books inc sells books to Texas Campus Store at $12 each. The marginal production

Question

books inc sells books to Texas Campus Store at $12 each. The marginal production cost for Books inc is $1 per book. Texas Campus Store prices the book to its customers at $24 and expects demand over the next two months to be normally distributed, with a mean of 20,000 and a standard deviation of 5,000. Texas Campus Store places a single order with books inc for delivery at the beginning of the two-month period. Currently, Texas Campus Store discounts any unsold books at the end of two months down to $3, and any books that did not sell at full price sell at this price.

A plan under discussion is for books inc to refund Texas Campus Store $5 per book that does not sell during the two-month period. As before, Texas Campus Store will discount them to $3 and sell any that remain.

Q4. Under this plan, how many books will Texas Campus Store order?

What is the expected profit for Texas Campus Store?

What is the expected profit for books inc?

Explanation / Answer

For the Texas Campus Store,

Cost of underage, Cu = Selling price - Purchase cost = $24 - $12 = $12
Cpst of overage, Co = Purchase cost - refund - salvage value = $12 - $5 - $3 = $4

So, the critical factor = Cu / (Co + Cu) = 12 / (4+12) = 0.75

For the optimal ordering policy, the in-stock probability should be equal to the critical factor i.e. 0.75.

So, a standard normal variable, Z ~ N(0,1) = NORMSINV(0.75) = 0.674

Optimal order quantity, Q = Mean demand + Z * Std. deviation = 20,000 + 0.674 * 5,000 = 23,372

For Z equal to 0.674, the normal loss function, L(Z) (use tables) = 0.149

Expected lost sales, L(Q) = L(Z) x Std. deviation = 0.149 x 5000 = 746

Expected Sales, S(Q) = D - L(Q) = 20000 - 746 = 19254

Expected left over, V(Q) = Q - S(Q) = 23372 - 19254 = 4118

Expected profit (for Texas Campus Store) = Cu*S(Q) - Co*V(Q) = 12*19254 - 4*4118 = $214,576

Expected profit (for books inc) = (12 - 1)*S(Q) + (12 - 1 - 5)*V(Q) = 11*19254 + 6*4118 = $236,502

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