unit- iv (simulation)

8/13/2019 UNIT- IV (Simulation)

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LESSON

15

SIMULATION

CONTENTS

15.0 Aims and Objectives

15.1 Introduction

15.2 Advantages and Disadvantages of Simulation

15.3 Monte Carlo Simulation

15.4 Simulation of Demand Forecasting Problem

15.5 Simulation of Queuing Problems

15.6 Simulation of Inventory Problems

15.7 Let us Sum Up

15.8 Lesson-end Activities

15.9 Keywords

15.10 Questions for Discussion

15.11 Terminal Questions

15.12 Model Answers to Questions for Discussion

15.13 Suggested Readings

15.0 AIMS AND OBJECTIVES

This is the last lesson of the QT which will discuss about the Mathematical analysis and

mathematical technique simulation technique is considered as a valuable tool because

wide area of applications.

15.1 INTRODUCTION

In the previous chapters, we formulated and analyzed various models on real-life problems.

All the models were used with mathematical techniques to have analytical solutions. In

certain cases, it might not be possible to formulate the entire problem or solve it through

mathematical models. In such cases, simulation proves to be the most suitable method,

which offers a near-optimal solution. Simulation is a reflection of a real system,

representing the characteristics and behaviour within a given set of conditions.

In simulation, the problem must be defined first. Secondly, the variables of the model are

introduced with logical relationship among them. Then a suitable model is constructed.

After developing a desired model, each alternative is evaluated by generating a series of

values of the random variable, and the behaviour of the system is observed. Lastly, theresults are examined and the best alternative is selected the whole process has been

summarized and shown with the help of a flow chart in the Figure 90.


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Quantitative Techniques

for ManagementSimulation technique is considered as a valuable tool because of its wide area of application.

It can be used to solve and analyze large and complex real world problems. Simulation

provides solutions to various problems in functional areas like production, marketing,

finance, human resource, etc., and is useful in policy decisions through corporate planning

models. Simulation experiments generate large amounts of data and information using a

small sample data, which considerably reduces the amount of cost and time involved in

the exercise.For example, if a study has to be carried out to determine the arrival rate of customers at

a ticket booking counter, the data can be generated within a short span of time can be

used with the help of a computer.

Figure 15.1: Simulation Process

15.2 ADVANTAGES AND DISADVANTAGES OF

SIMULATION

Advantages

Simulation is best suited to analyze complex and large practical problems when it is

not possible to solve them through a mathematical method.

Simulation is flexible, hence changes in the system variables can be made to select

the best solution among the various alternatives.

In simulation, the experiments are carried out with the model without disturbing the

system.

Policy decisions can be made much faster by knowing the options well in advance

and by reducing the risk of experimenting in the real system.

Disadvantages

Simulation does not generate optimal solutions.

It may take a long time to develop a good simulation model.

In certain cases simulation models can be very expensive.

The decision-maker must provide all information (depending on the model) aboutthe constraints and conditions for examination, as simulation does not give the

answers by itself.

Problem Definition

Introduction of Variables

Construction of Simulation Model

Testing of variables with values

Simulate

Examination of results

Selection of best alternative

Not Acceptable Not Acceptable

Acceptable


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Simulation15.3 MONTE CARLO SIMULATION

In simulation, we have deterministic models and probabilistic models. Deterministic

simulation models have the alternatives clearly known in advance and the choice is

made by considering the various well-defined alternatives. Probabilistic simulation model

is stochastic in nature and all decisions are made under uncertainty. One of the probabilistic

simulation models is the Monte Carlo method. In this method, the decision variables arerepresented by a probabilistic distribution and random samples are drawn from probability

distribution using random numbers. The simulation experiment is conducted until the

required number of simulations are generated. Finally, the best course of action is selected

for implementation. The significance of Monte Carlo Simulation is that decision variables

may not explicitly follow any standard probability distribution such as Normal, Poisson,

Exponential, etc. The distribution can be obtained by direct observation or from past

records.

Procedure for Monte Carlo Simulation:

Step 1: Establish a probability distribution for the variables to be analyzed.

Step 2: Find the cumulative probability distribution for each variable.

Step 3: Set Random Number intervals for variables and generate random numbers.

