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To investigate the influence of manufacturers' loss aversion preference and retailers' CSR behaviour on green supply chain decision making, a green CSR supply chain game model with three different Stackelberg structures for centralized decision making, manufacturers' rational preference and loss aversion preference are constructed using the Nash bargaining solution as the loss aversion reference point, and a combined revenue sharing-liability sharing contract is introduced to achieve green supply chain coordination under the two decentralized structures.The results show that the retailer's CSR responsibility is conducive to the green development of the supply chain, but the manufacturer's loss aversion not only damages the retailer's profits, but also damages its own profits;In addition, when the manufacturer's rational preference, with the improvement of the retailer's CSR level, the effective coordination interval length of the coordination factor under the combined contract increases, and the supply chain coordination flexibility increases; When the manufacturer's loss aversion preference increases, the upper and lower limits of the coordination factor of the combined contract will increase with the increase of the loss aversion preference, and the length of the coordination interval will also decrease with the increase of the loss aversion preference, which has a negative impact on the cooperation between the supply chain members.

PurposeIt is necessary to implement green supply chains. But green development needs to be gradual and coexist with ordinary products in the market. This paper aims to study the green and ordinary product pricing and green decision-making under chain-to-chain competition.Design/methodology/approachThis paper considers consumers' multiple preferences and takes two competitive supply chains with asymmetric channels as the research object. Through the construction of the game models involving different competitive situations, this paper studies the pricing, green decision-making and the supply chains' profits, and discusses the impact of consumer green preference, channel preference, green investment and competition on the decision-making and performance. Finally, this paper further studies the impact of the decision structure on the environmental and economic benefits of supply chains.FindingsThe results show that consumer green preference has an incentive effect on the green supply chain and also provides an opportunity for the regular supply chain to increase revenue. Specifically, consumers' preference for green online channels improves the product greenness, but its impact on the green retailer and regular supply chain depends on the green investment cost. Moreover, competition not only fosters product sustainability, but also improves supply chain performance. This paper also points out that the decentralization of the regular supply chain is conducive to the environmental attributes of the green product, while the environment-friendly structure of the green supply chain is different under different conditions. In addition, the profit of a supply chain under centralized decision is not always higher than that under decentralized decision.Originality/valueThe novelty of this paper is that it investigates the pricing of two heterogeneous alternative products and green decision-making for the green product under the competition between two supply chains with asymmetric channels, in which the green supply chain adopts dual channels and the regular supply chain adopts a single retail channel.

Enterprise-wide supply chain planning problems naturally exhibit a multi-level decision network structure, where the upper level of a hierarchy may have his objective function and decision space partly determined by other levels. In addition, each planner’s control instruments may allow him to influence the policies at other levels and thereby to improve his own objective function. As a tool, bi-level programming is applied for modeling decentralized decisions in which two decision makers make decisions successively. In this paper, we specifically address bi-level decision-making problems with budget constraint as an attractive feature in the context of enterprise-wide supply chain. We first describe the typical bi-level linear programming problem (BLLPP) and its optimal solution to the penalty function problem, and then, a cooperative decision-making problem in supply chain is modeled as BLLPP. A particle swarm optimization-based computational algorithm is designed to solve the problem, and the numerical example is presented to illustrate the proposed framework.

This paper focuses on the stochastic disturbance on quality control strategies with a two-echelon supply chain. A state equation is described to the change of product quality, and the stochastic differential game models with quality disturbance are designed according to different cooperation modes between the manufacturer and the retailer. The paper aims to investigate the relationship between the quality disturbance with quality control strategies of the supply chain members. After the construction and analysis of the evolutionary process of the product quality to stochastic differential model, the optimal equilibrium solutions and subsidy ratio is solved, by using the dynamic programming method. The result shows: The product quality is inversely proportional to the quality investment cost and the depreciation rate, and is proportional to the marginal revenue of the quality; The cooperative game model has the better performance in improving the input of the supply chain members to product quality.

