Allocating fixed costs and common revenue via data envelopment analysis

An issue of considerable importance involves the allocation of fixed costs or common revenue among a set of competing entities in an equitable way. Based on the data envelopment analysis (DEA) theory, this paper proposes new methods for (i) allocating fixed costs to decision making units (DMUs) and (ii) distributing common revenue among DMUs, in such a way that the relative efficiencies of all DMUs remain unchanged and the allocations should reflect the relative efficiencies and the input-output scales of individual DMUs. To illustrate our methods, numerical results for an example are described in this paper.     

A discrete competitive facility location model with variable attractiveness

We consider the discrete version of the competitive facility location problem in which new facilities have to be located by a new market entrant firm to compete against already existing facilities that may belong to one or more competitors. The demand is assumed to be aggregated at certain points in the plane and the new facilities can be located at predetermined candidate sites. We employ Huff's gravity-based rule in modelling the behaviour of the customers where the probability that customers at a demand point patronize a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. The objective of the firm is to determine the locations of the new facilities and their attractiveness levels so as to maximize the profit, which is calculated as the revenue from the customers less the fixed cost of opening the facilities and variable cost of setting their attractiveness levels. We formulate a mixed-integer nonlinear programming model for this problem and propose three methods for its solution: a Lagrangean heuristic, a branch-and-bound method with Lagrangean relaxation, and another branch-and-bound method with nonlinear programming relaxation. Computational results obtained on a set of randomly generated instances show that the last method outperforms the others in terms of accuracy and efficiency and can provide an optimal solution in a reasonable amount of time.     

Contract and product quality in platform selling

More and more e-tailers (platforms) are allowing manufacturers direct access to customers. Two common contracts are offered by platforms to manufacturers: the revenue sharing contract where a platform appropriates a portion of the manufacturer’s revenue, and the fixed fee contract where a platform charges a fixed rent for each sale. Using an analytical model, this paper studies the interrelationship between a platform’s contract choice and a manufacturer’s product quality decision. We find that if product quality is exogenously given, the platform will always adopt the revenue sharing contract. If the manufacturer endogenously decides the quality, however, the platform’s contract choice may be changed. This is because the revenue sharing contract, compared to fixed fee, leads to a lower selling price of the manufacturer, whereas the fixed fee contract can motivate a higher quality than does revenue sharing. As a result, a large (small) market heterogeneity induces the platform to adopt the revenue sharing (fixed fee) contract. We also extend the model to several directions, finding that longer product line, manufacturer competition, lower marginal production cost, and higher platform cost all tend to induce the platform to put forward a fixed fee contract; while if quality decision is less flexible than contract decision, the platform is more ready to embrace revenue sharing. Besides, when there are two platforms competing for the same market, they should differentiate their contract choices so as to mitigate competition.     

Fixed cost and resource allocation based on DEA cross-efficiency

In many managerial applications, situations frequently occur when a fixed cost is used in constructing the common platform of an organization, and needs to be shared by all related entities, or decision making units (DMUs). It is of vital importance to allocate such a cost across DMUs where there is competition for resources. Data envelopment analysis (DEA) has been successfully used in cost and resource allocation problems. Whether it is a cost or resource allocation issue, one needs to consider both the competitive and cooperative situation existing among DMUs in addition to maintaining or improving efficiency. The current paper uses the cross-efficiency concept in DEA to approach cost and resource allocation problems. Because DEA cross-efficiency uses the concept of peer appraisal, it is a very reasonable and appropriate mechanism for allocating a shared resource/cost. It is shown that our proposed iterative approach is always feasible, and ensures that all DMUs become efficient after the fixed cost is allocated as an additional input measure. The cross-efficiency DEA-based iterative method is further extended into a resource-allocation setting to achieve maximization in the aggregated output change by distributing available resources. Such allocations for fixed costs and resources are more acceptable to the players involved, because the allocation results are jointly determined by all DMUs rather than a specific one. The proposed approaches are demonstrated using an existing data set that has been applied in similar studies.     

A note on the paper ‘On dynamical multi-team Cournot game in exploitation of a renewable resource’

In a recent paper (Asker, 2007) [1] a dynamic Cournot oligopoly game is proposed and it is claimed that this model represents competition among firms that exploit a common access natural resource. According to the author’s claim, the feature that relates the model with renewable natural resource harvesting is given by the presence of a particular cost function where the total cost of each fisherman is proportional to the square of the own quantity of harvesting and inversely proportional to the total harvesting quantity. In contrast, the usual function used in the literature on the exploitation of natural resources (such as fisheries) is inversely proportional to the available resource stock, and not to the total harvesting. This, in some sense, assumes exactly the opposite (as the available resource is inversely proportional to the total harvesting). So, we believe that the paper (Asker, 2007) [1] contains an error which is probably due to a misunderstanding or a misreading and misinterpretation of the (well-established) literature on bioeconomic modelling, but nevertheless misleading to researchers interested in bioeconomic modelling. The aim of this short note is to explain the mistake and to summarize the correct derivation and interpretation of the cost function. Our goal is to avoid the propagation of a subtle (but nevertheless misleading) error.     

