5.1. Experimental Data Setup
The store has 20 parallel racks (equivalently 10 picking aisles) and 10 picking columns per rack. Instead of demand at the item level, we rather focus on category-level demand, equivalently on average 10 SKUs at each pick column and 10 pick columns per rack. The total number of SKUs equals 2000, conforming with the assortment size in a typical grocery store. The length of a picking face, ; the associated depth, ; and the aisle width, , are 2 m. The length of a parallel aisle, , is 20 m.
The demand of items per picking column/face is assumed to follow a negative binomial distribution, which is denoted by
[
3,
34]. The probability of success
for items within Classes A, B, and C are set as 0.96, 0.975, and 0.99, respectively. Let
and the average order size,
, ranging between 1.98 and 7.68, conforms to empirical observations and previous case studies [
33]. The maximum picking capacity is 20 items per basket. Notably, the largest order size should not be great than the basket capacity, as we assume no order split is permitted.
Furthermore, omni-channel grocery delivery service tends to provide a 30 min service guarantee. Given the various distance between the store and neighborhood consumers, we assume vehicle delivery time varies between 5 min and 20 min. Thus, the due time for order picking is thus assumed uniformly distributed .
Unlike storage assignment decisions in a traditional logistics warehouse, the random storage assignment policy that is commonly tested in previous literature is rarely observed in retail stores [
2]. We mimic the arrangement of ABC classes, in which the percentages of the number of storage spaces for each class are expressed as 10%:20%:70%. Because shelves in brick-and-mortar retail stores are typically arranged to prolong in-store customer shopping trips so as to increase the likelihood of pulse purchases [
43,
44], items in classes A, B, and C are placed from the rightmost aisles to the leftmost aisles.
The entire workforce available for omni-channel order fulfillment comprises specialized pickers and flexible store associates, such as checkout cashiers and salespersons, who are not trained or hired for in-store order picking but are capable of order picking. The number of specialized pickers available is three, which also serves as the benchmark in the computational experiments. The proficiency of flexible store associates in order picking varies in search time. We utilize the framework of a learning curve, as shown in (14), to characterize the individual proficiency in order picking.
denotes the initial search time for beginners.
indicates the learning rate of individual difference in search time, and
is the cumulated item picked previously. We let
and
to control systematic variance but manipulate the value of
to differentiate individual skills. We set
as 100 and 1000, indicating the specialists’ and the flexible workers’ proficiency. We note that differentiating individual proficiency in order picking is the sole purpose of applying this learning curve formula, and we have no intention to justify a specific learning curve formula.
Workforce composition is one of the significant characteristics that distinguish order fulfillment in retail stores from that in traditional warehouses. Besides the base workforce (i.e., three specialized order pickers), we additional consider six workforce compositions. As shown in
Table 4, we incrementally add more flexible or specialized workers to analyze the benefits of a heterogeneous workforce in order fulfillment. Furthermore, we randomly generate 40, 60, and 80 customer orders in the experiment. Test instances and results are deposited in the Harvard Dataverse repository (
https://doi.org/10.7910/DVN/HZI7CV). There are 10 replications for each group.
We set retrieve time, , as constant and homogeneous for items located on different layers of a picking column. The walking speed of each picker is 40 m/min.
Policies for order picking and picker scheduling define how customer orders are batched, sequenced, and allocated. The Earliest Start Date (ESD) rule that is also implemented by Henn [
15] and Scholz et al. [
16] is applied as the baseline for comparison. As shown in Algorithm 3, this priority-based algorithm firstly assigns a priority value for each customer order based on its due date in the ascending order. Two kinds of feasible picking positions are identified if (1) the current picking capacity is large enough for the selected customer order, or (2) a new picking position is created. Note that when the algorithm calculates the completion time of a picking position, the heterogeneous picking efficiency of a picker has been explicitly applied. Lastly, the algorithm assigns online customer orders iteratively to any feasible picking position that has the earliest start date.
Algorithm 3 Earliest Start Date Policy |
1: | sort ascendingly the set of customer orders according to due dates |
2: | while there are unassigned orders, |
3: | let be the first order in the sorted list; |
4: | for all order pickers, |
5: | if can be assigned to the last batch of picker then |
6: | = start time of the last batch of picker (CASE 1); |
7: | else |
8: | = completion time of the last batch of picker (CASE 2); |
9: | end if |
10: | end for |
11: | ; |
12: | assign to the last batch of picker (CASE 1), or open a new batch for picker and assign to it (CASE 2); |
13: | remove from the list of unassigned orders; |
14: | end while |
As an NP-hard problem, previous literature has shown that for instances with more than 20 orders, the commercial solver, such as CPLEX, cannot report a solution with a CPU time limit of several hours [
18,
19]. Hence, for the sake of brevity, this study only reports the effectiveness of GAVND compared with several classic heuristics reported in recent studies. Besides GAVND, we additionally consider an elitist-based GA policy and a VND algorithm.
