ISyE 2030 Test 2 Solutions

Transcription

1 1 NAME ISyE 2030 Test 2 Solutions Fall 2004 This test is open notes, open books. Do precisely 5 of the following problems your choice. You have exactly 55 minutes. 1. Suppose that we are simulating the sales of turkeys at a local market. The owner starts off with 10 turkeys. The daily demand for turkeys for the next 6 days is as follows Each turkey costs the owner $7 to buy, and can be sold to a customer for $13. It costs the owner $1 to store each turkey overnight (i.e., if it wasn t sold that day). It also costs the owner $2 per turkey if he is unable to fill a demand maybe the irate customer does damage to the store. If, at the end of a particular day, the owner has less than or equal to 3 turkeys in stock, he places an order to bring the inventory level back up to 10 the next morning. He gets the turkeys the first thing in the morning before any customers arrive. Compute his total profit/loss for this 6-day period. How many turkeys are left at the end of the last day? ANSWER: There are a number of interpretations for this problem, all of which are perfectly acceptable, as long as you state the assumptions you used. In the answer that I ll now display, let s assume: We have to pay for the initial 10 turkeys. We have no holding cost for the first day. We don t have to pay holding costs for any turkeys left over after the last day. So here s what happens...

2 2 Our earnings are therefore Day t Demand D t Inventory at Order Quantity End of Day I t Turkeys bought (7 units) $252 Holding cost (16 units) $16 Stockout cost (1 unit) $2 Turkeys sold (33 units) $429 TOTAL $159 Further, note that 3 turkeys are left.

3 3 2. Consider a queueing system with customers arriving according to a Poisson process at the rate of 5 per hour. Which of the following systems gives the lower steady-state expected time in system (i.e., waiting in line plus service time)? (A) Two slow parallel servers, each of whom can serve customers with Exp(3/hour) service times, or (B) One fast server, who can serve customers with Exp(6/hour) services? Show all work. ANSWER: (A) Here we have an M/M/2 queueing system with λ = 5, µ = 3, and c = 2. Thus, ρ = λ/(cµ) = 5/6. Going to the M/M/2 tables, we do a little calculation and come up with P 0 = 1/11, and with a little more algebra, we have L = 60/11. Finally, w = L/λ = 1.1 hours. (B) Meanwhile, now we have we have an M/M/1 queueing system with λ = 5 and µ = 6. Thus, ρ = λ/µ = 5/6 (still). Going to the M/M/1 tables, we do a teensy calculation and come up with w = 1 µ(1 ρ) = 1. Thus, choose the M/M/1.

4 4 3. The Reddy Mikks Company owns a small paint factory that produces both interior and exterior house paints for wholesale distribution. Two basic raw materials, A and B, are used to manufacture the paints. The maximum availability of A is 8 tons/day. The max availability of B is 6 tons/day. The daily requirements of the raw materials per ton of interior and exterior paints are summarized as follows: For a ton of exterior paint, you need 1 tons of A and 1.5 tons of B. For a ton of interior paint, you need 2 tons of A and 1 ton of B. A market survey has established that the daily demand for interior paint cannot exceed that of exterior paint by more than one ton. The survey also shows that the demand for interior paint is limited to 2 tons daily. The wholesale price per ton is $3000 for exterior paint and $2000 for interior paint. We are interested in finding the mix of interior and exterior that maximizes gross income. (a) Formulate this problem as a linear program. ANSWER: Let x denote the amount of interior; let y denote the amount of exterior. Here is the formulation. max 2000x y s.t. 2x + y 8 x y 6 x y + 1 x 2 x, y 0. (b) Draw a graph illustrating the relevant inequalities. (c) What is the best mix of paints to produce? ANSWER: Draw the picture, and check the corners, and you ll see that the answer is (x, y) = (2, 8/3) or (0,4) (or a linear combination of the two). The reason for the linear combination is that the optimal answer is actually a line segment. (d) Write out relevant Xpress code.

5 5 4. A newsvendor faces the following discrete demand distribution: X P (Demand = X) The newsvendor buys the papers for 60 cents and sells them for $1.00. Leftover papers are salvaged for 10 cents each. (a) What would the marginal benefit be to the newsvendor of going from 21 to 22 papers? ANSWER: Let p denote the probability of selling the xth unit, and let D denote the demand. x P(D = x) p 1 p 0.4p = 0.5(1 p) = Exp net profit Exp profit from Exp loss from not from stocking selling xth unit selling xth unit xth unit < > Thus, you get more going from 21 to 22. (b) What s the optimal number of papers to stock each day? ANSWER: 22.

6 6 5. The Village Butcher Shop makes its meat loaf from a combination of lean ground beef and ground pork. The beef contains 70% meat and 30% fat and costs the shop 80 cents per pound; the pork contains 60% meat and 40% fat, and costs 60 cents per pound. We are interested in determining how much of each kind of meat the shop should use in each pound of meat loaf if it wants to minimize its cost and to keep the fat content of the meat loaf to no more than 35%. I just want you to formulate this problem you don t have to solve it. ANSWER: Let B denote the amount of beef and P the amount of pork. Here is the formulation. min 0.8B + 0.6P s.t. 0.3B + 0.4P 0.35 B + P = 1 B, P 0.

7 7 6. Formulate the making change problem (i.e., cashier returns as few coins as possible) assuming the cashier will give back only coins of value 1 cents, 5 cents, 10 cents, 25 cents, and $1.00 (yes, there is a $1.00 coin), and the change you will receive is $5.67. Make sure to clearly define the variables. ANSWER: Let Here is the formulation. x 1 = number of pennies used x 2 = number of nickels used x 3 = number of dimes used x 4 = number of quarters used x 5 = number of dollars used min 5i=1 x i s.t. 0.01x x x x 4 + x 5 = 5.67 x i 0 and integer for all i.