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Målprogrammering (Goal Programming –GP) Introdusert av Charnes og Cooper i 1961 Flere mål som ønskes å optimeres eller tas hensyn til samtidig i motsetning til for eksempel profittmaksimeringsmodell med ett mål Irwin/McGraw-Hill 11.1 © The McGraw-Hill Companies, Inc., 2003 Eksempler på mål: 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 McGraw-Hill/Irwin 11.2 © The McGraw-Hill Companies, Inc., 2003 To hovedtilnærminger • Vektet målprogrammering - målene er grovt sett sammenlignbare • Leksikografisk målprogrammering - Hierarki av prioritetsnivåer for de ulike målene McGraw-Hill/Irwin 11.3 © The McGraw-Hill Companies, Inc., 2003 Tradisjonelt rammeverk for analyse av beslutningstaking forutsetter: 1. Èn beslutningstaker 2. En begrenset rekke av mulige valg 3. Et veldefinert kriterium (F.eks. nytte eller profitt) Kritikk mot tradisjonell tilnærming • Beslutningstakeren er vanligvis ikke interessert i å ordne de mulige beslutningene i henhold til ett enkelt kriterium, men prøver å finne et optimalt kompromiss blant flere ulike mål McGraw-Hill/Irwin 11.4 © The McGraw-Hill Companies, Inc., 2003 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 McGraw-Hill/Irwin 11.5 © The McGraw-Hill Companies, Inc., 2003 These goals are not commensurate (i.e., they cannot be directly combined or compared). 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 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. McGraw-Hill/Irwin 11.6 © The McGraw-Hill Companies, Inc., 2003 Several approaches to multiple objective models (also called multi-criteria decision making) have been developed: Multi-attribute utility theory Search for Pareto optimal solutions via multi-criteria linear programming Analytic Hierarchy Process (AHP) Developed by Thomas Saaty, AHP helps managers choose between many decision alternatives on the basis of multiple criteria. Goal Programming (GP) Introduced by A. Charnes and W.W. Cooper. GP is a heuristic approach to the multipleobjectives model. 11.7 Only Goal Programming will be discussed. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 GOAL PROGRAMMING Goal Programming is an extension of Linear Programming that enables the planner to come as close as possible to satisfying various goals and constraints. 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 11.8 “close enough.” McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 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. McGraw-Hill/Irwin 11.9 © The McGraw-Hill Companies, Inc., 2003 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 McGraw-Hill/Irwin 11.10 © The McGraw-Hill Companies, Inc., 2003 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. McGraw-Hill/Irwin 11.11 © The McGraw-Hill Companies, Inc., 2003 Data for Contribution to the Goals Unit Contribution of Product Factor 1 2 3 Goal Total profit (millions of dollars) 12 9 15 ≥ 125 Employment level (hundreds of employees) 5 3 4 = 40 Capital investment (millions of dollars) 5 7 8 ≤ 55 McGraw-Hill/Irwin 11.12 © The McGraw-Hill Companies, Inc., 2003 Penalty Weights Goal Factor Penalty Weight for Missing Goal 1 Total profit 5 (per $1 million under the goal) 2 Employment level 4 (per 100 employees under the goal) 2 (per 100 employees over the goal) 3 Capital investment 3 (per $1 million over the goal) McGraw-Hill/Irwin 11.13 © The McGraw-Hill Companies, Inc., 2003 Data for Contribution to the Goals Unit Contribution of Product Factor 1 2 3 Goal Total profit (millions of dollars) 12 9 15 ≥ 125 Employment level (hundreds of employees) 5 3 4 = 40 Capital investment (millions of dollars) 5 7 