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.     

Resource allocation based on cost efficiency

In this paper, we consider a resource allocation (RA) problem and develop an approach based on cost (overall) efficiency. The aim is to allocate some inputs among decision making units (DMUs) in such way that their cost efficiencies improve or stay unchanged after RA. We formulate a multi-objective linear programming problem using two different strategies. First, we propose an RA model which keeps the cost efficiencies of units unchanged. This is done assuming fixed technical and allocative efficiencies. The approach is based on the assumption that the decision maker (DM) may not have big changes in the structure of DMUs within a short term. The second strategy does not impose any restrictions on technical and allocative efficiencies. It guarantees that none of the cost efficiencies of DMUs get worse after RA, and the improvement for units is possible if it is feasible and beneficial. Two numerical examples and an empirical illustration are also provided.     

An equitable method for allocating fixed costs by using data envelopment analysis

This paper concerns the shared cost allocation problem by using Data Envelopment Analysis (DEA), which is observed in practical applications such as public services and production processes. In the management context, the cost allocation problem tries to balance the different desires of two management layers: central manager and each sector manager. The cost can be assigned in an equitable way to the various Decision Making Units (DMUs). To achieve this goal, we present a new DEA-based method for dividing a fixed cost among DMUs. In the proposed method, the fixed cost is assigned to DMUs such that the efficiency measures and the Returns to Scale classifications of all DMUs before and after assigning the fixed cost remain unchanged. Also, the gaps among the costs allocated to DMUs will be minimized. The proposed method has the flexibility to consider the management standpoints. Finally, numerical results of an elucidatory example are furnished to demonstrate the applicability and reliability of our scheme.     

基于DEA的二阶段网络系统的固定成本分摊方法

结合DEA和博弈的思想研究二阶段网络系统的固定成本分摊问题,将分摊成本作为新的投入,可以证明存在某种分摊使DMU整体效率达到最优,在此基础上考虑各个DMU之间以及DMU内部之间的博弈,首先建立讨价还价乘积最大化模型,求出各DMU唯一的分摊解,然后建立DMU子系统之间的讨价还价模型,给出子系统的分摊解,最终的分摊方案满足系统效率和子系统效率为1,与现有的方法相比具有一定的优势.     

The fair allocation of common fixed cost or revenue using DEA concept

The common fixed cost or revenue distribution amongst decision making units (briefly, DMUs) in an equitable way is one of the problems that can be solved by data envelopment analysis (DEA) concept. The motivation of this paper is common fixed cost or revenue allocation based on following three principles: First, allocation must be directly proportional to the elements (inputs and outputs) that are directly proportional to imposed common fixed cost or to obtained common fixed revenue. Second, allocation must be inversely proportional to the elements that are inversely proportional to common fixed cost or revenue. Finally, the elements that have no effect on common fixed cost or revenue must have no effect on allocation as well.     

Cost allocation and convex data envelopment

This paper considers allocation rules. First, we demonstrate that costs allocated by the Aumann–Shapley and the Friedman–Moulin cost allocation rules are easy to determine in practice using convex envelopment of registered cost data and parametric programming. Second, from the linear programming problems involved it becomes clear that the allocation rules, technically speaking, allocate the non-zero value of the dual variable for a convexity constraint on to the output vector. Hence, the allocation rules can also be used to allocate inefficiencies in non-parametric efficiency measurement models such as Data Envelopment Analysis (DEA). The convexity constraint of the BCC model introduces a non-zero slack in the objective function of the multiplier problem and we show that the cost allocation rules discussed in this paper can be used as candidates to allocate this slack value on to the input (or output) variables and hence enable a full allocation of the inefficiency on to the input (or output) variables as in the CCR model.     

Centralized DEA models with the possibility of downsizing

While traditional data envelopment analysis (DEA) models assess the relative efficiency of similar, independent decision making units (DMUs) centralized DEA models aim at reallocating inputs and outputs among the units setting new input and output targets for each one. This system point of view is appropriate when the DMUs belong to a common organization that allocates their inputs and appropriates their outputs. This intraorganizational perspective opens up the possibility that greater technical efficiency for the organization as a whole might be achieved by closing down some of the existing DMUs. In this paper, we present three centralized DEA models that take advantage of this possibility. Although these models involve some binary variables, we present efficient solution approaches based on Linear Programming. We also present some numerical results of the proposed models for a small problem from the literature.     

