The simulation problems and solutions in operations research are mentioned below
Example : An ice-cream parlor's record of previous month’s sale of a particular variety of ice cream as follows (see Table).
Simulation of Demand Problem
Simulate the demand for first 10 days of the month
Solution: Find the probability distribution of demand by expressing the frequencies in terms of proportion. Divide each value by 30. The demand per day has the following distribution as shown in table.
Probability Distribution of Demand
Find the cumulative probability and assign a set of random number intervals to various demand levels. The probability figures are in two digits, hence we use two digit random numbers taken from a random number table. The random numbers are selected from the table from any row or column, but in a consecutive manner and random intervals are set using the cumulative probability distribution as shown in Table.
Cumulative Probability Distribution
To simulate the demand for ten days, select ten random numbers from random number tables. The random numbers selected are, 17, 46, 85, 09, 50, 58, 04, 77, 69 and 74 The first random number selected, 7 lies between the random number interval 17-49 corresponding to a demand of 5 ice-creams per day. Hence, the demand for day one is 5. Similarly, the demand for the remaining days is simulated as shown in Table.
Example : A dealer sells a particular model of washing machine for which the probability distribution of daily demand is as given in Table.
Probability Distribution of Daily Demand
The simulation problems using random numbers are given below
Find the average demand of washing machines per day.
Solution: Assign sets of two digit random numbers to demand levels as shown in Table.
Random Numbers Assigned to Demand
Ten random numbers that have been selected from random number tables are 68, 47, 92, 76, 86, 46, 16, 28, 35, 54. To find the demand for ten days see the Table below.
Table 15.7: Ten Random Numbers Selected
Average demand =28/10 =2.8 washing machines per day. The expected demand /day can be computed as,
Expected demand per day
where, pi = probability and xi = demand
= (0.05 × 0) + (0.25 × 1) + (0.20 × 2) + (0.25 × 3) + (0.1 × 4) + (0.15 × 5)
= 2.55 washing machines.
The average demand of 2.8 washing machines using ten-day simulation differs significantly when compared to the expected daily demand. If the simulation is repeated number of times, the answer would get closer to the expected daily demand.
Example : A farmer has 10 acres of agricultural land and is cultivating tomatoes on the entire land. Due to fluctuation in water availability, the yield per acre differs. The probability distribution yields are given below:
a. The farmer is interested to know the yield for the next 12 months if the same water availability exists. Simulate the average yield using the following random numbers 50, 28, 68, 36, 90, 62, 27, 50, 18, 36, 61 and 21, given in table.
b. Due to fluctuating market price, the price per kg of tomatoes varies from Rs. 5.00 to Rs. 10.00 per kg. The probability of price variations is given in the Table below. Simulate the price for next 12 months to determine the revenue per acre. Also find the average revenue per acre. Use the following random numbers 53, 74, 05, 71, 06, 49, 11, 13, 62, 69, 85 and 69.
Table for Random Number Interval for Yield
Table for Random Number Interval for Price
Simulation for 12 months period
Average revenue per acre = 21330 / 12
= Rs. 1777.50
Example : J.M Bakers has to supply only 200 pizzas every day to their outlet situated in city bazaar. The production of pizzas varies due to the availability of raw materials and labor for which the probability distribution of production by observation made is as follows:
Simulate and find the average number of pizzas produced more than the requirement and the average number of shortage of pizzas supplied to the outlet.
Solution: Assign two digit random numbers to the demand levels as shown in table
Random Numbers Assigned to the Demand Levels
Selecting 15 random numbers from random numbers table and simulate the production per day as shown in table below.
Simulation of Production Per Day
The average number of pizzas produced more than requirement
= 0.8 per day
The average number of shortage of pizzas supplied
= 0.26 per day
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Quantitative Techniques – Introduction
Measures Of Central Tendency
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Waiting Model (queuing Theory)
Theoretical Probability Distributions
Probability Distribution Of A Random Variable
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