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# Lesson9.1 hpaa241

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• This is the lesson 9.1. This lesson discusses Decision Theory with Unknown State Probabilities.
• The reason for studying decision theory are that most management decisions are made in an environment of uncertainty. Decision theory provides a orderly way of choosing among several alternatives or alternative strategies when decisions are made under either uncertainty or risk.
• We classify these decisions either under uncertainty or under risk, where uncertainty exists when the decision maker is unable to ascertain or subjectively estimate the probabilities of the various states of nature that are going to occur. And risk exists when the decision maker does not know with certainty the states of nature, but can estimate the probabilities of various outcomes. So we assume in these analyses that the probabilities of the outcomes are not known.
• The first step in our analysis is to develop a payoff matrix, where we list various alternatives here we list three alternatives: A1, A2 and A3. In various states of nature that can occur and here we list four labeled S1, S2, S3 and S4.
• Our task then is to determine for each of the alternatives when each of the states of nature occurs what the consequences of that state of nature occurring if we had selected that alternative.
• So the C’s of i, j’s in this table, are the consequences state i occurring under alternative j. These consequences could be cost they could be profits or some other type of consequences. In our analysis we’ll simply use either profits or cost, but you could think of other consequences that you could value here such as the loss of light if we had an immunization program. Productive light here is lost if this was an analysis of various drug therapies and so forth. So the consequences depend on the context of the problem.
• For this series of lectures we will use a of home health care example. This example is drawn from a book by Chuck Austin and Stuart Boxeman Quantitative Analysis for Health Services Administration. It makes a nice case study to build the lectures around. So the idea is, suppose a home health agency is considering adding physical therapy services for its clients. And it’s considering three different ways of doing this. Option A is to contract with an independent practitioner and that would cost the agency \$60 per visit. Option B is to hire a staff PT at a monthly salary of \$4000 plus \$400/mo. for a leased car plus \$7/visit for supplies and travel expenses. And the third option, Option C, is to contract with an independent practitioner at \$35/visit but to pay for fringe benefits for that person at \$200/mo. and cover the car and other expenses as in Option B.
• For this example, our states of nature are these: That we can have monthly demand for patient services in terms of visits per month of 30, 90, 140, or 150. So we’ll just screetize this problem into having 4 possible states of nature as listed here. And we are going to assume initially that the probabilities of these states of nature unknown. We know that the four states of nature could occur but we’re not at this point willing to assign probabilities of occurrence of these states of nature.
• The next step is to develop a Payoff Matrix. We need to do some calculations to do that. If we contract with the independent contractor at \$60 per visit, and we assume the charge for the visit is \$75 then the net profit under this alternative is (75 minus 60) times the demand that occurs or 15 times D. So under alternative A the consequence is 15D and we’ll plug in the various values of D to determine what those net profits are.
• Second alternative, alternative A, we pay a monthly salary of \$4,000, the car allowance is \$400, and it’s \$7 a visit for expenses. And so we are out of pocket \$4,400 right away, so it’s minus \$4,400 and then it’s 75 minus 7 per visit, so the net profit is minus 4,400 plus 68 times D, depending on what D is.
• Finally, under alternative three, we contract for the services of our pt at \$35 a visit. And then we are paid a car allowance, with fringe benefits, and \$7 a visit. So we’re out \$400 and \$200 right away or minus 600 and then we have the \$75 charge minus \$35 that we paid a proprietor minus \$7 for the expenses. So the net profit is minus 600 plus (33 times D).
• I would begin to put that into the Payoff Matrix. So under alternative A, if we calculate for the 30 visit a month state of nature, S1 is equal to 30. That gives us a total profit of 15 times 30 of 450, similarly we fill out for the rest of that row, 90 visits a month, 140 visits a month, 150 visits a month. And the values for those are 1350, 2100, 2250 respectively.
• The second row in the table filled out similarly. Here the equation is minus 4,400 plus 68D. So the values are minus 2360, 1720, 5120, 5800 dollars.
• Under alternative three, the profit is minus 600 plus 33D and putting in the values 30, 90, 140, 150, we get 390, 2370, 4020, 4350.
• So our final Payoff Matrix, looks like this. Each of the alternatives is evaluated for each of the states of nature. And then we wanted to determine which of these three alternatives to choose.
