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DATA AND DECISION ANALYTICS
Practice for Decision Analysis Solutions Question 1 (a.) (5 pts) What are the decisions and the states of nature? Construct the payoff table for this decision problem. (b.) (5 pts) Draw a decision tree for this problem. Should you have the event indoors or outdoors? What is the expected value of the income from the event? What is the best case? The expected value is $69,750. If you choose inside, the best case is $75,000. If you could decide at the last minute the best case is $100,000. If you have perfect information, the expected value is $82,000.
(c.) (5 pts) Given your prior probability estimates about the weather, what is the chance that there will be an approaching cold front three days prior to the event? Here are the probability calculations: (d.) (5 pts) When you get to three days prior to the event, if you see an approaching cold front, what is the probability that it will be a sunny and warm day for the event? If there is no front, what is the probability that the day of the event it will be a rainy and miserable day? Here are the conditional probability calculations: (e.) (5 pts) Draw a decision tree for the event if you decide to take the option on the indoor space. What is the best decision policy now?
(c.) (5 pts) Which outcome causes the cost of the lease to be the greatest? What is the situation where the cost of the lease is the minimum? Does this help you choose between the three lease options? The greatest cost comes from the highest mileage under lease L3. The least cost comes from the lowest mileage, also under lease L3. So this lease could be the best or the worst outcome, and so might be a risky choice. Conversely, it appears that lease L2 is the least risky. (d.) (5 pts) Draw a decision tree for the lease choice problem. What lease option should you choose? Here is the exam 1 solution1: For each of the quiz versions, the best lease option is L2. (e.) (5 pts) Suppose you had perfect information and could know how many miles you will have to drive before you pick the lease option. What is the expected value of the lease cost? How much would you be willing to pay to get that perfect information? For each quiz, you first find the best lease option for each state of nature, then multiply each by the probability of the state to find the expected value. For the four quiz versions, the values are 3510(. 20) + 4840(. 50) + 5240(. 30) = 4694 3600(. 25) + 4780(. 45) + 5180(. 30) = 4605 3600(. 30) + 4960(. 45) + 5360(. 25) = 4652 3540(. 25) + 4840(. 40) + 5240(. 35) = 4655 The amount you’d be willing to pay would be $186 for version 1, $195 for version 2, $288 for version 3, and $225 for version 4.
(f.) (5 pts) Use your prior estimates of the driving you will be doing and the sample information from your company to determine the probabilities of ending up in a sales or administrative job. The formulas are Pr(sales) = Pr(sales | low) Pr(low) + Pr(sales | middle) Pr(middle)+ Pr(sales | high) Pr(high) Pr(admin) = Pr(admin | low) Pr(low) + Pr(admin | middle) Pr(middle)+ Pr(admin | high) Pr(high) (or you can remember that Pr(admin) = 1 − Pr(sales)) v.1 Pr(sales) =. 2(. 2) +. 25(. 5) +. 7(. 3) =. 375 Pr(sales) =. 625 v.2 Pr(sales) =. 2(. 25) +. 3(. 45) +. 7(. 3) =. 395 Pr(sales) =. 605 v.3 Pr(sales) =. 25(. 3) +. 3(. 45) +. 7(. 25) =. 385 Pr(sales) =. 615 v.4 Pr(sales) =. 2(. 25) +. 3(. 4) +. 75(. 35) =. 433 Pr(sales) =. 568 (g.) (5 pts) What is the revised probability that you will drive the high mileage amount if you have put in a sales track job? What is the revised probability that you will drive the low mileage amount if you are given a job in the administrative track? The formulas are Pr(high | sales) = Pr(sales^ |^ high) Pr(high) Pr(sales) Pr(low | admin) = Pr(admin | low) Pr(low) Pr(admin) version 1 .7(.3)^ =. 560 .8(.2)^ =. 256 .375. version 2 .7(.3)^ =. 53165 .8(.25)^ =. 33058 .395. version 3 .7(.25)^ =. 45455 .75(.3)^ =. 36585 .385. version 4 .75(.35)^ =. 60694 .8(.25)^ =. 35242 .433.
