Análisis de solidez y riesgo

6.2.1 Computational time and the total travel distance

The performance of the proposed RBP method is compared to FCFS, seed, CWII, and the LB to understand the relative performance. These problems are computationally difficult so the total travel distance, the run time and the percentage deviation from the lower bound are calculated and reported in Table 1. The RBP produced near-optimal solutions within about 2 minutes and outperformed the seed and the CW II algorithms. Moreover, RBP improvement over alternative methods was larger for scenarios in which the number of orders was smaller.

Table 1. Computational results over different algorithms

Specifically, in the sort-while-picking strategy, the seed algorithm requires a run time of 0.2 seconds. However, the LU gap is between 15 and 30%. CW II has a shorter total travel distance, but took a longer computational time (which was also noted by De Koster et al. (1999)). As the problem size increased, its computational time increased exponentially. When the number of orders was 2160, it took on average 137.30 seconds. RBP demonstrated a considerable improvement in travel distance. The LU gap ranged from 1.07 to 2.26% when the computational time was limited to 60 seconds, whereas the best approach identified in De Koster et al. (1999), CW II, showed a gap ranging from 9 to 14%.

The LU gap of RBP was larger under the sort-while-pick strategy. The increase in the gap is because RBP produced some batches that were not filled to capacity because of fixed non-uniform order sizes. Note that this has been partially improved by forming additional batches by merging these remaining batches using the CW II

algorithm. To investigate additional possibility and improve the solution quality, we conducted a neighborhood search considering different combinations of batches. We observed a small performance improvement, i.e., less than 0.2% of the total retrieval

Sort # FCFS Seed CW II RBP LB IB

Strategy orders Obj LU gap Obj CPU LU gap Obj CPU LU gap ObjL ObjU CPU LU gap Obj CPU Obj Sort- 360 5923.0 57.97% 3549.3 0.00 29.87% 2899.1 0.40 14.14% 2546.9 2546.9 11.47 2.26% 2489.3 0.77 2305.8 while- 720 11892.5 59.80% 6332.3 0.02 24.51% 5501.9 4.96 13.12% 4844.6 4844.6 40.33 1.33% 4780.3 1.83 4615.9 pick 1080 17915.3 60.48% 8970.1 0.05 21.06% 8033.3 16.20 11.86% 7177.2 7177.2 56.95 1.34% 7080.8 2.68 6938.6 1440 23961.0 60.82% 11573.1 0.09 18.88% 10505.0 39.09 10.63% 9504.9 9504.9 60.26 1.23% 9388.3 3.63 9256.0 1800 29989.7 60.95% 14122.7 0.14 17.08% 12942.6 75.68 9.52% 11849.0 11849.0 60.34 1.17% 11710.5 4.58 11587.2 2160 36033.8 61.06% 16605.7 0.21 15.50% 15412.0 137.30 8.96% 14183.3 14183.3 60.40 1.07% 14031.8 5.69 13916.0 Pick- 360 4645.5 55.74% 3147.4 0.01 34.67% 2476.9 0.46 16.98% 2128.7 2128.7 17.54 3.40% 2056.2 4.93 1897.4 then- 720 9342.6 57.37% 5539.1 0.02 28.09% 4659.0 4.79 14.51% 4107.7 4107.7 67.11 3.04% 3983.0 11.98 3814.4 sort 1080 14126.7 57.85% 7967.5 0.05 25.26% 6868.9 14.70 13.31% 6136.5 6160.5 75.30 3.34% 5955.0 12.87 5783.4 1440 18831.5 58.35% 10198.8 0.09 23.09% 8927.0 33.69 12.14% 8076.2 8145.3 96.46 3.70% 7843.7 18.14 7689.6 1800 23522.5 58.55% 12476.8 0.14 21.85% 10979.5 62.21 11.20% 10024.7 10100.9 105.02 3.47% 9750.3 22.80 9614.6 2160 28257.9 58.69% 14683.5 0.20 20.51% 13065.3 104.09 10.66% 12002.4 12108.5 140.54 3.60% 11672.5 27.71 11550.7

distance. The details and experimental results are summarized in Appendix A.3. While the computational time of RBP and CW II was almost equal under the sort-while-pick strategy, the run-time of RBP increased under the pick-then-sort strategy, because the batch packing stage was computationally intensive using the IP bin-packing algorithm. However, run-times were still smaller than 150 seconds for all cases. While the IP-based batch packing process may take slightly longer, this is not a significant computational burden. Note that in both RPP and BPr, the time limit for the branch-and-

bound procedure is 60 seconds, and the solution procedure requires multiple iterations of BPr.

The seed and CWII algorithms depend on having a large number of orders to improve performance. When the number of orders was 360 or 720, the algorithms experienced a large LU gap. Thus, the benefits of RBP are significant for large-size problems, but are even more prominent when the number of orders is small.

6.2.2 The average travel length per order

The average travel length per order is another metric that can evaluate the

performance of various batching methods, assuming all orders construct similar numbers of batches. With this objective, a large-size batching problem is preferred since larger problems can produce more efficient batches, thus reducing trip distance. The previous methods developed for batching demonstrate a significant improvement in average travel length per order as shown in Figure 9. The improvement declined as the number of orders increased. When the number of orders increased from 1800 to 2160, there were minimal gains in throughput of the order picking system. In all cases, RBP dominated

other heuristics in solution quality with very small gaps to IB and LB.

(a) (b)

Figure 9. The average travel length per order with the one-way traversal routing method: (a) sort-while-pick strategy; and (b) pick-then-sort strategy.

6.2.3 Impacts on picker blocking in narrow-aisle configuration

In narrow-aisle picking systems, the shorter travel length does not guarantee a shorter retrieval time due to picker blocking (Gue et al., 2006). Thus, we conduct a simulation study to quantify the effect on picker blocking on the various batching algorithms. Two situations are considered: a light congestion situation and a heavy congestion situation. A light congestion environment is defined as: the number of orders in a time window = 1080 orders, 4 time windows, pick:walk time ratio = 5:1, 5 pickers, setup time per batch = 120, and cart capacity = 10 orders or 20 items. A heavy

congestion environment is defined as: pick:walk time ratio = 10:1, 15 pickers, and cart capacity = 25 orders or 50 items.

Figure 10 depicts the comparison of the total retrieval time. RBP was relatively robust to picker blocking situation, while seed and CW II produced very poor results under heavy congestion. These findings emphasize the importance of picker blocking and selecting a batching algorithm that not only reduces travel distance, but also does not

create excessive picker blocking.

(a) (b)

Figure 10. The total retrieval time comparison via a simulation study: (a) light congestion case; and (b) heavy congestion case.

Other experimental results are summarized in Appendix A.4. RBP demonstrated consistent performance over other order picking profiles, including variations in both OPS sizes and storage policies.