• It turns out that no alternative dominates any other alternative. If one of these alternatives was better under all the states of nature, then we would simply choose that alternative and stop at that point. But this problem has been so designed that alternative 1 is better under certain states of nature, alternative 2 is better under other states of nature, and alternative three is better under others as well. So there is no single alternative that is absolutely the best under all the levels of demand that could occur. So we have to develop some sort of criterion to select which alternative is best.S
• The first criterion we look at is called a Maximin Criterion, the criterion that maximizes the minimum payoff for each decision. And what you do here is identify the minimum payoff for each alternative, that is, for each row. And you pick the largest minimum payoff for all of the alternatives.
• So that would look like this. Here we have our Payoff Matrix. Under alternative A1, the minimum is 45 so we’ll take and list it in the column on the right. Under alternative A2, the worst case or the minimum is minus 2360, so we’ll take that and list it on the column on the right. And under alternative A3, the worst case or minimum payoff is 390 so we’ll list that and then we choose that maximum of those minimums. That is 450. So our Maximin criterion says that under all the things that could go wrong, the one that goes the the least wrong is under alternative A1 and that’s the alternative we will choose. So the Maximin Criterion in this case would select alternative A1 with the maximum minimum payoff of 450.
• So the Maximin Decision Criterion is a very conservative criterion or risk adverse criterion. It is pessimistic. It assumes that nature will always vote against you. So that you try to minimize the worst thing that can happen to you and take the maximum of the minimum payoffs. Now this assumes that payoffs and the payoff tables were profits
• If the payoffs in the payoff matrix or table were costs, then the equivalent conservative or risk adverse criterion would be the Minimax Criterion. Since you would again take the smallest of the worst cases if they were costs you’d want to look at the maximum cost and take smallest of the worst case scenarios, or the alternative that gave you the Minimax. And that is also a pessimistic criterion.
• Another decision that we can use for decision making is called the Maximax Criterion. An that’s the criterion that maximizes the maximum payoff for each alternative. Here what we do is identify the maximum payoff for each alternative. And then we pick that alternative that has the maximum of the maximum payoffs
• So our Decision table or payoff Matrix would look like this. In each row we pick the greatest values. For A1 is 2250, for A2 is 5800, for A3 is 4350 and then we pick the maximums of those maximums, or 5800. In this case, our decision would be to take alternative 2 which gives you the maximum of the maximums.
• The Maximax Criterion is a very optimistic or risk seeking criterion. This would not be a criterion which preserves capital in the long run because while you are going out for these best case scenarios, nature may turn against you and clobber you with the worst possible cost. So if you chose A2, for example, seeking the maximum return, if state of nature one happened then you’re going to lose 2360 dollars which, as we indicate here, does not help you preserve capitol in the long run.
• Similarly if the values in the payoff matrix were costs, then the equivalent optimistic criterion is minimin. You would go for the minimum of the cost for all the various alternatives. And that again assumes nature is going vote in your behalf. And what you want to happen is actually going to happen. So that’s an optimistic way of looking at the decision making.
• Another criterion we can use is called the Minimax Regret Criterion. This criterion minimizes the loss incurred by not selecting the optimal or best alternative solution. What we do in this case, is take the largest element in each of the columns and we’ll start off with the first column. You subtract each element in that column from the largest element to compute the opportunity loss and continue on for all the other columns. And what you then do is identify the maximum regret for each alternative and then choose that decision that has the smallest maximum regret. So this is in a sense looking at the opportunity losses and minimizing your opportunity losses by selecting that alternative that gives you minimum of the opportunity losses or minimum maximum regret.
• So what we have to do here is to construct our regret table. So we’ll start of with column one. The maximum value in column one is 450. So we take 450
• And subtract each element in that column from 450. So 450 minus 450 gives you zero, 450 minus(minus2360) gives you plus 2810, and 450 minus 390 gives you 60. So those are three regret values under column one or state of nature one.
• We continue this process for state of nature two. The maximum value in column two is 2370.
• We’ll take 2370 and subtract from that all the other values in that column. So 2370 minus 1350 is 1020. 2370 minus 1720 is 650. 2370 minus 2370 is 0.
• Continuing on in this way we can compute all the other values in the table. That gives us the table of regrets.
• Which looks like this where we have a regret table with all those values filled in. And then for each row we’ll take the maximum regret. So under alternative one the maximum regret is 3559, under alternative two the maximum regret is 2810, and under alternative three the maximum regret is 1450.
• And we choose of those maximums, the minimum so we choose 1450. So that’s the minimum maximum regret. That’s the one you choose in order to choose the alternative that minimizes your opportunity costs. Now, cleverly, we have worked up an example here such that under each of the criteria that we have chosen you get a different answer. Maximin shows alternative one, Maximax shows alternative two, and Minimax Regret shows alternative three.
