1. Mountain Ski Sports

[This problem is a modern variant of the Newsboy Problem, a classic exercise in operations management. It’s adapted from Jeffrey D. Camm & James R. Evans’s excellent text, Management Science: Modeling, Analysis and Interpretation (1996), which lamentably is out of print.]

Mountain Ski Sports, a chain of ski equipment shops in Colorado, purchases skis from a manufacturer each summer for the coming winter season. The most popular model costs $150 and sells for $260. Any skis left over at the end of the winter are sold at the store’s spring sale for half-price ($130). Sales over the years have been quite stable. Gathering data from all its stores, Mountain Ski developed the following probability distribution for demand:

Demand:           150       175      200     225      250
Probability:     0.10     0.30    0.40    0.15     0.05

The manufacturer will take orders only for multiples of 20 (that is, in lots of 20 pairs of skis per lot), so Mountain Ski is considering the number of lots to order. Company policy calls for ordering within the historical demand range, that is, not less than 150 pairs and not more than 250 pairs for this particular model.

A. Determine the optimal decision for Mountain Ski Sports under the following criteria:
    1. Optimist criterion
    2. Pessimist criterion
    3. Hurwicz criterion with α = 0.25
    4. Hurwicz criterion with α = 0.50
    5. Hurwicz criterion with α = 0.75
    6. Minimax Regret criterion
    7. Modal criterion
    8. Laplace criterion (ignore for this exercise the given probability distribution)
    9. Expected Monetary Value (EMV) criterion
  10. Expected Opportunity Loss (EOL) criterion

B. Determine the expected demand. Explain why the expected demand is not equal to the optimal order quantity. (Expected demand is the expected value of the demand schedule.)

C. Determine the expected value of perfect information (EVPI) from the EMV calculations. Verify its correctness with the optimal EOL.

2. Wile E. Coyote & Friend

Wile E. Coyote, a hunter-gatherer of some repute, wants to set up a system of traps to capture his favorite desert wildlife, the celebrated Roadrunner. Wile E. is considering two alternative systems: the ACME SuperTrap and Beep Beep Associates’ DumboTrap. Both systems are leased on a one-year basis. ACME’s terms call for a $600 leasing fee plus a required yearly maintenance contract. The cost of maintenance will be $200 if the SuperTrap is used not more than 100 times during the year. Otherwise, the maintenance fee will be $400. BBA’s terms call for a $500 leasing fee plus a flat yearly maintenance contract of $400, regardless of frequency of use. Wile E. believes the probability of using the trap more than 100 times in a year is 0.4. He would also like to optimize his operating costs for hunting and gathering activities.

A. Determine the optimal decision for Wile E. under the following criteria:
     1. Optimist criterion 
     2. Pessimist criterion
     3. Hurwicz criterion with α = 0.4
     4. Minimax Regret criterion
     5. Laplace criterion (ignore the given distribution)
     6. Expected Monetary Cost (EMC) criterion
     7. Expected Opportunity Loss (EOL) criterion

B. Show that the optimist criterion is the same as the Hurwicz criterion with α = 1.

C. Show that the pessimist criterion is the same as the Hurwicz criterion with α = 0.

D. Show why the EVPI for Wile E. must equal his minimum EOL.

3. Julia's First Car

Julia wants to buy her first car. A friend of hers is selling his old but impeccably restored Triumph Spitfire two-seater convertible for $9,000. However, Julia can expect to incur in a $250 monthly expense to maintain the Spitfire in top condition. In addition, if the Spitfire were to develop mechanical problems, Julia would face a $5,000 repair job. Julia plans to keep the car for two years, and has no idea of the probability that the car will develop mechanical problems during any given year. But if it does malfunction during the first year, she believes that the likelihood of a resurgence of mechanical problems in the second year would be 70%. It is highly unlikely that major mechanical problems will recur in the same year, so Julia has dismissed that possibility as unrealistic.

Julia’s other option is to buy a brand-new Mazda Miata for $20,000. The Miata is fully guaranteed for mechanical problems during the first two years, and maintenance costs are expected to be $50 per month. Operating costs for the two cars are expected to be about the same. At the end of the two-year period, Julia plans to sell the car. She believes the Spitfire will hold its value, but the Miata would fetch only $12,000 in the used car market.

A. Assume that Julia would be equally pleased with either car and that she is not concerned about the time value of money in this particular problem. Which car should Julia buy?

