Supplement CIntroduction to Simulation

Operations Managementby

R. Dan Reid & Nada R. Sanders2nd Edition © Wiley 2005

PowerPoint Presentation byRoger B. Grinde, University of New Hampshire

Learning Objectives

Explain why simulation is a valuable tool for decision making.

Define the steps involved in the simulation modeling process

Generate random numbers from various distributions in Excel

Develop and run a simulation model in Excel Analyze the results from a simulation in

Excel

Computer Simulation A model that mimics what might happen in reality. Examples

Weather forecasting Rocket simulators Military war-game simulations

Business Examples Capital projects Business process redesign Production process analysis Service system analysis

Computer Simulation (continued)

Much of the time, uncertainty is present in the system we wish to study.

Simulation provides a way to directly model the uncertainty and/or dynamic behavior of the system.

Monte-Carlo simulation is the focus of this chapter. It focuses on assessing the uncertainty and risk of a particular situation or decision.

Discrete-Event simulation is another major branch of simulation. It focuses on studying the dynamic behavior of systems as they operate over time.

Monte-Carlo Simulation Schematic

Some inputs are fixed, or known with certainty (e.g., the price of our product).

Some inputs are uncertain, or random (e.g., the demand for our product at a given price).

We may have decision variables (e.g., quantity to produce). Since some of the inputs are random, the output of the simulation is

also random.

Simulation Modeling Process

Develop a deterministic spreadsheet model. Determine the appropriate probability

distributions and parameters to use for the random inputs.

Modify the deterministic model by incorporating the random inputs.

Re-calculate the model many times (each calculation is called a replication or trial).

Analyze the probability distribution of the output using statistical concepts.

Probability Distributions Probability Distributions are used to model the random inputs. You’re already probably familiar with some probability

distributions, even if you don’t know their names. Normal (“bell” curve) Bernoulli (e.g., flip of a coin) Discrete Uniform (e.g., roll of a die)

A very important part of simulation is modeling the random behavior of a situation using probability distributions.

As noted before, since some of the inputs are random, the output of a simulation is also random.

However, the probability distribution of the output of a simulation does not necessarily look like one of the standard probability distributions.

The next few slides show how we can generate random values in Excel from several different probability distributions.

Random Number Generation In Excel

Key to random number generation is generating random values that are “uniformly” distributed between 0 and 1.

It turns out that if we can do this, then we can transform this value into a sample value from any probability distribution.

“Uniformly” distributed simply means that any value in between 0 and 1 is equally likely.

Fortunately, Excel has a built-in function, =RAND(), which does exactly this.

=RAND() (empty parentheses are required!) returns a value between 0 and 1.

If you enter this function in a cell, and then copy it to some other cells, all the values will be different!

Also, if you hit the F9 key (which recalculates the worksheet), the values of cells with =RAND() in them will change!

This is simulation at work. Just like in real life, some things are uncertain (e.g., commuting time, waiting time at the food court).

=RAND()

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A B C D E50 Random Numbers between 0 and 1Using =RAND() function

0.9945 0.5342 0.3149 0.6358 0.53340.3848 0.5566 0.8701 0.2780 0.38220.4480 0.6078 0.5301 0.8625 0.39560.3980 0.1668 0.2904 0.8226 0.86380.5517 0.6018 0.8232 0.5490 0.95840.8009 0.2063 0.8286 0.0530 0.96140.2333 0.8206 0.5036 0.3716 0.18580.4665 0.4733 0.9441 0.3001 0.82610.8273 0.6341 0.3681 0.0971 0.54930.1276 0.2094 0.9277 0.7601 0.9193

A4: =RAND()

=RAND() entered in one cell, then copied.

All values are different.

If you do this, your values will be different too.

Hit F9 to re-calculate all 50 values.

RAND() produces a “U(0,1)” random number.

How “Uniform” are the values from RAND?

Histogram(50 Random Values from =RAND() function)

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Histogram(5000 Random Values from =RAND() function)

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As more values are generated, histograms become more uniform.

Just as in sampling, a bigger sample brings more precision.

Bernoulli Distribution Bernoulli Distribution: 2 outcomes

Example: Flip of a coin How to convert a value from RAND() to a

“heads” or a “tails”? Equal probability =IF(RAND()<0.5, “heads”, “tails”) Nothing special about the 0.5…could

simulate an unfair coin, or defective/non-defective parts, complaining or non-complaining customers, etc.