Step 4: Simulate the experiment by selecting random numbers from random numbers

tables until the required number of simulations are generated.

Step 5: Examine the results and validate the model.

15.4 SIMULATION OF DEMAND FORECASTING

PROBLEM

Example 1:An ice-cream parlor's record of previous months sale of a particular variety

of ice cream as follows (see Table 15.1).

Table 15.1: Simulation of Demand Problem

Simulate the demand for first 10 days of the month

Solution:Find the probability distribution of demand by expressing the frequencies in

terms of proportion. Divide each value by 30. The demand per day has the followingdistribution as shown in Table 15.2.

Table 15.2: Probability Distribution of Demand

Find the cumulative probability and assign a set of random number intervals to various

demand levels. The probability figures are in two digits, hence we use two digit randomnumbers taken from a random number table. The random numbers are selected from

the table from any row or column, but in a consecutive manner and random intervals are

set using the cumulative probability distribution as shown in Table 15.3.

Demand (No. of Ice-creams) No. of days

4 5

5 10

6 6

7 8

8 1

Demand Probability

4 0.17

5 0.33

6 0.20

7 0.27

8 0.03


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for ManagementTable 15.3: Cumulative Probability Distribution

To simulate the demand for ten days, select ten random numbers from random numbertables. The random numbers selected are,

17, 46, 85, 09, 50, 58, 04, 77, 69 and 74

The first random number selected, 7 lies between the random number interval 17-49corresponding to a demand of 5 ice-creams per day. Hence, the demand for day oneis 5. Similarly, the demand for the remaining days is simulated as shown in Table 15.4.

Table 15.4: Demand Simulation

Example 2:A dealer sells a particular model of washing machine for which the probabilitydistribution of daily demand is as given in Table 15.5.

Table 15.5: Probability Distribution of Daily Demand

Find the average demand of washing machines per day.

Solution: Assign sets of two digit random numbers to demand levels as shown in

Table 15.6.

Table 15.6: Random Numbers Assigned to Demand

Ten random numbers that have been selected from random number tables are 68, 47, 92,76, 86, 46, 16, 28, 35, 54. To find the demand for ten days see the Table 15.7 below.

Table 15.7: Ten Random Numbers Selected

Day 1 2 3 4 5 6 7 8 9 10

Random Number 17 46 85 09 50 58 04 77 69 74

Demand 5 5 7 4 6 6 4 7 6 7

Demand/day - 0 1 2 3 4 5

Demand - 0.05 0.25 0.20 0.25 0.10 0.15

Demand Probability Cumulative Probability Random Number Intervals

0 0.05 0.05 00-04

1 0.25 0.30 05-29

2 0.20 0.50 30-49

3 0.25 0.75 50-74

4 0.10 0.85 75-84

5 0.15 1.00 85-99

Trial No Random Number Demand / day

1 68 3

2 47 2

3 92 5

4 76 4

5 86 5

6 46 2

7 16 1

8 28 1

9 35 2

10 54 3

Total Demand 28

Demand Probability Cumulative Probability Random Number Interval

4 0.17 0.17 00-16

5 0.33 0.50 17-49

6 0.20 0.70 50-69

7 0.27 0.97 70-96

8 0.03 1.00 97-99


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SimulationAverage demand =28/10 =2.8 washing machines per day.

The expected demand /day can be computed as,

Expected demand per day = =

.......................(1)

where, pi= probability and xi = demand

= (0.05 0) + (0.25 1) + (0.20 2) + (0.25 3) + (0.1 4) + (0.15 5)

= 2.55 washing machines.

The average demand of 2.8 washing machines using ten-day simulation differs significantly

when compared to the expected daily demand. If the simulation is repeated number of

times, the answer would get closer to the expected daily demand.

Example 3:A farmer has 10 acres of agricultural land and is cultivating tomatoes on

the entire land. Due to fluctuation in water availability, the yield per acre differs. The

probability distribution yields are given below:

a. The farmer is interested to know the yield for the next 12 months if the same wateravailability exists. Simulate the average yield using the following random numbers

50, 28, 68, 36, 90, 62, 27, 50, 18, 36, 61 and 21, given in Table 15.8.

