Expected value of perfect information

11. The Hillblom Estate: Answer SheetNAME: _______________________________________ SECTION:___________a) (6 points) Junior Larry’s expected value for continuing the case: _______________________
Should Junior Larry accept this settlement offer?Attach your decision tree, labeled as Exhibit 1a.
_______________________
b) (2 points) Expected value of perfect information on finding DNA sample: __________________Attach the work you did to support this answer, labeled as Exhibit 1bc) (2 points) Risk tolerance R that makes Junior Larryindifferent between continuing the case and settling: ___________________________Attach the work you did to support this answer, labeled as Exhibit 1c.d) (2 points) Lawyer’s expected value for continuing the case: ___________________________
Should the lawyers prefer to settle?Attach your work, labeled as Exhibit 1d.
___________________________
e) (4 points) Expected value of continuing the case withpossibility of challenging the Hillblom law: ___________________________Attach your work, labeled as Exhibit 1e.f) (4 points) What is Junior Larry’s lawyers’ expected value
for this case with this additional information?Attach your work, labeled as Exhibit 1f.
___________________________
Please Initial:I pledge that I have neither received nor given unauthorized assistance on this exam. _________21. The Hillblom Estate (20 Points)When Larry Hillblom died in a plane crash in 1995, he left behind a lot of money and a legal mess.Hillblom was the “H” in DHL – he was one of the founders of the company, and, after buying out his twopartners (“D” and “L”), he oversaw DHL’s growth into the world’s largest courier delivery company.Hillblom was a millionaire before he turned 30 and died at the age of 52, leaving an estate worth anestimated $750 million.To avoid U.S. income taxes, Hillblom moved from the San Francisco Bay Area to the island of Saipan.Saipan is a tropical tax haven located in the middle of the Pacific Ocean (about 1500 miles east of thePhilippines) and part of the Commonwealth of Northern Mariana Islands (CNMI). Hillblom was a veryinfluential and powerful person in Micronesia and started dozens of businesses and financed projects inthe Philippines, Hawaii, and Vietnam. In addition to his mansion in Saipan, he maintained residences inManila, Hawaii, and Half Moon Bay in California. His hobbies included high‐end stereo equipment,boats, vintage airplanes, fancy cars, and, allegedly, illicit relationships with young Asian women.Hillblom and two business associates were killed on May 21, 1995, when Hillblom’s twin‐engineseaplane crashed on a flight returning to Saipan after a business trip to the nearby Pagan Island. Thenext morning a search party located parts of the plane and the bodies of Hillblom’s companions, butHillblom’s body was never found.Hillblom never married and had no legitimate children. His will was executed in 1982 and left the bulk ofhis estate to a trust with the primary beneficiary being the medical schools of the University of Californiasystem. Importantly, Hillblom’s will failed to make any provision, either pro or con, for any children hemay later have.Shortly after Hillblom’s death, an alleged illegitimate child, Junior Larry Hillbroom, came forward andfiled a lawsuit claiming a share of the Hillblom estate. (Yes, it was Junior Larry Hillbroom; Junior Larry’smother misspelled Hillblom on the birth certificate. And, yes, his name is Junior Larry). Junior Larry wasborn in 1984 and was 12 years old when the lawsuit was filed. Hillblom was a resident of CNMI and,under Section 2702 of the CNMI probate code, a child born after a will was executed is entitled toreceive a share of the estate unless the will explicitly disallows such claims.Junior Larry’s claim to a share of the Hillblom estate faces several impediments. First, Junior Larry has toawait the outcome of a vote on a proposed law (known as the Hillblom Law) before the CNMIlegislature. The proposed law would require children born after a will was executed to have legallyestablished paternity before the death of the father in