1. DECISION MODELING WITH MICROSOFT EXCEL Chapter 12 Copyright 2001 Prentice Hall Publishers and Ardith E. Baker Multi-Objective Decision Making and Heuristics Part 2

2. MULTIPLE OBJECTIVES In many applications, the planner has more than one ___________. The presence of multiple objectives is frequently referred to as the problem of “_________ apples and oranges.” Consider a corporate planner whose long-range goals are to: 1. Maximize discounted__________ 2. ____________market share at the end of the planning period 3. Maximize existing physical _________at the end of the planning period

3. It is also clear that the goals are ____________(i.e., there are trade-offs in the sense that sacrificing the requirements on any one goal will tend to produce greater___________ on the others. These goals are not __________________(i.e., they cannot be __________combined or compared). These models, although not applied as often in practice as some of the other models (such as linear programming,_____________, inventory control, etc.), have been found to be especially useful on problems in the________________.

4. Several approaches to multiple objective models (also called ______________decision making) have been developed: Multi-attribute ___________theory AHP and Goal Programming will be discussed. Developed by Thomas Saaty, AHP helps managers choose between many decision _____________on the basis of multiple criteria. Search for _________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 ___________approach to the multiple- objectives model.

5. Goal Programming is an extension of ___________ Programming that enables the planner to come as close as possible to satisfying various _______and constraints. GOAL PROGRAMMING It allows the decision maker, at least in a heuristic sense, to incorporate his or her ___________system in dealing with multiple conflicting goals. GP is sometimes considered to be an attempt to put into a mathematical _________________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 “___________” or “close enough.”

6. Suppose that we have an ____________program design model with decision variables x 1 and x 2, where x 1 is the hours of _______________work x 2 is the hours of _______________work Assume the following ___________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. ___________constraints (so-called hard constraints) that cannot be violated. 2. _______constraints (so-called soft constraints) that may be violated if necessary.

7. Now, suppose that each hour of classroom work involves 12 minutes of ______________experience and 19 minutes of _____________problem solving Each hour of laboratory work involves 29 minutes of small-group experience and 11 minutes of individual problem solving The total _________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 _____________program time working in small groups and ¹/ 3 of the time on problem_____________.

8. 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 “__________________” to the right-hand side. ~= In order to satisfy the system constraint, at least one of the two goals will be_____________. To ___________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

9. Deviation Variables Variables u 1 and v 1 are called __________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 ______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 _______________: 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__________.

10. The complete (___________) 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 ________LP model and can be easily solved in Excel. The optimal decision variables will _______the system constraint (total program hours). Min u 1 + v 1 + u 2 + v 2 Note: Both u 1 and v 1 can’t be 0

11. Solver will _________that either u 1 or v 1 (or both) will be zero, and thus these variables _____________ satisfy this desired condition. Note that the ___________function is the sum of the deviation variables. This choice of an objective function indicates that there is no _________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 ___________variables.

12. For example, any of the following three decisions is acceptable: 1. A decision that _______________the group experience goal by 5 minutes and hits the problem-solving goal exactly, 2. A ___________that hits the group experience goal exactly and underachieves the problem- solving goal by 5 minutes, and 3. A decision that ______________each goal by 2.5 minutes.

13. There is no ___________among the following three solutions because each of these yields the same value (i.e., 5) for the objective____________. 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)

14. Weighting the Deviation Variables Differences in units alone could produce a ____________among the deviation variables. One way of expressing a preference among the various goals is to assign different ___________ (weights) to the deviation variables in the objective function. as the_______________. Since v 2 (over-achievement of problem solving) has the smallest coefficient, the program designers would rather have v 2 __________ than any of the other deviation variables (positive v 2 is _____________the least). Min 10u 1 + 2v 1 + 20u 2 + v 2 In the program-planning example, one might select

15. With this objective function it is better to be 9 minutes over the problem-solving _______than to ______________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.

