MCom I Semester Statistical Analysis Simulation Study Material Notes

MCom I Semester Statistical Analysis Simulation Study Material Notes:  Meaning and Definition of simulation Important Elements of Simulations  types of Simulations System and Simulations Models Advantages of simulation Classification of Simulation models Main Causes of Using the Simulation Technique Ratinale Simulation Why to Use Simulation :

BCom 2nd Year Advance Payment Tax Study Material Notes in Hindi

Simulation

It is evident that there are many problems of real life which cannot be represented mathematically due to the stochastic nature of the problem, the complexity in problem formulation, or the conflicting ideas needed to properly describe the problem under study. Under such circumstances simulation is often used when all else fail. This method is often viewed as a “method of last resort”.

Simulation is one of the most used O.R. techniques as it is a versatile tool which tends itself to the solution of a large variety of O.R. problems which are otherwise difficult to solve. It is a technique (quantitative or otherwise) for carrying out experiments for analysing the behaviour and evaluating the performance of a proposed system under assumed condition of reality.

The simulation technique has long been applied by the analysts and designers in physical sciences and it has now become an important tool for dealing with the complicated problems of managerial decision making.

The first important application of simulation was probably made by John Von Neumann and Stanislaw Ulam for determining the complicated behaviour of neutrons in a nuclear shielding problem being too complex for mathematical analysis. After getting the remarkable success of this technique on neutron problem, it became more popular and found many applications in business and industry. In early 1950s, the development of digital computer further increased the rapid progress in the simulation techniques.

Meaning and Definition of Simulation

In fact, simulation is the representative model for real situations. While visiting some trade fairs and exhibitions we often find a number of simulated environments therein. For example, a children’s cycling part with various signals and crossings in the exhibition is a simulated (represented) model of city-traffic in real system. Also, a simple example is the testing of an aircraft model in a wind tunnel from which the performance of the real aircraft is determined for being fit under real operating conditions. In the laboratories, we often perform a number of experiments on simulated models to predict the behaviour of the real system under true environments. The environments in a museum of natural history and in a geological garden are also good examples of simulation.

Another idea of simulation is involved in flight simulators for training pilots. A computer directs the student’s handling of the controls in a simulated aeroplane Ilight deck. The instruments are then operated by the computer to give the same readings which they would in a real flight. An instructor can intervene with catastrophes’ like an engine failure or a bad storm and a television camera is moved over a model of some countryside to give the trainee visual feedback of now the aircraft is behaving.

The combination of computing and simulation has also resulted in the production of TV games. Players interrupt the way a computer program moves various images around the screen from a keyboard or hand-held controller. The computer incorporates their responses into these movements in accordance with the rules of the particular game. Incidentally, such programs make intensive use of random numbers to find the deflection of tennis balls, the positioning of hostile space ships, etc.

In all these example, we have tried to represent the reality to observe-what would happen under real operating situations. Thus, such representation of reality, which may be either in physical form or in a mathematical equations form, may be called simulation.

Simulation is nothing more or less than the technique of performing sampling experiments on the model of the system. The experiments are done on the model rather than on the real system itself because the latter would be too inconvenient, expensive, time-consuming and cumbersome. The experiments on real system may disrupt the system unnecessarily causing enormous troubles to all concerned. In many cases necessary computations cannot be performed analytically at all. Simulation then is the only way out. For simulation, we need a model which represents an image of reality as we see it. Some important definitions of simulation are given below:

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

Shannon “Simulation is a numerical technique for conducting experiments on a digital computer, which involves certain types of mathematical and logical relationships necessary to describe the behaviour and structure of a complex real-world system over extended period of time”

-T.H. Taylor “X simulates Y’ is true if and only if (a) X and Y are formal systems; (b) Y is taken to be the real system; (c) X is taken to be an approximation to the real system and (d) the rules of validity in X are non-error-free’ otherwise X will become the real system.”

