Simulation Models 16

Introduction Simulation can be used to analyze a wide variety of problems. The applications can be grouped into four general areas: Operations models Financial models Marketing models Games of chance Simulation models can yield important insights in all of these areas.

Operations Models In the operations of both manufacturing and service companies, there is likely to be uncertainty that can be modeled with simulation. Examples include: Bidding for a government contract (uncertainty in the bids by competitors) Warranty costs (uncertainty in the time until failure of an appliance) Drug production (uncertainty in the yield and timing)

Bidding for Contracts In situations where a company must bid against competitors, simulation can often be used to determine the company’s optimal bid. Usually the company does not know what its competitors will bid, but it might have an idea about the range of the bids its competitors will choose. Simulation can be used to determine a bid that maximizes the company’s expected profit.

Example 16.1: Contract Bidding.xlsx (slide 1 of 3) Objective: To simulate the profit to Miller from any particular bid, and to see which bid amount is best. Solution: Miller Construction Company must decide whether to make a bid on a construction project. Miller assesses that the cost to complete the project has a triangular distribution with minimum, most likely, and maximum values $9000, $10,000, and $15,000. Miller also assesses that the cost to prepare the bid has a triangular distribution with parameters $300, $350, and $500. Four potential competitors are going to bid against Miller, and the lowest bid wins the contract. Miller believes that each potential competitor will bid, independently of the others, with probability 0.5. Miller also believes that each competitor’s bid will be a multiple of Miller’s most likely cost to complete the project, where this multiple has a triangular distribution with minimum, most likely, and maximum values 0.9, 1.3, and 1.8, respectively. If Miller decides to prepare a bid, its bid amount will be a multiple of $500 in the range $10,500 to $15,000.

Example 16.1: Contract Bidding.xlsx (slide 2 of 3) First, simulate the number of competitors who will bid and then simulate their bids. Then for any bid Miller makes, see whether Miller wins the contract, and if so, what its profit is. The simulation model is shown below.

Example 16.1: Contract Bidding.xlsx (slide 3 of 3) To run the simulation, set the number of iterations to 1000, and set the number of simulations to 10 because there are 10 bid amounts Miller wants to test. The summary results and a histogram of profit with a $13,000 bid are shown below.

Warranty Costs When you buy a new product, it usually carries a warranty. A typical warranty might state that if the product fails within a certain period such as one year, you will receive a new product at no cost, and it will carry the same warranty. If the product fails after the warranty period, you have to bear the cost of replacing the product. Due to random lifetimes of products, the manufacturer needs a way to estimate the warranty costs of a product.

Example 16.2: Warranty Costs.xlsx (slide 1 of 4) Objective: To use simulation to estimate the number of replacements under warranty and the total NPV of profit from a given sale, using a discount rate of 8%. Solution: Yakkon Company sells a popular camera for $400. This camera carries a warranty such that if the camera fails within 1.5 years, the company gives the customer a new camera for free. If the camera fails after 1.5 years, the warranty is no longer in effect. Every replacement camera carries exactly the same warranty as the original camera, and the cost to the company of supplying a new camera is always $225.

Example 16.2: Warranty Costs.xlsx (slide 2 of 4) Yakkon estimates the distribution of time until failure from historical data, which indicates a right-skewed distribution, as shown in the figure to the right. This is a commonly used distribution called the gamma distribution. It is characterized by two parameters, α and β. These determine its shape and location.

Example 16.2: Warranty Costs.xlsx (slide 3 of 4) The simulation model is shown below.

Example 16.2: Warranty Costs.xlsx (slide 4 of 4) The @RISK setup is typical. Run 1000 iterations of a single simulation (because there is no RISKSIMTABLE function). The @RISK summary statistics and histograms for the two outputs are shown below.

Drug Production with Uncertain Yield In many manufacturing settings, products are produced in batches, and the usable yields from these batches are uncertain. This is particularly true in the drug industry.

Example 16.3: Drug Production.xlsx (slide 1 of 3) Objective: To use simulation to determine when Wozac should begin production for this order so that there is a high probability of completing it by the due date. Solution: Wozac Company has recently accepted an order from its best customer for 8000 ounces of a new miracle drug, and Wozac wants to plan its production schedule to meet the customer’s promised delivery date of December 1. The drug must be produced in batches, and there is uncertainty in the time required to produce a batch, which could be anywhere from 5 to 11 days. This uncertainty is described by the discrete distribution in the table below. Wozac believes the yield can be modeled by a triangular distribution with minimum, most likely, and maximum values equal to 600, 1000, and 1100 ounces. All batches must go through a rigorous inspection once they are completed. The probability that a typical batch passes inspection is only 0.8. If the batch fails inspection, none of it can be used to fill the order.

