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EMGT 5995 HW2--Problem 1

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EMGT 5995 HW2--Problem 1

Janet Condor owns a sporting goods store in a small resort area. She has not had much tennis-related business in the past but knows that this will change when a local athletic club holds a big name tournament. Condor believes interest in tennis will increase during and after the tournament and so she has contacted a distributor to make arrangements for a shipment of racquets. The distributor requires orders be made in blocks of 100 racquets, $23 per racquet, and no discount regardless of order size. At a selling price of $30, Condor estimates demand to be as follows:

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The distributor’s demand that orders be in blocks of 100 bothers Condor because of the losses that will incur on unsold racquets. She has found a buyer who agrees to buy any leftover racquets at the end of the tennis season for $20 per racquet. Nationwide interest in tennis has increased to the point that orders are taken months ahead of time, and Condor must order now so that the racquets will be on hand when the tournament takes place. In order to simplify her decision problem, Janet is willing to assume zero cost of turning down a tennis buff when all racquets have been sold.
  • Construct a payoff matrix showing profits earned in this situation.

(b) What act would condor choose under the Maximax, Maximin, maximum probability and Minimax of regret criteria?

(c) How many racquets should be ordered to maximize expected profits?

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Solution

  • The payoff matrix (net profit)

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b. Maximax= order 400

Maximin= order 200

Max Prob.= order 300

Regret Matrix

So order : 400

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E (ordering 200) = $1400

E (ordering 300) = 0.1*1100+0.2*1600+0.4*2100+0.2*2100+0.1*2100

= $1900

E(ordering 400) = 0.1*800+0.2*1300+0.4*1800+0.2*2300+0.1*2800

=$1800

Order: 300

EMGT 5995

emgt5995 hw2 problem 2
EMGT5995 HW2-Problem 2

A farm implement manufacturer has discontinued a particular tractor engine. Most parts in the engines are interchangeable with other engine parts with the exception of the crankshaft. The manufacturer must decide now how many replacement crankshafts to stock in the inventory before tooling the machine shop is changed for the new line. Crankshafts cost the manufacturer $210 to make and are sold for $300. They cannot be modified to fit another engine. If unsold, their scrap salvage value is $30 : estimates of demand for replacement crankshafts are as follows:

Assume that the cost of turning downing a customer who needs a crankshaft when none is available is zero.

(a) Construct a profit matrix for the manufacturer’s decision problem.

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(b) How many replacement units should be stocked to maximize expected profits?

(c) Construct the conditional loss matrix for this problem.

(d) Show that the expected profits plus the expected losses for any act sum to the same figure. Interpret this figure in words.

(e) How much should the manufacturer be willing to pay for a perfect forecast of demand for replacement crankshafts?

EMGT 5995

solution

MAX

Solution

a and b:

For (stocked)(demand); profit=(stocked)(300-210)

For (stocked)>(demand); profit=(demand)(90)-(stocked-demand)(210-30)

Payoff Matrix(x$1000)

Optimal: Act is to stock 1500 units.

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c. Conditional loss

(Regret) Matrix (*$1000)

Demand

Optimal Act is to stock 1500 units

MIN

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d. For any act the sum of EL and EPUU equals $171000. This is EPUC, the expected profit under certainty.If the manufacturer had a perfect forecast of demand and made optimal decisions the expected profit would equal $171000. As a cross check on this figure, EPUC may be computed by use of the optimal profits found on the diagonal in part(a):

EPUC=0.15*90000+0.3*135000+0.25*180000+0.2*225000+0.1*27000 =$171000

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If $171000 is the Expected profit under certainty and $114750 is the

expected profit of the optimal act under uncertainty then perfect information would be worth the difference or $56250. This is of course, the EL of the optimal act. (stock 1500)

EMGT 5995