Simulation

1. Simulation Summer 2013 Simulation

2. Many definitions. It is the process of studying the behavior of a real system using a computer-based model that replicates the behavior of that system. Used in situations involving probabilistic elements (e.g. random arrivals, service times and process yields) Used in situations where the complexity or the size of the problem makes it difficult to use optimizing models. Useful in both service or manufacturing systems. Simulation Simulation Process Simulation

3. Simulation characteristics In a simulation model we have “transactions” (customers, cars, ..) and “events” (arrivals, receiving service, departure,..). • When probability distributions are associated with events, we use a method called random deviate generation to get numbers from the probability distribution to simulate events. • Timing of event may or may not be important in a simulation. • For simulation of a warehouse operation, if inventory is charged on items at the end of the month, we do not need to know precise timing of withdrawal of items. We only need to know how many items were withdrawn during the month. • For simulation of toll booths, we need timings of two types of events – when each car arrives and how long it takes to pay. • When time is involved, simulation may be done by changing simulation clock in fixed increment or by changing clock from one event to the next (this is the preferred method). Simulation

4. Simulation characteristics..cont Simulation is not an optimization tool, rather we try to establish values of performance measures to arrive at better decision making. Here is an example. Suppose we would like improve customer satisfaction at a bank drive-in facility. We study arrival patterns, service times etc., and simulate the operation with one drive-in window, with two and may be with three drive-in windows. Obviously three windows will be most satisfactory from the customer point of view. But then we take into account the cost (initial and operating) and other factors to make the final decision. We will use EXCEL for some simple simulation exercises. In EXCEL we will use a function RAND() quite frequently. This function is volatile (it recalculates all the time). You should disable automatic calculations (switch to manual). Press F9 key and all values are recalculated. Simulation

5. Ex. 1 Simulate the tossing of a coin. • Model construction: No simulation clock is involved • Each transaction will be the toss of a coin. We will generate 500 transactions (an arbitrary decision). • We will assume that the coin is “fair”. The random variable X takes two values (0, 1 for T and H) with equal probabilities. • To generate of a transaction, we need a very simple formula. Generate a random number (RN). • If RN < 0.5, it’s a head; otherwise it’s a tail. • In Excel, we will use the following formula in 500 cells =IF(RAND()<0.5,"H","T“). Performance measure: We will compute the probability of tails based on our simulation to see if it is close to 0.5. Simulation

6. Ex. 1.. Excel: Simulate the tossing of a coin. Simulation

7. Generating random deviates (variates) • For every probability distribution, as the variable X goes the minimum value to the maximum value, the cumulative probability increases from 0 to 1. • Random number generators produce numbers between 0 to 1 (uniform distribution). Thus for any value of a random number, there is one matching point on the cumulative distribution of X. We match the value and generate X.       A A+1 … B-1 B  X • RAND() generates a random number (say RN, 0  RN < 1). function automatically. Discrete uniform distribution For the discrete uniform distribution , when X varies between integers A and B, Excel has a special function = RANDBETWEEN(A, B). Simulation

8. Empirical distribution We need cumulative probabilities. The logic is simple. We match RN with cumulative probability. If RN < 0.3, X = 300. If 0.3  RN < 0.7, X = 400. If 0.7  RN < 0.9, X = 500. If 0.90  RN < 1.0, X = 600 F(X) 1.0 0.0 300 400 500 600 X Suppose we pick RN = 0.632. 0.632  We get corresponding X value (= 400). Simulation

9. Using LOOKUP What is the value of X if RN = 0.632? • Suppose we use HLOOKUP to find demand corresponding to F(X) = 0.632 F(X) 1.0 0.0 • We need to use the function without exact match. • But since 0.3 < 0.632 < 0.7, Excel will match the value equal to 300. 0.632  300 400 500 600 X • To avoid this, we need to replace F(X) with some variable G(X), in which F(X) values are shifted to the right. Simulation

