Answer to Question #312988 in Quantitative Methods for Khan

Question

The buyer for Payless Shoe store decided to order a woman’s shoe at a buyers’ meeting in Miami. The shoe  will be part of the company’s Christmas promotion. New designs are coming out after Christmas, so the shoes have to be sold during the Christmas season. Payless plans to hold a special December clearance sale, in an attempt to sell all shoes not sold  by November 31st. The shoes will retail at $50 per pair and the company makes a profit of $15 on each pair. At the sale price of $21 per pair, all surplus shoes can be expected to sell during the December sale. The expected demand for the shoes is 700 pairs with a standard deviation of 300 pairs. How many pairs of the shoe should the buyer order?

Answer 1

As the profit is $15 at the price of $50, then the cost of the shoes is $35. Therefore sold them for $21 the company lost $14 on each pair.

Let X is the real shoe demand, and N is the order number. Then the total profit will be 15*N if N<X and 15*X -14*(N-X) = 29*X - 14*N if N>X.

So we should find the maximum of $\int_0^N(29 X-14 N) p(X)dX$ , where p(X) is the probability density function. Substitute normal distribution with mean 700 and standard deviation 300, we get

$F(N) = \int_0^N(29 X-14 N) \frac{e^{\frac{-(X-700)^2}{2 \cdot 300^2}}}{\sqrt{2 \pi} 300}dX$, which should be maximized. To do so we should solve the equation $\frac{dF(N)}{dN} = 0$ . This equation has a root near N = 1011.

So company should order 1010 pairs of shoes.