Session 5b

1. Session 5b

2. Overview Evolutionary Solver (Genetic Algorithm) • Advertising Example • Product Design Example • Conjoint Analysis Decision Models -- Prof. Juran

3. Nonlinear Problems Some nonlinear problems can be formulated in a linear fashion (i.e. some network problems). Other nonlinear functions can be solved with our basic methods (i.e. smooth, continuous functions that are concave or convex, such as portfolio variances). However, there are many types of nonlinear problems that pose significant difficulties. Decision Models -- Prof. Juran

4. Nonlinear Problems The linear solution to a nonlinear (say, integer) problem may be infeasible. The linear solution may be far away from the actual optimal solution. Some functions have many local minima (or maxima), and Solver is not guaranteed to find the global minimum (or maximum). Decision Models -- Prof. Juran

5. 3 Solvers • Simplex LP Solver • GRG Nonlinear Solver • Evolutionary Solver Decision Models -- Prof. Juran

6. Radio Advertising Example Music radio WABC has commercials of the following lengths (in seconds): 15, 15, 20, 25, 30, 35, 40, 57 The commercials must be assigned to 60-second breaks. What is the fewest number of breaks that are needed to air all of the commercials? Decision Models -- Prof. Juran

7. Managerial Problem Definition Decision Variables Which commercials get assigned to which programming breaks. Objective Minimize the total number of breaks. Constraints Every advertisement must be aired. No break can be longer than 60 seconds. Decision Models -- Prof. Juran

8. Formulation Decision Variables Define xi to be an integer variable identifying the break to which commercial i is assigned. For example, if commercial 1 is assigned to break 4, then x1 = 4. It should be clear that we won’t need any more than eight breaks, because there are only eight commercials. These eight x variables are the decision variables. Decision Models -- Prof. Juran

9. Formulation Define yj to be a binary variable, such that yj = 0 if no commercials are assigned to break j, and yj = 1 if any commercials are assigned to break j. Define vij to be a binary variable such that vij = 1 if commercial i is assigned to break j, and vij = 0 otherwise. Define wi to be the duration of commercial i. Decision Models -- Prof. Juran

10. Formulation Note that the duration of break j is equal to Let tj be the amount of “overtime” in break j. That is, Decision Models -- Prof. Juran

11. Formulation Objective Our objective, then, is to: Minimize Z = where m is a “large number”. Decision Models -- Prof. Juran

12. Formulation This is a good example of an advanced optimization trick: taking a constraint and building it into the objective function. It doesn’t really matter what value we use for m, as long as it is sufficiently large as to prevent any tj > 0. As it happens, in this problem m = 100 works fine. Decision Models -- Prof. Juran

13. Formulation Constraints For all xi, 1≤ xi ≤ 8. For all xi, xi is an integer. For all yj, yj is binary. Decision Models -- Prof. Juran

14. Solution Methodology Decision Models -- Prof. Juran

15. Solution Methodology The objective function is in B25, including a penalty of 100 units per second if any breaks go over 60 seconds. The decision variables (xi) are in the range E5:E12 (in the spreadsheet shown, all commercials are assigned to break 1, so xi = 1 for all i). The range B5:B12 contains the durations of each commercial (wi), and the range B16:B23 uses the Excel SUMIF function to calculate the duration of each commercial break. Decision Models -- Prof. Juran

16. Solution Methodology The range C16:C23 keeps track of which breaks have any assignments (the yi variables), while the range D16:D23 keeps track of how much the breaks go over the maximum limit (the tj variables). Recall that the yi and tj variables are the basic ingredients of the productive function. Notice how the use of the IF function in C16:C23 precludes the need to have an explicit binary constraint in Solver for the yi variables. Decision Models -- Prof. Juran

17. Solution Methodology The standard simplex algorithm (Solver’s default method) won’t work on this problem. The GRG Nonlinear algorithm will make an honest effort, but is likely to give up without finding the optimal solution. This is because of our use of MAX, IF, and SUMIF functions, resulting in discontinuities in our productive function and constraints as functions of the decision variables. However, the Evolutionary Solver, a genetic algorithm, can do a good job with a problem like this. Decision Models -- Prof. Juran

18. Solution Methodology Decision Models -- Prof. Juran

19. Solution Methodology The Evolutionary Solver operates in a completely different way from the other types. Instead of searching in a structured way guaranteed to reach the optimal solution, genetic algorithms operate somewhat like biological evolutionary processes, with some degree of randomness in the steps taken from one solution to the next. In a finite period of time, the Evolutionary Solver is not guaranteed to find the optimal solution, but it will find very good solutions and try to improve upon them. Decision Models -- Prof. Juran

20. Optimal Solution Decision Models -- Prof. Juran

21. Conclusions The solution indicates that commercials 1, 2, and 5 should go in one break, 3 and 7 should go in another, 4 and 6 should go in another, and 8 should go by itself. A reasonably bright person could solve this problem in their head, of course. The trick here was to set it up so that a computer could solve it, providing a method for the solution of much larger problems with the same basic structure. Decision Models -- Prof. Juran

