Unit Load Storage Problem Solution
ISyE 6202---Fall 2003
Block Stacking Analysis
Let’s calculate the footprint sq. ft. Since the width of the pallet is 52 in. and the required clearance between lanes of storage is 6 in. the width along the pick aisle per storage row is (52 + 6) = 58 in. Since the depth of a pallet is 48 in., if x pallets are to be stored in a row, the depth due to pallet storage is 48x. To assess the true cost of space, we add one-half of the required aisle space, since this space must be in front of each storage row. The calculated storage depth is therefore (48x + 48) in. Therefore, the footprint sq. ft. is: 58 (48x + 48) /144 = 19.33 + 19.33x.
The average number of footprints is dependent on how deep we store the pallets in a storage row, namely, the variable x, and the rate of pallet withdrawal. The pallets are withdrawn at a uniform rate. Since there are 47 loads to store, and the loads will be stored 4 high, there are 47/4 “=” 12 stacks to be stored. (The last stack initially has only 3 pallets in it.) The maximum number of storage rows, M, equals (Q/3x) rounded up to the nearest integer. For example, suppose we choose to configure a storage row 3-deep, i.e., choose x to be 3. There will be 3 x 4 = 12 loads per storage row, and thus 47/12 “=“ 4 storage rows required initially. The first 3 storage rows will have 12 loads in them, and the last or 4th storage row will have 11 loads initially.
How do we estimate the average number of storage rows used? Consider once again the case when x = 3. It will take 47 time units to empty the load. Given the assumption of uniform withdrawal, for the first 11 time units there will be 4 storage rows; for the next 12 time units there will be 3 storage rows; for the next 12 time units there will be 2 storage rows; and for the final 12 time units there will be 1 storage row. The average number of storage rows will be 12/47(3 + 2 + 1) + 11/47 (4) = 2.47 storage rows. We can also use conditional expectation. Suppose we had started with 3 full rows consisting of a total of 36 pallet loads. Then clearly the average number of rows occupied would be 2. Now 36/47 = 76.6% of the time we will be in this situation. The other 11/47 = 23.4% of the time we will be occupying the full 4 storage rows. The average value is simply the average of the two averages (2 and 4) weighted by their respective probabilities.
Now we can develop the following table. The best choice is 3 or 4 deep storage.
xM(x)Footprint area (sq. ft.)Avg. # of rowsAvg. sq. ft.
11238.676.38247
2658.003.45200
3477.332.47191
4396.671.98191
53116.01.72200
62135.31.49202
Single-deep and Double-deep pallet rack analysis
The width along the pick aisle to store 2 pallet loads in single-deep or double-deep pallet rack is 104 in. for the 2 pallets plus 1 rack member clearance of 3 in. and 3 horizontal clearances of 3 in. totaling 116 in. Thus, the billable width for one pallet position is 116/2 = 58 in.
The depth of single-deep pallet rack is the pallet depth of 48 in. plus one-half of the sum of the aisle width (96 in.) and the flue space (8 in.) totaling 100 in. For double-deep pallet rack it will be 100 + 48 = 148 in. since 2 pallets are stored in a lane.
Thus, the footprint in sq. ft. on the floor for single-deep pallet rack is: (58)(100)/144 = 40.28 and (58)(148)/144 = 59.61 for double-deep pallet rack. Since each LAYER of the storage row is independently supported, to arrive at a correct sq. ft. number we simply divide the floor footprint by the number of storage levels z, which in this case is 5. Thus, the billable sq. ft. for single-deep pallet rack is 40.28/5 = 8.06 sq. ft. and for double-deep pallet rack is 59.61/5=11.92 sq. ft.
As before, we now need to calculate the average number of footprints. For single-deep pallet rack, we initially need 47 pallet rack positions. Since withdrawal takes place uniformly, the average number of single-deep pallet rack positions is simply (47 + 1)/2 = 24. For double-deep rack, we initially need 24 pallet rack positions (with the last pallet rack position initially containing only 1 pallet); thus, the average number of double-deep pallet rack positions is 2/47(1 + 2 + ... + 23) + 1/47(24) = 12.26. Note that the sum 1 + 2 + ... + 23 = 23*24/2 so average number can be alternatively expressed as 12*(46/47) + 24 (1/47). This latter equation is an expectation. that is, 46/47 of the time we have anywhere from 1 to 23 full double-deep rows, and given this knowledge, the average of the numbers from 1 to 23 is 12, and 1/47 of the time we have 24 rows.
The billable sq. ft. for single-deep pallet rack is therefore 8.06*24 = 193, and the billable sq. ft. for double-deep pallet rack is therefore 11.92*12.26 = 146.
Economic analysis:Selecting the best alternative
MethodEstimated sq. ft.Cost per sq. ft.Total annual cost
Block Stacking191$5.00$955
Racking Alternatives:
Single-deep193$7.00$1351
Double-deep146$8.00$1168
The recommended storage alternative is block stacking. Double deep rack saves about 25% in space, but costs 60% more per sq. ft. and therefore is not worth it.
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