10. Population Dock 1 Arrival Queue Departure Ex. 2 Ships arrive in the night at a facility with two docks. If a dock is available in the morning, it is assigned to a waiting ship for the whole day and the ship leaves in the evening. If a dock cannot be assigned, there is a fee of $10,000 per day per ship. Simulate the operation and estimate the annual fee. Dock 2 Model construction: We will start with a flow chart Our model is simpler because both docks take 1 day to process. Simulation

11. Population Arrival Docks: Service 1 or 2 Departure Queue Ex. 2.. Dock simulation • Every day, we will generate new arrivals with HLOOKUP. Assume ship arrive between midnight and 6 a.m. These ships will be added to the queue. • We will assign up to 2 ships from the queue (assumed FIFO – First In First Out) and calculate remaining ships waiting. These waiting ships will incur fee for that day. • We will simulate the operation of a year and calculate the fee. • We can replicate the experiment many times. Performance measure: We will compute the annual fee. Simulation

12. Ex. 2… Excel: Dock simulation • ? • Run for 365 days • Simulation generated Simulation

13. Ex. 2…. Excel: Dock simulation Simulation

14. Ex. 2….. Excel: Dock simulation • Arrivals are generated with HLOOKUP. Simulation

15. Random Deviates: Continuous distributions Suppose X has continuous probability distribution (range 100 to 500) and we can find the cumulative distribution F(X). Every F(X) varies between 0 and 1. 1.0 0.0 F(X) 0.52 100 X 500 264 We can use random numbers (RN) to generate X values because RN also vary between 0 and 1 and there is a one to one correspondence. Suppose we pick RN = 0.52. We can find corresponding X value (say 264). Simulation

16. A X B A X B C Random Deviates…… Triangular: Let p = (B - A) / (C – A) =IF(RAND() ≤ p, X, Y) where X = A + SQRT(RAND() * (C – A) * (B – A)), Y = C - SQRT((1-RAND()) * (C – A) * (C – B)) Uniform: = A + RAND()*(B – A) Triangular distribution Uniform distribution Simulation

17. Ex. 3 Retirement Planning • You are 30 years old, and would like to invest 3000 dollars at the end of each year from now till you reach 60. • Assume interest paid to be N(12, 2) meaning normally distributed with mean = 12% and std. dev. = 2%; interest is paid at the end of year. • You would like to estimate probability of reaching the target of one million dollars at the age 60. • You would like to know chances of achieving the target if you increase the annual amount invested. Simulation

18. Ex. 3.. =B12*(1+NORMINV(RAND(), Mean_R,STDV_R))+ Annual_contr Simulation

19. Ex. 3… Retirement Planning sensitivity analysis Effect of changing contribution on the probability of achieving the desired outcome. Simulation

20. Ex. 4 An IPO is to be launched with the opening price expected to be from the distribution shown. For the next five years, the stock price is expected to increase by an amount given by a lognormal distribution with mean of 1.5% and standard deviation of 0.5% provided the company does not fail. The probability of failure is 40%, 30%, 20%, 20% and 10% during the next five years. Estimate the following using simulation: Price of the stock at the end of the 5 year period assuming the company has not failed. Probability of survival at the end of 5 years. Theoretical answer to part (b) is (1 – 0.4) * (1 – 0.3) * (1 – 0.2) * (1 – 0.2) * (1 – 0.1) = 0.24192 Simulation

21. Ex. 4.. IPO Launching Simulation

22. Ex. 4… IPO Launching How? Count failures through Y1, then Y2 – Y1, etc. Fail in Year 4: =IF(K3="Y","Y",IF(RAND()<F$15,"Y","N")) Simulation

23. Ex. 4…. IPO Launching Stock Price in Year 4: =IF(L4="N",Q4*(1+0.01*LOGINV(RAND(),Log_mean, Log_Stdv)),0) Simulation

24. Population Arrival Queue Service Departure Time Based Event Oriented Simulation We will consider the following: Transactions (customers) enter the system in a single line and are processed at a single facility (server) on a FIFO basis also called “First Come First Served” (FCFS). We will consider several different situations. First, the dock example appeared to have same flow chart but it was somewhat different. Docks were open only during day time (say from 7). This means ships arrive in the night could be considered as arriving at 7 and using the docks for fixed amount of time. Each row generated new arrivals for a new day. Simulation

