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- Question : 1E - Value of the stochastic solution Assume the farmer allocates his land according to the solution of Table 2, i.e., 120acresforwheat,80acresforcorn,and300acresforsugarbeets.Showthatif yields are random(20% below average,average,and 20% aboveaveragefor all crops with equal probability one third), his expected annual pro?t is $107,240. Todothis observethatplantingcosts arecertainbutsalesandpurchasesdepend ontheyield.Inotherwords,?ll ina tablesuchas Table5butwiththe?rst-stage decisions given here.
- Question : 2E - Price effect When yields are good for the farmer, they are usually also good for many other farmers. The supply is thus increasing, which will lower the prices. As an example, we may consider prices going down by 10% for corn and wheat when yields are above average and going up by 10% when yields are below average. Formulate the model where these changes in prices affect both sales and purchases of corn and wheat. Assume sugar beet prices are not affected by yields.
- Question : 3E - Binary ?rst stage Consider the case where the farmer possesses four ?elds of sizes 185, 145, 105,and65acres,respectively.Observethatthetotalof500acresisunchanged. Now,the?eldsareunfortunatelylocatedindifferentpartsofthevillage.Forreasons of ef?ciency the farmer wants to raise only one type of crop on each ?eld. Formulate this model as a two-stage stochastic program with a ?rst-stage program with binary variables.
- Question : 4E - Integer second stage Consider the case where sales and purchases of corn and wheat can only be obtained through contracts involving multiples of hundred tons. Formulate the model as a stochastic programwith a mixed-integersecond stage.
- Question : 5E - Consider any one of Exercises 2 to 4. Using standard mixed integer programmingsoftware,obtainanoptimalsolutionoftheextensiveformofthestochastic program. Compute the expected value of perfect information and the value of the stochastic solution.
- Question : 6E - Multistage program It is typicalin farmingtoimplementcroprotationinorderto maintaingoodsoil quality.Sugarbeetswould,forexample,appearintriennialcroprotation,which means they are planted on a given ?eld only one out of three years. Formulate a multistage programto describe this situation. To keep things simple, describe the case when sugar beets cannot be planted two successive years on the same ?eld, and assume no such rule applies for wheat and corn. (On a two-year basis, this exercise consists purely of formulation: with the basic data of the example, the solution is clearly to repeat the optimal solution in Table 5, i.e., to plant 170 acres of wheat, 80 acres of corn, and 250 acres of sugarbeets.Theproblembecomesmorerelevantonathree-yearbasis. Itis also relevant on a two-year basis with ?elds of the sizes given in Exercise 1. In terms of formulation, it is suf?cient to consider a three-stage model. The ?rst stage consists of ?rst-year planting. The second stage consists of ?rstyear purchases and sales and second-year planting. The third-stage consists of second-year purchases and sales. Alternatively, a four-stage model can be built, separating ?rst-year purchases and sales from second-year planting. Also discuss the question of discounting the revenues and expenses of the various stages.)
- Question : 7E - Risk aversion Economic theory tells us that, like many other people, the farmer would normally act as a risk-averse person. There are various ways to model risk aversion. One simple wayis to plan forthe worst case. More precisely,it consists of maximizingthe pro?tundertheworst situation.Note that forsome models,it is not known in advance which scenario will turn out to induce the lowest pro?t. In our example, the worst situation corresponds to Scenario 3 (below average yields). Planning for the worst case implies the solution of Table 4 is optimal. (a) Compute the loss in expected pro?t if that solution is taken. (b) A median situation would be to require a reasonable pro?t under the worst case. Find the solution that maximizes the expected pro?t under the constraint that in the worst case the pro?t does not fall below $58,000.What is now the loss in expected pro?t? (c) Repeat part (b) with other values of minimal pro?t: $56,000, $54,000, $52,000, $50,000, and $48,000. Graph the curve of expected pro?t loss. Also compare the associated optimal decisions
- Question : 8E - Data ?uctuations Table 1 contains mean data over a relatively long period, from the late nineties till 2006. Yield ?uctuations have been treated through random yields. What about other data
- Question : 9E - . If prices are also random variables, the news vendor
- Question : 10E - In the news vendor
- Question : 11E - Suppose c=10, q=25, r=5,anddemandisuniformon [50,150].Findthe optimal solution of the news vendorproblem.Also, ?nd the optimal solution of the deterministic model obtained by assuming a demand of 100. What is the value of the stochastic solution?
- Question : 1E - Suppose you consider just a ?ve-year planning horizon. Choose an appropriate target and solve over this horizon with a single ?rst-period decision.
- Question : 2E - Suppose you implement a buy-and-holdstrategy and make a single investment decision without any additional trading until the end of the time horizon. Formulate and solve this problem to determine an optimal allocation.
- Question : 3E - Suppose that goal G is also a random parameter and could be $75,000 or $85,000 with equal probabilities. Formulate and solve this problem. Compare this solution to the solution for the problemwith a known target
- Question : 4E - Suppose that every trade (purchase or sale) of an asset involves a transaction cost that is equal to 1% of the amount traded. Re-formulate the problem with this transaction cost and solve for the optimal solution.
