Cost Allocation Problem Research Articles

On the submodularity of multi-depot traveling salesman games

The Steiner traveling salesman problem is the problem of finding a minimum cost tour for a salesman that must visit a set of locations while traveling along costly streets before returning to his starting point at the depot. A solution to the problem is a minimum cost Steiner tour that both starts and ends at the depot and visits all the required locations. If different players are associated with the locations to be visited, the problem induces a cooperative traveling salesman (TS) game that poses the question of how to allocate the total cost of the tour between the different players involved. This cost allocation problem can be tackled using tools and solutions from cooperative game theory.The purpose of this paper is to generalize the notion of a traveling salesman game to allow for multiple depots in the underlying traveling salesman problem (TSP) and to analyze the submodularity of such multi-depot TS games. A multi-depot TSP can be represented by a connected (di)graph in which a fixed set of vertices are denoted depots, and a non-negative weight function is defined on the edges of the graph. The submodularity of multi-depot TS games is analyzed by characterizing graphs and digraphs that induce submodular multi-depot TS games for any choice of depots and for at least one choice of depots, respectively.

On Analyzing Cost Allocation Problems: Cooperation Building Structures and Order Problem Representations

To analyze cost allocation problems, this paper identifies associated cooperation building structures, with joint cost functions, and corresponding efficient order problem representations, with individualized cost functions. This paper presents an approach that, when applicable, offers a way not only to adequately model a cost allocation problem by means of a cooperative cost game, but also to construct a core element of such a game by means of a generalized Bird allocation. We apply the approach to both existing and new classes of cost allocation problems related to operational research problems: sequencing situations without initial ordering, maintenance problems, minimum cost spanning tree situations, permutation situations without initial allocation, public congestion network situations, traveling salesman problems, shared taxi problems and traveling repairman problems.

Bi-Objective-Based Cost Allocation for Cooperative Demand-Side Resource Aggregators

This paper presents a cooperative game theoretic approach to tackle the cost allocation problem for a virtual power plant which consists of multiple demand-side resource aggregators (DRAs) participating in the short-term two settlement electricity market. Given the considered game is balanced, we propose to employ the cooperative game theory's core cost allocation concept to efficiently allocate the bidding cost to the DRAs. Since the nonempty core contains many potential solutions, we develop a bi-objective optimization framework to determine the core cost allocation solution that can achieve efficient tradeoff between stability and fairness. To solve this problem, we jointly employ the ε-constraint and row constraint generation methods to construct the Pareto front, based on which we can specify a desired operation point with reasonable computation effort. Numerical studies show that our proposed design can efficiently exploit the nonempty core to find a cost allocation for the participants, achieve the desirable tradeoff between stability and fairness, and can address the practical DRAs' large-scale cooperation design.

Sharing Loading Costs for Multi Compartment Vehicles

Supply chains for goods that must be kept cool—cold chains—are of increasing importance in world trade. The goods must be kept within well-defined temperature limits to preserve their quality. One technique for reducing logistics costs is to load cold items into multiple compartment vehicles (MCVs), which have several spaces within that can be set for different temperature ranges. These vehicles allow better consolidation of loads. However, constructing the optimal load is a difficult problem, requiring heuristics for solution. In addition, the cost determined must be allocated to the different items being shipped, most often with different owners who need to pay, and this should be done in a stable manner so that firms will continue to combine loads. We outline the basic structure of the MCV loading problem, and offer the view that the optimization and cost allocation problems must be solved together. Doing so presents the opportunity to solve the problem inductively, reducing the size of the feasible set using constraints generated inductively from the inductive construction of minimal balanced collections of subsets. These limits may help the heuristics find a good result faster than optimizing first and allocating later.

