# 1 1 Slide © 2005 Thomson/South-Western Lesson 10 Multicriteria Decisions within LP Framework n Goal Programming n Goal Programming: Formulation and Graphical.

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1 1 Slide © 2005 Thomson/South-Western Lesson 10 Multicriteria Decisions within LP Framework n Goal Programming n Goal Programming: Formulation and Graphical Solution and Graphical Solution n Scoring Model for Job Selection

2 2 Slide © 2005 Thomson/South-Western Goal Programming n Goal programming may be used to solve linear programs with multiple objectives, with each objective viewed as a "goal". n In goal programming, d i + and d i -, deviation variables, are the amounts a targeted goal i is overachieved or underachieved, respectively. n The goals themselves are added to the constraint set with d i + and d i - acting as the surplus and slack variables.

3 3 Slide © 2005 Thomson/South-Western Goal Programming n One approach to goal programming is to satisfy goals in a priority sequence. Second-priority goals are pursued without reducing the first-priority goals, etc. n For each priority level, the objective function is to minimize the (weighted) sum of the goal deviations. n Previous "optimal" achievements of goals are added to the constraint set so that they are not degraded while trying to achieve lesser priority goals.

4 4 Slide © 2005 Thomson/South-Western Goal Programming Formulation Step 1: Decide the priority level of each goal. Step 2: Decide the weight on each goal. If a priority level has more than one goal, for each goal i decide the weight, w i, to be placed on the deviation(s), d i + and/or d i -, from the goal. If a priority level has more than one goal, for each goal i decide the weight, w i, to be placed on the deviation(s), d i + and/or d i -, from the goal.

5 5 Slide © 2005 Thomson/South-Western Goal Programming Formulation Step 3: Set up the initial linear program. Min w 1 d 1 + + w 2 d 2 - s.t. Functional Constraints, s.t. Functional Constraints, and Goal Constraints and Goal Constraints Step 4: Solve the current linear program. If there is a lower priority level, go to step 5. Otherwise, a final solution has been reached. If there is a lower priority level, go to step 5. Otherwise, a final solution has been reached.

6 6 Slide © 2005 Thomson/South-Western Goal Programming Formulation Step 5: Set up the new linear program. Consider the next-lower priority level goals and formulate a new objective function based on these goals. Add a constraint requiring the achievement of the next-higher priority level goals to be maintained. The new linear program might be: Min w 3 d 3 + + w 4 d 4 - Min w 3 d 3 + + w 4 d 4 - s.t. Functional Constraints, s.t. Functional Constraints, Goal Constraints, and Goal Constraints, and w 1 d 1 + + w 2 d 2 - = k w 1 d 1 + + w 2 d 2 - = k Go to step 4. (Repeat steps 4 and 5 until all priority levels have been examined.)

7 7 Slide © 2005 Thomson/South-Western Example: Conceptual Products Conceptual Products is a computer company that produces the CP400 and CP500 computers. The computers use different computers use different mother boards produced mother boards produced in abundant supply by the company, but use the same cases and disk drives. The CP400 models use two floppy disk drives and no zip disk drives whereas the CP500 models use one floppy disk drive and one zip disk drive.

8 8 Slide © 2005 Thomson/South-Western Example: Conceptual Products Conceptual Products is a computer company that produces the CP400 and CP500 computers. Many of the components used in the two computer models are produced in computer models are produced in abundant supply by the company. However, the memory modules, external hard drives, and cases are bought from suppliers. The CP400 model uses two memory modules and no external hard drive, whereas the CP500 uses one memory module and one external hard drive. Both models use one case.

9 9 Slide © 2005 Thomson/South-Western Example: Conceptual Products Suppliers can provide Conceptual Products with 1000 memory modules, 500 external hard drives, and 600 cases on a weekly basis. It takes one hour to manufacture a CP400 and its profit is \$200 and it takes one and one-half hours to manufacture a CP500 and its profit is \$500.