8 ≤ 55 McGraw-Hill/Irwin 11.14 © The McGraw-Hill Companies, Inc., 2003 Weighted Goal Programming Formulation for the Dewright Co. Problem Let Pi = 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 Deviations Goal Goal 1: 12P1 + 9P2 + 15P3 – (Over Goal 1) + (Under Goal 1) = 125 Goal 2: 5P1 + 3P2 + 4P3 – (Over Goal 2) + (Under Goal 2) = 40 Goal 3: 5P1 + 7P2 + 8P3 – (Over Goal 3) + (Under Goal 3) = 55 and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) McGraw-Hill/Irwin 11.15 © The McGraw-Hill Companies, Inc., 2003 Weighted Goal Programming Spreadsheet B 3 4 5 6 7 8 9 10 11 12 13 14 15 Goal 1 (Prof it) Goal 2 (Employ ment) Goal 3 (Inv estment) Units Produced C D E Contribution per Unit Produced Product 1 Product 2 Product 3 12 9 15 5 3 4 5 7 8 Product 1 8.33333333 McGraw-Hill/Irwin Product 2 0 Product 3 1.66666667 F G Goals H Lev el Achiev ed 125 >= 48.333333 = 55 <= Goal 125 40 55 Penalty Weights Prof it Employ ment Inv estment 11.16 I J K Deviations Amount Amount Ov er Under 0 0 8.333333 0 0 0 Ov er Goal 2 3 Under Goal 5 4 L M Constraints Balance (Lev el - Ov er + Under) 125 40 55 N O = = = Goal 125 40 55 Weighted Sum of Dev iations 16.66666667 © The McGraw-Hill Companies, Inc., 2003 Suppose that we have an educational program design model with decision variables x1 and x2, where x1 is the hours of classroom work x2 is the hours of laboratory work Assume the following constraint on total program hours: x1 + x2 < 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 violated11.17 if necessary. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 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 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 11.18 ¹/3 of the time on problem solving. These conditions are: 12x + 29x ~ = 1500 (small-group experience) 1 2 19x1 + 11x2 ~ = 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: 12x1 + 29x2 + u1 – v1 = 1500 (u1 > 0, v1 > 0) Where u1 = the amount by which total small-group experience falls short of 1500 v1 = the amount by which total small-group 11.19 experience exceeds 1500 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 Deviation Variables Variables u1 and v1 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 u1 or v1 (or both) to be zero because it is impossible to simultaneously exceed and fall short of 1500. In order to make 12x1 + 29x2 as close as possible to 1500, it suffices to make the sum u1 + v1 small. The individual problem-solving condition is written as the goal constraint: 19x1 + 11x2 + u2 – v2 = 2000 (u2 > 0, v2 > 0) As before, the sum of u2 +11.20 v2 should be small. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 The complete (illustrative) model is: Min u1 + v1 + u2 + v2 s.t. x1 + x2 < 100 (total program hours) 12x1 + 29x2 + u1 – v1 = 1500 (small-group experience) 19x1 + 11x2 + u2 – v2 = 2000 (problem solving) x1, x2 , u1, v1, u2, v2 > 0 Note: Both u1 and v1 can be 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). McGraw-Hill/Irwin 11.21 © The McGraw-Hill Companies, Inc., 2003 Solver will guarantee that either u1 or v1 (or both) will be zero, and thus these variables automatically satisfy this desired condition. The same statement holds for u2 and v2 and in general for any pair of deviation variables. 