Ski Jacket Production1 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. 14,00 0 13,00 0 14,00 0 14,00 0 15,50 0 10,50 0 16,00 0 8,000 5,000 11,00 0 8,000 15,00 0 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): Selling price per unit (S): Salvage value per unit (V): Fixed production cost (F): Table 2: Monetary Values $80 $100 $30 $100,000 1 From Practical Management Science (2nd ed., Winston and Albright, 2001 Duxbury Press, p. 614). Questions 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 = 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? 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 estimated mean demand or not? Explain. 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? This sheet is where a user sets parameter values Variable production cost per unit (C): Selling price per unit (S): Salvage value per unit (V): Fixed production cost (F): 80 100 30 100000 Demand Estimates 14000 13000 14000 14000 15500 10500 16000 8000 5000 11000 8000 15000 This sheet contains the solution to the project a) b) The appropriate parameter values are: Mean: Standard deviation: 12,000 3,496.75 Run Random Number Simulated Demand 1 0.9247699132 17,028 2 0.7096310504 13,931 3 0.2103196155 9,184 4 0.9362117637 17,328 5 0.2516837281 9,660 6 0.7091910909 13,927 7 0.0462757549 6,118 8 0.0826357473 7,148 9 0.3512093779 10,664 10 0.1646935222 8,589 11 0.7621131784 14,494 12 0.9529265286 17,853 13 0.8630509134 15,826 14 0.4076235033 11,183 15 0.9594872288 18,101 16 0.2907737003 10,073 17 0.4579267174 11,631 18 0.6748332675 13,585 19 0.2571683607 9,720 20 0.7426086443 14,278 21 0.8127413444 15,105 22 0.740593605 14,256 23 0.4207814241 11,301 24 0.9117441565 16,726 25 0.5868011945 12,767 26 0.9989326077 22,738 27 0.7562552201 14,428 28 0.2496493771 9,638 29 0.2814811862 9,977 30 0.6798343843 13,634 31 0.0294693219 5,396 32 0.5585582561 12,515 33 0.5436899888 12,384 34 0.0072860485 3,458 35 0.128351328 8,034 36 0.0730060383 6,917 37 0.635134263 13,208 38 0.7224930712 14,064 39 0.1421302843 8,256 40 0.9474175926 17,666 41 0.0349617163 5,662 42 0.2010244134 9,070 43 0.9974463 21,792 44 0.6641330246 13,482 45 0.7669317298 14,548 46 0.6825051785 13,660 47 0.1538944841 8,434 48 0.341993506 10,577 49 0.9630656581 18,250 50 0.0663937464 6,744 51 0.2480616595 9,620 52 0.3571014308 10,719 53 0.2679423269 9,835 54 0.1361355437 8,161 55 0.0133116257 4,248 56 0.7176547593 14,014 57 0.0603223942 6,573 58 0.7429822473 14,282 59 0.3079925453 10,246 60 0.6528741256 13,375 61 0.793710161 14,865 62 0.5642257722 12,565 63 0.2576978251 9,726 64 0.6279679456 13,142 65 0.0568464001 6,469 66 0.163143314 8,568 67 0.4308814886 11,391 68 0.7659540144 14,537 69 0.4625261049 11,671 70 0.7834332512 14,741 71 0.5418741944 12,368 72 0.1471697328 8,333 73 0.2889841704 10,055 74 0.1113837932 7,737 75 0.4721734877 11,756 76 0.9785732484 19,081 Surplus Demand Normal Demand Total Revenue Total Cost Profit Q = 7800 Q = 12000 Q = 7800 Q = 12000 Q = 7800 Q = 12000 Q = 7800 Q = 12000 Q = 7800 Q = 12000 9,228 7,800 17,028 1,056,840.21 1,702,800.71 724,000.00 1,060,000.00 332,840.21 642,800.71 6,131 7,800 13,931 963,938.43 1,393,128.12 724,000.00 1,060,000.00 239,938.43 333,128.12 1,384 7,800 9,184 821,520.64 918,402.13 724,000.00 1,060,000.00 97,520.64 -141,597.87 9,528 7,800 17,328 1,065,843.04 1,732,810.13 724,000.00 1,060,000.00 341,843.04 672,810.13 1,860 7,800 9,660 835,799.14 965,997.13 724,000.00 1,060,000.00 111,799.14 -94,002.87 6,127 7,800 13,927 963,803.73 1,392,679.11 724,000.00 1,060,000.00 239,803.73 332,679.11 6,118 6,118 611,815.10 611,815.10 724,000.00 1,060,000.00 -112,184.90 -448,184.90 7,148 7,148 714,805.20 714,805.20 724,000.00 1,060,000.00 -9,194.80 -345,194.80 2,864 7,800 10,664 865,921.20 1,066,404.00 724,000.00 1,060,000.00 141,921.20 6,404.00 789 7,800 8,589 803,683.37 858,944.58 724,000.00 1,060,000.00 79,683.37 -201,055.42 6,694 7,800 14,494 980,807.75 1,449,359.15 724,000.00 1,060,000.00 256,807.75 389,359.15 10,053 7,800 17,853 1,081,598.15 1,785,327.16 724,000.00 1,060,000.00 357,598.15 725,327.16 8,026 7,800 15,826 1,020,776.98 1,582,589.93 724,000.00 1,060,000.00 296,776.98 522,589.93 3,383 7,800 11,183 881,488.21 1,118,294.03 724,000.00 1,060,000.00 157,488.21 58,294.03 10,301 7,800 18,101 1,089,030.45 1,810,101.49 724,000.00 1,060,000.00 365,030.45 750,101.49 2,273 7,800 10,073 848,185.49 1,007,284.97 724,000.00 1,060,000.00 124,185.49 -52,715.03 3,831 7,800 11,631 894,916.17 1,163,053.90 724,000.00 1,060,000.00 170,916.17 103,053.90 5,785 7,800 13,585 953,552.22 1,358,507.40 724,000.00 1,060,000.00 229,552.22 298,507.40 1,920 7,800 9,720 837,593.06 971,976.85 724,000.00 1,060,000.00 113,593.06 -88,023.15 6,478 7,800 14,278 974,334.43 1,427,781.44 724,000.00 1,060,000.00 250,334.43 367,781.44 7,305 7,800 15,105 999,158.04 1,510,526.79 724,000.00 1,060,000.00 275,158.04 450,526.79 6,456 7,800 14,256 973,680.67 1,425,602.22 724,000.00 1,060,000.00 249,680.67 365,602.22 3,501 7,800 11,301 885,030.52 1,130,101.75 724,000.00 1,060,000.00 161,030.52 70,101.75 8,926 7,800 16,726 1,047,783.54 1,672,611.81 724,000.00 1,060,000.00 323,783.54 612,611.81 4,967 7,800 12,767 929,007.65 1,276,692.18 724,000.00 1,060,000.00 205,007.65 216,692.18 14,938 2,938 7,800 19,800 1,228,135.88 2,068,135.88 724,000.00 1,060,000.00 504,135.88 1,008,135.88 6,628 7,800 14,428 978,834.60 1,442,782.00 724,000.00 1,060,000.00 254,834.60 382,782.00 1,838 7,800 9,638 835,128.52 963,761.72 724,000.00 1,060,000.00 111,128.52 -96,238.28 2,177 7,800 9,977 845,319.43 997,731.45 724,000.00 1,060,000.00 121,319.43 -62,268.55 5,834 7,800 13,634 955,014.22 1,363,380.74 724,000.00 1,060,000.00 231,014.22 303,380.74 5,396 5,396 539,585.73 539,585.73 724,000.00 1,060,000.00 -184,414.27 -520,414.27 4,715 7,800 12,515 921,453.71 1,251,512.36 724,000.00 1,060,000.00 197,453.71 191,512.36 4,584 7,800 12,384 917,511.42 1,238,371.39 724,000.00 1,060,000.00 193,511.42 178,371.39 3,458 3,458 