B. Now assume that Julia has a preference for the Spitfire. Determine what the equivalent monetary value per year of that preference must be for her to choose the Spitfire over the Miata.

4. Oxxymoron — Prior Analysis

Oxxymoron PetroCorp has secured rights to drill for oil in the notoriously treacherous Thai Phoon Sea. Preliminary studies reveal that the Thai Phoon field could be either dry (no oil would be commercially recoverable), wet (the yield would be commercially viable), or happy-days (the field would dwarf currently proven global reserves). The company estimates that the chances of these three possibilities are 0.45, 0.54, and 0.01, respectively. Construction and positioning of a drilling platform call for an up-front investment of $1 billion (a thousand million). Annual operating costs of the platform amount to $200 million per year, and are expected to remain constant on a net present value (NPV) basis for the next 20 years (the time horizon for the decision). If the field turns out to be wet, the total 20-year revenue is expected to be $50 billion in NPV. If it turns out happy-days, the expected revenue would be $500 billion. It takes one full year of operations to determine the actual state of the oil field. Should Oxxymoron go ahead with the Thai Phoon project?

5. Oxxymoron — Posterior Analysis

Oxxymoron’s research department, GeoLab, can conduct a geologic test that would indicate if the Thai Phoon field were in a dry, wet or happy-days state with reasonable confidence. The reliability of the test, known from historical records, is as follows:

The GeoLab cost estimate for conducting the test in the Thai Phoon Sea is $100 million. Determine Oxxymoron’s decision strategy for the project given the possibility of conducting the test. (Naturally, you should first determine the EVPI to check if the test is worth considering. It is suggested you keep four-digit precision on the decimal portion of all computations to avoid significant round-off errors.)

6. Diego's First Job

Diego has just completed a summer internship with the Bureau for Bureaucratic Affairs in Washington, DC, and has been offered regular employment after he graduates next year with a starting salary of $36,000. The employment offer is valid for 90 days. If Diego does not accept it by then, the offer will be withdrawn. But Diego has other options. His aunt, for one, has told him that he can always work as a salesperson in her insurance agency if he so desires. This position typically pays $24,000 in salary and commissions in the first year, but the earning potential increases considerably with time. Diego, however, would rather work with the prestigious firm of Goldberg & Cash, whose starting salary is $50,000 and dress in three-piece suits. Diego believes there is an 80% chance that G&C will hire him if he achieves an A average in his senior year. If G&C does not hire him, he could apply again to the BBA in DC, but Diego has no idea if the position will be available then. On the other hand, if he fails to achieve an A average in his senior year, Diego believes that the likelihood that G&C would hire him drops to a mere 10%. He assesses the probability of achieving an A average as 0.7. Assuming that Diego’s sole criterion is to maximize his first year’s earnings, determine his optimal employment-decision strategy. (Hint: remember Diego’s aunt.)

7. Chicas Náuticas

Nuestra Señora de Atocha
(shipwrecked in 1622 off Dry Tortugas, Florida Keys)

Nautical Chicks is a club of women college students who learn sailing and open-sea navigation in the Caribbean. One good day while anchored near Gran Tortuga they discovered the remains of what seemed to be a very old shipwreck. At first they thought it was a sunken World War II cargo steamer, which was of no major interest. However, one of the girls, a history major, suggested it might be none other than the legendary Spanish galleon, El Conde del Chorizo, which sank in these waters in the midst of the terrible hurricane Santa Rabieta in 1622 with $100 million in gold doubloons.

After several reconnaissance dives, the Chicks concluded that the probability that the wreck was that of El Conde  was only one in a hundred. A quick marine-radio call to the Caribbean Salvage Company revealed that the cost of retrieving the contents of a wreck such as they had described was about $1 million. The Chicks then contacted Shipwrecks @ Key West, who are known for their expertise in identifying ancient vessels throughout the seven seas. The manager of Shipwrecks told them that the likelihood of their correctly identifying El Conde  was 80% whereas that of misidentifying any other vessel as El Conde  was 10%. The cost of conducting the identification in the high seas was $300,000.

A. Determine Nautical Chicks’ decision strategy. (Four-digit precision recommended.)

B. There are 20 ladies in Nautical Chicks. To raise the money needed to conduct the identification and recover the contents of the wreck, if any, each Chick would have to contribute $65,000. Compare this number to the net expected value given imperfect information accruing to each Chick. Would you invest your money in this venture? If your answer conflicts with the model’s recommendation, explain the discrepancy.

