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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 • The Analytical Hierarchy Process Chapter 9 - Multicriteria Decision Making**Overview**• Study of problems with several criteria, multiple criteria, instead of asingle 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 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, 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 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 9 - Multicriteria Decision Making**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 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- “important to achieve the goal for mugs”: • 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 Altered 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 Altered Problem Using Excel (2 of 6) Exhibit 9.7 Chapter 9 - Multicriteria Decision Making**Goal Programming**Solution for Altered Problem Using Excel (3 of 6) Exhibit 9.8 Chapter 9 - Multicriteria Decision Making**Goal Programming**Solution for Altered Problem Using Excel (4 of 6) Exhibit 9.9 Chapter 9 - Multicriteria Decision Making**Goal Programming**Solution for Altered Problem Using Excel (5 of 6) Exhibit 9.10 Chapter 9 - Multicriteria Decision Making**Goal Programming**Solution for Altered Problem Using Excel (6 of 6) Exhibit 9.11 Chapter 9 - Multicriteria Decision Making**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 9 - Multicriteria Decision Making**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 9 - Multicriteria Decision Making**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 9 - Multicriteria Decision Making**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 9 - Multicriteria Decision Making**Analytical Hierarchy Process**Pairwise Comparisons • In a pairwise 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 9 - Multicriteria Decision Making**Analytical Hierarchy Process**Pairwise Comparisons (2 of 2) Table 9.1 Preference Scale for Pairwise Comparisons Chapter 9 - Multicriteria Decision Making**Analytical Hierarchy Process**Pairwise Comparison Matrix • A pairwise comparison matrix summarizes the pairwise comparisons for a criteria. Income Level Infrastructure Transportation A B C Chapter 9 - Multicriteria Decision Making**Analytical Hierarchy Process**Developing Preferences Within Criteria (1 of 3) • In synthesization, decision alternatives are prioritized with each criterion and then normalized: Chapter 9 - Multicriteria Decision Making**Analytical Hierarchy Process**Developing Preferences Within Criteria (2 of 3) Table 9.2 The Normalized Matrix with Row Averages Chapter 9 - Multicriteria Decision Making**Analytical Hierarchy Process**Developing Preferences Within Criteria (3 of 3) Table 9.3 Criteria Preference Matrix Chapter 9 - Multicriteria Decision Making**Analytical Hierarchy Process**Ranking the Criteria (1 of 2) Pairwise Comparison Matrix: Table 9.4 Normalized Matrix for Criteria with Row Averages Chapter 9 - Multicriteria Decision Making**Analytical Hierarchy Process**Ranking the Criteria (2 of 2) Preference Vector: Market Income Infrastructure Transportation Chapter 9 - Multicriteria Decision Making**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: Chapter 9 - Multicriteria Decision Making**Analytical Hierarchy Process**Summary of Mathematical Steps • Develop a pairwise comparison matrix for each decision alternative for each criteria. • Synthesization • Sum the values of each column of the pairwise 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 pairwise 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 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**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 9 - Multicriteria Decision Making**Scoring Model**Example Problem • Mall selection with four alternatives and five criteria: • 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 9 - Multicriteria Decision Making**Scoring Model**Excel Solution Exhibit 9.16 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