The integrated orderline batching, batch scheduling, and picker routing problem with multiple pickers: the benefits of splitting customer orders

Abstract

Fast delivery is one of the most popular services in e-commerce retail. It consists in shipping the items ordered on-line in short times. Customer orders in this segment come with deadlines, and respecting this latter is pivotal to ensure a high service quality. The most time-consuming process in the warehouse is order picking. It consists in regrouping orders into batches, assigning those batches to order pickers, sequencing the batches assigned to each order picker such that the orders deadlines are satisfied, and the picking time is minimized. To speed up the order picking operations, e-commerce warehouses implement new logistical practices. In this paper, we study the impact of splitting the orders (assigning the orderlines of an order to multiple pickers). We thus generalize the integrated orders batching, batch scheduling, and picker routing problem by allowing the orders splitting and propose a route first-schedule second heuristic to solve the problem. In the routing phase, the heuristic divides the orders into clusters and constructs the picking tours that retrieve the orderlines of each cluster using a split-based procedure. In the scheduling phase, the constructed tours are assigned to pickers such that the orders deadlines are satisfied using a constraint programming formulation. On a publicly available benchmark, we compare our results against a state-of-the-art iterated local search algorithm designed for the non-splitting version of the problem. Results show that splitting the customer orders using our algorithm reduces the picking time by 30% on average with a maximum reduction of 60%.

Appendices

Appendix A: Mathematical formulation

We propose a mixed-integer linear formulation (MILP) to model the problem. The formulation is defined on a new directed graph $\mathcal {G}' = (\mathcal {V'}, \mathcal {A'})$. The node set $\mathcal {V}' = \{0,n+1\} \cup {\mathcal {L}}$ contains two copies of the depot ($\{0,n+1\}$) in addition to the pick locations set ${\mathcal {L}}$. The arc set $\mathcal {A}'$ is composed of arcs from 0 to the nodes in $\mathcal {V}' - \{0\}$, arcs from the nodes in $\mathcal {V}' - \{0,n+1\}$ to $n+1$, and two directed arcs between each pair of nodes in $\mathcal {V}' - \{0,n+1\}$. Arc $(0,n+1)$ models an empty tour with null processing time $t_{0,n+1} = 0$. Note that the travel time between pick location i and the two depot copies are equivalents ($t_{0,i} = t_{i,n+1}$). Moreover, the induced sub-graph $\mathcal {G}'[{\mathcal {L}}]$ is complete and symmetric ($t_{i,j} = t_{j,i} | \forall i,j \in {\mathcal {L}}$). The Table 5 presents the sets, parameters and variables used in the following MILP formulation.

Table 5 Notations used in the MILP formulation

$$\begin{aligned}&\min \sum _{k \in {\mathcal {K}}} at_{n+1}^{k,P} \end{aligned}$$

(6)

$$\begin{aligned}&\sum _{k \in {\mathcal {K}}}\sum _{p \in {\mathcal {P}}} x_{m,o}^{k,p} = 1&\forall o \in {\mathcal {O}}, \forall m \in {\mathcal {M}}_o \end{aligned}$$

(7)

$$\begin{aligned}&\sum _{o \in {\mathcal {O}}} \sum _{ m \in {\mathcal {M}}_o} q_{m} \cdot x_{m,o}^{k,p} \le W&\forall k \in {\mathcal {K}}, \forall p \in {\mathcal {P}} \end{aligned}$$

(8)

$$\begin{aligned}&\sum _{i \in {\mathcal {L}}\cup \{0\}} y_{i,j}^{k,p} - \sum _{i \in {\mathcal {L}}\cup \{n+1\}} y_{j,i}^{k,p} = 0&\forall j \in {\mathcal {L}}, \forall k \in {\mathcal {K}}, \forall p \in {\mathcal {P}} \end{aligned}$$

(9)

$$\begin{aligned}&\sum _{i \in {\mathcal {V}}' - \{0\}} y_{0,i}^{k,p} = 1&\forall k \in {\mathcal {K}}, \forall p \in {\mathcal {P}} \end{aligned}$$

(10)

$$\begin{aligned}&\sum _{i \in {\mathcal {V}}'- \{n+1\}} y_{i,n+1}^{k,p} = 1&\forall k \in {\mathcal {K}}, \forall p \in {\mathcal {P}} \end{aligned}$$

