An airline offers economy and business class tickets. For the airline to be profitable, it must sell a minimum of 25 business class tickets and a minimum of 40 economy tickets. The airline makes a profit of P3500 for each economy ticket and P6000 for each business class ticket. Let x and y be the number of economy and business class tickets, respectively. Which of the following is not a constraint in the problem? A x20 B y 2 25 x> 40 D 3500x + 6000y
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- You now have 10,000, all of which is invested in a sports team. Each year there is a 60% chance that the value of the team will increase by 60% and a 40% chance that the value of the team will decrease by 60%. Estimate the mean and median value of your investment after 50 years. Explain the large difference between the estimated mean and median.Based on Kelly (1956). You currently have 100. Each week you can invest any amount of money you currently have in a risky investment. With probability 0.4, the amount you invest is tripled (e.g., if you invest 100, you increase your asset position by 300), and, with probability 0.6, the amount you invest is lost. Consider the following investment strategies: Each week, invest 10% of your money. Each week, invest 30% of your money. Each week, invest 50% of your money. Use @RISK to simulate 100 weeks of each strategy 1000 times. Which strategy appears to be best in terms of the maximum growth rate? (In general, if you can multiply your investment by M with probability p and lose your investment with probability q = 1 p, you should invest a fraction [p(M 1) q]/(M 1) of your money each week. This strategy maximizes the expected growth rate of your fortune and is known as the Kelly criterion.) (Hint: If an initial wealth of I dollars grows to F dollars in 100 weeks, the weekly growth rate, labeled r, satisfies F = (I + r)100, so that r = (F/I)1/100 1.)A common decision is whether a company should buy equipment and produce a product in house or outsource production to another company. If sales volume is high enough, then by producing in house, the savings on unit costs will cover the fixed cost of the equipment. Suppose a company must make such a decision for a four-year time horizon, given the following data. Use simulation to estimate the probability that producing in house is better than outsourcing. If the company outsources production, it will have to purchase the product from the manufacturer for 25 per unit. This unit cost will remain constant for the next four years. The company will sell the product for 42 per unit. This price will remain constant for the next four years. If the company produces the product in house, it must buy a 500,000 machine that is depreciated on a straight-line basis over four years, and its cost of production will be 9 per unit. This unit cost will remain constant for the next four years. The demand in year 1 has a worst case of 10,000 units, a most likely case of 14,000 units, and a best case of 16,000 units. The average annual growth in demand for years 2-4 has a worst case of 7%, a most likely case of 15%, and a best case of 20%. Whatever this annual growth is, it will be the same in each of the years. The tax rate is 35%. Cash flows are discounted at 8% per year.
- W. L. Brown, a direct marketer of womens clothing, must determine how many telephone operators to schedule during each part of the day. W. L. Brown estimates that the number of phone calls received each hour of a typical eight-hour shift can be described by the probability distribution in the file P10_33.xlsx. Each operator can handle 15 calls per hour and costs the company 20 per hour. Each phone call that is not handled is assumed to cost the company 6 in lost profit. Considering the options of employing 6, 8, 10, 12, 14, or 16 operators, use simulation to determine the number of operators that minimizes the expected hourly cost (labor costs plus lost profits).Six months before its annual convention, the American Medical Association must determine how many rooms to reserve. At this time, the AMA can reserve rooms at a cost of 150 per room. The AMA believes the number of doctors attending the convention will be normally distributed with a mean of 5000 and a standard deviation of 1000. If the number of people attending the convention exceeds the number of rooms reserved, extra rooms must be reserved at a cost of 250 per room. a. Use simulation with @RISK to determine the number of rooms that should be reserved to minimize the expected cost to the AMA. Try possible values from 4100 to 4900 in increments of 100. b. Redo part a for the case where the number attending has a triangular distribution with minimum value 2000, maximum value 7000, and most likely value 5000. Does this change the substantive results from part a?A college student works in both the school cafeteria and library. She works no more than 12 hours per week at the cafeteria, and no more than 16 hours per week at the library. She must work at least 20 hours each week. Write a system of inequalities that describes all the given conditions. Write a system of inequalities letting x= number of hours worked at the cafeteria per week and y = number of hours worked at the library per week. x+yz x≤ ys
- Consider the following game of ’divide the dollar.’ There is a dollar to be split between two players. Player 1 can make any offer to player 2 in increments of 25 cents; that is, player 1 can make offers of 0 cents, 25 cents, 50 cents, 75 cents, and $1. An offer is the amount of the original dollar that player 1 would like player 2 to have. After player 2 gets an offer, she has the option of either accepting or rejecting the offer. If she accepts, she gets the offered amount and player 1 keeps the remainder. If she rejects, neither player gets anything. Draw the game tree.Consider the following variant of the ’divide the dollar’ game. Players 1 and 2 move simultaneously; 1 makes an offer to 2 and 2 specifies what would be an acceptable offer. For instance, player 1 might make an offer of 50 cents and player 2 might simultaneously set 25 cents as an acceptable offer. If player 1’s offer is at least as large as what is acceptable to player 2, then we will say that there is an agreement…For the remaining questions, consider the following problem description: An oil company is considering exploring new well sites S₁, S2, ..., S10 with respective costs C1, C2, C10. And in particular they want to find the least-cost selection of 5 out of the 10 possible sites. The binary decision variables x₁,x2,..., X10 denote the decision to explore the corresponding site.Dickie Hustler has $2 and is going to toss an unfair coin(probability .4 of heads) three times. Before each toss, hecan bet any amount of money (up to what he now has). Ifheads comes up, Dickie wins the number of dollars he bets;if tails comes up, he loses the number of dollars he bets.Use dynamic programming to determine a strategy thatmaximizes Dickie’s probability of having at least $5 afterthe third coin toss.
- Apply Linear Programming to the Folling Question: Dan Reid, chief engineer at New Hampshire Chemical, Inc., has to decide whether to build a new state-of-art processing facility. If the new facility works, the company could realize a profit of $200,000. If it fails, New Hampshire Chemical could lose $150,000. At this time, Reid estimates a 60% chance that the new process will fail. The other option is to build a pilot plant and then decide whether to build a complete facility. The pilot plant would cost $10,000 to build. Reid estimates a fifty-fifty chance that the pilot plant will work. If the pilot plant works, there is a 90% probability that the complete plant, if it is built, will also work. If the pilot plant does not work, there is only a 20% chance that the complete project (if it is constructed) will work. Reid faces a dilemma. Should he build the plant? Should he build the pilot project and then make a decision? Help Reid by analyzing this problemPlayer A and B play a game in which each has three coins, a 5p, 10p and a 20p. Each selects a coin without the knowledge of the other’s choice. If the sum of the coins is an odd amount, then A wins B’s coin. But, if the sum is even, then B wins A’s coin. Find the best strategy for each player and the values of the game.Suppose that Pizza King and Noble Greek stopadvertising but must determine the price they will chargefor each pizza sold. Pizza King believes that Noble Greek’sprice is a random variable D having the following massfunction: P(D $6) .25, P(D $8) .50, P(D $10) .25. If Pizza King charges a price p1 and NobleGreek charges a price p2, Pizza King will sell 10025( p2 p1) pizzas. It costs Pizza King $4 to make a pizza.Pizza King is considering charging $5, $6, $7, $8, or $9 fora pizza. Use each decision criterion of this section todetermine the price that Pizza King should charge.