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Optimal Policy Problem

My professor gave us the solution to one of our homework problems, and I don't understand. The attached file contains his solution, and my questions in red. Can you please help me to understand?

A vendor sells sweatshirts at football games. They are equally likely (0.5,0.5) to sell 200 or 400 sweatshirts per game. Each order placed to the supplier costs $500 plus $5 per quantity ordered. The vendor sells each sweatshirt for $8. The is a holding cost of $2 (inventory costs) for each shirt leftover after each game. The maximum inventory that the vendor can store is 400. The number of shirts that can be ordered from the supplier must be a multiple of 100. Determine an ordering policy that maximizes expected profits earned during the first three games of the season. Assume that any leftover sweatshirts have a value of $6.

IE 522 - SolutionLet ft(x) = maximum profit earned during games t, t + 1,...3 given that x shirts are on hand at the beginning of game t(before an order is placed) . x may equal 0, 100, 200, 300, or 400. We assume that before placing an order for game 1 no shirts are on hand. Let c(0) = 0 and for s>0,c(s) = 500 + 5s. Then

f3(x) = max { c(s) 1/2[2(x + s 200)+ + 2(x + s 400)+]
s
+ 1/2[6(x + s 200)+ + 6(x + s 400)+]

+ 1/2[8min(x + s, 200) + 8min(x + s, 400)]}

Here x+ = max(x, 0) and s(the number of shirts ordered before game 3) must satisfy x + s 200<_400 or s<_600 x. For t = 1 and 2

ft(x) = max { c(s) 1/2[2(x + s 200)+ + 2(x + s 400)+]
s
+ 1/2[8min(x + s, 200) + 8min(x + s, 400)]
+ 1/2ft+1(x + s 200)+ + 1/2ft+1(x + s 400)+}
Again s must satisfy s<_600 x. We list the computations in tabular form

f3(0) Computations What does f3(0) represent?

s Order Holding Salvage Expected Total Profit
Cost Cost Value Sales Revenue
0 0 0 0 0 0
---------------------------------------
100 1000 0 0 800 200
--------------------------------------
200 1500 0 0 1600 100
-------------------------------------
300 2000 2(.5(100) 6(.5(100)) 8(.5(200) 200
+.5(0)) +.5(0)) +.5(300))
------------------------------------
400 2500 2(.5(200) 6(.5(200) 8(.5(200) 300*
+.5(0)) +.5(0)) +.5(400))
------------------------------------
500 3000 2(.5(300) 6(.5(300) 8(.5(200) 200
+.5(100)) +.5(100)) +.5(400))
------------------------------------
600 3500 2(.5(400) 6(.5(400) 8(.5(200) 100
+.5(200)) +.5(200)) +.5(400))
--------------------------------------

f3(100) Computations
What does f3(100) represent?

s Order Holding Salvage Expected Total Profit
Cost Cost Value Sales Revenue

0 0 0 0 800 800*

100 1000 0 0 1600 600

200 1500 2(.5(100) 6(.5(100) 8(.5(200) 700
+.5(0)) +.5(0)) +.5(300))

300 2000 2(.5(200) 6(.5(200) 8(.5(200) 800*
+.5(0)) +.5(0)) +.5(400))

400 2500 2(.5(100) 6(.5(100) 8(.5(200) 700
+.5(300)) +.5(300)) +.5(400))

500 3000 2(.5(200) 6(.5(200) 8(.5(200) 600
+.5(400)) +.5(400)) +.5(400))

f3(200) Computations
What does f3(200) represent?
s Order Holding Salvage Expected Total Profit
Cost Cost Value Sales Revenue

0 0 0 0 1600 1600*

100 1000 2(.5(100) 6(.5(100) 8(.5(200) 1200
+.5(0)) +.5(0)) +.5(300))

200 1500 2(.5(200) 6(.5(200) 8(.5(200) 1300
+.5(0)) +.5(0)) +.5(400))

300 2000 2(.5(100) 6(.5(100) 8(.5(200) 1200
+.5(300)) +.5(300)) +.5(400))

400 2500 2(.5(200) 6(.5(200) 8(.5(200) 1100
+.5(400)) +.5(400)) +.5(400))

I don't understand why the problem is not complete at this point. Each of the three games has an optimal order quantity???
f3(300) Computations
What does f3(300) represent?

s Order Holding Salvage Expected Total Profit
Cost Cost Value Sales Revenue

0 0 2(.5(100) 6(.5(100) 8(.5(200) 2200*
+.5(0)) +.5(0)) +.5(300))

100 1000 2(.5(200) 6(.5(200) 8(.5(200) 1800
+.5(0) +.5(0)) +.5(400))

