Chapter 6 Multicriteria Decision Making

Transcription

1 Chapter 6 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 Scoring Models 2

2 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. 3 Goal Programming Example Problem Data ( of 2) Beaver Creek Pottery Company Example: Maximize Z = $40x 50x 2 subject to: x 2x 2 40 hours of labor 4x 3x 2 20 pounds of clay x, x 2 0 Where: x = number of bowls produced x 2 = number of mugs produced 4 2

3 Goal Programming Example Problem Data (2 of 2) 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 $,600 per day. Prefers not to keep more than 20 pounds of clay on hand each day. Would like to minimize the amount of overtime. 5 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 the respective goals in the order of the goal priorities. 6 3

4 Goal Programming Model Formulation Goal Constraints ( of 3) Labor goal: x 2x 2 d d = 40 (hours/day) Profit goal: 40x 50 x 2 d 2 d 2 =,600 ($/day) Material goal: 4x 3x 2 d 3 d 3 = 20 (lbs of clay/day) 7 Goal Programming Model Formulation Objective Function (2 of 3) Labor goals constraint (priority less than 40 hours labor; priority 4 minimum overtime): Minimize P d, P 4 d Add profit goal constraint (priority 2 achieve profit of $,600): Minimize P d, P 2 d 2, P 4 d Add material goal constraint (priority 3 avoid keeping more than 20 pounds of clay on hand): Minimize P d, P 2 d 2, P 3 d 3, P 4 d 8 4

5 Goal Programming Model Formulation Complete Model (3 of 3) Complete Goal Programming Model: Minimize P d, P 2 d 2, P 3 d 3, P 4 d subject to: x 2x 2 d d = 40 (labor) 40x 50 x 2 d 2 d 2 =,600 (profit) 4x 3x 2 d 3 d 3 = 20 (clay) x, x 2, d, d, d 2, d 2, d 3, d Goal Programming Alternative Forms of Goal Constraints ( of 2) Changing fourthpriority goal limits overtime to 0 hours instead of minimizing overtime: d d 4 d 4 = 0 minimize P d, 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 d 5 = 30 bowls x 2 d 6 = 20 mugs minimize P d, P 2 d 2, P 3 d 3, P 4 d 4, 4P 5 d 5 5P 5 d 6 0 5

6 Goal Programming Alternative Forms of Goal Constraints (2 of 2) Complete Model with Added New Goals: Minimize P d, P 2 d 2, P 3 d 3, P 4 d 4, 4P 5 d 5 5P 5 d 6 subject to: x 2x 2 d d = 40 40x 50x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 d d 4 d 4 = 0 x d 5 = 30 x 2 d 6 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3, d 4, d 4, d 5, d 6 0 Goal Programming Graphical Interpretation ( of 6) Minimize P d, P 2 d 2, P 3 d 3, P 4 d subject to: x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9. Goal Constraints 2 6

7 Goal Programming Graphical Interpretation (2 of 6) Minimize P d, P 2 d 2, P 3 d 3, P 4 d subject to: x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9.2 The FirstPriority Goal: Minimize 3 Goal Programming Graphical Interpretation (3 of 6) Minimize P d, P 2 d 2, P 3 d 3, P 4 d subject to: x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9.3 The SecondPriority Goal: Minimize 4 7

8 Goal Programming Graphical Interpretation (4 of 6) Minimize P d, P 2 d 2, P 3 d 3, P 4 d subject to: x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9.4 The ThirdPriority Goal: Minimize 5 Goal Programming Graphical Interpretation (5 of 6) Minimize P d, P 2 d 2, P 3 d 3, P 4 d subject to: x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Figure 9.5 The FourthPriority Goal: Minimize 6 8

9 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 d, P 2 d 2, P 3 d 3, P 4 d subject to: x 2x 2 d d = 40 40x 50 x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3 0 Solution: x = 5 bowls x 2 = 20 mugs d = 5 hours 7 Goal Programming Computer Solution Using Excel ( of 3) Exhibit

10 Goal Programming Computer Solution Using Excel (2 of 3) Exhibit Goal Programming Computer Solution Using Excel (3 of 3) Exhibit