In recent years, statistics show that the percentage of household consumption in GDP is getting higher and higher. One of the subsequent influences is that the transport demand for packages has increased sharply. Hence it is very necessary to propose diverse transport organization modes in accordance with the different situation in different countries. A kind of new transportation mode, multi-package multi-train pickup and delivery problem with time window (MPMT-PDPTW) is proposed in this paper. It aims to achieve express package transportation with minimum cost and maximize utilization of luggage car of a passenger train at the same time. Considering the multiple constraints, such as limited carrying capacity, fixed departing and accepting time, we define the quality of service (QoS) as the evaluation criteria for package express transportation by the passenger train. A kind of meta-heuristic algorithm, greedy randomized adaptive search procedure (GRASP), is designed to make full use of the redundant capacity of railway luggage car. An example is designed according to the railway passenger train timetable released by China Railway Corporation (CRC), the assignment schemes of packages at different railway stations reveal that the new transportation mode proposed in this paper is feasible and practical in package express transportation, and also has advantages in speed comparing with the highway and in transport cost over the airline.

This paper investigates the constraints handling technique (CHT) in algorithms of the constrained multi-criteria optimization problem (CMOP). The CHT is an important research topic in constrained multi-criteria optimization (MO). In this paper, two simple and practicable CHTs are proposed, where one is a nonequivalent relaxation approach which is much suitable for the constrained multi-criteria discrete optimization problem (MDOP), and the other is an equivalent relaxation approach for the general CMOP. By using these CHTs, a CMOP (i.e., the primal problem ) can be transformed into an unconstrained multi-criteria optimization problem (MOP) (i.e., the relaxation problem ). Based on the first CHT, it is theoretically proven that the efficient set of the primal CMOP is a subset of the strictly efficient set E ¯ of the relaxation problem and can be extracted from E ¯ by simply checking the dominance relation between the solutions in E ¯ . Follows from these theoretical results, a three-phase based idea is given to effectively utilize the existing algorithms for the unconstrained MDOP to solve the constrained MDOP. In the second CHT, the primal CMOP is equivalently transformed into an unconstrained MOP by a special relaxation approach. Based on such a CHT, it is proven that the primal problem and its relaxation problem have the same efficient set and, therefore, general CMOPs can be solved by utilizing any of the existing algorithms for the unconstrained MOPs. The implementing idea, say two-phase based idea , of the second CHT is illustrated by implanting a known MOEA. Finally, the two-phase based idea is applied to some of the early MOEAs and the application performances are comprehensively tested with some benchmarks of the CMOP.

This paper investigates the algorithms of the multi-modal shortest path problem (M-SPP) under static and certain environment and their extension in urban transit network (UTN). The related M-SPP is one of the important and practical problems in several fields such as urban transportation system and freight transportation. Existing algorithms of various M-SPP are usually based on building a new network in terms of the hypergraph or transfer graph and implemented by extending the label correcting algorithm and label setting algorithm (LSA) in such a new network. The UTN is composed of multiple modes (e.g., automobile, bus, subway, light rail, pedestrian and so on) and, to get their destination, the users can alternate between different modes. As a special demand, the time-window is usually correlated with the M-SPP. In contrast to the algorithms for the classical M-SPP, the algorithms for the M-SPP in UTN are much complicated, particularly, these for the M-SPP with the switching delay and arriving time-window. In this paper, we consider the M-SPP with switching delay and that with both of the switching delay and arriving time-window. Firstly, a new approach is adopted to simplify the mathematical formulas of thes M-SPP with switching delay and then an improved LSA for the M-SPP with switching delay in the UTN is proposed. Afterwards, a reverse LSA is given to solve the M-SPP with both of switching delay and arriving time-window in the UTN. Finally, some examples are given to illustrate the labeling process of the designed algorithms.

This paper investigates the solution algorithms for the multi-criteria multi-modal shortest path problem (M-SPP), which belongs to the set of problems known as NP-hard, in urban transit network (UTN). The related M-SPP is one of the important and practical problems in several fields such as urban transportation system and freight transportation. The UTN is composed of multiple modes (e.g., automobile, bus, subway, light rail, pedestrian and so on). To get their destination, the passengers can alternate between different modes. As a special demand, the time-window is usually associated with the M-SPP. Because of the service time-limit of modes, the available modes at a stop are varied with the time. So the optimal M-SPP with arriving time-window cannot be simply obtained by finding the optimal M-SPP firstly and then reversely deducing the leaving time-window of the origin according to the arriving time-window of destination. In this paper, the M-SPP with arriving time-window is firstly proposed. To solve the multi-criteria M-SPPs (MM-SPP) with transfer delaying, an improved exact label correcting algorithm (LCA) is designed and, to solve the proposed MM-SPPs with both of transfer delaying and arriving time-window, an exact reverse LCA is designed. Finally, some computing examples are given to test the effectiveness of the designed algorithms.

