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.

Attached file(s):

policy problem.doc View File

Attachment Content Summary (Note: view attachment at the above link before purchasing. Actual attachment content may vary slightly from that shown below.)

policy problem.doc

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 eq \O(<,_) 400 or s eq \O(<,_) 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 eq \O(<,_) 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.

IE522 Solutions: HW#4

Page PAGE 7 of NUMPAGES 7

Questions about an optimal policy problem are answered.

What is this?

By OTA - Overall OTA Rating

Jiong Tu, PhD (IP) - 4.8/5

Purchase Cost Now

$2.19 CAD (was ~$11.97)

Included in Download

Plain text response

Why you can trust BrainMass.com

Your Information is Secure
Best Online Academic Help Service
Students find real academic Success

Linear Programming - problem 1 consider the following problem: Min x +2y s.t. x+4y<=21 2x+y>=7 3x+111.5y<=21 x,y >= 0 using the graphical method, determine the optimal solution and the optimal objective funtion va ...
Optimal Solution - (See attached file for full problem description with proper symbols) --- Consider the linear program: Min x + y St x + 2y =  x, y > 0 a) Find (with any method you'd like) ...
Optimal assignment problem for an array - How do you solve the optimal assignment problem for the following array by solving two dual problems? Also, how many optimal solutions are there for the array and why? J1 | J2 | J3 | J4 | J5 ...
Linear Programming Problem - Max 8X + 7Y s.t. 15X + 4Y < 75 10X + 7Y < 70 X + Y < 8 X, Y > 0 a.Use a graph to show each constraint and the feasible region. b.Ident ...
Proof condition that gives an optimal solution-Duality Theory - (See attached file for full problem description)

Business · Chemistry · Statistics · Physics · Computer Science · Math · Biology · Economics · Psychology · Law · Health Sciences · more

Legal Terms and Conditions · Privacy Policy · Copyright Notification Policy · Non-Payment Policy
Reference Desk · Advertise on BrainMass · Contact Us · Homework Help Solutions · Student Center