1. Decision-Making November 23, 2012

2. Question 2: Making firecrackers

3. A firecracker factory can produce up to 70 000 firecrackers. The materials needed to produce 10 000 firecrackers cost 100 000 yuan. You need at least two workers to operate the plant, and for every 10 000 firecrackers beyond the basic production of 30 000, you need to hire another worker. You can sell firecrackers for 72 yuan each. But customers will only buy firecrackers in the month before New Year, and the total number you can sell is somewhere between 25 000 and 50 000. You can store unsold firecrackers for sale in subsequent years, but it will cost 200 000yuan/year to store each 10 000 firecrackers. It is not possible to wait till next New Year's to find how many you can sell. If you decide to store some firecrackers for a year, you will pay the storage costs at the beginning of that year. Your MARR is 20%. Do a scenario analysis for various levels of production and various levels of customer demand. On the basis of this analysis, you have several choices to make. First, should you buy the production apparatus at all? Secondly, if you do buy it, how many firecrackers should you produce? Suppose a market research study would allow you to find out exactly how many firecrackers the customers would buy each year. What would it be worth paying for the results of such a study?

4. First, should you buy the production apparatus at all? Secondly, if you do buy it, how many firecrackers should you produce? Suppose a market research study would allow you to find out exactly how many firecrackers the customers would buy each year. What would it be worth paying for the results of such a study?

6. Total Made Fixed cost Labour cost Material cost Total Present Cost Cost/Crac ker Cost/Cracker, Year2 Cost/Cracker, Year3 Unit Price 1 Unit Price 2 Unit Price 3 Profit 1 Profit 2 Profit 3 11 000 200 1001 300 1 4671 606600500417-700-967-1189 21 000 200 1 400 700 8671 006600500417-100-367-589 31 000 200 3001 500 500 667 806600500417100-167-389 41 000 300 4001 700 425 592 731600500417175- 92-314 51 000 400 5001 900 380 547 686600500417220- 47-269 61 000 500 6002 100 350 517 656600500417250- 17-239 71 000 600 7002 300 329 495 634600500417271 5-217 Demand (10 000):2.53.03.54.04.55.0Average Total Made 2- 200 3 167 300 278 4 300 433 567 700 567 5 433 567 700 833 9671 100767 6 344 700 833 9671 1001 233863 7 256 611 9671 1001 2331 367922

8. First, should you buy the production apparatus at all? Secondly, if you do buy it, how many firecrackers should you produce? Suppose a market research study would allow you to find out exactly how many firecrackers the customers would buy each year. What would it be worth paying for the results of such a study?

9. Given these results, what do we decide to do? We can apply the ideas of game theory, treating our choice of production level as our move in a game, and the level of demand as the opponent’s response. Theory of Games and Economic Behavior, John von Neumann and Oskar Morgernstern, 1944

10. The Minimax Strategy: Assume things will turn out for the worst, plan to minimize your losses (or maximize your gains) under these circumstances.

11. The Maximax Strategy: Assume things will turn out for the best, plan to minimize your losses (or maximize your gains) under these circumstances.

12. The Principle of Minimax Regret …also known as the Savage Principle, is based on the psychologically plausible premise that, if you make a plan based on the assumption that A will happen, and then B happens instead, you will regret losing the benefits you could have had if you'd been smart enough to guess right. So you adopt the strategy of minimizing the regret you might otherwise be obliged to feel. To apply this, you first construct a regret matrix…

13. Demand2.53.03.54.0 Total Made 50133267 6890134133 71778900

14. The Laplace Principle, or Principle of Insufficient Reason: In the absence of information to the contrary, assume all outcomes are equally likely, and choose the one with the highest expected value.

15. Qualitative Analysis How to take non-monetary factors into account in decision-making

16. Argument: the only way non-monetary factors should influence decision-making is as a constraint: ``We’re not going to use baby seals as a raw material, however profitable it is.’’ If, on the contrary, we allow trade-offs between our non-monetary factors and cost, then there’s an implicit rate of exchange between the factor and a dollar amount, and we can go back to optimizing the final dollar amount.