order to have a valid claim under Section 2702 ofthe probate code. If the Hillblom law were passed by the legislature and signed by the governor, the lawwould invalidate Junior Larry’s claim to a share of the Hillblom estate and Junior Larry would receivenothing. Junior Larry’s lawyers estimate a 0.60 probability of the law being approved.If the Hillblom Law is not approved, Junior Larry would still have to present evidence that he is, in fact,the son of Larry Hillblom. Such claims of paternity are routinely proved or disproved by matching DNA.However, Hillblom disappeared almost without trace – his body was never found and his mansion inSaipan had been antiseptically cleaned shortly after his death and no Hillblom DNA was found. Thoughthey had not yet found a useful DNA sample, Junior Larry’s lawyers assigned a 0.80 probability that someuseful DNA would eventually be found.3Even if a useful sample of Larry Hillblom’s DNA were found, there is still the possibility that the DNAevidence would show that Junior Larry was not actually Larry Hillblom’s son. Junior Larry’s lawyers thinkthere is a 0.70 probability that Junior Larry is in fact Hillblom’s son. The DNA sample (if found) woulddefinitively prove that Junior Larry was Hillblom’s son or prove that Junior Larry was not Hillblom’s son.Finally, even if paternity can be established and the Hillblom law is not approved, there is uncertaintyabout how large a share of the Hillblom estate Junior Larry would obtain in court. The share is uncertain,in part, because there is uncertainty about how many other illegitimate children might be eligible toreceive a share of the estate. Before distributing anything to heirs or other beneficiaries, the estate mustfirst pay a 55% estate tax. With a $750 million estate, this leaves $337.5 million to distribute. JuniorLarry’s lawyers estimate a 1‐in‐25 (or 0.04) probability that Junior Larry would receive 100% of the after‐estate‐tax value of the estate (the full $337.5 million). They considered three intermediate possibilitiesas well: they estimate probabilities of 0.20, 0.40, and 0.16 that Junior Larry would receive 5%, 10%, or20% (respectively) of the after‐tax value of the estate. Finally, Junior Larry’s lawyers estimate a 0.20probability Junior Larry would receive nothing, even if they establish paternity and the Hillblom law isnot approved.While vehemently denying that Junior Larry was Hillblom’s son, in 1996 (before the vote on the Hillblomlaw in the CNMI legislature), the trustees of the Hillblom estate offered Junior Larry a settlement of $8million. Junior Larry and his lawyers had to decide whether to continue the case or accept thissettlement offer.Junior Larry’s lawyers will take 45% of any settlement or awards paid to Junior Larry in exchange forcovering all of the legal expenses associated with the case. We will first analyze this problem from JuniorLarry’s perspective.a) (6 points) Build a decision tree and calculate the expected value of continuing the case. If JuniorLarry is risk‐neutral (i.e., bases his decisions on expected values), should he accept thissettlement offer?b) (2 points) Suppose that, before deciding about the settlement offer, Junior Larry’s team couldobtain perfect information about whether or not they will find a useful sample of LarryHillblom’s DNA. How much would this information be worth?c) (2 points) The amounts involved in this case are huge sums for Junior Larry, a twelve‐year‐oldboy living in poverty on a small island in the Pacific Ocean. Suppose Junior Larry has anexponential utility function with risk tolerance parameter R. What value of R would make JuniorLarry indifferent between continuing the case and settling?4Though Junior Larry’s lawyers – a team including some high‐profile attorneys in Los Angeles as well as anattorney in Saipan – have a duty to act in the best interest of their client, they have somewhat differentinterests in the case. Specifically, the lawyers are liable for the costs of continuing the lawsuit and mayhave a different attitude towards risk. If they accept the settlement, they will incur no further expenses.The