25. M1 5 min M1 A simple example Population Arrival Queue Service Departure A machine take exactly 5 minutes to process a job. Our work load is only 10 jobs per hour. So we don’t need simulation, we can simply schedule a job every 6 minutes. Job 1 5 Job 2 11 Job 3 17 There is an idle time of 1 minute after every job. Machine utilization is (5/6)* 100 = 83.3%. There will be no queue. Simulation

26. M1 5 min Example 5 Population Arrival Queue Service Departure We are now going to consider different arrival and service time distributions. Note that the average time between arrival is 6 minutes and average service time is 5 minutes. Simulation

27. Enter service Set SS = 1 Set next Dep. Time Join Q Add 1 to Q When an arrival event happens, the following is checked at that point in time. Arrival Flow chart Arrival SS? SS=1 SS=0 Determine next arrival time Simulation

28. Remove first Tr. From Q , start service & set Dep. Time Shorten Q by 1 Departure Flow chart When a departure event happens, the following is checked at that point in time. Departure Q. Status Q not empty Q Empty Set SS = 0 Simulation

29. Example 5.. Single server system simulation – 4 cases Model Parameters: Each model may have many parameters. Examples are: arrival and service rates, capacities, etc. Statistics of interest /performance measures: Statistics on performance measures can be useful in validating the model and for decision making. Some examples. What’s the average waiting time? Maximum waiting time? What is the maximum queue length? What is the server utilization? How many people had to wait in queue before using the server? How many people waited more than X minutes? Simulation

30. Example 5… Single server system simulation When the simulation starts at time zero, the system is empty. Value of a performance measure such as server utilization keeps on changing as time progresses. Expected Parameter value After some time, when the process reaches a “steady state”, value of the performance measure comes close to the expected value of that measure. Parameter value 0 Time  t In all four cases since average time between arrivals is 6 minutes and the average service time (when the server is busy) is 5 minutes. The server utilization will stabilize at about 83% (=100*75/90). For better estimate of the performance measure values, we generally chop off initial observations (up to period t). For our example, we will start collecting data from observation 201 (to 1200). Simulation

31. Example 5A How did we get numbers in the table below? Simulation

32. Example 5A.. Simulation

33. Example 5B When the number of arrivals is Poisson (with 10 units / hr.), the time between arrivals is exponential (with average time = 6 minutes). Simulation

34. Example 5D.. The graph here shows that it is not easy to determine when steady state may be reached (how many observations to chop off), nor do we know how many total observations to collect in a simulation run or how many time to replicate. Simulation

35. Example 5: Comparison A B Simulation

36. Simulation: general comments One can use visual basic macros within EXCEL. Risk solver also includes some simulation capability. Many specialized simulation languages have been developed. These can handle even more complex situations. Examples: AutoMod, Arena, GASP, GPSS, Promodel, SIMSCRIPT, Simula. Many simulation software packages also come with “animation” capability (there are even “free” ones). This can make a tremendous impact in visualizing the operations. One note of caution. Impressive visual display may give some false impressions even though data used in the simulation or the simulation logic is faulty. Simulation

37. Simulation Advantages Disadvantages Flexibility. Can handle large and complex systems. Can answer “what-if” questions. Does not interfere with the real system. Allows study of interaction among variables. “Time compression” is possible. Handles complications that other methods can’t. • Can be expensive and time consuming. • Does not generate optimal solutions. • Managers must choose solutions they want to try (“what-if” scenarios). • Each model is unique. Simulation

38. ATM Simulation 1 vs. 2 Average number of arrival per hour: 40 (Poison) Average service time 75 sec. (Exponential) Simulation