16. Goal Interval Constraints Another type of goal constraint is called a ________________constraint. Such a constraint _________the goal to a range or interval rather than a specific ___________value. Suppose, for example, that in the previous illustration the designers were ____________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 ___________with two goal constraints: 19x 1 + 11x 2 – v 1 0) 19x 1 + 11x 2 + u 1 > 1800 (u 1 > 0)

17. When the terms u 1 and v 1 are included in the objective function, the LP ______will attempt to ___________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 ____________ deviation variables u i, v i. At optimality at least one of the pair u i, v i will always be________. u i represents underachievement; v i represents ______________. Whenever u i is used it is ___________to g i (x 1, …, x n ). Whenever v i is used it is ______________from g i (x 1, …, x n ).

18. Only __________variables appear in the objective function, and the objective is always to___________. The decision variables x i, i = 1, …, n do not appear in the_____________. Four types of goals have been discussed: 1. ________. 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)

19. 2. _________Underachievement. To do this, write and in the__________, minimize u i, the under- achievement. v i does not appear in the objective function and it is only in this_____________, 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__________, this constraint will be active, for otherwise u i * could be made smaller. If u i *>0 then, since v i * must equal________, it must be true that g i (x 1, …, x n ) + u i * = b i.

20. g i (x 1, …, x n ) + u i - v i = b i (u i > 0, v i > 0) 3. __________Overachievement. To do this, write and in the objective, minimize v i, the ______- 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_______, this constraint will be active. The argument is __________to that in item 2.

21. 4. Goal Interval___________. 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 “________ out” the interval by writing a i - u i 0, v i > 0) which is ____________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_____________. Variables u i and v i are merely ____________and slack, respectively. ^^

22. In some cases, managers do not wish to express their __________among various goals in terms of weighted deviation variables, for the process of assigning __________may seem too arbitrary or subjective. ABSOLUTE PRIORITIES In such cases, it may be more acceptable to state preferences in terms of ___________________(as opposed to weights) to a set of goals. This approach requires that goals be ___________in a specific order. Therefore, the model is solved in stages as a ___________of models.

23. 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_____________. The client wants to reach several audiences, however, radio and television are not equally ____________in reaching all audiences. Therefore, the agency will estimate the _________of the advertisements in terms of rated exposures (i.e., “people reached per month”) on the audiences of interest.

24. The following data represent the number of ____________per $1000 expenditure: TV RADIO Total14,0006,000 Upper Income 1,2001,200 The following are the campaign goals, listed in order of absolute___________. 1. Total exposures will hopefully be at least ___________. 2. In order to maintain effective contact with the leading radio station, no more than _________ will be spent on TV advertising.

25. 3. The campaign should achieve at least _______ upper-income exposures. 4. If all other goals are satisfied, the total number of exposures would come as close as possible to being___________. Note that if all of the $120,000 is spent on TV advertising, then the maximum __________ exposures would be 1,680,000 (120*14,000). To model the problem, the following notation will be used: x 1 = dollars spent on _____( in thousands) x 2 = dollars spent on ______(in thousands) The objective function will be to maximize total ___________and the other goals will be treated as _______________.

26. An Infeasible Model The formulation and spreadsheet solution is shown below:

27. 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.

28. Swenson’s Goal Programming Model Note that the first goal (total exposures will be at least 840,000), if violated, will be___________________. The second goal (no more than $90,000 will be spent on TV advertising), if violated, will be_____________, etc. Employing this reasoning, the goals are restated, in _____________priority, as: 1. ___________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

29. 2. Minimize _________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____________ 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 ____ 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

30. Note that the goals are now stated in terms of either _____________underachievement (i.e., min. u i ) or minimizing _________________(i.e., min. v i ). In addition, the goals have been expressed as ______________. This method will facilitate a graphical analysis. Given that the priorities are formulated correctly, we must now distinguish between 1. _________constraints (all constraints that may not be violated) 2. _________constraints The only system constraint is: Total expenditures will be no _________than $120,000 x 1 + x 2 < 120

31. 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 __________variables and is of the ______form. In the objective function, P 1 denotes the highest _________, and so on.

