## 1. Develop a simulation model for 52 weeks of operation

1. Develop a simulation model for 52 weeks of operation at Ebony. Graph the inventory of soap over time. What is the total cost (inventory cost plus production change cost) for the 52 weeks?

2. Run the simulation for 500 iterations to estimate the average 52-week cost with values of U ranging from 30 to 80 in increments of 10. Keep L = 30 throughout.

3. Report the sample mean and standard deviation of the 52-week cost under each policy. Using the simulated results, is it possible to construct valid 95% confidence intervals for the average 52-week cost for each value of U? In any case, graph the average 52-week cost versus U. What is the best value of U for L 5 30?

4. What other production policies might be useful to investigate?

Management of Ebony, a leading manufacturer of bath soap, is trying to control its inventory costs. The weekly cost of holding one unit of soap in inventory is \$30 (one unit is 1000 cases of soap). The marketing department estimates that weekly demand averages 120 units, with a standard deviation of 15 units, and is reasonably well modeled by a normal distribution. If demand exceeds the amount of soap on hand, those sales are lost—that is, there is no backlogging of demand. The production department can produce at one of three levels: 110, 120, or 130 units per week. The cost of changing the production level from one week to the next is \$3000. Management would like to evaluate the following production policy. If the current inventory is less than L 5 30 units, they will produce 130 units in the next week. If the current inventory is greater than U 5 80 units, they will produce 110 units in the next week. Otherwise, Ebony will continue at the previous week’s production level. Ebony currently has 60 units of inventory on hand. Last week’s production level was 120.

## At the beginning of each week, a machine is in

At the beginning of each week, a machine is in one of four conditions: 1 = excellent; 2 = good; 3 5 average; 4 5 bad. The weekly revenue earned by a machine in state 1, 2, 3, or 4 is \$100, \$90, \$50, or \$10, respectively. After observing the condition of the machine at the beginning of the week, the company has the option, for a cost of \$200, of instantaneously replacing the machine with an excellent machine. The quality of the machine deteriorates over time, as shown in the file P10_41.xlsx. Four maintenance policies are under consideration:

■ Policy 1: Never replace a machine.

■ Policy 2: Immediately replace a bad machine.

■ Policy 3: Immediately replace a bad or average machine.

■ Policy 4: Immediately replace a bad, average, or good machine.

Simulate each of these policies for 50 weeks (using at least 250 iterations each) to determine the policy that maximizes expected weekly profit. Assume that the machine at the beginning of week 1 is excellent.

## 1. Egress management believes that a normal distribution is a

1. Egress management believes that a normal distribution is a reasonable model for the unknown demand in the coming year. What mean and standard deviation should Egress use for the demand distribution?

2. Use a spreadsheet model to simulate 1000 possible outcomes for demand in the coming year. Based on these scenarios, what is the expected profit if Egress produces Q = 7800 ski jackets? What is the expected profit if Egress produces Q = 12,000 ski jackets? What is the standard deviation of profit in these two cases?

3. Based on the same 1000 scenarios, how many ski jackets should Egress produce to maximize expected profit? Call this quantity Q.

4. Should Q equal mean demand or not? Provide an argument that includes a comparison of the marginal revenue of producing one more ski jacket versus the marginal cost.

5. Create a histogram of profit at the production level Q. Create a histogram of profit when the production level Q equals mean demand. What is the probability of a loss greater than \$100,000 in each case?

Egress, Inc., is a small company that designs, produces, and sells ski jackets and other coats. The creative design team has labored for weeks over its new design for the coming winter season. It is now time to decide how many ski jackets to produce in this production run. Because of the lead times involved, no other production runs will be possible during the season. Predicting ski jacket sales months in advance of the selling season can be quite tricky. Egress has been in operation for only three years, and its ski jacket designs were quite successful in two of those years. Based on realized sales from the last three years, current economic conditions, and professional judgment, 12 Egress employees have independently estimated demand for their new design for the upcoming season. Their estimates are listed in Table 10.2.

Estimated Demands

14,000 …………………………. 16,000
13,000 …………………………. 8000
14,000 …………………………. 5000
14,000 …………………………. 11,000
15,500 …………………………. 8000
10,500 …………………………. 15,000

To assist in the decision on the number of units for the production run, management has gathered the data in Table 10.3. Note that S is the price Egress charges retailers. Any ski jackets that do not sell during the season can be sold by Egress to discounters for V per jacket. The fixed cost of plant and equipment is F. This cost is incurred regardless of the size of the production run.

