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Chapter 15

To accompanyQuant i tat ive Analysis for Management, Tenth Edit io n,

by Render, Stair, and HannaPower Point slides created by Jeff Heyl

Simu lat ion Model ing

2009 Prentice-Hall, Inc.

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Learn ing Ob ject ives

1. Tackle a wide variety of problems bysimulation

2. Understand the seven steps of conducting asimulation3. Explain the advantages and disadvantages of

simulation4. Develop random number intervals and use

them to generate outcomes5. Understand alternative simulation packages

available

After completing this chapter, students will be able to:

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Chapter Outl ine

15.1 Introduction

15.2 Advantages and Disadvantages of Simulation

15.3 Monte Carlo Simulation

15.4 Simulation and Inventory Analysis

15.5 Simulation of a Queuing Problem

15.6 Fixed Time Increment and Next EventIncrement Simulation Models

15.7 Simulation Model for a Maintenance Policy

15.8 Two Other Types of Simulation

15.9 Verification and Validation

15.10 Role of Computers in Simulation

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In t roduct ion

Simulat ionis one of the most widely usedquantitative analysis tools

It is used by over half of the largest UScorporations in corporate planning

To simulateis to try to duplicate the features,appearance, and characteristics of a real system

We will build a mathematical modelthat comesas close as possible to representing the reality

of the system You can also build physical models to test

systems

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In t roduct ion

The idea behind simulation is to imitate a real-world situation mathematically

Study its properties and operating characteristics

Draw conclusions and make action decisions

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In t roduct ion

Using simulation, a manager should

1. Define a problem

2. Introduce the variables associated with theproblem

3. Construct a numerical model

4. Set up possible courses of action for testing

5. Run the experiment

6. Consider the results

7. Decide what courses of action to take

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Process of Simulat ion

Define Problem

Introduce ImportantVariables

Construct SimulationModel

Specify Values ofVariables to Be Tested

Conduct the

Simulation

Examine theResults

Select Best Courseof ActionFigure 15.1

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Advantages and Disadvantages

of Simulat ion

Simulation is useful because

1. It is relatively straightforward and flexible

2. Recent advances in computer software makesimulation models very easy to develop

3. Can be used to analyze large and complexreal-world situations

4. Allows what-if? type questions

5. Does not interfere with the real-world system

6. Enables study of interactions betweencomponents

7. Enables time compression

8. Enables the inclusion of real-world

complications

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Advantages and Disadvantages

of Simulat ion

The main disadvantages of simulation are

1. It is often expensive as it may require a long,complicated process to develop the model

2. Does not generate optimal solutions, it is atrial-and-error approach

3. Requires managers to generate all conditionsand constraints of real-world problem

4. Each model is unique and the solutions and

inferences are not usually transferable toother problems

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Monte Carlo Simu lat ion

When systems contain elements that exhibitchance in their behavior, the Monte Carlo methodof simulation can be applied

Some examples are

1. Inventory demand

2. Lead time for inventory

3. Times between machine breakdowns

4. Times between arrivals

5. Service times

6. Times to complete project activities

7. Number of employees absent

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Monte Carlo Simu lat ion

The basis of the Monte Carlo simulation isexperimentation on the probabilistic elementsthrough random sampling

It is based on the following five steps

1. Setting up a probability distribution forimportant variables

2. Building a cumulative probability distributionfor each variable

3. Establishing an interval of random numbersfor each variable

4. Generating random numbers

5. Actually simulating a series of trials

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Harrys Auto Tire Example

A popular radial tire accounts for a large portionof the sales at Harrys Auto Tire

Harry wishes to determine a policy for managingthis inventory

He wants to simulate the daily demand for anumber of days

Step 1: Establ ishing prob abi l i ty distr ibut ion s

One way to establish a probability distribution fora given variable is to examine historical outcomes

Managerial estimates based on judgment andexperience can also be used

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Historical daily demand for radial tires

DEMAND FOR TIRES FREQUENCY (DAYS)