32. The previous problem statement precisely means: 1. Find the set of decision variables that satisfies the system __________(S) and that also gives the ____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____________, all of the points in FR I are “optimal” and (again considering only the highest goal), we are ____________as to which of these points are selected.

33. 2. Find the _______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 _________ranking of the two highest-priority goals, all of the points in FR II are “_______,” 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 _________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 ____________(4) and u 4 > 0. Any point in FR IV is an optimal solution to the model.

34. Graphical Analysis and Spreadsheet Implementation of the Solution Procedure Since there are only two decision variables, we can use the ________method of LP. 1. Both the spreadsheet output and the __________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 __________. Alternative __________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

35. First goal:

36. 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.

37. 2. Now enter the constraints defining FR I, together with the new goal constraint (2)

38. 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.

39. This worksheet shows the third goal.

40. 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

41. The optimal solution is shown in this worksheet.

42. Recall that the fourth goal is to ____________ underachievement of the maximum possible number of___________, 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 ______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.

43. In reviewing the results of the _________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 14,000 6,000 EXPOSURE GROUP Note that radio and TV exposures are not equally _________in generating exposures in this segment of the population.

44. If there were no other considerations, then we would like as many _____________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 __________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 _______________________. To resolve this conflict of goals, use a _________ sum of the deviation variables as the objective in the final ________of the absolute priorities approach.

45. It is decided that underachievement in the ________ (960,000 exposures to the 50-and-over group) is three times as _________as underachievement in the fourth goal (1,680,000 total exposures).

46. Note that the new objective function has moved the ________solution from one end of FR III to the other. This optimal solution is as close as possible to the more heavily weighted_________. __________analysis on the weights in the objective function could be used to see when the solution changes from point B to point D.

47. ANALYTICAL HIERARCHY PROCESS This section deals with the real-world topic of making a decision when there are ________ objectives or criteria to consider. For example: Choosing which employment offer to accept. Picking which computer (or car, etc.) to buy. Deciding which new product to launch first. Selecting a site for a new restaurant, hotel, etc. Rating the best cities in which to live. Choosing a new software package for your company.

48. A simple way to attack such a decision would be to assign __________to each of the criteria that were to be considered in making the decision. Then, _____each decision alternative on a scale from 1 (worst) to 10 (best). Finally, you would _________the weights times the rankings for each criterion and sum them up. The ___________with the highest score would be the most preferred.

49. For example, you are in charge of purchasing the next computer for the office. You have to choose between the following three computers: 1. Model A runs an AMD K6-II chip at 400 MHz 2. Model B runs a Celeron chip at 333 MHz 3. Model C runs a Pentium II chip at 450 MHz The important criteria and their weights are: Criteria Weight Price 50% Price 50% Speed 15% Speed 15% Hard-disk Size 20% Hard-disk Size 20% Warranty/Support 15%

50. Now, rank each of the three models on these four _______. Rank them on a scale from 1 to 10 as described earlier. Model B has the highest weighted _______and thus would be the best computer to purchase.

51. This approach is quite ___________and there are difficulties in setting the ranking scales on such different criteria. Analytic hierarchy process (AHP) also uses a weighted ________approach idea, but it uses a method for assigning ratings (or rankings) and weights that is considered more ___________and consistent. (AHP) is based on ________comparisons between the decision alternatives on each of the criteria. Then, a similar set of ______________are made to determine the relative importance of each criterion and thus produces the___________.

