The Solution of Homework #1



The Solution of Project

Design of a fast-pick area : An office-products wholesaler

Assumption

➢ Picking and restocking costs are proportional to the time to pick and restock.

➢ The fast pick area is bin-shelving, 5 shelves to a section, with each shelve 12.75 inches high x 18 inches deep x 41.25 inches wide.

➢ We are considering several options for configuring the fast pick area: 5 sections of shelving, 10, 15, and 20.

➢ Over this range, the economics of pick and restocks do not change significantly.

➢ Because of the shapes of the storage units, we expect to actually fill only about 60% of the available volume and the rest is lost to imperfect fit of boxes in the shelves.

Parameters

➢ N = then number of sku’s considered(N = 100)

➢ Li = the length of the storage unit of sku i, in inches

➢ Wi = the width of the storage unit of sku i, in inches

➢ Hi = the height of the storage unit of sku i, in inches

➢ xi = pieces per case of sku i

➢ yi = pieces sold of sku i during recent reporting period

➢ bi = the maximum number of cases on hand

➢ pi = the number of picks of sku i during recent reporting period

➢ cf = the cost for picking from the fast pick area = 0.5

➢ cs = the cost for picking from the reserve area = 1

➢ cr = the cost for restocking = 1.75

➢ s = the savings by picking from the fast pick area rather than the reserve area = cs – cf = 0.5

Then, we can compute the following quantities for the analysis.

➢ Zi = cubic inches per case of sku i = Li x Wi x Hi

➢ UBi = the maximum space allocation of sku i to be held in the warehouse = Zi * bi

➢ fi = the flow of sku i during recent reporting period = [pic]

➢ Q = the cubic inches of a shelve = (12.75)(18)(41.25) = 9466.875 inches3

➢ k = option for the number of sections such that

|k |# of sections |

|1 |5 |

|2 |10 |

|3 |15 |

|4 |20 |

➢ Vk = the available space for the allocation for option k

| |the available space |

|V1 |(5)(5)(0.6)Q = (5)(5)(0.6)(9466.875) = 142,003.125 inches3 |

|V2 |(10)(5)(0.6)Q = (10)(5)(0.6)( 9466.875) = 284,006.25 inches3 |

|V3 |(15)(5)(0.6)Q = (15)(5)(0.6)( 9466.875) = 426,009.375 inches3 |

|V4 |(20)(5)(0.6)Q = (20)(5)(0.6)( 9466.875) = 568,012.5 inches3 |

Question 1 : For each of the options, estimate the total pick-savings, the total restocks costs, and the total net benefit of storing all these sku’s in the fast pick area.

We can compute as follows:

➢ The total pick-savings = [pic] for all k

➢ The total restock costs for option k = [pic] with [pic].

➢ The total net benefit for option k = [pic]

Followings are the results of the computation.

