Research Article | Open Access
Design and Profit Allocation in Two-Echelon Heterogeneous Cooperative Logistics Network Optimization
In modern supply chain, logistics companies usually operate individually and optimization researches often concentrate on solving problems related to separate networks. Consequences like the complexity of urban transportation networks and long distance deliveries or pickups and pollution are leading problems to more expenses and more complaints from environment protection organizations. A solution approach to these issues is proposed in this article and consists in the adoption of two-echelon heterogeneous cooperative logistics networks (THCLN). The optimization methodology includes the formation of cooperative coalitions, the reallocation of customers to appropriate logistics facilities, and the determination of the best profit allocation scheme. First, a mixed integer linear programing model is introduced to minimize the total operating cost of nonempty coalitions. Thus, the Genetic Algorithm (GA) and the Particle Swarm Optimization (PSO) algorithm are hybridized to propose GA-PSO heuristics. GA-PSO is employed to provide good solutions to customer clustering units’ reallocation problem. In addition, a negotiation process is established based on logistics centers as coordinators. The case study of Chongqing city is conducted to verify the feasibility of THCLN in practice. The grand coalition and two heterogeneous subcoalitions are designed, and the collective profit is distributed based on cooperative game theory. The Minimum Cost Remaining Savings (MCRS) model is used to determine good allocation schemes and strictly monotonic path principles are considered to evaluate and decide the most appropriate coalition sequence. Comparisons proved the combination of GA-PSO and MCRS better as results are found closest to the core center. Therefore, the proposed approach can be implemented in real world environment, increase the reliability of urban logistics network, and allow decision makers to improve service efficiency.
Cooperative game theory consists in the formation of coalitions and the effective allocation of the collective payoff to each participant. In modern supply chain, coalitions can be made up of facilities from either the same level or different echelons. Therefore, in a network comprising logistics centers (LCs), Distribution Centers (DCs), Pickup Centers (PCs), and customers, coalitions are generally formed vertically with LCs, DCs, and PCs or horizontally among DCs or PCs. A Two-echelon Heterogeneous Cooperative Logistics Network (THCLN) is a collaborative network with vertical synergies among LCs, DCs, and PCs, homogeneous horizontal synergies among DCs or PCs, and heterogeneous synergies among LCs and DCs or PCs. Properly optimizing THCLN can not only enhance the operational efficiency and avoid waste of resources, but also maximize players’ profits. In addition, the implementation of a coordinated transportation system can contribute to achieve more stable coalitions . However, well-developed negotiation mechanisms and a fair allocation of payoffs are necessary to increase players’ willingness to participate .
The optimization of THCLN is a complex process since multiple customers, transportation resources, and profit allocations problems are combined to enhance practicability. Previous researches have focused on establishing a mixed integer programming model and designing efficient algorithms for similar optimization purposes. But due to the number of customers, evolutionary algorithms are usually confronted to problems modeling the amount of available data. Thus, a commonly used approach consists in clustering customers into small units before the application of intelligent algorithms [3, 4]. Clustering can also facilitate the management of companies’ activities and allow decision makers to obtain more effective results by assigning customer clustering units to suitable suppliers [5, 6].
Solving customer clustering units’ assignment problem is a basic network optimization problem and first appeared as the Quadratic Assignment Problem (QAP) in Dantzig et al. (1959). The standard assignment problem is a traditional binary programming problem usually solved by exact methods. However, as the number of tasks had increased, QAPs became more complicated to solve. Therefore, researchers like Li et al.  concentrated on applying heuristic algorithms to assess large-scale location and layout problems. Recently, Sghir et al.  proposed an optimization method based on intelligent multiple agents to solve QAPs and provided the algorithm with the capability of learning and dynamically selecting agents. The efficiency of QAP solutions facilitates the optimization of vehicle routing problems or any logistics transportation-related subject. In other words, the reassignment of customers units not only does have significant impacts on the configuration of the network but also influences future decisions.