legal costs associated with continuing the case depend on how far the case goes. The lawyersestimate they will spend $300,000 before the Hillblom Law is voted on. If the Hillblom law does not pass,they will continue the case and incur an additional $400,000 in costs when trying to find a useful sampleof Hillblom’s DNA. Given a sample of Hillblom’s DNA, testing for paternity is not expensive (for thisanalysis, you can assume it is costless). However, if they succeed in establishing paternity, the lawyerswill face additional costs of $600,000 to prepare the case for court.d) (2 points) What is the lawyers’ expected value for continuing the case? If the lawyers were risk‐neutral, should they prefer to take the settlement offer?e) (4 points) Junior Larry’s lawyers believe that if the Hillblom Law passes, the law might besuccessfully challenged on constitutional grounds. A challenge of the law would be expensive(estimated cost of $1 million), but Junior Larry’s lawyers thought they had a reasonable chance –a probability of 0.40 – of succeeding in having the law thrown out, thereby allowing them tocontinue the case. How does the possibility of such a challenge affect the lawyers’ expectedvalue for continuing the case?f) (4 points) For this part of the question, you should ignore the possibility of challenging theHillblom law as described in part (e).Junior Larry’s lawyers were approached by Larry Hillblom’s mother, who offered to sell them asample of her own DNA for $1 million. She would receive the $1 million only if Junior Larry issuccessful in obtaining some money from the Hillblom estate, either in court or a settlement. Inthe case where there is a settlement or an award, the $1 million payment would be deductedfrom the award or settlement directly. Junior Larry and his lawyers would then divide theremainder as before: 45% to the lawyers, 55% to Junior Larry. (This payment would be made byJunior Larry and the lawyers and hence would be after the estate tax.)Though a DNA sample from Larry Hillblom’s mother could be useful, such an indirect DNA testhas two problems: First, the indirect test is somewhat less reliable as a test of paternity. Second,the indirect test would not be accepted by the court as evidence of paternity: Junior Larry’steam would still have to find DNA from Larry Hillblom himself to establish paternity.Junior Larry’s team believes that, if Larry Hillblom is, in fact, Junior Larry’s biological father, thereis a 0.85 probability that the indirect test based on Hillblom’s mother’s DNA would yield a“strong match” and a 0.15 probability that it would not show a “strong match.” If Larry Hillblomis in fact not Junior Larry’s biological father, the indirect DNA test would certainly (i.e., withprobability 1.0) not show a “strong match.”The results of this DNA test would be revealed before Junior Larry’s team has to decide on thesettlement offer. Larry Hillblom’s mother agreed to keep this DNA test secret, so the test and itsresults would not affect the settlement offer proposed by the estate.What is Junior Larry’s lawyers’ expected value for this case with this additional information?5EpilogueJunior Larry turned down the initial settlement offer.Regarding DNA, a sample of Hillblom’s DNA was located: the University of California at San Francisco (UCSF)medical center (one of the primary beneficiaries of the Hillblom estate!) had preserved a mole removed from LarryHillblom’s face during reconstructive surgery following an earlier plane crash that Hillblom survived. Under a courtorder, UCSF delivered a mole to the plaintiffs for DNA testing. UCSF announced shortly after that that there was amistake (the delivered mole did not actually belong to Larry Hillblom) and then delivered a different mole. Thechain of evidence had been broken and the plaintiffs no longer trusted the mole to be a reliable source of LarryHillblom’s DNA.In June 1996, the Hillblom Law was in fact passed by the CNMI legislature and signed by the governor. In 1997,before the law was challenged, the Hillblom estate reached a settlement with Junior Larry and three other of LarryHillblom’s (alleged) illegitimate children. A total of 8 children came forward to claim a share of the estate, eachaccompanied by their own team of lawyers. The lawyers used a novel approach to establish paternity – theycompared the DNA of the children with each other’s DNA and concluded that four of them (including Junior Larry)had the same father. The DNA matches were confirmed with DNA from Larry Hillblom’s mother, for which hergrandchildren paid her $1 million and deeded her title to a villa in France. DNA experts have called this “themother of all paternity cases.”The amounts of the settlements were not publicly revealed. Press reports varied widely, saying that the childrenreceived $10 to $90 million each. Later court filings suggest Junior Larry received $50 million, with his lawyerstaking a large portion of that amount in fees.Junior Larry is a now a United States citizen and splits his time between Idaho and the Pacific islands. Junior Larryhas had many legal problems of his own. In 2005, 21‐year‐old Junior Larry Hillbroom was arrested on drugpossession and bribery charges in Guam. (The magistrate’s complaint states Hillbroom reportedly told police “Ihave enough money to own you.”) In June 2012, he was charged with felony domestic battery in Idaho. Currently(April 2018), Junior Larry is being held on the island of Palau and facing drug trafficking charges.The story of Larry Hillblom is told in the 2012 book “King Larry: The Life and Ruins of a Billionaire Genius,” by JamesD. Scurlock and published by Scribner. The mysterious circumstances of his death are documented in the 2009movie “Shadow Billionaire.” This problem is based on the facts of the case as recounted in Steve Lippman andKevin McCardle’s article “Sex, Lies and the Hillblom Estate.” Of course, the estimates of probabilities and costs areour own, not those of Junior Larry or his lawyers or anybody involved in the case.62. Dupont University Technology Fund: Answer Sheet
NAME: _______________________________________
SECTION:___________
Remember to attach and label the appropriate support for each of your answers. If you are includingoutput from Crystal Ball to support your answer, please circle the relevant numbers on the CB output.
a) (4 points)
i) Expected total payoff:
_____________
ii) 95% confidence interval for expected total payoff: _____________
iii) P(Total Payoff > Amount Invested):
_____________
b) (2 points) i) Expected number of successes: _____________ii) Probability of 10 or more successes: _____________c) (4 points) i) Expected payment to DU: _____________ii) Probability limited partners get any money: _____________d) (4 points) i) Expected value of Simmons’s share: _____________ii) Simmons’s certainty equivalent: _____________e) (4 points) New fee payment for Wolfe Ventures: _____________f) (2 points) i) Effect on expected payment to DU?____________________________________________________________ii) Effect on expected payment to limited partners?____________________________________________________________Explanation:____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Please Initial:I pledge that I have neither received nor given unauthorized assistance on this exam. _________72. Dupont University Technology Fund (20 Points)Many universities have formed venture capital funds to help turn campus research discoveries intomarketable products. Here we consider the fund of a hypothetical university, Dupont University, whichis managed by a New York‐based venture‐capital firm, Wolfe Ventures. Wolfe Ventures serves asgeneral partner for the fund and recruited additional investors who are limited partners in the fund. Wewill look at evaluating the risks and returns associated with this fund, after a few years of operation,when the fund’s assets have been fully invested in start‐up companies.Suppose the Dupont University Technology Fund (DUTF) has investments in 14 companies, as describedin Table 1 below. (Note: This data is available for download on Canvas.)Table 1: All amounts are $ millions
AmountInvestedProbabilityof aLiquidityEventCompany
Value ofLiquidity Event:Lognormal withMean Std. Dev.