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. McGraw-Hill/Irwin 11.22 © The McGraw-Hill Companies, Inc., 2003 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 problemsolving goal by 5 minutes, and 3. A decision that underachieves each goal by 2.5 minutes. McGraw-Hill/Irwin 11.23 © The McGraw-Hill Companies, Inc., 2003 There is no preference among the following three solutions because each of these yields the same value (i.e., 5) for the objective function. (1) u1 = 0 v1 = 5 u2 = 0 v2 = 0 McGraw-Hill/Irwin (2) u1 = 0 v1 = 0 u2 = 5 v2 = 0 11.24 (3) u1 = 2.5 v1 = 0 u2 = 2.5 v2 = 0 © The McGraw-Hill Companies, Inc., 2003 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. In the program-planning example, one might select Min 10u1 + 2v1 + 20u2 + v2 as the objective function. Since v2 (overachievement of problem solving) has the smallest coefficient, the program designers would rather have v2 positive than any of the other deviation variables (positive v2 is penalized the least). McGraw-Hill/Irwin 11.25 © The McGraw-Hill Companies, Inc., 2003 With this objective function it is better to be 9 minutes over the problem-solving goal than to underachieve by 1 minute the small-groupexperience goal. To see this, note that for any solution in which u1 > 1, decreasing u1 by 1 and increasing v2 by 9 would yield a smaller value for the objective function. McGraw-Hill/Irwin 11.26 © The McGraw-Hill Companies, Inc., 2003 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 < 19x1 + 11x2 < 2100 In this situation the interval goal is captured with two goal constraints: 19x1 + 11x2 – v1 < 2100 (v1 > 0) 19x1 + 11x2 + u111.27> 1800 (u1 > 0) McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 When the terms u1 and v1 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 gi(x1, …, xn), and a right-hand side, bi. Goal constraints are written by using nonnegative deviation variables ui, vi. At optimality at least one of the pair ui, vi will always be zero. ui represents underachievement; vi represents overachievement. Whenever ui is used it is added to gi(x1, …, xn). Whenever vi is used it is subtracted from 11.28 gi(x1, …, xn). McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 Only deviation variables appear in the objective function, and the objective is always to minimize. The decision variables xi, i = 1, …, n do not appear in the objective. Four types of goals have been discussed: 1. Target. Make gi(x1, …, xn) as close as possible as possible to bi. To do this write the goal constraint as gi(x1, …, xn) + ui - vi = bi McGraw-Hill/Irwin 11.29 (ui > 0, vi > 0) © The McGraw-Hill Companies, Inc., 2003 2. Minimize Underachievement. To do this, write gi(x1, …, xn) + ui - vi = bi (ui > 0, vi > 0) and in the objective, minimize ui, the underachievement. vi does not appear in the objective function and it is only in this constraint, hence, the constraint can be equivalently written as gi(x1, …, xn) + ui > bi (ui > 0) If the optimal ui is positive, this constraint will be active, for otherwise ui* could be made smaller. If ui*>0 then, since vi* must equal zero, it must be true that gi(x1, …,11.30xn) + ui* = bi . McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 3. Minimize Overachievement. To do this, write gi(x1, …, xn) + ui - vi = bi (ui > 0, vi > 0) and in the objective, minimize vi, the overachievement. ui does not appear in the objective function, the constraint can be equivalently written as gi(x1, …, xn) - vi < bi (vi > 0) If the optimal vi is positive, this constraint will be active. The argument is analogous to that in item 2. McGraw-Hill/Irwin 11.31 © The McGraw-Hill Companies, Inc., 2003 4. Goal Interval Constraint. In this instance, the goal is to come as close as possible to satisfying ai < gi(x1, …, xn) < bi In order to write this as a goal, first “stretch out” the interval by writing ai - ui < gi(x1, …, xn) < bi + vi (ui > 0, vi > 0) which is equivalent to the two constraints gi(x1, …, xn) + ui > ai ^ +a gi(x1, …, xn) + ui - v i i ^ > 0) (ui > 0, v i gi(x1, …, xn) - ui > bi ^ -v +b gi(x1, …, xn) + u i i i ^ > 0, v > 0) (u i i The objective function ui + vi is minimized. ^ ^ Variables ui and vi are merely surplus and McGraw-Hill/Irwin slack, respectively. 