Churchman Conclusion : Simulation is the name given to “a quantitative technique used for evaluating alternative courses of action based upon facts and assumptions with a computerised mathematical model in order to represent actual decision making under conditions of uncertainty”.

Important Elements of Simulation

(1) Technique of Decision-making under Uncertainty : Simulation is a quantitative technique for decision-making under the conditions of uncertainty.

(2) Tool of Operation Research : Infact, simulation is the important tool of operation research.

(3) Used in Complex Problems : This technique is used to solve the such complicated problems which cannot be solved by mathematical, analytical and iterative methods.

(4) Setting up of a Model : Simulation is a technique that involves setting up a model of real situation and then performing experiments on the model.

(5) Appropriate Experiments : Simulation amounts to a trial and error, heuristic and muddling through process of making decisions on very complex problems 101 which there are no optimal solution.

(6) Use of Digital Computer : Under this technique computers are widely used to set up a model and for complicated calculations.

(7) Use of Random Numbers : Under this technique, probability distribution and random numbers are used.

(8) Evaluation of Alternative Courses of Action and Continuous Improvement : Through simulation, one can study the effects of certain information, organisational and environmental changes of the operation of a system by making alternatives in the model of the system and by observing the effects of these alternatives on the system’s behaviour.

(9) Approximate Solution : By applying this technique, we can get only approximate solution. It is not possible to get the optimal solution.

(10) Use in Various Fields : In modern time, this technique is used in various fields like Aerodynamic, Space travel, Nuclear science, Biological sciences and Socio-economic fields etc. But this technique is mostly applied to take the best possible managerial decisions in business management.

Types of Simulation

Simulation is mainly of following two types :

(1) Analogue Simulation (Environmental Simulation) : When we simulate the real situations in physical form, it is known as “Analogue Simulation’. Since this simulation is accordance to the real environment, hence it is also known as Environmental Simulation’. The model of space travel, aerodynamic testing, pilots training and video games are the main examples of ‘Analogue Simulation’. This type of simulation is not only much expensive but also more time consuming.

(II) Computer Simulation (System Simulation): For the complex and intricate problems of managerial decision making, the analogue simulation may not be applicable, and the actual experimentation with the system may be uneconomical also. Under these situations, the complex system is formulated into a mathematical model for which a computer programme is developed, and then the problem is solved by using high speed electronic computer. Such type of simulation is called a computer simulation or system simulation.

System and Simulation Models

System is a collection of interacting distinct objects or components whereas simulation models are mathematical in nature. Simulation models consist of (i) components, (ii) variables, (iii) parameters, and (iv) functional relationships. Components of the model depends upon the nature of system being simulated. Variables in models relate component to another. These variables could be classified as input variables, which may be controllable or non-controllable; output variables. which are controllable in nature; and, status variables, which tell the system status at a point in time. Parameters define the probability density function of input variables. Functional relationships describc the interaction of variables and components of model.

Donald G. Malcolm, “a simulated model may be defined as one which depicts the working of a large scale system of men, machines, mater information operating over a period of time in a simulated environment of the actual real world conditions.”

Hence, during the course of a simulation, the model mimics the important elements of what is being simulated. A simulation model may be a physical or mathematical model, a mental conception, or a combination. Many simulations involve physical model. Examples include a scaled down model of an aeroplane or ship constructed of wood or other material. Since physical models are relatively expensive to build, mathematical models are often preferred. In such a model, mathematical symbols or equations are used to represent the relationships in the system.

Classification of Simulation

Models Broadly speaking, simulation models can be classified on the following basis :

(1) Classification of simulation models on the basis of nature of environment.

(2) Classification of simulation models on the basis of behaviour during period of time.

(1) Classification of Simulation Models on the basis of Nature of Environment: On the basis of nature of environment, the simulation models can be classified into following two categories :

(a) Deterministic Models : In these models, input and output variable are not permitted to be random variables and models are described by exact functional relationship.

