 (lbs of clay/day) 97

8 Model Formulation Objective Function (2 of 3) 1. Labor goals constraint (priority 1 less than 40 hours labor; priority 4 minimum overtime): Minimize P 1 d 1, P 4 d 1 2. Add profit goal constraint (priority 2 achieve profit of $1,600): Minimize P 1 d 1, P 2 d 2, P 4 d 1 3. Add material goal constraint (priority 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 98

9 Model Formulation Complete Model (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 2x 2 d 1 d 1 = 40 40x 1 50 x 2 d 2 d 2 = 1,600 4x 1 3x 2 d 3 d 3 = 120 x 1, x 2, d 1, d 1, d 2, d 2, d 3, d 3 0 (labor) (profit) (clay) 99

10 Alternative Forms of Goal Constraints (1 of 2) Changing fourthpriority 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 fifthpriority goal important to achieve the goal for mugs : 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, 4P 5 d 5 5P 5 d 6 910

11 Alternative Forms of Goal Constraints (2 of 2) Complete Model with Added New Goals: Minimize P 1 d 1, P 2 d 2, P 3 d 3, P 4 d 4, 4P 5 d 5 5P 5 d 6 subject to: x 1 2x 2 d 1 d 1 = 40 40x 1 50x 2 d 2 d 2 = 1,600 4x 1 3x 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

12 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 2x 2 d 1 d 1 = 40 40x 1 50 x 2 d 2 d 2 = 1,600 4x 1 3x 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 912

13 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 2x 2 d 1 d 1 = 40 40x 1 50 x 2 d 2 d 2 = 1,600 4x 1 3x 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 FirstPriority Goal: Minimize d1 913

14 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 2x 2 d 1 d 1 = 40 40x 1 50 x 2 d 2 d 2 = 1,600 4x 1 3x 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 SecondPriority Goal: Minimize d2 914

15 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 2x 2 d 1 d 1 = 40 40x 1 50 x 2 d 2 d 2 = 1,600 4x 1 3x 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 ThirdPriority Goal: Minimize d3 915

16 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 2x 2 d 1 d 1 = 40 40x 1 50 x 2 d 2 d 2 = 1,600 4x 1 3x 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 FourthPriority Goal: Minimize d1 916

17 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 2x 2 d 1 d 1 = 40 40x 1 50 x 2 d 2 d 2 = 1,600 4x 1 3x 2 d 3 d 3 = 120 x 1, x 2, d 1, d 1, d 2, d 2, d 3, d 3 0 Solution: x 1 = 15 bowls x 2 = 20 mugs d 1 = 15 hours 917

18 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 2x 2 d 1 d 1 = 40 40x 1 50 x 2 d 2 d 2 = 1,600 4x 1 3x 2 d 3 d 3 = 120 x 1, x 2, d 1, d 1, d 2, d 2, d 3, d 3 0 Exhibit

19 Computer Solution Using QM for Windows (2 of 3) Exhibit

20 Computer Solution Using QM for Windows (3 of 3) Exhibit

21 Computer Solution Using Excel (1 of 3) Exhibit

22 Computer Solution Using Excel (2 of 3) Exhibit

23 Computer Solution Using Excel (3 of 3) Exhibit

24 Solution for Altered Problem Using Excel (1 of 6) Minimize P 1 d 1, P 2 d 2, P 3 d 3, P 4 d 4, 4P 5 d 5 5P 5 d 6 subject to: x 1 2x 2 d 1 d 1 = 40 40x 1 50x 2 d 2 d 2 = 1,600 4x 1 3x 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

25 Solution for Altered Problem Using Excel (2 of 6) Exhibit

26 Solution for Altered Problem Using Excel (3 of 6) Exhibit

27 Solution for Altered Problem Using Excel (4 of 6) Exhibit

28 Solution for Altered Problem Using Excel (5 of 6) Exhibit

29 Solution for Altered Problem Using Excel (6 of 6) Exhibit