The elitist-based GA policy is obtained by running GAVND except for the part of the VND operator.
The VND algorithm is implemented by firstly obtaining the initial solution based on the ESD rule and then running Algorithm 2.
To compare policy performance, we calculate the relative gap between two policies in terms of their tardiness values for each of the 210 instances. As shown below,
,
, and
denote the performance of VND, GA, and the proposed GAVND policies, relative to the ESD rule, which also serves as the benchmark in previous literature [
15,
16].
Moreover, we need to verify the performance of the proposed GAVND policy relative to the other two heuristics. As shown below,
and
indicate the performance of the proposed GAVND policy relative to the VND and GA policies.
We run numerous pre-test trials to find the best parameters for the hybrid GAVND algorithm. The algorithm parameter is determined so that the GAVND reaches steady states to obtain promising solutions within acceptable computing time, i.e., five minutes. We set the crossover and mutation rates to 0.5 and 0.2. The size of the initial population is 10, and there are 50 iterations. The GAVND and the other heuristics are coded in MATLAB R2021a and run on a computer with the 11th-Gen Intel(R) Core(TM) i7-1165G7 @2.8 GHz, 8 GB RAM, under Microsoft Windows 10 operating system.
5.2. Computational Study Results
We run computational tests to compare the performance of three heuristics with the benchmark policy applied in previous studies [
15,
16], as shown in
Figure 4. All solutions generated by the three heuristics outperform those yielded by the ESD rule, evidenced by the positive average gap values. For each group characterized by order size and workforce configuration, GAVND yields the best performance, followed by GA and VND. The average gap between GAVND and the ESD rule,
, exceeds 10% in all groups. It thus can be concluded that GAVND can generate considerably good quality solutions for various order sizes and workforce configurations.
Besides the illustrative description in
Figure 4, we are further interested in the effectiveness of the proposed GAVND compared with VND, GA, and ESD policies. We ran statistics check using IBM SPSS Statistics 22. Significance values in the Kolmogorov–Smirnov test and Levene’s test are less than 0.05, violating the assumptions of the
t-test. (Detailed reports for checking assumptions are intentionally omitted in the research due to page limit but available from the corresponding author). Thus, the Wilcoxon rank-sum test is applied to verify whether the proposed GAVND outperforms VND, GA, and ESD policies. The results of the Wilcoxon rank-sum test are shown in
Table 5 and
Table 6. Note that a sample with zero values, denoted as 0 in the column of Group in
Table 5, is created to construct two samples for the Wilcoxon rank-sum test. The sample with calculated gap values per instance is denoted as 1 in the column of Group in
Table 5. The two-sided
p-values from asymptotic 2-tailed are 0.000, suggesting the median value of
,
, and
significantly deviate positively from that in the sample of zeros. It thus can be concluded that the proposed GAVND outperforms ESD, VND, and GA. Moreover, we further run the Wilcoxon rank test for groups of three order sizes and groups of seven workforce configurations. The superiority of the proposed GAVDN holds as well.
In what follows, we explore the effect of order size and workforce configurations individually.
5.5. Discussions and Managerial Insights
In this paper, we have characterized a practice-oriented problem, namely the integrated order picking and heterogeneous picker scheduling problem in omni-channel retail stores. To the best of our knowledge, the jointed problem of order batching, sequencing, and allocating a multi-skilled workforce is not yet addressed in the literature. Previous studies oftentimes assume a homogeneous workforce, which is commonly observed in warehouse operations, but this assumption is widely challenged in retail store operations. As shown in
Table 1, this study is the first to investigate the effect of a heterogeneous workforce on the classic order picking problem. Furthermore, the proposed GAVND is verified to be effective compared with benchmark heuristic in recent articles, such as Henn [
15] and Scholz et al. [
16], and classic mate-heuristics, such as VND and GA. However, as there are no direct benchmark instances applied for both retail stores and warehouses, we do not conclude whether the proposed GAVND outperforms heuristics in the setting of warehouse operations, such as the summarized literature in
Table 1 [
15,
16,
17,
18,
19].
Moreover, the discussions on the heterogeneous workforce, which are missed in previous studies, yield some managerial insights. A workforce constituted of specialized and flexible multi-skilled employees can generate almost the same performance as that of all specialized pickers. This observation is crucial for omni-channel retail store managers. Training store associates with skillsets for order fulfillment is easy to roll out, and the store manager can then allocate these flexible cross-trained employees to help facilitate in-store order fulfillment in real time. Thus, dynamic allocation of cross-trained employees can be a cost-effective approach to guaranteeing timely omni-channel order fulfillment service. The savings because of hiring fewer specialized pickers can be utilized for other operations.