10. X A merchant stocks a certain perishable item. She knows that on any given d

Question

10. X A merchant stocks a certain perishable item. She knows that on any given day she will have a demand for either two, three or four of these items with probabilities 0.1, 0.4, 0.5 respectively. She buys the items for $1.00 each and sells them for $1.20 each. If any are left at the end of the day, they represent a total loss. How many items should the merchant stock in order to maximize her expected daily profit? Note: Work must be shown. Hint: For each number of items stocked, find the p.m.f. of the daily profit.

Explanation / Answer

We know that the maximum demand in a day is 4 and minimum demand is 2.

For the three stock values 2, 3, 4, the profits for each probability here is computed as:

The expected daily profit for each of the 3 stock values are computed here as:

For Stock = 2, Expected profit = 0.4
For Stock = 3, Expected profit = -0.1*0.6 + 0.4*0.6 + 0.5*0.6 = 0.48
For Stock = 4, Expected profit = -0.1*1.6 - 0.4*0.4 + 0.5*0.8 = 0.08

Therefore Stock should be 3 to maximize the expected profit here.

Demand = 2 Demand = 3 Demand = 4 Stock = 2 0.2*2 = 0.4 0.2*2 = 0.4 0.2*2 = 0.4 Stock = 3 0.2*2 - 1 = - 0.6 0.2*3 = 0.6 0.2*3 = 0.6 Stock = 4 0.2*2 - 2 = -1.6 0.2*3 - 1 = - 0.4 0.2*4 = 0.8

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