This paper investigates a simulated annealing (SA) for multi-criteria optimization problem (MOP) which incorporates the concept of archive in order to provide a set of tradeoff solutions for the problem under consideration. To determine the acceptance probability of a new solution, an evaluation measure of multi-criteria objective function is proposed which takes into account the dominant relation between the new and current solutions, as well as those in the archive. In addition, a mutation operator is proposed in which the constraints of MOP can be partially handled. To efficiently apply the proposed SA into constrained MOP (CMOP), an equivalent relaxation approach is proposed by which the CMOP, say primal problem , can be equivalently transformed into an unconstrained MOP, say relaxation problem . Based on such a constraints handling technique (CHT), it is theoretically proved that the primal problem and the relaxation problem have the same efficient set (ES) and the proposed CHT is then further extended into two existing multi-criteria optimization (MO) evolutionary algorithms (EA) (MOEA), say AMOSA and NSGA-II, respectively. Two performance metrics are proposed to evaluate the performance of the MOEAs. Finally, a comprehensive comparative study among AMOSA, NSGA-II and the proposed SA is performed on some popular benchmarks for MOP algorithms to show the effectiveness of the proposed SA and CHT. The comparative study shows that the overall performance of the proposed SA is generally superior to both of the AMOSA and NSGA-II, particularly, for the MOPs with three objectives the performance of the former is far better than the latter, and the proposed CHT can be very conveniently extended into any of the existing EAs for MOP such that they can be applied to solve the CMOPs.

This paper investigates an oriented spanning tree (OST) based simulated annealing (SA) for solving the multi-criteria shortest path problem (MSPP) as well as the multi-criteria constrained shortest path problem (MCSPP), especially for those with nonlinear objectives. As a popular search algorithm, because of "search-from-a-point" searching mechanism, there have been only a few attempts in extending SA to multi-criteria optimization, particularly, for various MSPPs. In contrast with the existing evolutionary algorithms (EAs), by representing a path as an OST, the designed SA provides an entirely new searching mechanism in sense of "search from a paths set to another paths set" such that both of its local and global search capabilities are greatly improved. Because the possibility of existing a feasible path in a paths set is usually larger than that of one path being feasible, the designed SA has much predominance for solving MCSPPs. Some computational comparisons are discussed and the test results are compared with those obtained by a recent EA of which the representing approach and the ideas of evolution operators such as mutation and crossover are adopted in most of the existing EAs for the shortest path problems. The test results indicate that the new algorithm is available for both of MSPPs and MCSPPs.

The well-known generalized assignment problem (GAP) is to minimize the costs of assigning n jobs to m capacity constrained agents (or machines) such that each job is assigned to exactly one agent. This problem is known to be NP-hard and it is hard from a computational point of view as well. In this paper, follows from practical point of view in real systems, the GAP is extended to the equilibrium generalized assignment problem (EGAP) and the equilibrium constrained generalized assignment problem (ECGAP). A heuristic equilibrium strategy based genetic algorithm (GA) is designed for solving the proposed EGAP. Finally, to verify the computational efficiency of the designed GA, some numerical experiments are performed on some known benchmarks. The test results show that the designed GA is very valid for solving EGAP.

This paper investigates an oriented spanning tree (OST) based genetic algorithm (GA) for the multi-criteria shortest path problem (MSPP) as well as the multi-criteria constrained shortest path problem (MCSPP). By encoding a path as an OST, in contrast with the existing evolutionary algorithms (EA) for shortest path problem (SPP), the designed GA provides a ''search from a paths set to another paths set'' mutation mechanism such that both of its local search and global search capabilities are greatly improved. Because the possibility to find a feasible path in a paths set is usually larger than that of only one path is feasible, the designed GA has more predominance for solving MCSPPs. Some computational tests are presented and the test results are compared with those obtained by a recent EA of which the encoding approach and the ideas of evolution operators such as mutation and crossover are adopted in most of the existing EAs for shortest path problems. The test results indicate that the new algorithm is available for both of MSPP and MCSPP.

