Simulation Chapter 16 of Quantitative Methods for Business, by Anderson, Sweeney and Williams Read sections 16.1, 16.2, 16.3, 16.4, and Appendix 16.1

COMPUTER SIMULATION What is it? Creating a computer model of a system Subjecting the model to various scenarios Statistically analyzing the results in order to predict how the real system will behave

EXAMPLE #1--coin tossing A gambling game where a coin is tossed repeatedly until the number of heads and number of tails differ by 3. You pay $100 for each toss. You collect $800 at the end of each game. Do you want to play?

EXAMPLE #1 By hand simulation of the game: Heads Tails Payoff

EXAMPLE #1 Suppose we simulate 10 games and find the number of tosses to be: 5,13,7,5,9,5,3,11,5, and 7. Avg. # of tosses / game = 7 Do you want to play?

EXAMPLE #1 To simulate this game on Excel, we need to get Excel to generate “heads” and “tails” randomly. The RAND() function =RAND() causes Excel to generate a (uniformly distributed) random value between 0 and 1 in that cell.

EXAMPLE #1 How could we randomly generate “Heads” or “Tails”? The IF() function =IF(logical condition,value if true,value if false) Using these tools, create a model which will simulate the gambling game.

Homework Status At this point you should be able to complete the following homework problems in Chapter 16: #1

Generating random values What about generating the weather on a given day if the probability of “rainy” is .5, the probability of “cloudy” is .3, and the probability of “clear” is .2? How could we use Excel to generate outcome from the throw of a single die? How could we use Excel to generate the outcome from the throw of a single die, if the die had “A”, “B”, “C”, “D”, “E”, “F” on its faces?

Generating random values To generate a new random value, just press F9. F9 actually “recalculates” all values on the workbook. This is done automatically whenever you change any cell in the workbook.

Building a Monte Carlo Simulation Model Identify the random values of interest, i.e. the things you model will need to randomly generate. (Outcome of flipping a coin, time a customer requires to complete a transaction, number of machine failures on a given day, etc.)

Building a Monte Carlo Simulation Model Define a mapping for the desired random values, in such a way that the probabilities are observed. May map from “the numbers on [0,1)”, if using Excel’s rand() function, or May map from “the X-digit integers” if using a random number table or multisided die

Building a Monte Carlo Simulation Model Define the logic associated with the system (and model).

Example #2-- Machine failures You are asked to recommend whether your company should purchase a maintenance agreement for the copy machine. Maintenance agreement costs $1000 per year and covers service call charges for all unscheduled maintenance. Without agreement, unscheduled maintenance calls cost $50 each.

Example #2-- Machine failures You find that the copy machine has breakdowns in this manner: In any given day, the probability of one machine failure is .03, the probability of two failures is .01, and the probability of 3 or more failures is 0.

Example #2-- Machine failures 1. What are the RANDOM VALUES in this situation? # of failures on any given day

Example #2-- Machine failures 2. MAPPING: Since the probabilities have at most 2 significant digits, we might generate 2-digit random numbers. We might map them to the “number of failures in a day” like this: 00-96 map to 0 failures 97-97 map to 1 failure 99 map to 2 failures

Example #2-- Machine failures 3. LOGIC (TO SIMULATE ONE YEAR OF OPERATION): Start Let DAY =1 (DAY will tell what day of the year it is) Generate and record # of failures for the day (using mapping). Is DAY = 365? If no, then add one to DAY and go to step 3. If yes, then stop and compute the total number of failures.

Example #2-- Machine failures Once the model is built, we run it a number of times, and perform statistical analysis of the results in order to predict how the real system will behave.

Example #2-- Machine failures To run the system, we generate random numbers (using a computer or a table or whatever). For our first run, we might choose to generate 2-digit random numbers from, a pair of 10-sided dice. Alternatively we could use, the numbers from some section of a Random Number table like Table 16.2 (p. 720 of your text)

Example #2-- Machine failures We can keep track of our observations in a table or chart. Day # random # # of failures 1 2 3 4 5 and so on...

Example #2-- Machine failures After 365 (simulated) days, we could count the number of failures, and compute the cost of service calls with and without maintenance agreement. We could run the model a number of times and then average the observations to reach a conclusion. (In actuality, we’d probably build a confidence interval for the mean # of failures per year.)