11 Goal Programming Solution for Altered Problem Using Excel ( of 6) Minimize P d, P 2 d 2, P 3 d 3, P 4 d 4, 4P 5 d 5 5P 5 d 6 subject to: x 2x 2 d d = 40 40x 50x 2 d 2 d 2 =,600 4x 3x 2 d 3 d 3 = 20 d d 4 d 4 = 0 x d 5 = 30 x 2 d 6 = 20 x, x 2, d, d, d 2, d 2, d 3, d 3, d 4, d 4, d 5, d Goal Programming Solution for Altered Problem Using Excel (2 of 6) Exhibit

12 Goal Programming Solution for Altered Problem Using Excel (3 of 6) Exhibit Goal Programming Solution for Altered Problem Using Excel (4 of 6) Exhibit

13 Goal Programming Solution for Altered Problem Using Excel (5 of 6) Exhibit Goal Programming Solution for Altered Problem Using Excel (6 of 6) Exhibit

14 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. 27 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 28 4

15 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. 29 Analytical Hierarchy Process General Mathematical Process Mathematically determine preferences for sites with respect to each criterion. Mathematically determine preferences for criteria (rank order of importance). Combine these two sets of preferences to mathematically derive a composite score for each site. Select the site with the highest score. 30 5

16 Analytical Hierarchy Process Pairwise Comparisons ( of 2) 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. 3 Analytical Hierarchy Process Pairwise Comparisons (2 of 2) Table 9. Preference Scale for Pairwise Comparisons 32 6

17 Analytical Hierarchy Process Pairwise Comparison Matrix A pairwise comparison matrix summarizes the pairwise comparisons for a criteria. Customer Market Site A B C A B C /3 / /5 A B C Income Level Infrastructure Transportation / /3 /9 3 /3 / /3 /4 / Analytical Hierarchy Process Developing Preferences Within Criteria ( of 3) In synthesization, decision alternatives are prioritized with each criterion and then normalized: Customer Market Site A B C A B C /3 /2 / /5 6/5 Customer Market Site A B C A 6/ 3/9 5/8 B C 2/ 3/ /9 5/9 /6 5/6 34 7

18 Analytical Hierarchy Process Developing Preferences Within Criteria (2 of 3) The row average values represent the preference vector Table 9.2 The Normalized Matrix with Row Averages 35 Analytical Hierarchy Process Developing Preferences Within Criteria (3 of 3) Preference vectors for other criteria are computed similarly, resulting in the preference matrix Table 9.3 Criteria Preference Matrix 36 8

19 Analytical Hierarchy Process Ranking the Criteria ( of 2) Pairwise Comparison Matrix: Criteria Market Income Infrastructure Transportation Market Income Infrastructure Transportation 5 /3 /4 /5 /9 /7 3 9 / Table 9.4 Normalized Matrix for Criteria with Row Averages 37 Analytical Hierarchy Process Ranking the Criteria (2 of 2) Preference Vector for Criteria: Market Income Infrastructure Transportation

20 Analytical Hierarchy Process Developing an Overall Ranking Overall Score: Site A score =.993(.502).6535(.289).0860(.790).062(.56) =.309 Site B score =.993(.85).6535(.0598).0860(.6850).062(.696) =.595 Site C score =.993(.3803).6535(.6583).0860(.360).062(.2243) =.534 Overall Ranking: Site Charlotte Atlanta Birmingham Score 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

21 Analytical Hierarchy Process: Consistency ( of 3) Consistency Index (CI): Check for consistency and validity of multiple pairwise comparisons Example: Southcorp s consistency in the pairwise comparisons of the 4 site selection criteria Step : Multiply the pairwise comparison matrix of the 4 criteria by its preference vector Market Income Infrastruc. Transp. Criteria Market / Income X Infrastructure /3 / Transportation /4 /7 / ()(.993)(/5)(.6535)(3)(.0860)(4)(.062) = (5)(.993)()(.6535)(9)(.0860)(7)(.062) = (/3)(.993)(/9)(.6535)()(.0860)(2)(.062) = (/4)(.993)(/7)(.6535)(/2)(.0860)()(.062) = Analytical Hierarchy Process: Consistency (2 of 3) Step 2: Divide each value by the corresponding weight from the preference vector and compute the average /0.993 = / = / = /0.062 = Average = 6.257/4 = Step 3: Calculate the Consistency Index (CI) CI = (Average n)/(n), where n is no. of items compared CI = (4.5644)/(4) = (CI = 0 indicates perfect consistency) 42 2