345,798.64 345,798.64 724,000.00 1,060,000.00 -378,201.36 -714,201.36 234 7,800 8,034 787,017.52 803,391.74 724,000.00 1,060,000.00 63,017.52 -256,608.26 6,917 6,917 691,655.24 691,655.24 724,000.00 1,060,000.00 -32,344.76 -368,344.76 5,408 7,800 13,208 942,242.02 1,320,806.74 724,000.00 1,060,000.00 218,242.02 260,806.74 6,264 7,800 14,064 967,920.17 1,406,400.57 724,000.00 1,060,000.00 243,920.17 346,400.57 456 7,800 8,256 793,670.63 825,568.76 724,000.00 1,060,000.00 69,670.63 -234,431.24 9,866 7,800 17,666 1,075,975.09 1,766,583.63 724,000.00 1,060,000.00 351,975.09 706,583.63 5,662 5,662 566,246.49 566,246.49 724,000.00 1,060,000.00 -157,753.51 -493,753.51 1,270 7,800 9,070 818,095.05 906,983.49 724,000.00 1,060,000.00 94,095.05 -153,016.51 13,992 1,992 7,800 19,800 1,199,746.11 2,039,746.11 724,000.00 1,060,000.00 475,746.11 979,746.11 5,682 7,800 13,482 950,454.50 1,348,181.66 724,000.00 1,060,000.00 226,454.50 288,181.66 6,748 7,800 14,548 982,450.83 1,454,836.11 724,000.00 1,060,000.00 258,450.83 394,836.11 5,860 7,800 13,660 955,798.89 1,365,996.29 724,000.00 1,060,000.00 231,798.89 305,996.29 634 7,800 8,434 799,012.78 843,375.93 724,000.00 1,060,000.00 75,012.78 -216,624.07 2,777 7,800 10,577 863,301.67 1,057,672.22 724,000.00 1,060,000.00 139,301.67 -2,327.78 10,450 7,800 18,250 1,093,505.54 1,825,018.45 724,000.00 1,060,000.00 369,505.54 765,018.45 6,744 6,744 674,368.30 674,368.30 724,000.00 1,060,000.00 -49,631.70 -385,631.70 1,820 7,800 9,620 834,603.11 962,010.35 724,000.00 1,060,000.00 110,603.11 -97,989.65 2,919 7,800 10,719 867,582.86 1,071,942.87 724,000.00 1,060,000.00 143,582.86 11,942.87 2,035 7,800 9,835 841,060.27 983,534.24 724,000.00 1,060,000.00 117,060.27 -76,465.76 361 7,800 8,161 790,833.00 816,109.99 724,000.00 1,060,000.00 66,833.00 -243,890.01 4,248 4,248 424,770.96 424,770.96 724,000.00 1,060,000.00 -299,229.04 -635,229.04 6,214 7,800 14,014 966,412.18 1,401,373.95 724,000.00 1,060,000.00 242,412.18 341,373.95 6,573 6,573 657,278.65 657,278.65 724,000.00 1,060,000.00 -66,721.35 -402,721.35 6,482 7,800 14,282 974,455.94 1,428,186.46 724,000.00 1,060,000.00 250,455.94 368,186.46 2,446 7,800 10,246 853,386.27 1,024,620.91 724,000.00 1,060,000.00 129,386.27 -35,379.09 5,575 7,800 13,375 947,236.32 1,337,454.41 724,000.00 1,060,000.00 223,236.32 277,454.41 7,065 7,800 14,865 991,953.21 1,486,510.69 724,000.00 1,060,000.00 267,953.21 426,510.69 4,765 7,800 12,565 922,961.90 1,256,539.66 724,000.00 1,060,000.00 198,961.90 196,539.66 1,926 7,800 9,726 837,765.17 972,550.57 724,000.00 1,060,000.00 113,765.17 -87,449.43 5,342 7,800 13,142 940,248.18 1,314,160.61 724,000.00 1,060,000.00 216,248.18 254,160.61 6,469 6,469 646,880.08 646,880.08 724,000.00 1,060,000.00 -77,119.92 -413,119.92 768 7,800 8,568 803,025.46 856,751.53 724,000.00 1,060,000.00 79,025.46 -203,248.47 3,591 7,800 11,391 887,733.28 1,139,110.92 724,000.00 1,060,000.00 163,733.28 79,110.92 6,737 7,800 14,537 982,115.93 1,453,719.77 