# Week 2 Assignment: Simulation problems

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The purpose of the assignment is to gather the information needed to set up problems, run simulations, and interpret the results to solve the problems.

Using specified data files, chapter example files, and templates from the “Topic 2 Student Data, Template, and Example Files” topic material, complete Chapter 15, Problems 2, 7, 9, 32 (parts a and b only), 33, and 35 from the textbook. Use the Palisade DecisionTools Excel software to complete these problems where requested and applicable. Problem 2 only requires the use of Microsoft Excel; however, you may optionally use the Palisade DecisionTools Excel software to generate the histogram.

Use the Palisade DecisionTools Excel software to complete the rest of the problems and ensure that all Palisade software output is included in your files. Unless otherwise stated, run simulations with 1,000 trials. Use 500 trials if the software will not allow you to run 1,000 trials. Ensure that only one Excel file is open when running a simulation.

To receive full credit on the assignment, complete the following.

- Ensure that the Palisade software output is included with your submission.
- Ensure that Excel files include the associated cell functions and/or formulas if functions and/or formulas are used.
- Include a written response to all narrative questions presented in the problem by placing it in the associated Excel file.
- Include screenshots of all simulation distribution results for output variables.
- Place each problem in its own Excel file. Ensure that your first and last name are in your Excel file names.

Chapter 15 Questions:

2. Use Excel’s functions (not @RISK) to generate 1000 random numbers from a normal distribution with mean 100 and standard deviation 10. Then freeze these random numbers.

a. Calculate the mean and standard deviation of these random numbers. Are they approximately what you would expect?

b. What fraction of these random numbers are within k standard deviations of the mean? Answer for k = 1; for k = 2; for k = 3. Are the answers close to what they should be (according to the empirical rules you learned in Chapters 2 and 5)?

c. Create a histogram of the random numbers using 10 to 15 categories of your choice. Does this histogram have approximately the shape you would expect?

7. Use @RISK’s Define Distributions tool to draw a triangular distribution with parameters 200, 300, and 600. Then superimpose a normal distribution on this drawing, choosing the mean and standard deviation to match those from the triangular distribution. (Click the Add Overlay button at the bottom of the window and then choose the distribution to superimpose.)

a. What are the 5th and 95th percentiles for these two distributions?

b. What is the probability that a random number from the triangular distribution is less than 400? What is this probability for the normal distribution?

c. Experiment with the sliders to answer questions similar to those in part b. Would you conclude that these two distributions differ most in the extremes (right or left) or in the middle? Explain.

9. A company is about to develop and then market a new product. It wants to build a simulation model for the entire process, and one key uncertain input is the development cost. For each of the following scenarios, choose an appropriate distribution together with its parameters, justify your choice in words, and use @RISK’s Define Distributions tool to show your chosen distribution.

a. Company experts have no idea what the distribution of the development cost is. All they can state is “we are 95% sure it will be at least $450,000, and we are 95% sure it will be no more than $650,000.”

b. Company experts can still make the same statement as in part a, but now they can also state: “We believe the distribution is symmetric, reasonably bell-shaped, and its most likely value is about $550,000.”

c. Company experts can still make the same statement as in part a, but now they can also state: “We believe the distribution is skewed to the right, and its most likely value is about $500,000.”

32. A hardware company sells a lot of low-cost, high volume products. For one such product, it is equally likely that annual unit sales will be low or high. If sales are low (60,000), the company can sell the product for $10 per unit. If sales are high (100,000), a competitor will enter and the company will be able to sell the product for only $8 per unit. The variable cost per unit has a 25% chance of being $6, a 50% chance of being $7.50, and a 25% chance of being $9. Annual fixed costs are $30,000.

a. Use simulation to estimate the company’s expected annual profit.

b. Find a 95% interval for the company’s annual profit, that is, an interval such that about 95% of the actual profits are inside it.

33. W. L. Brown, a direct marketer of women’s clothing, must determine how many telephone operators to schedule during each part of the day. W. L. Brown estimates that the number of phone calls received each hour of a typical eight-hour shift can be described by the probability distribution in the file P15_33.xlsx. Each operator can handle 15 calls per hour and costs the company $20 per hour. Each phone call that is not handled is assumed to cost the company $6 in lost profit. Considering the options of employing 6, 8, 10, 12, 14, or 16 operators, use simulation to determine the number of operators that minimizes the expected hourly cost (labor costs plus lost profits).

35. Lemington’s is trying to determine how many Jean Hudson dresses to order for the spring season. Demand for the dresses is assumed to follow a normal distribution with mean 400 and standard deviation 100. The contract between Jean Hudson and Lemington’s works as follows. At the beginning of the season, Lemington’s reserves x units of capacity. Lemington’s must take delivery for at least 0.8x dresses and can, if desired, take delivery on up to x dresses. Each dress sells for $160 and Hudson charges $50 per dress. If Lemington’s does not take delivery on all x dresses, it owes Hudson a $5 penalty for each unit of reserved capacity that is unused. For example, if Lemington’s orders 450 dresses and demand is for 400 dresses, Lemington’s will receive 400 dresses and owe Jean 400($50) + 50($5). How many units of capacity should Lemington’s reserve to maximize its expected profit?

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