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Operations Problem Set

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Problem Set

Problem 1

The local distribution center of a major grocery chain sources frozen vegetable packets from a supplier in Lexington, Kentucky. The vegetable packets are packed in 500-packet cartons and the distribution center sources them at the rate of $375 per carton. The distribution center estimates the annual demand for the frozen vegetables to be 60000 cartons. The fixed cost of placing an order, including transportation costs, is $700. Given the warehousing and cold storage costs, the grocery chain estimates the holding cost of inventory to be about 8% of the unit cost (in cartons). Determine the economic order quantity, the time between placement of orders and the total annual cost (i.e., sum of holding and ordering costs).

Solution to problem 1

EOQ =2 * Annual Demand*Ordering CostUnit Holding Cost =2 * 60000 *7000.08*375 = 1673.32

Thus, EOQ (Q*) = 1673.32 cartons

The time between order placements = EOQ / Annual Demand = 1673.32 / 60000 = 0.028 years
Annual Holding Cost = (Q* /2) * Unit Annual Holding Cost = (1673.32/2) * (0.08 * 375) = 25099.8
Annual Ordering Cost = (Annual Demand / Q* )*Ordering Cost = (60000/1673.32)*700 = 25099.8
Total Annual Cost = Annual Holding Cost + Annual Ordering Cost =$50199.6
.
Problem 2

Shoe House is a footwear retail chain operating in the Pittsburgh area. The company needs to plan for the upcoming winter and one of the decisions they need to make in the upcoming weeks is regarding how many snow boots to stock. Based on historic demand data, Shoe House estimates that the demand for snow boots to be normally distributed with a mean of 10000 units and a standard deviation of 1500 units. The company procures the snow boots at a price of $40 and sells it for $ 75. After February, unsold boots would be marked down to $25. How many snow boots should Shoe House stock to maximize expected profits?

Solution to problem 2

The cost parameters are as follows
Underage cost, Cu = $75 - $40 = $ 35
Overage cost, Co = $40 - $25 = $15

Critical ratio = Cu/ (Cu + Co) = 35/ (35 + 15) = 0.7 Thus, Shoe House’s order should be sufficient to satisfy the total demand with probability 0.7. The optimal order quantity Q is the 70th percentile of the demand distribution. Using the normal tables, we find the corresponding standardized value of z to be 0.525.

Therefore, Q = mean + (standard deviation) * (z value) = μ + σz = 10000+ (1500) * (0.525) = 10787.5 ~ 10788 snow boots.

Problem 3

Alpine Valley Bagels makes different types of bagels including everything, Jalapeno, and dried tomato bagels. Each pack contains 6 bagels and the table below provides the daily demand information for everything bagel packets. It costs $12 to make a batch of everything bagels consisting of 100 bagels and each packet (containing 6 bagels) is sold at the rate of $3/packet. Bagels that remain unsold at the end of the day have to be sold at a discounted price of 50 cents/packet the next day. How many packets of everything bagels should Alpine Valley make every morning in order to maximize expected profits?

Daily demand (for bagel packets) | Probability | D=400 | 0.2 | D=450 | 0.4 | D=550 | 0.2 | D=750 | 0.2 |

Solution to problem 3

Underage cost, Cu=$3-$0.72=$2.28
Overage cost, Co=$0.72-$0.5=$0.22

Critical ratio= Cu / Cu+ C0 = 2.28/2.50=0.912

It is optimal to make 750 packets since the smallest number Q such that P (D ≤ Q) ≥ 0.912 is 750.

To verify that Q = 750 is indeed the optimal production quantity, we compare the expected profits if Q = 550 and Q = 750 since P (D ≤ 550) ≤ 0.95 ≤ P (D ≤ 750).

Q=550

Expected profit = revenue+ salvage value-production cost
=3*(400* P (D ≤ 400) +450* P (D = 450)+550*P(D≥550)) + 0.50*(150* P (D ≤ 400) +100* P (D = 450)) – 0.72*550
=1440+35-396 = 1079

Q=750

Expected profit = revenue+ salvage value-production cost
=3*(400* P (D ≤ 400) +450* P (D = 450)+550*P(D=550)+750*P(D=750)) + 0.50*(350* P (D ≤ 400) +300* P (D = 450)+200*P(D=550)) – 0.72*750
=1560+115-540 = 1135

Therefore, Q = 750 maximizes the expected profit.

