Upload
ram84322
View
231
Download
0
Embed Size (px)
Citation preview
8/13/2019 UNIT- IV (Simulation)
1/18
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.
8/13/2019 UNIT- IV (Simulation)
2/18
49 6
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
8/13/2019 UNIT- IV (Simulation)
3/18
49 7
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
8/13/2019 UNIT- IV (Simulation)
4/18
49 8
Quantitative Techniques
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
8/13/2019 UNIT- IV (Simulation)
5/18
49 9
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.
Table 15.9: Simulation Problem
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
8/13/2019 UNIT- IV (Simulation)
6/18
50 0
Quantitative Techniques
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:
Table 15.13: Simulation Problem
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
8/13/2019 UNIT- IV (Simulation)
7/18
50 1
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
8/13/2019 UNIT- IV (Simulation)
8/18
50 2
Quantitative Techniques
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
8/13/2019 UNIT- IV (Simulation)
9/18
50 3
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
8/13/2019 UNIT- IV (Simulation)
10/18
8/13/2019 UNIT- IV (Simulation)
11/18
50 5
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 -
8/13/2019 UNIT- IV (Simulation)
12/18
50 6
Quantitative Techniques
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
8/13/2019 UNIT- IV (Simulation)
13/18
8/13/2019 UNIT- IV (Simulation)
14/18
50 8
Quantitative Techniques
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.
8/13/2019 UNIT- IV (Simulation)
15/18
8/13/2019 UNIT- IV (Simulation)
16/18
51 0
Quantitative Techniques
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
8/13/2019 UNIT- IV (Simulation)
17/18
51 1
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.
8/13/2019 UNIT- IV (Simulation)
18/18
51 2
Quantitative Techniques
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