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887654Decision Support and Intelligent Systems

Monte Carlo Simulation

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Monte Carlo

Monte Carlo is a technique for selecting numbersrandomly 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 theoperation of a gambling casino in Monaco.

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Case Study: Computer World Company

The manager of Computer World Company is attemptingto 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 eachweek and the average weekly revenue generated from thesale of laptop PCs.

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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 100weeks. From this frequency distribution, a probabilitydistribution of demand can be developed.

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Probability distribution of demand for laptop PCsPCs Demanded perWeek Frequency of Demand Probability of Demand,P(x)

0 20 .201 40 .402 20 .203 10 .104 10 .10

100 1.00

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A roulette wheel for demand

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Number roulette wheel

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Generating demand from random number

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Randomly generated demand for 15 weeksWeek r Demand, x Revenue ($)1 39 1 4,3002 73 2 8,6003 72 2 8,6004 75 2 8,6005 37 1 4,3006 02 0 07 87 3 12,9008 98 4 17,2009 10 0 010 47 1 4,30011 93 4 17,20012 21 1 4,30013 95 4 17,20014 97 4 17,20015 69 2 8,600

Sum = 31 $133,300

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Compute the estimated average weeklydemand and revenue

The manager can compute the estimated average weeklydemand and revenue.

Estimated average demand = 3115

= 2.07 laptop PCs per week

Estimated average revenue = $133,300 15= $8,886.67

The manager can then use this information to help

determine the number of PCs to order each week.

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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.

()=1

where = demand value 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

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Calculate Analytically vs. Simulation

Simulation results will not equal analytical results unlessenough 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 thatsimulation is generally useful).

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Computer simulation with Excel

Random numbers generated by a mathematical processinstead 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 inExcel by entering the formula =RAND( ).

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Computer World Store using Excel

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Simulate demand for 100 weeks

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Decision Making with Simulation

From the simulation, the manager knows that the averageweekly demand for laptop PCs will be approximately

1.49.

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.

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Decision Making with Simulation

If too few laptops are on hand to meet demand duringthe 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 eachweek 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 greatestaverage weekly revenue.

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Scenario 1: The manager order one laptop

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Scenario 2: The manager order two laptops

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Make a decision

If the manager order one laptop (scenario 1), the totalweekly revenue is $3,875.

If the manager order two laptops (scenario 2), the total

weekly revenue is $4,927.50.

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.