11.32 © The McGraw-Hill Companies, Inc., 2003 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). McGraw-Hill/Irwin 11.33 © The McGraw-Hill Companies, Inc., 2003 ABSOLUTE PRIORITIES 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. 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. McGraw-Hill/Irwin 11.34 © The McGraw-Hill Companies, Inc., 2003 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. McGraw-Hill/Irwin 11.35 © The McGraw-Hill Companies, Inc., 2003 Preemptive Goal Programming for Dewright The goals in the order of importance are: 1. 2. 3. 4. • Start with the objective of achieving the first goal (or coming as close as possible): – • Achieve a total profit (net present value) of at least $125 million. Avoid decreasing the employment level below 4,000 employees. Hold the capital investment down to no more than $55 million. Avoid increasing the employment level above 4,000 employees. 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 McGraw-Hill/Irwin 11.36 © The McGraw-Hill Companies, Inc., 2003 Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 1) Let Pi = 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 Goal 1: 12P1 + 9P2 + 15P3 Goal 2: 5P1 + 3P2 + 4P3 Goal 3: 5P1 + 7P2 + 8P3 Deviations – (Over Goal 1) + (Under Goal 1) = – (Over Goal 2) + (Under Goal 2) = – (Over Goal 3) + (Under Goal 3) = Goal 125 40 55 and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) McGraw-Hill/Irwin 11.37 © The McGraw-Hill Companies, Inc., 2003 Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 2) Let Pi = 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 Goal 1: 12P1 + 9P2 + 15P3 Goal 2: 5P1 + 3P2 + 4P3 Goal 3: 5P1 + 7P2 + 8P3 Deviations – (Over Goal 1) + (Under Goal 1) = – (Over Goal 2) + (Under Goal 2) = – (Over Goal 3) + (Under Goal 3) = Goal 125 40 55 (Under Goal 1) = (Level Achieved in Step 1) and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) McGraw-Hill/Irwin 11.38 © The McGraw-Hill Companies, Inc., 2003 Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 3) Let Pi = 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 Goal 1: 12P1 + 9P2 + 15P3 Goal 2: 5P1 + 3P2 + 4P3 Goal 3: 5P1 + 7P2 + 8P3 Deviations – (Over Goal 1) + (Under Goal 1) = – (Over Goal 2) + (Under Goal 2) = – (Over Goal 3) + (Under Goal 3) = Goal 125 40 55 (Under Goal 1) = (Level Achieved in Step 1) (Under Goal 2) = (Level Achieved in Step 2) and Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) McGraw-Hill/Irwin 11.39 © The McGraw-Hill Companies, Inc., 2003 Preemptive Goal Programming Formulation for the Dewright Co. Problem (Step 4) Let Pi = 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 Goal 1: 12P1 + 9P2 + 15P3 Goal 2: 5P1 + 3P2 + 4P3 Goal 3: 5P1 + 7P2 + 8P3 Deviations – (Over Goal 1) + (Under Goal 1) = – (Over Goal 2) + (Under Goal 2) = – (Over Goal 3) + (Under Goal 3) = Goal 125 40 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 Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3) McGraw-Hill/Irwin 11.40 © The McGraw-Hill Companies, Inc., 2003 Preemptive Goal Programming Spreadsheet Step 1: Minimize (Under Goal 1) A 1 2 3 4 5 6 7 8 9 10 11 12 B C D E F G H I J K L M N O = = = Goal 125 40 55 Dewright Co. Goal Programming (Preemptive Priority 1: Minimize Under Goal 1) Goal 1 (Prof it) Goal 2 (Employ ment) Goal 3 (Inv estment) Contribution per Unit Produced Product 1 Product 2 Product 3 12 9 15 5 3 4 5 7 8 Units Produced Product 1 