200 1500 2(.5(100) 6(.5(100) 8(.5(200) 1700
+.5(300)) +.5(300)) +.5(400))

300 2000 2(.5(200) 6(.5(200) 8(.5(200) 1600
+.5(400)) +.5(400)) +.5(400))

f3(400) Computations What does f3(400) represent?
a

s Order Holding Salvage Expected Total Profit
Cost Cost Value Sales Revenue

0 0 2(.5(200) 6(.5(200) 8(.5(200) 2800*
+.5(0)) +.5(0)) +.5(400))

100 1000 2(.5(300) 6(.5(100) 8(.5(200) 2200
+.5(100)) +.5(300)) +.5(400))

200 1500 2(.5(200) 6(.5(200) 8(.5(200) 2100
+.5(400)) +.5(400)) +.5(400))

f2(0) Computations What does f2(0) represent?

s Order Expected Expected Expected Total Profit
Cost Holding Sales Future
Cost Revenue Profit

0 0 0 0 .5(300 + 300) 300

100 1000 0 800 .5(300 + 300) 100

200 1500 0 1600 .5(300 +300) 400

300 2000 100 2000 .5(800 + 300) 450

400 2500 200 2400 .5(300 + 1600) 650*

500 3000 400 2400 .5(800 + 2200) 500

600 3500 600 2400 .5(1600 + 2800) 500

f2(100) Computations What does f2(100) represent?

s Order Expected Expected Expected Total Profit
Cost Holding Sales Future
Cost Revenue Profit
0 0 0 800 .5(300 + 300) 1100
-------------------------------------
100 1000 0 1600 .5(300 + 300) 900
--------------------------------------
200 1500 100 2000 .5(800 + 300) 950
--------------------------------------
300 2000 200 2400 .5(300 + 1600) 1150*
--------------------------------------
400 2500 400 2400 .5(800 + 2200) 1000
--------------------------------------
500 3000 600 2400 .5(1600 + 2800) 1000
--------------------------------------

f2(200) Computations What does f2(200) represent?

s Order Expected Expected Expected Total Profit
Cost Holding Sales Future
Cost Revenue Profit
0 0 0 1600 .5(300 + 300) 1900*

100 1000 100 2000 .5(800 + 300) 1450

200 1500 200 2400 .5(300 + 1600) 1650

300 2000 400 2400 .5(800 + 2200) 1500

400 2500 600 2400 .5(1600 + 2800) 1500

f2(300) Computations What does f2(300) represent?

s Order Expected Expected Expected Total Profit
Cost Holding Sales Future
Cost Revenue Profit
0 0 100 2000 .5(800 + 300) 2450*

100 1000 200 2400 .5(300 + 1600) 2150

200 1500 400 2400 .5(800 + 2200) 2000

300 2000 600 2400 .5(1600 + 2800) 2000

f2(400) Computations What does f2(400) represent?

s Order Expected Expected Expected Total Profit
Cost Holding Sales Future
Cost Revenue Profit
0 0 200 2400 .5(300 + 1600) 3150*

100 1000 400 2400 .5(800 + 2200) 2500

200 1500 600 2400 .5(1600 + 2800) 2500

Computations for f1(0) What does f1(0) represent?

s Order Expected Expected Expected Total Profit
Cost Holding Sales Future
Cost Revenue Profit
0 0 0 0 .5(650 + 650) 650

100 1000 0 800 .5(650 + 650) 450

200 1500 0 1600 .5(650 + 650) 750

300 2000 100 2000 .5(650 + 1150) 800

400 2500 200 2400 .5(650 + 1900) 975*

500 3000 400 2400 .5(1150 + 2450) 800

600 3500 600 2400 .5(1900 + 3150) 825

To illustrate the determination of an optimal ordering policy, suppose that during game 1 400 shirts are demanded and during game 2 200 shirts are demanded. Let xt(s) be number of shirts that should be ordered before game t if s shirts are on hand before beginning of game t. Then before game 1 we order x1(0) = 400 shirts. Then before game 2 we order x2(0 + 400 400) = 400 shirts. Before game 3 we order x3(0 + 400 200) = 0 shirts.

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Solution Preview

f3(0) represents the maximum profit earned during games 3 given that

no shirts are on hand at the beginning of game 3;

Likewise, f3(100) represent the maximum profit earned during games 3

if 100 shirts are on hand at the beginning of game 3;

Therefore, f3(200) represent the maximum profit earned during games 3

if 200 shirts are on hand at the beginning of game ...

Solution Summary

Questions about an optimal policy problem are answered.

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