A cooperative game approach for coordinating multi-microgrid operation within distribution systems

This paper focuses on simulating the potential cooperative behaviors of multiple grid-connected microgrids to achieve higher energy efficiency and operation economy. Motivated by the cooperative game theory, a group of individual microgrids is treated as one grand coalition with the aim of minimizing the total operation cost. Next, given that each microgrid operator is an independent and autonomous entity with the aim of maximum self-interest, a cost allocation method based on the concept of core in the cooperative game is implemented to ensure a fair cost share among microgrid coalition members, which guarantees the economic stability of the coalition. Considering the combinatorial explosive characteristic of the cost allocation problem, Benders Decomposition (BD) algorithm is applied to locate the core solution with computational efficiency. In addition, since microgrid coalition is formed at the distribution system level, network losses is not negligible. After considering network losses, the coalition operation model of multi-microgrid becomes an optimal power flow problem. A linearized optimal power flow for distribution (LOPF-D) model is applied instead of the conventional ACOPF model to reduce computation burden, meanwhile maintaining adequate accuracy. Case studies on standard IEEE systems demonstrate the advantages of multi-microgrid cooperation and the robustness of the formulated grand coalition. In addition, comparisons with the conventional ACOPF model verifies the high performance of the proposed LOPF-D model.

A new data envelopment analysis based approach for fixed cost allocation

In many real applications, there exist situations where some independent and decentralized entities will construct a common platform for production processes. A natural and essential problem for the common platform is to allocate the fixed cost or common revenue across these entities in an equitable way. Since there is no powerful central decision maker, each decision-making unit (DMU) might propose an allocation scheme that will favor itself, giving itself a minimal cost and/or a maximal revenue. It is clear that such allocations are egoistic and unacceptable to all DMUs except for the distributing DMU. In this paper, we will address the fixed cost allocation problem in this decentralized environment. For this purpose, we suggest a non-egoistic principle which states that each DMU should propose its allocation proposal in such a way that the maximal cost would be allocated to itself. Further, a preferred allocation scheme should assign each DMU at most its non-egoistic allocation and lead to efficiency scores at least as high as the efficiency scores based on non-egoistic allocations. To this end, we integrate a goal programming method with data envelopment analysis methodology to propose a new model under a set of common weights. The final allocation scheme is determined in such a way that the efficiency scores are maximized for all DMUs through minimizing the total deviation to goal efficiencies. Finally, both a numerical example from prior literature and an empirical study of nine truck fleets are provided to demonstrate the proposed approach.

Allocating a fixed cost across the decision making units with two-stage network structures

A prominent issue in many organizations involves the fair allocation of a total fixed cost among a group of entities. This paper extends the traditional fixed cost allocation problem to situations where the decision making units (DMUs) have a two-stage network structure. To this end, this paper first uses the data envelopment analysis (DEA) methodology to determine the relative efficiency while taking the internal structure and possible allocated costs into account. It shows that each DMU can separately maximize its relative efficiency to one through determining a series of allocations and selecting a set of relative weights. Next, we demonstrate that there exists an efficient allocation set based on a set of common weights, using which all DMUs and their two sub-stages can be simultaneously efficient. However, there are alternative allocation plans in the efficient allocation set. According to this non-uniqueness problem, we further optimize the allocation plans by taking the size of operation units into account, such that the allocation result is proportional to current input usages and output productions from a size point of view. In addition, we suggest a min–max model and a feasible computation algorithm for it to generate the final allocation plan in a way that minimizes the deviation between the efficient allocations and size allocations. More importantly, by repeatedly minimizing the maximum deviation, our proposed method can guarantee a unique allocation plan for all DMUs and sub-stages. Finally, both a numerical example modified from previous literature and an empirical application of bank activities are used to demonstrate the efficacy and usefulness of the proposed approach.