Monetary Values
Variable production cost per unit (C): …………………………… \$80
Selling price per unit (S): ………………………………………………. \$100
Salvage value per unit (V): ………………………………………………. \$30
Fixed production cost (F): ………………………………………. \$100,000

## In Example 11.3, we commented on the 95th percentile on

In Example 11.3, we commented on the 95th percentile on days required and the corresponding date to start production. If the company begins production on this date, then it is 95% sure to complete the order by the due date. We found this date to be August 2. Do you always get this answer? Find out by (1) running the simulation 10 more times, each with 1000 iterations, and finding the 95th percentile and corresponding date in each, and (2) running the simulation once more, but with 10,000 iterations. Comment on the difference between simulations (1) and (2) in terms of accuracy. Given these results, when would you recommend that production should begin?

## See how sensitive the results in Example 11.2 are to

See how sensitive the results in Example 11.2 are to the following changes. For each part, make the change indicated, run the simulation, and comment on any differences between your outputs and the outputs in the example.

a. The cost of a new camera is increased to \$500.

b. The warranty period is decreased to one year.

c. The terms of the warranty are changed. If the camera fails within one year, the customer gets a new camera for free. However, if the camera fails between 1 year and 1.5 years, the customer pays a pro rata share of the new camera, increasing linearly from 0 to full price. For example, if it fails at 1.2 years, which is 40% of the way from 1 to 1.5, the customer pays 40% of the full price.

d. The customer pays \$50 up front for an extended warranty. This extends the warranty to three years. This extended warranty is just like the original, so that if the camera fails within three years, the customer gets a new camera for free.

## Referring to Example 11.1, if the average bid for each

Referring to Example 11.1, if the average bid for each competitor stays the same, but their bids exhibit less variability, does Miller’s optimal bid increase or decrease? To study this question, assume that each competitor’s bid, expressed as a multiple of Miller’s cost to complete the project, follows each of the following distributions.

a. Triangular with parameters 1.0, 1.3, and 2.4

b. Triangular with parameters 1.2, 1.3, and 2.2

c. Use @RISK’s Define Distributions window to check that the distributions in parts a and b have the same mean as the original triangular distribution in the example, but smaller standard deviations. What is the common mean? Why is it not the same as the most likely value, 1.3?

## 1. You should develop the simulation model just described. Only

1. You should develop the simulation model just described. Only one @RISK output is required, the total retained impressions. Based on a run of 10 simulations (one for each number of ad agencies solicited) of 1000 iterations each, you should recommend to Joel whether his company should indeed start soliciting competing ads from several ad agencies.

2. Joel believes something has been omitted from the simulation model, although he has no idea how important it is. As it stands, on each iteration the simulation chooses the ad with the highest simulated performance measure. In reality, however, when Joel and his group view the draft ads and try to choose the most effective, there is an element of unreliability in their choice. The one they choose as apparently best might not be the one that is actually best. One way to model this is to have two correlated performance measures, each with the same lognormal distribution, for each ad agency. One will be the apparent performance measure and the other will be the actual. On each iteration, the simulation should identify the ad agency with the highest apparent performance measure, but then this agency’s actual performance measure should be used in the objective. You should use a correlation of 0.8, although you might want to experiment with other correlations. The real question is whether this more complex model makes a difference.

This fixed cost covers the process of reviewing the ads and choosing the best. Joel is not nearly as confident about his ability to measure the key performance measure of any ad, the retained impressions per media dollar spent on the ad. He has seen studies where this performance measure varies widely, with a right-skewed distribution. Therefore, he decides to model this performance measure with a right-skewed lognormal distribution. After experimenting with the parameters of this distribution, using the @RISK Define Distributions tool as a guide, he has chosen a mean of 30 and a standard deviation of 25. The mean appears to be consistent with his past experience with advertising effectiveness, and the standard deviation provides the type of skewness he wants. Joel decides to use this distribution in an @RISK simulation, where he will vary the number of ad agencies solicited from 1 to 10 with a RISKSIMTABlE function. Each ad agency will generate an independent performance measure from this lognormal distribution and, on each iteration, the ad agency with the highest performance measure will be chosen. Then this highest performance measure will be multiplied by the dollars left for media spending to calculate the real objective, total retained impressions. Joel has asked you to develop, run, and interpret the @RISK simulation. Specifically, you should do the following.

