1. Suppose we are considering the question of how much capacity to build in the face of uncertain demand. Assume that the cost is $20 per unit of lost sales due to insufficient capacity. Also assume that there is a cost of $7 for each unit of capacity built. The probability of various demand levels is as follows: Demand-X Units 0 1 2 3 4 5 6 7 Probability of X .05 .10 .15 .20 .20 .15 .10 .05 a. How many units of capacity should be built to minimize the total cost of providing capacity plus lost sales? b. State a general rule regarding the amount of capacity to build. c. What principle does this problem illustrate?
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- At the beginning of each week, a machine is in one of four conditions: 1 = excellent; 2 = good; 3 = average; 4 = bad. The weekly revenue earned by a machine in state 1, 2, 3, or 4 is 100, 90, 50, or 10, respectively. After observing the condition of the machine at the beginning of the week, the company has the option, for a cost of 200, of instantaneously replacing the machine with an excellent machine. The quality of the machine deteriorates over time, as shown in the file P10 41.xlsx. Four maintenance policies are under consideration: Policy 1: Never replace a machine. Policy 2: Immediately replace a bad machine. Policy 3: Immediately replace a bad or average machine. Policy 4: Immediately replace a bad, average, or good machine. Simulate each of these policies for 50 weeks (using at least 250 iterations each) to determine the policy that maximizes expected weekly profit. Assume that the machine at the beginning of week 1 is excellent.A firm must decide whether to construct a small, medium, or large stamping plant. A consultant’sreport indicates a .20 probability that demand will be low and an .80 probability that demand willbe high.If the firm builds a small facility and demand turns out to be low, the net present value will be$42 million. If demand turns out to be high, the firm can either subcontract and realize the net present value of $42 million or expand greatly for a net present value of $48 million.The firm could build a medium-size facility as a hedge: If demand turns out to be low, its netpresent value is estimated at $22 million; if demand turns out to be high, the firm could do nothingand realize a net present value of $46 million, or it could expand and realize a net present value of$50 million.If the firm builds a large facility and demand is low, the net present value will be – $20 million,whereas high demand will result in a net present value of $72 million.a. Analyze this problem using a decision…3. A toy manufacturer makes stuffed kittens and puppies which have relatively lifelike motions. There are three different mechanisms that can be installed in these "pets." These toys will sell for the same price regardless of the mechanism installed, but each mechanism has its own variable cost and setup cost. Profit, therefore, is dependent upon the choice of mechanism and upon the level of demand. The manufacturer has in hand a forecast of demand that suggests a 0.2 probability of light demand, a 0.45 probability of moderate demand, and a probability of 0.35 of heavy demand. Payoffs for each mechanism-demand combination appear in the table below. Wind-up action Pneumatic action Electronic action Demand Light Moderate $250,000 $90,000 -$100,000 400,000 440,000 400,000 Heavy 650,000 740,000 780,000 Construct the appropriate decision tree to analyze this problem. Use standard symbols for the tree. Analyze the tree to select the optimal decision for the manufacturer.
- Alternative Alternative 1 Alternative 2 Alternative 3 State of Nature Outcome 1 ($) 800 500 700 0.62 Probability according to the payoff table? OEMV (Alternative 1) = $900 EMV (Alternative 3) is the highest EMV EMV (Alternative 2) = EMV (Alternative 3) EMV (Alternative 1) is the highest EMV Outcome 2 ($) 1000 1200 900 0.38 Which of the following statements is correctThe number of hybrid cars that have be requested for rental from a car rental office during a 50 day sample period is given in the following table. Demand for Cars Number of Days Probability 3 3 0.06 4 7 0.14 5 12 0.24 6 14 0.28 7 10 0.20 8 4 0.08 50 1.00What is the expected (mean) number of hybrid rentals per day? Compute to one decimal place.11. Bakery Products is considering the introduction of a new line of pastries. In order to produce the new line, the bakery is considering either a major or a minor renovation of its current plant. Bill Wicker, head of operations, has developed the following conditional values table: Alternatives Favorable Market Unfavorable Market Major renovation $100,000 -$90,000Minor renovation $40,000 -$20,000 Do nothing $0 $0 Assume that the probability of a favorable market is equal to the probability of an unfavorable market.Part 2a) Choose the appropriate decision tree showing payoffs and probabilities.A.MinorFavorable40,000Unfavorable-20,000UnfavorableFavorableMajor100,000-90,000Do…
- Explain or define each of these terms:a. Laplace criterionb. Minimax regretc. Expected valued. Expected value of perfect informationA. A company wants to produce a souvenir with a marketing life of six months. Uncertainty surrounds the likely sales volume as well as the fixed costs of the venture as shown below: Sales units Probability Contrn. /unit Probability Fixed cost K7 K5 100 000 0.3 80 000 0.6 60 000 0.1 1.0 0.5 0.5 1.0 Determine the expected value of the contribution K400 000 K450 000 K500 000 Probability 0.2 0.5 0.3 1.0c. From the following decision tree, develop a payoff table and calculate: * Maximax, Minimax regret, Maximin, and EMV. ORs. 50,000 Good conditions (0.60) Poor conditions (0.40) -O Rs. 30,000 Apartment Building Good conditions (0.60) O Rs. 100,000 Office building Poor conditions (0.40) Purchase ORs -40,000 Warchouse Good conditions (0.60) Rs.30, 000 Poor conditions (0.40) O Rs. 10,000
- Stocks A and B have the following probability distributions ofexpected future returns:Probability A B0.1 (10%) (35%)0.2 2 00.4 12 200.2 20 250.1 38 45a. Calculate the expected rate of return, r⁄B, for Stock B (r⁄A =12%).b. Calculate the standard deviation of expected returns, σA, for Stock A (σB =20.35%).Now calculate the coefficient of variation for Stock B. Is it possible that most investorswill regard Stock B as being less risky than Stock A? Explain.c. Assume the risk-free rate is 2.5%. What are the Sharpe ratios for Stocks A and B? Arethese calculations consistent with the information obtained from the coefficient ofvariation calculations in part b? ExplainAn expensive piece of equipment is used in the masking operation for semi-conductor manufacture. A capacitor in the equipment fails randomly. The capacitor cost $7.50, but if it burns out while the machine is in use, the production process must be halted. Here the replacement cost is estimated to be $150. Based on past experience, the lifetime distribution of the capacitor is estimated to be Number of Months of Service 1 2 3 4 5 6 Probability of Failure 0.08 0.12 0.16 0.26 0.22 0.16 How often should the capacitors be replaced in order to minimize the expected monthly cost of planned and unplanned replacement?#6) A group of medical professionals is considering constructing a private clinic. If a patient demand for the clinic is high, the physicians could realize a net profit of $120,000. If the demand is low, they could lose $55,000. Of course, they do not have to proceed at all, in which case there is no cost. In the absence of any market data, the best the physicians can guess is that there is a 50-50 chance the demand would be high. a) Create a decision tree. b) What should the medical professionals do? What is the payoff? c) The physicians have been approached by a market research firm that offers to perform a study of the market at a fee of $5,000. The market researchers claim that their experience enables them to use Bayes’ theorem to make the following statements of probability: -probability of high demand given a positive survey result = 0.82 -probability of low demand given a positive survey result = 0.18 -probability of high demand given a negative survey result = 0.11…