1. Simulation Prepared by Amani Salah AL-Saigaly Supervised by Dr. Sana’a Wafa Al-Sayegh University of Palestine

2. Simulation  Meaning & Purpose of Simulation  Why Use Simulation?  Characteristics of the process of simulation  Simulation & Optimization  Simulation Procedures  Monte Carlo Simulation Technique  Application  Examples.

3. Meaning & Purpose of Simulation  Simulation is a quantitative approach, using a model, adopted for the purpose of verification, by appropriate experimentation, in terms of the adjustability and agreeability of the circumstantial facts of the real phenomenon to the assumed model.  Hence the process of simulation refers to evolving an artificial environment to cater to a real life situation. in this respect, simulation can be termed as imitation just like the method of mimicry that may be adopted by a person to imitate the voice or action of another person.

4. Con …  It cannot be considered as the best standard, an ideal or precise technique, but only a procedure of describing a situation through an artificial, imitative procedure.

5. Why Use Simulation? Experimental arm of operations research.  No optimization method available  Optimization algorithm takes too long  Run program to simulate system  system too big or complicated.

6. Characteristics of the process of simulation The following are the Characteristic features of the process of simulation:  To begin with, a given problem has to be clearly stated with explicit objectives.  In accordance with the defined problem and its objectives, an appropriate model has to be devised.  It is necessary to verify the devised model through repeated experimentation.  The result of the experiments should be evaluated to ascertain the appropriateness of the simulation process.  The simulation model can be either mathematical, physical.

7. Con …  It is absolutely necessary to collect data pertaining to the defined problem before starting the process of simulation.  The model developed should be capable of providing appropriate results for decision making.  A simulation model can be deterministic, in which case, the parameters are determined and constant.  There may be cases where models may be of astochastic type where the parameters are subject to variation in a random manner. This would necessitate a number of iterations for identifying the exact nature of the model.

8. Simulation & Optimization Optimization  Finds optimal answer  Is a calculation Monte Carlo Simulation  Finds feasible answer  Is an experiment.

9. Basis of Simulation Experimental arm of operations research.  Set up a mathematical model of a system  Write computer program  Develop animation routines (if required)  Run program to simulate system

10. Simulation Procedures  When in a given situation, the elements of chance prevail then the process can be covered under the probability technique. In such random behaviour situations, the method of Monte Carlo explained below can be used for dependable results.  Monte Carlo Simulation Mean: is a technique that involves using random numbers and probability to solve problems, A problem solving technique used to approximate the probability of certain outcomes by running multiple trial runs, called simulations, using random variables.

11. Monte Carlo Simulation Technique  This technique is useful for predicting the nature of behaviour of a system Monte Carlo’s Simulation procedure is essentially based on the element of chance. It is therefore called ‘Probabilistic Simulation’. Accordingly, the data involved in a problem is simulated by using random number generators.  Hence, the basic characteristics of Monte Carlo Simulation Technique are: - supposition of a model in the form of a probability distribution of the concerned variable. - it is a mechanical process based on random number generators.  The procedure of simulation on the basis of Monte Carlo Simulation Technique, can be explained with the help of the following examples.

12. Application  The process of simulation can be applied to a vriety of problems that are subject to quantification and estimation.  The technique of simulation can be advantageously used in inventory control policies.  In matters of financial planning can be applied.  It is a very useful tool that is popularly and reliably used for solving business problems.  It is applications in decision making problems is of much significance especially in avoiding unexpected risk. computer software packages can also be evolved suitably.

13. Example 1  A milk dairy’s records of sales of one liter milk packets during 100 days are as under: 171615141312111098Demand 781411813151095Number of Days Using the following random numbers simulate the demand for the first 15 days. Solution: Table 1 171615141312111098Demand.07.08.14.11.08.13.15.10.09.05 Probability(n/100) 52432839758111854467196186423Random

14. Con … Table 2 Random Number Random Number Interval Cumulative Probability ProbabilityDemand -00-04.05 8 8,1105-13.14.09 9 23,1814-23.24.10 10 2824-38.39.15 11 64,39,4339-51.52.13 12 54,5252-59.60.08 13 6460,70.71.11 14 71,81.7571-84.85.14 15 -85-92.93.08 16 9693-100.100.07 17

15. Con …  The following is the schedule of simulated demand for the first 15 days, in the order of the given random number. 151413121110987654321Day 1312111215 9913121517101410Demand Form Table 2: read the demand against each random number.

16. Example 2  In the first year M.Com. Class of a certain commerce college, the first lecture starts at 9 a.m.,Following is the probability distribution regarding number of students who are late comers for the first lecture each day. 2520151005NO. of Student coming late 0.050.100.200.300.35Probability Using the following sequence of random number, simulate the pattern for next 12 days and find average number of students coming late per day. 484445920286619565122395 Random

17. Solution : Random Number Random Number Interval Cumulative Probability ProbabilityNO. of Student coming late 23,12,0200-34.0.35.35 05 61,45,44,4835-64.65.30 10 6565-840.85.20 15 86,9285-940.95.10 20.05 25 Quiz: Complete the blanks in the table above and simulate the pattern for next 12 days and find average number of students coming late per day.

18. Solution : Random Number Random Number Interval Cumulative Probability ProbabilityNO. of Student coming late 23,12,0200-34.0.35.3505 61,45,44,4835-64.65.3010 6565-840.85.2015 86,9285-940.95.1020 95,9595-991.00.0525 121110987654321Day 10 20 05 2010251505 25 NO. of Student coming late The following is the simulate the pattern for next 12 days Total number of Student coming late =160 Average number of late comers= 160/12=13 approximately per day.