With the rising need of intelligent approaches for logistics network optimization, heuristic algorithms like the Particle Swarm Optimization (PSO) and the Genetic Algorithm (GA) have been successfully applied in many domains [9, 10]. Shimizu and Miura  proposed a multipopulation based discrete PSO algorithm in order to handle binary decision variables and optimize large-scale logistics networks. Cupić and Teodorović  proposed an integer programing model to maximize companies’ total profits and service quality and employed the Genetic Algorithm to solve a multiobjective hub location-allocation problem. Haddar et al.  proposed a 0-1 mixed interlinear programming model and combined the quantum PSO with an iterative heuristic algorithm to solve the Knapsack sharing problem. Guo et al.  used GA and PSO to study forward and reverse flows of fresh food supply chains in Shanghai and verified the validity of the model under low-carbon emission environments. Wang et al. [15, 16] combined an improved ant colony algorithm with GA to optimize networks through the reassignment of customers to proper logistics facilities. Alshamsi and Diabat  introduced a GA capable of solving complex problems with thousands continuous and binary variables, constraints, and multiple integer variables. The proposed algorithm was applied and proved many times faster than GAMS. Atashpendar et al.  extended the speed-constrained multiobjective PSO algorithm to increase computation speed, convergence, and solution quality.
In logistics network cooperation, game theoretic methods are usually applied to model participants’ behaviors in different scenarios and evaluate their contribution to the coalition as well as the proportion of allocated profits [19, 20]. Hernández and Peeta  evaluated the influences of collaboration mechanism on carriers and explored the differences with noncollaborative environments. Findings expressed high degree collaboration as beneficial to increase participants’ profits. Guajardo and Rönnqvist  addressed the collaborative logistics network optimization problem and studied coalitions’ layouts under core stability and equilibrium constraints. Applications in the forest transportation domain and for oil operations have been used to verify the relevance of the proposed optimization model. Wang et al. [15, 16] introduced collaboration as means of optimizing a single echelon multiple centers vehicle routing problem and applied an improved version of the Shapley value model to allocate profit.
Extensive articles have solved variants of network optimization problems, but there are still some shortcomings in the literature. (a) Researchers had significant achievements in the optimization of multicenter logistics network. However, few articles studied networks where PCs and DCs can independently collaborate with LCs. (b) The design of cooperative networks with different types of logistics facilities has not been studied particularly from aspects like coalitions’ homogeneity or heterogeneity. (c) Collaboration among facilities of the same coalitions should incorporate sharing of transportation resources in order to yield more cost savings.
To alleviate computational complexity, our paper will first examine similar characteristics of customers and cluster them into groups. Therefore, large logistics areas will be divided into smaller and easily manageable units to be further assigned to distribution or pickup centers. For the allocation of customer clustering units and vehicles, we combine Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithm to increase solution efficiency. In order to achieve stability, the consultative process among the participants is coordinated by logistics centers owing to their capability at performing distribution and collection activities. In such a position, LCs could be more influential and better coordinate DCs and PCs. Furthermore, we compute the collective payoff of each coalition and propose a profit-sharing mechanism using the Minimum Cost Remaining Savings (MCRS) model to ensure a fair distribution. Before the formation of a coalition, DCs and PCs used to independently operate at the second echelon of the logistics network and caused the unreasonable exploitation of resources, half loaded vehicles, and the increase of traffic density. By including the sharing transportation fleets in collaboration’s contracts, companies would significantly save cost and rationally use available resources. In addition, an effective customer clustering units’ assignment procedure should reduce the complexity of the logistics network and make the optimization process conducive to obtaining more efficient customer assignment results and fair profit distribution.
In summary, our research mainly contributes to the domain in the following directions: a mixed integer programming model is established to interpret the two-echelon heterogeneous cooperative logistics network (THCLN) and minimize the total cost of nonempty coalitions; variations in facilities transportation needs, routes, and required fleet size are considered as a result of cooperation, and a model is built to calculate payoffs based on members participation in the collective profit; this paper combines the Genetic Algorithm and the Particle Swarm Optimization heuristics to propose a GA-PSO hybrid algorithm and find the most appropriated logistics facility for each customer clustering unit. A practical study is adopted to support the effectiveness of the implemented approach in real world.
2. Problem Statement and Related Definition
2.1. Problem Description
THCLN increases the sustainability of logistics networks through the partitioning of customers, the sharing of transportation resources, and the reduction of greenhouse gas emissions. Figure 1 shows the network’s structural changes which occur as logistics facilities agree to cooperate. Owing to their comprehensive functions, this article assigns the coordination of coalitions to LCs. For the synergy purposes, customer units can be redistributed among LCs, DCs, and PCs in accordance with the required type of service and their geographical locations. Therefore, distribution units can only be reassigned to LCs or DCs, whereas pickup units should be allocated to LCs or PCs. Transportation at the first echelon is assured by semitrailer trucks, and small vehicles are used to serve customers.