17. …but in fact we don’t usually make decisions this way. Consider the question of getting married, for example.

18. Accepting that we are going to base our decisions on multiple criteria that can’t always be traded off against each other, there are some ways of simplifying the problem: Suppose we have N criteria and that A and B are two possible courses of action. Then if: Criterion i (A ) ≥ Criterion i (B) for 1 ≤ i ≤ N we say A dominates B, and we can eliminate B from our list of possible actions. If A is not dominated by any other course of action, A is efficient. So a first step is to reduce our alternatives to an efficient set.

19. Example: which choices are dominated with respect to the two criteria of hard-working and intelligent? HW IQ Bob Andy Mary Angus Sally Peter Amy Jill Moe

20. Once we have eliminated the dominated choices, we are left with an efficient set. HW IQ Bob Mary Sally

21. Decision matrices: Having reduced the problem to an efficient set, we can give each option a weighted score against a number of different criteria, thus forming a decision matrix.

22. Algorithm for setting up decision matrices: 1.Pick your criteria (e.g., industriousness, intelligence, charisma) 2.Give each a weight to indicate its importance. Let the weights sum to 10 3.Rank each alternative (Bob, Sally and Mary) against each criterion, with 10 being the highest. 4.Find the weighted totals for each candidate and pick the highest. 5.Do sensitivity analysis to see if the decision will change with minor changes in our preferences.

23. CriterionWeightBobSallyMary Industry5137 Intelligence3579 Charisma2972 Score10385066 How do we get the individual scores?

24. Individual scores can be obtained by normalization or by subjective evaluation. For example, to get a 1-10 score for intelligence, we could take the individual’s IQ score and divide by 20. This normalises with respect to the total population, but we might also normalise with respect to the candidate population. For subjective evaluation, we might want to average the evaluations from two or three evaluators.

25. A variant on the decision matrix is to look at the product of the individual scores as well as the sum. What is the effect of this, and why might we do it? CriterionBobSallyMary Industry137 Intelligence579 Charisma972 Sum151718 Product45147126

26. Some observations about decision-making: It’s easier to make pairwise comparisons: ``Is Andy smarter than Bob?’’ versus ``How smart is Andy?’’

27. Some observations about decision-making: It’s easier to compare specific criteria: ``Is Andy taller than Bob?’’ versus ``Is Andy a better person than Bob?’’

28. From these observations comes the Analytic Hierarchy Process: Make pairwise comparisons between candidates with respect to a criterion. Convert the results of these comparisons into numerical scores Make pairwise comparisons between the strength of the criteria Convert the results of these comparisons into numerical scores Combine the numerical scores to get an overall ranking of candidates Apply sensitivity analysis

29. Example: choice of an electricity-generating technology for BC: Criteria: Cost, Carbon Emissions, Unsightliness, Risk Candidates: Hydro, nuclear, natural gas, oil, coal.

30. Pairwise comparison: How does hydro compare to coal with respect to price? The preferred candidate gets a score of n, 1 ≤ n ≤ 9 The other candidate gets a score of 1/n

31. In this case, Hydro gets a `2’ compared to coal. We record this in a pairwise comparison matrix. HydroCoal-fired Hydro12 Coal-fired1/21 Comparison with respect to cost

32. Carry on and complete the matrix Comparison with respect to cost HydroNuclearNatural Gas OilCoal Hydro13242 Nuclear1/311/221 Natural Gas 1/22121 Oil1/41/2 1 Coal1/21121

33. Next, normalize the matrix: sum each column, then divide each entry by its column sum. Comparison with respect to cost HydroNuclearNatural Gas OilCoal Hydro1 0.3873242 Nuclear1/3 0.12911/221 Natural Gas ½ 0.1932121 Oil¼ 0.0971/2 1 Coal½ 0.1931121 Sums2.583 1