10 Slide © 2005 Thomson/South-Western Example: Conceptual Products The company has four goals: The company has four goals: Priority 1: Meet a state contract of 200 CP400 machines weekly. (Goal 1) Priority 1: Meet a state contract of 200 CP400 machines weekly. (Goal 1) Priority 2: Make at least 500 total computers weekly. (Goal 2) Priority 2: Make at least 500 total computers weekly. (Goal 2) Priority 3: Make at least \$250,000 weekly. (Goal 3) Priority 3: Make at least \$250,000 weekly. (Goal 3) Priority 4: Use no more than 400 man-hours per week. (Goal 4) Priority 4: Use no more than 400 man-hours per week. (Goal 4)

11 Slide © 2005 Thomson/South-Western n Variables x 1 = number of CP400 computers produced weekly x 1 = number of CP400 computers produced weekly x 2 = number of CP500 computers produced weekly x 2 = number of CP500 computers produced weekly d i - = amount the right hand side of goal i is deficient d i - = amount the right hand side of goal i is deficient d i + = amount the right hand side of goal i is exceeded d i + = amount the right hand side of goal i is exceeded n Functional Constraints Availability of memory modules: 2 x 1 + x 2 < 1000 Availability of external hard drives: x 2 < 500 Availability of cases: x 1 + x 2 < 600 Goal Programming: Formulation

12 Slide © 2005 Thomson/South-Western n Goals (1) 200 CP400 computers weekly: x 1 + d 1 - - d 1 + = 200 (2) 500 total computers weekly: (2) 500 total computers weekly: x 1 + x 2 + d 2 - - d 2 + = 500 x 1 + x 2 + d 2 - - d 2 + = 500 (3) \$250(in thousands) profit: (3) \$250(in thousands) profit:.2 x 1 +.5 x 2 + d 3 - - d 3 + = 250.2 x 1 +.5 x 2 + d 3 - - d 3 + = 250 (4) 400 total man-hours weekly: (4) 400 total man-hours weekly: x 1 + 1.5 x 2 + d 4 - - d 4 + = 400 x 1 + 1.5 x 2 + d 4 - - d 4 + = 400 Non-negativity: Non-negativity: x 1, x 2, d i -, d i + > 0 for all i x 1, x 2, d i -, d i + > 0 for all i Goal Programming: Formulation

13 Slide © 2005 Thomson/South-Western n Objective Functions Priority 1: Minimize the amount the state contract is not met: Min d 1 - Priority 1: Minimize the amount the state contract is not met: Min d 1 - Priority 2: Minimize the number under 500 computers produced weekly: Min d 2 - Priority 2: Minimize the number under 500 computers produced weekly: Min d 2 - Priority 3: Minimize the amount under \$250,000 earned weekly: Min d 3 - Priority 3: Minimize the amount under \$250,000 earned weekly: Min d 3 - Priority 4: Minimize the man-hours over 400 used weekly: Min d 4 + Priority 4: Minimize the man-hours over 400 used weekly: Min d 4 + Goal Programming: Formulation

14 Slide © 2005 Thomson/South-Western n Formulation Summary Min P 1 ( d 1 - ) + P 2 ( d 2 - ) + P 3 ( d 3 - ) + P 4 ( d 4 + ) s.t. 2 x 1 + x 2 < 1000 s.t. 2 x 1 + x 2 < 1000 + x 2 < 500 + x 2 < 500 x 1 + x 2 < 600 x 1 + x 2 < 600 x 1 + d 1 - - d 1 + = 200 x 1 + d 1 - - d 1 + = 200 x 1 + x 2 + d 2 - - d 2 + = 500 x 1 + x 2 + d 2 - - d 2 + = 500.2 x 1 +.5 x 2 + d 3 - - d 3 + = 250.2 x 1 +.5 x 2 + d 3 - - d 3 + = 250 x 1 +1.5 x 2 + d 4 - - d 4 + = 400 x 1 +1.5 x 2 + d 4 - - d 4 + = 400 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +, d 4 -, d 4 + > 0 x 1, x 2, d 1 -, d 1 +, d 2 -, d 2 +, d 3 -, d 3 +, d 4 -, d 4 + > 0 Goal Programming: Formulation

15 Slide © 2005 Thomson/South-Western n Iteration 1 To solve graphically, first graph the functional constraints. Then graph the first goal: x 1 = 200. Note on the next slide that there is a set of points that exceed x 1 = 200 (where d 1 - = 0). Goal Programming: Graphical Solution

