(Monté Carlo) Simulation MGS 3100 – Chapter 4

What is Simulation? A definition of simulation from Dictionary.com Imitation or representation, as of a potential situation or in experimental testing. The basic idea is to build an experimental device, or simulator, that will “act like” the system of interest … in a quick, cost-effective manner. Representation of the operation or features of one process or system through the use of another: computer simulation of an in-flight emergency. 3

Simulation in Business Analysis Uses mathematical models Probabilistic (as opposed to deterministic) Uses the entire range of possible values of a variable in the model Imitates a system or situation (like a coin-flip, or how long a person might have to wait in a line at a restaurant)

Why Simulate? Safety – flight simulator Cost – easier to simulate adding a new runway and find out effects than to implement in reality and then find out Time – Boeing uses simulated manufacturing before the real thing, with tremendous savings in time and money – can discover parts that do not fit and fix them before actual production

How does it work? Simulation requires you to know What variable is to be simulated Is the variable discrete or continuous? The distribution of the variable – values it can take on and the probabilities of those values occurring. Step 1: Generate a variable containing uniformly distributed random variables between 0 and 1 (the rand() function in Excel). Step 2: Create a rule to map the random numbers to values of the variable desired in the right proportion, and apply the rule.

Types of Variables in Simulation Models Discrete Used for simulating specific values or specific points. Example: Number of people in a waiting line (queue). Continuous Used for simulating any value (between specific points) Example: The amount of time a person spends in a queue. 8

Discrete Example 1– coin toss Variable to be simulated is “Outcome of a coin toss”. It takes on values “Heads” and “Tails”, each with 0.5 probability. Generate 100 random numbers (100 tosses of coin). Make a rule like – if random number > 0.5, then “Heads”, else “Tails”. This will create the right distribution of outcomes.

Discrete Example 2: Machine Failures Simulate machine failures based on this historical data Number of Failures per month Frequency (# of months this occurred) 1 2 3 36 20 Total 60

Discrete Example 2: Machine Failures Create the following cumulative probability table. Number of Failures per month Frequency (# of months this occurred) Probability Cumulative 1 2 3 36 20 0.600 0.333 0.050 0.016 0.933 0.983 1.000 Total 60 1.00

Discrete Example 2: Machine Failures Now map the random numbers between 0 and 1 using the cumulative prob. Column as the cutoffs. Random numbers between 0 and 0.6 represent 0 failures, between 0.6 and 0.933 represent 1 failure, and so on. 0.60 0.93 0.98 0 failures 1 failure 2 3 failures

Discrete Example 2 Solution – Random Number Mapping The random numbers are now mapped to number of failures as follows. Random # Number of Failures 0.345 0.008 0.985 0.878 3 1

Continuous Example – arrival time Variable to be simulated is arrival time at a restaurant which can literally take on infinite individual values For example someone could arrive at: 12:09:37 12:09:37:52 12:09:37:52:14, etc.

Continuous Example – arrival time con’t To simulate this situation, we must specify intervals At the restaurant the intervals could be all people arriving between 11am and 12pm, 12pm and 1pm, or 1pm and 2pm. As with the coin toss, generate random numbers in Excel ( =RAND() ) Make a rule – if random number: <=.333, then =11am-12pm >.333 up to =.666, then 12pm-1pm >.666 up to 1, then 1pm to 2pm each category is equally likely

Continuous Example – arrival time con’t If the random number is.47, then this would fall in the 12pm to 1pm category, If the random number is .88, then this would fall in the 1pm to 2pm category, etc. Because each category is equally likely, if we run enough trials, each category will contain about the same number of random numbers, which will tell the restaurant owner that it is equally likely that a person will arrive at any of the three times.

Continuous Example – arrival time con’t The owner looks at historical information and says that on an average day, 225 people eat lunch at his restaurant , and that typically 47 people arrive between 11am and 12pm 112 people arrive between 12pm and 1pm 66 people arrive between 1pm and 2pm How do we map these numbers?

