product pricing through simulationhaving studied

Question

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