Case: CONTROL Suspects Nothing *


   Larabee led the two agents to the inner office, knocked twice, and opened the door. A mature man, looking smart, looked up from his desk and said, "Ah! Max, 99."

   "Hello, Chief. You wanted to see us?"

   "Yes, Max, yes. Please, sit down. I have a very important assignment for you and 99."

   Sitting down in front of the desk, Max replied, "Wait, Chief, I request the cone of silence."

   "Oh, Max, not the cone of silence. We're in a secure office, for heaven's sake."

   "Can't take any chances, Chief."

   The Chief sighed, raised his brows, and nodded to Larabee, who lowered from the ceiling with a flick of a switch a pair of connected plexiglass spheres. An alluring and stylishly dressed 99 sat outside the cone on the desk.

   "Owrght hrswhtwknw," mumbled the Chief inside his sphere.


   "Chttr hzit thtgeyoz stlatn oyllwkk frmdhrhssns."

   "Ayknt hiryu chf."

   "The Chief's saying that KAOS stole a ton of yellowcake uranium from the Russians, Max," chirped 99 as the Chief motioned Larabee to raise the cone of silence.

   "Thank you, 99. Missed it by that much."

   "Alright, here's what we know," resumed the Chief. "Siegfried, the evil underboss of KAOS, plans to assign one of three agents to smuggle the yellowcake out of Russia and into their base of operations at Die Kleinereichkolonie in Argentina."

   "Wait a minute, Chief. I thought their base was Camp Gitchee Goommee Noonee Wa-Wa in New Jersey."

   "That's a cover, Max. All three agents will depart St. Petersburg at the same time, separately, but only one will be running the contraband. The other two are decoys."

   "So we intercept all three, right Chief?"

   "I'm afraid we can't do that, Max. You two are the only agents we have available, and you won't get a chance to take on all three. So if you happen to go after a decoy, KAOS will have no trouble getting the yellowcake to the Kolonie."

   "Then we'll have to devise a counter-strategy to outwit KAOS," interjected 99 eagerly.

   "99, please," objected Max, the duo's senior agent. "If you don't mind, I'd like to handle this myself."

   "Right, Max."

   Max turned to the Chief. "Then we'll have to devise a counter-strategy to outwit KAOS."

   The Chief, open-mouthed, stared at Max, who waited impassively for a response. Shaking his head, the Chief finally said, "Yes. Now, our analysts have done their research and here's what they've come up with." The Chief opened a manila folder containing documents. "Siegfried's agents are Shtarker, his longtime assistant," he said while presenting a photograph; ". . . Dalip, his formidable auxiliary," he went on, now showing another photo as Max dropped his jaw at the sight of the palpable formidableness; ". . . and a new guy, a Hungarian national from the Kingdom of Caronia named Yarmy, who's a dead ringer for you, Max."

   "Wow! You gotta be kidding me!" exclaimed Max, transfixed by the uncanny likeness of the image. 99, too, gazed at the picture in astonishment.

   "Siegfried," continued the Chief, "has three alternatives for transport at his disposal. Each agent will be assigned a different means of conveyance to Argentina: a chartered air express flight, commercial marine surface shipping, or U-boat consignment."

   "U-boat consignment?" reacted the three, in unison.

   "KAOS still operates their old Rub-a-Dub sub and Siegfried's adamant in skippering the thing again," explained the Chief.

   "I remember that mission well," said 99, reminiscing. "Same old Siegfried."

   "Reminds me of the midget subs at Pearl Harbor," muttered Max, the history buff.

   "Precisely," concurred the Chief, pointing at both agents. "Anyway, the mission suitability ratings given for the transports are 80%, 10%, and 70%, for the aircraft, merchantman, and U-boat, respectively. These ratings reflect the inherent difficulty of intercepting the transports with an undercover operation."

   "The merchant ship looks pretty bad," commented Max.

   "Yes, it does. In fact, it's the only transport susceptible to a second-round interception attempt."

   "Oh, really?" mulled 99.

   The Chief nodded, pursed lips, and pressed on: "Our analysts are also convinced that the skills and experience of each KAOS agent will prove more useful in certain transports than in others. They believe that Shtarker, a tough fellow and El Alamein veteran, can still handle an open sea voyage, but he is prone to airsickness and is a bit claustrophobic. So they've assessed his potential for mission performance to be (on a scale from 0 to 10 where 10 is tops) 1, 6, and 2, for the aircraft, merchantman, and U-boat assignments, respectively."