Table 15.8: Simulation Problem

b. Due to fluctuating market price, the price per kg of tomatoes varies from Rs. 5.00

to Rs. 10.00 per kg. The probability of price variations is given in the Table 216

below. Simulate the price for next 12 months to determine the revenue per acre.

Also find the average revenue per acre. Use the following random numbers 53, 74,

05, 71, 06, 49, 11, 13, 62, 69, 85 and 69.


Solution:

Table 15.10: Table for Random Number Interval for Yield

Yield of tomatoes per acre (kg) Probability

200 0.15

220 0.25

240 0.35

260 0.13

280 0.12

Price per kg (Rs) Probability

5.50 0.05

6.50 0.15

7.50 0.30

8.00 0.25

10.00 0.15

Yield of tomatoes

per acre

Probability Cumulative Probability Random Number

Interval

200

220

240

260

280

0.15

0.25

0.35

0.13

0.12

0.15

0.40

0.75

0.88

1.00

00 14

15 39

40 74

75 87

88 99


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for ManagementTable 15.11: Table for Random Number Interval for Price

Table 15.12: Simulation for 12 months period

Average revenue per acre = 21330 / 12

= Rs. 1777.50

Example 4:J.M Bakers has to supply only 200 pizzas every day to their outlet situated

in city bazaar. The production of pizzas varies due to the availability of raw materials and

labor for which the probability distribution of production by observation made is as follows:


Simulate and find the average number of pizzas produced more than the requirement

and the average number of shortage of pizzas supplied to the outlet.

Solution: Assign two digit random numbers to the demand levels as shown in

Table 15.14

Table 15.14: Random Numbers Assigned to the Demand Levels

Price Per Kg Probability Cumulative Probability Random Number Interval

5.00

6.50

7.50

8.0010.00

0.05

0.15

0.30

0.250.25

0.05

0.20

0.50

0.751.00

00 04

05 19

20 49

50 7475 99

Month

(1)

Yield

(2)

Price

(3)Revenue / Acre (4) = 2 3 (Rs)

1

2

3

4

5

6

7

8

9

10

11

12

240

220

240

220

250

240

220

240

220

220

240

220

8.00

8.00

6.50

8.00

6.50

7.50

6.50

6.50

8.00

8.00

10.00

8.00

1960

1760

1560

1760

1820

1800

1430

1560

1760

1760

2400

1760

Production per day 196 197 198 199 200 201 202 203 204

Probability 0.06 0.09 0.10 0.16 0.20 0.21 0.08 0.07 0.03

Demand Probability Cumulative Probability No of Pizzas shortage

196 0.06 0.06 00-05

197 0.09 0.15 06-14

198 0.10 0.25 15-24

199 0.16 0.41 25-40

200 0.20 0.61 41-60

201 0.21 0.82 61-81

202 0.08 0.90 82-89

203 0.07 0.97 90-96

204 0.03 1.00 97-99


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SimulationSelecting 15 random numbers from random numbers table and simulate the production

per day as shown in Table 15.15 below.

Table 15.15: Simulation of Production Per Day

The average number of pizzas produced more than requirement

= 12/15

= 0.8 per day

The average number of shortage of pizzas supplied

= 4/15

= 0.26 per day

Check Your Progress 15.1

1. Discuss the role of simulation in demand forecasting.

2. What is Monte Carlo simulation?

Notes: (a) Write your answer in the space given below.

(b) Please go through the lesson sub-head thoroughly you will get your

answers in it.

(c) This Check Your Progress will help you to understand the lesson

better. Try to write answers for them, but do not submit your answers

to the university for assessment. These are for your practice only.

_____________________________________________________________________

____________________________________________________________________________________________________________________

_____________________________________________________________________

__________________________________________________________________

_____________________________________________________________

15.5 SIMULATION OF QUEUING PROBLEMS

Example 5:Mr. Srinivasan, owner of Citizens restaurant is thinking of introducing

separate coffee shop facility in his restaurant. The manager plans for one service counter

for the coffee shop customers. A market study has projected the inter-arrival times atthe restaurant as given in the Table 15.16. The counter can service the customers at the

following rate:

Trial Number Random Number Production Per

day

No of Pizzas over

produced

No of pizzas

shortage

1 26 199 - 1

2 45 200 - -3 74 201 1 -

4 77 201 1 -

5 74 201 1 -

6 51 200 - -

7 92 203 3 -

8 43 200 - -

9 37 199 - 1

10 29 199 - 1

11 65 201 1 -

12 39 199 - 1

13 45 200 - -

14 95 203 3 -

15 93 203 3 -

Total 12 4


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for ManagementTable 15.16: Simulation of Queuing Problem

Mr. Srinivasan will implement the plan if the average waiting time of a customers in thesystem is less than 5 minutes.