A 19.7 0.90B 8.2 0.90C 8.6 0.80D 1.5 0.25E 2.9 0.65F 19.4 0.90G 1.6 0.35H 0.2 0.15I 7.3 0.65J 4.0 0.50K 0.9 0.50L 5.9 0.80M 0.4 0.10N 0.1 0.20
30.00 20.0010.00 2.5015.00 7.508.50 6.004.40 1.2620.00 10.005.70 2.302.20 1.1012.50 12.507.50 2.502.00 2.2513.40 7.505.00 3.751.00 2.25
Total 80.7The second column in Table 1 shows how much DUTF has invested in each of these companies; theseamounts have been committed to the companies and are now sunk costs.The third column shows the probability that each DUTF company has a “liquidity event” where DUTF’sinterest in the company is bought out by some other investors. If there is no liquidity event for acompany, the payoff to DUTF is assumed to be zero and the entire amount invested by DUTF is lost.The payoff to DUTF associated with a liquidity event is quite uncertain. Some companies may show greatpromise and generate a large payoff while others may have a small payoff, perhaps less than DUTF’sinitial investment. The payoff from a liquidity event is modeled as a lognormal distribution where themeans and standard deviations of the lognormal distribution are specific to the company. These meansand standard deviations are listed in the fourth and fifth columns of Table 1. (The “locations” for thelognormal distributions are all equal to zero.) To simplify the model, assume that the occurrence ofliquidity events for the different companies are independent and the (potential) payoffs associated witha liquidity event for the different companies are independent.8Please run your model for at least 10,000 trials to ensure a reasonable level of accuracy in your results.Build a simulation model to answer the following questions:a) (4 points)i) What is the expected total payoff of the DUTF portfolio?ii) Provide a 95% confidence interval for the expected total payoff of the DUTF portfolio:iii) What is the probability that the total payoff is more than the total invested?b) (2 points) We call a DUTF company a “success” if it has a liquidity event and the payoff exceedsthe amount invested by DUTF in the company.i) What is the expected number of DUTF companies that will be successes by this standard?ii) What is the probability that 10 or more companies will be successful by this standard?Payoffs from the DUTF companies are distributed to the fund managers (Wolfe Ventures), to DU, and tothe fund’s limited partners. Wolfe Ventures will be paid a fee equal to 10% of all payoffs from the DUTFcompanies. The remaining proceeds are distributed to DU and the fund’s limited partners (the outsideinvestors). In recognition of DU’s contribution of much of the intellectual property underlying the DUTFcompanies (as well as their financial investment), DU’s claims to DUTF proceeds have been granted ahigher priority than the limited partners’ claims. Specifically, DU will receive the first $75 milliongenerated by the fund, after Wolfe Ventures’ fees. The limited partners will not receive any paymentuntil after DU has received its $75 million. However, DU’s share of the proceeds is capped at $75 million;if the fund generates more than that amount, DU will not further benefit and the limited partners willdivide the proceeds.For example, if the total payoff of DUTF is $100 million, Wolfe Ventures would receive $10 million, DUwould receive their full $75 million, and the limited partners would receive the remaining $15 million. Ifthe total payoff is $80 million, Wolfe Ventures would receive $8 million, DU would receive $72 millionand the limited partners would receive nothing.c) (4 points)i) What is the expected payment to DU from DUTF?ii) What is the probability that the limited partners receive any money at all?d) (4 points) One of the limited partners is a wealthy DU alumnus named Charlotte Simmons; sheholds fifty percent (50%) of the limited partners’ share. Simmons is risk averse and has anexponential utility function with a risk tolerance of $100 million.i) What is the expected payoff of Simmons’s investment in DUTF?ii) What is Simmons’s certainty equivalent for her investment in the fund?9e) (4 points) To better align incentives, the DUTF managers and investors are contemplatingchanging Wolfe Ventures’ management fee structure from the current 10% fee on grossproceeds to a new fee that depends on the net proceeds for the fund, i.e., what they earnbeyond the $80.7 million that was invested in the fund. The question is how to determine a newpercentage that is equitable to Wolfe Ventures in that it gives them the same expected value asthe original deal.For example, with this new fee structure, if the new fee is set at f% and the proceeds from thefund total $100 million, Wolfe Ventures would receive f% × ($100$80.7). If the total proceedsfall short of $80.7 million, Wolfe Ventures would receive nothing.What fee f would be equitable to Wolfe Ventures in that it would lead them to have the sameexpected value as the original deal? (You can approximate f within 1%, if necessary.)f) (2 points) For this part of the question, assume that Wolfe Ventures’ original fee structure is inplace, not the one contemplated in part e.For simplicity, we have assumed that the occurrence of liquidity events for the differentcompanies are independent and the (potential) payoffs associated with a liquidity event for thedifferent companies are independent. Because the occurrence of a liquidity event is drivenprimarily by technological factors (e.g., does this new drug work?), it may be reasonable toassume that the liquidity events are independent. However, the assumption that the payoffsgiven a liquidity event are independent is not a good assumption: the payoffs for the portfoliocompanies may be affected by general market conditions and hence would likely be positivelycorrelated.Suppose the payoffs given a liquidity event for individual companies retain the samedistributions (and same means and standard deviations) as given in Table 1, but rather thanbeing independent, they are all positively correlated with each other. How would this positivecorrelation affect:i) The expected payments to DU?ii) The expected payments to the limited partners?We are looking for a qualitative description (e.g., would the expected payments increase,decrease, or stay the same?) and a brief explanation. You do not need to build a new model toprovide a precise answer for this question.103. Southstrom’s Staffing Problem: Answer SheetNAME: _______________________________________ SECTION:___________
a) (6 points) What is the total cost of the optimal staffing plan?Attach a copy of your model showing the optimal plan. Label this Exhibit 3a.