0 101 20

2 40

3 60

4 40

5 30

200

Table 15.1

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Step 2: Bu i ld ing a cumulative probabi l i ty distr ibut ion

for each variable

Converting from a regular probability to a

cumulative distribution is an easy job A cumulative probability is the probability that a

variable will be less than or equal to a particularvalue

A cumulative distribution lists all of the possible

values and the probabilities Tables 15.2 and 15.3 show these distributions

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Probability of demand for radial tires

DEMAND VARIABLE PROBABILITY OF OCCURRENCE

0 10/200 = 0.05

1 20/200 = 0.10

2 40/200 = 0.20

3 60/200 = 0.30

4 40/200 = 0.20

5 30/200 = 0.15

200/200 = 1.00

Table 15.2

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Cumulative probability for radial tires

DAILY DEMAND PROBABILITY CUMULATIVE PROBABILITY

0 0.05 0.05

1 0.10 0.15

2 0.20 0.35

3 0.30 0.65

4 0.20 0.85

5 0.15 1.00

Table 15.3

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Step 3: Sett ing random num ber intervals

We assign a set of numbers to represent eachpossible value or outcome

These are random number intervals A random numberis a series of digits that have

been selected by a totally random process

The range of the random number intervalscorresponds exactlyto the probability of the

outcomes as shown in Figure 15.2

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Graphical representation of the cumulativeprobabilitydistributionfor radialtires

00

8685

6665

3635

1615060501

Random

Numbers

Represents 4Tires Demanded

Represents 1Tire Demanded0.05

0.15

0.35

0.65

0.85

1.001.00

0.80

0.60

0.40

0.20

0.000 1 2 3 4 5

Daily Demand for Radials

Cumula

tiveProbability

Figure 15.2

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Assignment of random number intervals forHarrys Auto Tire

DAILY DEMAND PROBABILITY CUMULATIVEPROBABILITY

INTERVAL OFRANDOM NUMBERS

0 0.05 0.05 01 to 05

1 0.10 0.15 06 to 15

2 0.20 0.35 16 to 35

3 0.30 0.65 36 to 65

4 0.20 0.85 66 to 85

5 0.15 1.00 86 to 00

Table 15.4

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Step 4: Generat ing random numbers

Random numbers can be generated in severalways

Large problems will use computer program to

generate the needed random numbers For small problems, random processes like

roulette wheels or pulling chips from a hat maybe used

The most common manual method is to use arandom number table

Because everyth ingis random in a randomnumber table, we can select numbers fromanywhere in the table to use in the simulation

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Table of random numbers (partial)

52 06 50 88 53 30 10 47 99 37

37 63 28 02 74 35 24 03 29 60

82 57 68 28 05 94 03 11 27 79

69 02 36 49 71 99 32 10 75 21

98 94 90 36 06 78 23 67 89 85

96 52 62 87 49 56 59 23 78 71

33 69 27 21 11 60 95 89 68 48

50 33 50 95 13 44 34 62 64 39

88 32 18 50 62 57 34 56 62 31

90 30 36 24 69 82 51 74 30 35

Table 15.5

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Step 5: Simu lat ing th e experiment

We select random numbers from Table 15.5

The number we select will have a correspondingrange in Table 15.4

We use the daily demand that corresponds to theprobability range aligned with the randomnumber

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Ten-day simulation of demand for radial tires

DAY RANDOM NUMBER SIMULATED DAILY DEMAND

1 52 3

2 37 3

3 82 4

4 69 4

5 98 5

6 96 5

7 33 2

8 50 3

9 88 5

10 90 5

39 = total 10-day demand

3.9 = average daily demand for tires

Table 15.6

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Note that the average demand from thissimulation (3.9 tires) is different from theexpecteddaily demand

Expected

dailydemandtiresofDemandtiresofyProbabilit

5

0ii

i(0.05)(0) + (0.10)(1) + (0.20)(2) + (0.30)(3)+ (0.20)(4) + (0.15)(5)

2.95 tires If this simulation were repeated hundreds or

thousands of times it is much more likely theaverage simulateddemand would be nearly thesame as the expecteddemand

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Using QM for Windows for Simulat ion

Monte Carlo simulation of Harrys Auto Tire usingQM for Windows

Program 15.1

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Simulation w ith Excel Spreadsheets

