Chapter 9 - Multicriteria Decision Making 1 Chapter 9 Multicriteria Decision Making Introduction to Management Science 8th Edition by Bernard W. Taylor III

Chapter 9 - Multicriteria Decision Making 2 Goal Programming Graphical Interpretation of Goal Programming Computer Solution of Goal Programming Problems with QM for Windows and Excel The Analytical Hierarchy Process Chapter Topics

Chapter 9 - Multicriteria Decision Making 3 Study of problems with several criteria, multiple criteria, instead of a single 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. Overview

Chapter 9 - Multicriteria Decision Making 4 Beaver Creek Pottery Company Example: Maximize Z = $40x x 2 subject to: 1x 1 + 2x 2  40 hours of labor 4x 2 + 3x 2  120 pounds of clay x 1, x 2  0 Where: x 1 = number of bowls produced x 2 = number of mugs produced Goal Programming Model Formulation (1 of 2)

Chapter 9 - Multicriteria Decision Making 5 Adding objectives (goals) in order of importance, 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. Goal Programming Model Formulation (2 of 2)

Chapter 9 - Multicriteria Decision Making 6 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. Goal Programming Goal Constraint Requirements

Chapter 9 - Multicriteria Decision Making 7 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 + Goal Programming Goal Constraints and Objective Function (1 of 2)

Chapter 9 - Multicriteria Decision Making 8 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 d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Goal Programming Goal Constraints and Objective Function (2 of 2)

Chapter 9 - Multicriteria Decision Making 9 Changing fourth-priority goal limits overtime to 10 hours instead of minimizing overtime: d d 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- “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 - Goal Programming Alternative Forms of Goal Constraints (1 of 2)

Chapter 9 - Multicriteria Decision Making 10 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 -, 4P 5 d 5 -, 5P 5 d 6 - subject to: x 1 + 2x 2 + d d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d d 3 + = 120 d d 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 11 Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d 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 Goal Programming Graphical Interpretation (1 of 6)

Chapter 9 - Multicriteria Decision Making 12 Figure 9.2 The First-Priority Goal: Minimize Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Goal Programming Graphical Interpretation (2 of 6)

Chapter 9 - Multicriteria Decision Making 13 Figure 9.3 The Second-Priority Goal: Minimize Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Goal Programming Graphical Interpretation (3 of 6)

Chapter 9 - Multicriteria Decision Making 14 Figure 9.4 The Third-Priority Goal: Minimize Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Goal Programming Graphical Interpretation (4 of 6)

Chapter 9 - Multicriteria Decision Making 15 Figure 9.5 The Fourth-Priority Goal: Minimize Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Goal Programming Graphical Interpretation (5 of 6)

Chapter 9 - Multicriteria Decision Making 16 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 d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d 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 Goal Programming Graphical Interpretation (6 of 6)

Chapter 9 - Multicriteria Decision Making 17 Exhibit 9.1 Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 +, P 4 d 1 + subject to: x 1 + 2x 2 + d d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d d 3 + = 120 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +  0 Goal Programming Computer Solution Using QM for Windows (1 of 3)

Chapter 9 - Multicriteria Decision Making 18 Exhibit 9.2 Goal Programming Computer Solution Using QM for Windows (2 of 3)

Chapter 9 - Multicriteria Decision Making 19 Exhibit 9.3 Goal Programming Computer Solution Using QM for Windows (3 of 3)

Chapter 9 - Multicriteria Decision Making 20 Exhibit 9.4 Goal Programming Computer Solution Using Excel (1 of 3)

Chapter 9 - Multicriteria Decision Making 21 Exhibit 9.5 Goal Programming Computer Solution Using Excel (2 of 3)

Chapter 9 - Multicriteria Decision Making 22 Exhibit 9.6 Goal Programming Computer Solution Using Excel (3 of 3)

Chapter 9 - Multicriteria Decision Making 23 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 d 1 + = 40 40x x 2 + d d 2 + = 1,600 4x 1 + 3x 2 + d d 3 + = 120 d d 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 Goal Programming Solution for Altered Problem Using Excel (1 of 6)

Chapter 9 - Multicriteria Decision Making 24 Exhibit 9.7 Goal Programming Solution for Altered Problem Using Excel (2 of 6)

Chapter 9 - Multicriteria Decision Making 25 Exhibit 9.8 Goal Programming Solution for Altered Problem Using Excel (3 of 6)

Chapter 9 - Multicriteria Decision Making 26 Exhibit 9.9 Goal Programming Solution for Altered Problem Using Excel (4 of 6)

Chapter 9 - Multicriteria Decision Making 27 Exhibit 9.10 Goal Programming Solution for Altered Problem Using Excel (5 of 6)

Chapter 9 - Multicriteria Decision Making 28 Exhibit 9.11 Goal Programming Solution for Altered Problem Using Excel (6 of 6)

Chapter 9 - Multicriteria Decision Making 29 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. Analytical Hierarchy Process Overview

Chapter 9 - Multicriteria Decision Making 30 Southcorp Development Company shopping mall site selection. Three potential sites: Atlanta Birmingham Charlotte. Criteria for site comparisons: Customer market base. Income level Infrastructure Analytical Hierarchy Process Example Problem Statement

Chapter 9 - Multicriteria Decision Making 31 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. Analytical Hierarchy Process Hierarchy Structure

Chapter 9 - Multicriteria Decision Making 32 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. Analytical Hierarchy Process General Mathematical Process

Chapter 9 - Multicriteria Decision Making 33 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. Analytical Hierarchy Process Pairwise Comparisons

Chapter 9 - Multicriteria Decision Making 34 Table 9.1 Preference Scale for Pairwise Comparisons Analytical Hierarchy Process Pairwise Comparisons (2 of 2)

Chapter 9 - Multicriteria Decision Making 35 Income Level Infrastructure Transportation A B C Analytical Hierarchy Process Pairwise Comparison Matrix A pairwise comparison matrix summarizes the pairwise comparisons for a criteria.

