Ski Jacket Production[1] 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, twelve Egress employees have independently estimated demand for their new design for the upcoming season. Their estimates are shown in Table 1. 14,000 16,000 13,000 8,000 14,000 5,000 14,000 11,000 15,500 8,000 10,500 15,000 Table 1: Estimated Demands To assist in the decision on the number of units for the production run, management has gathered the data in Table 2. 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 irrespective of the size of the production run. 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 Table 2: Monetary Values Questions Egress management believes that a normal distribution is a reasonable model for the unknown demand in the coming year. Use the mean and standard deviation of the employees’ estimates for the demand distribution. 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 = 7,800 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? Based on the same 1000 scenarios, how many ski jackets should Egress produce to maximize expected profit? Call this quantity Q* Should Q* equal estimated mean demand or not? Explain. Create a histogram of profit at the production level Q*. Create a histogram of profit when the production level Qequals mean demand. What is the probability of a loss greater than $100,000 in each case?

Ski Jacket Production[1] 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, twelve Egress employees have independently estimated demand for their new design for the upcoming season. Their estimates are shown in Table 1. 14,000 16,000 13,000 8,000 14,000 5,000 14,000 11,000 15,500 8,000 10,500 15,000 Table 1: Estimated Demands To assist in the decision on the number of units for the production run, management has gathered the data in Table 2. 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 irrespective of the size of the production run. 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 Table 2: Monetary Values Questions Egress management believes that a normal distribution is a reasonable model for the unknown demand in the coming year. Use the mean and standard deviation of the employees’ estimates for the demand distribution. 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 = 7,800 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? Based on the same 1000 scenarios, how many ski jackets should Egress produce to maximize expected profit? Call this quantity Q* Should Q* equal estimated mean demand or not? Explain. Create a histogram of profit at the production level Q*. Create a histogram of profit when the production level Qequals mean demand. What is the probability of a loss greater than $100,000 in each case?

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Ski Jacket Production[1]

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, twelve Egress employees have independently estimated demand for their new design for the upcoming season. Their estimates are shown in Table 1.

14,000	16,000
13,000	8,000
14,000	5,000
14,000	11,000
15,500	8,000
10,500	15,000

Table 1: Estimated Demands

To assist in the decision on the number of units for the production run, management has gathered the data in Table 2. 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 irrespective of the size of the production run.

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

Table 2: Monetary Values

Questions

Egress management believes that a normal distribution is a reasonable model for the unknown demand in the coming year. Use the mean and standard deviation of the employees’ estimates for the demand distribution.
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 = 7,800 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?
Based on the same 1000 scenarios, how many ski jackets should Egress produce to maximize expected profit? Call this quantity Q*
Should Q* equal estimated mean demand or not? Explain.
Create a histogram of profit at the production level Q*. Create a histogram of profit when the production level Qequals mean demand. What is the probability of a loss greater than $100,000 in each case?

[1] From Practical Management Science (2nd ed., Winston and Albright, Duxbury Press, p. 614).

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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