Problem 4

Cherish Tree Co. Ltd is a wholesale supplier of Christmas trees in Michigan. The cost of growing and maintaining a tree is $15, on average and each tree sells for $48, on average. The following table provides relevant sales information for the Christmas tree in years 2000-2011.

Year | Sales (in thousands) | 2000 | 4374 | 2001 | 3876 | 2002 | 4335 | 2003 | 4850 | 2004 | 4660 | 2005 | 4915 | 2006 | 5210 | 2007 | 5085 | 2008 | 4882 | 2009 | 5425 | 2010 | 4673 | 2011 | 5520 |

a. Forecast the demand for Christmas trees (in thousands) for the years 2003-2012 using a 3-period simple moving average technique. How much is the mean forecast error (bias) for the years 2003-2011? Calculate the mean absolute deviation and mean absolute percent error for the years 2003-2011. Would you consider these forecasts to be accurate or inaccurate? Explain why. b. If Cherish Tree decides to grow the exact number of trees as their forecast suggests, how much total loss would the firm incur? Trees cut for a Christmas season perish at the end of the season and hence cannot be carried over to the next year, i.e., they have to be salvaged.

Solution to problem 4

a. Forecast the demand for Christmas trees (in thousands) for the years 2003-2012 using a 3-period simple moving average technique. How much is the mean forecast error (bias) for the years 2003-2011? Calculate the mean absolute deviation and mean absolute percent error for the years 2003-2011. Would you consider these forecasts to be accurate or inaccurate? Explain why.

Year | Sales(in thousands) | Forecast(in thousands) | Error(in thousands) | Absolute
Error(in thousands) | Absolute percent error | Underage
(in thousands) | Overage
(in thousands) | 2000 | 4374 | | | | | | | 2001 | 3876 | | | | | | | 2002 | 4335 | | | | | | | 2003 | 4850 | 4195.00 | 655.00 | 655.00 | 13.51 | 21615.00 | 0.00 | 2004 | 4660 | 4353.67 | 306.33 | 306.33 | 6.57 | 10109.00 | 0.00 | 2005 | 4915 | 4615.00 | 300.00 | 300.00 | 6.10 | 9900.00 | 0.00 | 2006 | 5210 | 4808.33 | 401.67 | 401.67 | 7.71 | 13255.00 | 0.00 | 2007 | 5085 | 4928.33 | 156.67 | 156.67 | 3.08 | 5170.00 | 0.00 | 2008 | 4882 | 5070.00 | -188.00 | 188.00 | 3.85 | 0.00 | 2068000.00 | 2009 | 5425 | 5059.00 | 366.00 | 366.00 | 6.75 | 12078.00 | 0.00 | 2010 | 4673 | 5130.67 | -457.67 | 457.67 | 9.79 | 0.00 | 5034333.33 | 2011 | 5520 | 4993.33 | 526.67 | 526.67 | 9.54 | 17380.00 | 0.00 | 2012 | | 5206.00 | | | | | | MFE | | | 229.63 | | | | | MAD | | | | 373.11 | | | | MAPE | | | | | 7.43 | | | Total loss | | | | | | 7191840.33 |

Where
3-period simple moving average
Error et = Dt - SMAt
Absolute error=|et |
Absolute percent error=
Mean forecast error, MFE =
Mean absolute deviation, MAD =
Mean absolute percent error, MAPE =

b. If Cherish Tree decides to grow the exact number of trees as their forecast suggests, how much total loss would the firm incur? Trees cut for a Christmas season perish at the end of the season and hence cannot be carried over to the next year, i.e., they have to be salvaged.

If Cherish Tree decides to grow the exact number of trees as their forecast suggests, then sales = minimum {demand, forecast}.

Overage cost = cost of growing and maintaining a tree=$15 (incurred when forecast > demand)
Underage cost =sales price – production cost = $48-$15=$33 (incurred when forecast < demand)

Total loss incurred=total underage cost + total overage cost=$7191840.33
.
Problem 5

The following table shows the quarterly sales (in tonnes) of CapeGod brand of minestrone soups in years 2008 to 2011.