- Question : 1E - The detailed-level decisions can be found quite easily according to an order of merit rule. In this case, one begins with Mode 1 and uses the least expensive equipment until its capacity is exhausted or demand is satis?ed. One continues to exhaust capacity or satisfy demand in order of increasing unit operating cost and mode. Show that this procedure is indeed optimal for determining the ytij values.
- Question : 2E - Provethat,inthecaseofnoserialcorrelation(?t and ?t+1 stochasticallyindependent),anoptimal solutionhas the same valuefor wt and xt forall ? . Give an example where this does not occur with serial correlation.
- Question : 3E - For the examplein (3.11), supposewe add a reliability constraint of the form in (3.14)to the expectedvalue problem,but we use a right-handside of 11 instead of 12. What is the stochastic programexpected value of this solution?
- Question : 1E - For theexamplegiven,what is theprobabilityof exceedingthe stress constraint foranaxledesignedaccordingtothestochasticprogramoptimalspeci?cations?
- Question : 2E - Again, for the example given, what is the probability of exceeding the stress constraint for an axle designed according to the deterministic program
- Question : 3E - Supposethateveryaxlecanbetestedbeforebeingshippedatacostof s pertest. The test completely determines the dimensions of the product and thus informs the producerof the risk of failure. Formulate the new problemwith testing.
- Question : 1E - Northam Airlines is trying to decide how to partition a new plane for its Chicago
- Question : 2E - Tomatoes Inc. (TI) produces tomato paste, ketchup, and salsa from four resources: labor, tomatoes, sugar, and spices. Each box of the tomato paste requires 0.5 labor hours, 1.0 crate of tomatoes, no sugar,and 0.25 can of spice. A ketchup box requires 0.8 labor hours, 0.5 crate of tomatoes, 0.5 sacks of sugar, and 1.0 can of spice. A salsa box requires 1.0 labor hour, 0.5 crate of tomatoes, 1.0 sack of sugar, and 3.0 cans of spice. Thecompanyisdecidingproductionforthenextthreeperiods.Itisrestricted to using200 hoursof labor,250crates of tomatoes, 300sacks ofsugar,and 100 cans of spices in each period at regular rates. The company can, however, pay for additional resources at a cost of 2.0 per labor hour, 0.5 per tomato crate, 1.0 per sugar sack, and 1.0 per spice can. The regular productioncosts for each product are 1.0 for tomato paste, 1.5 for ketchup, and 2.5 for salsa. Demandisnotknownwithcertaintyuntilaftertheproductsaremadeineach period. TI forecasts that in each period two possibilities are equally likely, corresponding to a good or bad economy. In the good case, 200 boxes of tomato paste, 40 boxes of ketchup, and 20 boxes of salsa can be sold. In the bad case, these valuesare reducedto 100, 30, and 5, respectively.Any surplusproduction is stored at costs of 0.5, 0 .25, and 0.2 per boxfortomato paste, ketchup, and salsa, respectively. TI also considers unmet demand important and assigns costs of 2.0, 3 .0, and 6 .0 per box for tomato paste, ketchup, and salsa, respectively, for any demandthat is not met in each period.
- Question : 3E - TheClearLakeDamcontrolsthewaterlevelinClearLake,awell-knownresort in Dreamland.The Dam Commission is trying to decide how much water to releaseineachofthenextfourmonths.TheLakeiscurrently150mmbelow?ood stage. The dam is capable of lowering the water level 200 mm each month, but additional precipitationand evaporationaffect the dam. The weather near Clear Lake is highly variable.The Dam Commission has dividedthe months into two two-month blocks of similar weather. The months within each block have the same probabilities for weather, which are assumed independent of one another. In each month of the ?rst block, they assign a probability of 1/2 to having a natural 100-mm increase in water levels and probabilities of 1/4 to having a 50-mm decrease or a 250-mm increase in water levels. All these ?gures correspond to natural changes in water level without dam releases. In each month of the second block, they assign a probability of 1/2 to having a natural 150-mm increase in water levels and probabilities of 1/4 to having a 50-mm increase or a 350-mmincrease in water levels. If a ?ood occurs, then damageis assessed at $10,000per mm above?ood level. A water level too low leads to costly importation of water. These costs are $5000 per mm less than 250 mm below ?ood stage. The commission ?rst considers an overall goal of minimizing expected costs. They also consider minimizing the probabilityof violating the maximum and minimum water levels. (This makes the problem a special form of chanceconstrained model.)Consider both objectives
- Question : 4E - The Energy Ministry of a medium-size country is trying to decide on expenditures for new resources that can be used to meet energy demand in the next decade. Thereare currentlytwo major resources to meet energydemand.These resources are, however, exhaustible. Resource 1 has a cost of 5 per unit of demand met and a total current availability equal to 25 cumulative units of demand. Resource 2 has a cost of 10 per unit of demand met and a total current availabilityof10demandunits.Anadditionalresourcefromoutsidethecountry is always available at a cost of 16.7 per unit of demand met. Someinvestmentis consideredin each ofResources 1 and 2 to discovernew supplies andbuildcapital. Resource1 is, however,elusive.A unitofinvestment in new sources of Resource 1 yields only 0.1 demand unit of Resource 1 with probability 0.5 and yields 1 demand unit with probability 0.5 . For Resource 2, investmentis well known.Eachunit ofinvestmentyields a demandunitequivalent of Resource 2. Cumulative demand in the current decade is projected to be 10, while demandin the next decade will be 25. The ministry wants to minimize expected costs of meeting demands in the current and following decade assuming that the results of Resource 1 investment will only be known when the current decade ends. Next-decade costs are discounted to 60% of their future real values (which should not change).