Continuous Example – arrival time con’t To complete the mapping, we need to make a cumulative distribution function (CDF)

Continuous Example – arrival time con’t Make a new rule – if random number: <=.21, then =11am-12pm >.21 up to =.71, then 12pm-1pm >.71 up to 1, then 1pm to 2pm

Note on Random Numbers in Excel Spreadsheets Once entered in a spreadsheet, a random number function remains “live.” A new random number is created whenever the spreadsheet is re-calculated. To re-calculate the spreadsheet, use the F9 key. Note, almost any change in the spreadsheet causes the spreadsheet to be recalculated! If you do not want the random number to change, you can freeze it by selecting: tools, options, calculations, and checking “manual.”

Evaluating Results Simulation measures the quality of a solution because it gives the probability of a certain event occurring Simulation also shows the variability Simulation does not necessarily give the best possible answer. It gives the most likely answer. 6

Appendix Some useful information on very popular probability distribution function 6

Probability Distributions A probability distribution defines the behavior of a variable by defining its limits, central tendency and nature Mean Standard Deviation Upper and Lower Limits Continuous or Discrete Examples are: Normal Distribution (continuous) Binomial (discrete) Poisson (discrete) Uniform (continuous or discrete) Custom (created to suit a specific purpose)

Uniform Distribution All values between minimum and maximum occur with equal likelihood Conditions Minimum Value is Fixed Maximum Value is Fixed All values occur with equal likelihood Excel function: RAND() – returns a uniformly distributed random number in the range (0,1)

Normal Distribution Conditions: Use when: Uncertain variable is symmetric about the mean Uncertain variable is more likely to be in vicinity of the mean than far away Use when: Distribution of x is normal (for any sample size) Distribution of x is not normal, but the distribution of sample means (x-bar) will be normally distributed with samples of size 30 or more (Central Limit Theorem) Excel function: NORMSDIST() – returns a random number from the cumulative standard normal distribution with a mean of zero and a standard deviation of one [e.g., NORMSDIST(1) = .84]

Simulation Continuous Variables

Distributions Variables to be simulated may be normal (e.g. height) or exponential (e.g. service time) or various other distributions. Task is to convert uniform distribution to the required distribution. Freq 1 Freq infinity

Application - Queuing Systems A queuing system is any system where entities (people, trucks, jobs, etc.) wait in line for service (processing of some sort) – retail checkout lines, jobs on a network server, phone switchboard, airport runways, etc.

Queuing System Inputs Queuing (waiting line) systems are characterized by: Number of servers / number of queues SSSQ – Single Server Single Queue SSMQ – Single Server Multiple Queue MSSQ – Multiple Server Single Queue MSMQ - Multiple Server Multiple Queue Arrival Rate (Arrival Intervals) Service Rate (Service Times)

Performance Variables (outcome) Performance of a queuing system is measured by Average time waiting in queue/system Average number of entities in queue/system Time in Queue Service Time Arrival time Service Begins Service Ends Time in System

Distributions in Queuing Arrival Intervals (time between two consecutive arrivals) and Service Time (time to serve one customer) are exponentially distributed. Confirm it yourself by watching cars on a street!

Sample Problem A loading dock (SSSQ) has trucks arriving every 36 minutes (0.6 hrs) on average, and the average service (loading / unloading) time is 30 minutes (0.5 hrs). A new conveyer belt system can reduce that to 15 minutes (0.25 hours). Simulate the arrival of 200 trucks to see how performance would be affected by the new system.

Simulating Exponential Distributions To convert the uniform distribution of the random numbers to an exponential distribution, take the negative natural log of the random numbers. This creates an exponential distribution with an average of 1.00. To get an average of 0.6 (to represent average arrival interval in hours), simply multiply result by 0.6. Thus, the conversion formula is: –ln(rand())*µ where µ is the mean of the exponential distribution desired.

Sample Conversion Average: 1 infinity ….. ….. Random -ln(rand()) 0.500645 1.040982 Random -ln(rand()) 0.449796 0.798962 0.858464 0.15261 0.828061 0.188668 0.938751 0.063206 0.84637 0.166798 0.428408 0.847678 0.357574 1.028412 0.63932 0.447351 Average: 1 infinity ….. …..