| |V1 |V2 |V3 |V4 |

|Total pick-savings |3,769 |3,769 |3,769 |3,769 |

|Total restock costs |3,333.881533 |1,666.940767 |1,111.294027 |833.4705201 |

|Total net benefit |435.1184669 |2,102.059233 |2,657.705973 |2,935.52948 |

|ID |fi |Ui with V1 |Ui with V2 |Ui with V3 |Ui with V4 |

|11038 |7513.0368 |748.3429154 |1496.685831 |2245.028378 |2993.37117 |

|12990 |547.584 |202.0311186 |404.0622372 |606.0932563 |808.1243417 |

|12991 |821.376 |247.4365764 |494.8731527 |742.3096072 |989.7461429 |

|12992 |4791.6 |597.6310753 |1195.262151 |1792.892932 |2390.523909 |

|12993 |4364.8 |570.3941567 |1140.788313 |1711.182189 |2281.576252 |

|12994 |828.733 |248.542241 |497.084482 |745.6266005 |994.1688007 |

|12995 |3043.656 |476.3110715 |952.622143 |1428.93298 |1905.243973 |

|15004 |33942.9376 |1590.624276 |3181.248552 |4771.872045 |6362.496059 |

|15006 |49093.044 |1912.94672 |3825.89344 |5738.839218 |7651.785624 |

|15014 |18590.4576 |1177.166606 |2354.333212 |3531.499238 |4708.665651 |

|15016 |71648.2624 |2310.97754 |4621.95508 |6932.931482 |9243.908643 |

|15024 |154793.6 |3396.79544 |6793.59088 |10190.38465 |13587.17953 |

|15026 |177920.652 |3641.715999 |7283.431999 |10925.1462 |14566.86161 |

|15034 |160087.8144 |3454.395277 |6908.790554 |10363.18413 |13817.57884 |

|15036 |113905.056 |2913.831235 |5827.66247 |8741.49227 |11655.32303 |

|15038 |54273.408 |2011.344397 |4022.688794 |6034.032201 |8045.376268 |

|15044 |132642.426 |3144.373458 |6288.746916 |9433.118825 |12577.49177 |

|15046 |97471.0704 |2695.446779 |5390.893558 |8086.339009 |10781.78535 |

|15048 |143222.144 |3267.367455 |6534.734909 |9802.100755 |13069.46767 |

|15054 |36800.568 |1656.228122 |3312.456243 |4968.683549 |6624.911399 |

|15056 |56814.912 |2057.899031 |4115.798062 |6173.696079 |8231.594772 |

|15663 |26837.7186 |1414.378646 |2828.757293 |4243.135242 |5657.513656 |

|15665 |36016.596 |1638.491624 |3276.983248 |4915.474066 |6553.965421 |

|15669 |14956.956 |1055.879849 |2111.759697 |3167.639026 |4223.518701 |

|16004 |43234.88 |1795.188355 |3590.37671 |5385.564181 |7180.752241 |

|16006 |21423.0016 |1263.668191 |2527.336382 |3791.003951 |5054.671934 |

|16014 |12624.2688 |970.0542216 |1940.108443 |2910.162187 |3880.216249 |

|16016 |11615.7184 |930.4990432 |1860.998086 |2791.496671 |3721.995562 |

|16024 |52588.2 |1979.871693 |3959.743386 |5939.614104 |7919.485473 |

|16026 |78443.4 |2418.082411 |4836.164822 |7254.246043 |9672.328057 |

|16034 |62607.888 |2160.266944 |4320.533889 |6480.799769 |8641.066359 |

|16036 |66629.808 |2228.57459 |4457.149179 |6685.722672 |8914.296896 |

|16038 |50490.2475 |1939.977278 |3879.954556 |5819.930878 |7759.907838 |

|16044 |49662.4128 |1924.00768 |3848.01536 |5772.022092 |7696.029456 |

|16046 |66007.8848 |2218.149438 |4436.298877 |6654.447223 |8872.596297 |

|16048 |41239.044 |1753.263485 |3506.52697 |5259.789592 |7013.052789 |

