Question 5 (15 points). A retailer sells Christmas trees at $50 each and can sal

Question

Question 5 (15 points). A retailer sells Christmas trees at $50 each and can salvage unsold trees for $10/unit. Because of long lead-times the retailer must commit orders to the farm at least three months before the holiday season begins, and pays $22 per unit. a. Compute the overage and underage cost per unit. b. Based on forecasts and marketing studies, the retailer estimates the demand for Christmas trees with a normal distribution with mean 6000 and standard deviation 2000. How much should the retailer order? c. If the demand is uniform with mean 5000 and range 14000, 6000] what is the optimal order quantity? What is the optimal ordering quantity if the retailer uses the following discrete distribution for demand? d. quantity 2000 3000 4000 5000 6000 7000 8000 9000 10000 probability 0.02 0.03 0.10.15 0.30 0.15 0.15 0.05 0.05

Explanation / Answer

a. Here,

Selling price P = $ 50

Cost price C = $ 22

Salvage price S = $ 10

Overage Cost per unit = C - S = 22 -10 = $ 12

Underge cost per unit = P - C = 50 - 22 = $ 28

(b) Mean demand = 6000

Stadandard deviation of demand = 2000

Here F(Qoptimum) = underage cost/ (underage cost + overage cost)

F(Qoptimum)  = 28/(28 + 12) = 28/40 = 0.7

so as the demand is normal distirubtion, here

F(Qoptimum)  = 0.7

Z = F-1 (0.7) = NORMSINV(0.7) or using Z table

Z = 0.5244

Z = (Qoptimum - 6000)/2000 = 0.5244

Qoptimum = 6000 + 2000 * 0.5244 = 7048.8 or 7049

(c) Here also

FOr uniform distribution

F(X) = X -4000)/ (6000 - 4000) = (X -4000)/2000

SO as we got F(X) = 0.7

(X -4000)/2000 = 0.7

X = 4000 + 2000 * 0.7 = 5400

Qoptimum = 5400

(d) Here also

F(Q = Qoptiomum ) = 0.7

so here we will get cumulative frequency distribution table.

so we will get P(Q) = 0.70 at 7000 as P(7000) = 0.75 so we will demand 7000 units

Q P(Q) CDF 2000 0.02 0.02 3000 0.03 0.05 4000 0.1 0.15 5000 0.15 0.3 6000 0.3 0.6 7000 0.15 0.75 8000 0.15 0.9 9000 0.05 0.95 10000 0.05 1

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