Intersection is the fundamental unit of urban traffic network. Thus the delay caused by traffic light at intersection is non-ignorable when regarding the evacuation routes. Because an intersection with traffic light consists of a repeated sequence of time windows with designated duration, different starting time of evacuation corresponds to different intersection delay and evacuation routes. Considering the intersection delay and capacity, superior evacuation routes and the flow assigned to each route are obtained by adopting the compound labeling algorithm to find the minimum cost flow of the traffic network. In the numerical example, evacuation routes with different starting time are compared, which indicates that the proposed method can get a good result with consideration of evacuation time and route optimization.

In this paper, the equilibrium optimization problem is proposed and the assignment problem is extended to the equilibrium multi-job assignment problem, equilibrium multi-job quadratic assignment problem and the minimum cost and equilibrium multi-job assignment problem. Furthermore, the mathematical models of the equilibrium multi-job assignment problem and the equilibrium multi-job quadratic assignment problem with fuzzy parameters are formulated. Finally, a genetic algorithm is designed for solving the proposed programming models and some numerical examples are given to verify the efficiency of the designed algorithm.

This paper mainly investigates the fixed charge solid transportation problem under fuzzy environment, in which the direct costs, the fixed charges, the supplies, the demands and the conveyance capacities are supposed to be fuzzy variables. As a result, several new models, i.e., expected value model, chance-constrained programming model and dependent-chance programming model, are constructed on the basis of credibility theory. After that, the crisp equivalences are also discussed for different models. In order to solve the models, hybrid intelligent algorithm is designed based on the fuzzy simulation technique and tabu search algorithm. Finally, two application results are given to show the applications of the models and algorithm.

A matching of a graph is an independent subset of the edges set and a maximum matching is a matching with as many edges in it as possible. The maximum weighted matching problem is to find a maximum matching in a given graph such that the sum of the weights of the edges in it is maximum. In this paper, the concepts of expected maximum fuzzy weighted matching, the α-maximum fuzzy weighted matching and the most maximum fuzzy weighted matching are initialized. According to various decision criteria, the maximum fuzzy weighted matching problem is formulated as expected value model, chance-constrained programming and dependent-chance programming by using the credibility theory, and the crisp equivalents are also given. Furthermore, a hybrid genetic algorithm is designed for solving the proposed fuzzy programming models. Finally, a numerical example is given.

The assignment problem is to find the total costs optimal jobs assignment schedule where n jobs are allocated to n workers, and each worker receives exactly just one job, such that the total cost is optimal. The quadratic assignment problem with penalty takes three types of costs into consideration: direct cost, interactive cost an penalty. In this paper, the fuzzy quadratic assignment problem with penalty is formulated as expected value model, chance-constrained programming and dependent-chance programming according to various decision criteria, and the crisp equivalents are given. Furthermore, hybrid genetic algorithm is designed for solving the proposed fuzzy programming models.

In this paper, some properties of Ramsey numbers are studied, and the following results are presented. 1. (1) For any positive integers k1, k2, …, km l1, l2, …, lm (m > 1), we have r ∏i=1m ki + 1, ∏i=1m li + 1 ≥ ∏i=1m [ r (ki + 1,li + 1) − 1] + 1 . 2. (2) For any positive integers k1, k2, …, km, l1, l2, …, ln , we have r ∑i=1m ki + 1, ∑j=1n lj + 1 ≥ ∑i=1m∑j=1n r (ki + 1,lj + 1) − mn + 1 . Based on the known results of Ramsey numbers, some results of upper bounds and lower bounds of Ramsey numbers can be directly derived by those properties.

Let G = (V, E) be a graph. For a function f : V → {−1, 1}, the weight of f is w(f) = Σv ∈ V f(v). For a vertex v in V, we define f[v] = Σu∈N[v] f(u). A signed dominating function of G is a function f : V → {−1, 1} such that f[v] ⩾ 1 for all v ∈ V. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. A signed dominating function of a weight γs(G) we call a γs-function of G. In this paper, we study the signed domination problem of general graph, and obtain some lower bounds of the signed domination number of a graph, and show that these lower bounds are sharp, and extend a result in Dunbar et al. (1995).

In this paper, we prove that χT ( G ) = 5 for any Halin graph G with Δ ( G ) = 4, where Δ ( G ) and χT ( G ) denote the maximal degree and the total chromatic number of G , respectively.

LetG be a planar graph with δ(G)≥3,f 0 be a face ofG. In this paper it is proved that for any Halin graph with Δ(G)≥6,x c (G)=Δ(G)+1, where Δ(G),x c (G) denote the maximum degree and the complete chromatic number ofG, respectively.

