Problem 12: BatchingYou order two products from the same supplier. The annual demand for Product 1 is 10,000 units andthe annual demand for Product 2 is 20,000 units. Note that demand for both products is constantthroughout the year. The holding cost is the same for both products, $1 per unit per year. However, youincur a fixed cost of $200 each time you order Product 1, and a fixed cost of $100 each time you orderProduct 2. These fixed order costs are independent of the size of the order.a.How many units of Product 1 and Product 2 should be ordered at a time in order to minimizetotal holding + order cost?Number of Units of Product 1 to Order _____________Number of Units of Product 2 to Order _____________Supporting Work:b. Suppose that the supplier insists that orders for Product 1 and Product 2 be coordinated so thatthey can be shipped at the same time (still incurring the fixed cost of $200 and $100,respectively). Given this requirement, how many units of Product 1 and Product 2 (to the nearestinteger) should be ordered at a time in order to minimize total holding + order cost? (Hint: theproducts must be ordered the same number of times per year)Number of Units of Product 1 to Order _____________Number of Units of Product 2 to Order _____________Supporting Work:Problem 3: QueuingA professor holds online office hours all day Saturday and Sunday the weekend prior to final exams. He isavailable from 8AM to 8PM both days to answer student questions. Students who arrive while theprofessor is busy with another student simply wait until their turn comes up. Students are processed in theorder they arrive, and all students tolerate whatever wait is necessary to get their questions answered. Theprofessor notes from past experience that students arrive randomly with questions â the average timebetween arrivals is 36 minutes and the coefficient of variation of interarrival times is 1. Similarly, the timerequired to answer student questions is randomly distributed with an average of 24 minutes and acoefficient of variation of 1.a. On average how long does a student have to wait to get in to see the professor?Average waiting time (minutes) ___________Supporting Work:b. Suppose the professor would prefer the average waiting time to be no more than 40 minutes. Byhow much would the average interarrival time have to grow in order to meet this standard?Increase in Average Interarrival Time (minutes) __________Supporting Work:c. Assume again an average interarrival time of 36 minutes and suppose the professor is consideringreducing student waiting time by answering questions faster. How much faster would the professorhave to answer questions in order to reduce the average waiting time to 40 minutes?Decrease in the Average Processing Time (minutes) __________Supporting Work:d. Again assume an average interarrival time of 36 minutes and an average processing time of 24minutes. What would be the average waiting time if the professor could clone himself (therebycreating a system with two servers and a single queue)?Average Waiting Time (minutes) __________Supporting Work:Problem 14: NewsboySusan sells snow cones from a pushcart. Snow cones come in two flavors, Redeye Raspberry (RR) andBoozy Banana (BB). Susan’s cost for each cone is the same, $0.50/unit, and she charges $2.00/unit forcones of either flavor. From experience, Susan knows that the daily demand for RR cones is normallydistributed with mean 100 and standard deviation 30, while demand for BB cones is normally distributedwith mean 120 and standard deviation 60.Assume that the demand for RR cones is independent of the demand for BB cones (and vice versa), and thatdemand in excess of supply is lost (no substitutions). Leftover snow cones are discarded at the end of theday.a. How many RR and BB cones should be stocked at the beginning of the day to maximize Susan’sexpected profit? What is the expected profit of this policy?Number of RR Cones ____________Number of BB Cones ____________Expected Profit ____________Supporting Work:b. If Susan can stock no more than 250 snow cones, how many RR and BB cones should be stocked atthe beginning of the day to maximize Susan’s expected profit? What is the expected profit of thispolicy (Hint: the maximum order size makes the cost of overstocking cones of either flavorincrease)?Number of RR Cones ____________Number of BB Cones ____________Expected Profit ____________Supporting Work:Problem 5: Mismatch CostsPurchasing road salt for towns in the Northeast is a challenging task. The town of Homer, New York hascalculated a forecast of their annual salt needs using historical data. The forecast is summarized in the tablebelow (Q is the quantity needed):For example, there is a 60.6% chance they will need 50,000 tons or fewer, there is a 3.3% chance they willneed exactly 100,000 tons and there is a very small chance they will need more than 200,000 tons. SupposeHomer wants to minimize the amount of inventory it purchases while at the same time having no more thana 6% change of running out of salt (which would force it to purchase salt on the spot market for a premium).a. At the start of the season, how much salt (in tons) should Homer have available in its storage sheds?Assume salt must be purchased in increments of 10,000.Order Quantity (tons) __________Supporting Work:b. Now suppose Homer has been offered the following deal from American Salt Mine (ASM). ASMwill sell Homer salt options for $30 per option with an exercise price of $40 for each option Homerpurchases in advance of the season, Homer can âexerciseâ an option during the season to receive oneton of salt during the season. For example, if Homer purchases 100,000 options before the seasonstarts, then it pays ASM $30 x 100,000 = $3,000,000 for those options. As Homer needs salt duringthe season, ASM will deliver up to 100,000 tons for a price of $40 per ton. Options are good only forthis season â any unexercised options at the end of the season have no value. Finally, if Homerexercises all of its options and still needs more salt, then it will have to purchase salt in the spotmarket, for an estimated $80 per ton. Given this deal, how many options should Homer purchasefrom ASM? Assume options must be purchased in increments of 10,000 tons.Number of Options to Purchase __________Supporting Work:Problem 26: Risk PoolingThe FAMU Bookstore stocks two types of cashmere sweaters. The two sweaters are identical in everyway except on the first sweater is stitched FAMU FOOTBALL while on the second is stitchedRATTLERS FOOTBALL (weâll refer to these two types as FAMU sweaters and RATTLERS sweaters).Both sweaters retail for $100 apiece and cost the Bookstore $40 to procure. Because the procurementlead time is long relative to the length of the football season, the Bookstore places a single order to coveranticipated sales for the entire season. Any sweaters left over at the end of the season are shipped to areseller for $20 apiece. The demand for FAMU sweaters is normally distributed with a mean of 1000and a standard deviation of 400. The demand for RATTLERS sweaters is normally distributed with amean of 800 and a standard deviation of 300. Itâs been noted that in previous years when the demandfor one type of sweater is high, the demand for the other type of sweater is low, leading the Bookstore toestimate the correlation between the two sweaters at -0.40.a. How many FAMU sweaters should the Bookstore order for the season to maximize expectedprofit? What is the expected profit?Number of FAMU Sweaters to Order (units) __________________Expected Profit ($) __________________Supporting work:b. How many RATTLERS sweaters should the Bookstore order for the season to maximizeexpected profit? What is the expected profit?Number of RATTLERS Sweaters to Order (units)__________________(4 points)Expected Profit ($) __________________Supporting work:c. The Bookstoreâs manager is offered the opportunity to replace the FAMU and RATTLERSsweaters with a new sweater on which FAMU RATTLERS FOOTBALL is stitched (which weâllrefer to as a FAMU RATTLERS sweater). The manager believes that all customers who wouldotherwise have demanded a FAMU or a RATTLERS sweater will instead buy the FAMURATTLERS sweater, i.e., no sales will be lost by stocking FAMU RATTLERS sweaters anddiscontinuing the FAMU and RATTLERS lines. In addition, all of the cost and demandinformation previously given remains the same. How many FAMU RATTLERS sweatersshould the Bookstore order for the season to maximize expected profit? What is the expectedprofit?Number of FAMU RATTLERS sweaters to Order (units) __________________Expected Profit ($) __________________
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