Before implementing the plan, Mr. Srinivasan would like to know the following:

i. Mean waiting time of customers, before service.

ii. Average service time.

iii. Average idle time of service.

iv. The time spent by the customer in the system.

Simulate the operation of the facility for customer arriving sample of 20 cars when therestaurant starts at 7.00 pm every day and find whether Mr. Srinivasan will go for theplan.

Solution: Allot the random numbers to various inter-arrival service times as shown inTable 15.17.

Table 15.17: Random Numbers Allocated to Various Inter-Arrival Service Times

i. Mean waiting time of customer before service = 20/20 = 1 minute

ii. Average service idle time = 17/20 = 0.85 minutes

iii. Time spent by the customer in the system = 3.6 + 1 = 4.6 minutes.

Example 6:Dr. Strong, a dentist schedules all his patients for 30 minute appointments.Some of the patients take more or less than 30 minutes depending on the type of dentalwork to be done. The following Table 15.18 shows the summary of the various categoriesof work, their probabilities and the time actually needed to complete the work.

Waiting TimeSl.

No.

Random

Number(Arrival)

Inter

ArrivalTime

(Min)

Arrival

Time at

Service

Starts at

Random

Number(service)

Service

Time(Min)

Service

Ends at

CustomerService

(Min)

1 87 6 7.06 7.06 36 4 7.10 - 6

2 37 3 7.09 7.10 16 3 7.13 1 -

3 92 6 7.15 7.15 81 5 7.20 - 2

4 52 4 7.19 7.20 08 2 7.22 1 -

5 41 4 7.23 7.23 51 4 7.27 - 1

6 05 2 7.25 7.27 34 3 7.30 2 -

7 56 4 7.29 7.30 88 6 7.36 1 -

8 70 5 7.34 7.36 88 6 7.42 2 -

9 70 5 7.39 7.42 15 3 7.45 3 -

10 07 2 7.41 7.45 53 4 7.49 4 -

11 86 6 7.47 7.49 01 2 7.51 2 -

12 74 5 7.52 7.52 54 4 7.56 - 1

13 31 3 7.55 7.56 03 2 7.58 1 -

14 71 5 8.00 8.00 54 4 8.04 1 2

15 57 4 8.04 8.04 56 4 8.08 - -

16 85 6 8.10 8.10 05 2 8.12 - 2

17 39 3 8.13 8.13 01 2 8.15 - 1

18 41 4 8.17 8.17 45 4 8.21 - 2

19 18 3 8.20 8.21 11 3 8.24 1 -

20 38 3 8.23 8.24 76 5 8.29 1 -

Total 83 72 20 17

Interarrival times Service times

Time between two

consecutive arrivals (minutes)Probability

Service time

(minutes)Probability

2 0.15 2 0.10

3 0.25 3 0.25

4 0.20 4 0.30

5 0.25 5 0.2

6 0.15 6 0.15


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SimulationTable 15.18: Simulation Problem

Simulate the dentists clinic for four hours and determine the average waiting time for

the patients as well as the idleness of the doctor. Assume that all the patients show up at

the clinic exactly at their scheduled arrival time, starting at 8.00 am. Use the following

random numbers for handling the above problem: 40,82,11,34,25,66,17,79.

Solution:Assign the random number intervals to the various categories of work asshown in Table 15.19.

Table 15.19: Random Number Intervals Assigned to the Various Categories

Assuming the dentist clinic starts at 8.00 am, the arrival pattern and the service category

are shown in Table 15.20.

Table 15.20: Arrival Pattern of the Patients

Table 15.21: The arrival, departure patterns and patients waiting time are tabulated.

