[Basic Newsvendor] A book publisher sells the latest self-help title to Boundaries, a retail bookstore chain, at $10 per copy.The marginal production cost for the publisher is $5 per copy.The retailer prices the book at $22 to its customers.The books are kept on the regular rack for a one-month period, after which they are discounted down to $4 and all remaining copies will sell at this price. The retailer places a one-time order when the book is first released (there will be no reprinting and no opportunity for reordering). They forecast that demand will be a random variable with mean 10,000 and standard deviation 5,000. (Assume that demand follows the Normal distribution.)

a.How many copies should Boundaries order?

b.How much is its expected profit?

c.How many books does it expect to sell at a discount?

d.How much profit does the publisher make given Boundaries' action? How much is the total profit for both publisher and retailer?

e.What is the maximum possible profit if this supply chain was vertically integrated, i.e. the publisher and retailer are a single entity?

[Buy-Back Contract] The publisher is considering a new contractual arrangement: The wholesale price will be increased from $10 to $12.25; in return, the publisher will buy back all leftover inventory at a price of $12.68.However the retailer is responsible for shipping the books back to the publisher at a cost of $1 per copy. The publisher is able to sell all the returned books on their website at a discounted price of $5.

f.Under this plan, how many copies should Boundaries order?

g.How much is the expected profit for Boundaries?

h.How much is the expected profit for the publisher?

i.How much is the total expected profit for both publisher and retailer? How does this total compare to Part e?

j.What should the publisher do, i.e. should they offer the buy-back contract? Why, or why not?

Newsvendor Formulas

Critical Ratio: CR = C_u / [C_u + C_o]

Excel formula for Z: = NORMSINV(CR)

Order Quantity: Q = + Z

Once Q and Z are known:

Expected Lost Sales (i.e. understock): ELS = X Loss(Z)

To find Loss(Z), you can use the Standard Normal Loss table, or

in Excel: Loss(Z): = NORMDIST(Z,0,1,0) - Z*(1-NORMSDIST(Z))

Expected Sales: ES = - ELS

Expected Leftover Inventory (i.e. overstock): ELI = Q - ES

Expected Profit for the Retailer: EP_Ret = C_u X ES - C_o X ELI

Expected Profit for the Manufacturer: EP_Mfg = (w - c) X Q [without any contract]

EP_Mfg = (w - c) X Q - b X ELI + s X ELI [with a buy-back contract]