Transshipment Problem Research Articles

Solving transshipment problems with mixed constraints

In this article we have considered a generalization of the standard transshipment model in which the origin and destination constraints consist not only of equality but also of greater than or equal to or less than or equal to type constraints The proposed algorithm solves the original transshipment problem by transforming the problem into an equivalent transportation problem by introducing an additional row and a column. The optimal solution of the original transshipment problem is obtained from the optimal solution of the transformed transportation problem. We have discussed the case of both balanced as well unbalanced transshipment problems. The proposed algorithm is easy to understand and apply. It can serve the managers by providing the solution to a variety of distribution problems with mixed constraints and in paradoxical situations. A detailed numerical example to illustrate the proposed algorithm is also presented.

Stochastic optimization for transshipment problems with positive replenishment lead times

Abstract Transshipments, monitored movements of material at the same echelon of a supply chain, represent an effective pooling mechanism. Earlier papers dealing with transshipments either do not incorporate replenishment lead times into their analysis, or only provide a heuristic algorithm where optimality cannot be guaranteed beyond settings with two locations. This paper uses infinitesimal perturbation analysis by combining with a stochastic approximation method to examine the multi-location transshipment problem with positive replenishment lead times. It demonstrates the computation of optimal base stock quantities through sample path optimization. From a methodological perspective, this paper deploys a duality-based gradient computation method to improve computational efficiency. From an application perspective, it solves transshipment problems with non-negligible replenishment lead times. A numerical study illustrates the performance of the proposed approach.

Heuristic solutions for transshipment problems in a multiple door cross docking warehouse

Cross docking is a practice in logistics with the main operations of unloading products from an incoming truck, regrouping them with respect to their destinations and loading them directly into an outbound truck with minimum storage in between these operations. In this article, we study the transshipment scheduling problem in a multiple inbound and outbound dock configuration. The operations manager has several decisions to make: he can decide to transship products directly from inbound to an outbound truck, if an outbound truck is available; he can temporarily store certain products and have them loaded later on; or he can replace an outbound truck to facilitate direct loading. The objective is to find the best schedule of transshipment operations to minimize the sum of inventory holding and truck replacement costs. In this article, we present several heuristics to attain this objective. Numerical experiments are presented and the results are compared with the optimal solution to evaluate the performance of the heuristics.

On the benefits of operational flexibility in a distribution network with transshipment

We consider a centralized distribution network with multiple retailers who receive replenishment inventory to satisfy customer demand of the local markets. The operational flexibility of the network is defined as the opportunity that one retailer's excess inventory can be transferred to satisfy other retailers’ unmet customer demand due to stock-outs. A general modeling framework is developed to optimize retailers’ order quantities under any possible flexibility level of a stylized two-stage distribution network. We apply the framework to formulate and solve the transshipment problem of a distribution network with three retailers. Six typical flexibility levels are investigated to make the comparison study on the firm's profit performance under three ordering quantity policies: average demand, newsvendor order quantity, and optimal order quantity. We find that the operational flexibility and system optimization are complements to the firm's performance. The ordering policy with newsvendor ordering quantity can perform fairly well with moderate flexibility level when compared with the optimized ordering policy with full flexibility.

A heuristic for balancing the inventory level of different locations through lateral shipments

Solving transshipment problems to optimality is difficult, unless several simplifying hypotheses are assumed (such as unit-sized customer demands and replenishments, negligible replenishment lead time, etc.). For this reason, some heuristics have been recently proposed in order to provide rules, which incorporate relevant factors of the problem, to find conditions under which it makes sense to transship a certain number of units from one retailer to another. Most of these studies concern emergency transshipment, which means that shipments between locations can occur only when a shortage happens, and shipments are assumed to be fast enough to satisfy the location in shortage. When this assumption is not feasible, as in many real cases, transshipments between locations have to be performed before a shortage happens. The paper addresses this case, which can be named ‘preventive’ transshipment, where the inventory level of different locations at the same echelon is balanced through lateral shipments, before a shortage happens. A heuristic for deciding on transshipment policy (when to transship and how much), trying to minimise overall expected costs, is presented. A simulation study considering different scenarios is performed and results confirm the effectiveness of the heuristic.