22. Product Design Example Conjoint Analysis is a multivariate technique used specifically to understand how respondents develop preferences for products or services. It is based on the simple premise that consumers evaluate the value or utility of a product (real or hypothetical) by combining the utility provided by each attribute characterizing the product. -- Prof. Pradeep Chintagunta, Univ. of Chicago Decision Models -- Prof. Juran

23. Conjoint Analysis Conjoint Analysis is a decompositional method. Respondents provide overall evaluations of products that are presented to them as combinations of attributes. These evaluations are then used to infer the utilities of the individual attributes comprising the products. In many situations, this is preferable to asking respondents how important certain attributes are, or to rate how well a product performs on each of a number of attributes. Decision Models -- Prof. Juran

24. Conjoint Analysis • After determining the contribution of each attribute to the consumer’s overall evaluation, one could • Define the product with the optimal combination of features • Predict market shares of different products with different sets of features • Isolate groups of customers who place differing importances on different features • Identify marketing opportunities by exploring the market potential for feature combinations not currently available • Show the relative contributions of each attribute and each level to the overall evaluation of the product Decision Models -- Prof. Juran

25. Product Design Example Decision Models -- Prof. Juran

26. Product Design Example Assume that a consumer's purchase decision on an electric razor is based on four attributes, each of which can be set at one of three levels (1, 2, or 3). Using conjoint analysis, our analysts have divided the market into five segments (labeled as customers 1, 2, 3, 4, and 5) and have determined the "part-worth" that each customer gives to each level of each attribute. Decision Models -- Prof. Juran

27. We assume here that all customers within a particular segment view electric razors more or less the same in terms of which levels of which attributes constitute an attractive product. We also assume that customers in a segment conduct a sort of mathematical analysis (perhaps unconsciously) in which they weigh the various attributes of a product to come up with an overall value with respect to competing products. Conjoint analysis usually assumes the customer buys the product yielding the highest total part-worth. Decision Models -- Prof. Juran

28. Decision Models -- Prof. Juran

29. For example, consider Segment 1 and these two products: We assume that customers in Segment 1 will not buy Product B, because they value Product A at 1 + 4 + 4 + 4 = 13 and Product B at 1 + 1 + 1 + 2 = 5. Decision Models -- Prof. Juran

30. Decision Models -- Prof. Juran

31. Currently there is a single product in the market that sets all four attributes equal to 1 (call it Razor 0). We want to introduce two new types of electric razors, and capture as much of the market as possible. We want to design a two-product line that maximizes the number of market segments that will buy one of our two products. Assume that in the case of a tie, the consumer does not purchase our product. Decision Models -- Prof. Juran

32. Managerial Formulation Decision Variables Which levels of each attribute to design into each of our two products. Objective Maximize the number of customer segments who will buy one of our products. Constraints There are only four attributes, each of which must be assigned to one of three existing levels for each product. (In other words, no product can have more or less than one level per attribute.) Decision Models -- Prof. Juran

33. Formulation: Preliminaries There are five customer segments, and we will index them from 1 to 5 with the subscript letter i. There are three products (the existing Razor 0, plus our Razors 1 and 2 to be designed), and we will index them from 1 to 3 with the subscript letter j. There are four product attributes, and we will index them from 1 to 4 with the subscript letter k. There are three possible levels for each attribute, and we will index them from 1 to 3 with the subscript letter l. Decision Models -- Prof. Juran

34. x ij v ij = 1 x ij > v v products not ij i j a jkl b ikl Symbol Variable Description A binary variable; 1 if segment will buy product , i j 0 otherwise. The total “value” that segment places on product . i j For our two products ( = 1, 2), if j A binary variable; 1 if product has level of j l attribute , 0 otherwise. k There are 1 2 of these per product; 36 total in this problem. The “value” placed by segment on level of i l attribute . k There are 12 of these per segment; 60 total for this problem (as shown in the table on slide 27 ). Decision Models -- Prof. Juran

35. Example: Using the example on slide 28, consider Segment 1’s evaluation of Razors A and B. For Razor A: Decision Models -- Prof. Juran

36. For Razor B: Since , , and . In English, customer segment 1 will buy Razor A and not buy Razor B. Decision Models -- Prof. Juran

37. Formulation Decision Models -- Prof. Juran

38. Solution Methodology Decision Models -- Prof. Juran

39. Solution Methodology Decision Models -- Prof. Juran

40. Solution Methodology Decision Models -- Prof. Juran

41. Solution Methodology Decision Models -- Prof. Juran

42. Optimal Solution Decision Models -- Prof. Juran

43. Conclusions It turns out that there is a line of two products that can capture all five segments! Razor 1, with attribute levels (1, 1, 3, 1), captures segments 4 and 5. Razor 2, with attribute levels (3, 1, 1, 1), captures segments 1, 2, and 3. Decision Models -- Prof. Juran

44. Summary Evolutionary Solver (Genetic Algorithm) • Advertising Example • Integer and Binary tricks • Moving Constraints into the productive Function • MAX, IF, SUMIF • Product Design Example • Conjoint Analysis • VLOOKUP, MAX, IF Decision Models -- Prof. Juran