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.     

Characterizing an equitable allocation of shared costs: A DEA approach

In many applications to which DEA could be applied, there is often a fixed or common cost which is imposed on all decision making units. This would be the case, for example, for branches of a bank which can be accessed via the numerous automatic teller machines scattered throughout the country. A problem arises as to how this cost can be assigned in an equitable way to the various DMUs. In this paper we propose a DEA approach to obtain this cost allocation which is based on two principles: invariance and pareto-minimality. It is shown that the proposed method is a natural extension of the simple one-dimensional problem to the general multiple-input multiple-output case.     

DEA models for minimizing weight disparity in cross-efficiency evaluation

Cross-efficiency evaluation is a commonly used approach for ranking decision-making units (DMUs) in data envelopment analysis (DEA). The weights used in the cross-efficiency evaluation may sometimes differ significantly among the inputs and outputs. This paper proposes some alternative DEA models to minimize the virtual disparity in the cross-efficiency evaluation. The proposed DEA models determine the input and output weights of each DMU in a neutral way without being aggressive or benevolent to the other DMUs. Numerical examples are tested to show the validity and effectiveness of the proposed DEA models and illustrate their significant role in reducing the number of zero weights.     

An ellipsoidal frontier model: Allocating input via parametric DEA

This paper presents the ellipsoidal frontier model (EFM), a parametric data envelopment analysis (DEA) model for input allocation. EFM addresses the problem of distributing a single total fixed input by assuming the existence of a predefined locus of points that characterizes the DEA frontier. Numeric examples included in the paper show EFM’s capacity to allocate shares of the total fixed input to each DMU so that they will all become efficient. By varying the eccentricities, input distribution can be performed in infinite ways, gaining control over DEA weights assigned to the variables in the model. We also show that EFM assures strong efficiency and behaves coherently within the context of sensitivity analysis, two properties that are not observed in other models found in the technical literature.     

Centralized resource allocation with stochastic data

Data Envelopment Analysis (DEA) is a technique based on mathematical programming for evaluating the efficiency of homogeneous Decision Making Units (DMUs). In this technique inefficient DMUs are projected on to the frontier which constructed by the best performers. Centralized Resource Allocation (CRA) is a method in which all DMUs are projected on to the efficient frontier through solving just one DEA model. The intent of this paper is to present the Stochastic Centralized Resource Allocation (SCRA) in order to allocate centralized resources where inputs and outputs are stochastic. The concept discussed throughout this paper is illustrated using the aforementioned example.     

Spherical frontier DEA model based on a constant sum of inputs

In this paper, a Data Envelopment Analysis (DEA) model in which a fixed input needs to be assigned to a group of Decision-Making Units (DMUs) is presented. This is performed by assuming the existence of a geometric place represented by a sphere that characterizes the DEA frontier. It is shown that, under this assumption, it becomes relatively easy to find a way to distribute the fixed input to all DMUs, by considering that the individual assignments will be fair through the requirement that all DMUs be efficient or, in other words, be located on the spherically shaped efficiency frontier. A model is presented and results are compared to those obtained by using two different methods proposed in the literature within the same context.     

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.     

DEA model with shared resources and efficiency decomposition

Data envelopment analysis (DEA) has proved to be an excellent approach for measuring performance of decision making units (DMUs) that use multiple inputs to generate multiple outputs. In many real world scenarios, DMUs have a two-stage network process with shared input resources used in both stages of operations. For example, in hospital operations, some of the input resources such as equipment, personnel, and information technology are used in the first stage to generate medical record to track treatments, tests, drug dosages, and costs. The same set of resources used by first stage activities are used to generate the second-stage patient services. Patient services also use the services generated by the first stage operations of housekeeping, medical records, and laundry. These DMUs have not only inputs and outputs, but also intermediate measures that exist in-between the two-stage operations. The distinguishing characteristic is that some of the inputs to the first stage are shared by both the first and second stage, but some of the shared inputs cannot be conveniently split up and allocated to the operations of the two stages. Recognizing this distinction is critical for these types of DEA applications because measuring the efficiency of the production for first-stage outputs can be misleading and can understate the efficiency if DEA fails to consider that some of the inputs generate other second-stage outputs. The current paper develops a set of DEA models for measuring the performance of two-stage network processes with non splittable shared inputs. An additive efficiency decomposition for the two-stage network process is presented. The models are developed under the assumption of variable returns to scale (VRS), but can be readily applied under the assumption of constant returns to scale (CRS). An application is provided.     