Example 16.3: Drug Production.xlsx (slide 2 of 3) Simulate successive batches and keep a running total of the usable ounces obtained so far. IF functions can then be used to check whether the order is complete or another batch is required. Keep track of the days required to produce all of the batches needed to meet the order and then “back up” to see when production must begin to meet the due date. The completed model appears below.

Example 16.3: Drug Production.xlsx (slide 3 of 3) Set the number of iterations to 1000 and the number of simulations to 1, and then run the simulation as usual. After running the simulation, obtain the histograms of the number of batches required and the number of days required, as shown below.

Financial Models There are many financial applications where simulation can be applied. Future cash flows, future stock prices, and future interest rates are some of the many uncertain variables financial analysts must deal with.

Financial Planning Models Many companies use simulation in their capital budgeting and financial planning processes. Simulation can be used to model the uncertainty associated with future cash flows, including questions such as: What are the mean and variance of a project’s net present value (NPV)? What is the probability that a project will have a negative NPV? What are the mean and variance of a company’s profit during the next fiscal year? What is the probability that a company will have to borrow more than $2 million during the next year?

Example 16.4: New Car Development.xlsx (slide 1 of 3) Objective: To simulate the cash flows from the new car model, from the development time to the end of its life cycle, so that GF can estimate the NPV of after-tax cash flows from this car. Solution: General Ford (GF) Auto Corporation is developing a new model of compact car. This car is assumed to generate sales for the next five years. GF has gathered the following information about the car:

Example 16.4: New Car Development.xlsx (slide 2 of 3) The model is like most financial multiyear spreadsheet models. The completed model extends several years to the right, but most of the work is for the first year or two. From that point, copy to the other years to complete the model. The completed model for GF appears below.

Example 16.4: New Car Development.xlsx (slide 3 of 3) Set the number of iterations to 1000 and the number of simulations to 1, and then run the simulation as usual. After running @RISK, obtain the histogram shown below. The second slider has been positioned at its default 5th percentile setting. Financial analysts often call this percentile the value at risk at the 5% level, or VaR 5%, because it indicates nearly the worst possible outcome.

Cash Balance Models All companies track their cash balance over time. As specific payments come due, companies sometimes need to take out short-term loans to keep a minimal cash balance.

Example 16.5: Cash Balance.xlsx (slide 1 of 3) Objective: To simulate Entson’s cash flows and the loans the company must take out to meet a minimum cash balance. Solution: Entson Company believes that its monthly sales from November of the current year to July of next year are normally distributed with the means and standard deviations given in the table below. Each month Entson incurs fixed costs of $250,000. In March, taxes of $150,000 and in June taxes of $50,000 must be paid. Dividends of $50,000 must also be paid in June. Entson estimates that its receipts in a given month are a weighted sum of sales from the current month, the previous month, and two months ago, with weights 0.2, 0.6, and 0.2: Materials and labor needed to produce a month’s sales must be purchased one month in advance, and the cost of these averages to 80% of the product’s sales.

Example 16.5: Cash Balance.xlsx (slide 2 of 3) At the beginning of January, Entson has $250,000 in cash, and the company wants to ensure that each month’s ending cash balance never falls below $250,000. This means that Entson might have to take out short-term (one-month) loans. The interest rate on a short-term loan is 1% per month. At the beginning of each month, Entson earns interest of 0.5% on its cash balance. The completed simulation model is shown to the right.

Example 16.5: Cash Balance.xlsx (slide 3 of 3) Set the number of iterations to 1000 and the number of simulations to 1. Then run the simulation in the usual way. After running the simulation, obtain the summary results and the summary trend chart shown below.

Investment Models Individual investors typically want to choose investment strategies that meet some pre-specified goal, such as a retirement goal.

Example 16.6: Retirement Planning.xlsx (slide 1 of 3) Objective: To use simulation to estimate the value of Sally’s future investments, in today’s dollars, from several investment strategies in T-bills, T-bonds, and stocks. Solution: At age 25, Sally Evans has 40 years until retirement. She plans to invest $1000 at the beginning of each of the next 40 years. Each year, she plans to put fixed percentages—the same each year—of this $1000 into stocks, Treasury bonds (T-bonds), and Treasury bills (T-bills). These percentages are called investment weights. She has the historical data shown in the table below (with some rows hidden).