Example #2-- Machine failures For example suppose the number of failures per (simulated) year were: 24, 20, 19, 18, 19, 16, 14, 17, 14,17 Estimated average number of failures per year (from this simulation) = 17.8

Example #2-- Machine failures Cost of paying $50 per maintenance call times average of 17.8 failures per year = $890 per year. Cost of maintenance contract = $1000. per year. Conclusion: Don’t buy the maintenance contract.

Example #2-- Machine failures Why didn’t each (simulated) year get the same number of failures? What conclusion would we have reached if we’d stopped after 2 (simulated years)? How many years should we simulate?

How many times to run a simulation model? Make several (N0) preliminary runs of the model, recording the value of the quantity you wish to predict, and then computing the sample standard deviation (s). The total estimated number of runs required to get a (1- α) confidence interval of width ≤ W is given by: N ≥ (t2 * s2) / (W/2)2 Where t is the t-statistic for N0-1 degrees of freedom and α/2 , where α=level of significance

Homework At this point you should be able to complete the following homework problems in Chapter 16: #3 #6 #8 #13

Random Number Options in Excel VLOOKUP to generate samples from a discrete distribution To generate samples from a discrete distribution, we store the description of the distribution in 2 columns: The first holds the cumulative probability distribution. (offset by one row and starting with zero) The second holds the values that the random variable can take

Random Number Options in Excel In the cells which are to hold the randomly generated values, we specify =VLOOKUP(RAND(), area where table describing distribution is stored, 2) The first argument is a randomly generated value between 0 and 1 The third argument is the column number (in the table) where the values are stored

EXAMPLE #3--Inventory Consider an inventory system which uses a reorder quantity model. Suppose the assumptions of EOQ, EPLS, Shortages, or Quantity Discounts models are not satisfied. Specifically suppose demand is not “known and constant”. How to determine a good lot size, Q?

EXAMPLE #3--Inventory Items cost $10 each and annual holding rate is 45%. Fixed cost to place an order is $15. Weekly demand is distributed in this way: Weekly Demand Probability 9 .1 10 .2 11 .4 12 .2 13 .1 The company is considering using lot sizes of 50, 75, and 100. Use simulation to determine which lot size tends to give the lowest annual costs.

Homework At this point you should be able to complete the following homework problems in Chapter 16: #15

Random Number Options in Excel How to generate samples from a Normal distribution with mean μ and standard deviation σ? =NORMINV(R, μ, σ) How to generate samples from a Uniform distribution on [a,b) =a+(b-a)*R where R is a random number on [0,1)

EXAMPLE #4--Inventory Items cost $10 each and annual holding rate is 45%. Fixed cost to place an order is $15. Weekly demand is normally distributed with mean of 22 units and a standard deviation of 5 units. The company is considering using lot sizes of 50, 75, and 100. Use simulation to determine which lot size tends to give the lowest annual costs.

EXAMPLE #4--Inventory Items cost $10 each and annual holding rate is 45%. Fixed cost to place an order is $15. Weekly demand is uniformly distributed between 15 and 20 units. The company is considering using lot sizes of 50, 75, and 100. Use simulation to determine which lot size tends to give the lowest annual costs.

Homework At this point you should be able to complete the following homework problems in Chapter 16: #17 #21 Dates assignment (on Blackboard)

Waiting Line Models Up until now, the examples we have considered could be considered “static” models, in that they have involved independent trials In the Machine Failure problem, the number of failures on one day did not affect the number of failures on any other day. In the Inventory examples, the demand in one week did not affect the demand in another week. How can we create a model of a system in which that’s not the case? That is, what if the state of the system changes as a result of one event, so that the state of the system is different when the next event occurs?

Waiting Line Models Interarrival time Probability .5 minutes 10% Consider a customer service system with one server Customers arrive with interarrival times distributed like this: Service times are distributed normally with mean=1.5 minutes, and standard deviation = .5 minutes Create a simulation model which simulates 1000 customer arrivals and computes Average wait time # of customers who must wait % of customers who must wait Interarrival time Probability .5 minutes 10% 1 minute 20% 1.5 minutes 40% 2 minutes 2.5 minutes

Waiting Line Models Challenge: Modify your model for 2 servers