With the increase of traffic demand, urban traffic problems become more and more serious. Owing to the complexity of traffic system, the fuzzy control theory demonstrates powerful advantage. In this paper, taking the signal control problem of the intersections group as objective, we provide a distributed control system, which comprises some local fuzzy controllers and a special case controller. Each local fuzzy controller controls the traffic flows at its designated intersection according to the traffic flows of related intersections in addition to its own. When the current traffic situation is beyond the ability of local traffic controller, the special case controller will be activated. By adopting the designed simulated annealing algorithm, the special case controller optimizes the green time extension of all the intersections in the road network. In this algorithm, success index is given to evaluate the efficiency of each feasible solution taking the system optimization into account. Finally, to evaluate the performance of this system, it is compared with a general fuzzy control system, and its availability is validated by system simulation.

The fixed charge transportation problem is an extension of the traditional transportation problem in which two kinds of costs, say direct cost and fixed charge, are taken into consideration. The direct cost is the cost associated with each source-destination pair, and the fixed charge occurs when the transportation activity takes place in the corresponding source-destination pair. In this paper, the fuzzy fixed charge transportation problem is modeled as chance-constrained programming by using the possibility measure and credibility measure firstly. Then a heuristic genetic algorithm is designed.

In the urban traffic system, the traffic congestion is a common phenomena. One main reason of traffic congestion is that the traffic flow and the traffic path is not equilibrium. In this paper, the concept of equilibrium network models include the equilibrium flow and the equilibrium path are firstly initialized and then the mathematical models are formulated and genetic algorithm is designed for solving the proposed equilibrium path model.

The automatic design algorithm of the bottleneck is the critical problem to realize computer-aided design of railway yard. Each station bottleneck can be decomposed two components: a core switches group and several ladder tracks that are composed by six basic equipments. The structure of the sketch of a bottleneck can be input and then recognized by means of optimized mathematic model. And all the arrangement style of the ladder track can be enumerated by means of different morphology of a binary tree. The combination method of the sketch input and structural enumeration developed in the paper makes it available to design railway yard in an automatic and optimized method based on constrains reasoning optimum mathematical models.

The solid transportation problem is a generalization of the traditional transportation problem in which three kinds of constraint are taken into account instead of two. In general, the three kinds of constraint are understood as source, destination, and transport mode. The fixed charge transportation problem is an extension of the traditional transportation problem in which two kinds of costs, say direct cost and fixed charge, are taken into consideration. In this paper, the fixed charge solid transportation problem with fuzzy data is modeled as a chance-constrained programming by using the credibility measure. Finally, a genetic algorithm and an illustrated example is given.

The k-tree problem is to find a tree with k vertices in a given graph such that the total cost is minimum and is known to be NP-hard. In this paper, the k-tree problem with fuzzy weights is firstly formulated as the chance-constrained programming by using the possibility measure and the credibility measure. Then an oriented tree and knowledge-based hybrid genetic algorithm is designed for solving the proposed fuzzy programming models.

This paper investigates a minimum spanning tree (MST) problem with fuzzy costs and quadratic cost structure, which we call the fuzzy quadratic minimum spanning tree problem (FQMST). After formulating the FQMST problem as a chance-constrained programming model based on a credibility measure, the deterministic equivalent is proposed when the fuzzy direct costs and fuzzy interactive costs are characterized by trapezoidal fuzzy numbers. Then, a genetic algorithm is designed for solving FQMST problems. Finally, a numerical example is provided for illustrating the effectiveness of the genetic algorithm.

The derivative problems from the classical shortest path problem (SPP) are becoming more and more important in real life[1]. The dissimilar shortest paths problem is a typical derivative problem. In Vehicles Navigation System(VNS),it is necessary to provide drivers alternative paths to select. Usually, the path selected is a dissimilar path to the jammed path. In fact, ”dissimilar” is fuzzy. Considering traffic and transportation networks in this paper, we put forward to the definition of dissimilar paths measure that takes into account the decision maker's preference on both the road sections and the intersections. The minimum model is formulated in which not only the length of paths but also the paths dissimilar measure is considered. And a genetic algorithm also is designed. Finally, we calculate and analyze the dissimilar paths in the traffic network of the middle and east districts of Lanzhou city in P.R. of China by the method proposed in this paper.