Dominance stochastic models in data envelopment analysis

In this paper stochastic models in data envelopment analysis (DEA) are developed by taking into account the possibility of random variations in input-output data, and dominance structures on the DEA envelopment side are used to incorporate the modelbuilder's preferences and to discriminate efficiencies among decision making units (DMUs). The efficiency measure for a DMU is defined via joint dominantly probabilistic comparisons of inputs and outputs with other DMUs and can be characterized by solving a chance constrained programming problem. Deterministic equivalents are obtained for multivariate symmetric random errors and for a single random factor in the production relationships. The goal programming technique is utilized in deriving linear deterministic equivalents and their dual forms. The relationship between the general stochastic DEA models and the conventional DEA models is also discussed.     

An equitable DEA-based approach for assigning fixed resources along with targets

A typical problem in organization management is how to divide a fixed resource along with a target among decision making units (DMUs) of an organization equitably. By using the data envelopment analysis technique, this paper concerns the problem from the perspective of efficiency analysis and proposes a new sharing model. In the proposed method, the fixed resource and target are divided among DMUs such that the efficiencies of DMUs remain unchanged after assigning the fixed cost and target. The proposed method is unit-invariant; it eliminates resource waste and target insufficiency brought by slacks. Also, every DMU is assigned a positive resource and a positive target under this method. Two corresponding algorithms are designed to yield a unique allocation. The proposed approach can be developed under both constant returns to scale and variable returns to scale. Two examples are presented to illustrate the validity and superiorities of our method.     

Resource allocation for branch network system with considering heterogeneity based on DEA method

In this paper, we proposed a new DEA approach to allocate the resource in branch network system which is not covered by the existing resource allocation works under a centralized decision-making environment. The branch network system is typically appears in multi-national or multi-regional corporations, which has many branches across multiple locations. Given the spatial distribution of the production, we imposed additional restrictions on resource allocation and divided the resource inputs into three groups: fixed inputs, regional inputs that allocated to the branches in the same area and common resource that an additional resource allocated to all the branches. Then, we generalize the model further to accommodate technological heterogeneity due to the difference in the geographical locations of the branches. And the objective of the proposed models is to maximize the gross profits of the entire organization, which is a natural assumption for a for-profit organization. Finally, an example was presented to illustrate the proposed approach with heterogeneous technology is more practically feasible and superior than the prior approach with homogeneous technology.     

A generalized DEA model for inputs/outputs estimation

In this paper we discuss the question: among a group of decision making units (DMUs), if a DMU changes some of its input (output) levels, to what extent should the unit change outputs (inputs) such that its efficiency index remains unchanged? In order to solve this question we propose a solving method based on Data Envelopment Analysis (DEA) and Multiple Objective Linear Programming (MOLP). In our suggested method, the increase of some inputs (outputs) and the decrease due to some of the other inputs (outputs) are taken into account at the same time, while the other offered methods do not consider the increase and the decrease of the various inputs (outputs) simultaneously. Furthermore, existing models employ a MOLP for the inefficient DMUs and a linear programming for weakly efficient DMUs, while we propose a MOLP which estimates input/output levels, regardless of the efficiency or inefficiency of the DMU. On the other hand, we show that the current models may fail in a special case, whereas our model overcomes this flaw. Our method is immediately applicable to solve practical problems.     

Centralized resource allocation DEA models based on revenue efficiency under limited information

In this paper, we extend the centralized DEA models by Lozano et al (2011) to allocate resources based on revenue efficiency across a set of DMUs under a centralized decision-making environment. The aim is to allocate resources so as to maximize the total output revenue produced by all the DMUs under limited information. To uncover the sources of total revenue increase from the centralized resource allocation model, we further decompose the aggregate revenue efficiency into three components: the aggregate output-oriented technical efficiency, the aggregate output allocative efficiency and the aggregate revenue re-allocative efficiency. Finally, two empirical data sets are presented to show that our proposed approach is not only an efficient tool to allocate the resources among the DMUs based on the revenue efficiency but additionally provides the central DM with guidance on how to identify the weak areas where more effort should be devoted to improve the total outputs.