(a) Pickup and delivery before logistics network optimization
(b) Product flow among facilities before logistics network optimization
(c) After logistics network optimization
From a global point of view, Figure 1(a) illustrates a logistics networks characterized by the complexity of both distribution and pickup flows. In fact, companies usually invest in marketing events like advertisements and offer discounts in order to attract more customers to buy their products or require their services. Owing to these events, several customers overlook factors like distance or road conditions and make decisions which might prove irrational over a long period of time. Consequently, with the development of city, the increase in population and the number of logistics facilities, the entire network gains in complexity and gives rise to phenomena like traffic congestion and pollution. Despite the existence of cases where customers change suppliers as new facilities are established closer to them, others maintain partnerships due to loyalty entertained for years. As our proposed initial network presents a two-echelon transportation pattern, we also take into account semitrailer truck trips between PCs and DCs in order to highlight the existence of prior partnerships among centers. Those semitrailer truck transportations in Figure 1(b) can be interpreted as follows: PCs usually collect semifinished products from customers and process them before supplying DCs.
2.2. Logistics Network Optimization Architecture
Solving the THCLN consists in encouraging logistics facilities to synergize and share customers, semitrailer trucks, and vehicles. In addition, agreements should be achieved between PCs and DCs to use LCs as upper-level facility and redirect initial product flows through logistics centers. After the optimization, the network would be more organized, as shown in Figure 1(c), and long distance distributions or pickups can be avoided within the same city. Products at the first echelon are still transported by means of semitrailer trucks and distribution or pickup operations carried out with a homogeneous fleet of vehicle.
The logistics network optimization architecture includes a three-phase solution approach shown in Figure 2. First, customer units’ location and related demand data are collected and clustered, and a mixed integer linear mathematical model is proposed to interpret real world operations. Then, considering the available type of vehicle fleet as well as their capacities, GA and PSO are integrated to address the two-echelon heterogeneous logistics network optimization problem. Results obtained after running GA-PSO are used to compute the gap between objective function values before and after collaboration. Further, the Minimum Cost Remaining Savings method and the strictly monotonic path selection principles are consecutively applied. Their objectives are to distribute the cost savings among logistics facilities and evaluate the stability of one or multiple coalitions in the network.
2.3. Related Definition
In order to interpret the proposed problem into a mathematical model for further computations, some related variables are defined as follows: denotes the set of PCs, and is the total number of PCs. denotes the set of DCs, and is the total number of DCs. denotes the set of pickup units in the two-echelon heterogeneous logistics network, and is the total number of pickup units. denotes the set of distribution units in the two-echelon heterogeneous logistics network, and is the total number of distribution units. expresses that LCs only apply their pickup capacities in the two-echelon heterogeneous logistics network. expresses that LCs only apply their delivery capacities in the two-echelon heterogeneous logistics network. expresses the pickup quantity from the pickup unit to LC or PC with in a working period, . denotes the delivery demand transported from LC or DC to the distribution unit within a working period, . denotes the distance from the pickup unit to LC or PC, . denotes the distance from LC or DC to the distribution unit , . expresses the transport quantity from pickup facility to within a working period, , and . expresses the transport quantity from distribution facility to within a working period, and . expresses the pickup capacity of the th PC or LC within a working period, where . expresses the delivery capacity of the th DC or LC within a working period, where . denotes the total pickup capacity of LC and DCs and can be expressed as , where . denotes the total delivery capacity of LC or PCs and can be expressed as , where . expresses the pickup quantity of the pickup unit within one working period. expresses the delivery demand of the distribution unit within one working period. expresses the total pickup quantity from all pickup units within one working period and can be expressed as , where . expresses the total delivery demand of all distribution units within one working period, and can be expressed as , where ; expresses the pickup relation matrix, where indicates that LC or PC collects goods from pickup unit , and implies that LC or PC does not collect goods from the pickup unit. expresses the distribution relation matrix, where indicates that LC or DC delivers goods to distribution unit and implies that LC or DC does not delivery goods to the distribution unit . denotes the loading capacity of a semitrailer truck. denotes the loading capacity of a vehicle. denotes the freight rate of a semitrailer truck (dollar/pallet·kilometer). denotes the freight rate of a vehicle (dollar/pallet·kilometer). denotes the average annual maintenance cost of a semitrailer truck (dollar/pallet·kilometer). denotes the average annual maintenance cost of a vehicle (dollar/pallet·kilometer). expresses the variable cost coefficient of LCs expresses the variable cost coefficient of DCs. expresses the variable cost coefficient of PCs. expresses the number of working periods. expresses the maximum delivery distance. expresses the cost savings provided by the coordinator when logistics facilities agree to cooperate. expresses the government incentive provided to logistics facilities in case of cooperation. denotes the distance from the LC or PC to , , . denotes the distance from the LC or DC to , , . denotes the number of trips of the vehicle within one working period. denotes the number of trips of the semitrailer truck within one working period. denotes the cooperation relation between PCs and the coordinator; if PC agrees to cooperate with LC, set ; otherwise set . denotes the cooperation relation between DCs and the coordinator; if DC agrees to cooperate with LC, set ; otherwise set . denotes the fixed cost of LCs. denotes the fixed cost of PC . denotes the fixed cost of DC . denotes the sequence set of LC and DCs, where with 0 representing LC. Notice that any DC can be the first node in coalitions without LC. denotes the variable transportation quantity between LCs and DCs, , . For example, denotes that the semitrailer truck departs from LC, visits and , and returns to LC; expresses the variable transport quantity from to .