(b) Stochastic or Probabilistic Models : In these models, input and output variables are allotted random numbers and at least one of the variables of functional relationship is given by probability functions.

(2) Classification of Simulation Models on the basis of Behaviour During Period of Time : On the basis of behaviour during period of time, the simulation models can be classified into following two categories :

(a) Static Model : These models do not take variable time into consideration.

(b) Dynamic Model : These models deal with time-varying interaction.

It should be noted that ‘Stochastic (Probabilistic) Models’ and ‘Dynamic Models’ are widely used in the field of business management.

Simulation Study Material Notes

Main Causes of Using the Simulation Technique

or

Rationale of Simulation

or

Why to Use Simulation ?

Whenever the characteristics like uncertainty, complexity, dynamic interaction between the decision and subsequent event and the need to develop detailed procedures and finally divided time intervals, all combined together in one situation. then model becomes too complex to be solved by any of the techniques of mathematical programming and probabilistic models. Then such complex model must be analysed by some other kind of quantitative technique, which may give quite accurate and reliable results. Many new techniques are investigated so far, but among all the best available is simulation’.

In general, the simulation technique is a dependable tool in situations where mathematical analysis is either too complex or too expensive. In short, following are the reasons for adopting simulation in place of other known mathematical techniques :

(1) Simulation techniques allow experimentation with a model of the system rather than the Actual Operating System: Sometimes experimenting with the actual system itself could prove to be too expensive and in several cases too disruptive. For example, if we compare two different ways of providing food service in a hospital, the confusion that may arise from operation of two different systems long enough to get valid observations might be too great. Similarly, the operation of a large computer centre under a number of different operating alternative might be too costly to be feasible.

(2) Sometimes there is no sufficient time to allow the system to operate extensively : Simulation models can be used to obtain operating characteristic estimates in much less time than require to gather the same operating data from a real system.

For example, if we want to study long term trends in world population, it is not possible to wait for desired number of years to see the results. Simulation allows to incorporate time into an analysis. In a computer situation of business operation, the manager can compress the result of several years or periods into a few minutes of running time.

(3) The non-technical manager can comprehend simulation more easily than a complex mathematical model : Simulation does not require simplifications and assumptions to the extend needed in analytical solutions. A simulation model is easier to explain to management personnel since it is a description of behaviour of some system or process.

(4) Simulation is useful in sharpening managerial decision-making skills through gaming : A descriptive model that relates managerial decisions to important operating characteristics such as profits, market share, etc. can be developed. Gaming also enables managers to experiment with new ideas without disrupting normal operations.

(5) Simulation models can be used to conduct experiments without disrupting real system : Experimenting with a real system can be very costly. It would be unreasonable to go through the expense of purchasing and installing a new flexible manufacturing system without first estimating its benefits in detail from an operating perspective. A simulation model can be used to conduct experiments for a fraction of the cost of installing such a system.

(6) The Use of Simulation enables a manager to provide insights into certain managerial problems where Analytical Solution of a Model is not possible or where the Actual Environment is difficult to observe : For example, simulation is widely used in spaceflights or the charting of satellite.

Simulation Study Material Notes

Advantages of Simulation

Some important advantages of the simulation techniques are as follows:

(1) Comparatively Easier Technique : The simulation is an easier technique to use than mathematical models and thus, considered quite superior to the mathematical analysis. Simulation technique has the advantage of being relatively free from complicated mathematics and thus, can be easily understood by the operating staff and also by non-technical managers.

(2) Flexible Technique : Simulation models are comparatively flexible and can be modified to adjust the variation in the environments of real situations.

(3) Availability of Valuable Informations : Simulation models provide the valuable informations with regards to different parameters related to operation of actual system and real problems.

(4) Lesser Cost and Lesser Risk : When the problems are of such a nature that it is risky to attempt straight optimal solutions or when it is not advisable to experiment with reality itself, simulative models are helpfui. Before making major complex decisions on commitment of resources, or before installing complex and expensive systems, simulative models can be constructed to visualise their characteristics and consequences and understand their implications more thoroughly. Simulation can serve as a ‘pre-service test’ to trace out new policies and decision rules for operating system before running the risk of experimenting on the real system.