|16054 |46443.6 |1860.612059 |3721.224119 |5581.835262 |7442.447016 |

|16056 |36934.632 |1659.242188 |3318.484376 |4977.725747 |6636.967663 |

|16665 |14227.2 |1029.799346 |2059.598693 |3089.397532 |4119.196709 |

|16669 |5833.4256 |659.4087794 |1318.817559 |1978.226014 |2637.634685 |

|17021 |37229.2536 |1665.846797 |3331.693594 |4997.53957 |6663.386094 |

|17022 |30386.2896 |1504.98347 |3009.966939 |4514.949668 |6019.93289 |

|17023 |40992.0896 |1748.006013 |3496.012025 |5244.017177 |6992.022903 |

|17025 |26885.2584 |1415.630793 |2831.261586 |4246.891682 |5662.522243 |

|17028 |45164.5656 |1834.813016 |3669.626031 |5504.438143 |7339.250857 |

|17041 |7498.4448 |747.6158374 |1495.231675 |2242.847144 |2990.462859 |

|17042 |11354.112 |919.9611351 |1839.92227 |2759.882952 |3679.843936 |

|17048 |17409.6384 |1139.167979 |2278.335958 |3417.503377 |4556.671169 |

|17921 |12241.49328 |955.2347053 |1910.469411 |2865.703645 |3820.938194 |

|17922 |3521.042 |512.304884 |1024.609768 |1536.9144 |2049.2192 |

|17923 |8517.59568 |796.8038133 |1593.607627 |2390.411048 |3187.21473 |

|17928 |15629.20704 |1079.347688 |2158.695376 |3238.042533 |4317.390044 |

|18099 |4244.625 |562.4870876 |1124.974175 |1687.460986 |2249.947981 |

|19021 |10561.6 |887.2739912 |1774.547982 |2661.821537 |3549.095382 |

|19022 |7264.32 |735.8518346 |1471.703669 |2207.555141 |2943.406855 |

|19023 |9404.39808 |837.2562449 |1674.51249 |2511.768322 |3349.02443 |

|19028 |20752.3296 |1243.730606 |2487.461212 |3731.191206 |4974.921608 |

|19041 |4694.52 |591.5459567 |1183.091913 |1774.637579 |2366.183439 |

|19048 |6225.83808 |681.2269241 |1362.453848 |2043.680437 |2724.907249 |

|19921 |7605.22752 |752.9202868 |1505.840574 |2258.76049 |3011.680653 |

|19922 |3226.46016 |490.406307 |980.812614 |1471.218679 |1961.624906 |

|19923 |2074.15296 |393.1997959 |786.3995919 |1179.599194 |1572.798926 |

|19928 |8166.97728 |780.2316567 |1560.463313 |2340.694586 |3120.926115 |

|25070 |25749.504 |1385.40684 |2770.813679 |4156.219837 |5541.626449 |

|25071 |29145.6 |1473.938604 |2947.877208 |4421.815086 |5895.753448 |

|25072 |25090.56 |1367.565294 |2735.130588 |4102.695208 |5470.260278 |

|25073 |61053.696 |2133.284969 |4266.569939 |6399.853857 |8533.138476 |

|25074 |21288.96 |1259.708664 |2519.417329 |3779.125373 |5038.83383 |

|25075 |8870.4 |813.1384463 |1626.276893 |2439.414939 |3252.553251 |

|25076 |20959.488 |1249.92291 |2499.845819 |3749.768113 |4999.690817 |

|25077 |13102.848 |988.2703024 |1976.540605 |2964.81042 |3953.080561 |

|25078 |63351.3408 |2173.055403 |4346.110806 |6519.165138 |8692.220184 |

|25079 |35659.008 |1630.337515 |3260.675031 |4891.011743 |6521.348991 |

|25970 |17860.62432 |1153.828365 |2307.656731 |3461.484528 |4615.312703 |

|25971 |47679.43488 |1885.204393 |3770.408786 |5655.612251 |7540.816334 |