(11)

$$\begin{aligned}&t_{i,j} - M (1 - y_{i,j}^{k,p}) ~~~~~~~~ \le ~ at_{j}^{k,p} - at_i^{k,p}&\forall (i,j) \in \mathcal {A}', \forall k \in {\mathcal {K}}, \forall p \in {\mathcal {P}}\nonumber \\&+ \left( \beta ^{s}\right) _{\mathbb {1}_{(i=0 ~\cap ~j \ne n+1)}} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~&\nonumber \\&~~ +\left( \beta ^{p} \cdot \sum _{o \in {\mathcal {O}}} \sum _{m \in {\mathcal {M}}_o | i = l_m} x_{m,o}^{k,p}\right) _{\mathbb {1}_{(i \ne 0)}} ~~~~~~&\end{aligned}$$

(12)

$$\begin{aligned}&at_{n+1}^{k,p} \le at_{0}^{k,p+1}&\forall k \in {\mathcal {K}}, \forall p \in {\mathcal {P}}- \{P\} \end{aligned}$$

(13)

$$\begin{aligned}&\sum _{j \in {\mathcal {L}}} y_{0,j}^{k,p} \ge \sum _{j \in {\mathcal {L}}} y_{0,j}^{k,p+1}&\forall k \in {\mathcal {K}}, \forall p \in {\mathcal {P}}- \{P\} \end{aligned}$$

(14)

$$\begin{aligned}&\sum _{j \in {\mathcal {L}}\cup \{n+1\}} y_{l_m,j}^{k,p} \ge x_{m,o}^{k,p}&\forall o \in {\mathcal {O}},\forall m \in {\mathcal {M}}_o, \forall k \in {\mathcal {K}}, \forall p \in {\mathcal {P}} \end{aligned}$$

(15)

$$\begin{aligned}&et_o \ge at_{n+1}^{k,p} - M \cdot (1 - x_{m,o}^{k,p} )&\forall o \in {\mathcal {O}}, \forall m \in {\mathcal {M}}_o, \forall k \in {\mathcal {K}}, \forall p \in {\mathcal {P}} \end{aligned}$$

(16)

$$\begin{aligned}&et_o ~~\le d_o ~~&\forall o \in {\mathcal {O}} \end{aligned}$$

(17)

$$\begin{aligned}&et_o \ge 0,~ at_{i}^{k,p}~\in \{0,1\}&\forall o \in {\mathcal {O}}, \forall i \in {\mathcal {V}}', \forall p \in {\mathcal {P}}, \forall k \in {\mathcal {K}} \end{aligned}$$

(18)

$$\begin{aligned}&x_{m,o}^{k,p} \in \{0,1\},~ y_{i,j}^{k,p} \in \{0,1\}&\forall o \in {\mathcal {O}}, \forall m \in {\mathcal {M}}_o, \forall (i,j) \in \mathcal {A}'\nonumber \\&\quad&\forall p \in {\mathcal {P}}, \forall k \in {\mathcal {K}} \end{aligned}$$

(19)

The objective function (6) is the sum of pickers completion time. Assuming that the horizon start time equals 0, it is equivalent to minimize the total processing time defined in Eq. (1) since there is no waiting times between tours and at picking point in the optimal solution. Constraints (7) assign each orderline to exactly one position of one picker. Constraints (8) ensure that each tour satisfies the cart capacity. Constraints (9), (10), (11) are flow constraints for each picker’s tour. Constraints (12) are an adaptation of the subtour elimination constraints of Miller–Tucker–Zemlin (Desrochers and Laporte 1991) that use nodes arrival time variables. Besides the travel time between the nodes of arc (i, j) in the current tour (p, k), a setup time is added to the constraint if $(i=0)$ and $(j\ne n+1)$. Furthermore, the search and pick time of all orderlines retrieved from the tail of the arc (i, j) are added to the constraint if $(i \ne 0)$. Constraints (13) prevent overlaps between the consecutive tours of each picker. Constraints (14) ensure that the non-empty tours of each picker are positioned at the begenning of the sequence to avoid the exploration of some symmetric solutions (i.e. symmetries that result in having an empty tour at different positions between two non-empty tours). Constraints (15) link the variable $x_{m,o}^{k,p}$ and $y_{i,j}^{k,p}$: A picker k must stop at the picking locations of all orderlines that he/she retrieves during his/her tour at position p. Constraints (16) define the completion time of each order as the completion time of the last tour that retrieves one of its orderlines. Constraints (17) force the satisfaction of the deadlines. Finally, constraints (18) and (19) define the domain of the variables.