Transmission Cost Allocation Based on Data Envelopment Analysis and Cooperative Game Method

Transmission cost allocation is a traditional but challenging problem in electric power industries. Many factors can affect the transmission cost allocation. But most traditional allocation methods focus on just one point. The Data Envelopment Analysis (DEA) approach has the characteristic of multi-criterion. It can overcome the drawbacks of traditional allocation methods. This paper combines the game theory and DEA approach to solve the problem of transmission cost allocation. In order to take various key points of power transmission into account, three criterions are proposed. These criterions reflect the effect of load demand, transmission line usage and marginal cost on transmission cost allocation. Nucleolus and Shapley value are used to solve the DEA game programming. For measuring the fairness of the transmission cost allocation methods, this paper proposes an algorithm to evaluate the satisfaction of users with the different allocation methods. The DEA game approach is applied to IEEE 14-bus system. The results which are compared with that of other three traditional allocation methods demonstrate the applicability of the proposed method.

Allocating a fixed cost based on a DEA-game cross efficiency approach

In many real managerial applications, an issue of considerable importance is allocating a total fixed cost across a set of competing decision making units (DMUs). The fixed cost allocation problem has also become one of the most important applications of the data envelopment analysis (DEA) methodology. In this paper, we will approach the fixed cost allocation problem by explicitly considering both competition and cooperation relationships among DMUs. To this end, we integrate cooperative game theory and the cross efficiency method to develop a DEA-game cross efficiency approach to generate a unique and fair allocation plan. With the proposed approach, each DMU is considered as a player and a super-additive characteristic function is defined for coalitions of DMUs. Then, the Shapley value is calculated for each DMU and accordingly associated common weights are optimized to determine the final allocation plan. Since the cross efficiency method considers peer appraisal and the cooperative game theory allows for equitable negotiations, all DMUs are supposed to reach a consensus on the equitable allocation scheme through our novel approach. From this perspective, our proposed approach is promising and attractive for allocating a fixed cost in large organizations. Finally, the DEA-game cross efficiency approach is demonstrated with a numerical example derived from previous literature and the results are compared to some existing methods. Additionally, we apply the proposed approach to an empirical application concerning city commercial bank activities in China.

Centralized fixed cost allocation for generalized two-stage network DEA

ABSTRACTMany studies have dealt with the problem of fixed cost allocation by using data envelopment analysis. However, existing models allocate the fixed cost by treating the decision-making units (DMUs) as black-boxes and ignore the internal production structures of DMUs. To our knowledge, only a few work has considered the fixed cost allocation problem for an elementary two-stage production structure without external inputs and outputs. This paper deals with the fixed cost allocation problem for a general two-stage network production structure, in which both external inputs and outputs exist. Specifically, additive two-stage models are first presented to evaluate the performance for each DMU when allocating the fixed cost. Then, by introducing the concepts of satisfaction degree and fairness degree, we propose an approach to obtain an optimal allocation plan under the control of the centralized authority. Finally, an application to 27 banks is utilized to illustrate the proposed approach.

The Refugee Clausee: Why Contractual Allocation of Rescue Costs is Critical During Periods of Mass Migration at Sea

In recent years merchant ships have been called to assist distressed migrant vessels at unprecedented rates. Pursuant to moral and legal obligations arising under international law, merchant shipmasters and crews have courageously responded by rescuing thousands of migrant seafarers. These rescues unfortunately are very expensive undertakings affecting shipping interests both in the form of direct expenditures and indirect losses flowing from commercial delays. With these challenges as the backdrop, this Article examines the commercial implications of merchant vessel participation in large-scale humanitarian rescues. First, it outlines the scope of the legal obligations imposed on private shipmasters to render assistance to distressed vessels. Next, it surveys the language of popularly used shipping contracts, such as the New York Produce Exchange (NYPE) time charterparty forms, and evaluates problems of rescue cost allocation by analysing relevant court opinions and arbitration awards from both the United States and England. Finally, drawing from the shipping industry’s experience in modifying contracts in response to pressing challenges such as maritime piracy, it argues new contract clauses should be adopted by industry participants to more precisely address rescue risk.