3. Model Formulation
The THCLN optimization problem is interpreted into a mixed integer linear programing model to minimize the total cost. The objective function is the aggregation of logistics facilities fixed and variable operations costs as well as transportation and maintenance costs of semitrailer trucks and vehicles. is the total transportation and maintenance cost among LCs and PCs (if cooperation occurs) within a working period; is the total transportation and maintenance cost among LCs and DCs (if cooperation occurs) within a working period; is the total transportation and maintenance cost of vehicles from LC or PC to each pickup unit and LC or DC to each distribution unit within a working period; and is the result of subtracting cooperation discounts from the sum of fixed and variable costs within a working period. The objective function can be expressed as follows:withsubject to
The objective function value is calculated using (3). Equations (4) are the subpart of the cost function and represent transportation, maintenance, and fixed and variable costs. Constraints (5)-(6) ensure that a customer unit is only served by a center. Constraint (7) introduces the binary relation variable between DCs, PCs, and their corresponding customers. Constraints (8)-(9) guarantee that delivery and pickup centers, respectively, meet their customers’ needs. Constraint (10) stipulates that pickup activities can either be performed by LCs or PCs, whereas constraint (11) allows only LCs and DCs to deliver products to customer units. Constraints (12)-(13) regulate the maximum travelling distance of pickup and delivery vehicles. Constraint (14) defines nonnegative variables.
4. The Hybrid Algorithm Solving Procedure
The Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithm are combined to design a GA-PSO hybrid algorithm used to address the two-echelon heterogeneous logistics network optimization model. GA is an evolutionary computing approach used to mimic the natural selection procedure and study combinatorial optimization problems [17, 23]. PSO is one of the swarm intelligence stochastic evolutionary metaheuristic approaches, proposed by Kennedy and Eberhart . PSO is inspired by observations of the social behavior of bird flocking, fish schooling, and so on, based on swarm theory, and can be used to optimize the local and global locations according to the particle’s position and velocity. Traditionally, these two algorithms have the merits of being population-based heuristics with the ability to evaluate the fitness of each possible solution. Through local and global search capabilities, GA and PSO have been successfully applied to solve problems in management, engineering, and pure science domains. Considering their individual abilities, we propose GA combined with PSO in a hybrid algorithm, which inherits the merits of both approaches and increases the probability of obtaining optimal solutions.
As a hybrid algorithm, GA-PSO is applied to meet the requirements of logistics network optimization for more complex algorithms. Existing traditional heuristics have already been proved efficient, but the increasing complexity of logistics networks limits their capacity at finding near-optimal solutions. In consequence, properly integrating different methods into a hybrid solution approach like GA-PSO can effectively improve the quality of optimization results. Other hybrid heuristics exist but some of them display weaknesses compared to GA-PSO in terms of performance. For example, Chen et al.  embedded a local search heuristic in the Ant Colony Optimization (ACO) algorithm to improve the solution search ability of ACO and address a task assignment problem. The proposed methodology had a satisfying global search capability but the local search presented insufficiencies. Combining GA and PSO on the contrary ensures good performances at both local and optimum search level. This integration can improve the optimization procedure and reduce the number of necessary iterations. The relevant operations and processes of the proposed hybrid algorithm are illustrated in sections below.
4.1. Hybrid Algorithm Encoding Scheme and Evaluation Function Design
Hybrid algorithm encoding and objective function designing are the key issues in the optimization operation processes . In the two-echelon heterogeneous logistics distribution network, the pickup/delivery capacity of each logistics facility and the transportation cost among logistics facilities should be taken into account. Therefore, a two-dimensional chromosome encoding is proposed in this study. The first dimension of the chromosomes is denoted as a sequence of natural numbers: , and is the total number of pickup units, and () is the total number of distribution units. The second dimension is the chromosome (particle) position. The chromosome position can be defined as the sequence number of logistics facilities including LC, DC, and PC that are assigned to serve each corresponding pickup unit and distribution unit. indicates the logistics facility including logistics center or pickup center assigned to the th pickup unit in the th chromosome, while represents the logistics facility including logistics center or distribution center assigned to the th distribution unit in the th chromosome. The two-dimensional chromosome encoding table is shown in Table 1. For example, if we set , , and , , denotes that the PC 2 is assigned to the third pickup unit in the second chromosome, if , then indicates that the DC 3 is assigned to the 35th distribution unit in the second chromosome.