Bernoulli Distribution Example

Formulas entered in row 4, then copied down to simulate 100 coin flips.

Some rows hidden here.

Count up number of heads, tails.

What would happen if we were to re-calculate this spreadsheet?

What would happen if we were to simulate 10,000 coin flips rather than 100?

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Coin Toss Simulation

U(0,1) H/T0.010 Heads0.031 Heads0.908 Tails0.936 Tails0.196 Heads Number Heads 430.667 Tails Number Tails 570.713 Tails Total 1000.355 Heads0.151 Heads0.757 Tails

A4: =RAND()

B4: =IF(A4<0.5,"Heads","Tails")

E8: =COUNTIF(B$4:B$103,"Heads")

E9: =COUNTIF(B$4:B$103,"Tails")

Discrete Uniform Distribution

Finite number of outcomes, each with the same probability. Coin flip is a simple example of this. Roll of a die is a more complex example. Practical example: selecting someone at random

from a known number of entries. Implementation

We could use an IF statement as before, but this becomes difficult because of the many possible outcomes. We would have to “nest” IF statements within others, and there is an Excel limit on this.

A better way is to use the VLOOKUP function in Excel.

Discrete Uniform: Roll a Die

Key: Divide up the range between 0 and 1 into 6 equal intervals. Assign the possible die rolls (1, 2, …, 6) to each one of these intervals.

Table in D5:G12 sets up the intervals.

Column A generates a U(0,1) value.

Column B converts the U(0,1) value into a die roll by looking up the U(0,1) value in the table to see which interval it falls into. Then it returns the corresponding die roll.

Results section tallies up the results from 100 die rolls.

Is there anything special about the equal probabilities? Could we simulate an “unfair” die just by changing the probabilities in D7:D12?

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Die Roll Simulation

U(0,1) Roll0.607 40.127 1 Cumulative Probability Distrib0.005 1 Probability Begin End Outcome0.911 6 0.167 0.000 0.167 10.533 4 0.167 0.167 0.333 20.505 4 0.167 0.333 0.500 30.496 3 0.167 0.500 0.667 40.167 2 0.167 0.667 0.833 50.198 2 0.167 0.833 1.000 60.294 20.631 40.924 6 Results of Simulation0.421 3 Roll Frequency Fraction0.245 2 1 18 0.180.453 3 2 15 0.150.556 4 3 24 0.240.663 4 4 20 0.200.077 1 5 10 0.100.711 5 6 13 0.130.776 5 1000.385 30.468 30.352 30.668 50.440 30.008 10.503 40.614 4

A4: =RAND()(copied down)

E17: =COUNTIF(B$4:B$103,D17)(copied down)

F17: =E17/E$23(copied down)

E8: =F7

F7: =E7+D7(copied down)

B4: =VLOOKUP(A4,E$7:G$12,3)

E23: =SUM(E17:E22)

General Discrete Distribution In the last example, is there anything

special about the equal probabilities? Could we simulate an “unfair” die just by

changing the probabilities in D7:D12? General Discrete Distributions are

handled in the same way. Examples

Demand for products or services Number of machines breaking down in a day

General Discrete Distribution Example1

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General Discrete Distribution

U(0,1) Demand0.034 1000.549 250 Cumulative Distribution0.039 100 Probability Begin End Demand0.356 200 0.05 0.00 0.05 1000.569 250 0.10 0.05 0.15 1500.207 200 0.25 0.15 0.40 2000.139 150 0.30 0.40 0.70 2500.049 100 0.15 0.70 0.85 3000.534 250 0.10 0.85 0.95 3500.477 250 0.05 0.95 1.00 4000.036 100 1.000.470 2500.772 3000.066 150 Results of Simulation0.682 250 Demand Frequency Fraction0.389 200 100 10 0.100.450 250 150 8 0.080.962 400 200 18 0.180.502 250 250 36 0.360.651 250 300 18 0.180.738 300 350 5 0.050.644 250 400 5 0.050.419 250 1000.545 2500.800 300

A4: =RAND()

B4: =VLOOKUP(A4,E$7:G$13,3)

Note the different probabilities for the different possible demand values.