|25972 |79108.00128 |2428.304247 |4856.608494 |7284.911545 |9713.215393 |

|25973 |69475.01792 |2275.659221 |4551.318442 |6826.976543 |9102.63539 |

|25974 |11779.32448 |937.0291167 |1874.058233 |2811.086889 |3748.115851 |

|25975 |14973.28448 |1056.456042 |2112.912085 |3169.367607 |4225.823476 |

|25976 |17937.27936 |1156.301741 |2312.603482 |3468.904653 |4625.206204 |

|25977 |9045.29472 |821.1155078 |1642.231016 |2463.346119 |3284.461492 |

|25978 |39707.31072 |1720.394837 |3440.789675 |5161.183665 |6881.57822 |

|25979 |41776.9968 |1764.661871 |3529.323741 |5293.984743 |7058.646324 |

|26012 |10570.77 |887.6590909 |1775.318182 |2662.976835 |3550.635781 |

|30071 |10869.12 |900.098598 |1800.197196 |2700.295351 |3600.393801 |

|30078 |14705.28 |1046.958682 |2093.917364 |3140.87553 |4187.83404 |

|32038 |13593.184 |1006.592039 |2013.184078 |3019.775621 |4026.367495 |

|33038 |13709.9136 |1010.904785 |2021.809571 |3032.713858 |4043.618478 |

|35072 |9218.664 |828.9472336 |1657.894467 |2486.841293 |3315.78839 |

|35073 |15797.322 |1085.137142 |2170.274284 |3255.410892 |4340.547856 |

|35079 |24364.64 |1347.636897 |2695.273793 |4042.910026 |5390.546701 |

|38048 |23365.44 |1319.714156 |2639.428313 |3959.141819 |5278.855759 |

|38600 |16663.65415 |1114.49472 |2228.989439 |3343.48361 |4457.978146 |

|38601 |19743.0129 |1213.108395 |2426.21679 |3639.324587 |4852.432783 |

|38602 |27889.758 |1441.834013 |2883.668026 |4325.50133 |5767.335106 |

|38603 |82641.7566 |2481.947915 |4963.89583 |7445.842523 |9927.790031 |

|38604 |5871.528 |661.5588165 |1323.117633 |1984.676124 |2646.234832 |

|38606 |17321.0076 |1136.264582 |2272.529164 |3408.793186 |4545.057581 |

|38607 |16601.4576 |1112.412867 |2224.825735 |3337.238054 |4449.650739 |

|38609 |67146.9669 |2237.206608 |4474.413216 |6711.618722 |8948.824963 |

Question 2 : For each of the options, if you can choose any subset of sku’s to go into the fast pick area, which should they be and in what amounts ? What is the total pick-savings, the total restock costs, and the total net benefit ?

We can compute them as follows:

1) Compute the viscosity of sku i = [pic]

2) Sort them in decreasing order

Now, we have N+1 alternatives of selecting sku’s according to the order of viscosity. Let S be the set of sku’s selected for fast pick area and R={1,…,N} be the set of remaining sku’s.

3) First select the most viscous sku i from R and S(S({i} and R(R\{i}. Compute [pic].

4) Compute [pic],[pic], and [pic].

5) Repeat step 3) and 4) while adding a new sku into S and removing from R until either we find a sku i such that [pic] which means that the net benefit of sku i becomes negative so that adding some additional sku’s into the fast pick area will decrease the net benefit, or |S| = N.

6) Select S as a set of sku’s with the maximum net benefit.

Followings are the results of the computation.