724,000.00 1,060,000.00 258,115.93 393,719.77 3,871 7,800 11,671 896,131.64 1,167,105.47 724,000.00 1,060,000.00 172,131.64 107,105.47 6,941 7,800 14,741 988,226.91 1,474,089.69 724,000.00 1,060,000.00 264,226.91 414,089.69 4,568 7,800 12,368 917,031.19 1,236,770.62 724,000.00 1,060,000.00 193,031.19 176,770.62 533 7,800 8,333 795,993.99 833,313.30 724,000.00 1,060,000.00 71,993.99 -226,686.70 2,255 7,800 10,055 847,636.96 1,005,456.54 724,000.00 1,060,000.00 123,636.96 -54,543.46 7,737 7,737 773,675.38 773,675.38 724,000.00 1,060,000.00 49,675.38 -286,324.62 3,956 7,800 11,756 898,677.03 1,175,590.09 724,000.00 1,060,000.00 174,677.03 115,590.09 11,281 7,800 19,081 1,118,441.83 1,908,139.44 724,000.00 1,060,000.00 394,441.83 848,139.44 Statistic Expected profit Standard deviation Q* = Q = 7800 Q = 12000 167,981.31 140,866.41 145,572.65 356,752.81 7800 ... the quantity Q* should not necessarily be equal to the mean dem stock that would be sold at the salvage value price. Statistic Pr(Profit < -$100000) Q = 7800 0.05 Q = 12000 0.25 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 0.8735020962 0.8111686936 0.5579121694 0.2032807705 0.9247020591 0.3638252692 0.1364378729 0.7286868388 0.2427597848 0.7075576946 0.8785554791 0.5728871094 0.9488340311 0.2921328583 0.7890427091 0.1072892083 0.2031250294 0.7504584988 0.575915606 0.62423553 0.6684130324 0.6791678507 0.4154431179 0.187120174 0.6245100674 0.9253663567 0.9125277174 0.8817308038 0.3153091339 0.6760681829 0.6909745852 0.6665727494 0.8916159596 0.8046978269 0.853015026 0.4041511168 0.9083088821 0.3130258523 0.8621125121 0.2890291059 0.2193023079 0.3343982572 0.9764485441 0.4470182283 0.1646429774 0.2900245584 0.0518891491 0.8827368833 0.1954342811 0.0833829026 0.0976557246 0.7800467511 0.2003707323 0.7504135126 0.8765763069 0.2969246033 0.9835497848 0.6492488565 0.8901625623 0.142326311 0.4851938872 0.6667728878 0.4171896373 0.1271698264 0.1546988611 0.5971279635 0.735189457 0.7230858605 0.7990417036 0.9123822626 0.4572797628 0.5395964673 0.9378807013 0.1176099328 0.8360508403 0.2668083287 0.2297827556 0.2134259288 0.3192006031 0.8478507519 0.7735872769 15,997 15,085 12,509 9,098 17,026 10,782 8,166 14,129 9,561 13,910 16,083 12,642 17,712 10,087 14,808 7,660 9,096 14,364 12,669 13,107 13,523 13,627 11,253 8,893 13,110 17,043 16,743 16,139 10,319 13,597 13,744 13,505 16,319 15,002 15,670 11,152 16,652 10,296 15,811 10,055 9,292 10,504 18,942 11,534 8,589 10,065 6,311 16,157 9,000 7,165 7,472 14,701 9,062 14,363 16,049 10,135 19,460 13,340 16,292 8,259 11,870 13,507 11,269 8,014 8,446 12,860 14,198 14,070 14,931 16,740 11,625 12,348 17,375 7,849 15,421 9,823 9,414 9,222 10,357 15,592 14,625 8,197 7,285 4,709 1,298 9,226 2,982 366 6,329 1,761 6,110 8,283 4,842 9,912 2,287 7,008 1,296 6,564 4,869 5,307 5,723 5,827 3,453 1,093 5,310 9,243 8,943 8,339 2,519 5,797 5,944 5,705 8,519 7,202 7,870 3,352 8,852 2,496 8,011 2,255 1,492 2,704 11,142 3,734 789 2,265 8,357 1,200 6,901 