34. Next, normalize the matrix: sum each column, then divide each entry by its column sum. Comparison with respect to cost HydroNuclear Natural Gas OilCoal Hydro 0.387 0.400 0.364 Nuclear 0.129 0.1330.1000.182 Natural Gas 0.193 0.2670.2000.182 Oil 0.97 0.0670.1000.091 Coal 0.193 0.1330.2000.182

35. Now work out the average value of each row: Comparison with respect to cost Cost Avg. 0.383 0.145 0.205 0.089 0.178 HydroNuclear Natural Gas OilCoal Hydro 0.387 0.400 0.364 Nuclear 0.129 0.1330.1000.182 Natural Gas 0.193 0.2670.2000.182 Oil 0.97 0.0670.1000.091 Coal 0.193 0.1330.2000.182

36. Go through the same process for each criterion to create a priority matrix. Priority Matrix Cost Avg. 0.383 0.145 0.205 0.089 0.178 Carbon Avg. 0.256 0.177 0.149 0.212 0.251 Generatio n Hydro Nuclear Gas Oil Coal Sightliness Avg. 0.166 0.067 0.319 0.112 0.311 Risk Avg. 0.066 0.237 0.209 0.322 0.111

37. CostCarbonUnsightlinessRisk Cost1241 Carbon½12½ Unsightliness¼½1¼ Risk1241 We ask, ``How much more important is cost than risk?’’, and similar questions, to create a pairwise comparison matrix for the criteria themselves.

38. CostCarbonUnsightlinessRisk Cost1 0.364241 Carbon½ 0.18212½ Unsightliness¼ 0.091½1¼ Risk1 0.364241 Then we normalise the columns and average the rows to create a vector of preferences. Column Total2.75 1 Criterion Average 0.412 0.123 0.098 0.435

39. Now we multiply the priority matrix by the preference vector... Criterion Average 0.412 0.123 0.098 0.435 Cost Avg. 0.383 0.145 0.205 0.089 0.178 Generatio n Hydro Nuclear Gas Oil Coal Carbon Avg. 0.256 0.177 0.149 0.212 0.251 Sightliness Avg. 0.166 0.067 0.319 0.112 0.311 Risk Avg. 0.066 0.237 0.209 0.322 0.111

40. …and the result gives us our final preferences. Criterion Average 0.412 0.123 0.098 0.435 Cost Avg. 0.383 0.145 0.205 0.089 0.178 Weighted score Hydro 0.234 Nuclear 0.191 Gas 0.224 Oil 0.214 Coal 0.183 Carbon Avg. 0.256 0.177 0.149 0.212 0.251 Sightliness Avg. 0.166 0.067 0.319 0.112 0.311 Risk Avg. 0.066 0.237 0.209 0.322 0.111 ×=

41. It is psychologically possible for a person to prefer coffee to hot chocolate, hot chocolate to tea, and tea to coffee.

42. Such people do not make good subjects for this procedure. Fortunately, there is a test to eliminate them.

43. If the decision-maker is consistent, each column in the pairwise-comparison matrix will be a multiple of every other. If this is the case, the PCM will have a single non-zero eigenvalue, λ. And λ ≈ n, where n is the rank of the matrix. CoalOilGas Coal124 Oil0.512 Gas0.250.51 (not λ > n as in the text)

44. So to determine whether the decision-maker is consistent, calculate the eigenvalues of the PCM and find the largest, λ max. Then calculate (λ max -n)/(n-1), the consistency index, or CI. If the decision-maker is perfectly consistent, this should be zero. Compare this with the same statistic for a random matrix. These have been helpfully tabulated by previous researchers.

45. Compare this with the same statistic for a random matrix. These have been helpfully tabulated by previous researchers. Size of MatrixRandom Index (RI) 20 30.58 40.90 51.12 61.24 71.32 81.41 91.45 101.49

46. Now compare CI for the PCM with RI for a random matrix. If CI/RI < 0.1 then the PCM is acceptably consistent.