To test our MILP formulation, we conducted preliminary experiments on a benchmark of small instances generated by Van Gils et al. (2019). In these instances, the number of orders is set to $\{18,12,6\}$ and the batch capacity is set to $\{4,8,12\}$. We ran the MILP formulation on some of those instances by setting a time limit of 2 h. We observed that the MILP is not able to return a feasible solution for most of the instances, even for the smallest ones (6 orders, $W = 15$). For the few instances where the MILP returned feasible solutions, the gaps were poor (no less than 100%). We thus conclude that those results are not exploitable in any comparative analysis.

Appendix B: Kruskal–Wallis test

Table 6 presents results of a Kruskal–Wallis H test on order picking time, on CPu time, and on gap $\varDelta _{ILS|RFSS^{+}}$. Tables 7, 8, and 9 present results of a pairwise-comparison between groups of each warehouse factor on order picking time, on CPu time, and on gap $\varDelta _{ILS|RFSS^{+}}$ using the test of Dunn. Note that the Test of Dunn is done when the $p\text {-}value$ returned by the Kruskal–Wallis’ test is significant (i.e.  $p\text {-}value < 0.05$).

Table 6 Kruskal–Wallis H test on order picking time, on CPU time, and on gap $\varDelta _{ILS|RFSS^{+}}$

Table 7 Dunn test on order picking time

Table 8 Dunn test on CPU time

Table 9 Dunn test on gap $\varDelta _{ILS|RFSS^{+}}$

Appendix C: Cconstraint programming formulation

We propose a natural CP formulation to model the problem. The formulation uses the parameters and variables summirized in Table 10.

Table 10 Notations used in the CP formulation

$$\begin{aligned}&\sum _{k = 1}^K (pos_v^k \ge 1 ) = 1,&\forall v \in \Pi \end{aligned}$$

(20)

$$\begin{aligned}&\sum _{v\in \Pi } (pos_v^k = t) \le 1,&\forall k \in \{1,\ldots ,K\}, \forall t \in \{1,\ldots , P\} \end{aligned}$$

(21)

$$\begin{aligned}&st_{1}^k = 0,&\forall k \in \{1,\ldots ,K\} \end{aligned}$$

(22)

$$\begin{aligned}&st_{t+1}^k = st_{t}^k + p_{t}^{k},&\forall k \in \{1,\ldots ,K\}, \forall t \in \{1,\ldots ,P-1\} \end{aligned}$$

(23)

$$\begin{aligned}&pos_v^k = t \implies p_{t}^k = p_v,&\forall v \in \Pi , \forall k \in \{1,\ldots ,K\}, \forall t \in \{1,\ldots , P\} \end{aligned}$$

(24)

$$\begin{aligned}&pos_v^k = t \implies st_t^k = st_v,&\forall v \in \Pi , \forall k \in \{1,\ldots ,K\}, \forall t \in \{1,\ldots , P\} \end{aligned}$$

(25)

$$\begin{aligned}&st_{v} + p_v \le \tilde{d}_v&\forall v \in \Pi \cup \{v^+\} \end{aligned}$$

(26)

$$\begin{aligned}&p_{t}^{k}, st_{v}, st_{t}^{k} \in {\mathbb {R}}^+,&\forall v \in \Pi , \forall k \in \{1,\ldots ,K\}, \forall t \in \{1,\ldots , P\} \nonumber \\&pos_{v}^{k} \in \{1,\ldots , P\},&\forall k \in \{1,\ldots ,K\}, \forall t \in \{1,\ldots , P\} \end{aligned}$$

(27)

Constraints (20) guarantee that each tour is sequenced once and only once. Constraints (21) ensure that no more then one tour is sequenced at the tth position of picker k. Constraints (22) and (23) sequence the tours assigned to each picker. Constraints (24) link the variables $pos_v^k$ and $p_{t}^k$ while constraints (25) synchronize $st_v$ and $st_t^k$ variables. Constraints (26) bound the end time of each tour by its deadline. Finally, constraints (27) define the domain of the variables.