Weighted Aumann-Shapley pricing

Cost allocation problems arise in many contexts in economics and management science. In a typical problem that we have in mind, a decision maker must decide how to allocate the joint cost of production among several commodities using prices. Furthermore, these prices must satisfy certain reasonable postulates among which is the requirement that total revenue associated with these prices must cover total cost. In this paper, we investigate a generalization of Aumann-Shapley pricing, called Weighted Aumann-Shapley pricing, that allows for asymmetric pricing of commodities even when those commodities affect costs in a symmetric fashion. Weighted AS pricing is a natural extension of (symmetric) Aumann-Shapley pricing, and may be considered a non-atomic analogue of Owen's modified diagonal formula (with respect to the multilinear extension) for the weighted TU Shapley Value. Received December 1993/Revised version June 1998     

A single-period inventory placement problem for a supply chain with the expected profit objective

Consider the expected profit maximizing inventory placement problem in an N-stage, supply chain facing a stochastic demand for a single planning period for a specialty item with a very short selling season. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy customer demand. Stocking decisions are made before demand occurs. Because of delays, only a known fraction of demand at a stage will wait for shipments. Unsatisfied demand is lost. The revenue, salvage value, ordering, shipping, processing, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. After characterizing an optimal solution, we propose an algorithm for its computation. For the zero fixed cost case, the computations can be done on a spreadsheet given normal demands. For the nonnegative fixed cost case, we develop an effective branch and bound algorithm.     

A cost allocation problem arising in hub–spoke network systems

This paper studies a cost allocation problem arising from hub–spoke network systems. When a large-scale network is to be constructed jointly by several agents, both the optimal network design and the fair allocation of its cost are essential issues. We formulate this problem as a cooperative game and analyze the core allocation, which is a widely used solution concept. The core of this game is not necessarily non-empty as shown by an example. A reasonable scheme is to allocate the cost proportional to the flow that an agent generates. We show that, if the demand across the system has a block structure and the fixed cost is high, this cost allocation scheme belongs to the core. Numerical experiments are given with real telecommunication traffic data in order to illustrate the usefulness of our analytical findings.     

A Markov Sensitivity Model for Examining the Impact of Cost Allocations in Hospitals

The cost allocation process in hospitals typically entails an accounting step-down procedure whereby costs are allocated from non-revenue producing service centres to revenue centres. The resulting revenue centre costs are then compared with the third party (Blue Cross, Medicare, Medicaid) allowable costs. Any costs in excess of the allowable costs are not reimbursable. This procedure has been conceptualized using a Markov chain in a recent journal article. The purpose of this paper is to demonstrate how the Markov model may be used to assess the impact of various changes in the original data without having to recalculate the entire step-down process via a Markov model or any other procedure. The changes include an alternate step-down model, a different cost allocation basis for one or more service centres, and the expansion or contraction of one or more service centres.     

Minimum Cost Compromise Mixed Allocation

When more than one (say p) characteristics in multivariate stratified population are defined on each unit of the population, the individual optimum allocations may differ widely and can not be used practically. Moreover, there may be a situation such that no standard allocation is advisable to all the strata, for one reason or another. In such a situation, Clark and Steel (J R Stat Soc, Ser D Stat 49(2):197–207, 2000) suggested that different allocations may be used for different groups of strata having some common characteristics for double sampling in stratification. Later on, Ahsan et al. (Aligarh J Stat 25:87–97, 2005) used the same concept in univariate stratified sampling. They minimized the variance of the stratified sample mean for a fixed cost to obtain an allocation and called this allocation “mixed allocation”. In the present paper, a “compromise mixed allocation” is worked out for the fixed precisions of the estimates of the p-population means of a multivariate stratified population. A numerical example is also presented.     

Cooperative games in facility location situations with regional fixed costs

A continuous single-facility location problem, where the fixed cost (or installation cost) depends on the region where the new facility is located, is studied by mean of cooperative Game Theory tools. Core solutions are proposed for the total cost allocation problem. Sufficient conditions in order to have a nonempty core are given, then the Weber problem with regional fixed costs is studied.     

Allocating fixed costs or resources and setting targets via data envelopment analysis

In many applications of data envelopment analysis (DEA), there is often a fixed cost or input resource which should be imposed on all decision making units (DMUs). Cook and Zhu [W.D. Cook, J. Zhu, Allocation of shared costs among decision making units: A DEA approach, Computers and Operations Research 32 (2005) 2171-2178] propose a practical DEA approach for such allocation problems. In this paper, we prove that when some special constraints are added, Cook and Zhu’s approach probably has no feasible solution. The research of this paper focuses on two main aspects: to obtain a new fixed costs or resources allocation approach by improving Cook and Zhu’s approach, and to set fixed targets according to the amount of fixed resources shared by individual DMUs. When such special constraints are attached, our model is proved to be able to achieve a feasible costs or resources allocation. Numerical results for an example from the literature are presented to illustrate our approach.     