22 Analytical Hierarchy Process: Consistency (3 of 3) Step 4: Compute the Ratio CI/RI where RI is a random index value obtained from Table 9.5 Table 9.5 Random Index Values for n Items Being Compared CI/RI = 0.052/0.90 = Note: Degree of consistency is satisfactory if CI/RI < Analytical Hierarchy Process Excel Spreadsheets ( of 4) Exhibit

23 Analytical Hierarchy Process Excel Spreadsheets (2 of 4) Exhibit Analytical Hierarchy Process Excel Spreadsheets (3 of 4) Exhibit

24 Analytical Hierarchy Process Excel Spreadsheets (4 of 4) Exhibit Scoring Model Overview Each decision alternative graded in terms of how well it satisfies the criterion according to following formula: S i = Σg ij w j where: w j = a weight between 0 and.00 assigned to criterion j;.00 important, 0 unimportant; sum of total weights equals one. g ij = a grade between 0 and 00 indicating how well alternative i satisfies criteria j; 00 indicates high satisfaction, 0 low satisfaction

25 Scoring Model Example Problem Mall selection with four alternatives and five criteria: Grades for Alternative (0 to 00) Weight Decision Criteria (0 to.00) Mall Mall 2 Mall 3 Mall 4 School proximity Median income Vehicular traffic Mall quality, size Other shopping S = (.30)(40) (.25)(75) (.25)(60) (.0)(90) (.0)(80) = S 2 = (.30)(60) (.25)(80) (.25)(90) (.0)(00) (.0)(30) = S 3 = (.30)(90) (.25)(65) (.25)(79) (.0)(80) (.0)(50) = S 4 = (.30)(60) (.25)(90) (.25)(85) (.0)(90) (.0)(70) = Mall 4 preferred because of highest score, followed by malls 3, 2, 49 Scoring Model Excel Solution Exhibit

26 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,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. 5 Analytical Hierarchy Process Example Problem Problem Statement Purchasing decision, three model alternatives, three decision criteria. Pairwise comparison matrices: Price Bike X Y Z X Y Z /3 /6 3 /2 6 2 Gear Action Bike X Y Z X Y Z 3 7 /3 4 /7 /4 Weight/Durability Bike X Y Z X Y Z /3 3 2 /2 Prioritized decision criteria: Criteria Price Gears Weight Price 3 5 Gears /3 2 Weight /5 /

27 Analytical Hierarchy Process Example Problem Problem Solution ( of 4) Step : Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Price Bike X Y Z Row Averages X Y Z Gear Action Bike X Y Z Row Averages X Y Z Analytical Hierarchy Process Example Problem Problem Solution (2 of 4) Step continued: Develop normalized matrices and preference vectors for all the pairwise comparison matrices for criteria. Weight/Durability Bike X Y Z Row Averages X Y Z Criteria Bike Price Gears Weight X Y Z

28 Analytical Hierarchy Process Example Problem Problem Solution (3 of 4) Step 2: Rank the criteria. Criteria Price Gears Weight Row Averages Price Gears Weight Price Gears Weight Analytical Hierarchy Process Example Problem Problem Solution (4 of 4) Step 3: Develop an overall ranking. Bike X Bike Y Bike Z Bike X score =.6667(.6479).0853(.2299).4429(.222) =.5057 Bike Y score =.2222(.6479).232(.2299).698(.222) =.238 Bike Z score =.(.6479).704(.2299).3873(.222) =.2806 Overall ranking of bikes: X first followed by Z and Y (sum of scores equal.0000)