Chapter 9 - Multicriteria Decision Making 36 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 37 Table 9.2 The Normalized Matrix with Row Averages Analytical Hierarchy Process Developing Preferences Within Criteria (2 of 3)

Chapter 9 - Multicriteria Decision Making 38 Table 9.3 Criteria Preference Matrix Analytical Hierarchy Process Developing Preferences Within Criteria (3 of 3)

Chapter 9 - Multicriteria Decision Making 39 Table 9.4 Normalized Matrix for Criteria with Row Averages Analytical Hierarchy Process Ranking the Criteria (1 of 2) Pairwise Comparison Matrix:

Chapter 9 - Multicriteria Decision Making 40 Analytical Hierarchy Process Ranking the Criteria (2 of 2) Preference Vector: Market Income Infrastructure Transportation

Chapter 9 - Multicriteria Decision Making 41 Overall Score: Site A score =.1993(.5012) (.2819) (.1790) (.1561) =.3091 Site B score =.1993(.1185) (.0598) (.6850) (.6196) =.1595 Site C score =.1993(.3803) (.6583) (.1360) (.2243) =.5314 Overall Ranking: Analytical Hierarchy Process Developing an Overall Ranking

Chapter 9 - Multicriteria Decision Making 42 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 43 Exhibit 9.12 Goal Programming Excel Spreadsheets (1 of 4)

Chapter 9 - Multicriteria Decision Making 44 Exhibit 9.13 Goal Programming Excel Spreadsheets (2 of 4)

Chapter 9 - Multicriteria Decision Making 45 Exhibit 9.14 Goal Programming Excel Spreadsheets (3 of 4)

Chapter 9 - Multicriteria Decision Making 46 Exhibit 9.15 Goal Programming Excel Spreadsheets (4 of 4)

Chapter 9 - Multicriteria Decision Making 47 Each decision alternative graded in terms of how well it satisfies the criterion according to following formula: S i =  g ij w j where: w j = a weight between 0 and 1.00 assigned to criteria j; 1.00 important, 0 unimportant; sum of total weights equals one. g ij = a grade between 0 and 100 indicating how well alternative i satisfies criteria j; 100 indicates high satisfaction, 0 low satisfaction. Scoring Model Overview

Chapter 9 - Multicriteria Decision Making 48 Mall selection with four alternatives and five criteria: S 1 = (.30)(40) + (.25)(75) + (.25)(60) + (.10)(90) + (.10)(80) = S 2 = (.30)(60) + (.25)(80) + (.25)(90) + (.10)(100) + (.10)(30) = S 3 = (.30)(90) + (.25)(65) + (.25)(79) + (.10)(80) + (.10)(50) = S 4 = (.30)(60) + (.25)(90) + (.25)(85) + (.10)(90) + (.10)(70) = Mall 4 preferred because of highest score, followed by malls 3, 2, 1. Scoring Model Example Problem

Chapter 9 - Multicriteria Decision Making 49 Exhibit 9.16 Scoring Model Excel Solution

Chapter 9 - Multicriteria Decision Making 50 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 51 Step 1: Model Formulation: Minimize P 1 d 1 -, P 2 d 2 -, P 3 d 3 - subject to: 80x x 2 + d d 1 + = 3,000 interviews 50x x 2 + d d 2 + = $2,500 budget 80x 1 + d d 3 + = 1,000 telephone interviews where: x 1 = number of telephone interviews x 2 = number of personal interviews Goal Programming Example Problem Solution (1 of 2)

Chapter 9 - Multicriteria Decision Making 52 Goal Programming Example Problem Solution (2 of 2) Step 2: QM for Windows Solution:

Chapter 9 - Multicriteria Decision Making 53 Purchasing decision, three model alternatives, three decision criteria. Pairwise comparison matrices: Prioritized decision criteria: Analytical Hierarchy Process Example Problem Problem Statement

Chapter 9 - Multicriteria Decision Making 54 Step 1: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Analytical Hierarchy Process Example Problem Problem Solution (1 of 4)

Chapter 9 - Multicriteria Decision Making 55 Step 1 continued: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Analytical Hierarchy Process Example Problem Problem Solution (2 of 4)

Chapter 9 - Multicriteria Decision Making 56 Step 2: Rank the criteria. Price Gears Weight Analytical Hierarchy Process Example Problem Problem Solution (3 of 4)

Chapter 9 - Multicriteria Decision Making 57 Step 3: Develop an overall ranking. Bike X Bike Y Bike Z Bike X score =.6667(.6479) (.2299) (.1222) =.5057 Bike Y score =.2222(.6479) (.2299) (.1222) =.2138 Bike Z score =.1111(.6479) (.2299) (.1222) =.2806 Overall ranking of bikes: X first followed by Z and Y (sum of scores equal ). Analytical Hierarchy Process Example Problem Problem Solution (4 of 4)