- Question : 5E - Paci?c Pulp and Paper is deciding how to manage their main forest. They have trees at a variety of ages, which we will break into Classes 1 to 4. Currently, they have8000acres in Class 1, 10,000 acres in Class 2, 20,000 in Class 3, and 60,000 in Class 4. Each class corresponds to about 25 years of growth. Thecompanywouldliketodeterminehowtoharvestineachofthenextfour25year periods to maximize expected revenue from the forest. They also foresee the company
- Question : 6E - A hospital emergency room is trying to plan holiday weekend staf?ng for a Saturday, Sunday, and Monday. Regular-time nurses can work any two days of the weekend at a rate of $300 per day. In general, a nurse can handle 10 patients during a shift. The demand is not known, however. If more patients arrive than the capacity of the regular-time nurses, they must work overtime at an average cost of $50 per patient overload. The Saturday demand also gives a good indicator of Sunday
- Question : 7E - After winning the pole at Monza, you are trying to determine the quickest way to get through the ?rst right-hand turn, which begins 200 meters from the start and is 30 meters wide. You are through the turn at 100 meters past the beginningofthenextstretch(seeFigure15).Asinthe?gure,youwillattempttostay 10 meters inside the barrier on the starting stretch (maintaining this distance from each barrier as accelerate as fast as possible until point d1 . At this distance, you will start braking as hard as possible and take the turn at the current velocity reached at some point d2 . (Assume a circular turn with radius equal to the square of velocity divided by maximum lateral acceleration.) Obviously, you do not want to go off the course. The problemis that you can never be exactly sure of the car and track speed untilyoustartbrakingatpoint d1 .Atthatpoint,youcantellwhetherthetrackis fast,medium,orslow,andyoucanthendeterminethepoint d2 whereyouenter the turn. You suppose that the three kinds of track/car combinationsare equally likely. If fast, you accelerate at 27 m/sec 2 , decelerate at 45 m/sec2 , and have a maximum lateral acceleration of 1.8 g (= 17.5 m/sec 2 ). For medium, these values are 24, 42, and 16; for slow, the values are 20, 35, and 14. You want to minimize the expected time through this section. You also assume that Fig. 15 Opening straight and turn for Problem 7. ifyoufollowanoptimalstrategy,othercompetitorswillnotthrowyououtofthe race (although you may not be sure of that). After ?nding the optimal strategy for any feasible position on the second straight-away, ?nd an optimal strategy with a constraint to remain no more than 10 meters from the inside wall after completing the turn and comparethe results.
- Question : 8E - In training for the Olympic decathlon, you are trying to choose your takeoff pointforthelongjumptomaximizeyourexpectedof?cialjump.Unfortunately, when you aim at a certain spot, you have a 50/50 chance of actually taking off 10 cm beyond that point. If that violates the of?cial takeoff line, you foul and lose that jump opportunity. Assume that you have three chances and that your longest jump counts as your of?cial ?nish. Youthenwanttodetermineyouraimingstrategyforeachjump.Assumethat your actual takeoff is independentfrom jump to jump. Initially you are equally likely to hit a 7.4- or 7.6-meter jump from your actual takeoff point. If you hit a long ?rst jump, then you have a 2/3 chance of another 7.6-meter jump and 1/3 chance of jumping 7.4 meters. The probabilities are reversed if you jumped 7.4 meters the ?rst time. You always seem to hit the third jump the same as the second. First,?ndastrategytomaximizetheexpectedof?cialjump.Then,maximize decathlon points from the followingTable 8. Table 8 Decathlon Points for Problem 8. Distance Points Distance Points 7.30 886 7.46 925 7.31 888 7.47 927 7.32 891 7.48 930 7.33 893 7.49 932 7.34 896 7.50 935 7.35 898 7.51 937 7.36 900 7.52 940 7.37 903 7.53 942 7.38 905 7.54 945 7.39 908 7.55 947 7.40 910 7.56 950 7.41 913 7.57 952 7.42 915 7.58 955 7.43 918 7.59 957 7.44 920 7.60 960 7.45 922 7.61 962

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