Category of work Probability Cumulative probability Random Number Interval

Filling 0.40 0.40 00-39

Crown 0.15 0.55 40-54

Cleaning 0.15 0.70 55-69

Extraction 0.10 0.80 70-79

Check-up 0.20 1.00 80-99

Patient Number Scheduled Arrival Random Number Service category Service Time

1 8.00 40 Crown 60

2 8.30 82 Check-up 15

3 9.00 11 Filling 45

4 9.30 34 Filling 45

5 10.00 25 Filling 45

6 10.30 66 Cleaning 15

7 11.00 17 Filling 45

8 11.30 79 Extraction 45

Time Event (Patient Number) Patient Number (Time to go) Waiting (Patient Number)

8.00 1 arrives 1 (60) -

8.30 2 arrives 1 (30) 2

9.00 1 departure, 3 arrives 2 (15) 3

9.15 2 depart 3 (45) -

9.30 4 arrive 3 (30) 4

10.00 3 depart, 5 arrive 4 (45) 5

10.30 6 arrive 4 (15) 5,6

10.45 4 depart 5 (45) 6

11.00 7 arrive 5 (30) 6,7

11.30 5 depart, 8 arrive 6 (15) 7,8

11.45 6 depart 7 (45) 8

12.00 End 7 (30) 8

Category Time required (minutes) Probability of category

Filling 45 0.40

Crown 60 0.15

Cleaning 15 0.15

Extraction 45 0.10Check-up 15 0.20


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SimulationTable 15.26: Random Numbers Assigned for Lead-time

Table 15.27: Simulation Work-sheet for Inventory Problem (Case 1)

Reorder Quantity = 35 units, Reorder Level = 20 units, Beginning Inventory = 45 units

Lead Time (Days) Probability Cumulative probability Random Number Interval

1 0.20 0.20 00-19

2 0.30 0.50 20-49

3 0.35 0.85 50-84

4 0.15 1.00 85-99

Day

Random

Number

(Demand)

Demand

Random

Number

(Lead

Time)

Lead

Time

(Days)

Inventory

at end of

day

Qty.

Recei-

ved

Order-

ing

Cost

Holding

Cost

Short-

age

Cost

0 - - - - 45 - - - -

1 58 7 - - 38 - - 38 -

2 45 6 - - 32 - - 32 -

3 43 6 - - 26 - - 26 -

4 36 6 73 3 20 - 50 20 -

5 46 6 - - 14 - - 14 -

6 46 6 - - 8 - - 8 -

7 70 7 - - 1 35 - 36 -

8 32 5 - - 31 - - 31 -

9 12 4 - - 27 - - 27 -

10 40 6 - - 21 - - 21 -

11 51 6 21 2 15 - 50 15 -

12 59 7 - - 8 - - 8 -

13 54 6 - - 37 35 - 37 -

14 16 4 - - 33 - - 33 -

15 68 7 - - 26 - - 26 -

16 45 6 45 2 20 - 50 20 -

17 96 10 - - 10 - - 10 -

18 33 5 - - 40 35 - 40 -

19 83 8 - - 32 - - 32 -

20 77 8 - - 24 - - 24 -

21 05 3 - - 21 - - 21 -

22 15 4 76 3 17 - 50 17 -

23 40 6 - - 11 - - 11 -

24 43 6 - - 5 - - 5 -

25 34 5 - - 35 35 - 35 -

26 44 6 - - 29 - - 29 -

27 89 9 96 4 20 - 50 20 -

28 20 4 - - 16 - - 16 -

29 69 7 - - 9 - - 9 -

30 31 5 - - 4 - - 4 -

31 97 10 - - 29 35 - 29 -

32 05 3 - - 26 - - 26 -

33 59 7 94 4 19 - 50 19 -

34 02 2 - - 17 - - 17 -

35 35 5 - - 12 - - 12 -

Total 300 768 -


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for ManagementTable 15.28: Simulation Work-sheet for Inventory Problem (Case II)

Reorder Quantity = 30 units, Reorder Level = 20 units, Beginning Inventory = 45 units

The simulation of 35 days with an inventory policy of reordering quantity of 35 units at

the time of inventory level at the end of day is 20 units, as worked out in Table 10.27. The

table explains the demand inventory level, quantity received, ordering cost, holding cost

and shortage cost for each day.