A particle swarm optimization-based hybrid algorithm for minimum concave cost network flow problems

Traditionally, minimum cost transshipment problems have been simplified as linear cost problems, which are not practical in real applications. Some advanced local search algorithms have been developed to solve concave cost bipartite network problems. These have been found to be more effective than the traditional linear approximation methods and local search methods. Recently, a genetic algorithm and an ant colony system algorithm were employed to develop two global search algorithms for solving concave cost transshipment problems. These two global search algorithms were found to be more effective than the advanced local search algorithms for solving concave cost transshipment problems. Although the particle swarm optimization algorithm has been used to obtain good results in many applications, to the best of our knowledge, it has not yet been applied in minimum concave cost network flow problems. Thus, in this study, we employ an arc-based particle swarm optimization algorithm, coupled with some genetic algorithm and threshold accepting method techniques, as well as concave cost network heuristics, to develop a hybrid global search algorithm for efficiently solving minimum cost network flow problems with concave arc costs. The proposed algorithm is evaluated by solving several randomly generated network flow problems. The results indicate that the proposed algorithm is more effective than several other recently designed methods, such as local search algorithms, genetic algorithms and ant colony system algorithms, for solving minimum cost network flow problems with concave arc costs.

A heuristic approach for designing a distribution network in a supply chain system

One of the most important problems in supply chain management is the distribution network design problem system which involves locating production plants and distribution warehouses, and determining the best strategy for distributing the product from the plants to the warehouses and from the warehouses to the customers. Unlike most of past research, our study allows for multiple levels of capacities available to the warehouses and plants. We developed a mixed integer programming model for the problem and solved it by a heuristic procedure which contains 2 sub-procedures. We used harmony-search meta-heuristic as the main procedure and linear programming to solve a transshipment problem as a subroutine at any iteration of the main procedure. Key words: Distribution network design, harmony search, mixed integer programming, supply chain management, transshipment problem.

A two-step transshipment model with fuzzy demands and service level constraints

We consider a distribution network composed of one supplier and several non-identical locations characterized by fuzzy customer demands and service level constraints. These locations could cooperate together via product transfer known as transshipment. The transshipment problem consists in determining the replenishment quantities that minimize the total inventory cost where a specific transfer policy is practiced. Our objectives in this paper are to identify an inter-location transfer policy that participates to satisfy the service level constraints and to determine the approximate replenishment quantities. To achieve these objectives we propose: (1) a new transshipment model based on the chance constrained programming, (2) a two-step transshipment policy that differs from classic ones by product transfer from locations in need to others also in need and (3) a hybrid algorithm based on fuzzy simulation and genetic algorithms to approximate replenishment quantities. (Received in March 2009, accepted in September 2009. This paper was with the authors 2 months for 1 revision.)

The transshipment fund mechanism: Coordinating the decentralized multilocation transshipment problem

AbstractThe multilocation replenishment and transshipment problem is concerned with several retailers facing random demand for the same item at distinct markets, that may use transshipments to eliminate excess inventory/shortages after demand realization. When the system is decentralized so that each retailer operates to maximize their own profit, there are incentive problems that prevent coordination. These problems arise even with two retailers who may pay each other for transshipped units. We propose a new mechanism based on a transshipment fund, which is the first to coordinate the system, in a fully noncooperative setting, for all instances of two retailers as well as all instances of any number of retailers. Moreover, our mechanism strongly coordinates the system, i.e., achieves coordination as the unique equilibrium. The computation and information requirements of this mechanism are realistic and relatively modest. We also present necessary and sufficient conditions for coordination and prove they are always satisfied with our mechanism. Numerical examples illustrate some of the properties underlying this mechanism for two retailers. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010

AN INAPPROXIMABILITY OF TRANSSHIPMENT PROBLEMS WITH PERMUTABLE TRANSIT VECTORS

We investigate the hardness of approximating a transshipment problem with a permutable transit vector. We have already known that the transshipment problem of minimizing the total shipping cost is NP-hard. In this paper, we prove that even if the unit shipping cost associated with every arc is given as a positive integer, there is no polynomial time constant factor approximation algorithm for the transshipment problem under the hypothesis that PWe obtain the inapproximability result by a gap-introducing reduction from an NP-complete problem by the name of 3DM. On the other hand, we propose a preparatory heuristic algorithm to the transshipment problem, which is based on a relaxation into the minimum cost flow problem and the minimum weight perfect matching on a bipartite graph. We also provide an upper bound of the total shipping cost obtained by the proposed heuristic.

