Product Pricing Through Simulation
Having studied a simpler problem, let us revert to our earlier illustration regarding fixing a price.
Let us suppose that we want to simulate the strategy of pricing two rupees above competition in our illustration.
1. From a table of uniform random numbers over the interval 0 to 1, select a random number x. Suppose x = 0.32 in the first trial.
2. Value of the competition price Pc is given by the following equation:
Pc = 4x + 8
The reason why we define Pc = 4x + 8 is that according to the data given in our illustration P_{c} is uniformly distributed between 8 and 12 while x lies between 0 and 1. So, to translate a random number from the unit interval 0 to 1 into a uniform random number between 8 and 12, we must add 4x to 8 which is the lower limit.
P_{c} = 4x + 8 = [4(0.32) + 8]
= 9.28
3. Firm's price, since it should be two rupees above the competitors' price, it will be determined as
9.28 + 2 = 11.28
4. Mean sales volume Qm is obtained from,
Qm = 18,000  500P + 100P_{c}
Q_{m}= 18,000  500(11.28) + 100(9.28)
= 13,288
5. To arrive at the actual sales volume, we must apply a deviation value to the mean value obtained above. We know that the standard deviation of the distribution of quantity sold is 2,000. To obtain the deviation value, we have to randomly assume the number of standard deviations. The actual value is away from the mean. We consult a table of random normal numbers with mean zero and standard deviation 1. Suppose the standard random normal number z =  0.75 for the first trial.
6. The actual sales volume + Q
= 2000Z + 13,288
= 2000 ( 0.75) + 13,288
= 11,788
7. The sales revenue = 11.28 x 11,788
= Rs.1,32,968.64
8. We must try to take a value for the unit cost. We know that the mean unit cost is 7 and that the actual cost is normally distributed around this mean with a standard deviation of Re.1. As in the case of the sales quantity, we must assume a figure for the number of standard deviations the actual value is away from the mean.
Let us assume Z = 0.52 in the first trial.
9. The actual unit cost will be,
C = 1 x 0.52 + 7 = 7.52
10. The total cost will be,
7.52 x 11,788 = Rs.88,645.76
11. The profit will be,
PQ  CQ = Rs.(1,32,968.64  88,645.76)
= Rs.44322.88
Now we have completed a simulated trial or a 'run'. Steps 1 to 11 will have to be repeated again and again with other randomly selected values. Suppose this is done 500 times. For each of these runs a profit value would be obtained and the average of the 500 profit values would represent the expected value of the profit from adopting the strategy of pricing the product two rupees above the competitors' average price. To complete the simulation similar trials would have to be conducted for the other two pricing strategies as well. Then the firm can choose the strategy which yields the highest expected profit.
In some situations the variance of the performance criterion is also of interest in addition to its expected value. The decision maker might be willing to sacrifice some expected gain if he can have less variance (hence less risk) in the gain. In the above example, the variance of profit from following the high price strategy can be estimated simply by calculating the sample variance in the 500 profit values from the 500 runs.
As you might have guessed, the quality of estimates of expected profit and variance of profit improves as the number of runs increases.
Since the approach involves large numbers of simple repetitive calculations, computers are used to conduct large simulation runs.
Let us now consider one more problem.
Zenith Limited is planning to introduce a new product. The following data has been collected during the feasibility study.
The variable cost per unit, total fixed expenses, selling price per unit, sales volume, and the probabilities associated with these random variables are as follows:
Variable Cost
Variable cost per unit Rs.

Probability

4.50
4.80
5.00
5.20
5.80

0.10
0.25
0.30
0.20
0.15

Fixed Expenses (Total)
Fixed Expenses

Probability

10,000
12,000
15,000
20,000
22,000

0.10
0.15
0.25
0.30
0.20

Sales price per unit and sales volume
Sales price per unit Rs.

Sales volume

Probability

7.00
7.30
8.00
8.50
9.00

15,000
14,800
14,500
14,000
13,000

0.15
0.25
0.30
0.20
0.10

From the above data, simulate 15 trials and calculate the expected profit. Use random numbers from the table of random digits given. Assume that there is a fixed sales volume for each of the sales prices. In other words the sales volume need not be considered as a separate random variable.
Construction of cumulative probability distribution and assignment of random numbers:
Variable Cost
Variable cost per unit Rs.

Probability

Cumulative probability

Random Nos. allotted

4.50
4.80
5.00
5.20
5.80

0.10
0.25
0.30
0.20
0.15

0.10
0.35
0.65
0.85
1.00

00 to 09
10 to 34
35 to 64
65 to 84
85 to 99

Fixed Expenses
Total fixed expenses

Probability

Cumulative probability

Random Nos. allotted

10,000
12,000
15,000
20,000
22,000

0.10
0.15
0.25
0.30
0.20

0.10
0.25
0.50
0.80
1.00

00 to 09
10 to 24
25 to 49
50 to 79
80 to 99

Sales Price Per Unit
Selling Price per unit

Probability

Cumulative probability

Random Nos. allotted

7.00
7.30
8.00
8.50
9.00

0.15
0.25
0.30
0.20
0.10

0.15
0.40
0.70
0.90
1.00

00 to 14
15 to 39
40 to 69
70 to 89
90 to 99

(It is given that there is a sales volume corresponding to each sales price and therefore the sales volume need not be considered as a separate variable.)
Total net profit = 381340
Expected profit = 381340/15 = Rs.25,422.67
Note
Total Cost = Total Fixed Expenses + Total Variable Cost
Net Profit = Total Sales Value  Total Cost
We can use random numbers in any order. Here let us consider them horizontally from left to right. We will use a different set of random numbers for selling price per unit, variable cost per unit and the total fixed expenses.
Simulation based on 15 Trials
Trial no.

Random no. generated

Selling price per unit

Sales volume

Total sales value

Random no. generated

Variable cost per unit

Total variable cost

Random no. generated

Total fixed expenses

Total cost

Net profit

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

1

97

9.00

13000

117000

58

5.00

65000

66

20000

85000

32000

2

95

9.00

13000

117000

39

5.00

65000

24

12000

77000

40000

3

12

7.00

15000

105000

22

4.80

72000

72

20000

92000

13000

4

11

7.00

15000

105000

13

4.80

72000

57

20000

92000

13000

5

90

9.00

13000

117000

02

4.50

58500

32

15000

73500

43500

6

49

8.00

14500

116000

80

5.20

75400

15

12000

87400

28600

7

57

8.00

14500

116000

67

5.20

75400

49

15000

90400

25600

8

13

7.00

15000

105000

14

4.80

72000

63

20000

92000

13000

9

86

8.50

14000

119000

99

5.80

81200

00

10000

91200

27800

10

81

8.50

14000

119000

16

4.80

67200

04

10000

77200

41800

11

02

7.00

15000

105000

89

5.80

87000

96

22000

10900

() 4000

12

92

9.00

13000

117000

96

5.80

75400

76

20000

95400

21600

13

75

8.50

14000

119000

63

5.00

70,000

20

12000

82000

37000

14

91

9.00

13000

117000

67

5.20

67600

26

15000

82600

34400

15

24

7.30

14800

108040

60

5.00

74000

72

20000

94000

14040

