Monte Carlo Simulation

Monte Carlo Simulation

What is simulation?

To simulate is to imitate.Simulation involves developing a model of a real

phenomenon and then experimenting. A given system is imitated and the variables and constants associated with it are manipulated in the artificial environment to examine the behavior of the system.

There are 4 phases of a simulation process- Definition of problem and statement of objective. Construction of appropriate model. Experimentation with model created Evaluation of results of simulation

Monte Carlo Simulation

Imitate the behavior of the stochastic process.

Technique of solving problems through random numbers.

Used to solve a variety of problems involving stochastic situations (one where some or all parameters of a problem are described by random variables)

Used for solving problems involving decision making under uncertainty

Monte Carlo Analysis

Monte Carlo analysis is a technique that computes, or iterates, the project cost or project schedule many times using input values selected at random from probability distribution of possible costs or duration to calculate a distribution of possible total project cost or completion dates.

The benefit of simulation from the viewpoint of the analyst stems from the fact that the results of taking a particular course of action can be estimated prior to the implementation in the real world. Instead of using hunches and intuition to determine what may happen, the analyst using simulation can test and evaluate various alternatives and select the one that gives the best results.

Steps-

From the given probability of occurrence of events, establish cumulative probability.

Assign tag nos. to events in such a way that tag nos. represent cumulative probability.

Obtain random nos. from a random no. table.

Correlate random nos. with tag nos. assigned to the events and identify the value for respective events.

Random numbers

A random number generator (often abbreviated as RNG) is a computational or physical device designed to generate a sequence of numbers or symbols that lack any pattern, i.e. appear RANDOM.

Hardware-based systems for random number generation are widely used, but often fall short of this goal, though they may meet some of the statistical tests for randomness intended to ensure that they do not have any easily discernible patterns.

Methods for generating random results have existed since ancient times, including dice, coin flipping, the shuffling of playing cards, and many other techniques.

It is observed that demand for a product varies in a random fashion. The demand per day is observed to have the following probability.

Demand 25 33 42 51Probability .15 .25 .45 .15

Simulate the demand for the ensuing 15 days using Monte Carlo Simulation.

The random nos. are 40, 92, 47, 01,60,05, 69,79,09, 66,77,

69,45,18,93

Demand varies in a random fashion. Demand may get influenced by both internal

and external factors.Demand Probability Cum P R.N interval.

25 .15 .15 0 to 1433 .25 .40 15 to 39

42 .45 .85 40 to 84 51 .15 1.00 85 to 99 Random nos. are assigned from 0-99, both nos.

inclusive. There will be 100 tag nos. representing a cumulative P of 1.

Choose a set of 2 digit random nos. for conducting a series of trials. More the no. of random nos., more the accuracy.

Trial no.

Random no.

Simulated demand/day

Trial no.

Random no.

Simulated demand/ day

1 40 42 11 77 42

2 92 51 12 69 42

3 47 42 13 45 42

4 01 25 14 18 33

5 60 42 15 93 51

6 05 25

7 69 42

8 79 42

9 09 25

10 66 42

Advantages of simulation

Has capacity to lend itself to problems that are cumbersome or impossible to handle mathematically using analytical methods.

The technique allows analysts to experiment with the system behavior without subjecting it to the risks that would be inherent in the experimenting with the real system.

Also compresses time to enable the manager to visualize long term effects in a quick manner.

Often used to test proposed analytical solutions.

Disadvantages of simulation

Does not represent a methodology for derivations of optimal solutions to the given problems.

This approach is designed merely to provide a characterization of the behavior of the system in general for a given set of inputs.

Yields only estimates which are subject to sampling error.

Is restricted to situations which contain elements which can be described by random variables.

Example

A company manufactures 30 units per day. The sale of these items depends on demand which has the following distribution.

Sales Distribution 27 0.10

28 0.1529 0.2030 0.3531 0.1532 0.05

The production cost and sale price of each unit are Rs. 40 and Rs. 50 respectively. Any unsold product is to be disposed off at a net loss of Rs.15 per unit. There is a penalty of Rs. 5 per unit if the demand is not met.

Using the following random numbers, estimate the total profit/loss for the company for the next ten days.10,99,65,99,95,01,79,11,16,20

If the company decides to produce 29 units per day, what is the advantage or disadvantage to the company?

Assignment of random nos.

Sales Prob Cum Prob RNI-------------------------------------------------------------27 0.10 0.10 00-0928 0.15 0.25 10-2429 0.20 0.45 25-4430 0.35 0.80 45-7931 0.15 0.95 80-9432 0.05 1.00 95-99

Now we simulate demand for next 10 days using the given random nos.

From the given information we have Profit per unit sold=Rs.50-40=Rs.10 Loss per unit unsold=Rs.15 Penalty for refusing demand=Rs.5 per unit Using these inputs, the profit/loss for ten days is

calculated , first when production is 30 units per day, and then when it is 29 units.

Total profit for 10 days=2695, when 30 units are produced.

If the company decides to produce 29 units per day, the total profit works out to be the same.=Rs. 2695

Day R.N Estimated sales

1 10 28

2 99 32

3 65 30

4 99 32

5 95 32

6 01 27

7 79 30

8 11 28

9 16 28

10 20 28

30 units 29 units

28*10-2*15=250 28*10-1*15=265

30*10-2*5=290 29*10-3*5=275

30*10=300 29*10-1*5=285

290 275

290 275

225 240

300 285

250 265

250 265

250 265

Assignment

Navguide corporation , a small electronic firm, manufactures a navigational instrument used on sailboats. Demand for the instrument is probabilistic, and a review of past records has yielded the weekly demand distribution as shown in table 1.

Navguide is considering the purchase of a sophisticated industrial robot to be used in the assembly of the instrument. Three different robots are being considered, each having different capacities, production efficiencies, and purchase costs. Table 2 summarizes the no. of instruments that could be manufactured each week on a regular time & overtime basis, the expected production cost per unit , and the overhead costs. Given that Navguide sells the instrument for $ 1,800, management wants to determine which robot to purchase so as to maximize weekly profit.

Demand Probability

10 0.10

20 0.14

30 0.26

40 0.24

50 0.18

60 0.08

TABLE 1

Robot 1 Robot 2

Regular time capacity(units)/week

30 40

Overtime cap. (units)/ week

30 40

Regular time cost/unit

$1,200 $1,100

Overtime cost/unit $1,600 $1,450

Overhead cost/week $10,000 $15,000

TABLE 2

Random Numbers: 44, 46, 85, 99, 09, 95, 22, 87, 64, 50.

Demand Probability

Cumulative probability

Montecarlo no. range

10 0.10 0.10 00-09

20 0.14 0.24 10-23

30 0.26 0.50 24-49

40 0.24 0.74 50-73

50 0.18 0.92 74-91

60 0.08 1.00 92-99

Simulated values for weekly demand

Mapping of random nos.

44, 46, 85, 99, 09, 95, 22, 87, 64, 50.

---------------------------------------------------------Week Demand Cost(Robot1) Robot2

1 30 (1200*30)+10,000=46,000 48,000

2 30 46,000 48,000

3 50 (1200*30)+(1600*20)+10,000=78,000 73,500

4 60 (1200*30)+(1600*30)+10,000=94,000 88,000

5 10 (1200*10)+10,000=22,000 26,000

6 60 94,000 88,000

7 20 (1200*20)+10,000=34,000 37,000

8 50 78,000 73,500

9 40 36000+16000+10,000=62,000 59,000

10 40 62,000 59,000

Weekly Profit= (1800*Weekly Demand)-Weekly Cost