Complexity of the Transshipment Problem with a Permutable Transit Vector

In this paper, we show that the transshipment problem with a permutable transit vector remains NP-hard even when each entry of the given transit vector takes either zero or two. We prove the hardness by a reduction from an NP-complete problem by the name of 3DM. We also show that the transshipment problem can be solved in polynomial time if each entry of the transit vector takes either zero or one. We obtain the time complexity by applying the successive shortest path algorithm for the minimum cost flow problem.

Transshipment of Inventories: Dual Allocations vs. Transshipment Prices

We study a newsvendor game with transshipments, in which n retailers face a stochastic demand for an identical product. Before the demand is realized, each retailer independently orders her initial inventory. After the demand is realized, the retailers select an optimal transshipment pattern and ship residual inventories in order to meet residual demands. Unsold inventories are salvaged at the end of the period. We compare two methods for distribution of residual profit -- transshipment prices (TP) and dual allocations (DA) -- that were previously analyzed in literature. Transshipment prices are selected a priori, before the demand is known, while dual allocations, which are obtained by calculating the dual prices for the transshipment problem, are calculated ex post, after observing the true demand. We first study the conditions for the existence of the Nash equilibria under DA, and then compare the performance of the two methods and show that neither allocation method dominates the other. Our analysis suggests that DA may yield higher efficiency among ``more asymmetric'' retailers, while TP works better with retailers who are alike'', but the difference in profits does not seem significant. We also link expected dual prices to TP, and use those results to develop heuristics for TP with more than two symmetric retailers. For general instances with more than two asymmetric retailers, we propose a TP agreement which uses a neutral central depot to coordinate the transshipments (TPND). While DA in general outperform TPND in our numerical simulations, its ease of implementation makes TPND an attractive alternative to DA when the efficiency losses are not significant (e.g., high critical fractiles or lower demand variances).

An efficient algorithm for the evacuation problem in a certain class of networks with uniform path-lengths

In this paper, we consider the evacuation problem in a network which consists of a directed graph with capacities and transit times on its arcs. This problem can be solved by the algorithm of Hoppe and Tardos [B. Hoppe, É. Tardos, The quickest transshipment problem, Math. Oper. Res. 25(1) (2000) 36–62] in polynomial time. However their running time is high-order polynomial, and hence is not practical in general. Thus, it is necessary to devise a faster algorithm for a tractable and practically useful subclass of this problem. In this paper, we consider a network with a sink s such that (i) for each vertex v ≠ s the sum of the transit times of arcs on any path from v to s takes the same value, and (ii) for each vertex v ≠ s the minimum v - s cut is determined by the arcs incident to s whose tails are reachable from v . This class of networks is a generalization of grid networks studied in the paper [N. Kamiyama, N. Katoh, A. Takizawa, An efficient algorithm for evacuation problem in dynamic network flows with uniform arc capacity, IEICE Trans. Infrom. Syst. E89-D (8) (2006) 2372–2379]. We propose an efficient algorithm for this network problem.

B34 Hardness of Approximating Transshipment Problems with Permutable Transit Vectors(Advanced machining technology)

B34 Hardness of Approximating Transshipment Problems with Permutable Transit Vectors(Advanced machining technology)

Complexity analysis for maximum flow problems with arc reversals

We provide a comprehensive study on network flow problems with arc reversal capabilities. The problem is to identify the arcs to be reversed in order to achieve a maximum flow from source(s) to sink(s). The problem finds its applications in emergency transportation management, where the lanes of a road network could be reversed to enable flow in the opposite direction. We study several network flow problems with the arc reversal capability and discuss their complexity. More specifically, we discuss the polynomial time algorithms for the maximum dynamic flow problem with arc reversal capability having a single source and a single sink, and for the maximum (static) flow problem. The presented algorithms are based on graph transformations and reductions to polynomially solvable flow problems. In addition, we show that the quickest transshipment problem with arc reversal capability and the problem of minimizing the total cost resulting from arc switching costs are \(\mathcal{NP}\) -hard.