The chromosome encoding method shown in Table 1 makes sure that each pickup/distribution unit can be served by a certain logistics facility, in other words, the pickup/distribution service can be correspondingly completed by LC, PC or DC. In addition, using the chromosome encoding method to initialize populations, the computational complexity can effectively be simplified. For example, assume there are four pickup units, five distribution units, one LC, two PCs, and three DCs in the two-echelon heterogeneous logistics distribution network. The first chromosome can be generated randomly as follows: 0 1 2 0 3 5 4 0 5, where pickup units 1, 4 and distribution unit 8 are served by LC, pickup units 2 and 3 are served by PC 1 and PC 2, respectively, and distribution units 5 and 7 are served by DC 3 and DC 5, respectively. The initial fitness function value can be computed based on the objective function. However, some randomly generated chromosomes in the initial population may not satisfy requirements like maximum delivery distance limit, missing the PCs or DCs, and so on. Therefore, new chromosomes need to be generated in order to substitute the previous unqualified chromosomes. To evaluate the effectiveness of our hybrid algorithm, the fitness function needs to be properly defined, and the unqualified chromosomes should be penalized. Set the unqualified chromosome number as , with the initial value of being equal to 0. The objective function of the th chromosome is , and the fitness function can be formulated as follows:where is the penalty weight for each unqualified chromosome (particle), and it can be defined relatively to the value of the objective function.
4.2. GA Updates and the Corresponding PSO Operations
GA updates mechanism including the corresponding PSO operations is presented in the hybrid algorithm procedure. The fitness function value and a cumulative probability are firstly calculated for the Genetic Algorithm. Afterwards, the selection, crossover, mutation, and PSO operations are conducted sequentially before the offspring is released into the new population. In the selection process, the roulette wheel selection is proposed based on the elevated chromosomes fitness function value. In the crossover process, a random vector including 0 and 1 is presented to choose the genes corresponding to 0 from the primary parent and the ones corresponding to 1 from the second parent; the offspring can be created by combinations . For example, consider a network where there are four pickup units, five distribution units, one LC, two PCs, and three DCs, the parents can be generated as PA1 = , PA2 = , and the random binary vector as EX1 = . We can get as one of the offspring: offspr1 = . Nevertheless, the offspring chromosomes may not satisfy all the constraints. Therefore, the binary vectors need to be regenerated until the new chromosomes are qualified. The process of mutation occurs after the crossover: two particular orders of genes are reversed for pickups and deliveries in a chromosome. The inversed chromosomes including genes only reserved for pickups, those only reserved for deliveries, and the ones reserved for both pickups and deliveries are compared with the original chromosome, and the optimal one is selected using the mutation operation.
The PSO operations update both location and velocity information in our proposed algorithm. The position of each particle (chromosome) is presented as where ( is population size) and is the iteration number. The corresponding velocity of particle is . The position is evaluated in each iteration, and if the fitness function value is better than that of previous iterations, the position is updated as the “pbest”. Meanwhile, the global best position of current iteration is updated as “gbest”. The initial individual best position “pbest” and the global best position “gbest” are determined based on the initial population after GA operations. The related formulas to update both the velocity and position are presented as follows:where formulas (16) and (17) are used to update both the velocity and position for pickup services and formulas (18) and (19) are used to update both the velocity and position for delivery services. and are two acceleration coefficients. rand(·) represents a random fraction between 0 and 1. denotes the individual best position of particle at the th iteration, and is the global best position of particle at the th iteration. fix(·) assures that the position of each particle is an integer. and represent allowable maximum velocity for pickup and delivery services, respectively. denotes an integer between 0 and , and denotes an integer between 0 and , where and represent the numbers of logistics facility set for pickup service and delivery service, respectively. is the inertia weight and can be formulated as follows:where indicates the maximum number of iterations, denotes the initial inertia weight, and denotes the inertia weight for the maximum number of iterations.
4.3. Hybrid Algorithm Procedure
Based on the features described above, the hybrid algorithm procedure is described in Figure 3 and the main steps can be detailed as follows.