16 Slide © 2005 Thomson/South-Western n Functional Constraints and Goal 1 Graphed 2 x 1 + x 2 < 1000 Goal 1: x 1 > 200 x 1 + x 2 < 600 x 2 < 500 PointsSatisfying Goal 1 x1x1x1x1 x2x2x2x2 Goal Programming: Graphical Solution 1000800600400200 200 400 600 800 1000 1200

17 Slide © 2005 Thomson/South-Western n Iteration 2 Now add Goal 1 as x 1 > 200 and graph Goal 2: x 1 + x 2 = 500. Note on the next slide that there is still a set of points satisfying the first goal that also satisfies this second goal (where d 2 - = 0). Goal Programming: Graphical Solution

18 Slide © 2005 Thomson/South-Western n Goal 1 (Constraint) and Goal 2 Graphed 2 x 1 + x 2 < 1000 Goal 1: x 1 > 200 x 1 + x 2 < 600 x 2 < 500 Points Satisfying Both Goals 1 and 2 x1x1x1x1 x2x2x2x2 Goal 2: x 1 + x 2 > 500 Goal Programming: Graphical Solution 200 400 600 800 1000 1200 1000800600400200

19 Slide © 2005 Thomson/South-Western n Iteration 3 Now add Goal 2 as x 1 + x 2 > 500 and Goal 3:.2 x 1 +.5 x 2 = 250. Note on the next slide that no points satisfy the previous functional constraints and goals and satisfy this constraint. Thus, to Min d 3 -, this minimum value is achieved when we Max.2 x 1 +.5 x 2. Note that this occurs at x 1 = 200 and x 2 = 400, so that.2 x 1 +.5 x 2 = 240 or d 3 - = 10. Goal Programming: Graphical Solution

20 Slide © 2005 Thomson/South-Western n Goal 2 (Constraint) and Goal 3 Graphed 2 x 1 + x 2 < 1000 Goal 1: x 1 > 200 x 1 + x 2 < 600 x 2 < 500 Points Satisfying Both Goals 1 and 2 x1x1x1x1 x2x2x2x2 Goal 2: x 1 + x 2 > 500 Goal 3:.2 x 1 +.5 x 2 = 250 (200,400) Goal Programming: Graphical Solution 200 400 600 800 1000 1200 1000800600400200

21 Slide © 2005 Thomson/South-Western Scoring Model for Job Selection A graduating college student with a double major in Finance and Accounting has received the following three job offers: financial analyst for an investment financial analyst for an investment firm in Chicago firm in Chicago accountant for a manufacturing accountant for a manufacturing firm in Denver firm in Denver auditor for a CPA firm in Houston auditor for a CPA firm in Houston

22 Slide © 2005 Thomson/South-Western Scoring Model for Job Selection n The student made the following comments: “The financial analyst position “The financial analyst position provides the best opportunity for my long-run career advancement.” “I would prefer living in Denver “I would prefer living in Denver rather than in Chicago or Houston.” “I like the management style and “I like the management style and philosophy at the Houston CPA firm the best.” n Clearly, this is a multicriteria decision.

23 Slide © 2005 Thomson/South-Western Scoring Model for Job Selection n Considering only the long-run career advancement criterion: advancement criterion: The financial analyst position in The financial analyst position in Chicago is the best decision alternative. n Considering only the location criterion: The accountant position in Denver The accountant position in Denver is the best decision alternative. n Considering only the style criterion: The auditor position in Houston is the best alternative. The auditor position in Houston is the best alternative.

24 Slide © 2005 Thomson/South-Western Steps Required to Develop a Scoring Model n Step 1: List the decision-making criteria. n Step 2: Assign a weight to each criterion. n Step 3: Rate how well each decision alternative satisfies each criterion. n Step 4: Compute the score for each decision alternative. n Step 5: Order the decision alternatives from highest score to lowest score. The alternative with the highest score is the recommended alternative.