3.7037 Goals Lev el Achiev ed Goal 125 >= 125 40 = 40 61.481 <= 55 Deviations Amount Amount Ov er Under 0 0 0 0 6.481 0 Constraints Balance (Lev el - Ov er + Under) 125 40 55 Minimize (Under Goal 1) McGraw-Hill/Irwin Product 2 0 Product 3 5.3704 11.41 © The McGraw-Hill Companies, Inc., 2003 Preemptive Goal Programming Spreadsheet Step 3: Minimize (Over Goal 3) A 1 2 3 4 5 6 7 8 9 10 11 12 B C D E F G H I J K L M N O = = = Goal 125 40 55 Dewright Co. Goal Programming (Preemptive Priority 3: Minimize Over Goal 3) Goals Goal 1 (Prof it) Goal 2 (Employ ment) Goal 3 (Inv estment) Contribution per Unit Produced Product 1 Product 2 Product 3 12 9 15 5 3 4 5 7 8 Units Produced Product 1 8.333 McGraw-Hill/Irwin Product 2 0 Lev el Achiev ed 125 48.333 55 >= = <= Goal 125 40 55 Deviations Amount Amount Ov er Under 0 0 8.333333 0 0 0 Constraints Balance (Lev el - Ov er + Under) 125 40 55 Minimize (Ov er Goal 3) (Under Goal 1) = 0 (Under Goal 2) = 0 Product 3 1.667 11.42 © The McGraw-Hill Companies, Inc., 2003 Preemptive Goal Programming Spreadsheet Step 4: Minimize (Over Goal 2) A 1 2 3 4 5 6 7 8 9 10 11 12 13 B C D E F G H I J K L M N O = = = Goal 125 40 55 Dewright Co. Goal Programming (Preemptive Priority 4: Minimize Over Goal 2) Goals Goal 1 (Prof it) Goal 2 (Employ ment) Goal 3 (Inv estment) Contribution per Unit Produced Product 1 Product 2 Product 3 12 9 15 5 3 4 5 7 8 Units Produced Product 1 8.333 McGraw-Hill/Irwin Product 2 0 Lev el Achiev ed 125 48.333 55 >= = <= Goal 125 40 55 Deviations Amount Amount Ov er Under 0 0 8.333 0 0 0 Constraints Balance (Lev el - Ov er + Under) 125 40 55 Minimize (Ov er Goal 2) (Under Goal 1) = 0 (Under Goal 2) = 0 (Ov er Goal 3) = 0 Product 3 1.667 11.43 © The McGraw-Hill Companies, Inc., 2003 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. McGraw-Hill/Irwin 11.44 © The McGraw-Hill Companies, Inc., 2003 The following data represent the number of exposures per $1000 expenditure: Total Upper Income TV RADIO 14,000 1,200 6,000 1,200 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 11.45 advertising. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 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: x1 = dollars spent on TV ( in thousands) x2 = dollars spent on radio (in thousands) The objective function will be to maximize total exposures and the other goals will be treated as 11.46 constraints. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 Infeasible LP model Media Selection TV Radio Maximize X1 X2 OF Total Exposures (thousands) 14 6 840 Expenditures (th's) 15 105 Technological coeffisients LHS RHS Slack or surplu s Max Expenditures (th's) 1 1 120 <= 120 1.6485E-11 Min Exposures (th's) 14 6 840 >= 840 -3.7517E-10 Max TV (th's) 1 15 <= 90 75 144 >= 168 -24 Min Upper-income Exposures (th's) McGraw-Hill/Irwin 1.2 1.2 11.47 © The McGraw-Hill Companies, Inc., 2003 Since there are only two decision variables in this model, the graphical approach can be used. x2 140 The graph shows that there are no points that satisfy all the constraints. 120 X1 = 90 X1 + X2 = 120 < 1200X1 +1200X2 = 168,000 > < McGraw-Hill/Irwin 120 140 11.48 x1 © The McGraw-Hill Companies, Inc., 2003 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 u1 subject to the condition 14,000x1 + 6,000x2 + u1 > 840,000; u1 > 0 McGraw-Hill/Irwin 11.49 © The McGraw-Hill Companies, Inc., 2003 2. Minimize expenditures in excess of $90,000 on TV Min v2 subject to the condition x1 – v2 < 90,000; v2 > 0 3. Minimize underachievement of 168,000 upperincome exposures Min u3 subject to the condition 1,200x1 + 1,200x2 + u3 > 168,000; u3 > 0 4. Minimize underachievement of 1,680,000 total exposures (the maximum possible) Min u4 subject to the condition 14,000x1 + 6,000x2 + u4 > 1,680,000; u4 > 0 McGraw-Hill/Irwin 11.50 © The McGraw-Hill Companies, Inc., 2003 Note that the goals are now stated in terms of either minimizing underachievement (i.e., min. ui) or minimizing overachievement (i.e., min. vi). 