Simulation : Stochastic Programming

Simulation : Stochastic Programming 

DEFINITION

Simulation is a representation of reality through the use of model or other device, which will react in the same manner as reality under a given set of conditions.

CLASSIFICATION OF SIMULATION MODELS

Simulation models are classified as:

(a) Simulation of Deterministic models:

In the case of these models, the input and output variables are not permitted to be random variables and models are described by exact functional relationship.

(b) Simulation of Probabilistic models:

In such cases method of random sampling is used. The techniques used for solving these models are termed as Monte-Carlo technique.

(c) Simulation of Static Models:

These models do not take variable time into consideration.

(d) Simulation of Dynamic Models:

These models deal with time varying interaction.

ADVANTAGES OF SIMULATION

Simulation is a widely accepted technique of operations research due to the following reasons:

* It is straightforward and flexible.

* It can be used to analyze large and complex real world situations that cannot be solved by conventional quantitative analysis models.

* It is the only method sometimes available.

* It studies the interactive effect of individual components or variables in order to determine

which ones are important.

* Simulation model, once constructed, may be used over and again to abnalyze all kinds of different situations.

* It is the valuable and convenient method of breaking down a complicated system into subsystems and their study. Each of these subsystems works individually or jointly with others.

LIMITATIONS OF SIMULATION TECHNIQUE

* Since simulation model mostly deals with uncertainties, the results of simulation are only reliable approximations involving statistical errors, optimum results cannot be produced by simulation.

* In many situations, it is not possible to identify all the variables, which affect the behaviour of the system.

* In very large and complex problems, it is very difficult to make the computer program in view of the large number of variables and the involved inter–relationship among them.

* For problems requiring the use of computer, simulation may be comparatively costlier and time consuming in many cases.

* Each solution model is unique and its solutions and inferences are not usually transferable to other problems, which can be solved by other techniques.

MONTE-CARLO SIMULATION

The Monte-Carlo method is a simulation technique in which statistical distribution functions are created using a series of random numbers. Working on the digital computer for a few minutes we can create

data for months or years. The method is generally used to solve problems which cannot be adequately represented by mathematical models or where solution of the model is not possible by analytical method.

Monte-Carlo simulation yields a solution, which should be very close to the optimal, but not necessarily the exact solution. But this technique yields a solution, which converges to the optimal solution as the number of simulated trials tends to infinity. The Monte-Carlo simulation procedure can be summarized in the following steps:

Step 1: Clearly define the problem:

(a) Identify the objectives of the problem.

(b) Identify the main factors, which have the greatest effect on the objective of the problem.

Step 2: Construct an approximate model:

(a) Specify the variables and parameters of the mode.

(b) Formulate the appropriate decision rules, i.e. state the conditions under which the experiment is to be performed.

(c) Identify the type of distribution that will be used. Models use either theoretical distributions or empirical distributions to state the patterns of the occurrence associated with the variables.

(d) Specify the manner in which time will change.

RANDOM NUMBERS

Random number is a number in a sequence of numbers whose probability of occurrence is the same as that of any other number in that sequence

Pseudo-Random Numbers

Random numbers which are generated by some deterministic process but which satisfy statistical test for randomness are called Pseudo-random number

 Generation of Random Numbers

Using some arithmetic operation one can generate Pseudo-random numbers. These methods most commonly specify a procedure, where starting with an initial number called seed is generates the second number and from that a third number and so on. A number of recursive procedure are available, the most common being the congruence method or the residue method. This method is described by the expression:
ri + 1 = (ari + b) (modulo m),
Where a, b and m are constants, ri and ri +1 are the ith and (i + 1)th random numbers. The expression implies multiplication of a by ri and addition of b and then division by m. Then ri +1 is the remainder or residue. To begin the process of random number generation, in addition to a, b and m, the value of r0 is also required. It may be any random number and is called seed.

Examples 

Numerical 1) 

A sample of 100 arrivals of automobiles at a toll booth is found to be according the following distribution.

 

Step 4

Sum of waiting time of driver = 2.5 min

Sum of idle time = 8.5 min

1)  Total elapse time = 24.5 min

2) % Driver waiting time = (2.5/24.5)*100 = 10.20

3) %Idle time of service = (8.5/24.5)*100 = 34.69

4) Average waiting time of driver = 2.5/10 = 0.25 min

5) Average idle time of service = 8.5/10 = 0.85 min