1. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.1 GOAL PROGRAMMING 1. Opprettholde stabil profitt 2. Øke eller opprettholde gitte markedsandeler 3. Diversifisering av produkter 4. Holde arbeidsstokken på et gitt nivå 5. Forurense minst mulig Målene kan ofte ikke sammenlignes eller kombineres direkte Ulike mål er ofte i konflikt med hverandre

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.2

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.3 Two approaches Weighted goal programming - goals are roughly comparable Preemptive goal programming – Hierarchy of priority levels for the different goals

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.4 MULTIPLE OBJECTIVES In many applications, the planner has more than one objective. The presence of multiple objectives is frequently referred to as the problem of “combining apples and oranges.” Consider a corporate planner whose long-range goals are to: 1. Maximize discounted profits 2. Maximize market share at the end of the planning period 3. Maximize existing physical capital at the end of the planning period

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.5 It is also clear that the goals are conflicting (i.e., there are trade-offs in the sense that sacrificing the requirements on any one goal will tend to produce greater returns on the others. These goals are not commensurate (i.e., they cannot be directly combined or compared). These models, although not applied as often in practice as some of the other models (such as linear programming, forecasting, inventory control, etc.), have been found to be especially useful on problems in the public sector.

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.6 Several approaches to multiple objective models (also called multi-criteria decision making) have been developed: Multi-attribute utility theory Only Goal Programming will be discussed. Developed by Thomas Saaty, AHP helps managers choose between many decision alternatives on the basis of multiple criteria. Search for Pareto optimal solutions via multi-criteria linear programming Analytic Hierarchy Process (AHP) Goal Programming (GP) Introduced by A. Charnes and W.W. Cooper. GP is a heuristic approach to the multiple- objectives model.

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.7 Goal Programming is an extension of Linear Programming that enables the planner to come as close as possible to satisfying various goals and constraints. GOAL PROGRAMMING It allows the decision maker, at least in a heuristic sense, to incorporate his or her preference system in dealing with multiple conflicting goals. GP is sometimes considered to be an attempt to put into a mathematical programming context, the concept of satisficing. Coined by Herbert Simon, it communicates the idea that individuals often do not seek optimal solutions, but rather solutions that are “good enough” or “close enough.”

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.8 Weighted Goal Programming A common characteristic of many management science models (linear programming, integer programming, nonlinear programming) is that they have a single objective function. It is not always possible to fit all managerial objectives into a single objective function. Managerial objectives might include: –Maintain stable profits. –Increase market share. –Diversify the product line. –Maintain stable prices. –Improve worker morale. –Maintain family control of the business. –Increase company prestige. Weighted goal programming provides a way of striving toward several objectives simultaneously.

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.9 Weighted Goal Programming With weighted goal programming, the objective is to –Minimize W = weighted sum of deviations from the goals. –The weights are the penalty weights for missing the goal. Introduce new changing cells, Amount Over and Amount Under, that will measure how much the current solution is over or under each goal. The Amount Over and Amount Under changing cells are forced to maintain the correct value with the following constraints: Level Achieved – Amount Over + Amount Under = Goal

10. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.10 The Dewright Company The Dewright Company is one of the largest producers of power tools in the United States. The company is preparing to replace its current product line with the next generation of products—three new power tools. Management needs to determine the mix of the company’s three new products to best meet the following three goals: 1.Achieve a total profit (net present value) of at least $125 million. 2.Maintain the current employment level of 4,000 employees. 3.Hold the capital investment down to no more than $55 million.

11. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.11 Data for Contribution to the Goals Unit Contribution of Product Factor123Goal Total profit (millions of dollars)12915≥ 125 Employment level (hundreds of employees)534= 40 Capital investment (millions of dollars)578≤ 55

12. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.12 Penalty Weights GoalFactorPenalty Weight for Missing Goal 1Total profit5 (per $1 million under the goal) 2Employment level4 (per 100 employees under the goal) 2 (per 100 employees over the goal) 3Capital investment3 (per $1 million over the goal)

13. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.13 Data for Contribution to the Goals Unit Contribution of Product Factor123Goal Total profit (millions of dollars)12915≥ 125 Employment level (hundreds of employees)534= 40 Capital investment (millions of dollars)578≤ 55

14. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.14 Weighted Goal Programming Formulation for the Dewright Co. Problem LetP i = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize W = 5(Under Goal 1) + 2Over Goal 2) + 4 (Under Goal 2) + 3 (Over Goal 3) subject to Level Achieved DeviationsGoal Goal 1: 12P 1 + 9P 2 + 15P 3 – (Over Goal 1) + (Under Goal 1) =125 Goal 2:5P 1 + 3P 2 + 4P 3 – (Over Goal 2) + (Under Goal 2) = 40 Goal 3:5P 1 + 7P 2 + 8P 3 – (Over Goal 3) + (Under Goal 3) =55 and P i ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

15. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.15 Weighted Goal Programming Spreadsheet

16. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.16 Suppose that we have an educational program design model with decision variables x 1 and x 2, where x 1 is the hours of classroom work x 2 is the hours of laboratory work Assume the following constraint on total program hours: x 1 + x 2 < 100 (total program hours) Two Kinds of Constraints In the goal programming approach, there are two kinds of constraints: 1. System constraints (so-called hard constraints) that cannot be violated. 2. Goal constraints (so-called soft constraints) that may be violated if necessary.

17. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.17 Now, suppose that each hour of classroom work involves 12 minutes of small-group experience and 19 minutes of individual problem solving Each hour of laboratory work involves 29 minutes of small-group experience and 11 minutes of individual problem solving The total program time is at most 6,000 minutes (100 hr * 60 min/hr). There are two goals: Each student should spend as close as possible to ¼ of the maximum program time working in small groups and ¹/ 3 of the time on problem solving.

18. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.18 These conditions are: 12x 1 + 29x 2 1500 (small-group experience) ~= 19x 1 + 11x 2 2000 (individual problem solving) ~= Where means that the left-hand side is desired to be “as close as possible” to the right-hand side. ~= In order to satisfy the system constraint, at least one of the two goals will be violated. To implement the goal programming approach, the small-group experience condition is rewritten as the goal constraint: 12x 1 + 29x 2 + u 1 – v 1 = 1500 (u 1 > 0, v 1 > 0) Where u 1 = the amount by which total small-group experience falls short of 1500 v 1 = the amount by which total small-group experience exceeds 1500

19. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.19 Deviation Variables Variables u 1 and v 1 are called deviation variables since they measure the amount by which the value produced by the solution deviates from the goal. Note that by definition, we want either u 1 or v 1 (or both) to be zero because it is impossible to simultaneously exceed and fall short of 1500. In order to make 12x 1 + 29x 2 as close as possible to 1500, it suffices to make the sum u 1 + v 1 small. The individual problem-solving condition is written as the goal constraint: 19x 1 + 11x 2 + u 2 – v 2 = 2000 (u 2 > 0, v 2 > 0) As before, the sum of u 2 + v 2 should be small.

20. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.20 The complete (illustrative) model is: 12x 1 + 29x 2 + u 1 – v 1 = 1500 (small-group experience) 19x 1 + 11x 2 + u 2 – v 2 = 2000 (problem solving) s.t. x 1 + x 2 < 100 (total program hours) x 1, x 2, u 1, v 1, u 2, v 2 > 0 Now this is an ordinary LP model and can be easily solved in Excel. The optimal decision variables will satisfy the system constraint (total program hours). Min u 1 + v 1 + u 2 + v 2 Note: Both u 1 and v 1 can be 0

21. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.21 Solver will guarantee that either u 1 or v 1 (or both) will be zero, and thus these variables automatically satisfy this desired condition. Note that the objective function is the sum of the deviation variables. This choice of an objective function indicates that there is no preference among the various deviations from the stated goals. The same statement holds for u 2 and v 2 and in general for any pair of deviation variables.

22. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.22 For example, any of the following three decisions is acceptable: 1. A decision that overachieves the group experience goal by 5 minutes and hits the problem-solving goal exactly, 2. A decision that hits the group experience goal exactly and underachieves the problem- solving goal by 5 minutes, and 3. A decision that underachieves each goal by 2.5 minutes.

23. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.23 There is no preference among the following three solutions because each of these yields the same value (i.e., 5) for the objective function. u 1 = 0 v 1 = 5 u 2 = 0 v 2 = 0 (1) u 1 = 0 v 1 = 0 u 2 = 5 v 2 = 0 (2) u 1 = 2.5 v 1 = 0 u 2 = 2.5 v 2 = 0 (3)

24. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.24 Weighting the Deviation Variables Differences in units alone could produce a preference among the deviation variables. One way of expressing a preference among the various goals is to assign different coefficients (weights) to the deviation variables in the objective function. as the objective function. Since v 2 (over- achievement of problem solving) has the smallest coefficient, the program designers would rather have v 2 positive than any of the other deviation variables (positive v 2 is penalized the least). Min 10u 1 + 2v 1 + 20u 2 + v 2 In the program-planning example, one might select

25. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.25 With this objective function it is better to be 9 minutes over the problem-solving goal than to underachieve by 1 minute the small-group- experience goal. To see this, note that for any solution in which u 1 > 1, decreasing u 1 by 1 and increasing v 2 by 9 would yield a smaller value for the objective function.

26. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.26 Goal Interval Constraints Another type of goal constraint is called a goal interval constraint. Such a constraint restricts the goal to a range or interval rather than a specific numerical value. Suppose, for example, that in the previous illustration the designers were indifferent among programs for which 1800 < [minutes of individual problem solving] < 2100 i.e., 1800 < 19x 1 + 11x 2 < 2100 In this situation the interval goal is captured with two goal constraints: 19x 1 + 11x 2 – v 1 0) 19x 1 + 11x 2 + u 1 > 1800 (u 1 > 0)

27. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.27 When the terms u 1 and v 1 are included in the objective function, the LP code will attempt to minimize them. Summary of the Use of Goal Constraints Each goal constraint consists of a left-hand side, say g i (x 1, …, x n ), and a right-hand side, b i. Goal constraints are written by using nonnegative deviation variables u i, v i. At optimality at least one of the pair u i, v i will always be zero. u i represents underachievement; v i represents overachievement. Whenever u i is used it is added to g i (x 1, …, x n ). Whenever v i is used it is subtracted from g i (x 1, …, x n ).

28. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.28 Only deviation variables appear in the objective function, and the objective is always to minimize. The decision variables x i, i = 1, …, n do not appear in the objective. Four types of goals have been discussed: 1. Target. Make g i (x 1, …, x n ) as close as possible as possible to b i. To do this write the goal constraint as g i (x 1, …, x n ) + u i - v i = b i (u i > 0, v i > 0)

29. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.29 g i (x 1, …, x n ) + u i - v i = b i (u i > 0, v i > 0) 2. Minimize Underachievement. To do this, write and in the objective, minimize u i, the under- achievement. v i does not appear in the objective function and it is only in this constraint, hence, the constraint can be equivalently written as g i (x 1, …, x n ) + u i > b i (u i > 0) If the optimal u i is positive, this constraint will be active, for otherwise u i * could be made smaller. If u i *>0 then, since v i * must equal zero, it must be true that g i (x 1, …, x n ) + u i * = b i.

30. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.30 g i (x 1, …, x n ) + u i - v i = b i (u i > 0, v i > 0) 3. Minimize Overachievement. To do this, write and in the objective, minimize v i, the over- achievement. u i does not appear in the objective function, the constraint can be equivalently written as g i (x 1, …, x n ) - v i 0) If the optimal v i is positive, this constraint will be active. The argument is analogous to that in item 2.

31. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.31 4. Goal Interval Constraint. In this instance, the goal is to come as close as possible to satisfying a i < g i (x 1, …, x n ) < b i In order to write this as a goal, first “stretch out” the interval by writing a i - u i 0, v i > 0) which is equivalent to the two constraints g i (x 1, …, x n ) + u i > a i g i (x 1, …, x n ) + u i - v i + a i (u i > 0, v i > 0) ^^ g i (x 1, …, x n ) - u i > b i g i (x 1, …, x n ) + u i - v i + b i (u i > 0, v i > 0) ^^ The objective function u i + v i is minimized. Variables u i and v i are merely surplus and slack, respectively. ^^

32. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.32 Weighted vs. Preemptive Goal Programming Weighted goal programming is designed for problems where all the goals are quite important, with only modest differences in importance that can be measured by assigning weights to the goals. Preemptive goal programming is used when there are major differences in the importance of the goals. –The goals are liested in the order of their importance. –It begins by focusing solely on the most important goal. –It next does the same for the second most important goal (as is possible without hurting the first goal). –It continues the the following goals (as is possible without hurting the previous more important goals).

33. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.33 In some cases, managers do not wish to express their preferences among various goals in terms of weighted deviation variables, for the process of assigning weights may seem too arbitrary or subjective. ABSOLUTE PRIORITIES In such cases, it may be more acceptable to state preferences in terms of absolute priorities (as opposed to weights) to a set of goals. This approach requires that goals be satisfied in a specific order. Therefore, the model is solved in stages as a sequence of models.

34. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.34 Preemptive Goal Programming Introduce new changing cells, Amount Over and Amount Under, that will measure how much the current solution is over or under each goal. The Amount Over and Amount Under changing cells are forced to maintain the correct value with the following constraints: Level Achieved – Amount Over + Amount Under = Goal Start with the objective of achieving the first goal (or coming as close as possible): –Minimize (Amount Over/Under Goal 1) Continue with the next goal, but constrain the previous goals to not get any worse: –Minimize (Amount Over/Under Goal 2) subject to Amount Over/Under Goal 1 = (amount achieved in previous step) Repeat the previous step for all succeeding goals.

35. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.35 Preemptive Goal Programming for Dewright The goals in the order of importance are: 1.Achieve a total profit (net present value) of at least $125 million. 2.Avoid decreasing the employment level below 4,000 employees. 3.Hold the capital investment down to no more than $55 million. 4.Avoid increasing the employment level above 4,000 employees. Start with the objective of achieving the first goal (or coming as close as possible): –Minimize (Under Goal 1) Then, if for example goal 1 is achieved (i.e., Under Goal 1 = 0), then –Minimize (Under Goal 2) subject to (Under Goal 1) = 0

36. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.36 Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 1) LetP i = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Under Goal 1) subject to Level Achieved DeviationsGoal Goal 1: 12P 1 + 9P 2 + 15P 3 – (Over Goal 1) + (Under Goal 1) =125 Goal 2:5P 1 + 3P 2 + 4P 3 – (Over Goal 2) + (Under Goal 2) = 40 Goal 3:5P 1 + 7P 2 + 8P 3 – (Over Goal 3) + (Under Goal 3) =55 and P i ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

37. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.37 Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 2) LetP i = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Under Goal 2) subject to Level Achieved DeviationsGoal Goal 1: 12P 1 + 9P 2 + 15P 3 – (Over Goal 1) + (Under Goal 1) =125 Goal 2:5P 1 + 3P 2 + 4P 3 – (Over Goal 2) + (Under Goal 2) = 40 Goal 3:5P 1 + 7P 2 + 8P 3 – (Over Goal 3) + (Under Goal 3) =55 (Under Goal 1) = (Level Achieved in Step 1) and P i ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

38. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.38 Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 3) LetP i = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Over Goal 3) subject to Level Achieved DeviationsGoal Goal 1: 12P 1 + 9P 2 + 15P 3 – (Over Goal 1) + (Under Goal 1) =125 Goal 2:5P 1 + 3P 2 + 4P 3 – (Over Goal 2) + (Under Goal 2) = 40 Goal 3:5P 1 + 7P 2 + 8P 3 – (Over Goal 3) + (Under Goal 3) =55 (Under Goal 1) = (Level Achieved in Step 1) (Under Goal 2) = (Level Achieved in Step 2) and P i ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

39. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.39 Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 4) LetP i = Number of units of product i to produce per day (i = 1, 2, 3), Under Goal i = Amount under goal i (i = 1, 2, 3), Over Goal i = Amount over goal i (i = 1, 2, 3), Minimize (Over Goal 2) subject to Level Achieved DeviationsGoal Goal 1: 12P 1 + 9P 2 + 15P 3 – (Over Goal 1) + (Under Goal 1) =125 Goal 2:5P 1 + 3P 2 + 4P 3 – (Over Goal 2) + (Under Goal 2) = 40 Goal 3:5P 1 + 7P 2 + 8P 3 – (Over Goal 3) + (Under Goal 3) =55 (Under Goal 1) = (Level Achieved in Step 1) (Under Goal 2) = (Level Achieved in Step 2) (Over Goal 3) = (Level Achieved in Step 3) and P i ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)

40. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.40 Preemptive Goal Programming Spreadsheet Step 1: Minimize (Under Goal 1)

41. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.41 Preemptive Goal Programming Spreadsheet Step 3: Minimize (Over Goal 3)

42. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.42 Preemptive Goal Programming Spreadsheet Step 4: Minimize (Over Goal 2)

43. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.43 Example: Swenson’s Media Selection Model J. R. Swenson is an advertising agency which has just completed an agreement with a pharmaceutical manufacturer to mount a radio and television campaign to introduce a new product, Mylonal. The total expenditures for the campaign are not to exceed $120,000. The client wants to reach several audiences, however, radio and television are not equally effective in reaching all audiences. Therefore, the agency will estimate the impact of the advertisements in terms of rated exposures (i.e., “people reached per month”) on the audiences of interest.

44. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.44 The following data represent the number of exposures per $1000 expenditure: TV RADIO Total14,0006,000 Upper Income 1,2001,200 The following are the campaign goals, listed in order of absolute priority. 1. Total exposures will hopefully be at least 840,000. 2. In order to maintain effective contact with the leading radio station, no more than $90,000 will be spent on TV advertising.

45. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.45 3. The campaign should achieve at least 168,000 upper-income exposures. 4. If all other goals are satisfied, the total number of exposures would come as close as possible to being maximized. Note that if all of the $120,000 is spent on TV advertising, then the maximum obtainable exposures would be 1,680,000 (120*14,000). To model the problem, the following notation will be used: x 1 = dollars spent on TV ( in thousands) x 2 = dollars spent on radio (in thousands) The objective function will be to maximize total exposures and the other goals will be treated as constraints.

46. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.46 Infeasible LP model Media Selection TVRadioMaximize X1X2OF Total Exposures (thousands)146840 Expenditures (th's)15105 Technological coeffisientsLHS RHS Slack or surplu s Max Expenditures (th's)11120<=1201.6485E-11 Min Exposures (th's)146840>=840-3.7517E-10 Max TV (th's)1 15<=9075 Min Upper-income Exposures (th's)1.2 144>=168-24

47. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.47 Since there are only two decision variables in this model, the graphical approach can be used. < 140 x2x2 x1x1 120 140 X 1 = 90 X 1 + X 2 = 120 1200X 1 +1200X 2 = 168,000 < > The graph shows that there are no points that satisfy all the constraints.

48. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.48 Swenson’s Goal Programming Model Note that the first goal (total exposures will be at least 840,000), if violated, will be underachieved. The second goal (no more than $90,000 will be spent on TV advertising), if violated, will be overachieved, etc. Employing this reasoning, the goals are restated, in descending priority, as: 1. Minimize the underachievement of 840,000 total exposures. Min u 1 subject to the condition 14,000x 1 + 6,000x 2 + u 1 > 840,000; u 1 > 0

49. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.49 2. Minimize expenditures in excess of $90,000 on TV Min v 2 subject to the condition x 1 – v 2 0 3. Minimize underachievement of 168,000 upper- income exposures Min u 3 subject to the condition 1,200x 1 + 1,200x 2 + u 3 > 168,000; u 3 > 0 4. Minimize underachievement of 1,680,000 total exposures (the maximum possible) Min u 4 subject to the condition 14,000x 1 + 6,000x 2 + u 4 > 1,680,000; u 4 > 0

50. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.50 Note that the goals are now stated in terms of either minimizing underachievement (i.e., min. u i ) or minimizing overachievement (i.e., min. v i ). In addition, the goals have been expressed as inequalities. This method will facilitate a graphical analysis. Given that the priorities are formulated correctly, we must now distinguish between 1. system constraints (all constraints that may not be violated) 2. goal constraints The only system constraint is: Total expenditures will be no greater than $120,000 x 1 + x 2 < 120

51. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.51 The model can now be expressed as: Min P 1 u 1 + P 2 v 2 + P 3 u 3 + P 4 u 4 s.t. x 1 + x 2 < 120(S) 14,000 x 1 + 6,000x 2 + u 1 > 840,000(1) 14,000 x 1 + 6,000x 2 + u 1 > 840,000(1) x 1 - v 2 < 90(2) x 1 - v 2 < 90(2) 1,200 x 1 + 1,200x 2 + u 3 > 168,000(3) 1,200 x 1 + 1,200x 2 + u 3 > 168,000(3) 14,000 x 1 + 6,000x 2 + u 4 >1,680,000(4) 14,000 x 1 + 6,000x 2 + u 4 >1,680,000(4) x 1, x 2, u 1, v 2, u 3, u 4 > 0 Note that the objective function consists only of deviation variables and is of the Min form. In the objective function, P 1 denotes the highest priority, and so on.

52. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.52 The previous problem statement precisely means: 1. Find the set of decision variables that satisfies the system constraint (S) and that also gives the Min possible value to u 1 subject to constraint (1) and x 1, x 2, u 1 > 0. Call this set of decisions FR I (i.e., feasible region I). Considering only the highest goal, all of the points in FR I are “optimal” and (again considering only the highest goal), we are indifferent as to which of these points are selected.

53. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.53 2. Find the subset of points in FR I that gives the Min possible value to v 2, subject to constraint (2) and v 2 > 0. Call this subset FR II. Considering only the ordinal ranking of the two highest-priority goals, all of the points in FR II are “optimal,” and in terms of these two highest-priority goals, we are indifferent as to which of these points are selected. 3. Let FR III be the subset of points in FR II that minimize u 3, subject to constraint (3) and u 3 > 0. 4. FR IV is the subset of points in FR III that minimize u 4, subject to constraint (4) and u 4 > 0. Any point in FR IV is an optimal solution to the model.

54. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.54 Graphical Analysis and Spreadsheet Implementation of the Solution Procedure Since there are only two decision variables, we can use the graphical method of LP. 1. Both the spreadsheet output and the geometry reveal the the Min of u 1 s.t. (S), (1), and x 1, x 2, u 1 > 0 is u 1 * = 0. The important information is that u 1 = 0 which tells us that the first goal can be completely attained. Alternative optima for the current model are provided by all values of (x 1, x 2 ) that satisfy the conditions x 1 + x 2 < 120 14,000x 1 + 6,000x 2 > 840,000 x 1, x 2 > 0 FR I

55. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.55 Media Selection - GP model TVRadio Minimiz e X1X2U1V2U3U4OF Total Exposures (thousands) 111154 Expenditures (th's)1200030240 Technological coeffisientsLHS RHS Slack or surplus Max Expenditures (th's)11 120 <=<= 0 Min Exposures (th's)1461 1680 >=>=840 Max TV (th's)1 90 <=<= 0 Min Upper-income Exposures (th's)1.2 1 168 >=>= 0 Exposures Target (th's)146 11680 >=>= 0

56. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.56 Ra nk Media Selection -Step 1 TVRadio Minimize X1X2U1V2U3U4OF Penalty 1 10 Expenditures (th's)6000000 Technological coeffisientsLHS RHS Slack or surplus Max Expenditures (th's)11 60<=12060 1Min Exposures (th's)1461 840>=8402.121E-10 2 3 4

57. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.57 u 1 = 0 At any such point, the goal is attained (u 1 * = 0) so that, in terms of only the first goal, these decisions are equally preferable. Thus FR I is the shaded area ABC.

58. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.58 Ra nk Media Selection -Step 2 TVRadio Minimize X1X2U1V2U3U4OF Penalty 1 10 Expenditures (th's)6000000 Technological coeffisientsLHS RHS Slack or surplus Max Expenditures (th's)11 60<=12060 1Min Exposures (th's)1461 840>=8400 2Max TV (th's)1 60<=9030 3 4 Value of U1 found in step 1 1 0=00

59. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.59 u 1 = 0 v 1 = 0 We see that: Min v 2 such that x in FR I, goal (2) and v 2 > 0 is v 2 * = 0. x 1, x 2 > 0 Thus, FR II is defined by x 1 + x 2 < 120 14,000x 1 + 6,000x 2 > 840,000 x 1 < 90 x 1, x 2 > 0 FR II The shaded area ABDE is a subset of FR I and as expected, the size of the feasible region is smaller.

60. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.60 Ra nk Media Selection - Step 3 TVRadio Minimize X1X2U1V2U3U4OF Penalty 1124 Expenditures (th's)1510500240 Technological coeffisientsLHS RHS Slack or surplus Max Expenditures (th's)11 120<=1200 1Min Exposures (th's)1461 840>=8400 2Max TV (th's)1 15<=9075 3Min Upper-income Exposures (th's)1.2 1 168>=1680 Value of U1 found in step 1 1 0=00 Value of V2 found in step 2 1 0=00 4

61. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.61 FR III is the line segment BD. In this case u 3 * = 24,000. Although the first two goals were completely attained (since u 1 * = v 2 * = 0), the third goal cannot be completely attained because u 3 * > 0. x 1 + x 2 < 120 14,000x 1 + 6,000x 2 > 840,000 x 1 < 90 1,200x 1 + 1,200x 2 > 168,000 – 24,000 = 144,000 FR III

62. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.62 Ra nk Media Selection - Step 4 TVRadio Minimize X1X2U1V2U3U4OF Penalty 1240 Expenditures (th's)90300024240 Technological coeffisientsLHS RHS Slack or surplus Max Expenditures (th's)11 120<=1200 1Min Exposures (th's)1461 1440>=840600 2Max TV (th's)1 90<=900 3Min Upper-income Exposures (th's)1.2 1 168>=1680 4Exposures Target (th's)146 11680>=16800 Value of U1 found in step 1 1 0=00 Value of V2 found in step 2 1 0=00 Value of U3 found in step 3 1 24= 0

63. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.63 Recall that the fourth goal is to minimize underachievement of the maximum possible number of exposures, which is 1,680,000. 14,000x 1 + 6,000x 2 + u 4 > 1,680,000 Thus, we wish to minimize the underachievement u 4 where The unique optimum is x 1 * = 90, x 2 * = 30 (i.e., spend $90,000 on TV ads & $30,000 on radio ads). Since u 4 = 240,000, we achieve 1,680,000 - 240,000 = 1,440,000 exposures.

64. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.64 In reviewing the results of the absolute priority study, the older members of the Mylonal market begins to take on importance. COMBINING WEIGHTS AND ABSOLUTE PRIORITIES The exposures per $1000 of advertising are: TV RADIO 50 and over 3,000 8,000 EXPOSURE GROUP Note that radio and TV exposures are not equally effective in generating exposures in this segment of the population.

65. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.65 If there were no other considerations, then we would like as many 50-and-over exposures as possible. Since radio yields such exposures at a higher rate than TV (8000 > 3000), the maximum possible number of 50-and-over exposures would be achieved by allocating all of the $120,000 available to radio. Thus, the maximum number of 50-and-over exposures is 120 x 8000 = 960,000. Once the first three goals are satisfied, we would like to come as close as possible to minimizing underachievement. To resolve this conflict of goals, use a weighted sum of the deviation variables as the objective in the final phase of the absolute priorities approach.

66. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.66 Ra nk Wei ght Media Selection - Weighted Step 4 TVRadio Minimize X1X2U1V2U3U4U5OF Penalty 131065 Expenditures (th's)15105002484075 Technological coeffisientsLHS RHS Slack or surplu s Max Expenditures (th's)11 120<=1200 1Min Exposures (th's)1461 840>=8400 2Max TV (th's)1 15<=9075 3Min Upper-income Exposures (th's)1.2 1 168>=1680 4Exposures Target (th's)146 1 1680>=16800 4Exposures > 50 years (th's)38 1960>=9600 Value of U1 found in step 1 1 0=00 Value of V2 found in step 2 1 0=00 Value of U3 found in step 3 1 24= 0

67. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.67 Note that the new objective function has moved the optimal solution from one end of FR III to the other. This optimal solution is as close as possible to the more heavily weighted goal. Sensitivity analysis on the weights in the objective function could be used to see when the solution changes from point B to point D.

68. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.68 Multi-Objective Decision Making Many problems have multiple objectives: –Planning the national budget save social security, reduce debt, cut taxes, build national defense –Admitting students to college high SAT or GMAT, high GPA, diversity –Planning an advertising campaign budget, reach, expenses, target groups –Choosing taxation levels raise money, minimize tax burden on low-income, minimize flight of business –Planning an investment portfolio maximize expected earnings, minimize risk Techniques –Preemptive goal programming –Weighted goal programming