Example 16.6: Retirement Planning.xlsx (slide 2 of 3) Think of each historical year as a possible scenario, where the scenario specifies the returns and inflation factor for that year. Then for any future year, randomly choose one of these scenarios. Because more recent scenarios should have a greater chance of being chosen, give a weight to each scenario, starting with 1 for 2007. Then the weight for any year is a damping factor multiplied by the weight from the next year. The simulation model, with damping factor 0.98, is shown to the right.

Example 16.6: Retirement Planning.xlsx (slide 3 of 3) Set the number of iterations to 1000 and the number of simulations to 3 (one for each set of investment weights to be tested). Then run the simulation as usual. Summary results and the histogram for simulation 1 (put 80% in stocks) is shown below.

Marketing Models There are plenty of opportunities for marketing departments to use simulation. They face uncertainty in the brand-switching behavior of customers, the entry of new brands into the market, customer preferences for different attributes of products, the effects of advertising on sales, and so on.

Models of Customer Loyalty What is a loyal customer worth to a company? This is an extremely important question for companies. Companies know that if customers become dissatisfied with the company’s product, they are likely to switch and never return. Marketers refer to this customer loss as churn. The loss in profit from churn can be large, particularly because long-standing customers tend to be more profitable in any given year than new customers.

Example 16.7: Customer Loyalty.xlsx (slide 1 of 3) Objective: To use simulation to find the NPV of a customer and to see how this varies with the retention rate. Solution: CCAmerica is a credit card company. The first year a customer signs up for service typically results in a loss to the company because of various administrative expenses. However, after the first year, the profit from a customer is typically positive, and this profit tends to increase through the years. The company has estimated the mean profit from a typical customer to be as shown in column B to the right. Assume that the actual profit from a customer in the customer’s nth year of service is normally distributed with mean shown in the table to the right and standard deviation equal to 10% of the mean.

Example 16.7: Customer Loyalty.xlsx (slide 2 of 3) At the end of each year, the customer leaves the company, never to return, with probability 0.15, the churn rate, or stays with probability 0.85, the retention rate. The company wants to estimate the NPV of the net profit from any such customer who has just signed up for service at the beginning of year 1, at a discount rate of 15%, assuming that the cash flow occurs in the middle of the year. It also wants to see how sensitive this NPV is to the retention rate. Keep simulating profits for the customer until the customer churns. The simulation model is shown below. It simulates 30 years of potential profits, but this could be varied.

Example 16.7: Customer Loyalty.xlsx (slide 3 of 3) Set the number of iterations to 1000 and the number of simulations to 5 (one for each potential retention rate). Then run the simulation as usual. Summary results for all five retention rates, as well as a line chart of mean NPV and years loyal as a function of the retention rate, are shown below.

Example 16.8: Free Maintenance.xlsx (slide 1 of 3) Objective: To use simulation to see whether it makes sense for Jamesons to give a free maintenance agreement to DVD player purchasers. Solution: Jamesons, which sells electronic appliances, is considering giving customers a free maintenance agreement with each purchase of a DVD player. The unit profit without free maintenance is currently $20. The company believes this will decrease to $16 with free maintenance. Prior to this year, 50,000 customers were loyal to Jamesons, and 100,000 customers were loyal to their competitors. The uncertain quantities and their parameters are given in the table below.

Example 16.8: Free Maintenance.xlsx (slide 2 of 3) Jamesons is hoping that the decrease in unit profit from the free maintenance agreement will be more than offset by the higher loyalty percentage. Using a 15-year planning horizon, does the NPV of profits with a 10% discount rate confirm the company’s hopes? Compare two simulations, one without free maintenance and one with it. Because they are so similar, you can use RISKSIMTABLE to run both simulations. The completed simulation model is shown below.

Example 16.8: Free Maintenance.xlsx (slide 3 of 3) Set up @ RISK to run 1000 iterations and 2 simulations, one for each maintenance decision to be tested. Then run the simulation as usual. The summary measures for the two simulations are shown below.

Marketing and Sales Models The next example is a model for marketing and selling condos. In this model, the main issue is the timing of sales. A deterministic model of this timing can provide very misleading results.

