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A Reorder problem-- Electronic Store

Module by: Paul E Pfeiffer. E-mail the author

Summary: As an example of a reorder problem we consider an electronic store which reorders a certain type of unit on a regular weekly basis. Two possible actions: 1. Order two units, at a cost of $150 each; 2. Order four units, at a cost of $120 each. An optimal demand policy, based on sales the previous week is sought.

A reorder problem

Example 1

An electronic store stocks a certain type of DVD player. At the end of each week, an order is placed for early delivery the following Monday. A maximum of four units is stocked. Let the states be the number of units on hand at the end of the sales week: E={0,1,2,3,4}E={0,1,2,3,4}. Two possible actions:

  • Order two units, at a cost of $150 each
  • Order four units, at a cost of $120 each

Units sell for $200. If demand exceeds the stock in hand, the retailer assumes a penalty of $40 per unit (in losses due to customer dissatisfaction, etc.). Because of turnover, return on sales is considered two percent per week, so that discount is α=1/1.02α=1/1.02 on a weekly basis.

In state 0, there are three possible actions: order 0, 2, or 4. In states 1 and 2 there are two possible actions: order 0 or 2. In states 3 and 4, the only action is to order 0. Customer demand in week n+1n+1 is represented by a random variable Dn+1Dn+1. The class is iid, uniformly distributed on the values 0, 1, 2, 3, 4. If Xn is the state at the end of week n, then {Xn,Dn+1}{Xn,Dn+1} is independent for each n.

Analyze the system as a Markov decision process with type 3 gains, depending upon current state, action, and demand. Determine the transition probability matrix PA (properly padded) and the gain matrix (also padded). Sample calculations are as follows:

State 0, action 0: p00(0)=1(all otherp0k(0)=0 State 0, action 2:p00(2)=P(D2)=3/5,p01(2)=P(D=1)=1/5, etc. State 2, action 2: p2j(k)=1/5,k=0,1,2,3,4 State 0, action 0: p00(0)=1(all otherp0k(0)=0 State 0, action 2:p00(2)=P(D2)=3/5,p01(2)=P(D=1)=1/5, etc. State 2, action 2: p2j(k)=1/5,k=0,1,2,3,4

For state = i, action = a, and demand = k, we seek g(i,a,k)g(i,a,k)

g(0,0,k)=-40k0 -40 -80 -120 -160 g(0,2,k)=-300+200min{k,2}-40max{k-2,0}-300 -100 100 60 20 g(0,4,k)=-480+200k-480 -280 -80 120 320 g(0,0,k)=-40k0 -40 -80 -120 -160 g(0,2,k)=-300+200min{k,2}-40max{k-2,0}-300 -100 100 60 20 g(0,4,k)=-480+200k-480 -280 -80 120 320

  1. Complete the transition probability table and the gain table.
  2. Determine an optimum infinite-horizon strategy with no discounting.
  3. Determine an optimum infinite-horizon strateby with discounting (alpha = 1/1.02).
  4. The manager decides to set up a six-week strategy, after which new sales conditions may be established. Determine an optimum strategy for the six-week period.

Data file

% file orderdata.m
% Version of 4/5/94
% Data organized for computation
type = 3;
states = 0:4;
= [0  2  4 ...  % Actions (padded)
 0  2 02 ...
 0  2 02 ...
 0 00 00 ...
 0 00 00];
C  = [0 -300 -480 ... % Order costs (padded)
0 -300 -300 ...
0 -300 -300 ...
0  0  0 ...
0  0  0];
SP = 200; % Selling price
BP = 40;  % Backorder penalty
PD = 0.2*ones(1,5);  % Demand probabilities

Transition Probabilities and Gains

The procedure

% file reorder.m
% Version of 4/11/94
% Calculates PA and GA for reorder policy
states = input('Enter row vector of states ');
A = input('Enter row vector A of actions (padded)  ');
C = input('Enter row vector C of order costs (padded) ');
D = input('Enter row vector D of demand values ');
PD = input('Enter row vector PD of demand probabilities ');
SP = input('Enter unit selling price SP ');
BP = input('Enter backorder penalty cost BP ');
m = length(states');
q = length(A);
na = q/m;
N = length(D);
S = ones(na,1)*states;
S = S(:)';
[d,s] = meshgrid(D,S);
a = A'*ones(1,N);
ca = C'*ones(1,N);
TA = (s + a - d).*(s + a - d >= 0);
for i = 1:q
PA(i,:) = tdbn(states,TA(i,:),PD);
end
PA
GA = ca + SP*d - (SP + BP)*(d -s -a).*(d > s+a)

The calculations

orderdata
reorder
Enter row vector of states states
Enter row vector A of actions (padded) A
Enter row vector C of order costs (padded) C
Enter row vector D of demand values D
Enter row vector PD of demand probabilities PD
Enter unit selling price SP SP
Enter backorder penalty cost BP BP
PA =
1.0000  0       0       0       0
0.6000  0.2000  0.2000  0       0
0.2000  0.2000  0.2000  0.2000  0.2000
0.8000  0.2000  0       0       0
0.4000  0.2000  0.2000  0.2000  0
0.4000  0.2000  0.2000  0.2000  0
0.6000  0.2000  0.2000  0       0
0.2000  0.2000  0.2000  0.2000  0.2000
0.2000  0.2000  0.2000  0.2000  0.2000
0.4000  0.2000  0.2000  0.2000  0
0.4000  0.2000  0.2000  0.2000  0
0.4000  0.2000  0.2000  0.2000  0
0.2000  0.2000  0.2000  0.2000  0.2000
0.2000  0.2000  0.2000  0.2000  0.2000
0.2000  0.2000  0.2000  0.2000  0.2000
GA =
 0 -40 -80 -120 -160
 -300 -100 100 60 20
 -480 -280 -80 120 320
 0 200 160  120  80
-300 -100 100 300 260
-300 -100 100 300 260
0     200 400 360 320
-300 -100 100 300 500
-300 -100 100 300 500
0     200 400 600 560
0     200 400 600 560
0     200 400 600 560
0     200 400 600 800
0     200 400 600 800
0     200 400 600 800