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) The only system constraint is: Total expenditures will be no greater than $120,000 x1 + x2 < 120 2. goal constraints 11.51 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 The model can now be expressed as: Min P1u1 + P2v2 + P3u3 + P4u4 s.t. x1 + x2 < 120 14,000 x1 + 6,000x2 + u1 > 840,000 x1 - v2 < 90 1,200 x1 + 1,200x2 + u3 > 168,000 14,000 x1 + 6,000x2 + u4 > 1,680,000 (S) (1) (2) (3) (4) x1, x2, u1, v2 , u3, u4 > 0 Note that the objective function consists only of deviation variables and is of the Min form. In the objective function, P1 denotes the highest priority, and so on. 11.52 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 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 u1 subject to constraint (1) and x1, x2, u1 > 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. McGraw-Hill/Irwin 11.53 © The McGraw-Hill Companies, Inc., 2003 2. Find the subset of points in FR I that gives the Min possible value to v2, subject to constraint (2) and v2 > 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 u3, subject to constraint (3) and u3 > 0. 4. FR IV is the subset of points in FR III that minimize u4, subject to constraint (4) and u4 > 0. Any point in FR IV is an optimal 11.54 solution to the model. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 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 u1 s.t. (S), (1), and x1, x2, u1 > 0 is u1* = 0. The important information is that u1 = 0 which tells us that the first goal can be completely attained. Alternative optima for the current model are provided by all values of (x1, x2) that satisfy the conditions x1 + x2 < 120 FR I 14,000x1 + 6,000x2 > 840,000 11.55 x1, x2 > 0 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 Media Selection - GP model TV Radio X1 X2 Total Exposures (thousands) Expenditures (th's) 120 0 Minimiz e U1 V2 U3 U4 OF 1 1 1 1 54 0 30 24 0 Technological coeffisients Max Expenditures (th's) 1 1 Min Exposures (th's) 14 6 Max TV (th's) 1 1 -1 Min Upper-income Exposures (th's) 1.2 1.2 Exposures Target (th's) 14 6 McGraw-Hill/Irwin LHS 1 1 11.56 RHS Slack or surplus 120 < = 120 0 1680 > = 840 840 90 < = 90 0 168 > = 168 0 1680 > = 1680 0 © The McGraw-Hill Companies, Inc., 2003 Media Selection -Step 1 TV Radio X1 X2 Penalty Expenditures (th's) Minimize U1 V2 1 60 0 0 0 Technological coeffisients 1 U3 Max Expenditures (th's) 1 1 Min Exposures (th's) 14 6 1 0 U4 OF 1 0 0 LHS RHS Slack or surplus 60 <= 120 60 840 >= 840 2.121E-10 2 3 4 Ra nk McGraw-Hill/Irwin 11.57 © The McGraw-Hill Companies, Inc., 2003 At any such point, the goal is attained (u1* = 0) so that, in terms of only the first goal, these decisions are equally preferable. Thus FR I is the shaded area ABC. u1 = 0 McGraw-Hill/Irwin 11.58 © The McGraw-Hill Companies, Inc., 2003 Media Selection -Step 2 TV Radio X1 X2 Minimize U1 Penalty Expenditures (th's) V2 U3 1 60 0 0 0 Technological coeffisients Max Expenditures (th's) 1 1 1 Min Exposures (th's) 14 6 2 Max TV (th's) 1 1 -1 0 U4 OF 1 0 0 LHS RHS Slack or surplus 60 <= 120 60 840 >= 840 0 60 <= 90 30 0 = 0 0 3 4 Value of U1 found in step 1 1 Ra nk McGraw-Hill/Irwin 11.59 © The McGraw-Hill Companies, Inc., 2003 We see that: Min v2 such that x in FR I, goal (2) and v2 > 0 is v2* = 0. x1, x2 > 0 Thus, FR