Step 1 (algorithm initialization). In the hybrid algorithm procedure, 0 denotes that the logistics center, which can be used for pickup service and delivery service. An integer within or is randomly chosen and assigned to the gene position in each chromosome, and check whether the chromosomes fulfill the requirement or not. In case the condition is not respected, new chromosomes are generated until requirements are fulfilled, and calculate the fitness function value based on formula (15).
Step 2. Execute the roulette wheel selection based on the elevated chromosomes fitness function value, execute crossover and mutation operations based on the crossover probability pc and the mutation probability pm, respectively, and update the optimal solutions and new chromosomes.
Step 3. Select the chromosomes as the initial particles in the population, conduct the PSO operations as part of the hybrid algorithm, and calculate the individual optimal solution and the global optimal solution .
Step 4. Repeat the following steps until the maximum number of iterations is reached or the termination conditions are satisfied.
Step 4.1. Reexecute the selection, crossover, and mutation operations from GA; if the current fitness function value from the new offspring chromosomes is better than with the parents, update the optimal solutions and new chromosomes.
Step 4.2. Determine whether the number of iterations has achieved the periodic acc or not; if yes, replace the total GG of worst-fit chromosomes from GA with the corresponding number of best-fit particles from PSO algorithm; continue PSO with particles, and update and ; otherwise, the exchange procedure will not be conducted; execute the PSO algorithm on particles and update the individual and global optimal solutions.
Step 5. If the number of iterations achieves the maximum number of iterations, the loop procedure will be finished. The current best known particle (chromosome) and fitness function value will be chosen as the optimal solution; otherwise, return to Step 2 to continue.
Step 6. Calculate and select the optimal solution from all feasible chromosomes. The optimal solution will be the final result for the two-echelon heterogeneous logistics network.
In the above hybrid algorithm procedure, the GA operations and PSO operations are reasonably combined by exchanging worst-fit solutions and best-fit solutions between GA and PSO algorithms. Therefore, it enhances the optimization capability of the hybrid algorithm and provides a more robust local and global search capability.
5. Profit Allocation Strategy
5.1. Minimum Cost Remaining Savings Model
The MCRS model is a game theoretic method used to solve cost or profit allocation problems . The selection of MCRS to calculate logistics facilities profits is due to its well-known merits summarized as follows: the model is simple to compute and each to implement; results are generally fairly balanced and presented in a single vector, easy to visualize; MCRS guarantees both individual and collective profit rationality.
The method proceeds to a two-step distribution, where a portion of the collective profit is distributed before the allocation of the remaining part in order to measure members’ contributions. The upper and lower bound vectors and of the allocation scheme are first determined according to highest and lowest possible profit of each participant. denotes the maximum profit of logistics facility and denotes its minimum value. Assume that represents the actual profit of logistics facility , being the collective profit of alliance and the total benefit of the grand coalition. The value of the facility ’s profit is determined by the following:where where refers to the hyperplane and is a coefficient used to balance the results. The bound vector’s variables and can be solved by the following linear programming model:
5.2. Strictly Monotonic Path Principles
The determination of good coalitions is subject to the verification of strictly monotonic path (SMP) principles. The multitude of coalition possibilities generally provides a wide range of profit allocation schemes. A sequence can be adopted only if the cost reduction percentage of each participant increases whenever a new facility joins. If is a sequence of the coalition, can represent the position of facility in sequence . represents the cost reduction rate when facility joins the coalition as the th member and can be calculated as follows:
The process of selecting the SMP sequence is as follows.
Step 1. Select all sequence coalitions where the cost reduction percentage conforms to the principles of SMP, that is, monotonically increasing, and find the diagonal values in the matrix.
Step 2. Find the minimum value of the diagonal line and move to the next step or choose the second smallest value if the percentages are equal. Continue until all members are searched.
Step 3. The coalition sequence yielded from Step 2 is a candidate for the best profit distribution scheme. The process should be repeated for every sequence to ensure that all the possibilities have been evaluated.
6. Empirical Analyses
6.1. Data Description
We conduct a practical experimentation of the proposed network optimization methodology in Chongqing city, China, to evaluate the effects of collaboration and the applicability of our approach in real situations. Real locations of one LC, two PCs, three DCs, 50 pickup units, and 70 distribution units have been selected to illustrate irrational customer allocations observed in the urban logistics network. In the cooperative network, LC plays the roles of participant and coordinator. In order to clearly define the operations of the logistics center, we represent its pickup function by and the distribution function by . Figure 4 shows the logistics network before optimization. In order to facilitate the calculation, each distribution and pickup unit’s demand is converted to the standard roll pallet quantity. The characteristics of all the logistics facilities are summarized in Table 2.