_____________
b) (2 points)How many experienced sales representatives should Southstrom plan to hire? ___________How does this possibility affect the cost of the optimal plan?___________________Attach appropriate documentation, labeled Exhibit 3b.c) (2 points)How would a higher cost of hiring in November affect the optimal staffing plan?___________________________________________________________________________What is the cost of optimal plan? __________________________Attach appropriate documentation, labeled Exhibit 3c.d) (2 points) What is the cost of hiring a new employee in December? ____________________Attach appropriate documentation, labeled Exhibit 3d.e) (4 points)In addition to hiring staff, Southstrom can also lay off staff or ask staff to work overtime.What is total cost of the optimal staffing plan? __________________Attach a copy of your spreadsheet model showing your staffing plan, labeled Exhibit 3e.f) (4 points)With uncertainty about needs in December, what is the expected cost of Southstrom’s optimalstaffing plan?__________________Attach a copy of your spreadsheet model showing your staffing plan, labeled Exhibit 3f.Please Initial:I pledge that I have neither received nor given unauthorized assistance on this exam. ________113. Southstrom’s Staffing Problem (20 points)Southstrom, a new department store in the Southeastern United States, needs to hire and train salesrepresentatives to meet customer demand and, in particular, to prepare for the winter holiday shoppingseason. Suppose Southstrom has 131 trained staff members at the end of June/beginning of July. Theyneed to develop a schedule to hire and train additional staff members to meet the following minimumrequirements for the coming months.Table 1: Staffing requirements (person months)(Note: This data is available for download on Canvas.)New hires are trained by having them work with trained staff members for one month. After that time,the new hires are considered trained. Training the new hires requires effort on the part of the trainedstaff: each new hire requires 20% of the attention of a trained staff member. For example, if Southstromwere to train 10 new hires in a given month, this would reduce the number of trained staff membersavailable to work in sales in that month by 2. Thus, as we develop a plan for hiring and staffing, we needto take into account the time required to train new hires. An example staffing plan is shown in Table 2below. (Note that the plan in Table 2 does not meet the requirements above and is shown only forillustrative purposes.)Table 2: An example staffing plan (all units are person months)(Note: This model is available for download on Canvas.)Southstrom’s goal is to develop a staffing plan that minimizes total costs, while meeting the minimumsales staff requirements given in Table 1. New hires are paid $2,000 in their first month (when beingtrained) and trained staff members are paid $3,500 per month, regardless of whether they are workingin sales or training new hires. For now, you should assume that all new hires successfully completetraining and that staff members cannot be fired, do not quit, and do not work overtime. All new hiresstart at the beginning of the month.MonthMinimum Number ofTrained Staff MembersWorking in SalesJuly 120August 135September 150October 150November 170December 200MonthNew Hires inTraining
Total Trained StaffMembers Working onMembers Working
Members
Training
on Sales
Trained Staff
Trained Staff
