Question

Problem 2

Stacys is a local department store. They need to determine the purchase quantity of winter jackets for the upcoming winter. The unit purchase cost is $30 per jacket and the unit selling price is $45.99 per jacket. If the jacket cannot be sold out at the end of the season, they would be salvaged at a unit price of $10.99 per jacket. The following table shows the demand for the jacket for the winter season and associated probabilities based on historical data.

What is the optimal order quantity of the jacket to maximize the expected seasonal profit? What are the expected sales & profit?

Problem 2 Stacys is a local department store. They need to determine the purchase quantity of winter jackets for the upcoming winter. The unit purchase cost is $30 per jacket and the unit selling price is $45.99 per jacket. If the jacket cannot be sold out at the end of the season, they would be salvaged at a unit price of $10.99 per jacket. The following table shows the demand for the jacket for the winter season and associated probabilities based on historical data Seasonal Demand (jacket) Probability D 90 0.12 D 110 0.15 D 145 0.35 180 0.20 D = D 200 0.18 What is the optimal order quantity of the jacket to maximize the expected seasonal profit? What are the expected sales & profit?

Answer 1

Cost of shortage Cs= Price - cost = 45.99-30 = 15.99

Cost of excess Ce = cost - salvage value = 30-10.99 = 19.01

service level = Cs / Cs+Ce = 15.99 / 15.99+19.01 = 0.4568

This service level can be achieved at D=145, or where the cumulative probabiity exceeds the service level.

Demand / Supply	90	110	145	180	200
90	15.99x90	15.99x90-20x19.01	15.99x90-55x19.01	15.99x90-19.01x90	15.99x90-19.01x110
110	15.99x90	15.99x110	15.99x110-35x19.01	15.99x110-70x19.01	15.99x110-19.01x90
145	15.99x90	15.99x110	15.99x145	15.99x145-35x19.01	15.99x145-19.01x55
180	15.99x90	15.99x110	15.99x145	15.99x180	15.99x180-19.01x20
200	15.99x90	15.99x110	15.99x145	15.99x180	15.99x200

The payoff tale can be simplified as

						Prob.	Profit	net profit
Demand / Supply	90	110	145	180	200
90	1439	1059	393	-272	-652	0.12	1967	236.04
110	1439	1759	1094	428	52	0.15	4772	715.8
145	1439	1759	2318	1653	1272	0.35	8441	2954.35
180	1439	1759	2318	2878	2498	0.2	10892	2178.4
200	1439	1759	2318	2878	3198	0.18	11592	2086.56

It can be seen that the maximum profit of 2954.35 corresponds to D=145.