6.2. Improved ASO Parameter Setting and Optimization Results
In this section, we introduce the different parameters initialized for optimization purposes. They are separated in two categories such as objective function calculation parameters and algorithmic parameters and displayed as follows:(1)Objective function parameters: , , , , , , = 52, , , , , , , , , , . To encourage logistics facilities to cooperate, the local government provides the following as incentive: , , , , , , . The total cost savings provided if all the facilities form the grand alliance is .(2)Algorithm parameters: is the popular size used to increase the diversity of initial chromosomes; = 2 expresses the maximum velocity for pickup service; = 3 expresses the maximum velocity for delivery service; = 0.8 and = 0.02 indicate the crossover and mutation probability, respectively; indicates chromosomes as the initial particles in the population; expresses the number of chromosomes for exchanges between GA and PSO; denotes the number of iterations used for replacement between GA and PSO; and represent the numbers of logistics facility for pickup and delivery, respectively; is the penalty weight for each unqualified chromosome or particle; is the maximum number of iterations; is the initial inertia weight and is the inertia weight of the maximum evolution generation; are coefficients used for PSO speed calculation.
In this study, a working period consists of five working days. GA-PSO algorithm is used to assign customer units to corresponding logistics facilities and to compute the total cost in one working period. Cost savings are redistributed to participants of every coalition by applying the MCRS model. Optimization results are listed in Table 3, the initial customer units’ assignment results are displayed in Table 4, and the best customer allocation in the grand alliance is summarized in the Table 5.
Compared with the initial customer allocation presented in Table 4, the cooperative network illustrated in Table 5 shows that the number of assigned customers varies from a logistics facility to another. While some centers have observed an increase, others have been allocated less units. For example, DC2’s customers units decreased from 15 to 9, whereas PC1’s customer units increased from 14 to 22. As a consequence, the total distance travelled by each facility’s vehicles will decrease and savings will increase. Besides, the sharing of customers units allows LC, DCs, and PCs to reduce the number of unreasonable delivery and pickup trips, as well as the complexity of their corresponding transportation network.
For comparison purposes, we implement and test the proposed GA-PSO hybrid algorithm, CPTS (Cooperative Parallel Tabu Search), HGA (Hybrid Genetic Algorithm), and TLBO (Teaching–Learning-Based Optimization) with same data. CPTS utilizes the information interchange to accomplish the cooperation and boost both intensification and strategic diversification (James et al. 2009). HGA incorporates the elitism genetic operator and some improvement heuristic methods . TLBO considers the solutions proposed by individuals’ search, two-phase optimization process including teacher and learner phases to obtain the global best fitness values (Dokeroglu, 2015). Each algorithm is executed 20 times, and the best solution is selected as the optimal cost for convergence. The optimal total cost and the number of iterations can be used to measure the effectiveness of four algorithms shown in Table 6.
As shown in Table 6, the -test and value indicate that the total cost and the number of iterations are significantly different with the three algorithms. In addition, the number of times GA-PSO hybrid algorithm finds out the optimal solution is higher. For example, the best solution appears 12 out of 20 times with the GA-PSO, while it only appears four or five times with CPTS, HGA, and TLBO.
Compared with other three algorithms, the average optimization cost and number of iterations from our proposed algorithm are of better quality. Thus, we can conclude that GA-PSO has the following merits: exchanging best-fit chromosomes and worst-fit particles between the GA and the PSO algorithm enhances the GA-PSO’s capability to obtain better solutions. For example, the GA-PSO algorithm has a higher probability to obtain the best solution. The proposed algorithm combines GA and PSO algorithm’s global and local search capability. As illustration, the average cost from the proposed hybrid algorithm is lower than that of CPTS, HGA, and TLBO algorithms by 3.8%, 6.4%, and 1.8%, respectively. The termination and periodic iteration rules are reasonably designed in the GA-PSO hybrid algorithm procedure, which improves the efficiency of the hybrid algorithm. Therefore, the average number of iterations from GA-PSO is significantly less than that of the CPTS, HGA, and TLBO algorithms by 53, 35, and 105 iterations, respectively. These findings indicate that our hybrid algorithm has better capability to obtain the optimal solution than that of the other three algorithms.
