Monte-Carlo Simulation
Let us, for a shortwhile, leave the illustration for determining the price and consider a simpler illustration for understanding the Monte-Carlo method of simulation.
Example
A dealer in refrigerators wants to use a scientific method to reduce his investment in stock. The daily demand for a refrigerator is random and varies from day to day in an unpredictable pattern. From the past sales records, the dealer has been able to establish a probability distribution of the demand as given below:
|
Daily demand (units)
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
|
Probability
|
0.06
|
0.14
|
0.18
|
0.17
|
0.16
|
0.12
|
0.08
|
0.06
|
0.03
|
|
The dealer also knows from his past experience that the lead time is almost fixed at 5 days. The dealer would like to study the implications of a possible inventory policy of ordering 30 units, whenever the inventory at the end of the day is 20 units. The inventory on hand is 30 units and the simulation can be run for 25 days. Use the following random numbers.
|
Random Numbers
|
03
|
38
|
17
|
32
|
69
|
24
|
61
|
30
|
03
|
48
|
88
|
71
|
27
|
80
|
33
|
90
|
78
|
55
|
87
|
16
|
34
|
45
|
59
|
20
|
59
|
|
When we conduct simulation runs, we use random numbers to simulate the actual demand. How do we assign, say, two digit random numbers chosen for a particular demand and also take into account the probabilities known? This is done by calculating the cumulative probabilities at each level of demand as shown below:
|
Daily Demand (units)
|
Probability
|
Cumulative Probability
|
Random numbers allotted
|
|
2
3
4
5
6
7
8
9
10
|
0.06
0.14
0.18
0.17
0.16
0. 2
0.08
0.06
0.03
|
0.06
0.20
0.38
0.55
0.71
0.83
0.91
0.97
1.00
|
00 - 05
06 - 19
20 - 37
38 - 54
55 - 70
71 - 82
83 - 90
91 - 96
97 - 99
|
The random numbers have been allotted on the basis of the following logic. Looking at the cumulative probabilities we can say that a number between 0 and 5, or to be exact, the numbers 0, 1, 2, 3, 4 and 5 (six numbers in all) signify a demand level of 2 units. Similarly, the random numbers 6 to 19 (i.e. 14 numbers) correspond to the demand level of 3 units and so on. The result of simulation trials conducted for 25 days is tabulated below:
|
Day
|
Random no. generated
|
Inventory at the beginning of the day(units)
|
Daily demand (units)
|
Inventory at the end of the day (units)
|
Lost sales (units)
|
Stocks received
|
Qty. ordered
|
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
|
1
|
03
|
30
|
2
|
28
|
-
|
-
|
-
|
|
2
|
38
|
28
|
5
|
23
|
-
|
-
|
-
|
|
3
|
17
|
23
|
3
|
20
|
-
|
-
|
30
|
|
4
|
32
|
20
|
4
|
16
|
-
|
-
|
-
|
|
5
|
69
|
16
|
6
|
10
|
-
|
-
|
-
|
|
6
|
24
|
10
|
4
|
6
|
-
|
-
|
-
|
|
7
|
61
|
6
|
6
|
0
|
-
|
-
|
-
|
|
8
|
30
|
0
|
4
|
0
|
4
|
30
|
-
|
|
9
|
03
|
30
|
2
|
28
|
-
|
-
|
-
|
|
10
|
48
|
28
|
5
|
23
|
-
|
-
|
-
|
|
11
|
88
|
23
|
8
|
15
|
-
|
-
|
30
|
|
12
|
71
|
15
|
7
|
8
|
-
|
-
|
-
|
|
13
|
27
|
8
|
4
|
4
|
-
|
-
|
-
|
|
14
|
80
|
4
|
7
|
0
|
3
|
-
|
-
|
|
15
|
33
|
0
|
4
|
0
|
4
|
-
|
-
|
|
16
|
90
|
0
|
8
|
0
|
8
|
30
|
-
|
|
17
|
78
|
30
|
7
|
23
|
-
|
-
|
-
|
|
18
|
55
|
23
|
6
|
17
|
-
|
-
|
30
|
|
19
|
87
|
17
|
8
|
9
|
-
|
-
|
-
|
|
20
|
16
|
9
|
3
|
6
|
-
|
-
|
-
|
|
21
|
34
|
6
|
4
|
2
|
-
|
-
|
-
|
|
22
|
45
|
2
|
5
|
0
|
3
|
-
|
-
|
|
23
|
59
|
0
|
6
|
0
|
6
|
30
|
-
|
|
24
|
20
|
30
|
4
|
26
|
-
|
-
|
-
|
|
25
|
59
|
26
|
6
|
20
|
-
|
-
|
30
|
Column 2 of the table indicates the series of random numbers drawn from a random number table. The demand corresponding to the random number has been listed in column 4. Though the table contains the stock position, sales lost, quantities received and an order for each trial, how do we evaluate the financial implication of the inventory policy which has fixed the reorder point at 20 units and the ordering quantity at 30 units? To do this, we would have to gather details regarding ordering cost, carrying costs and storage costs and determine the total cost. The policy could then be varied and the total cost determined for alternative policies through simulation. The most acceptable policy would be the one that shows the least total cost (an alternative method would be to compare the average total cost for 25 days). Even without assigning any costs, we can observe from the table that the policy of ordering 30 units whenever stock falls to 20 units is not desirable as quite a number of lost sales units have arisen over a short period of 25 days.