II is defined by x1 + x2 < 120 14,000x1 + 6,000x2 > 840,000 FR II x1 < 90 x1, x2 > 0 The shaded area ABDE is a subset of FR I and as expected, the size of the feasible region is smaller. v1 = 0 u1 = 0 McGraw-Hill/Irwin 11.60 © The McGraw-Hill Companies, Inc., 2003 Media Selection - Step 3 TV Radio X1 X2 Minimize U1 V2 Penalty Expenditures (th's) 15 105 0 U4 OF 1 1 24 24 0 RHS Slack or surplus <= 120 0 840 >= 840 0 15 <= 90 75 168 >= 168 0 0 = 0 0 0 = 0 0 Technological coeffisients LHS 120 Max Expenditures (th's) 1 1 1 Min Exposures (th's) 14 6 2 Max TV (th's) 1 3 Min Upper-income Exposures (th's) 1.2 0 U3 1 -1 1.2 1 4 Value of U1 found in step 1 RaValue of V2 found in step 2 nk McGraw-Hill/Irwin 1 1 11.61 © The McGraw-Hill Companies, Inc., 2003 FR III is the line segment BD. In this case u3* = 24,000. Although the first two goals were completely attained (since u1* = v2* = 0), the third goal cannot be completely attained because u3* > 0. FR III McGraw-Hill/Irwin x1 + x2 < 120 14,000x1 + 6,000x2 > 840,000 x1 < 90 1,200x1 + 1,200x2 > 168,000 – 24,000 = 144,000 11.62 © The McGraw-Hill Companies, Inc., 2003 Media Selection - Step 4 TV Radio X1 X2 Minimize U1 V2 U3 Penalty Expenditures (th's) 90 30 0 0 24 U4 OF 1 240 240 Technological coeffisients Max Expenditures (th's) 1 1 1 Min Exposures (th's) 14 6 2 Max TV (th's) 1 3 Min Upper-income Exposures (th's) 1.2 1.2 4 Exposures Target (th's) 14 6 Value of U1 found in step 1 Value of V2 found in step 2 Ra Value of U3 found in step 3 nk McGraw-Hill/Irwin LHS 1 -1 1 1 1 1 1 11.63 RHS Slack or surplus 120 <= 120 0 1440 >= 840 600 90 <= 90 0 168 >= 168 0 1680 >= 1680 0 0 = 0 0 0 = 0 0 24 = 24 0 © The McGraw-Hill Companies, Inc., 2003 Recall that the fourth goal is to minimize underachievement of the maximum possible number of exposures, which is 1,680,000. Thus, we wish to minimize the underachievement u4 where 14,000x1 + 6,000x2 + u4 > 1,680,000 Since u4 = 240,000, we achieve 1,680,000 240,000 = 1,440,000 exposures. McGraw-Hill/Irwin 11.64 The unique optimum is x1* = 90, x2* = 30 (i.e., spend $90,000 on TV ads & $30,000 on radio ads). © The McGraw-Hill Companies, Inc., 2003 COMBINING WEIGHTS AND ABSOLUTE PRIORITIES In reviewing the results of the absolute priority study, the older members of the Mylonal market begins to take on importance. The exposures per $1000 of advertising are: EXPOSURE GROUP 50 and over TV RADIO 3,000 8,000 Note that radio and TV exposures are not equally effective in generating exposures in this segment of the population. McGraw-Hill/Irwin 11.65 © The McGraw-Hill Companies, Inc., 2003 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 11.66 phase of the absolute priorities approach. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 Media Selection - Weighted Step 4 TV Radio X1 X2 Minimize U1 V2 U3 Penalty Expenditures (th's) 15 105 Wei ght 0 0 24 U4 U5 OF 1 3 1065 840 75 Technological coeffisients Max Expenditures (th's) 1 1 1 Min Exposures (th's) 14 6 2 Max TV (th's) 1 3 Min Upper-income Exposures (th's) 1.2 1.2 4 Exposures Target (th's) 14 6 4 Exposures > 50 years (th's) 3 8 Value of U1 found in step 1 Value of V2 found in step 2 Ra Value of U3 found in step 3 nk McGraw-Hill/Irwin LHS 1 -1 1 1 1 1 1 1 11.67 RHS Slack or surplu s 120 <= 120 0 840 >= 840 0 15 <= 90 75 168 >= 168 0 1680 >= 1680 0 960 >= 960 0 0 = 0 0 0 = 0 0 24 = 24 0 © The McGraw-Hill Companies, Inc., 2003 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. McGraw-Hill/Irwin 11.68 © The McGraw-Hill Companies, Inc., 2003 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 McGraw-Hill/Irwin 11.69 © The McGraw-Hill Companies, Inc., 2003