Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 827021, 15 pages

http://dx.doi.org/10.1155/2015/827021

## A Methodology to Exploit Profit Allocation in Logistics Joint Distribution Network Optimization

^{1}School of Management, Chongqing Jiaotong University, Chongqing 400074, China^{2}School of Transportation Science and Engineering, Beihang University, Beijing 100191, China^{3}College of Transport & Communications, Shanghai Maritime University, Shanghai 201306, China^{4}Department of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195-2700, USA

Received 1 July 2014; Revised 9 September 2014; Accepted 26 September 2014

Academic Editor: Tsan-Ming Choi

Copyright © 2015 Yong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Logistics joint distribution network (LJDN) optimization involves vehicle routes scheduling and profit allocation for multiple distribution centers. This is essentially a combinational and cooperative game optimization problem seeking to serve a number of customers with a fleet of vehicles and allocate profit among multiple centers. LJDN routing optimization based on customer clustering units can alleviate the computational complexity and improve the calculation accuracy. In addition, the profit allocation mechanism can be realized based on cooperative game theory through a negotiation procedure by the Logistics Service Provider (LSP). This paper establishes a model to minimize the total cost of the multiple centers joint distribution network when each distribution center is assigned to serve a series of distribution units. An improved particle swarm optimization (PSO) algorithm is presented to tackle the model formulation by assigning distribution centers (DCs) to distribution units. Improved PSO algorithm combines merits of PSO algorithm and genetic algorithm (GA) with global and local search capabilities. Finally, a Shapley value model based on cooperative game theory is proposed to obtain the optimal profit allocation strategy among distribution centers from nonempty coalitions. The computational results from a case study in Guiyang city, China, suggest the optimal sequential coalition of distribution centers can be achieved according to Strictly Monotonic Path (SMP).

#### 1. Introduction

A logistics joint distribution network (LJDN) is usually composed of several logistics facilities (e.g., logistics centers and distribution centers) and a large number of customers [1–3]. Different from single-depot logistics distribution network optimization problems, the cooperation mechanism widely exists for LJDN. How to allocate the cost savings among different logistics participants in LJDN is considered as the most critical issue during the optimization procedure. Properly optimizing LJDN not only mitigates network-wide traffic congestion and reduces negative environmental effects (i.e., energy consumption and traffic pollution), but also promotes mutual cooperation for profit maximization [4, 5]. To achieve these goals, a necessary negotiation process between multiple participants is desired, and this process can be implemented by introducing a Logistics Service Provider (LSP) to coordinate or discussing within the existing participants [6–8]. The LSP can be defined as a Logistics Service Provider that performs corresponding logistics operations on behalf of other participants [9]. LSPs are actively looking for opportunities to increase both efficiency and profit for their own clients (i.e., participants) [10, 11]. In addition, logistics companies and manufacturers tend to outsource their noncore business to a third-party for financial savings, and this raises the transportation demand and stimulates resource integration in LSP market.

LJDN optimization is a strategic and tactical procedure with multiple complicated steps, such as multiple depots (or centers) vehicle routing optimization and profit allocation procedures. Most previous studies focused on designing efficient algorithms for solving Multiple Depot Vehicle Routing Problems (MDVRP) [2, 12–14]. To ensure successful delivery and pickup services in MDVRP, the coordination among multiple depots should be taken prior to vehicle routing. This incurs an interesting issue on how to allocate profit within multiple logistics entities in a cooperative manner [15, 16]. Traditional MDVRP neglected such a profit allocation procedure by assuming that each logistic entity is willing to cooperate. This assumption oversimplifies the realistic condition where the logistics activities are profit-driven, and thus individual benefit should be incorporated during horizontal and vertical cooperation [11], which can be achieved via a profit allocation mechanism to retain a mutually beneficial relationship. The stability of cooperation relies on the rationality of profit allocation. Therefore, the rationality of profit allocation is the core of the logistics joint distribution network optimization [17] and should be taken into account in this study.

The cost savings from multiple depots vehicle routing optimization will be served as the input to allocate profit among various logistics entities. Therefore, developing a robust solution algorithm to find optimal solutions for MDVRP is necessary. However, in metropolitan logistics distribution network with thousands of customers, traditional approach for MDVRP may not be effective to cope with such a sophisticated scenario [18, 19]. To alleviate the computational complexity and improve the calculation accuracy, customer clustering should be initially applied before VRP optimization. A large logistics region can be grouped into several smaller zones where customers share certain common features (i.e., geospatial location, demand, etc.). Then, customers within each zone form a clustering unit and request service from each depot. Many researchers have proposed a variety of clustering methods to study MDVRP [14, 20–24]. Thangiah and Salhi [21] presented a generalized clustering approach based on genetic algorithm, and their genetic clustering method can be further used to solve the MDVRP. Wu et al. [22] proposed a hybrid simulated annealing algorithm to solve MDVRP, where insertion and 2-swap operators were used to manipulate customers from one cluster to another cluster. Dondo and Cerdá [23] developed a three-stage heuristic approach for the multiple depots routing problem with time windows and heterogeneous vehicles: cluster generation, cluster assignment and sequencing, and nodes sequencing within clusters (i.e., vehicle scheduling). Mirabi et al. [14] presented three hybrid heuristics to solve the MDVRP. Yücenur and Demirel [24] developed a genetic algorithm based on the clustering technique for studying the multidepot vehicle routing problem, and the cluster first-route second algorithm was proposed. In their paper, clustering procedures were used to group customers with similar characteristics to reduce the calculation complexity.