Results for 100 trials roughly correspond to the input probabilities (obviously, more trials would result in a closer match).

Continuous Probability Distributions

So far, we’ve only dealt with discrete distributions. Discrete distribution is one where the outcome

can be only one of a finite number of possibilities (technically, a “countable” number possibilities).

Continuous distribution allow any possible value, possibly bounded above and/or below.

Actually, RAND() is an example of a continuous probability distribution, U(0,1).

Here we’ll look at the uniform distribution, the normal distribution, and the exponential distribution.

Continuous Uniform Distribution

Uniform distribution between a (minimum) and b (maximum) is designated U(a,b).

RAND() returns a U(0,1) random value. Convert value from RAND() into value from U(a,b).

X = a + (b−a)*RAND() If a=10, b=50, and RAND()=0.37, then X = 10 + (50-

10)*0.37 = 10 + 14.8 = 24.8 What if RAND()=0? What if RAND()=1?

Examples Time to complete a task, based on minimum and maximum

time estimates. Unit costs Demand

U(10,50) Example: Histogram based on 250 trials

Frequency (%, n=250)

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2.0%

4.0%

6.0%

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10.0%

12.0%

14.0%

12.08 16.04 19.99 23.95 27.90 31.86 35.82 39.77 43.73 47.68

Midpoint of Range

Discrete Uniform Distribution: A Reprise

Earlier we used a VLOOKUP function to simulate a Discrete Uniform distribution

If the possible values are integers, we can use what we’ve learned about the continuous uniform distribution to be more efficient.

Suppose we want a discrete uniform distribution between a and b, inclusive, designated DU(a,b).

Use =INT(a+(b−a+1)*RAND()) Example: DU(10,50), suppose RAND()=0.37

X=INT(10+(50−10+1)*0.37) = INT(25.17) = 25 INT returns the integer part of a number

The “+1” is needed to ensure nothing smaller than a is returned, and that b is an actual possibility.

This approach is easier than the VLOOKUP approach especially when there are many possibilities (e.g., choosing one person at random out of a list of 5000 entries).

Normal Distribution Normal Distribution characterized by a

mean (µ) and standard deviation (). Designated N(µ,)

Random Number Generation =NORMINV(RAND(),µ,) NORMINV is the “inverse” of the normal

distribution function. RAND acts like a cumulative probability value (between 0 and 1).

If µ=0 and =1, then =NORMINV(RAND(),0,1) essentially returns a random Z-value from a normal distribution.

N(80,10) Example: Histogram based on 250 trials

Frequency (%, n=250)

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5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

55.45 61.44 67.42 73.41 79.40 85.38 91.37 97.36 103.34 109.33

Midpoint of Range

Exponential Distribution Very common distribution when modeling customer arrivals to

service systems, machine breakdowns, etc. Characterized by a mean, denoted µ. We refer to an

exponential distribution as EXP(µ). For example, µ would represent the average time between

customer arrivals, the average time between machine breakdowns, etc.

Generating an EXP(µ) random value: =−µ*LN(RAND()) LN is the natural logarithm function (which is the mathematical

inverse of the EXP function. The minus sign is needed because the natural logarithm of a value

between 0 and 1 is negative. Example

Suppose µ=25, and RAND()=0.68 Then X = −25*LN(0.68) = 9.64

EXP(10) Example: Histogram of 250 trials

Frequency (%, n=250)

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2.45 7.30 12.15 17.00 21.86 26.71 31.56 36.41 41.26 46.12

Midpoint of Range

Example: DG Outerwear Decide number of coats to order

Place order in June, but demand not realized until Fall. No chance for another order.

If we order too few, we lose out on sales. If we order too many, we must sell the

remainder at a loss. Demand for coats at the regular price is

random. We believe we can sell any coats left over, but the salvage price itself is uncertain.

DG Outerwear Problem Parameters Unit Cost: $75 Regular Sales Price: $100 Demand at regular price: Uniformly distributed

between 20 and 40. Salvage Price: May be $15, $20, $25, or $30

with respective probabilities 0.05, 0.30, 0.50, and 0.15.

Assume all leftover coats will be sold at a single salvage price.

DG considering ordering 35 coats. Is this a good idea?