(k.) (5 pts) What do you suppose would happen if the mileage amount where the rate changes were to go up for lease option L2? What if that mileage amount were to go up for lease option L3? Describe what would happen to your decision policy. If the 6,000 miles go up for lease L2, then the cost of that lease will increase. If the 6,000 miles go up for lease L3, then the cost of that lease will decrease. If you use the original decision tree the decision policy changes in the following way. Since the best lease is L2, if the mileage decreases, L2 remains the best choice. But if it goes up, and some mileage level the best choice becomes L instead. If you use the tree with your job assignment as sample information, describing the policy is slightly more complicated. If you get the sales job, you will continue to select L2 if the mileage goes down (like with the simple tree). If the mileage goes up, at some point you will switch to L3. If you get the admin job, then for version 1 and 3 you will switch from L to L2 if the mileage level goes down or switch from L1 to L3 if the mileage goes up. For version 2, the answer is just like the simple tree (or getting the sales job). For version 4, if the mileage goes up, you will just stay with L3, but if the mileage goes down, you will switch to L2 at some point. Question 3 (a.) (5 pts) Construct a payoff table for this problem that determines the financial outcome for each decision and for each demand level. Demand Capacity Subcontract Price Increase 110,000 5405 5440 5300 120,000 5885 5880 5900 130,000 6365 6320 6200 (b.) (5 pts) What is the expected value if we had perfect information (EVPI) for this decision problem? If 110, choose subcontract. If 120, choose price increase. If 130, choose increase capacity. So the value is_._ 5(5440) +. 35(5900) +. 15(6365) = 5740
(c.) (5 pts) Represent the problem as a decision tree and determine the strategy that maximizes the expected monetary value. (d.) (5 pts) Suppose that there are a number of subcontractors and SRS might be able to negotiate a better price. Over what range of subcontracted prices does your decision from the decision tree remain the same? Over what prices would it change? If the subcontractor is cheaper, since it is already the best decision, we would just make more money. If the subcontractor becomes more expensive, then the second best solution (increase capacity) will become best. With some algebra you can find that if the subcontractor costs $0.55 more, then increasing capacity will be the best decision. (e.) (5 pts) What is the probability that the report will be for high demand? Pr( H ) = Pr( H |110) Pr(110) + Pr( H |120) Pr(120) + Pr( H |130) Pr(130) Pr( H ) = (. 2)(. 5) + (. 4)(. 35) + (. 8)(. 15) = 0_._ 36
(b.) (5 pts) Draw a decision tree for the problem and determine the expected value the best bowl game decision. Here are the four trees for the Blimpie Bowl decision. For version one of the exam, the right decision is to accept the bowl offer: (c.) (5 pts) Suppose that you could know for sure how the Camel game will turn out. What is the expected value of the bowl game decision? If you could know the game outcome, after a big win you would reject the Blimpie offer and take the big payoff. For a regular win or a loss, you would accept the Blimpie offer. The value of the decision with perfect information would be v1 v2 v3 v Big win 800,000 800,000 800,000 850, Win 400,000 400,000 300,000 400, Loss 200,000 200,000 150,000 300, Probabilities .4/.25/.35 .4/.2/.4 .45/.25/.3 .4/.25/. Expected value 490,000 480,000 480,000 545, (d.) (5 pts) Write the values of all of the conditional probabilities that characterize Lee’s sample information. This can be done either with algebraic formulas or with a clearly marked table. What is the probability that Lee will predict an Eagle victory over the Camels? Three conditional probabilities are given in the problem and the other three (when a team loses) are just the complements. These probabilities are Pr(Lee forecast|game outcome). The probability of Lee forecasting a win (LFW) is the formula: Pr( LF W ) = Pr( LF W | BW ) Pr( BW )+ Pr( LF W | W ) Pr( W )+ Pr( LF W | L ) Pr( L ) and here are tables with the results for exam version one:
(e.) (5 pts) If Lee does predict a win, what is the probability that the Eagles will then win the game by more than 14 points? If Lee predicts a loss, what is the probability the Camels will win? The required probabilities are Pr( BW | LF W ) = Pr( LF W | BW ) Pr( BW ) / Pr( LF W ) Pr( L | LF L ) = Pr( LF L | L ) Pr( L ) / Pr( LF L ) These are given in the top left and bottom right cells (along with all the other similar conditionals) for version one of the exam: (f.) (5 pts) Draw a decision tree for the problem if you decide to hire Lee and get his prediction for the game. How much is Lee’s expertise worth? Here are the trees for exam version one. The value of the tree is shown, and the value of Lee’s information is the difference between that value and the expected value of the original tree with no sample information:
(b.) (5 pts) What are the payoffs for each combination of decision and state of nature? If you ship and it is OK , then the payoff is $31,000. If you ship and it is NOK then the payoff is -$9,000. If you rework and the machine is OK then the payoff is $16,000. Finally, if you rework and the machine is NOK , the payoff is $10,000. (c.) (5 pts) Draw a decision tree for the problem and determine the expected value of the delivery and identify the best decision policy. The tree looks like this: The values for the nodes of the tree are: v1: a = 15 , 000 b = 15 , 400 c = 15 , 400 ;best policy is to rework v2: a = 15 , 000 b = 15 , 100 c = 15 , 100 ;best policy is to rework v3: a = 17 , 000 b = 15 , 400 c = 17 , 000 ;best policy is to ship v4: a = 17 , 000 b = 15 , 700 c = 17 , 000 ;best policy is to ship (d.) (5 pts) What is the probability that if Test 1 is run, it will result in satisfactory/unsatisfactory outcome? What is the probability that if Test 2 is run, it will result in satisfactory/unsatisfactory outcome? This is the marginal probability of the sample information, S or U , and is Bayes’ rule, using the marginal probabilities of the states of nature OK and NOK with the prior conditional probabilities. This is what the sum of the joint probabilities using the table calculation method also does, so either method computes: Pr( S ) = Pr( S | OK ) Pr( OK )+ Pr( S | NOK ) Pr( NOK ) Pr( U ) = Pr( U | OK ) Pr( OK ) + Pr( U | NOK ) Pr( NOK ) For Test 1 Pr( S ) =. 8 Pr( OK )+. 3 Pr( NOK ) Pr( U ) =. 2 Pr( OK ) +. 7 Pr( NOK )
For Test 2 Pr( S ) =. 6 Pr( OK )+. 1 Pr( NOK ) Pr( U ) =. 4 Pr( OK ) +. 9 Pr( NOK ) Pr( OK ) =. 6 for exams v1 and v2; Pr( OK ) =. 65 for exams v3 and v4, so v1 & v2: Test 1 Pr( S ) =. 8(. 6) +. 3(. 4) =. 60 v1 & v2: Test 2 Pr( U ) =. 4(. 6) +. 9(. 4) =. 60 v3 & v4: Test 1 Pr( U ) =. 2(. 65) +. 7(. 35) =. 375 v3 & v4: Test 2 Pr( S ) =. 6(. 65) +. 1(. 35) =. 425 (e.) (5 pts) If Test 1 is run and results in satisfactory/unsatisfactory outcome, what is the probability that the customer will discover that the machine is not okay ( NOK ) if the machine is shipped without rework? What is the probability if the outcome of the test is satisfactory/unsatisfactory This is just an application of Pr( A | B ) = Pr( B | A ) Pr( A ) Pr( B ) to find the revised conditional probabilities (the last column of the table used in the book). The two probabilities for Test 1: v1 & v2: Pr( NOK | S ) =. 3(. 4) /. 6 =. 2 v1 & v2: Pr( NOK | U ) =. 7(. 4) /. 4 =. 7 v3 & v4: Pr( NOK | S ) =. 3(. 35) /. 625 =
. 168 v3 & v4: Pr( NOK | U ) = . 7(. 35) /. 375 =. 653 (f.) (5 pts) If Test 2 is run and results in satisfactory/unsatisfactory outcome, what is the probability that the customer will discover that the machine is not okay ( NOK ) if the machine is shipped without extensive rework? What is the probability if the outcome of the test is satisfactory/unsatisfactory This is the same application of Pr( A | B ) =Pr( B | A ) Pr( A ) Pr( B ) to find the revised conditional probabilities (the last column of the table used in the book). The two probabilities for Test 2: v1 & v2: Pr( NOK | S ) =. 1(. 4) /. 4 =. 10 v1 & v2: Pr( NOK | U ) =. 9(. 4) /. 6 =. 60 v3 & v4: Pr( NOK | S ) =. 1(. 35) /. 425 = . 082 v3 & v4: Pr( NOK | U ) = . 9(. 35) /. 575 =. 548
With perfect information, the value increases by about $7,000. With the tests, the value increases by about $4,000. The cost of the test reduces the value to half that, and so the test is worth doing. (i.) (5 pts) Suppose the customer is in a hurry to use the machine and wants to maximize the probability that the machine is delivered and does not require the three week on-site repair. Would they prefer CFOB to use Test 1/2 or not? How could the customer induce CFOB to also want to maximize this probability, rather than the profit of the machine delivery? The profitability is always maximized if CFOB does rework but this also reduces the profit. If the customer knew that CFOB had the test then the probability would increase (because when the test was U , CFOB would do the rework and maximize the probability). To get them to always do the rework they would have offer a payoff better than the expected payoff for the test, so they could offer an additional $2,000 or $3,000 (since CFOB will spend $2,000 on the test). Question 6 (a.) (5 pts) What are the decisions and the states of nature? Construct the payoff table for this decision problem. Decisions are to choose Enzo or Ferruccio. The states of nature are hit or miss. The payoff table is Enzo Ferruccio Hit 2,400,000 675, Miss (300,000) 75, (b.) (5 pts) Draw a decision tree for this problem. Should your uncle invest in Enzo or Ferruccio? What is the expected value of the investment? Invest with Ferruccio, and the expected value is 285,000.