Emissions allocation in transportation routes

This article studies the allocation of CO2 emissions to a specific shipment in routing transportation. The authors show that this problem differs from a cost allocation problem specifically because the concavity condition does not hold necessarily in the CO2 allocation problem. This implies that a traditional cost allocation method cannot be straightforwardly translated into a CO2 allocation problem, and thus, new methods need to be developed. This study proposes four allocation mechanisms that are extensions from the literature and the common practice in industries. In doing so, the authors introduce the concept of carbon efficiency to assess if a particular allocation rule drives companies to carbon emissions reduction. They present analytical properties that show that the current practice of allocating CO2 emissions based on the greenhouse gas protocol, fails to be sensitive to drive companies to the sustainable practice of placing consolidated orders. The authors also conduct an experimental analysis using data of a logistics service provider that operates in Europe. Using the results of the experiments, they show that simple allocation methods can lead to a fair and carbon efficient allocation. In addition, the study provides insights by conducting a sensitivity analysis, and it shows that the CO2 allocations are not substantially susceptible to shipment size estimation errors.

A hybrid genetic algorithm and DEA approach for multi-criteria fixed cost allocation

This paper proposes a hybrid genetic algorithm and data envelopment analysis framework for solving the fixed cost allocation (FCA) problem. The proposed framework allows managers to incorporate different FCA sub-objectives for efficient and inefficient decision-making units (DMUs) and solves the FCA problem so that the total entropy of resource allocation for efficient DMUs is maximized, and correlation between resource allocation and efficiency scores of inefficient DMUs is minimized. The FCA sub-objectives and solutions are kept consistent with the overall management objective of rewarding efficient DMUs by allocating to them fewer fixed cost resources. We illustrate the application of our approach using an example from the literature. The results of our study indicate that the solution values obtained in our study are superior to those obtained in other studies under various criteria. Additionally, the relative gap between the solution obtained using our procedure, and the upper bound on the optimal value is approximately 1%, which indicates that our solution is very close to the optimal solution.

Social enterprise tree network games

We investigate the cost allocation strategy associated with the problem of providing service among network users residing at nodes of a social network. We assume that the social network platform is established in a symmetric complete network. There is a cost associated with each link and the service between any pair of nodes can be delivered via a directed path. The example of a cost efficient solution for such network is a (non-rooted) minimum cost directed spanning tree. The network cost should be distributed among users who might have conflicting interests. The objective of this paper is to formulate the above cost allocation problem as a cooperative game, to be referred to as a Social Enterprise Tree Network (SETN) game, and develop a “fair” and efficient cost allocation scheme. The SETN game is related to the minimum cost spanning tree game. The profound difference is that in the minimum cost spanning tree game the service is delivered from some common source node to the rest of the network, while under the social network platform there is no source and the service is established through the interaction among all participating nodes. The input to our cost allocation problem is the optimal non-rooted directed spanning tree SETN. We formulate several associated SETN games in characteristic function form. Then we construct a couple of efficient cost allocation algorithms that find some points in the core of those SETN games and thus result in subsidy-free cost allocations.

Centralized fixed cost and resource allocation considering technology heterogeneity: a DEA approach

Many studies have concentrated on fixed cost and resource allocation issues by using data envelopment analysis (DEA). The existing approaches allocate fixed cost or resource primary based on efficiency principle. They usually assume that all of the DMUs become efficient after fixed cost or resource allocation. However, due to the existing of technology heterogeneity among DMUs, it is impractical for all the DMUs to achieve a common technology level, especially when some DMUs are far from the efficient frontier. In this paper, under the centralized decision environment, we present a new approach to deal with fixed cost and resource allocation problems by considering the factor of technology heterogeneity. Specifically, the concepts of meta-efficiency and group efficiency as well as meta-technology ratio are firstly introduced to reflect the technology level of the DMUs. Then two centralized DEA models considering technology heterogeneity are proposed to allocate fixed cost and resources, respectively. Finally, two numerical examples are presented to illustrate the feasibility and superiority of the proposed approach compared with prior studies.

