Medium-Scale Expansion Profits Low Demand Medium High Annual Profit ($1000s) 50 150 200 Large-Scale Expansion Profits Annual Profit P(x) ($1000s) 20% 0 50% 100 30% 300 P(x) 20% 50% 30% Expected Profit ($1000s) Risk Analysis for Medium-Scale Expansion Annual Profit (x) Probability Demand $1000s P(x) (x - ) Low 50 20% Medium 150 50% High 200 30% (x - )2 (x - )2 * P(x) 2 = = Risk Analysis for Large-Scale Expansion Annual Profit (x) Probability Demand $1000s P(x) (x - ) Low 0 20% Medium 100 50% High 300 30% (x - )2 2 = = (x - )2 * P(x) Case Study - Bell Computer Company The Bell Computer Company is considering a plant expansion enabling the company to begin production of a new computer product. You have obtained your MBA from the University of Phoenix and, as a vicepresident, you must determine whether to make the expansion a medium- or large- scale project. The demand for the new product involves an uncertainty, which for planning purposes may be low demand, medium demand, or high demand. The probability estimates for the demands are 0.20, 0.50, and 0.30, respectively. Case Study - Kyle Bits and Bytes Kyle Bits and Bytes, a retailer of computing products sells a variety of computer-related products. One of Kyle's most popular products is an HP laser printer. The average weekly demand is 200 units. Lead time (lead time is defined as the amount of time between when the order is placed and when it is delivered) for a new order from the manufacturer to arrive is one week. If the demand for printers were constant, the retailer would re-order when there were exactly 200 printers in inventory. However, Kyle learned demand is a random variable in his Operations Management class. An analysis of previous weeks reveals the weekly demand standard deviation is 30. Kyle knows if a customer wants to buy an HP laser printer but he has none available, he will lose that sale, plus possibly additional sales. He wants the probability of running short (stock-out) in any week to be no more than 6%. Copyright 2017 by University of Phoenix. All rights reserved. Case Study 1 - Bell Computer Company The Bell Computer Company is considering a plant expansion enabling the company to begin production of a new computer product. The demand for the new product involves an uncertainty, which for planning purposes may be low demand, medium demand, or high demand. The probability estimates for the demands are 0.20, 0.50, and 0.30, respectively. The value of Annual Profit ($1000s) of Medium-Scale and Large-Scale Expansion Profits is given in the table below. Medium-Scale Expansion Profits Demand Low Medium High Annual Profit ($1000s) 50 150 200 P(x) 20% 50% 30% Large-Scale Expansion Profits Annual Profit ($1000s) 0 100 300 P(x) 20% 50% 30% Expectation is given by the formula sum (Xi*Pi). I used excel formula SUMPRODUCT to calculate the value of expectation of Annual Profit ($1000s) of Medium-Scale and Large-Scale Expansion Profits which is $145000 and $140000 respectively. Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2012). Decision of Investing in Medium-Scale Expansion Profits as compared to large scale expansion profits is preferred for the objective of maximizing the expected profit. As expected Medium-Scale Expansion Profits is higher being $145000 as compared to Large-Scale Expansion Profits (being $140000 only). The formula for variance is given by sum (x - ) 2 * P(x). For this I calculate (x - ), then (x - )2 and then (x - ) * P(x) and finally sum (x - ) * P(x). The formula for standard deviation is sqrt (variance). Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2012). The calculations for variance of Annual Profit ($1000s) of Medium-Scale is given below. Risk Analysis for Medium-Scale Expansion Demand Low Medium High Annual Profit (x) $1000s 50 150 200 Probability P(x) 20% 50% 30% (x - ) -83.33 16.67 66.67 (x - )2 6944.4444 277.7778 4444.4444 (x - )2 * P(x) 1388.89 138.89 1333.33 2 = = 2861.11 53.4893551 The calculations for variance of Annual Profit ($1000s) of Large-Scale is given below. Hossein, P. (2014). Risk Analysis for Large-Scale Expansion Demand Low Medium High Annual Profit (x) $1000s 0 100 300 Probability P(x) 20% 50% 30% (x - ) -133.33 -33.33 166.67 (x - )2 17777.7778 1111.1111 27777.7778 (x - )2 * P(x) 3555.56 555.56 8333.33 2 = = 12444.44 111.55467 Decision of Investing in Medium-Scale Expansion Profits as compared to large scale expansion profits is preferred for the objective of minimizing the risk or uncertainty. Hossein, P. (2014). Standard deviation and variance are measures of dispersion. Standard deviation measures spread of data in true units and variance in squared units. The low value of measures of dispersion implies that central tendency is reliable. Here expectation is a measure of central tendency. That is low value of Standard deviation and variance will imply that expectation of that Scale Expansion (medium or large) is more reliable. Croxton, F. E., & Cowden, D. J. (1939). The results for expectation, Standard deviation and variance for medium and large Scale Expansion is given below. Expected Profit ($1000s) 2 = = Medium-Scale Expansion Profits 145 2861.11 53.49 Large-Scale Expansion Profits 140 12444.44 111.55 Here clearly with low value of measures of dispersion (namely Standard deviation and variance) I can say that expectation of profits of medium Scale Expansion is more reliable as compared to that of Large Scale Expansion. And with high value of measures of dispersion (namely Standard deviation and variance) I can say that expectation of profits of large Scale Expansion is less reliable as compared to that of medium Scale Expansion. Hence Decision of Investing in Medium-Scale Expansion Profits as compared to large scale expansion profits is preferred for the objective of maximizing the expected profit as well as minimizing the risk or uncertainty. Hossein, P. (2014). Case 2: Kyle Bits and Bytes Kyle Bits and Bytes, a retailer of computing products sells a variety of computer-related products. One of Kyle's most popular products is an HP laser printer. The average weekly demand is 200 units. Lead time (lead time is defined as the amount of time between when the order is placed and when it is delivered) for a new order from the manufacturer to arrive is one week. If the demand for printers were constant, the retailer would re-order when there were exactly 200 printers in inventory. However, Kyle learned demand is a random variable in his Operations Management class. An analysis of previous weeks reveals the weekly demand standard deviation is 30. Kyle knows if a customer wants to buy an HP laser printer but he has none available, he will lose that sale, plus possibly additional sales. The formula for re-order point is given by: R = dL + z**L. Where, d = Average daily demand; L = lead time; = Standard deviation of daily demand; z = Number of standard deviations corresponding to the service level probability. Russell, R. S., Taylor, B. W. (2011). In the given scenario, d = 200/7 units L = 7 days = 30/7 Maximum accepted probability of stock out is 6%. It means, service level is 0.94 z = 1.56 Hence, Reorder point R = (200/7)*7 + 1.56*(30/7)* 7 = 200 + 17.69 = 217.69. That is Kelly's will place an order when inventory level reaches 218 units. Safety stock is given by: Safety stock = z**L. Where, L = lead time; = Standard deviation of daily demand; z = Number of standard deviations corresponding to the service level probability Russell, R. S., Taylor, B. W. (2011). In this scenario, L = 7 days = 30/7 Maximum accepted probability of stock out is 6%. It means, service level is 0.94 z = 1.56 Thus, Safety stock =1.56*(30/7)* 7 = 17.69 = 18 units. Thus, Kelly's should maintain 18 units safety stock of HP laser printer to avoid stock out. Reference Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2012). Basic business statistics: Concepts and applications. Pearson Higher Education AU. Croxton, F. E., & Cowden, D. J. (1939). Applied general statistics. Cumming, G. (n.d.). Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta Analysis. Hossein, P. (2014). Introduction to Probability, Statistics, and Random Processes. Kappa Research. Russell, R. S., Taylor, B. W. (2011). Operation Management. (7th ed.). Wiley Publication. John Wiley & Sons. Medium-Scale Annual Expansion Profits Profit ($1000s) P(x) Low Demand Medium High 50 150 200 Expected Profit ($1000s) Large-Scale Expansion Annual Profit Profits ($1000s) P(x) 20% 50% 30% 0 100 300 145 20% 50% 30% 140 Risk Analysis for Medium-Scale Expansion Demand Low Medium High Annual Profit (x) Probability $1000s P(x) 50 20% 150 50% 200 30% (x - ) -83.33 16.67 66.67 (x - )2 6944.4444 277.7778 4444.4444 (x - )2 * P(x) 1388.89 138.89 1333.33 2 = = 2861.11 53.48935512 Risk Analysis for Large-Scale Expansion Demand Low Medium High Annual Profit (x) Probability $1000s P(x) (x - ) 0 20% -133.33 100 50% -33.33 300 30% 166.67 (x - )2 17777.7778 1111.1111 27777.7778 (x - )2 * P(x) 3555.56 555.56 8333.33 2 = = 12444.44 111.5546702 Expected Profit ($1000s) 2 = = Medium-Scale Large-Scale Expansion Profits Expansion Profits 140 145 2861.11 12444.44 53.49 111.55 Case Study 1 - Bell Computer Company The Bell Computer Company is considering a plant expansion enabling the company to begin production of a new computer product. The demand for the new product involves an uncertainty, which for planning purposes may be low demand, medium demand, or high demand. The probability estimates for the demands are 0.20, 0.50, and 0.30, respectively. The value of Annual Profit ($1000s) of Medium-Scale and Large-Scale Expansion Profits is given in the table below. Medium-Scale Expansion Profits Demand Low Medium High Annual Profit ($1000s) 50 150 200 P(x) 20% 50% 30% Large-Scale Expansion Profits Annual Profit ($1000s) 0 100 300 P(x) 20% 50% 30% Expectation is given by the formula sum (Xi*Pi). I used excel formula SUMPRODUCT to calculate the value of expectation of Annual Profit ($1000s) of Medium-Scale and Large-Scale Expansion Profits which is $145000 and $140000 respectively. Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2012). Decision of Investing in Medium-Scale Expansion Profits as compared to large scale expansion profits is preferred for the objective of maximizing the expected profit. As expected Medium-Scale Expansion Profits is higher being $145000 as compared to Large-Scale Expansion Profits (being $140000 only). The formula for variance is given by sum (x - ) 2 * P(x). For this I calculate (x - ), then (x - )2 and then (x - ) * P(x) and finally sum (x - ) * P(x). The formula for standard deviation is sqrt (variance). Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2012). The calculations for variance of Annual Profit ($1000s) of Medium-Scale is given below. Risk Analysis for Medium-Scale Expansion Demand Low Medium High Annual Profit (x) $1000s 50 150 200 Probability P(x) 20% 50% 30% (x - ) -83.33 16.67 66.67 (x - )2 6944.4444 277.7778 4444.4444 (x - )2 * P(x) 1388.89 138.89 1333.33 2 = = 2861.11 53.4893551 The calculations for variance of Annual Profit ($1000s) of Large-Scale is given below. Hossein, P. (2014). Risk Analysis for Large-Scale Expansion Demand Low Medium High Annual Profit (x) $1000s 0 100 300 Probability P(x) 20% 50% 30% (x - ) -133.33 -33.33 166.67 (x - )2 17777.7778 1111.1111 27777.7778 (x - )2 * P(x) 3555.56 555.56 8333.33 2 = = 12444.44 111.55467 Decision of Investing in Medium-Scale Expansion Profits as compared to large scale expansion profits is preferred for the objective of minimizing the risk or uncertainty. Hossein, P. (2014). Standard deviation and variance are measures of dispersion. Standard deviation measures spread of data in true units and variance in squared units. The low value of measures of dispersion implies that central tendency is reliable. Here expectation is a measure of central tendency. That is low value of Standard deviation and variance will imply that expectation of that Scale Expansion (medium or large) is more reliable. Croxton, F. E., & Cowden, D. J. (1939). The results for expectation, Standard deviation and variance for medium and large Scale Expansion is given below. Expected