A DEA approach for fair allocation of common revenue

An issue of considerable importance, how to allocate a common revenue in an equitable manner across a set of competing entities. This paper introduces a new approach to obtaining allocation common revenue on all decision making units (DMUs) in such a way that the relative efficiency is not changed. In this method for determining allocation common revenue dose not need to solving any linear programming. A numerical example is provided to illustrate the results of the analysis.     

The generalized assignment problem with flexible jobs

The Generalized Assignment Problem (GAP) seeks an allocation of jobs to capacitated resources at minimum total assignment cost, assuming a job cannot be split among multiple resources. We consider a generalization of this broadly applicable problem in which each job must not only be assigned to a resource, but its resource consumption must also be determined within job-specific limits. In this profit-maximizing version of the GAP, a higher degree of resource consumption increases the revenue associated with a job. Our model permits a job’s revenue per unit resource consumption to decrease as a function of total resource consumption, which allows modeling quantity discounts. The objective is then to determine job assignments and resource consumption levels that maximize total profit. We develop a class of heuristic solution methods, and demonstrate the asymptotic optimality of this class of heuristics in a probabilistic sense.     

On optimal proportional reinsurance and investment in a Markovian regime-switching economy

In this paper, the surplus of an insurance company is modeled by a Markovian regimeswitching diffusion process. The insurer decides the proportional reinsurance and investment so as to increase revenue. The regime-switching economy consists of a fixed interest security and several risky shares. The optimal proportional reinsurance and investment strategies with no short-selling constraints for maximizing an exponential utility on terminal wealth are obtained.     

A Markov model for single-leg air cargo revenue management under a bid-price policy

In this paper, we consider the capacity allocation problem in single-leg air cargo revenue management. We assume that each cargo booking request is endowed with a random weight, volume and profit rate and propose a Markovian model for the booking request/acceptance/rejection process. The decision on whether to accept the booking request or to reserve the capacity for future bookings follows a bid-price control policy. In particular, a cargo will be accepted only when the revenue from accepting it exceeds the opportunity cost, which is calculated based on bid prices. Optimal solutions are derived by maximizing a reward function of a Markov chain. Numerical comparisons between the proposed approach and two existing static single-leg air cargo capacity allocation policies are presented.     

Proposing a method for fixed cost allocation using DEA based on the efficiency invariance and common set of weights principles

One of the applications of data envelopment analysis is fixed costs allocation among homogenous decision making units. In this paper, we first prove that Beasley’s method (Eur J Oper Res 147(1):198–216, 2003), whose infeasibility has been claimed by Amirteimoori and Kordrostami (Appl Math Comput 171(1):136–151, 2005), always has a feasible solution and the efficiency invariance principle does not necessarily satisfy in Amirteimoori and Kordrostami’s method (Appl Math Comput 171(1):136–151, 2005). Hence, we present two equitable methods for fixed cost allocation based on the efficiency invariance and common set of weights principles such that, if possible, they help meet these two principles. In the first method, the costs are allocated to DMU in such a way that the efficiency score of DMUs does not change, and simultaneously this allocation has the minimum distance from the allocation that has been obtained with a common set of weights. However, in the second method, the costs are allocated in such a way that input and output of all units have a common set of weights and it has the minimum distance from the allocation that satisfies the efficiency invariance principle. Moreover, both methods, consider the satisfaction of each unit of the allocated cost. Finally, the proposed method is illustrated by two real world examples.     

Cost,Revenue and Profit Efficiency Models in Generalized Fuzzy Data Envelopment Analysis

One of the most important information given by data envelopment analysis models is the cost, revenue and profit efficiency of decision making units (DMUs). Cost efficiency is defined as the ratio of minimum costs to current costs, while revenue efficiency is defined as the ratio of maximum revenue to current revenue of the DMU. This paper presents a framework where data envelopment analysis (DEA) is used to measure cost, revenue and profit efficiency with fuzzy data. In such cases, the classical models cannot be used, because input and output data appear in the form of ranges. When the data are fuzzy, the cost, revenue and profit efficiency measures calculated from the data should be uncertain as well. Fuzzy DEA models emerge as another class of DEA models to account for imprecise inputs and outputs for DMUs. Although several approaches for solving fuzzy DEA models have been developed, numerous deficiencies including the $\alpha$-cut approaches and types of fuzzy numbers must still be improved. This scheme embraces evaluation method based on vector for proposed fuzzy model. This paper proposes generalized cost, revenue and profit efficiency models in fuzzy data envelopment analysis. The practical application of these models is illustrated by a numerical example.     

The Convergence to Equilibrium of Neutral Genetic Models

This article is concerned with the long-time behavior of neutral genetic population models with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter.     

考虑公平性的两阶段系统固定成本分摊DEA模型

针对两阶段系统固定成本分摊问题,已有研究大多是基于效率不变性或构造满意度指标等提出不同方法,少有考虑固定成本分摊方案的公平性。本文首先通过非自利原则得到系统整体与各阶段的成本分摊上限,针对具有群组式组织结构的两阶段系统,提出新的固定成本分摊模型,所得分摊结果更为公平,最后通过实际算例验证了本文模型的可行性和有效性。