Customer clustering is usually an intermediate stage during the MDVRP optimization procedure. The distribution centers (DCs) can be assigned to a number of customer clustering units for delivery service, and this issue can be considered as a variant of the quadratic assignment problem (QAP) [25, 26]. The QAP was first introduced by Koopmans and Beckmann [27] and aimed to assign facilities to locations in such a way that each facility is assigned to one exact location. The goal of the QAP is to minimize the sum of the distances multiplied by the corresponding flows and the associated cost of allocating each facility to a certain location [28–30]. Within each clustered region, the mathematical programming model and solution algorithm can be further developed to solve MDVRP [12, 15, 31]. Ho et al. [13] developed two hybrid genetic algorithms based on customer clustering techniques for dealing with MDVRP. Liu et al. [32] presented a mathematical programming model and a two-phase greedy algorithm to study the full truckloads multidepot capacitated vehicle routing problem in carrier collaboration. Aras et al. [16] formulated two mixed-integer linear programming models (MILP) for selective MDVRP with pricing, and a Tabu Search on the basis of heuristic method was proposed to solve the MILP model. Bettinelli et al. [33] established an integer linear programming model and presented a branch-and-cut-and-price algorithm to solve the multidepot heterogeneous vehicle routing problem with time windows. Narasimha et al. [34] proposed an extension of ant-colony technique to solve the min-max MDVRP. Tu et al. [35] presented a bi-level Voronoi diagram-based metaheuristic to tackle the large-scale MDVRP.

Based on the aforementioned discussion, a natural thought is to incorporate the profit allocation paradigm into multiple depots’ vehicle routing optimization for logistics joint distribution network. However, only a small number of relevant studies have been conducted on this research domain. Özener and Ergun [36] presented cost allocation mechanisms based on the cooperative game theory, and then a set of new properties and several cost allocation schemes were proposed to study a collaborative transportation procurement network. Krajewska et al. [37] presented the profit margins resulting from horizontal cooperation among freight carriers, which is based on the cooperative game theory for a pickup and delivery problem with time windows. The possibilities of sharing these profit margins among the partners were also discussed. Wang et al. [38] constructed two mathematical models to study the optimal allocation of the module members for given garment assembly tasks in a modular production system. Frisk et al. [17] proposed a new cost allocation method based on economic models. These models include Shapley value, the nucleolus, shadow prices, and volume weights for collaborative forest transportation. Cruijssen et al. [7] proposed a novel “supplier-initiated outsourcing” procedure to exploit synergy in transportation. Lozano et al. [39] presented a linear model to allocate the cost savings among different companies when their transportation requirements are simultaneously considered. However, the above studies suffer from the following issues. (1) The network size in most studies is relatively small. When considering the large-scale logistics distribution network, customer clustering should be adopted prior to VRP optimization for reducing the calculation complexity. Consequently, the mathematical programming model and solution algorithm should be designed to optimize the logistics distribution network based on customer clustering units rather than customers. (2) Most studies focus on investigating the mechanism of profit allocation from economic perspectives but neglect the interaction between profit allocation and VRP optimization. To the best of our knowledge, no explicit architecture was developed to explain how optimized vehicle routings affect the profit distribution among multiple logistics participants. Therefore, a reasonable profit allocation approach based on cooperative game theory should be designed and combined with MDVRP for LJDN optimization.

This study aims to construct a multiple centers logistics joint distribution network and then develops a linear programming model with a solution algorithm to optimize the network for cost savings calculation. Based on the computed cost savings, a profit allocation approach is proposed to distribute the total profit within logistics participants and determine the optimal strategy for sequential coalitions. Compared with the previous studies, the main contributions of this paper lie in the following. (1) The model formulation is first established to minimize the total cost in the LJDN optimization procedure. (2) An improved particle swarm optimization (PSO) algorithm is proposed to assign distribution centers (DCs) to distribution units and resolve the model. (3) Shapley value model is utilized to study the profit allocation among multiple distribution centers in LJDN. (4) A real-world numerical study is undertaken to demonstrate the applicability of the proposed method.

#### 2. Logistics Joint Distribution Network

Logistics joint distribution network (LJDN) can be established through negotiation. The negotiation procedure is organized by either Logistics Service Provider (LSP) or players from the distribution network [1, 4, 40–42]. LJDN can reasonably integrate the resources together. Therefore, it can reduce the crisscross transportation phenomenon and realize information sharing. In our study, the logistics distribution network contains multiple DCs and a large amount of customer clustering units. Figure 1 presents a logistics network structure change before and after joint distribution. DCs are independent with each other before the LJDN is established. Each customer clustering unit is a group of customers with common features, such as similar temperature controlled goods and similar geographical conditions; the customer clustering unit is referred to as a distribution unit.