July 10 131 2 129August 10 141 2 139September 10 151 2 149October 10 161 2 159November 20 171 4 167December 0 191 0 191Trained Staff Members123. Southstrom’s (continued)To simplify the problem, you should assume that you may have non‐integer numbers of new hires orstaff members. *Please do NOT use any binary or integer constraints.*For all questions, write your answers on the answer sheet for this problem and attach the appropriatedocumentation to the answer sheet.a) (6 points) Formulate and solve an optimization model to compute an optimal staffing plan. What isthe total cost of this optimal plan? What is the plan?Parts (b)‐(d) represent independent variations on the basic problem of part (a): The changescontemplated in one part do not apply in later parts of the problem. Note that parts (b)‐(d) shouldbe answered using the Sensitivity report for part (a). In your documentation, describe how youranswer is related to the results given in the sensitivity report.b) (2 points) Southstrom is considering the possibility of hiring some experienced sales representativesto work in December. These sales representatives would work only in December, require notraining, and would command a premium monthly salary of $4,500. How many of these experiencedsales representatives should Southstrom plan to hire, if any? How does this possibility affect thetotal cost of the optimal plan?c) (2 points) It is sometimes difficult to hire new staff members in November because many otherretail stores are hiring at the same time. Suppose Southstrom had to pay new hires in November thefull monthly salary of $3,500 rather than the lower monthly salary of $2,000 that is paid to new hiresin other months. How would this higher cost affect Southstrom’s optimal staffing plan? How does itaffect the cost of this optimal plan?d) (2 points) A new hire in December not only costs Southstrom $2,000 in salary without meeting itsneeds for trained staff; the new hire also requires trained staff members to devote time to training.Generally, Southstrom would not want to consider any new hires in December. However,sometimes Southstrom may have to consider new hires in December for other reasons (e.g., thenew hire is the boss’s son who will be finishing college in December). What is the cost of a new hirein December, taking into account the need to train this new hire as well as the direct salary ($2000)paid to this new hire?133. Southstrom’s (continued)e) (4 points) Now consider the base case staffing problem of part (a) without any of the possiblechanges considered in parts (b), (c) and (d). Suppose that in addition to hiring new staffmembers, Southstrom can also lay off (i.e., fire) some staff members in any month and can asktrained staff members to work overtime.Laid‐off staff members would be terminated at the beginning of the month (thus they would notbe available to work that month) and would be paid $1,000 as a one‐time severance payment.Southstrom can also ask trained staff members to work overtime in any month. Staff memberswould be paid 150% of their regular salary (for a monthly rate of 1.5 x $3,500 = $5,250 permonth) for overtime work. The staff can work a maximum of 30% overtime in any given month.Reformulate the optimization problem to take into account the option to lay off staff membersand/or work overtime. What is the new optimal staffing plan with these options? What is thecost with this optimal plan?f) (4 points) Finally, Southstrom would like to modify their plan from part (e) to accommodateuncertainty in their future needs. Though they are certain about their needs for the months ofJuly through November, they are uncertain about their needs for December due to uncertaintyabout holiday sales. As in the model of part (e), Southstrom can hire and lay off in any monthand can have trained staff work overtime (up to a maximum of 30%).The minimum requirement of 200 trained staff members working in sales in December (given inTable 1) represents a “normal” anticipated need; there is a 50% chance that this will be theirstaffing need. Though this is the most likely scenario, Southstrom thinks there is also a 30%chance of a “busy” December need of 250 staff members working in sales, and a 20% chance ofa “reduced” December need of 180 staff members working in sales. Southstrom will not knowtheir actual need until the beginning of December.Southstrom wants to develop a staffing plan that minimizes expected costs, given this uncertainneed. What is Southstrom’s optimal staffing plan, given these uncertainties? What is theirexpected cost with this optimal plan?

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