A retail outlet sells seasonal products for $ 10 per unit. Production costs are

Question

A retail outlet sells seasonal products for $ 10 per unit. Production costs are $ 8 per unit. During the regular season unsold units will be sold at half the retail price at the end of the season. It is assumed that the demand for products is normally distributed with µ = 500 and = 100.

a. What is the recommended order quantity?

b. What is the probability of a stockout?

c. To make customers happy and return to the store, the owner feels that stockouts must be avoided if possible. What is the recommended quantity if the owner is willing to tolerate 0.15 probability of stockout?

d. Using your answer to section c, how much do you charge for goodwill for a stockout?

Explanation / Answer

Cu = cost per unit of demand underestimated = selling price - cost = 10 - 8 = 2
Co = cost per unit of demand overestimted = cost - salvage value = 8 - (10/2) = 3

Critical factor = Cu / (Co + Cu) = 2 / (3+2) = 2/5 = 0.40

(a)

The optimal production quantity occurs at the point where the marginal benefit of producing one additional unit is just less than its expected marginal cost. The probability of selling or the in-stock probability should be less than or equal to the critical ratio for finding the optimal order size

So, Z = NORMSINV(0.40) where Z ~ N(0,1)

or, Z = -0.2533

Optimal order (production) quantity, Q = µ + Z * = 500 - 0.2533*100 = 474.67 or 475 units

(b)

Stockout probability = 1 - in-stock probability = 1 - 0.40 = 0.60

(c)

If stockout probability is 0.15 then the corresponding in-stock probability = 0.85

Z = NORMSINV(0.85) = 1.036

Order (production) quantity, Q = µ + Z * = 500 + 1.036*100 = 603.6 or 604 units

(d)

The goodwill cost will affect the Cu. If 0.85 be the critical ratio then,

Cu / (Co + Cu) = 0.85

or, Cu / (3 + Cu) = 0.85

or, 0.15 Cu = 3 * 0.85

or, Cu = $17

Earlier Cu was $2 only. So, the additional $15 is due to goodwill. So, goodwill has been valued as $15 per unit of lost demand.

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