1,262 6,563 8,249 2,335 11,660 5,540 8,492 459 4,070 5,707 3,469 214 646 5,060 6,398 6,270 7,131 8,940 3,825 4,548 9,575 49 7,621 2,023 1,614 1,422 2,557 7,792 6,825 - 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,660 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 6,311 7,800 7,800 7,165 7,472 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 7,800 15,997 15,085 12,509 9,098 17,026 10,782 8,166 14,129 9,561 13,910 16,083 12,642 17,712 10,087 14,808 7,660 9,096 14,364 12,669 13,107 13,523 13,627 11,253 8,893 13,110 17,043 16,743 16,139 10,319 13,597 13,744 13,505 16,319 15,002 15,670 11,152 16,652 10,296 15,811 10,055 9,292 10,504 18,942 11,534 8,589 10,065 6,311 16,157 9,000 7,165 7,472 14,701 9,062 14,363 16,049 10,135 19,460 13,340 16,292 8,259 11,870 13,507 11,269 8,014 8,446 12,860 14,198 14,070 14,931 16,740 11,625 12,348 17,375 7,849 15,421 9,823 9,414 9,222 10,357 15,592 14,625 1,025,914.43 998,546.21 921,281.98 818,935.10 1,056,790.06 869,467.42 790,978.12 969,869.53 832,835.44 963,304.47 1,028,504.79 925,273.68 1,077,374.27 848,601.05 990,247.67 766,027.24 818,877.30 976,907.13 926,084.16 939,214.66 951,688.76 954,818.83 883,596.48 812,787.93 939,290.57 1,057,282.53 1,048,298.85 1,030,171.41 855,556.89 953,912.36 958,305.96 951,157.30 1,035,572.25 996,061.14 1,016,090.24 880,548.87 1,045,563.72 854,881.87 1,020,329.06 847,650.76 824,747.55 861,122.66 1,114,270.77 892,027.15 803,661.99 847,956.06 631,146.31 1,030,706.06 815,989.07 716,514.32 747,161.75 987,021.60 817,850.62 976,892.26 1,027,481.44 850,059.00 1,129,788.22 946,208.45 1,034,756.67 793,762.03 902,105.81 951,215.04 884,066.10 786,424.52 799,367.96 931,797.65 971,940.12 968,105.81 993,929.65 1,048,202.93 894,745.09 916,429.11 1,067,258.66 781,478.61 1,008,632.03 840,698.72 828,418.01 822,645.45 856,702.58 1,013,762.10 984,751.70 1,599,714.76 1,508,487.35 1,250,939.95 909,783.68 1,702,633.55 1,078,224.73 816,593.74 1,412,898.44 956,118.13 1,391,014.89 1,608,349.29 1,264,245.59 1,771,247.55 1,008,670.16 1,480,825.56 766,027.24 909,591.00 1,436,357.09 1,266,947.21 1,310,715.54 1,352,295.88 1,362,729.42 1,125,321.59 889,293.12 1,310,968.58 1,704,275.10 1,674,329.50 1,613,904.72 1,031,856.31 1,359,707.87 1,374,353.19 1,350,524.33 1,631,907.50 1,500,203.79 1,566,967.45 1,115,162.91 1,665,212.39 1,029,606.23 1,581,096.86 1,005,502.52 929,158.50 1,050,408.85 1,894,235.89 1,153,423.82 858,873.29 1,006,520.19 631,146.31 1,615,686.87 899,963.56 716,514.32 747,161.75 1,470,072.01 906,168.74 1,436,307.54 1,604,938.12 1,013,529.99 1,945,960.74 1,334,028.16 1,629,188.90 825,873.44 1,187,019.38 1,350,716.80 1,126,887.00 801,415.08 844,559.87 1,285,992.16 1,419,800.41 1,407,019.35 1,493,098.85 1,674,009.78 1,162,483.62 1,234,763.71 1,737,528.88 784,928.69 1,542,106.78 982,329.07 941,393.36 922,151.50 1,035,675.27 1,559,206.99 1,462,505.67 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 724,000.00 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