Simulation Model

Key Formulas B17, C17: =RAND() D17 = INT(E$4+(E$5−E$4+1)*B17) E17 = MIN(B$8,D17) F17 = B$8−E17 G17 = VLOOKUP(C17,E$10:G$13,3) H17 = B$5*E17 I17 = G17*F17 J17 = H17+I17−(B$8*B$4)

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DG Winter Coats

Fixed Inputs Demand Distribution (Discrete Uniform)Purchase Price of Coat $75 Minimum 20

Regular Sales Price $100 Maximum 40

Decision Variable Salvage Price Distribution (Discrete)Purchase Quantity 35 Cumulative Distribution

Probability Begin End Price0.05 0.00 0.05 $150.30 0.05 0.35 $200.50 0.35 0.85 $250.15 0.85 1.00 $301.00

Simulation Logic

Replication RN1 RN2 Demand

Reg Sales Qty

Salv Sales Qty

Salv Price

Reg Rev

Salv Rev Profit

1 0.003 0.992 20 20 15 $30 $2,000 $450 ($175)

Simulation Model (continued) This is for one replication, or trial. We

need to run many trials to get a good sense of the results of this decision.

We’ve used relative and absolute cell references in the model so that we can copy the formulas in Row 17 down.

We’ll copy this row down so that we have 250 trials of the simulation.

Then, we’ll calculate summary statistics of the resulting profit values.

Simulation Model Replicated, with Summary Statistics

Note: Many rows hidden (but calculations use all rows)

Summary Statistics for a Purchase Quantity of 35

Average Profit = $435 Std.Dev. = $392 Minimum = −$325 Maximum = $875 95% Confidence Interval

= ($386, $483)

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DG Winter Coats

Fixed Inputs Demand Distribution (Discrete Uniform)Purchase Price of Coat $75 Minimum 20

Regular Sales Price $100 Maximum 40

Decision Variable Salvage Price Distribution (Discrete)Purchase Quantity 35 Cumulative Distribution

Probability Begin End Price0.05 0.00 0.05 $150.30 0.05 0.35 $200.50 0.35 0.85 $250.15 0.85 1.00 $301.00

Simulation Logic

Replication RN1 RN2 Demand

Reg Sales Qty

Salv Sales Qty

Salv Price

Reg Rev

Salv Rev Profit

1 0.911 0.701 39 35 0 $25 $3,500 $0 $8752 0.279 0.724 25 25 10 $25 $2,500 $250 $1253 0.064 0.585 21 21 14 $25 $2,100 $350 ($175)4 0.595 0.355 32 32 3 $25 $3,200 $75 $6505 0.404 0.405 28 28 7 $25 $2,800 $175 $3506 0.932 0.302 39 35 0 $20 $3,500 $0 $875

249 0.544 0.795 31 31 4 $25 $3,100 $100 $575250 0.084 0.823 21 21 14 $25 $2,100 $350 ($175)

Average $435Standard Deviation $392

Minimum ($325)Maximum $875

95% Confidence Interval on AverageLower CL $386Upper CL $483

DG Coats Example: Comments

Purchasing 35 coats results in an average profit of $435. Can we do better? What do the standard deviation, minimum, and maximum values tell us?

95% confidence interval on mean profit We are 95% confident the true value of the mean

profit lies somewhere in the interval from $386 to $483.

What effect would more (or less) simulation trials have on this confidence interval?

What does the confidence interval say about an individual value of profit, such as what we will get this year if we order 35 coats?

DG: Finding Optimal Order Quantity

We can change the purchase quantity, press F9, and calculate the 250 trials and summary statistics for this new purchase quantity. This is a powerful tool!

Purchase Quantity Average

Standard Deviation Minimum Maximum

20 $500 $0 $500 $500

25 $568 $117 $225 $625

30 $551 $264 -$50 $750

35 $435 $392 -$325 $875

40 $230 $460 -$600 $1000

The highest average profit occurs when we purchase 25 coats. Thought Question: The average demand is 30 (remember demand was

uniformly distributed between 20 and 40). Why is the highest average profit at a purchase quantity less than this average?

Does making decisions based on the average always make sense?

DG: Histogram of Profit Values

Purchase Quantity = 25

Histogram of profit values

Why does this histogram shape make sense?