Evolutionary multiobjective optimization of the multi-location transshipment problem

We consider a multi-location inventory system where inventory choices at each location are centrally coordinated. Lateral transshipments are allowed as recourse actions within the same echelon in the inventory system to reduce costs and improve service level. However, this transshipment process usually causes undesirable lead times. In this paper, we propose a multiobjective model of the multi-location transshipment problem which addresses optimizing three conflicting objectives: (1) minimizing the aggregate expected cost, (2) maximizing the expected fill rate, and (3) minimizing the expected transshipment lead times. We apply an evolutionary multiobjective optimization approach using the strength Pareto evolutionary algorithm (SPEA2), to approximate the optimal Pareto front. Simulation with a wide choice of model parameters shows the different trades-off between the conflicting objectives.

Multi-location transshipment problem with capacitated transportation

We consider coordination among stocking locations through replenishment strategies that take explicitly into consideration transshipments, transfer of a product among locations at the same echelon level. We incorporate transportation capacity such that transshipment quantities between stocking locations are bounded due to transportation media or the location’s transshipment policy. We model different cases of transshipment capacity as a capacitated network flow problem embedded in a stochastic optimization problem. Under the assumption of instantaneous transshipments, we develop a solution procedure based on infinitesimal perturbation analysis to solve the stochastic optimization problem, where the objective is to find the policy that minimizes the expected total cost of inventory, shortage, and transshipments. Such a numerical approach provides the flexibility to solve complex problems. Investigating two problem settings, we show the impact of transshipment capacity between stocking locations on system behavior. We observe that transportation capacity constraints not only increase total cost, they also modify the inventory distribution throughout the network.

Transshipment of Inventories: Dual Allocations vs. Transshipment Prices

We study a newsvendor game with transshipments, in which n retailers face a stochastic demand for an identical product. Before the demand is realized, each retailer independently orders her initial inventory. After the demand is realized, the retailers select an optimal transshipment pattern and ship residual inventories to meet residual demands. Unsold inventories are salvaged at the end of the period. We compare two methods for distribution of residual profit—transshipment prices (TPs) and dual allocations (DAs)—that were previously analyzed in literature. TPs are selected before the demand is known, and DAs, which are obtained by calculating the dual prices for the transshipment problem, are calculated after observing the true demand. We first study the conditions for the existence of the Nash equilibria under DA and then compare the performance of the two methods and show that neither allocation method dominates the other. Our analysis suggests that DAs may yield higher efficiency among “more asymmetric” retailers, whereas TPs work better with retailers that are “more alike,” but the difference in profits does not seem significant. We also link expected dual prices to TPs and use those results to develop heuristics for TPs with more than two symmetric retailers. For general instances with more than two asymmetric retailers, we propose a TP agreement that uses a neutral central depot to coordinate the transshipments (TPND). Although DAs in general outperform TPND in our numerical simulations, its ease of implementation makes TPND an attractive alternative to DAs when the efficiency losses are not significant (e.g., high critical fractiles or lower demand variances).

Flows in dynamic networks with aggregate arc capacities

Dynamic networks are characterized by transit times on edges. Dynamic flow problems consider transshipment problems in dynamic networks. We introduce a new version of dynamic flow problems, called bridge problem. The bridge problem has practical importance and raises interesting theoretical issues. We show that the bridge problem is NP-complete. Traditional static flow techniques for solving dynamic flow problems do not extend to the new problem. We give a linear programming formulation for the bridge problem which is based on the time-expanded network of the original dynamic network.

A game theory approach for solving the transshipment problem: a supply chain management strategy teaching tool

PurposeTo describe how the well‐known Shapley Value concept from cooperative game theory can be used as an approach to solve the transshipment problem.Design/methodology/approachThis paper provides a numerical example for a three‐person game in a non‐cooperative environment, a selected cooperative environment, and a fully cooperative environment. The well‐known Shapely Value concept from cooperative game theory is used as an approach to solve the transshipment problem and show “how” the pooling coalitions should be formed. The appurtenance of the Shapley Value is verified to ensure that a stable solution exists.FindingsThe primary focus of this paper is to describe how the well‐known Shapley Value concept from cooperative game theory can be used as an approach to solve the transshipment problem and show “how” the pooling coalitions should be formed.Research limitations/implicationsWhile pooling coalitions are presented, this paper does not discuss revenue sharing.Practical implicationsThis paper helps students understand “how to” make the right decisions about where they want to position themselves with respect to supply chain strategies.Originality/valueThe principal contribution of this approach is for establishing a pooling coalition in order to provide a more stable and cooperative solution. This paper also presents application to recent supply chain management business scenarios found in industry.