Quarter | 2008 | 2009 | 2010 | 2011 | Winter | 1324 | 1430 | 1227 | 1310 | Spring | 1124 | 1276 | 1086 | 1178 | Summer | 820 | 670 | 875 | 754 | Fall | 843 | 934 | 994 | 890 |

a. Using a simple 4-point moving average, forecast the quarterly sales in 2009-2011. Compute the MFE, MAD, MSE and MAPE for 2009-2011 forecasts. What is the quarterly forecast for 2012? b. Generate demand forecasts for quarterly sales starting from Spring 2008 until Fall 2011, using simple exponential smoothing with α value of 0.6. Assume that the forecast for Winter 2008 is 1324. Compute the MFE, MAD, MSE, and MAPE for quarters Spring 2008 to Fall 2011. What are your forecasts for the quarters of 2012? c. Plot the actual demand and the forecasts you have calculated in parts (a) and (b) on the same graph. Do you think these forecasts are accurate? Explain why. If not, please propose a better approach and provide forecasts for 2012.

Solution to problem 5

a. Using a simple 4-point moving average, forecast the quarterly sales in 2009-2011. Compute the MFE, MAD, MSE and MAPE for 2009-2011 forecasts. What is the quarterly forecast for 2012?

Quarter | Sales | Forecast | Error | Absolute error | Squared error | Absolute
Percent
error | 1 | 1324 | | | | | | 2 | 1124 | | | | | | 3 | 820 | | | | | | 4 | 843 | | | | | | 5 | 1430 | 1027.75 | 402.25 | 402.25 | 161805.06 | 28.13 | 6 | 1276 | 1054.25 | 221.75 | 221.75 | 49173.06 | 17.38 | 7 | 670 | 1092.25 | -422.25 | 422.25 | 178295.06 | 63.02 | 8 | 934 | 1054.75 | -120.75 | 120.75 | 14580.56 | 12.93 | 9 | 1227 | 1077.50 | 149.50 | 149.50 | 22350.25 | 12.18 | 10 | 1086 | 1026.75 | 59.25 | 59.25 | 3510.56 | 5.46 | 11 | 875 | 979.25 | -104.25 | 104.25 | 10868.06 | 11.91 | 12 | 994 | 1030.50 | -36.50 | 36.50 | 1332.25 | 3.67 | 13 | 1310 | 1045.50 | 264.50 | 264.50 | 69960.25 | 20.19 | 14 | 1178 | 1066.25 | 111.75 | 111.75 | 12488.06 | 9.49 | 15 | 754 | 1089.25 | -335.25 | 335.25 | 112392.56 | 44.46 | 16 | 890 | 1059.00 | -169.00 | 169.00 | 28561.00 | 18.99 | 17 | | 1033.00 | | | | | 18 | | 1033.00 | | | | | 19 | | 1033.00 | | | | | 20 | | 1033.00 | | | | | MFE | | | 1.75 | | | | MAD | | | | 200 | | | MSE | | | | | 55443.06 | | MAPE | | | | | | 20.65 |

b. Generate demand forecasts for quarterly sales starting from Spring 2008 until Fall 2011, using simple exponential smoothing with α value of 0.8. Assume that the forecast for Winter 2008 is 263. Compute the MFE, MAD, MSE, and MAPE for quarters Spring 2008 to Fall 2011. What are your forecasts for the quarters of 2012?

Forecast for month 2= forecast for month 1.
Forecast for month 3 = 0.6*sales in month 2 + (1-0.6)*forecast for month 2.