25 Slide © 2005 Thomson/South-Western n Mathematical Model S j =  w i r ij S j =  w i r ij iwhere: r ij = rating for criterion i and decision alternative j S j = score for decision alternative j Scoring Model for Job Selection

26 Slide © 2005 Thomson/South-Western Scoring Model: Step 1 n List of Criteria Career advancement Career advancement Location Location Management Management Salary Salary Prestige Prestige Job Security Job Security Enjoyable work Enjoyable work

27 Slide © 2005 Thomson/South-Western Scoring Model: Step 2 n Five-Point Scale Chosen Importance Weight Importance Weight Very unimportant1 Very unimportant1 Somewhat unimportant2 Somewhat unimportant2 Average importance3 Average importance3 Somewhat important4 Somewhat important4 Very important5 Very important5

28 Slide © 2005 Thomson/South-Western Scoring Model: Step 2 n Assigning a Weight to Each Criterion Criterion Importance Weight Career advancementVery important5 Career advancementVery important5 LocationAverage importance3 LocationAverage importance3 ManagementSomewhat important4 ManagementSomewhat important4 SalaryAverage importance3 SalaryAverage importance3 PrestigeSomewhat unimportant2 PrestigeSomewhat unimportant2 Job securitySomewhat important4 Job securitySomewhat important4 Enjoyable workVery important5 Enjoyable workVery important5

29 Slide © 2005 Thomson/South-Western n Nine-Point Scale Chosen Level of Satisfaction Rating Level of Satisfaction Rating Extremely low1 Extremely low1 Very low2 Very low2 Low3 Low3 Slightly low4 Slightly low4 Average5 Average5 Slightly high6 Slightly high6 High7 High7 Very high8 Very high8 Extremely high9 Extremely high9 Scoring Model: Step 3

30 Slide © 2005 Thomson/South-Western n Rate how well each decision alternative satisfies each criterion. Decision Alternative Decision Alternative Analyst Accountant Auditor Analyst Accountant Auditor Criterion Chicago Denver Houston Criterion Chicago Denver Houston Career advancement8 6 4 Location3 8 7 Management5 6 9 Salary6 7 5 Prestige7 5 4 Job security4 7 6 Enjoyable work8 6 5 Scoring Model: Step 3

31 Slide © 2005 Thomson/South-Western n Compute the score for each decision alternative. Decision Alternative 1 - Analyst in Chicago Decision Alternative 1 - Analyst in Chicago Criterion Weight ( w i ) Rating ( r i 1 ) w i r i 1 Criterion Weight ( w i ) Rating ( r i 1 ) w i r i 1 Career advancement 5 x 8 =40 Location 3 3 9 Management 4 520 Salary 3 618 Prestige 2 714 Job security 4 416 Enjoyable work 5 840 Score 157 Scoring Model: Step 4

32 Slide © 2005 Thomson/South-Western n Compute the score for each decision alternative. S 1 = 5(8)+3(3)+4(5)+3(6)+2(7)+4(4)+5(8) = 157 S 2 = 5(6)+3(8)+4(6)+3(7)+2(5)+4(7)+5(6) = 167 S 3 = 5(4)+3(7)+4(9)+3(5)+2(4)+4(6)+5(5) = 149 Scoring Model: Step 4

33 Slide © 2005 Thomson/South-Western n Compute the score for each decision alternative. Decision Alternative Decision Alternative Analyst Accountant Auditor Analyst Accountant Auditor Criterion Chicago Denver Houston Criterion Chicago Denver Houston Career advancement403020 Location 92421 Management202436 Salary182115 Prestige1410 8 Job security162824 Enjoyable work403025 Score 157 167 149 Score 157 167 149 Scoring Model: Step 4

34 Slide © 2005 Thomson/South-Western n Order the decision alternatives from highest score to lowest score. The alternative with the highest score is the recommended alternative. The accountant position in Denver has the highest score and is the recommended decision alternative. The accountant position in Denver has the highest score and is the recommended decision alternative. Note that the analyst position in Chicago ranks first in 4 of 7 criteria compared to only 2 of 7 for the accountant position in Denver. Note that the analyst position in Chicago ranks first in 4 of 7 criteria compared to only 2 of 7 for the accountant position in Denver. But when the weights of the criteria are considered, the Denver position is superior to the Chicago job. But when the weights of the criteria are considered, the Denver position is superior to the Chicago job. Scoring Model: Step 5

35 Slide © 2005 Thomson/South-Western Scoring Model for Job Selection n Partial Spreadsheet Showing Steps 1 - 3

36 Slide © 2005 Thomson/South-Western Scoring Model for Job Selection n Partial Spreadsheet Showing Formulas of Step 4

37 Slide © 2005 Thomson/South-Western Scoring Model for Job Selection n Partial Spreadsheet Showing Results of Step 4

38 Slide © 2005 Thomson/South-Western End of Lesson 10

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