System Analysis System set of interdependent elements that interact in order to accomplish a one or more final outcomes. Constrained and affected by:

System Analysis

System Analysis• System – set of interdependent elements that interact in

order to accomplish a one or more final outcomes.

• Constrained and affected by:• Endogenous factors – internal limits on capabilities of

elements (resources), interactions, etc.

• Exogenous factors – outside (“environmental”) demands, supply, prices, etc.

• Need ways to anticipate behavior of system in order to make informed decisions.

System Analysis

System

Experiment w/ Actual System

Experiment w/ Model of System

Physical Model

Mathematical Model

SimulationAnalytical Solution

Adapted from: Law, A. M. (2007), Simulation Modeling and Analysis (4th ed.), McGraw-Hill, New York.

Our focus!

Simulation

Simulation• Analytic Model – mathematical representation of a

physical system.• Discrete or continuous• Dynamic or static• Deterministic or stochastic (random)

• Simulation – “running” analytical model to study operation of physical system.• May use exact model or an approximation• Usually computer driven

Simulation• What are some advantages of simulation?

• Does not affect system itself

• Can be used before system is implemented

• May reveal critical interaction while designing

• Provides solution when model is infeasible or intractable

• Allows for compression/expansion of time

• Possible to vary conditions

• Useful for “selling” idea or concept

• May be inexpensive (or may not)

Simulation• What are some disadvantages of simulation?

• Requires programmers/computers

• May be time consuming

• Can be abused and/or misunderstood

• Garbage In – Garbage Out

Simulation Process

Define Model

Adapted from: Law, A. M. (2007), Simulation Modeling and Analysis (4th ed.), McGraw-Hill, New York.

Collect Data

Assump. Valid?

Verify w/ Test Data

Program Simulation

Sim. Valid?

No

Yes

No

Yes

Design Experiments

Execute Runs

Analyze & Report

Monte Carlo Simulation

Monte Carlo Simulation• Traditionally, uncertain (random) values were often

replaced by their means.• Missed critical consequences of high/low values

• Monte Carlo Simulation – uses (pseudo) random numbers to vary uncertain inputs/assumptions.• Useful for risk analysis• Randomly samples from known distributions • In Excel, we utilize =rand( ) function.

• Generates value in (0,1)• Used as probability w/ CDF

Monte Carlo SimulationSimple example:

Each bulb produced on a line has a 5% chance of being defective.

A single bulb may be simulated in Excel as: =if(rand()<0.05,1,0)

Monte Carlo SimulationSimple example:

10rand()

10

x0.5

Monte Carlo Simulation

Note: your numbers will be different …It’s the nature of random values!

Monte Carlo Simulation

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Another example:

Monte Carlo Simulation

Weekly demand for bulbs is (approx.) normally distributed with mean of 100 and std. dev. of 20.

A single day’s demand may be simulated in Excel as: =norminv(rand(),100,20)

Note: for integer values, can use =round(norminv(rand(),100,20),0)

Monte Carlo Simulation

Let’s look at a “system”:

Monte Carlo Simulation

Weekly demand for bulbs is distributed as N(100,20).

Each week, we refill stock up to 105 bulbs.

How many do we sell?

# Sold = min{Demand, 105}

Monte Carlo Simulation

And the standard deviation using s.

But also …

Plus, create confidence intervals.

Monte Carlo Simulation

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Monte Carlo Simulation

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Frequency

And another layer:

Monte Carlo Simulation

Weekly demand for bulbs is distributed as N(100,20).Each week, we refill stock up to 105 bulbs.Each bulb has a 5% chance of being defective.How many do we sell?

(We can use the =binom.inv() function to get a random # defects.)

# Sold = min{Demand, 105 – # Defects }

Monte Carlo Simulation

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Monte Carlo Simulation• Additional questions:

• What if defect rate is not known w/ certainty?

• What is optimal stocking level?

• What if we have two (or more) types of bulbs to stock in a fixed space, each with its own costs, demands and defect rates?

We will need special software (e.g., Crystal Ball, @Risk) or to learn how to program in VBA!

Optimization

Optimization Process

Source: Gosavi, A. (2003), Simulation Based Optimization, Kluwer Academic Publishers, The Netherlands.

Simulator

• Simulator used to value current choice• Optimizer moves choice in “better” direction

Optimization Example• Consider the machine maintenance decision

• Simulator: given distribution of (random) time between breakdowns, cost of breakdown, current maintenance frequency and cost of maintenance, determines average cost.

• Optimizer: improves choice of maintenance frequency until stopping condition is reached.

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Optimization Process• Example of optimizer step:

Simulator provides estimated values (V) for xn – hn and xn + hn.

Slope is estimated by: V(xn – hn) – V(xn + hn)

2hn

If current choice is xn.

dn =

Choice is improved: xn+1 = xn + andn

(Note: an and hn get smaller as process continues.)

V

xn xn+1

V(xn + hn)V(xn – hn)