On the Submodularity of Multi-Depot Traveling Salesman Games

The Steiner traveling salesman problem (STSP) is the problem of finding a minimum cost tour for a salesman that must visit a set of locations while traveling along costly streets before returning to his starting point at the depot. A solution to the problem is a minimum cost tour that both starts and ends at the depot and visits all the required locations. If different players are associated with the destinations to be visited, the STSP induces a cooperative traveling salesman (TS) game that poses the question of how to allocate the total cost of the tour between the different players involved. This cost allocation problem can be tackled using tools and solutions from cooperative games. The purpose of this paper is to generalise the notion of a traveling salesman (TS) game to allow for multiple depots in the underlying STSP and to analyse the submodularity of such multi-depot TS games. A multi-depot STSP can be represented by a connected (di)graph in which a fixed set of nodes are denoted depots, and a non-negative weight function is defined on the edges of the graph. The submodularity of multi-depot TS games is analysed by characterising graphs and digraphs that induce submodular multi-depot TS games for any position of the depots and for at least one position of the depots, respectively.

Coordinated economic dispatch and cost allocation of cooperative multi‐microgrids

This study models the cooperative behaviours of multi-microgrid (MMG) in order to achieve higher energy efficiency and operation economy. Under the framework of cooperative game theory, microgrid members aggregate as one grand coalition and are scheduled via coordinated economic dispatch. Furthermore, a Nash–Harsanyi (N–H) cost allocation scheme is proposed to ensure a fair cost sharing among the players. Considering the combinatorial explosive characteristic of the cost allocation problem, a bi-level programming algorithm is designed to quickly pin the N–H share. Case studies with a MMG system demonstrate the advantages of microgrid cooperation and the robustness of the formulated grand coalition, as well as the computation efficiency of the proposed cost sharing method.

A cooperative game approach to cost allocation in a rapid-transit network

We consider the problem of allocating costs of a regional transit system to its users, who employ shortest path routes between all pairs of nodes in the system network. We provide an axiomatic set of conditions that a solution should satisfy and use cooperative game theory to model the cost allocation problem. We provide an allocation, called the equal cost share solution, which is efficient to compute and is the unique solution that satisfies the conditions. In addition, we show not only that the cost allocation game has a nonempty core, but further, that the game is concave, meaning that the Shapley value allocation, which coincides with the equal cost share solution, always lies in the core of the game. We provide an application of the equal cost share solution to the Washington, D.C. Metro transit network and compare it to the existing fare pricing structure. As compared to equal cost share pricing, the Metro overcharges for short downtown trips and undercharges for very long commutes. The equal cost share solution is easy to update in real time as the cost data and user distribution change, or when the transit network expands.

Solution to Toll Highway Cost Allocation Based on Shapley Value

Cooperative games theory has been widely used in the solution of Multi-participant cost allocation problem. This paper reports on the application of Shapley Value to procedure of finding the optimal solution to cost allocation problem among local governments at various levels in China. Based on the analysis of relationship among vehicle flowrate in different segments of highway, we deduce the formula to find a stable solution and raise reasonable solutions to cost allocation of roll highway among various governments involved. Eventually we put forward proper proposals for central government considering its dominant function.

Loss allocation in energy transmission networks

In this paper we study a cost allocation problem that is inherent to most energy networks: the allocation of losses. In particular, we study how to allocate gas losses between haulers in gas transmission networks. We discuss four allocation rules, two of them have already been in place in real networks and two others that are defined for the first time in this paper. We then present a comparative analysis of the different rules by studying their behavior with respect to a set of principles set forth by the European Union. This analysis also includes axiomatic characterizations of two of the rules. Finally, as an illustration, we apply them to the Spanish gas transmission network.