• Minimax regret is criterion that is also relatively conservative. It is trying to minimize the opportunity costs that occur f you chose the wrong alternative. It is not quite as pessimistic as the maximin criterion. So in some sense it is a happy medium. But what criterion you use in a given decision is some what, of course, context dependant. What we have done here is illustrate three different approaches which you might use in choosing an alternative. And in the next lesson we are going to go ahead and assume tat we have some probabilities associated with these alternatives so we can do expected monetary values. We are building a platform for our decision analysis and each lesson built a little bit more strength in that platform.
• ### Lesson9.1 hpaa241

1. 1. Lesson 9.1 Decision Theory with Unknown State Probabilities
2. 2. Decision Theory •Most management decisions are made in an environment of uncertainty. •Decision theory provides a orderly way of choosing among several alternative strategies when decisions are made under uncertainty or risk.
3. 3. Decision Theory • Uncertainty exists when the decision maker is unable to ascertain or subjectively estimate the probabilities of the various states of nature. • Risk exists when the decision maker does not know with certainty the state of nature, but the probabilities of various outcomes is known.
4. 4. Payoff Matrix a1 Alternativesi a2 a3 s1 States of Naturej s2 s3 s4
5. 5. Payoff Matrix Alternativesi a2 a3 s2 c12 s3 c13 s4 c14 c21 c22 c23 c24 c31 a1 s1 c11 States of Naturej c32 c33 c34
6. 6. Payoff Matrix Alternativesi a2 s2 c12 s3 c13 s4 c14 c21 c22 c23 c24 c31 a1 s1 c11 States of Naturej c32 c33 c34 Cij is the consequence of state I under alternative j a3
7. 7. Home Health Example Suppose a home health agency is considering adding physical therapy (PT) services for its clients. There are three ways to do this: Option A: contract with an independent practitioner at \$60 per visit. Option B: hire a staff PT at a monthly salary of \$4000 plus \$400/mo. for a leased car plus \$7/visit for supplies and travel. Option C: independent practitioner at \$35/visit but pay for fringe benefits at \$200/mo. and cover the car and expenses as in Option B. Source: Austin, CJ and Boxerman, SB, Quantitative Analysis for Health Services Administration, AUPHA/Health Administration Press, Ann Arbor, Michigan, 1995
8. 8. Payoff Matrix: Home Health Example s1 Demand of Patient Services: Visits/ mo. 30 States of Naturej s2 s3 90 140 s4 150 Assumption: Probabilities of States of Nature are unknown.
9. 9. Payoff Matrix: Home Health Example Alternativesi •Contract with independent Contractor at \$60/visit. a1 Net Profit = (75 - 60) * D = 15*D a2 a3 Assumption: Charge \$75 per visit.
10. 10. Payoff Matrix: Home Health Example Alternativesi •Contract with independent Contractor at \$60/visit. a1 Net Profit = (75 - 60) * D = 15*D •Pay monthly salary of \$4,000 •Car allowance \$400 •Expenses @\$7 a visit a2 Net Profit = - 4,000 - 400 + (75 - 7) * D = -4,400 + 68*D a3 Assumption: Charge \$75 per visit.
11. 11. Payoff Matrix: Home Health Example Alternativesi •Contract with independent Contractor at \$60/visit. a1 Net Profit = (75 - 60) * D = 15*D •Pay monthly salary of \$4,000 •Car allowance \$400 •Expenses @\$7 a visit a2 Net Profit = - 4,000 - 400 + (75 - 7) * D = -4,400 + 68*D •Contract @ \$35 per visit •Car allowance \$400 •Fringe benefits of \$200 •Expenses @\$7 a visit Net Profit = -400 -200+ (75 - 35 -7) * D = -600 + 33*D a3 Assumption: Charge \$75 per visit.
12. 12. Payoff Matrix Total Profit (Alt 1) = 15*D s1 s2 s3 s4 2250 30 a1 a2 90 140 450 1350 2100 150
13. 13. Payoff Matrix Total Profit (Alt 2) = -4,400 + 68D s1 s2 s3 s4 2250 30 a2 140 450 1350 2100 -2360 a1 90 1720 5120 150 5800
14. 14. Payoff Matrix Total Profit (Alt 3) = -600 + 33D s1 s2 s3 s4 2250 30 140 450 1350 2100 -2360 a1 90 150 1720 5120 5800 390 2370 4020 4350 a2
15. 15. Payoff Matrix s1 s2 s3 s4 2250 30 140 450 1350 2100 -2360 a1 90 150 1720 5120 5800 390 2370 4020 4350 a2
16. 16. Payoff Matrix No alternative dominates any other alternative s1 s2 s3 s4 2250 30 140 450 1350 2100 -2360 a1 90 150 1720 5120 5800 390 2370 4020 4350 a2
17. 17. Criteria for Decision Making Maximin Criterion- criterion that maximizes the minimum payoff for each alternative. Steps: 1) Identify the minimum payoff for each alternative. 2) Pick the largest minimum payoff.