Using Excel to simulate tire demand for HarrysAuto Tire Shop

Program 15.2A

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Excel simulation results for Harrys Auto TireShop showing a simulated average demand of 2.8tires per day

Program 15.2B

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Generating normal random numbers in Excel

Program 15.3A

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Excel output with normal random numbers

Program 15.3B

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Simulat ion and Invento ry Analys is

We have seen deterministic inventory models

In many real-world inventory situations, demandand lead time are variables

Accurate analysis is difficult without simulation

We will look at an inventory problem with twodecision variables and two probabilisticcomponents

The owner of a hardware store wants to establish

order quant i tyand reorder po intdecisions for aproduct that has probabilistic daily demand andreorder lead time

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Simkins Hardware Store

The owner of a hardware store wants to find agood, low cost inventory policy for an electricdrill

Simkin identifies two types of variables,

controllable and uncontrollable inputs The controllable inputs are the order quantity and

reorder points

The uncontrollable inputs are daily demand and

variable lead time The demand data for the drill is shown in Table

15.7

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Probabilities and random number intervals fordaily Ace drill demand

(1)DEMAND FORACE DRILL

(2)FREQUENCY(DAYS)

(3)PROBABILITY

(4)CUMULATIVEPROBABILITY

(5)INTERVAL OFRANDOM NUMBERS

0 15 0.05 0.05 01 to 05

1 30 0.10 0.15 06 to 15

2 60 0.20 0.35 16 to 35

3 120 0.40 0.75 36 to 75

4 45 0.15 0.90 76 to 90

5 30 0.10 1.00 91 to 00300 1.00

Table 15.7

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Probabilities and random number intervals forreorder lead time

(1)LEAD TIME(DAYS)

(2)FREQUENCY(ORDERS)

(3)PROBABILITY

(4)CUMULATIVEPROBABILITY

(5)RANDOM NUMBERINTERVAL

1 10 0.20 0.20 01 to 20

2 25 0.50 0.70 21 to 70

3 15 0.30 1.00 71 to 00

50 1.00

Table 15.8

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The third step is to develop a simulation model

A flow diagram, or flowchart, is helpful in thisprocess

The fourth step in the process is to specify the

values of the variables that we wish to test The first policy Simkin wants to test is an order

quantity of 10 with a reorder point of 5

The fifth step is to actually conduct thesimulation

The process is simulated for a 10 day period

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Flow diagramfor Simkinsinventoryexample

Figure 15.3 (a)

Start

Begin dayof simulation

Hasorder

arrived?

Increase currentinventory byquantity ordered

Select random numberto generate todaysdemand

Isdemand greaterthan beginning

inventory?

Recordnumber oflost sales

Yes

No

Yes

No

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Flow diagramfor Simkinsinventoryexample

Figure 15.3 (b)

Compute ending inventory= Beginning inventoryDemand

Record endinginventory = 0

No

Isending inventoryless than reorder

point?

Haveenough days

of this order policybeen simulated

?

Hasorder been

placed that hasntarrived yet

?

Placeorder

Select randomnumber togenerate leadtime

Compute average ending inventory,average lost sales, average numberof orders placed, and correspondingcosts

End

No

No

No

Yes

Yes

Yes

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Using the table of random numbers, thesimulation is conducted using a four-step process

1. Begin each day by checking whether an orderedinventory has arrived. If it has, increase the currentinventory by the quantity ordered.

2. Generate a daily demand from the demand probabilityby selecting a random number

3. Compute the ending inventory every day. If on-handinventory is insufficient to meet the days demand,satisfy as much as possible and note the number of

lost sales.4. Determine whether the days ending inventory has

reached the reorder point. If necessary place an order.