| |V1 |V2 |V3 |V4 |

|# of sku’s |56 |73 |85 |95 |

|Total pick-savings |2586.5 |3273.5 |3541 |3704 |

|Total restock costs |772.7096399 |884.6721663 |808.8085828 |749.6156761 |

|Total net benefit |1813.79036 |2388.827834 |2732.191417 |2954.384324 |

|ID |fi |Ui with V1 |Ui with V2 |Ui with V3 |Ui with V4 |

|11038 |7513.0368 |1554.41711 |2054.468795 |2631.561264 |3156.35904 |

|12990 |547.584 |419.6480265 |554.6476359 |710.4460467 |852.1263908 |

|12991 |821.376 |513.9617682 |679.3018475 |870.1151521 |1043.637427 |

|12992 |4791.6 |1241.366692 |1640.710923 |2101.580379 |2520.686985 |

|12993 |4364.8 |1184.791649 |1565.935846 |2005.801267 |2405.807172 |

|12994 |828.733 |0 |0 |0 |1048.300897 |

|12995 |3043.656 |989.3673932 |1307.644147 |1674.956413 |2008.98375 |

|15004 |33942.9376 |0 |4366.832201 |5593.458753 |6708.93144 |

|15006 |49093.044 |0 |0 |0 |8068.422309 |

|15014 |18590.4576 |0 |0 |4139.527452 |4965.050623 |

|15016 |71648.2624 |0 |0 |0 |9747.235794 |

|15024 |154793.6 |0 |9325.417656 |11944.892 |14326.99605 |

|15026 |177920.652 |7564.373958 |9997.812138 |12806.15953 |15360.02143 |

|15034 |160087.8144 |0 |9483.549797 |12147.44286 |14569.94051 |

|15036 |113905.056 |0 |7999.508279 |10246.53972 |12289.95073 |

|15038 |54273.408 |0 |5521.859319 |7072.928594 |8483.443807 |

|15044 |132642.426 |0 |8632.429088 |11057.24558 |13262.33119 |

|15046 |97471.0704 |5598.835115 |7399.964886 |9478.586871 |11368.84927 |

|15048 |143222.144 |0 |8970.091572 |11489.75617 |13781.09499 |

|15054 |36800.568 |0 |0 |5824.155851 |6985.635191 |

|15056 |56814.912 |0 |5649.668429 |7236.638797 |8679.801835 |

|15663 |26837.7186 |0 |3882.974949 |4973.68784 |5965.563026 |

|15665 |36016.596 |0 |0 |5761.785141 |6910.826233 |

|15669 |14956.956 |0 |0 |3713.020398 |4453.487615 |

|16004 |43234.88 |0 |0 |0 |0 |

|16006 |21423.0016 |0 |0 |4443.711825 |5329.896811 |

|16014 |12624.2688 |0 |0 |0 |4091.492482 |

|16016 |11615.7184 |0 |0 |3272.11655 |3924.656741 |

|16024 |52588.2 |0 |0 |6962.254267 |8350.698308 |

|16026 |78443.4 |0 |6638.500554 |8503.230105 |10198.98247 |

|16034 |62607.888 |0 |0 |7596.617399 |9111.568987 |

|16036 |66629.808 |0 |6118.233845 |7836.822457 |9399.676818 |

|16038 |50490.2475 |0 |0 |0 |8182.431734 |

|16044 |49662.4128 |0 |0 |6765.807463 |8115.0752 |

|16046 |66007.8848 |0 |0 |7800.162225 |9355.705638 |

|16048 |41239.044 |0 |0 |0 |0 |

|16054 |46443.6 |0 |0 |0 |7847.685297 |

|16056 |36934.632 |0 |4555.212897 |5834.75487 |6998.347913 |

|16665 |14227.2 |0 |0 |3621.307844 |4343.485332 |

|16669 |5833.4256 |0 |0 |0 |0 |

|17021 |37229.2536 |3460.206158 |4573.344909 |5857.980095 |7026.204817 |

|17022 |30386.2896 |3126.069624 |4131.717576 |5292.301324 |6347.715843 |

|17023 |40992.0896 |3630.862803 |4798.901324 |6146.894449 |7372.73577 |

|17025 |26885.2584 |2940.471115 |3886.41254 |4978.091037 |5970.844328 |

|17028 |45164.5656 |3811.173577 |5037.217576 |6452.152829 |7738.870149 |

|17041 |7498.4448 |1552.906864 |2052.472706 |2629.004481 |3153.292372 |

|17042 |11354.112 |1910.893121 |2525.622153 |3235.059807 |3880.209976 |

|17048 |17409.6384 |2366.217628 |3127.423295 |4005.9046 |4804.780102 |

|17921 |12241.49328 |1984.161458 |2622.460712 |3359.099949 |4028.986759 |

|17922 |3521.042 |1064.131778 |1406.460028 |1801.5293 |2160.79837 |

|17923 |8517.59568 |1655.077446 |2187.511283 |2801.97488 |3360.757305 |

