Decision Support and Intelligent Systems. Monte Carlo Simulation

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1 Decision Support and Intelligent Systems Monte Carlo Simulation

2 Monte Carlo Monte Carlo is a technique for selecting numbers randomly from a probability distribution. A mathematical process used within a simulation. The name Monte Carlo is appropriate because the basic principle behind the process is the same as in the operation of a gambling casino in Monaco.

3 Case Study: Computer World Company The manager of Computer World Company is attempting to determine how many laptop PCs the store should order each week. A primary consideration in this decision is the average number of laptop computers that the store will sell each week and the average weekly revenue generated from the sale of laptop PCs.

4 Case Study: Computer World Company A laptop sells for $4,300. The number of laptops demanded each week is a random variable (x), that ranges from 0 to 4. From past sale records, the manager has determined the frequency of demand for laptop PCs for the past 100 weeks. From this frequency distribution, a probability distribution of demand can be developed.

5 Probability distribution of demand for laptop PCs PCs Demanded per Week Frequency of Demand Probability of Demand, P(x)

6 A roulette wheel for demand

7 Number roulette wheel

8 Generating demand from random number

9 Randomly generated demand for 15 weeks Week r Demand, x Revenue ($) , , , , , , , , , , , , ,600 Sum = 31 $133,300

10 Compute the estimated average weekly demand and revenue The manager can compute the estimated average weekly demand and revenue. Estimated average demand = = 2.07 laptop PCs per week Estimated average revenue = $133, = $8, The manager can then use this information to help determine the number of PCs to order each week.

11 Calculate Analytically vs. Simulation Although this example is convenient for illustrating how simulation works, the average demand could have more appropriately been calculated analytically using formula for expected value. E x = n i=1 P(x i )x i where x i = demand value i P(x i ) = probability of demand n = the number of different demand values Therefore, E(x) = (.20)(0) + (.40)(1) + (.20)(2) + (.10)(3) + (.10)(4) = 1.5 PCs per week

12 Calculate Analytically vs. Simulation Simulation results will not equal analytical results unless enough trials of the simulation have been conducted to reach steady state. It is difficult to validate that the results of a simulation truly replicate reality. Simulation most often is employed whenever analytical analysis is not possible (this is one of the reasons that simulation is generally useful).

13 Computer simulation with Excel Random numbers generated by a mathematical process instead if physical process are pseudo-random number. A table of random numbers must be uniform, efficiently generated, and absent of patterns. Random numbers between 0 and 1 can be generated in Excel by entering the formula =RAND( ).

14 Computer World Store using Excel

15 Simulate demand for 100 weeks

16 Decision Making with Simulation From the simulation, the manager knows that the average weekly demand for laptop PCs will be approximately However, the manager cannot order 1.49 laptop PCs each week (because fractional laptops are not possible). Thus, the manager wants to include some additional information in the model that will affect the decision.

17 Decision Making with Simulation If too few laptops are on hand to meet demand during the week, then not only will there be a loss of revenue, but there will also be shortage cost of $500 per unit incurred because the customer will be unhappy. However, each laptop still in stock at the end of each week that has not been sold will incur an inventory or storage cost of $50. The manger wants to order either one or two laptops, depending on which order size will result in the greatest average weekly revenue.

18 Scenario 1: The manager order one laptop

19 Scenario 2: The manager order two laptops

20 Make a decision If the manager order one laptop (scenario 1), the total weekly revenue is $3,875. If the manager order two laptops (scenario 2), the total weekly revenue is $4, Thus, the correct decision, based on weekly revenue, would be to order two laptops per week. In fact, one of the main attributes of simulation is its usefulness as a model for experimenting, called what-if? analysis.