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Simkin Hardwares first inventory simulation

Table 15.9

ORDER QUANTITY = 10 UNITS REORDER POINT = 5 UNITS

(1)DAY

(2)UNITSRECEIVED

(3)BEGINNINGINVENTORY

(4)RANDOMNUMBER

(5)DEMAND

(6)ENDINGINVENTORY

(7)LOSTSALES

(8)ORDER

(9)RANDOMNUMBER

(10)LEADTIME

1 10 06 1 9 0 No

2 0 9 63 3 6 0 No

3 0 6 57 3 3 0 Yes 02 1

4 0 3 94 5 0 2 No

5 10 10 52 3 7 0 No

6 0 7 69 3 4 0 Yes 33 2

7 0 4 32 2 2 0 No

8 0 2 30 2 0 0 No

9 10 10 48 3 7 0 No

10 0 7 88 4 3 0 Yes 14 1

Total 41 2

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Analyzing Simkins Inventory Cost

The objective is to find a low-cost solution soSimkin must determine what the costs are

Equations for average daily ending inventory,average lost sales, and average number of orders

placedAverageendinginventory

dayperunits4.1days10

unitstotal41

Averagelost sales dayperunits0.2days10

lostsales2

Averagenumber oforders placed

dayperorders0.3days10

orders3

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Simkins store is open 200 days a year Estimated ordering cost is $10 per order

Holding cost is $6 per drill per year

Lost sales cost $8

Daily order cost = (Cost of placing one order)x (Number of orders placed per day)

= $10 per order x 0.3 orders per day = $3

Daily holding cost = (Cost of holding one unit for one day) x(Average ending inventory)

= $0.03 per unit per day x 4.1 units per day

= $0.12

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Simkins store is open 200 days a year Estimated ordering cost is $10 per order

Holding cost is $6 per drill per year

Lost sales cost $8

Daily stockout cost = (Cost per lost sale)x (Average number of lost sales perday)

= $8 per lost sale x 0.2 lost sales per day

= $1.60Total daily

inventory cost = Daily order cost + Daily holding cost+ Daily stockout cost

= $4.72

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For the year, this policy would cost approximately$944

This simulation should really be extended formany more days, perhaps 100 or 1,000 days

Even after a larger simulation, the model must beverified and validated to make sure it trulyrepresents the situation on which it is based

If we are satisfied with the model, additionalsimulations can be conducted using other values

for the variables After simulating all reasonable combinations,

Simkin would select the policy that results in thelowest total cost

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Simulat ion o f a Queuing Prob lem

Modeling waiting lines is an important applicationof simulation

The assumptions of queuing models are quiterestrictive

Sometimes simulation is the only approach thatfits

In this example, arrivals do not follow a Poissondistribution and unloading rates are notexponential or constant

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Port o f New Orleans

Fully loaded barges arrive at night for unloading The number of barges each night varies from 0 - 5

The number of barges vary from day to day

The supervisor has information which can be

used to create a probability distribution for thedaily unloading rate

Barges are unloaded first-in, first-out

Barges must wait for unloading which is

expensive The dock superintendent wants to do a simulation

study to enable him to make better staffingdecisions

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Overnight barge arrival rates and random numberintervals

NUMBER OFARRIVALS PROBABILITY

CUMULATIVEPROBABILITY

RANDOMNUMBER INTERVAL

0 0.13 0.13 01 to 13

1 0.17 0.30 14 to 30

2 0.15 0.45 31 to 45

3 0.25 0.70 46 to 70

4 0.20 0.90 71 to 905 0.10 1.00 91 to 00

Table 15.10

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Unloading rates and random number intervals

DAILY UNLOADINGRATE PROBABILITY


RANDOMNUMBER INTERVAL

1 0.05 0.05 01 to 05

2 0.15 0.20 06 to 20

3 0.50 0.70 21 to 70

4 0.20 0.90 71 to 90

5 0.10 1.00 91 to 001.00

Table 15.11

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Queuing simulation of barge unloadings

Table 15.12

(1)

DAY

(2)

NUMBER DELAYEDFROM PREVIOUS DAY

(3)

RANDOMNUMBER

(4)

NUMBER OFNIGHTLY ARRIVALS

(5)

TOTAL TO BEUNLOADED

(6)

RANDOMNUMBER

(7)

NUMBERUNLOADED

1 52 3 3 37 3

2 0 06 0 0 63 0

3 0 50 3 3 28 3

4 0 88 4 4 02 15 3 53 3 6 74 4

6 2 30 1 3 35 3

7 0 10 0 0 24 0

8 0 47 3 3 03 1

9 2 99 5 7 29 3

10 4 37 2 6 60 3

11 3 66 3 6 74 4

12 2 91 5 7 85 4

13 3 35 2 5 90 4

14 1 32 2 3 73 3

15 0 00 5 5 59 3

20 41 39

Total delays Total arrivals Total unloadings

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Three important pieces of information