Day

Random

Number

(Demand)

Demand

Random

Number

(Lead

Time)

Lead

Time

(Days)

Inventory

at end of

day

Qty.

Received

Ordering

Cost

Holding

Cost

Shortage

Cost

0 - - - - 45 - - - -

1 58 7 - - 38 - - 38 -

2 45 6 - - 32 - - 32 -

3 43 6 - - 26 - - 26 -

4 36 6 73 3 20 - 50 20 -

5 46 6 - - 14 - - 14 -

6 46 6 - - 8 - - 8 -

7 70 7 - - 31 30 - 31 -

8 32 5 - - 29 - - 29 -

9 12 4 - - 25 - - 25 -

10 40 6 - - 19 - 50 19 -

11 51 6 21 2 13 - - 13 -

12 59 7 - - 38 - - 38 -

13 54 6 - - 32 30 - 32 -

14 16 4 - - 21 - - 21 -

15 68 7 - - 21 - - 21 -

16 45 6 45 2 15 - 50 15 -

17 96 10 - - 5 - - 5 -

18 33 5 - - 30 - - 30 -

19 83 8 - - 22 - - 22 -

20 77 8 - - 14 - 50 14 -

21 05 3 - - 11 - - 11 -

22 15 4 76 3 7 - - 7 -

23 40 6 - - 31 30 - 31 -

24 43 6 - - 14 - - 14 -

25 34 5 - - 20 - 50 20 -

26 44 6 - - 14 - - 14 -

27 89 9 96 4 5 - - 5 -

28 20 4 - - 1 - - 1 -

29 69 7 - - 24 30 - 24 -

30 31 5 - - 19 - 50 19 -

31 97 10 - - 9 - - 9 -

32 05 3 - - 6 - - 6 -

33 59 7 94 4 0 - - - 20

34 02 2 - - 28 30 - 28 -

35 35 5 - - 23 - - 23 -

Total 300 683 20


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for Management 15.7 LET US SUM UP

By going through this lesson it is very true and clear that simulation is a reflection of a

real system representing the characteristics and behaviour within a given set of conditions.

The most important point in simulation is that simulation technique is considered as a

valuable tool because of its wide area of application. The most important approach to

solving simulation is the Monte Carlo Simulation which can be solved with the help ofprobabilistic and deterministic model. The deterministic simulation mode, have the

alternatives clearly known in advance where as the probabilistic model is stochastic in

nature and all decisions are made under uncertainty.

15.8 LESSON-END ACTIVITIES

1. Apply the Monte Carlo Simulation technique weather in forecasting.

2. In the corporate the top Bosses use to take major decisions apply the Simulation

techniques in designing and performing organisations take an industry like Reliance,

Tata, Infosys to support your answer.

15.9 KEYWORDS

Simulation : A management science analysis that brings into play a

construction and mathematical model that represents a real-

world situation.

Random number : A number whose digits are selected completely at random.

Flow chart : A graphical means of representing the logic of a simulation

model.

15.10 QUESTIONS FOR DISCUSSION

1. Write True or False against each statement

(a) Simulations models are built for management problems and require

management input.

(b) All simulation models are very expensive.

(c) Simulation is best suited to analyse complex & large practical problem

(d) Simulation-generate optimal solution.

(e) Simulation model can not be very expensive.

2. Fill in the blank

(a) Simulation is one of the most widely used ________ analysis book.

(b) Simulation allow, for the ________ of real world complications.

(c) System ________ in similar to business gaming.

(d) Monte Carlo method used ________ number.

(e) Simulation experiments generate large amount of ________ and information.

3. Briefly comment on the following

(a) The problem tackled by simulation may range from very simple to extremely

complex.(b) Simulations allows us to study the interactive effect of individual components

or variables in order to determine which one is important.


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for Management4. The materials manager of a firm wishes to determine the expected mean demand

for a particular item in stock during the re-order lead time. This information is

needed to determine how far in advance to re-order, before the stock level is

reduced to zero. However, both the lead time, and the demand per day for the item

are random variables, described by the probability distribution.