(5) Prediction of Unknown Difficulties and Bottlenecks : When new elements are introduced into a system, simulation can be used to anticipate bottlenecks and other problems that may arise in the behaviour of the system. In other words, by simulation technique, the management can forsee the difficulties and bottlenecks that may arise due to addition of new machines, equipment or process. Thus, this technique eliminates the need of costly trial and error methods of trying out the new concept on real methods and equipment.

(6) Saving of Time in Computer Simulation : Computer simulation can compress the performance of a system over several years and involving large calculations into a few minutes of computer running time.

(7) Useful in Training : It is always advantageous to train the people on simulated models before putting into their hands the costly real system. Simulated exercises have been developed to make the trainee expert and experienced. On account of this personal attachment, the trainee gains sufficient confidence, and moreover becomes familiar with data processing on electronic computer.

(8) Verification of Results : Through simulation, one can study the effects of a certain informational organizational and environmental changes on the operation of a system by making alterations in the model of the system and by observing the effects of these alterations on the system’s behavior.

(9) Not Interfere with the Real-world System : Simulation do not interfere with the real-world system because with simulation, experiments are done with the model, not on the system itself.

(10) Useful in Complex Real-world Situations : Simulation can be used to analyse large and complex real world situations that cannot be solved by conventional quantitative analysis models. Sometimes simulation is the only available method.

Simulation Study Material Notes

Limitations or Drawbacks of Simulation Technique

Although many operations research analysts consider the simulation as a method of last resort and use it only when all other techniques fail. If the problem can be well represented by a mathematical model, the analytical method is considered to be more economical, accurate and reliable. But in the case of very large and complex problems simulation may suffer the similar drawbacks as other mathematical

models. The limitations of the simulation technique can be briefly outlined as follows:

(1) Simulation Technique does not provide Optimal Solution : Optimum results cannot be produced by simulation. Since the model mostly deals with uncertainties, the results of simulation are only reliable approximations involving statistical errors.

(2) Effective Only under Conditions of Uncertainty : All situations cannot be evaluated using simulation. Only the situations involving uncertainties can be tackled by simulation. Because, without a random component, all simulated experiments would provide the same answer.

(3) Difficulty in Quantification of Variables : The another difficulty lies in the quantification of the variables. In many situations, it is not possible to quantify all the variables which affect the behavior of the system.

(4) Costly and Time Consuming Technique : Simulation is comparatively costlier and time consuming method in many situations.

(5) Unsuitable in a Large Number of Complex Variable : In very large and complex problems, it becomes difficult to make the computer program on account of large number of variables and the involved inter-relationships among them. The number of variables may be too large and may exceed beyond the capacity of the available computer.

Simulation Study Material Notes

Monte Carlo Simulation

The Monte Carlo technique has become so much important part of simulation models that the terms are often assumed to be synonymous. “Monte Carlo” is the code name given by Von Neumann and S.M. Ulam to the technique of solving problems which are quite expensive for experimental solution and too much difficult for analytical treatment.

Monte Carlo method uses random numbers and is used to solve problems which involve conditions of uncertainty and where mathematical formulation is impossible. Monte Carlo simulation is a substitute for the mathematical evaluation of a model.

This technique is restricted for application involving random numbers to solve deterministic and stochastic problems. The principle of this technique is replacement of actual statistical universe by another universe described by some assumed probability distribution and then sampling from this theoretical population by means of random numbers.

In fact, this process is the generation of simulated statistics (random variables) that can be explained in simple terms as choosing a random number and substituting this value in standard probability density function to obtain random variable or simulated statistics. When probability density function is not standard for a given process we build empirical probability density functions alongwith the likely values or process parameters. The random number is generated either on a computer or is picked up from a table and then the value is compared with cumulative probability and likely value of process parameter is obtained.