Quarter | Sales | Forecast | Error | Absolute error | Squared error | Absolute
Percent
error | 1 | 1324 | 1324.00 | | | | | 2 | 1124 | 1324.00 | -200.00 | 200.00 | 40000.00 | 17.79 | 3 | 820 | 1204.00 | -384.00 | 384.00 | 147456.00 | 46.83 | 4 | 843 | 973.60 | -130.60 | 130.60 | 17056.36 | 15.49 | 5 | 1430 | 895.24 | 534.76 | 534.76 | 285968.26 | 37.40 | 6 | 1276 | 1216.10 | 59.90 | 59.90 | 3588.49 | 4.69 | 7 | 670 | 1252.04 | -582.04 | 582.04 | 338768.70 | 86.87 | 8 | 934 | 902.82 | 31.18 | 31.18 | 972.48 | 3.34 | 9 | 1227 | 921.53 | 305.47 | 305.47 | 93314.28 | 24.90 | 10 | 1086 | 1104.81 | -18.81 | 18.81 | 353.83 | 1.73 | 11 | 875 | 1093.52 | -218.52 | 218.52 | 47752.82 | 24.97 | 12 | 994 | 962.41 | 31.59 | 31.59 | 997.95 | 3.18 | 13 | 1310 | 981.36 | 328.64 | 328.64 | 108001.71 | 25.09 | 14 | 1178 | 1178.55 | -0.55 | 0.55 | 0.30 | 0.05 | 15 | 754 | 1178.22 | -424.22 | 424.22 | 179961.10 | 56.26 | 16 | 890 | 923.69 | -33.69 | 33.69 | 1134.83 | 3.79 | 17 | | 903.47 | | | | | 18 | | 903.47 | | | | | 19 | | 903.47 | | | | | 20 | | 903.47 | | | | | MFE | | | -46.73 | | | | MAD | | | | 219 | | | MSE | | | | | 84355.14 | | MAPE | | | | | | 23.49 |

c. Plot the actual demand and the forecasts you have calculated in parts (a) and (b) on the same graph. Do you think these forecasts are accurate? Explain why. If not, please propose a better approach and provide forecasts for 2012.

The high values of MAD, MAPE and MSE values indicate that the forecasts are not accurate. MFE is not a good indicator of forecast accuracy. The key issue here is that the data contains both seasonal and trend characteristics. Therefore, using a trend adjusted – seasonal forecasting technique can be expected to produce better forecasts.

We fit the trend line using Sales as “Known Y” values and Quarter as “Known X” values in the Excel functions “Slope” and “Intercept”. We obtain the following regression parameters

Regression Slope: -9.85
Regression Intercept: 1129.63
Therefore, Y=1129.63 - 9.85*X.

Quarter | Sales | Regression
Forecast | Index
=Actual/forecast | 1 | 1324 | 1119.78 | 1.18 | 2 | 1124 | 1109.93 | 1.01 | 3 | 820 | 1100.09 | 0.75 | 4 | 843 | 1090.24 | 0.77 | 5 | 1430 | 1080.40 | 1.32 | 6 | 1276 | 1070.55 | 1.19 | 7 | 670 | 1060.71 | 0.63 | 8 | 934 | 1050.86 | 0.89 | 9 | 1227 | 1041.01 | 1.18 | 10 | 1086 | 1031.17 | 1.05 | 11 | 875 | 1021.32 | 0.86 | 12 | 994 | 1011.48 | 0.98 | 13 | 1310 | 1001.63 | 1.31 | 14 | 1178 | 991.79 | 1.19 | 15 | 754 | 981.94 | 0.77 | 16 | 890 | 972.10 | 0.92 |

Next, we divide the actual values with regression forecast to find the seasonal adjustment of each quarter. We observe that each quarter shows the same seasonal properties. For this data, quarters are the seasons. We take the average of adjustments that belong to the same season to find an average seasonal index as follows:

| 2008 | 2009 | 2010 | 2011 | Average | Winter | 1.18 | 1.32 | 1.18 | 1.31 | 1.25 | Spring | 1.01 | 1.19 | 1.05 | 1.19 | 1.11 | Summer | 0.75 | 0.63 | 0.86 | 0.77 | 0.75 | Fall | 0.77 | 0.89 | 0.98 | 0.92 | 0.89 |

Finally, we make our forecast for 2012 given the regression line. We adjust the regression forecast with the calculated seasonal indices and get the final forecasts for 2012 as follows:

Quarter | Period | Regression
Forecast | Updated forecast | 2012 Winter | 17 | 962.25 | 1201.00 | 2012 Spring | 18 | 952.40 | 1058.48 | 2012 Summer | 19 | 942.56 | 707.31 | 2012 Fall | 20 | 932.71 | 830.18 |

Where regression forecast =1129.63 - 9.85*quarter and updated forecast = regression forecast * seasonal index.

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