|17928 |15629.20704 |2241.962181 |2963.195213 |3795.545475 |4552.470215 |

|18099 |4244.625 |0 |1544.228115 |1977.995917 |2372.456754 |

|19021 |10561.6 |1842.997168 |2435.884259 |3120.11488 |3742.342215 |

|19022 |7264.32 |1528.471318 |2020.176314 |2587.636155 |3103.674188 |

|19023 |9404.39808 |1739.103032 |2298.567667 |2944.22658 |3531.377479 |

|19028 |20752.3296 |2583.409417 |3414.485081 |4373.600951 |5245.804112 |

|19041 |4694.52 |0 |0 |0 |2495.021186 |

|19048 |6225.83808 |1415.007432 |1870.211409 |2395.546679 |2873.277365 |

|19921 |7605.22752 |1563.924977 |2067.035316 |2647.657673 |3175.665467 |

|19922 |3226.46016 |0 |1346.340607 |1724.522561 |2068.434602 |

|19923 |2074.15296 |816.7331821 |1079.473988 |1382.694124 |1658.437202 |

|19928 |8166.97728 |0 |2142.014789 |2743.698594 |3290.859301 |

|25070 |25749.504 |2877.691567 |3803.436984 |4871.807963 |5843.365806 |

|25071 |29145.6 |3061.584921 |4046.488322 |5183.131498 |6216.774879 |

|25072 |25090.56 |2840.63208 |3754.455563 |4809.067848 |5768.113775 |

|25073 |61053.696 |4431.143249 |5856.629776 |7501.7348 |8997.764474 |

|25074 |21288.96 |2616.598169 |3458.350562 |4429.788078 |5313.196329 |

|25075 |8870.4 |1689.006855 |2232.355689 |2859.415909 |3429.653483 |

|25076 |20959.488 |2596.271732 |3431.485167 |4395.376296 |5271.921995 |

|25077 |13102.848 |2052.7812 |2713.155234 |3475.270217 |4168.324226 |

|25078 |63351.3408 |4513.752226 |5965.813832 |7641.588241 |9165.508117 |

|25079 |35659.008 |3386.448215 |4475.859239 |5733.111071 |6876.433851 |

|25970 |17860.62432 |2396.669383 |3167.671295 |4057.458111 |4866.614645 |

|25971 |47679.43488 |3915.8438 |5175.559919 |6629.355011 |7951.410786 |

|25972 |79108.00128 |5043.941211 |6666.563147 |8539.175373 |10242.09611 |

|25973 |69475.01792 |4726.875284 |6247.497989 |8002.396406 |9598.270268 |

|25974 |11779.32448 |1946.345802 |2572.479863 |3295.07967 |3952.199269 |

|25975 |14973.28448 |2194.412902 |2900.349463 |3715.046593 |4455.917884 |

|25976 |17937.27936 |2401.806944 |3174.461595 |4066.15578 |4877.046843 |

|25977 |9045.29472 |1705.576372 |2254.255574 |2887.467389 |3463.299115 |

|25978 |39707.31072 |3573.510375 |4723.098778 |6049.799256 |7256.277418 |

|25979 |41776.9968 |3665.459443 |4844.627607 |6205.465072 |7442.98681 |

|26012 |10570.77 |0 |0 |0 |0 |

|30071 |10869.12 |1869.635742 |2471.092389 |3165.21284 |3796.433812 |

|30078 |14705.28 |2174.685502 |2874.275813 |3681.648955 |4415.859938 |

|32038 |13593.184 |2090.838112 |2763.45495 |3539.698932 |4245.601604 |

|33038 |13709.9136 |2099.796314 |2775.294981 |3554.864782 |4263.791896 |

|35072 |9218.664 |1721.844006 |2275.756461 |2915.007793 |3496.33175 |

|35073 |15797.322 |2253.987719 |2979.089334 |3815.904194 |4576.888961 |

|35079 |24364.64 |2799.2379 |3699.744989 |4738.989282 |5684.059826 |

|38048 |23365.44 |2741.238306 |3623.087087 |4640.798466 |5566.287356 |

|38600 |16663.65415 |2314.967678 |3059.686378 |3919.140641 |4700.713284 |

|38601 |19743.0129 |0 |3330.416166 |4265.917396 |5116.645819 |

|38602 |27889.758 |0 |0 |0 |6081.36421 |

|38603 |82641.7566 |0 |6813.834191 |8727.814293 |10468.35425 |

|38604 |5871.528 |0 |0 |2326.383426 |2790.321266 |

|38606 |17321.0076 |0 |0 |0 |4792.534159 |

|38607 |16601.4576 |0 |0 |0 |4691.932453 |

|38609 |67146.9669 |0 |0 |0 |0 |

Question 3 : Revisit your solutions to the two previous questions:

1) How many sku’s were stored in amounts less than a single storage unit?