Infinite-horizon strategy (no discounting)

polit
Data needed:
- - - - - - - - - - - - - - -
Enter case number to show gain type case
Enter row vector of states states
Enter row vector A of possible actions A
Enter value of alpha (= 1 for no discounting) 1
Enter matrix PA of transition probabilities PA
Enter matrix GA of gains GA
Enter row vector PD of demand probabilities PD
Index Action Value
0 -80
2 2 -44
3 4 -80
4 0 112
5 2 52
6 2 52
7 0 256
8 2 100
9 2 100
10 0 352
11 0 352
12 0 352
13 0 400
14 0 400
15 0 400
Initial policy: action numbers
2 1 1 1 1
Policy: actions
2 0 0 0 0
New policy: action numbers
3 2 2 1 1
Policy: actions
4 2 2 0 0
 
Long-run distribution
0.2800 0.2000 0.2000 0.2000 0.1200
Test values for selecting new policy
Index Action Test Value
1.0000    0         -248.0000
2.0000    2.0000    -168.8000
3.0000    4.0000    -41.6000
4.0000    0         -48.8000
5.0000    2.0000    -5.6000
6.0000    2.0000    -5.6000
7.0000    0         131.2000
8.0000    2.0000    138.4000
9.0000    2.0000    138.4000
10.0000   0         294.4000
11.0000   0         294.4000
12.0000   0         294.4000
13.0000   0         438.4000
14.0000   0         438.4000
15.0000   0         438.4000
Optimum policy
State Action Value
0         4.0000    -168.0000
1.0000    2.0000    -132.0000
2.0000    2.0000    12.0000
3.0000    0         168.0000
4.0000    0         312.0000
Long-run expected gain per period G
126.4000

Infinite-horizon strategy (with discounting)

polit
Data needed:
- - - - - - - - - - - - - - -
Enter type number to show gain type
Enter row vector of states states
Enter row vector A of possible actions A
Enter value of alpha (= 1 for no discounting) 1/1.02
Enter matrix PA of transition probabilities PA
Enter matrix GA of gains GA
Enter row vector PD of demand probabilities PD
Index Action Value
1 0 -80
2 -44
3 -80
4 0 112
5 2 52
6 2 52
7 0 256
8 2 100
9 2 100
10 0 352
11 0 352
12 0 352
13 0 400
14 0 400
15 0 400
Initial policy: action numbers
2 1 1 1 1
Policy: actions
2 0 0 0 0
 
New policy: action numbers
3 2 2 1 1
Policy: actions
4 2 2 0 0
Test values for selecting policy
Index Action Test Value
1.0e+03 *
0.0010 0  6.0746
0.0020 0.0020 6.1533
0.0030 0.0040 6.2776
0.0040 0      6.2740
0.0050 0.0020 6.3155
0.0060 0.0020 6.3155
0.0070 0      6.4533
0.0080 0.0020 6.4576
0.0090 0.0020 6.4576
0.0100 0      6.6155
0.0110 0      6.6155
0.0120 0      6.6155
0.0130 0      6.7576
0.0140 0      6.7576
0.0150 0      6.7576
Optimum policy
State Action Value
1.0e+03 *
0      0.0040 6.2776
0.0010 0.0020 6.3155
0.0020 0.0020 6.4576
0.0030 0      6.6155
0.0040 0      6.7576

Finite-horizon calculations

dpinit
Initialize for finite horizon calculations
Matrices A, PA, and GA, padded if necessary
Enter case number to show gain type case
Enter vector of states states
Enter row vector A of possible actions A
Enter matrix PA of transition probabilities PA
Enter matrix GA of gains GA
Enter row vector PD of demand probabilities  PD
Call for dprog
dprog
States and expected total gains
0    1    2    3    4
-44  112  256  352  400
States Actions
0  2
1  0
2  0
3  0
4  0
dprog
States and expected total gains
0          1.0000    2.0000    3.0000   4.0000
135.2000   178.4000  315.2000  478.4000 615.2000
States Actions
0  4
1  2
2  2
3  0
4  0
dprog
States and expected total gains
0          1.0000     2.0000      3.0000     4.0000
264.4800   300.4800   444.4800    600.4800   744.4800
States Actions
0  4
1  2
2  2
3  0
4  0
dprog
States and expected total gains
1.0000  2.0000  3.0000  4.0000
390.8800 426.8800 570.8800 726.8800  870.8800
States  Actions
    0     4
    1     2
    2     2
    3     0
    4     0
dprog
States and expected total gains
        0    1.0000    2.0000    3.0000    4.0000
 517.2800  553.2800  697.2800  853.2800  997.2800
States  Actions
    0     4
    1     2
    2     2
    3     0
    4     0
dprog
States and expected total gains
  1.0e+03 *
        0    0.0010    0.0020    0.0030    0.0040
   0.6437    0.6797    0.8237    0.9797    1.1237
States  Actions
    0     4
    1     2
    2     2
    3     0
    4     0

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