Minimizing makespan in two-stage hybrid cross docking scheduling problem

Abstract

This paper studies the two-stage hybrid cross docking scheduling problem. In which, the job in the second stage cannot be processed until its precedent subset jobs in the first stage have all been completed and at least one stage contains more than one machine. The objective is to minimize the makespan. Firstly, a mixed integer programming is presented and solved by CPLEX for small scale instances. Secondly, four heuristics are proposed to investigate the performance for moderate and large scale instances. Furthermore, one lower bound is given to compare with the four heuristics. Finally, computational experiments are carefully designed to illustrate and compare these approaches and computational results are reported in detail.

Introduction

In order to increase agility and decrease inventory, more logistics companies are turning to cross docking which brings significant benefits including little or no inventory, low handling costs, low space requirement, centralized processing and low transportation costs. In cross docking, the products on incoming shipments have been packed so that they can be easily sorted in cross docking center (CDC) or for outgoing shipments based on customers’ requirements (Fig. 1). Items are carried from incoming-vehicle docking point to outgoing-vehicle docking point without being stored as inventory at CDC. Shipments typically stay less than 24 h at CDC, sometimes less than an hour. Buffa [1] showed that logistics cost could be reduced by integrating inbound and outbound vehicles in distribution system. Cross docking operation can achieve this function actually.

The most famous application of cross docking is Wal-Mart. Cross docking has helped Wal-Mart to improve its market share and profitability [2]. Besides, cross docking also make great commercial success in Home Depot, Costco, Canadian Tire, FedEx and so on. In perishable products or environments with little or no storage, cross docking operation can provide low inventory and transportation cost for companies by integrating goods in distribution center. Many 3PL companies gain great reduction of transportation cost by employing cross docking operation, which changes LTL (less than truckload) to FTL (full truckload) from suppliers to CDC and from CDC to customers after cross docking, as shown in Fig. 2.

The cross docking scheduling model has been first proposed by Chen and Lee from the scheduling points of view [3]. Two-machine cross docking flow shop problem can be described as follows: there are two sets of jobs, $J_1=\{j_{11},j_{12},\dots ,j_{1n}\}$ and $J_2=\{j_{21},j_{22},\dots ,j_{2m}\}$, two machines, $M_1$ and $M_2$, and a set of subsets $J_1$ of $\{S_1,S_2,\dots ,S_m\}$, each subset $S_j$ corresponding to a job $j_{2j}\in J_2$. Jobs in $J_1$ are processed in $M_1$ and jobs in $J_2$ are processed on machine $M_2$. Each job $j_{2j}\in J_2$ can be processed on $M_2$ only after all jobs in the corresponding precedent subset $S_j$ of $J_1$ have been processed on $M_1$. Job $j_{1i}$ and $j_{2j}$ require processing time $p_{1i}$ and $p_{2j}$ on $M_1$ and $M_2$, respectively. The problem is denoted as $F2\left|\text{<mi mathvariant="italic" is="true">CD</mi>}\right|C_{\max }$ and the objective is to minimize the makespan. Chen and Lee [3] showed the problem is NP-hard in strong sense, and developed approximation algorithm and branch and bound algorithm based on the several characteristics of its feasible solution. Chen and Song [4] studied cross docking scheduling problem with total completion time occurring in just-in-time logistics. Based on the different characteristics of the problem, several heuristics and branch and bound algorithm were proposed. Ma and Chen [5] also studied the cross docking scheduling problem with total completion time, a dynamic programming was designed with computational complexity of $O(\text{<mi mathvariant="italic" is="true">nm</mi>}{2}^m)$.

This paper will focus on minimizing makespan in two-stage hybrid cross docking scheduling problem. The model proposed by Chen and Lee [3] is a special form of this problem. In their model, there are two stages, and only one machine exists at each stage. In this paper, there are also two stages, while at least one stage contains more than one parallel machine, so we call it hybrid cross docking scheduling problem. By three field notation in [6] and the notation in [7], our problem is denoted as $F2(P)\left|\text{<mi mathvariant="italic" is="true">CD</mi>}\right|C_{\max }$. Since $F2\left|\text{<mi mathvariant="italic" is="true">CD</mi>}\right|C_{\max }$ is strongly NP-hard, it is not difficulty to see that $F2(P)\left|\text{<mi mathvariant="italic" is="true">CD</mi>}\right|C_{\max }$ is also strongly NP-hard.

Because $F2(P)\left|\text{<mi mathvariant="italic" is="true">CD</mi>}\right|C_{\max }$ is a new problem, there is no literature considering the problem. We will review the most relative problem of hybrid flow shop problem without cross docking constraints denoted as $F2(P)\parallel C_{\max }$, which occurs in many different environments, including automobile manufacture [8] and printed circuit board manufacture [9]. Brah and Hunsucker [10] presented a branch and bound algorithm to solve hybrid flow shop scheduling problems. Sriskandarajah and Sethi [11] analyzed a special two-stage hybrid flow shop problem, in which the first stage contains either one machine or the same number of machines as the second stage. They also analyzed the worst case behavior of several heuristic algorithms. Lee and Vairaktarakis [12] developed heuristics with worst-case error bounds for multistage hybrid flow shop scheduling problem by extending results for a two-stage hybrid flow shop and aggregating machines at each stage. Gupta [13] applied Johnson's rule [14] to the case with one machine in the first stage and multiple machines in the second stage. Bertel and Billaut [15] provided an MIP model for the three-stage hybrid flow shop scheduling problem and heuristic approaches to solve it. Ruiz et al. [16] contribute to the recent research efforts to bridge the gap between the theory and the practice of scheduling by modeling a realistic manufacturing environment and analyzing the effect of the inclusion of several characteristics.

$F2(P)\left|\text{<mi mathvariant="italic" is="true">CD</mi>}\right|C_{\max }$ problem is a general form of $F2\left|\text{<mi mathvariant="italic" is="true">CD</mi>}\right|C_{\max }$, and is more common in real logistics operations. The two stage of our problem are equal to inbound and outbound of cross docking operation and the machines in two stages are equal to the vehicles in inbound and outbound. In fact, this is just many 3PL companies’ real situation.

The rest of this paper is organized as follows: Section 2 provides the mixed integer programming model. Section 3 develops four heuristic approaches, whereas Section 4 gives the lower bound (LB) of optimal value. Computational experiments are described and results are reported in Section 5. Section 6 concludes the paper.