Profit ($1000s) 2 = = Medium-Scale Expansion Profits 145 2861.11 53.49 Large-Scale Expansion Profits 140 12444.44 111.55 Here clearly with low value of measures of dispersion (namely Standard deviation and variance) I can say that expectation of profits of medium Scale Expansion is more reliable as compared to that of Large Scale Expansion. And with high value of measures of dispersion (namely Standard deviation and variance) I can say that expectation of profits of large Scale Expansion is less reliable as compared to that of medium Scale Expansion. Hence Decision of Investing in Medium-Scale Expansion Profits as compared to large scale expansion profits is preferred for the objective of maximizing the expected profit as well as minimizing the risk or uncertainty. Hossein, P. (2014). Case 2: Kyle Bits and Bytes Kyle Bits and Bytes, a retailer of computing products sells a variety of computer-related products. One of Kyle's most popular products is an HP laser printer. The average weekly demand is 200 units. Lead time (lead time is defined as the amount of time between when the order is placed and when it is delivered) for a new order from the manufacturer to arrive is one week. If the demand for printers were constant, the retailer would re-order when there were exactly 200 printers in inventory. However, Kyle learned demand is a random variable in his Operations Management class. An analysis of previous weeks reveals the weekly demand standard deviation is 30. Kyle knows if a customer wants to buy an HP laser printer but he has none available, he will lose that sale, plus possibly additional sales. The formula for re-order point is given by: R = dL + z**L. Where, d = Average daily demand; L = lead time; = Standard deviation of daily demand; z = Number of standard deviations corresponding to the service level probability. Russell, R. S., Taylor, B. W. (2011). In the given scenario, d = 200/7 units L = 7 days = 30/7 Maximum accepted probability of stock out is 6%. It means, service level is 0.94 z = 1.56 Hence, Reorder point R = (200/7)*7 + 1.56*(30/7)* 7 = 200 + 17.69 = 217.69. That is Kelly's will place an order when inventory level reaches 218 units. Safety stock is given by: Safety stock = z**L. Where, L = lead time; = Standard deviation of daily demand; z = Number of standard deviations corresponding to the service level probability Russell, R. S., Taylor, B. W. (2011). In this scenario, L = 7 days = 30/7 Maximum accepted probability of stock out is 6%. It means, service level is 0.94 z = 1.56 Thus, Safety stock =1.56*(30/7)* 7 = 17.69 = 18 units. Thus, Kelly's should maintain 18 units safety stock of HP laser printer to avoid stock out. Reference Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2012). Basic business statistics: Concepts and applications. Pearson Higher Education AU. Croxton, F. E., & Cowden, D. J. (1939). Applied general statistics. Cumming, G. (n.d.). Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta Analysis. Hossein, P. (2014). Introduction to Probability, Statistics, and Random Processes. Kappa Research. Russell, R. S., Taylor, B. W. (2011). Operation Management. (7th ed.). Wiley Publication. John Wiley & Sons. Medium-Scale Annual Expansion Profits Profit ($1000s) P(x) Low Demand Medium High 50 150 200 Expected Profit ($1000s) Large-Scale Expansion Annual Profit Profits ($1000s) P(x) 20% 50% 30% 0 100 300 145 20% 50% 30% 140 Risk Analysis for Medium-Scale Expansion Demand Low Medium High Annual Profit (x) Probability $1000s P(x) 50 20% 150 50% 200 30% (x - ) -83.33 16.67 66.67 (x - )2 6944.4444 277.7778 4444.4444 (x - )2 * P(x) 1388.89 138.89 1333.33 2 = = 2861.11 53.48935512 Risk Analysis for Large-Scale Expansion Demand Low Medium High Annual Profit (x) Probability $1000s P(x) (x - ) 0 20% -133.33 100 50% -33.33 300 30% 166.67 (x - )2 17777.7778 1111.1111 27777.7778 (x - )2 * P(x) 3555.56 555.56 8333.33 2 = = 12444.44 111.5546702 Expected Profit ($1000s) 2 = = Medium-Scale Large-Scale Expansion Profits Expansion Profits 140 145 2861.11 12444.44 53.49 111.55