Example 16.9: Selling Condos.xlsx (slide 1 of 4) Objective: To develop a simulation model that allows us to see how the uncertain timing affects the monetary outcomes for Pletcher, and to compare this simulation model to a deterministic model with no uncertainty about the timing of sales. Solution: Blackstone Development Company has just finished building 12 high-end condos, each priced at $300,000. Blackstone has hired Pletcher Marketing to market and sell these condos. Pletcher will incur all of the marketing and maintenance costs, assumed to be $800 per unsold condo per month, and it will receive a 10% commission ($30,000) from Blackstone at the time of each condo sale. Because Blackstone wants these condos to be sold in a timely manner, it has offered Pletcher a $200,000 bonus at the end of the first year if at least half of the condos have been sold, and an extra $500,000 bonus at the end of the second year if all of the condos have been sold. Pletcher estimates that it can sell five condos per month on average, so it should be able to collect the bonuses. However, it also realizes that there is uncertainty about the number of sales per month.

Example 16.9: Selling Condos.xlsx (slide 2 of 4) To make a fair comparison between a deterministic model with five sales per month and a simulation model with uncertainty in the timing of sales, we need a discrete distribution for monthly sales that has mean 5. Use the Poisson distribution, which is discrete and has only one parameter, the mean. As shown below, the Poisson distribution with mean 5 has virtually no probability of values larger than 15.

Example 16.9: Selling Condos.xlsx (slide 3 of 4) The deterministic model (not shown) sells all condos by the end of year 2, receives both bonuses, and realizes an NPV (including bonuses) of $2,824,333—but it is not very realistic. The simulation model, modeled through 40 months, is shown below.

Example 16.9: Selling Condos.xlsx (slide 4 of 4) Set @RISK to run 1000 iterations for a single simulation. Then run the simulation in the usual way. Distributions of the three outputs are shown below.

Simulating Games of Chance It is instructive to see how simulation can be used to analyze games of chance, including sports contests. Many analysts refer to Monte Carlo simulation, which comes from the gambling casinos of Monte Carlo.

Simulating the Game of Craps Most games of chance are great candidates for simulation because they are driven by randomness. One such game is the game of craps, which is played as follows: A player rolls two dice and observes the sum of the two sides turned up. If the sum is 7 or 11, the player wins immediately. If the sum is 2, 3, or 12, the plays loses immediately. If the sum is any other number, that number becomes the player’s point. Then the dice are thrown repeatedly until the sum is the player’s point or 7. If the player’s point occurs before a 7, the player wins. If a 7 occurs before the point, the player loses.

Example 16.10: Craps.xlsx Objective: To use simulation to find the probability of winning a single game of craps. Solution: There are no input numbers, only the rules of the game. The simulation model, shown below, is for a single game. Simulate 40 tosses and use only those that are necessary to determine the outcome of a single game. Set the number of iterations to 10,000 (to obtain a very accurate answer) and the number of simulations to 1. Then run the simulation as usual. The estimate of the probability of winning is shown in cell J8.

Example 16.11: March Madness Men 2009.xlsm (slide 1 of 3) Objective: To simulate the NCAA basketball tournament and keep a tally on the number of times each team wins the tournament. Solution: At the time this example was written, the most recent NCAA Basketball Tournament was the 2013 tournament. On the Sunday evening when the 68-team field was announced, all that was known were the pairings (which teams would play which other teams) and the team ratings, based on Jeff Sagarin’s rating system, portions of which are shown to the right. Sagarin predicts that the actual point differential in a game will be the difference between the ratings of the two teams. Assume that the actual point differential is normally distributed with mean equal to Sagarin’s prediction and standard deviation 10. If the actual point differential is positive, team A wins. If it is negative, team B wins.

Example 16.11: March Madness Men 2009.xlsm (slide 2 of 3) A portion of the simulation model sheet is shown below. (Of course, Florida did not win the tournament; this figure just shows one possible scenario.)

Example 16.11: March Madness Men 2009.xlsm (slide 3 of 3) Some of the results based on 1000 iterations are shown below.

An Automated Template for @RISK Models (slide 1 of 2) The macro language for Excel®, VBA, can be used to automate @RISK. An automated template that can be used for any simulation is shown below. (See the file Simulation Template.xlsm.)

An Automated Template for @RISK Models (slide 2 of 2) The text boxes provide the motivation and instructions, but there are two basic ideas: First, you often have particular inputs you would like to vary in a sensitivity analysis. Once you specify these in the Inputs section, the program will run a separate simulation for each combination of the input values. Second, you typically have outputs that you want to summarize in certain ways. The Outputs section lets you specify the summary measures you want for each of your outputs. The program then lists the results on separate worksheets. It is still up to you to develop the logic of the simulation, but you no longer need to worry about RISKSIMTABLE functions or statistical functions.