➢ When all of sku’s are put into the fast pick area

| |V1 |V2 |V3 |V4 |

|# of sku’s |6 |1 |0 |0 |

➢ When some of sku’s are put into the fast pick area

| |V1 |V2 |V3 |V4 |

|# of sku’s |0 |0 |0 |0 |

2) How many sku’s were stored in amounts exceeding their maximum on-hand inventory?

➢ When all of sku’s are put into the fast pick area

| |V1 |V2 |V3 |V4 |

|# of sku’s |1 |3 |9 |18 |

➢ When some of sku’s are put into the fast pick area

| |V1 |V2 |V3 |V4 |

|# of sku’s |2 |6 |10 |21 |

| | | | | |

3) How does the quality of your solution change if you were to round each volume stored to one that is a multiple of the volume of a storage unit?

Suppose we have a set of sku’s that will be put into the fast pick area with volume V. Then we know that the space allocation to each sku that maximizes the total net benefit, is [pic]. However, given the fact that each of the considered SKU’s will be stored in the fast-pick area in a quantity that corresponds to an integral number of cases, the aforementioned space allocation is only an approximation of the actual optimal solution, resulting from the fluid-based relaxation of the original model. In other words, this first solution addresses only a relaxation of the actual problem, and therefore, it will over-estimate the resulting net benefit; any rounding of this solution to a set of SKU volumes [pic] that are multiples of the corresponding case volumes, will tend to decrease the total net benefit.

There are several ways to perform the rounding of the original relaxed solution, obtained from the fluid-based model, to a near-optimal feasible solution that also observes the granularity of the SKU’s under consideration. One heuristic that tries to control the net benefit losses due to this rounding is as follows:

A Greedy heuristic

Suppose that we have selected a certain set of n sku’s to be put into the fast pick area. Then,

1. Compute the optimal volume for each sku resulting from the fluid-based model, [pic].

2. Round all sku’s that their currently assigned volume is less than a case to a whole case volume Zi.

3. Re-compute the optimal [pic] for the remaining sku’s.

4. Round all sku’s that their assigned volume is exceeding the maximum volume to its maximum volume UBi.

5. Re-compute the optimal [pic] for the remaining sku’s.

6. Identify the sku such that when its volume is reduced to the closest integer multiple of its case volume Zi, it has the smallest net benefit decrease.

7. Set this sku to that volume and repeat steps 5 and 6 for the remaining sku’s.

Question 4 : For each of the options, estimate how much an additional section of rack would be worth to you.

The space of an additional section is W = (5)(0.6)Q = (5)(0.6)(9466.875) = 28400.625 inches3. So new available space for each option is as follows: Vk + W.

| |new available space |

|V1+W |142,003.125 + 28400.625 = 170403.75 inches3 |

|V2+W |284,006.25 + 28400.625 = 312406.875 inches3 |

|V3+W |426,009.375 + 28400.625 = 454410 inches3 |

|V4+W |568,012.5 + 28400.625 = 596413.125inches3 |

We can compute the new total net benefit for each option according the procedures described in Question 1 and 2. Then the worth of an additional section of rack can be computed by (Net benefit with Vk+W) – (Net benefit with Vk).

➢ When all sku’s are put into the fast pick area

| |k=1 |k=2 |k=3 |k=4 |

|Net benefit with Vk |435.1184669 |2,102.059233 |2,657.705973 |2,935.52948 |

|Net benefit with Vk+W |990.7653891 |2,253.599303 |2,727.16185 |2,975.218552 |

|Worth of W |555.6469222 |151.54007 |69.455877 |39.689072 |

➢ When some of sku’s are put into the fast pick area

| |k=1 |k=2 |k=3 |k=4 |

|Net benefit with Vk |1,813.79036 |2,388.827834 |2,732.191417 |2,954.384324 |

|Net benefit with Vk+W |1,956.084955 |2,472.093893 |2,783.458695 |2,990.080308 |

|Worth of W |142.294589 |83.266059 |51.267278 |35.695984 |

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