Question

You are a wholesaler specialized in organic, locally produced vegetables. You maintain a record of the quantity, in pounds, of the demand of vegetables per day. You have determined that the demand approximate a normal distribution with a mean of 22.55 and a standard deviation 2.64. You buy the vegetables at $12 /lb and sell them at $51 /lb.

Currently, you order the mean demand. Any left over is thrown away at the end of the day. What is your expected units short and profit per day following this ordering strategy?

You noticed that your reliability to provide supplies affects your future sales, you will lose some customers when you stock out. You estimate the penalty of stocking out to be $24 per lb. What should your new optimal order size be? What will your expected units short and expected profit values be?

You decided to check the empirical distribution of the demand. In your record (the table below), demand has been rounded up to the nearest pound (lb).

Based on this empirical distribution, what is the new optimal Q assuming that, like in Part 3, there is stock out penalty of $24? What are your new expected units short and expected profit values? (For Part 4, you may assume that the vegetables are supplied, consumed and resold in discrete pound (lb) units.)

Answer 1

part 1) Given data

Cost of overage (Co) = Cost - Salvage value = 12 - 0 = 12
Cost of underage (Cu) = Selling price - Cost = 51 - 12 = 39

Present order size (Q) = mean demand = 22.55 lb.

Corresponding standard normal variable value, Z = (22.55 - 22.55) / 2.64 = 0
Corresponding normal loss variable L(Z) = 0.399 (use table)

Expected lost sales (units short), L(Q) = Std. dev. * L(Z) = 2.64 * 0.399 = 1.05 lb.

Expected Sales, S(Q) = Average Demand - L(Q) = 22.55 - 1.05 = 21.5 lb.

Expcted left over, V(Q) = Q - S(Q) = 22.55 - 21.5 = 1.05 lb.

Expected profit = Cu * S(Q) - Co * V(Q) = 39 * 21.5 - 12 * 1.05 = $825.9

Part 2)

Cost of Shortage (Cs) = Revenue per unit - Cost per unit - Penalty per unit = 51 - 12 - 24 = 15$

Cost of excess (Ce) = Cost per unit - Salvage = 12 - 0 = 12$

Service level (SL) = Cs/(Cs+Ce) = 15/(15+12) = 0.555

Mean = 22.55

SD = 2.64

a) Optimal order size = Mean + SL*SD = 22.55 + 2.64*0.555 = 24.015

b) Expected unit short = Mean + SD - Optimal order size = 22.55 + 2.64 - 24.015 = 1.175

c) Expected Profit = The same will be exactly equal order size. Thus Profit per pound = Revenue - Cost = 51 - 12 = 39$

Profit = 24.015*39 = 936.58$

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