Average number of bargesdelayed to the next day

dayperdelayedbarges1.33

days15

delays20

Average number ofnightly arrivals

arrivals2.73days15

arrivals41

Average number of bargesunloaded each day unloadings2.60days15

unloadings39

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Using Excel to Simulate the Port of

New Orleans Queuing Problem

An Excel model for the Port of New Orleansqueuing simulation

Program 15.4A

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Using Excel to Simulate the Port of

New Orleans Queuing Problem

Output from the Excel formulas in Program 15.4A

Program 15.4B

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Fixed Time Increment and Next Event

Inc rement Simulation Models

Simulation models are often classified into f ixedt ime increment modelsand next event incrementmodels

The terms refer to the frequency in which the

system status is updated Fixed time increments update the status of the

system at fixed time intervals

Next event increment models update only whenthe system status changes

Fixed event models randomly generate thenumber of events that occur during a time period

Next event models randomly generate the timethat elapses until the next event occurs

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Simulation Model for a

Maintenance Pol icy

Simulation can be used to analyze differentmaintenance policies before actuallyimplementing them

Many options regarding staffing levels, parts

replacement schedules, downtime, and laborcosts can be compared

This can including completely shutting downfactories for maintenance

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Three Hil ls Power Company

Three Hills provides power to a large city througha series of almost 200 electric generators

The company is concerned about generatorfailures because a breakdown costs about $75

per generator per hour Their four repair people earn $30 per hour and

work rotating 8 hour shifts

Management wants to evaluate the

1. Service maintenance cost2. Simulated machine breakdown cost

3. Total cost

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There are two important maintenance systemcomponents

Time between successive generator breakdownswhich varies from 30 minutes to three hours

The time it takes to repair the generators whichranges from one to three hours in one hourblocks

A next event simulation is constructed to studythis problem

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Three Hillsflow diagram

Start

Generate random numberfor Time BetweenBreakdowns

Record actual clock timeof breakdown

Examine time previousrepair ends

Is therepairpersonfree to begin

repair?

Wait until previousrepair is completed

No

YesFigure 15.4 (a)

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Yes

Generate random numberfor repair time required

Compute time repaircompleted

Compute hours of machinedowntime = Time repaircompletedClock timeof breakdown

Enoughbreakdownssimulated?

Compute downtime andcomparative cost data End

No


Three Hillsflow diagram

Figure 15.4 (b)

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Time between generator breakdowns at ThreeHills Power

TIME BETWEENRECORDEDMACHINE

FAILURES (HRS)

NUMBEROF TIMES

OBSERVED PROBABILITY

CUMULATIVE

PROBABILITY

RANDOMNUMBER

INTERVAL0.5 5 0.05 0.05 01 to 05

1.0 6 0.06 0.11 06 to 11

1.5 16 0.16 0.27 12 to 27

2.0 33 0.33 0.60 28 to 60

2.5 21 0.21 0.81 61 to 81

3.0 19 0.19 1.00 82 to 00

Total 100 1.00

Table 15.13

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Generator repair times required

REPAIR TIMEREQUIRED (HRS)

NUMBEROF TIMESOBSERVED PROBABILITY


RANDOMNUMBERINTERVAL

1 28 0.28 0.28 01 to 28

2 52 0.52 0.80 29 to 80

3 20 0.20 1.00 81 to 00

Total 100 1.00

Table 15.14

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Simulation of generator breakdowns and repairs

(1)

BREAKDOWNNUMBER

(2)

RANDOMNUMBER FORBREAKDOWNS

(3)

TIMEBETWEENBREAKDOWNS

(4)

TIME OFBREAKDOWN

(5)

TIME REPAIR-PERSON ISFREE TOBEGIN THISREPAIR

(6)

RANDOMNUMBERFORREPAIRTIME

(7)

REPAIRTIMEREQUIRED

(8)

TIMEREPAIRENDS

(9)