## 1. Starting with the model from Case 2.2, you should

1. Starting with the model from Case 2.2, you should replace all of the inputs with probability distributions. Specifically, based on their knowledge of the new product and the market, Jim and Catherine have suggested using the distributions with the parameters listed in the file C10_04.xlsx. Note that the most likely values for the triangular distributions and the means for the normal distributions are the values from Case 2.2.

2. Jim and Catherine are again interested in adjusting the 0-1 variables from Case 2.2 so that they can examine different scenarios. However, they realize that the number of combinations of the two 0-1 variables they have control of, low-end versus high-end and mildly aggressive marketing versus very aggressive, is a small number, 2×2 = 4. Therefore, they want you to use a RISKSIMTABLE function, with an index from 1 to 4, so that a single @RISK run with 4 simulations can be made. You will have to use lookup functions so that the two 0-1 variables change appropriately as the index of RISKSIMTABLE varies from 1 to 4. As for the third 0-1 variable, whether the competition will introduce a competing product, they want you to model this probabilistically, using a RISKBERNOULLI function with parameter 0.3. This simply indicates that there is a 30% chance of a competing product.

3. Once you have the model set up correctly, you should designate the NPV of eTech sales as an @RISK output cell. You should also designate @RISK output ranges for each of the annual cash flow series requested in Case 2.2: net revenues from existing products, marketing costs for the ePlayerX, and so on. Then you should run @RISK with at least 1000 iterations for each of 4 simulations. In your memo to management, you can decide which results appear most interesting and should be reported. However, Jim and Catherine are especially interested in which of the uncertain inputs have the most effect on the bottom line, NPV of eTech sales, so you should definitely include one or more tornado graphs. Of course, they also want some help in deciding which of the four simulations—that is, which of the four possible eTech strategies—is best in terms of the NPV output.

4. Although Jim and Catherine are pleased with the simulation model, a big improvement over the deterministic model from Chapter 2, they still think something is missing: correlations. They have estimated correlations between several inputs, also listed in the file C10_04.xlsx, and they would like you to run the analysis again with these correlations included. Does this make any difference in the results, or was this extra complexity unnecessary?

This is an extension of Case 2.2 from Chapter 2, so you should read that case first. It asks you to develop a spreadsheet model for Jim Simons, VP for Production at eTech, and Catherine Dolans, VP for Marketing, so that they can better understand the implications of their decisions regarding the company’s new product ePlayerX. The model is supposed to contain 0-1 variables that Jim and Catherine can adjust to see (1) the effects of the ePlayerX product type (low-end or high-end), (2) the eTech marketing strategy for the ePlayerX (mildly aggressive or very aggressive), and (3) whether another company introduces a competing product to the market. The model from Case 2.2 has provided Jim and Catherine plenty of insight, but they both agree that it lacks a very important ingredient: uncertainty. Therefore, they have asked you to introduce uncertainty explicitly into the model and to quantify its effects. Specifically, you have been asked to do the following and then write your findings in a memo to management.

## The customer loyalty model in Example 11.9 assumes that once

The customer loyalty model in Example 11.9 assumes that once a customer leaves (becomes disloyal), that customer never becomes loyal again. Assume instead that there are two probabilities that drive the model, the retention rate and the rejoin rate, with values 0.75 and 0.15, respectively. The simulation should follow a customer who starts as a loyal customer in year 1. From then on, at the end of any year when the customer was loyal, this customer remains loyal for the next year with probability equal to the retention rate. But at the end of any year the customer is disloyal, this customer becomes loyal the next year with probability equal to the rejoin rate. During the customer’s nth loyal year with the company, the company’s mean profit from this customer is the nth value in the mean profit list in column B. Keep track of the same two outputs as in the example, and also keep track of the number of times the customer rejoins.

## If you own a stock, buying a put option on

If you own a stock, buying a put option on the stock will greatly reduce your risk. This is the idea behind portfolio insurance. To illustrate, consider a stock that currently sells for \$56 and has an annual volatility of 30%. Assume the risk-free rate is 8%, and you estimate that the stock’s annual growth rate is 12%.

a. Suppose you own 100 shares of this stock. Use simulation to estimate the probability distribution of the percentage return earned on this stock during a one‑year period.

b. Now suppose you also buy a put option (for \$238) on the stock. The option has an exercise price of \$50 and an exercise date one year from now. Use simulation to estimate the probability distribution of the percentage return on your portfolio over a one‑year period. Can you see why this strategy is called a portfolio insurance strategy?

c. Use simulation to show that the put option should, indeed, sell for about \$238.