Bin Frequency$225 3$252 13$278 1$305 3$332 5$358 1$385 5$412 6$438 2$465 4$492 4$518 0$545 6$572 4$598 0

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Histogram of Profit Values for Purchase Quantity = 25 (250 total replications)

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Simulation Using Data Tables

In the DG example we simply copied the formulas down to replicate the model 250 times. This only works when the logic of the model is simple enough to arrange in a single row of the spreadsheet.

For more complex models, we need to use Excel’s Data Table feature (first used in Supplement A).

With a Data Table approach, we build the logic of the simulation for one single trial. Then the Data Table effectively recalculates the model for as many times as we wish.

Data Table Approach for DG Problem

We only need the logic for a single trial (Row 17).

Next slide gives steps for Data Table.

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A B C D E F G H I JSimulation Logic

RN1 RN2 Demand

Reg Sales Qty

Salv Sales Qty

Salv Price

Reg Rev

Salv Rev Profit

0.001 0.250 20 20 15 $20 $2,000 $300 ($325)

Data Table for Simulation ReplicationsReplications Profit

(325)$ Summary Statistics1 875$ Average 413$ 2 35$ StdDev 403$ 3 795$ Minimum (400)$ 4 (325)$ Maximum 875$ 5 75$ 6 (100)$ 95% Confidence Interval7 200$ Lower CL 363$ 8 200$ Upper CL 463$ 9 (35)$

10 (250)$ 11 (400)$ 249 200$ 250 715$

B21: =J 17

Data Table Steps for DG Problem

Steps1. Two columns needed, for replications and profit. In the replications

column, from A22 to A271, enter the numbers 1…250. Do not put anything in Cell A21.

2. Cell B21, enter “=J17.” This references the profit value from the simulation logic.

3. Select A21:B271.4. Keeping A21:B271, go to Data/Table, “Row Input Cell” blank, and

click on A21 (or any blank cell on the worksheet) for the “Column Input Cell.” Click “OK.” The results from 250 replications should now be showing in B22:B271.

5. If all the values in B22:B271 are the same, press the F9 key to force recalculation of the worksheet.

6. Compute summary statistics from the results.7. If desired, freeze the results from the simulation in Cells B22:B271

using Edit/Copy, Edit/Paste Special/Values. Comments

The “column input cell” must not have anything in it. This is a different use of the Data Table than in Supplement A. Here, the Data Table is being “faked” into recalculating the simulation output measure 250 times.

Supplement C Highlights A computer simulation is a model that mimics what might happen in reality. Computer simulations model the uncertainty present in a system by

generating random numbers from known probability distributions. Simulation is a valuable tool because it can simultaneously consider the

uncertainty present in many factors of a problem, and provide outputs that show how theis “input” uncertainty translates into uncertainty in the output measure.

Monte-Carlo Simulation can be conducted using Excel without any aAdd -Iins. Commercial aAdd -iIns such as Crystal Ball and @Risk provide additional functionality that is more difficult to employ using stand-alone Excel.

Simple Discrete- Event Simulations can be conducted in Excel, but separate software products, such as ProModel, ProcessModel, and Extend are better suited to modeling of systems whose state and behavior change over time.

The simulation modeling process in spreadsheets consists of developing a deterministic model with correct logic, determining the appropriate probability distributions to use for the random inputs, incorporating those distributions in the model itself, running many replications of the simulation model, any analyzing the simulation results by computing and interpreting summary statistical measures.

Supplement C Highlights (continued)

Each time Excel’s RAND() function calculates, it generates a uniformly -distributed random number between 0 and 1, denoted U(0,1).

Random numbers from probability distributions (e.g., Bernoulli, discrete uniform, general discrete, continuous uniform, normal, and exponential, among others) are derived from a U(0,1) random number through mathematical calculations.

Replications of simulation models in Excel can be performed by copying the entire logic itself or by using Excel’s Data Table feature. For simple models where the logic fits into a single row, copying the logic itself is acceptable. However, for more complex models, the Data Table feature should be used.

At a minimum, one should consider basic summary statistics such as the average, standard deviation, minimum, and maximum when interpreting results from a simulation. One should also compute a confidence interval to assess the precision of the estimate for the mean, and to determine whether additional replications should be run. It is also a good idea to generate a histogram of the results to see the actual probability distribution of the output measure.

The EndCopyright © 2005 John Wiley & Sons, Inc. All

rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United State Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.