A publisher sells books to Barnes & Noble at $16 each. The unit production cost

Question

A publisher sells books to Barnes & Noble at $16 each. The unit production cost for the publisher is $4 per book. Barnes & Noble prices the book to its customers at $30 and expects demand over the next two months to be normally distributed, with a mean of 16,000 and a standard deviation of 5,000. Barnes & Noble places a single order with the publisher for delivery at the beginning of the two-month period. Currently, Barnes & Noble discounts any unsold books at the end of two months down to $5, and any books that did not sell at full price sell at this price. A publisher sells books to Barnes & Noble at $16 each. The unit production cost for the publisher is $4 per book. Barnes & Noble prices the book to its customers at $30 and expects demand over the next two months to be normally distributed, with a mean of 16,000 and a standard deviation of 5,000. Barnes & Noble places a single order with the publisher for delivery at the beginning of the two-month period. Currently, Barnes & Noble discounts any unsold books at the end of two months down to $5, and any books that did not sell at full price sell at this price.

Explanation / Answer

(a)

For Barnes and Noble,
Cu = cost of underage = Selling price - cost = 30 -16 = 14
Co = cost of overage = Cost - salvage value = 14 - 5 = 9

So, critical factor = Cu / (Co+Cu) = 14/23 = 0.609

At optimality, the in-stock proobability must match critical factor, so, Z = NORMSINV(0.609) = 0.276

So, optimal order quantity(Q*) = 16,000 + 0.276*5,000 = 17,380

Normal loss function corresponding to the above Z value = 0.276
So, lost sales, L(Q) = 0.276 x 5,000 = 1,380

Expected sales, S(Q) = 16,000 - 1,380 = 14,620

Expected left over, V(Q) = 17,380 - 14,620 = 2,760 (this is the number of books to be sold at discount)

Expected profit = Cu*S(Q) - Co*V(Q) = 14*14620 - 9*2760 = $179,840

(b)

Profit to the publsher = Q* x (16 - 4) = 17,380 x 12 = $208,560

(c)

Cu = cost of underage = Selling price - cost = 30 -16 = 14
Co = cost of overage = Effective cost - salvage value = (14 - 4) - 5 = 5

So, critical factor = Cu / (Co+Cu) = 14/19 = 0.737

At optimality, the in-stock proobability must match critical factor, so, Z = NORMSINV(0.737) = 0.634

So, optimal order quantity(Q*) = 16,000 + 0.634*5,000 = 19,170

Normal loss function corresponding to the above Z value = 0.16
So, lost sales, L(Q) = 0.16 x 5,000 = 800

Expected sales, S(Q) = 16,000 - 800 = 15,200

Expected left over, V(Q) = 19,170 - 15,200 = 3,970 (this is the number of books to be sold at discount)

Expected profit = Cu*S(Q) - Co*V(Q) = 14*15200 - 5*3970 = $192,950

(d)

Profit of the publisher = Q* x (16 - 4) - V(Q) x 4 = 19,170*12 - 3970*4 = $214,160

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