6.3. MCRS Model Application and Coalition Sequence Selection
Previous studies have already shown the efficiency of MCRS at profit distribution in multiple players’ games . Indeed, this method can be used to incorporate the coordinator’s payoff into the design of the collective profit allocation plan and guarantee the stability of coalitions. Based on the fact that our case study’s coordinator (LC) is also part of the network and receives the highest incentive from the government, we assume there is no extra payoff and set the synergy requirement . In this study, we first consider the formation of a grand coalition before splitting the group into two subcoalitions and evaluate the influences of each of these scenarios. The combined form of all nonempty alliances is shown in Table 7.
The sum of the revenue of and constitute the total profit of LC. Profit allocation results display differences in facilities’ benefits as they appear in different coalitions. For example, PC1’s initial profit while being alone is 1158, but, after sharing customers and transportation resources with , PC1 could save 1432. However, in certain circumstances, the profit can decrease as another participant enters the coalition. DC2 can be considered as an example to the extent that its initial profit (769) decreases after joining and DC1 and becomes 724. In fact, rational decision making requires every participant to give priority to coalitions where the increase of benefits is guaranteed. Therefore, we can conclude that the stability of any coalition depends on the profit of the members before and after new members adhere. In the process to form the grand coalition, cost reduction percentages have been calculated, presented in Figure 5, and illustrated in Figure 6.
The logistics center is denoted by and in Figure 5, while the legend in Figure 6 refers to it as LCP and LCD. We can notice at the fourth step in Figure 6 the simultaneous entrance of LCD and DC1 to emphasize that the delivery capacities of LC should not be applicable unless there is a DC in the coalition.
The examination of coalition sequences is critical to the profit distribution strategy and to participants’ willingness to become member. In other words, the order in which members are added to the coalition affects the distribution of profits and the satisfaction of SMP principles. Nevertheless, following the examination of every possible combination, the SMP based coalition sequence is . Table 8 shows the cost reduction matrix representing members’ behaviors as a new facility enters .
In the design of the grand coalition, we have considered joining first the coalition followed by PC1 and PC2 in order to assure that pickup activities are completed before deliveries and to conform to the proposed THCLN. The percentages of total operating costs reduced by LC, PC1, and PC2, at the level of unique pickup operations in the coalition, are, respectively, 26.3%, 29.2%, and 32.0%. After the adhesion of distribution centers in the order , the final sequence for the grand coalition yields as cost reduction percentage sequence , respectively. In the formation process, the profits of PCs have increased with the addition of DCs. In brief, we can conclude that LC as coordinator and the upper-level facility of PCs and DCs can better optimize the entire logistics network. In this article, we required LC to be first member to enter the coalition. However in other circumstances, LC may not consider the value of its profit and allow another facility to make the first move for the sake of a more stable coalition.
6.4. Alliance Stability
In this section, we examine the accuracy of the MCRS method at providing the best profit distribution schemes in the investigated case. In this regard, three different methods are used to calculate appropriate profits of each facility [29, 30]: the Shapley value model, the Cost Gap Allocation (CGA) model, and the Game Quadratic Programming (GQP) method. Moreover, the Euclidean distance between each distribution scheme and core center of the grand coalition will be evaluated to verify each method’s performance. In accordance with the snowball theory , the best strategy should be the closest to the center. The vector at the core is determined using formula (25) where the collective profit of the grand coalition is , represents a member and is an analytic expression to control the scope of the core. In addition, Table 9 presents the results of the four methods, and Table 10 gives the core center and corresponding distances.
Despite the allocation of lowest profits to PC2, LCD, and DC2, as shown in Table 9, MCRS yields the lowest distance (354) to the core center in Table 10. Therefore, we can notice that MCRS is the closest to the center and confirms our profit distribution scheme the most appropriate. Besides, findings allow us to conclude that, in a cooperative coalition, individual profit should not be the most important criteria. Decision makers and coordinators have to observe the collective impacts of cooperation on the network.
6.5. Analysis of Two Heterogeneous Subcoalitions’ Network
Segmentation is a strategic management approach used to control complex networks and consists in dividing the components into small parts. Some researchers have studied cooperation in logistics network optimization but concentrated on the design of multiple coalitions. Thus, in addition to the grand coalition, this paper has also considered other cases where the network would consist in two heterogeneous coalitions. In other words, PCs can independently cooperate with LC and a DC can form a coalition with LC and other DCs. It is important to notify that possibilities of constituting groups with different types of centers are numerous, but this article only evaluates circumstances where the same type of facility joins with LC in a single coalition. Our purpose is to examine the influence of multiple and single heterogeneous coalition in logistics network. Therefore, based on our solution methodology, the cost reduction percentages of possible sequences under two subcoalitions scenarios are shown in Tables 11 and 12.