Simulation Study Material Notes

Conclusion : Monte Carlo technique is a simulation technique which is based on the use of random numbers and standard or empirical probability distributions and which is used for solving those complex problems where physical experimentation is impracticable and where mathematical evaluation is impossible.

Simulation Study Material Notes

Generation of Random Numbers

A random number is a number in a sequence of numbers whose probability of occurrence is same as that of any other number in the sequence. Suppose we are interested in one digit numbers 0, 1, 2, …,9. There are in all ten numbers. We want to generate numbers such that any of the ten numbers is equally likely to be generated. In other words, the chance (probability) that a given number is generated on any trial should be 1/10.

Monte Carlo simulation requires the generation of a sequence of random numbers that is an integral part of the simulation model. This sequence of random numbers help in choosing random observations (samples) from the probability distribution. Random numbers are assigned in such a manner that their proportion is exactly equal to the probability distribution.

Simulation Study Material Notes

Methods of Generation of Random Numbers : Random numbers may be obtained by the following methods :

(1) Manual Methods: These methods usually involve such devices as roulette wheels, dice rolling, card shuffling, dice rolling etc. One way to generate random numbers is to fix up a spinning arrow on a common clock. When the arrow is spun, the number on which it stons would be taken to be random number for that trial. Naturally, any number of spinnings of the arrow would result in an equal number ul random numbers. Although simple, these are very slow methods and cannot meet the practical requirements where a large number of random numbers may be needed.

(2) Use of Tables : A most fast and convenient method is to make use of the published tables of random numbers, like the one published by the Rand Corporation (of USA): A Million Random Digits. A random number table is a very efficient way to generate random data in most situations. The numbers in these table are considered to be truely random numbers because these were generated using some random physical process.

While choosing the random numbers from the table, the starting point on the table is immaterial. We may start with any number in any column or row, and proceed in the same column or row to the next number, but a consistent, unvaried pattern should be used in drawing random numbers. We should not jump from one number to another indiscriminately.

(3) Computer-generated Random Numbers: Perhaps the most common method to obtain random numbers is to generate them by a computer program. These numbers lie between 0 and 1 (0 and 100%) in conjunction with the cumulative probability distribution of a random variable, including 0, but not 1.

Pseudo-Random Numbers True randomness implies complete dependence on chance process and total lack of predictability. Hence, when the random numbers are generated by deterministic mathematical process, they cannot be truely random numbers.

Random numbers are called pseudo-random numbers when they are generated by some deterministic process but they qualify the pre-determined statistical test for randomness. Infact, these numbers are always predictable, repetitive and reproducible. The sequence of numbers generated by such process is completely determined by the input data or the first random number) used for the method.

“Pseudo random numbers are numbers that are generated in such a way that they act like true random numbers and cannot be detected as non-random by any reasonable statistical test. The advantage of pseudo random numbers is that a given sequence can be reproduced for the purpose of checking results (of simulation) and making comparisons among alternative models and procedures.”-Ronald C. Gulezian

For example, suppose that we want to generate four digit integers on digital computers by mid-square method and the last number generated was 4,567. To obtain the next number, in the sequence, we will square the last one number generated and use the middle four digits of the product. In this case the product is 20857489 so that the next pseudo number is 8,574. The next new numbers in the sequence are 5134, 3579, 8092, … etc. Thus, using this method, having drawn up a suitable computer programme, a four-digit number may be fed into the computer and a list of pseudo-random numbers is obtained.