   "Those aren't particularly spectacular numbers, Chief," observed Max.

   "Well, Siegfried's assistant is not particularly spectacular," returned the Chief while inadvertently glancing at Larabee, who stoically blinked before uttering: "Good enough for government work."

   "Ahem, as for Dalip," the Chief carried on, "he is truly a class apart, but is simply too big for the U-boat. So he's rated at 10, 10, and zero."

   "Is Dalip the notorious Mr. Big?" asked Larabee.

   "Au contraire," answered 99, "his antithesis." Max chuckled at her apropos characterization.

   "Finally," said the Chief, "we have Yarmy. He served valiantly in the Marine Corps at Guadalcanal. No small feat."

   "Guadalcanal?" marveled Max.

   "The Marine Corps. Yarmy knows naval vessels like an old salt. If anyone can pull off a long-range submarine mission, it's him. So our analysts rated him a 9, 9, 9."

   "All nines?" inquired Max pointedly.

   "Dunno why. He just seems to attract nines." 99 stole a furtive glance at the Chief. "At any rate, the possible variation for the agents' mission performance ratings are 40%, 0%, and 10%, for Shtarker, Dalip, and Yarmy, respectively."

   "Zero percent?" questioned 99. "Isn't that overly optimistic?"

   "Like I say, 99, Dalip is a class apart: the top KAOS agent, perfect mission track record, WWE World Heavyweight Champion, two-time Mr. India, international movie star—"

   "Dang!" The Chief looked at Max. "Sorry about that, Chief."

   "Okay, any suggestions?"

   "Certainly, Chief. The solution is to blockade St. Petersburg with the Seventh Fleet."

   The Chief made a face. "Max, the Seventh Fleet is based in Japan."

   "Oh. Would you believe the U.S.S. Ward?"


   "How about two cops in a rowboat?"

   "Chief," interposed 99, "I've got it! According to my calculations—" But suddenly she held back, concealed her notes, and stared at Max.

   Max looked at his ever loyal partner of so many years. Tenderly. "Please go on, 99."

   Agent 99 showed that if the CONTROL analysts' assessments were correct, possible variation included, Siegfried should rationally assign the yellowcake operation to Dalip using the air express flight. But Siegfried would surely also know that CONTROL could figure that out as well and would therefore target the aircraft. Hence, Siegfried should actually assign the mission to Yarmy in the U-boat, which is in fact the preferred realistic scenario for CONTROL. But Siegfried could count on CONTROL to cleverly figure out this stratagem, too, and he would therefore switch back to Dalip. And so on and on and on. Consequently, in order to prevent Siegfried from endlessly cycling through the alternatives and eventually choosing at random, to CONTROL's detriment, CONTROL had to make Siegfried believe they resolutely intended to intercept the aircraft, thereby forcing his hand.

   "Excellent analysis, 99!" hailed the Chief. "We'll get our field operatives to spread the dis via chatter."

   Max joined in: "Of course, the old «Midway is short of water» trick. Good thinking, 99." She beamed at his warm congratulation.

   When they left the Chief's office, 99 spoke to her partner: "Max, thank you for your understanding."

   "Understanding? 99, I love you better than the whole world. Would you believe it? The whole world."

   "Yes, Max, I believe it."

A. Develop a model that shows why 99's conclusions are correct.

B. Show why the U-boat is in fact the preferred realistic scenario for CONTROL.

C. Explain the opportunity loss to Siegfried if the merchant ship option were to be eliminated from his set of decision alternatives.

D. Determine Siegfried's optimal strategy. Should the chatter disinformation be relevant to his optimal decision?

*  This case is dedicated to Barbara Feldon and in memory of Don Adams, Edward Platt, and Robert Karvelas.
     Case narrative based on the Get Smart characters created by Mel Brooks and Buck Henry.
     © 2010 Diego Azeta — All rights reserved

Jurassic Park logo © Universal Studios / Amblin Entertainment
Coyote and Roadrunner cartoons © Warner Bros. Entertainment
Silver Mountain Ski Resort (Idaho) image © Destination360
CONTROL and KAOS logos are the property of the Get Smart organization
Dalip Singh image © Warner Bros. Entertainment
Fair Use doctrine invoked for noncommercial educational purposes

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