The solution of Project(ISyE 6202)
The Solution of Project
Design of a fast-pick area : An office-products wholesaler
Assumption
Ø Picking and restocking costs are proportional to the time to pick and restock.
Ø 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.
Ø We are considering several options for configuring the fast pick area: 5 sections of shelving, 10, 15, and 20.
Ø Over this range, the economics of pick and restocks do not change significantly.
Ø 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
Ø N = then number of sku’s considered(N = 100)
Ø Li = the length of the storage unit of sku i, in inches
Ø Wi = the width of the storage unit of sku i, in inches
Ø Hi = the height of the storage unit of sku i, in inches
Ø xi = pieces per case of sku i
Ø yi = pieces sold of sku i during recent reporting period
Ø bi = the maximum number of cases on hand
Ø pi = the number of picks of sku i during recent reporting period
Ø cf = the cost for picking from the fast pick area = 0.5
Ø cs = the cost for picking from the reserve area = 1
Ø cr = the cost for restocking = 1.75
Ø 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.
Ø Zi = cubic inches per case of sku i = Li x Wi x Hi
Ø UBi = the maximum space allocation of sku i to be held in the warehouse = Zi * bi
Ø fi = the flow of sku i during recent reporting period =
Ø Q = the cubic inches of a shelve = (12.75)(18)(41.25) = 9466.875 inches3
Ø k = option for the number of sections such that
k / # of sections1 / 5
2 / 10
3 / 15
4 / 20
Ø Vk = the available space for the allocation for option k
the available spaceV1 / (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:
Ø The total pick-savings = for all k
Ø The total restock costs for option k = with .
Ø The total net benefit for option k =
Followings are the results of the computation.
V1 / V2 / V3 / V4Total 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 =
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¬SÈ{i} and R¬R\{i}. Compute .
4) Compute ,, and .
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 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