Some Important Applications of Simulation in Business Management

Monte Carlo simulation has been applied to a wide diversity of problems ranging from queuing process, inventory problem, risk analysis concerning a major capital investment such as the introduction of a new product, expansion of the capacity and many other problems. Budgeting is another area where simulation can be very useful. In fact, the system of flexible budgeting is an exercise in simulation. Simulation can as well be used for preparing the master budget through functional budgets. We now discuss a few applications in detail :

(1) Application in Complex Queuing/Waiting Line Problems : Perhaps the major use of computer based Monte Carlo simulation model has been in the solution of complex queuing problem. The GPSS (General Purpose Systems Simulator) program developed by IBM (International Business Machines) is a most popular queuing simulation tool. As the assumptions required for solving queuing problems analytically are quite restrictive, for most realistic queuing systems, simulation may actually be the only approach available.

Simulation Study Material Notes

(2) Application to Inventory Control : In order to provide adequate service to customers, the reorder point must be chosen with proper consideration of demand during lead time. If both the lead time and demand of inventory per unit of time are random variables, then simulation technique can be used to me estigate the effect of different inventory policies (e.g., different combinations of order quantity and reorder point) on a probabilistic inventory system.

(3) Application to Capital Budgeting : An important decision of financial analysis is to select the optimum alternative among various capital investment policies and evaluation of risk involved with the specific decision. The main purpose of evaluating the risk is to determine the effect of various factors (e.g., selling price, market growth rate, market size etc.) on financial parameters. The samples from the probability distributions of the factors involved can be drawn and analysed to determine the rate of return on investment.

While evaluating the rising investments, Prof. David B. Hertz (1964) proposed the use of a simulation model to obtain the expected return for an investment proposal. The method involves the following steps:

Step 1. To develop the probability distribution of uncertain factors.

The probability distributions are developed on the basis of assessment of the probable outcomes. The following factors are taken into consideration for evaluating an investment proposal.

(i) Market size

(ii) Selling price

(iii) Market growth rate

(iv) Share of market

(v) Investment required

(vi) Residual value of investment

(vii) Operating costs

(viii) Fixed costs

(ix) Useful life of facilities

Step 2. To generate a set of random numbers.

Simulation Study Material Notes

A set of random numbers is generated and respective probability distributions are assigned to each factor in the system for calculating the expected values of these factors. Then the values of all the factors are combined to determine the rate of return for the combination.

Step 3. To simulate the process.

Above process is repeated several times to simulate a clear portrayal of investment risk.

(4) Application in Project Management-PERT/CPM: A number of network simulation models related to project management have also been developed. For example, we can determine the critical path even if we are given randomly selected activity-time for each activity. Repeating this process large number of times, the probability distribution for project completion time can be determined. Also we can determine the probability that each given activity is on the critical path.

(5) Other Applications : Simulation has a variety of other applications except the applications in business management. It can be used for planning military strategy, traffic control, medical diagnosis, hospital emergency facilities, gambling and analysis, location analysis, e.g., determining optimal location for plants and warehouses etc. Thus there is a wide range of applications of simulation technique.

Simulation Study Material Notes

Simulation Procedure or Methodology for Simulation

A simulation study involves several stages of activities. In short, simulation study involves the following steps:

(1) Identify or Define the Problem : The first step in problem solving of any situation is to identify and clearly define the problem and list the objective(s) that the solution is intended to achieve. This is true of simulation as well. A clear statement not only facilitates the development of an appropriate model but also provides a basis for evaluation of the simulation results.

(2) Analyse Costs and Benefits : Since a simulation study can be very expensive, it is advisable to review the probable costs and possible benefits of such a study before proceeding too far. Some solution method rather than simulation might be quicker or less expensive to use. The remaining steps should be taken if a simulation study is found to be desirable.

(3) Abstract the real system into a model i.e., Construct a Simulation Model: The next step in simulation is the development of a suitable model. During the course of a simulation, the model mimics the important elements of what is being simulated. A simulation model may be a physical or mathematical model, a mental conception, or a combination. Many simulations involve physical models. Examples include a scaled down model of an aero plane or ship constructed out of wood or other material. Since physical models are relatively expensive to build, mathematical Models are offer Preferred In such a model mathematical Symbols or equations are used to Represent the Relationships in the System :

Simulation Study Material Notes