52. The basic procedure is as follows: 1. Develop the __________for each decision alternative for each criterion by developing a pairwise comparison _____ for each criteriondeveloping a pairwise comparison _____ for each criterion __________the resulting matrix__________the resulting matrix _________the values in each row to get the corresponding rating_________the values in each row to get the corresponding rating calculating and checking the __________ ratiocalculating and checking the __________ ratio

53. 2. Develop the ________for the criteria by developing a pairwise comparison matrix for each___________developing a pairwise comparison matrix for each___________ normalizing the __________matrixnormalizing the __________matrix averaging the values in each _____to get the corresponding ratingaveraging the values in each _____to get the corresponding rating calculating and ________the consistency ratiocalculating and ________the consistency ratio 3. Calculate the _______average rating for each decision alternative. Choose the one with the __________score.

54. Consider the following example: Sleepwell Hotels is looking for some help in selecting the “best” revenue management software package from among several vendors. The director of revenue management for this chain of hotels has been given this task. Three vendors have been identified whose software meets the following basic needs: Revenue Technology Corporation (RTC) PRAISE Strategic Solutions (PSS) El Cheapo (EC)

55. The important criteria are: 1. The ____________of the installed system 2. The follow-up _________provided over the coming year 3. The sophistication of the ___________math engines 4. The amount of _____________for Sleepwell

56. The first step in the AHP procedure is to make pairwise ___________between the vendors for each criterion. Here is the ________scale for making these comparisons: DESCRIPTION 1Equally preferred 3Moderately preferred 5Strongly preferred 7Very strongly preferred 9Extremely strongly preferred RATING Values 2, 4, 6, or 8 may also be assigned and represent ___________halfway between the integers on either side.

57. Start with the total cost __________and generate the following data in a spreadsheet: The _______in the row is being compared to the vendor in the column. A value between 1 and 9 indicates that the vendor in the row is __________to the vendor in the column. If the vendor in the ______is preferred to the vendor in the row, then the inverse of the rating is given.

58. The next step is to _________the matrix. This is done by totaling the numbers in each column. Each entry in the column is then ___________by the column sum to yield its normalized score.

59. Now, calculate the _________________and check its value. The purpose for doing this is to make sure that the original preference ratings were__________. There are 3 steps to arrive at the consistency ratio: 1. Calculate the consistency ________for each vendor. 2. Calculate the consistency ________(CI). 3. Calculate the consistency ________(CI/RI where RI is a random index). To calculate the consistency measure, we can take advantage of Excel’s matrix _____________function =MMULT().

60. Multiply the average ______for each vendor times the scores in the first row one-at-a-time, sum these products up and divide this _____by the average rating for the first vendor.

61. RANDOM INDEX 20.00 20.00 30.58 30.58 40.90 40.90 51.12 51.12 61.24 61.24 71.32 71.32 81.41 81.41 91.45 91.45 101.51 N

62. If we are perfectly___________, then the consistency measures will equal n and therefore, the CIs will be equal to ______and so will the consistency_______. If this ratio is very ________(Saaty suggests > 0.10), then we are not consistent enough and the best thing to do is go back and _______the comparisons. Now, continue for the other three criteria. You can easily do this by copying the “________” sheet into three other sheets (“Service,” “Sophistication,” and “Custom”) and then simply changing the _______ comparisons.

63. Consistency ratio for “Service.”

64. Consistency ratio for “Sophistication.”

65. Consistency ratio for “Customization.”

66. In all three cases, the CR value ranges from 0.0 to 0.047 which means that we are being___________. Note also that PSS is the winner on the Service criterion, RTC and PSS are tied for the best in terms of Sophistication, and PSS is considered the best on Customization. All of this work concludes the first step in the procedure. The next step is to use similar ______ comparisons to determine the appropriate _______ for each of the criteria. The process is the same in that we make ______________, except that now we make the comparisons between the criteria not the vendors.

67. Consistency ratio for weights on criterion.

68. The final step is to ________the weighted average ratings of each decision alternative and use the ________to decide from which vendor to purchase the software. These results are pulled from all the other ___________. From these results, we find that RTC barely edges out PSS for the software____________.