(c.) (5 pts) For designers like Enzo, if his line was going to be a hit, there would be a positive reaction at the show 90% of the time, but if the line were going to be a miss, the show would only have a positive reaction 20% of the time. What is the probability that Enzo would get a positive reaction at the show? After a positive reaction, what is the probability his line will be a hit? After a negative reaction, what is the probability his line will be a hit? Pr( pos ) = (. 2)(. 9) + (. 8)(. 2) =. 34 Pr( hit | pos ) = (. 2)(. 9) /. 34 =. 529 Pr( hit | neg ) = (. 2)(. 1) /. 66 =. 03 (d.) (5 pts) For a designer like Ferruccio, the probability of a positive reaction if his line were going to be a hit is 80%, and if his line were to be a miss, there is a 40% chance that there will be a positive reaction at the show. What is the probability of a positive reaction at the Atlanta show? After a positive reaction, what is the probability his line will be a hit? After a negative reaction, what is the probability his line will be a hit? Pr( pos ) = (. 3)(. 8) + (. 7)(. 4) =. 52 Pr( hit | pos ) = (. 3)(. 8) /. 52 =. 462 Pr( hit | neg ) = (. 3)(. 2) /. 48 =. 125 (e.) (5 pts) You show these probability results to your uncle and ask him if we should consider showing either designer’s line at the Atlanta show. He says the answer is obvious, even without doing any further calculations. Which designer is he thinking about and how does he know? Right now you choose Ferruccio. But if you test Enzo and the test comes back positive, the probability of a hit increases from 20% to 53% and if that happens you get $2,500,000. So the expected value will jump dramatically. If the test is negative, you just make the same decision you do now. If you test Ferruccio, and it comes back positive, you make the same decision you did before. If it is negative, then you might switch to Enzo, but because you did not test Enzo, the chance of the big payoff is still just 20%.
(b.) (5 pts) Construct a payoff table for this decision problem. Here are the payoff calcuations for exam version 1: (c.) (5 pts) Draw a decision tree for this problem. What is the best decision for the company? Here are the decision trees for exam version 1. The best version is marked with an arrow: (d.) (5 pts) What is the expected value of this supply chain project? What outcomes are possible? What is the probability and monetary value of each outcome? Is there a probability of losing money on the supply chain project? The expected value for the four exam versions are $108,000, $119,750, $112,500 and $119,000. The outcomes and probabilities are v1 v2 v3 v Strong 180, 50% 140, 45% 180, 55% 200, 60% Average 90, 35% 140, 40% 90, 30% 65, 25% Weak (90), 15% 5, 15% (90), 15% (115) 15% lose money 15% 0% 15% 15%
(e.) (5 pts) Suppose that the selling price of the product is affected by substitute food products that could be more or less available. Over what range of the sales price ($8) would your decision remain the same? If H is chosen, then if the price increases, it becomes more attractive, so the breakeven price is less than $8 (v1,v3,v4). If L is chosen, then if the price decreases, it becomes more attractive, so the breakeven price is more than $8. Can compute the value by looking to equate the expected value of the two options, using price times the expected demand minus expected cost. So the price is the difference of the expected costs divided by the difference between the expected demands: v1 v2 v3 v H expected demand 62,000 63,750 62,500 66, L expected demand 48,500 52,750 44,250 48, H expected cost 388,000 411,250 387,500 409, L expected cost 281,500 302,250 260,750 281, breakeven price 7.89 9.91 6.95 7. (f.) (5 pts) The expert has given you the probabilities for an up market for the substitute product. What are the probabilities for his predictions in a down market for the substitute product? Here are the probabilities for each exam version: (g.) (5 pts) What is the probability that the expert will predict a down market for the substitute product? Here are the probabilities for each exam version:
