SIMULATION

Simulation is to imitate reality to represent reality.

Simulation is a technique for conducting experiments

Simulation is descriptive and not optimizing technique

Simulation is a process often consists of repetition of an experiment in many, many times to obtain an estimate of the overall effect of certain actions.

Simulation is usually called for only when the problem under investigation is too complex to be treated by analytical models or by numerical optimization techniques.

In a simulation, a given system is copied and the variables and constants associated with it are manipulated in that artificial environment to examine the behaviour of the system.

In general terms, Simulation involves developing a model of some real life phenomenon and then performing experiments on the model evolved.

Often we do not find a mathematical technique that; a model once constructed may permit us to predict what will be the consequences of taking a certain action. In particular we could ‘experiment’ on the model by ‘trying’ alternative actions or parameters and compare their consequences. This ‘experimentation’ allow us to answer ‘what if’ questions relating the effects of your assumption on the model response.

The availability of the computers makes it possible for us to deal with an extraordinary large quantity of details which can be incorporated into a model and the ability to manipulate the model over many experiments (i.e. replicating all the possibilities that may be imbedded in the external world and events would seem to recur).

For example,

· Testing of an aircraft model in a wind tunnel to test the aerodynamic properties of an the model

· A model of a traffic signal system

· Military war games

· Business games for training

· Planetarium etc.

Simulation defined

A simulation of a system or an organism is the operation of a model or simulator which is a representation of the system or organism. The model is amenable to manipulation which would be impossible, too expensive or unpractical to perform on the entity it portrays. The operation of the model can be studied and for it, properties concerning the behaviour of the actual system can be inferred.

-Shubik

Simulation is the process of designing a model of a real system and conducting the experiments with this model for the purpose of understanding the behaviour (within the limits imposed by a criterion or set of criterion) for the operation of the system.

- Shannon

Steps of Simulation process Step 1: Identify the problem.

If an inventory system is being simulated, then the problem may concern the determination of the size of the order (number of units to be ordered) when the inventory falls up to the reorder level (point).

Step 2: Identify the decision variables, performance criterion (objective) and the decision rules.

In the context of the above defined inventory problem, the demand (consumption rate), lead time and safety stock are identified as the decision variable. These variables shall be responsible to measure the performance of the system in terms of total inventory cost under the decision rule- when to order.

Step 3: Construct a numerical model.

Numerical model is constructed to be analyzed on the computer. Some times the model is written in a particular simulation language which is suited for the problem under the analysis.

Step 4: Validate the model

Validation is necessary to ensure whether it is truly representing the system being analyzed and the results will be reliable.

Step 5: Design the experiments

Conduct experiments with the simulation model by listing specific values of variables to be tested (i.e. list courses of action for testing) at each trail (run).

Step 6: Run the simulation model.

Run the model on the computer to get the results in form of operating characteristics. Step 7: Examine the results.

Examine the results of the problem as well as their reliability and correctness. If the simulation is complete, then select the best course of action (or alternative) otherwise make the desired changes in the model decision variables, parameters or design and return to step 3.

Advantages of simulation

1. Simulation is a straight forward and simple technique

2. The technique is very useful to analyze large and complex problems which are not amenable to mathematical or quantitative methods.

3. It is an interactive method, which enables the decision maker to study the changes and their effects on the performance of the system.

4. The experiments in a simulation are run on the model and not on the system itself.

Limitations of simulation

1. At times simulation models can be very costly and expensive

2. It is trail and error technique to produce different solutions in repeated runs.

3. The solution obtained from the simulation may not be optimal.

4. The simulation model needs to be examined and analyzed for decision making. It only creates an alternative and not an optimal solution by itself.

Monte-Carlo techniques or Monte-Carlo simulation.

The Monte-Carlo method is a simulation technique in which statistical distribution functions are created by using series of random numbers Random numbers.

The underlying theory in random number is that, each number has an equal opportunity of being selected.

There are various ways in which random numbers may be generated. These could be: result of some device like coin or die; published table of random numbers, mid-square method, or some other sophisticated method.

It may be mentioned here that random numbers generated by some method may not be really random in nature. In fact such numbers are called pseudo-random-numbers.

Rand corporation (of USA): A million random digits, is a random number table used in simulation situations. The numbers in these tables are in random arrangement.

The Monte-Carlo simulation technique consists of the following steps.

1. Setting up a probability distribution for variables to be analyzed.

2. Building a cumulative probability distribution for each random variable.

3. Generating random numbers. Assign an appropriate set of random numbers to represent value or range (interval) of values for each random variable.

4. Conduct the simulation experiment by means of random sampling

5. Repeat Step 4 until the required number of simulation runs has been generated.

6. Design and implement a course of action and maintain control.

Example 1.

A bakery keeps stock of popular brand of cake. Previous experience shows the daily demand pattern for the item with associated probabilities, as given below:

Also estimate the daily average demand for the cakes on the basis of simulated data. lution:

Using the daily demand distribution, we obtain a probability distribution as shown in the following Table.

Example 2.

XYZ spare parts company wishes to determine the level of stock it should carry for items in its range. Demand is not certain and there is a lead time for stock replenishment. For one item X, the following information is obtained.

Stock on hand at the beginning of the simulation exercise was 20 units.

Carry out a simulation run over a period of 10 days with the objective of evaluating the inventory rule:

Order 15 units when present inventory plus outstanding falls below 15 units.

The sequence of random numbers to be used is :0,9,1,1,5,1,8,6,3,5,7,1,1,2,9 using the first number for day one.

Solution:

Let us begin the simulation by assuming that

i) orders are placed at the end of the day and received after 3 days at the end of the day.

ii) back orders are accumulated in case of short supply and are supplied when stock is available.

The cumulative probability distribution and the random number range for daily demand is shown in the table.

Example 3.

A small retailer deals in a perishable commodity, the daily demand and supply of which are random variables. The past 500 days data show the following:

 

The retailer buys the commodity at Rs.20 per kg and sells at Rs.30 per kg. If any commodity remains at the end the day it has no resale value and is a dead loss. Moreover, the loss on any unsatisfied demand is Rs.8 per kg

Given the following random numbers. Simulate six day sales.

31

18

63

84

15

79

07

32

43

75

81

27

Using the random numbers alternatively, for example, first pair (31) to simulate supply and second pair (18) to simulate demand, etc.

Solution:

Probability and Random Number Interval for Daily Demand and Supply

A study of the time required to service customers by adding up the bills, receiving payments and placing packages, yields the following distribution

Example 5.

A tourist car operator finds that during the past few months the cars use has varied so much that the cost of maintaining the car varied considerably. During the past 200 days the demand for the car fluctuated as below.

The simulated demand for the cars for the next 10 weeks period is given in the table below

Example 6.

An automobile production line turns out about 100 cars a day, but deviations occur owing to many causes. The production is more accurately described by the probability distribution given below.

Finished cars are transported across the bay at the end of each day by the ferry. If the erry has space for only 101 cars, what will be the average numbers of waiting to be shipped and what will be the average number of empty space on the ship?

Solution:

Example 7.

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

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 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 10 days.

Random Numbers: 10, 99, 65,99,95,01,79,11,16 and 20.

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

Set up the simulation for next 10 days

Since the profit is same for both the cases there is no disadvantage to the company.