NUMBEROFHOURSMACHINEDOWN

1 57 2 02:00 02:00 07 1 03:00 1

2 17 1.5 03:30 03:30 60 2 05:30 2

3 36 2 05:30 05:30 77 2 07:30 2

4 72 2.5 08:00 08:00 49 2 10:00 2

5 85 3 11:00 11:00 76 2 13:00 2

6 31 2 13:00 13:00 95 3 16:00 3

7 44 2 15:00 16:00 51 2 18:00 3

8 30 2 17:00 18:00 16 1 19:00 2

9 26 1.5 18:30 19:00 14 1 20:00 1.5

10 09 1 19:30 20:00 85 3 23:00 3.5

11 49 2 21:30 23:00 59 2 01:00 3.5

12 13 1.5 23:00 01:00 85 3 04:00 5

13 33 2 01:00 04:00 40 2 06:00 5

14 89 3 04:00 06:00 42 2 08:00 4

15 13 1.5 05:30 08:00 52 2 10:00 4.5

Total 44

Table 15.15

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Cost Analys is o f Simulat ion

The simulation of 15 generator breakdownscovers 34 hours of operation

The analysis of this simulation is

Service

maintenancecost

= 34 hours of worker service timex $30 per hour

= $1,020

Simulated machinebreakdown cost = 44 total hours of breakdown

x $75 lost per hour of downtime

= $3,300

Total simulatedmaintenance cost ofthe current system

= Service cost + Breakdown cost

= $1,020 + $3,300

= $4,320

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Cost Analys is o f Simulat ion

The cost of $4,320 should be compared withother alternative plans to see if this is a goodvalue

The company might explore options like adding

another repairperson Strategies such as prevent ive maintenancemight

also be simulated for comparison

B i ld i E l Si l i M d l

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Bui ld ing an Excel Simulat ion Model

for Three Hi lls Power Company

An Excel spreadsheet model for simulating theThree Hills Power Company maintenance problem

Program 15.5A

B i ld i E l Si l t i M d l

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Bui ld ing an Excel Simulat ion Model

for Three Hi lls Power Company

Output from Excel spreadsheet in Program 15.5A

Program 15.5B

T Oth T f

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Two Other Types o f

Simulat ion Models

Simulation models are often broken intothree categories The Monte Carlo method

Operational gaming

Systems simulation

Though theoretically different,computerized simulation has tended to

blur the differences

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Operat ional Gam ing

Operat ional gamingrefers to simulation involvingtwo or more competing players

The best examples of this are military games andbusiness games

These types of simulation allow the testing ofskills and decision-making in a competitiveenvironment

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Sys tems Simulat ion

Systems simu lat ionis similar in that allows usersto test various managerial policies and decisionsto evaluate their effect on the operatingenvironment

This models the dynamics of largesystems

A co rporate operating s ystemmight model sales,production levels, marketing policies,investments, union contracts, utility rates,financing, and other factors

Econom ic simulat ions, often called econometricmodels, are used by governments, bankers, andlarge organizations to predict inflation rates,domestic and foreign money supplies, andunemployment levels

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Gross NationalProduct

Inflation Rates

UnemploymentRates

Monetary

SuppliesPopulationGrowth Rates

Sys tems Simulat ion

Inputs and outputs of a typical economic systemsimulation

Econometric Model(in Series of

MathematicalEquations)

Income TaxLevels

Corporate TaxRates

Interest Rates

Government

SpendingForeign Trade

Policy

Figure 15.5

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Veri f icat ion and Val idat ion

It is important that a simulation model be checkedto see that it is working properly and providinggood representation of the real world situation

The veri f icat ionprocess involves determining

that the computer model is internally consistentand following the logic of the conceptual model

Verification answers the question Did we buildthe model right?

Validationis the process of comparing a

simulation model to the real system it representsto make sure it is accurate

Validation answers the question Did we build theright model?

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Role of Compu ters in Simulat ion

Computers are critical in simulating complextasks

Three types of computer programming languagesare available to help the simulation process

General-purpose languages Special-purpose simulation languages

1. These require less programming

2. Are more efficient and easier to check for errors

3. Have random number generators built in

Pre-written simulation programs built to handle awide range of common problems

Excel and add-ins can also be used forsimulation problems

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