Manually simulate the problem for 30 re-orders, to estimate the demand during

lead time.

5. A company has the capacity to produce around 300 bikes per day. Daily production

varies from 295 to 304 depending upon getting the clearance from the final inspection

department. The probability distribution of bikes passed through final inspection

per day is given below:

The finished bikes are transported in a long trailer lorry sufficient to accommodate

300 mopeds. Simulate the process for 10 days and find:

a. The average number of bikes waiting in the factory yard.

b. The average empty space in the lorry.

6. In a single pump petrol station, it was observed that the inter-arrival times and

service times are as given in the table. Using the random numbers given, simulate

the queue behaviour for a period of 30 minutes and estimate the probability of thepump being idle and the mean time spent by a customer waiting to fill petrol.

Use the following random numbers: 93, 14, 72, 10, 21, 81, 87, 90, 38, 10, 29, 17, 11,

68, 10, 51, 40, 30, 52 & 71.

Production per day Probability

295 0.03

296 0.04

297 0.10

298 0.20

299 0.25

300 0.15

301 0.09

302 0.07

303 0.05

304 0.02

Inter-arrival time Service time

Minutes Probability Minutes Probability

1 0.10 2 0.10

3 0.17 4 0.23

5 0.35 6 0.35

7 0.23 8 0.22

9 0.15 10 0.10

Lead time (days) Probability Demand / day (units) Probability

1 0.45 1 0.15

2 0.30 2 0.25

3 0.25 3 0.40

4 4 0.20


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Simulation

ServiceNo. of TV sets requiring service

Frequency of request

1 15

2 15

3 20

4 25

5 25

Servicing doneNo. of TV sets serviced

Frequency of service

1 10

2 30

3 20

4 15

5 25

Type of bowling Probability of hitting a boundary

Over pitched 0.1

Short-Pitched 0.3

Outside off stump 0.2

Outside leg stump 0.15

Bouncer 0.20

Attempted Yorker 0.05

7. A one-man TV service station receives TV sets for repair. TV sets are repaired on

a first come, first served basis. The observations of the study made over a 100

day period are given below.

Simulate a 10 day period of arrival and service pattern.

8. ABC company stocks certain products. The following data is available:

a. No. of Units: 0 1 2 3

Probability: 0.1 0.2 0.4 0.3

b. The variation of lead time has the following distribution

Lead time (weeks): 1 2 3

Probabilities: 0.30 0.40 0.30

The company wants to know (a) how much to order? and (b) when to order ?

Assume that the inventory in hand at the start of the experiment is 20 units and 15

units are ordered closed as soon as inventory level falls to 10 units. No back orders

are allowed. Simulate the situation for 25 weeks.

9. A box contains 100 balls of which 20 percent are white, 30 percent are black and

the remaining are red. Simulate the process for drawing balls at random from the

box, identify and note the colour and then replace. Use the following 10 random

numbers to simulate: 52, 60, 02, 3379, 79, 30, 36, 58 and 43.

10. Rahul, the captain of the cricket team, has the following observations on the number

of runs scored against type of ball. The bowling probability of a bowler for the type

of balls bowled are given below.


18/18

51 2


for ManagementThe number of runs scored off each type of ball is shown in the table given below:

Simulate the game for 3 overs (6 balls per over) and calculate the batting average

of Rahul.

15.12 MODEL ANSWERS TO QUESTIONS FOR

DISCUSSION

1. (a) True (b) False (c) True (d) False (e) False2. (a) Quantitative (b) Inclusion (c) Simulation (d) Random (e) Data

15.13 SUGGESTED READINGS

Ernshoff, J.R. & Sisson, R.L. Computer Simulations Models, New York Macmillan

Company.

Gordon G.,System Simulation, Englewood cliffs N.J. Prentice Hall.

Chung, K.H. Computer Simulation of Queuing SystemProduction & Inventory

Management Vol. 10.

Shannon, R. I. Systems Simulation. The act & Science. Englewood Cliffs, N.J. Prentice

Hall.

Type of bowling Probability of hitting a boundary

Over pitched 1

Short-Pitched 4

Outside off stump 3

Out side leg stump 2

Bouncer 2

Attempted Yorker 0

unit- iv (simulation)

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