18. 18. Maximin Decision Criterion s1 s2 Maximin s3 s4 2250 450 30 140 450 1350 2100 -2360 a1 90 150 1720 5120 5800 -2360 390 2370 4020 4350 390 a2
19. 19. Maximin Decision Criterion The maximin criterion is a very conservative or risk adverse criterion. It is a pessimistic criterion. It assumes nature will vote against you.
20. 20. Minimax Decision Criterion If the values in the payoff matrix were costs, the equivalent conservative or risk adverse criterion would be the minimax criterion. It is a pessimistic criterion.
21. 21. Criteria for Decision Making Maximax Criterion- criterion that maximizes the maximum payoff for each alternative. Steps: 1) Identify the maximum payoff for each alternative. 2) Pick the largest maximum payoff.
22. 22. Maximax Decision Criterion s1 s2 s3 s4 2250 2250 Maximax 30 140 450 1350 2100 -2360 a1 90 150 1720 5120 5800 5800 390 2370 4020 4350 4350 a2
23. 23. Maximax Decision Criterion The maximax criterion is a very optimistic or risk seeking criterion. It is not a criterion which preserves capital in the long run.
24. 24. Minimin Decision Criterion If the values in the payoff matrix were costs, the equivalent optimistic criterion is minimin. It assumes nature will vote for you.
25. 25. Criteria for Decision Making Minimax Regret Criterion- criterion that minimizes the loss incurred by not selecting the optimal alternative. Steps: 1) Identify the largest element in the first column. 2) Subtract each element in the column from the largest element to compute the opportunity loss and repeat for each column. 3) Identify the maximum regret for each alternative and then choose that alternative with the smallest maximum regret.
26. 26. Minimax Regret: Regretj = Max [cij] - cij s1 s2 s3 s4 30 140 150 450 1350 2100 2250 -2360 a1 90 1720 5120 5800 a2 390 a3 2370 4020 4350
27. 27. Minimax Regret: Regretj = Max [cij] - cij s1 s2 s3 s4 30 a1 a2 a3 90 140 150 450 450 - 450 0 1350 2100 2250 -2360 450 - (-2360) 2810 1720 390 450 - 390 60 2370 5120 4020 5800 4350
28. 28. Minimax Regret: Regretj = Max [cij] - cij s1 s2 s3 s4 30 a1 a2 a3 90 140 150 450 450 - 450 0 1350 2100 2250 -2360 450 - (-2360) 2810 1720 390 450 - 390 60 2370 5120 4020 5800 4350
29. 29. Minimax Regret: Regretj = Max [cij] - cij s1 30 a1 a2 a3 450 450 - 450 0 s2 90 1350 2370 - 1350 1020 s3 s4 140 150 2100 2250 -2360 450 - (-2360) 2810 1720 2370 - 1720 650 5120 390 450 - 390 60 2370 2370 - 2370 0 4020 5800 4350
30. 30. Minimax Regret: Regretj = Max [cij] - cij s1 30 a1 a2 a3 450 450 - 450 0 s2 90 1350 2370 - 1350 1020 s3 140 2100 5120 - 2100 3020 s4 150 2250 5800 - 2250 3550 -2360 450 - (-2360) 2810 1720 2370 - 1720 650 5120 5120 - 5120 0 5800 5800 - 5800 0 390 450 - 390 60 2370 2370 - 2370 0 4020 5120 - 4020 1100 4350 5800 - 4350 1450
31. 31. Minimax Regret: Regretj = Max [cij] - cij s1 s2 s3 30 90 140 s4 150 3550 0 1020 3020 2810 a1 3550 Max Regret 650 0 0 2810 1100 1450 1450 a2 60 a3 0
32. 32. Minimax Regret: Regretj = Max [cij] - cij s1 s2 s3 30 90 140 s4 150 3550 0 1020 3020 2810 a1 3550 Max Regret 650 0 0 2810 1100 1450 1450 a2 60 a3 0
33. 33. Minimax Regret Decision Criterion The minimax regret criterion is also a conservative criterion. It is not as pessimistic as the maximin criterion.