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.     

Allocating the fixed cost as a complement of other cost inputs: A DEA approach

In cost allocation problem, traditional DEA approaches allocate the fixed cost among a group of decision making units (DMUs), and treat the allocated cost as an extra input of each DMU. If costs except for the fixed cost are regarded as inputs in the cost allocation problem, then it is obvious that the fixed cost is a complement of other inputs rather than an extra independent input. Therefore it is necessary to combine the allocated cost with other cost measures in cost allocation problem. Based on this observation, this paper investigates the relationship between the allocated cost and the DEA efficiency score and develops a DEA-based approach to allocate the fixed cost among various DMUs. An example of allocating advertising expenditure between a car manufacturer and its dealers is presented to illustrate the method proposed in this paper.     

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 new method for ranking non-extreme efficient units in data envelopment analysis

Data envelopment analysis (DEA) evaluates the performance of decision making units (DMUs). When DEA models are used to calculate efficiency of DMUs, a number of them may have the equal efficiency 1. In order to choose a winner among DEA efficient candidates, some methods have been proposed. But most of these methods are not able to rank non-extreme efficient DMUs. Since, the researches performed about ranking of non-extreme efficient units are very limited, incomplete and with some difficulties, we are going to develop a new method to rank these DMUs in this paper. Therefore, we suppose that DMU _o is a non-extreme efficient under evaluating DMU. In continue, by using “Representation Theorem”, DMU _o can be represented as a convex combination of extreme efficient DMUs. So, we expect the performance of DMU _o be similar to the performance of convex combination of these extreme efficient DMUs. Consequently, the ranking score of DMU _o is calculated as a convex combination of ranking scores of these extreme efficient DMUs. So, the rank of this unit will be determined.     

A note on DEA vs principal component analysis: An improvement to Joe Zhu's approach

This paper considers a previous article published by Zhu in the European Journal of Operational Research which describes a joint use of data envelopment analysis (DEA) and principal component analysis (PCA) in ranking of decision making units (DMUs). In Zhu's empirical study, DEA and PCA yield a consistent ranking. However, this paper finds that in certain instances, DEA and PCA may yield inconsistent rankings. The PCA procedure adopted by Zhu is slightly modified in this article by incorporating other important features of ranking that Zhu has not considered. Numerical results reveal that both approaches show a consistency in ranking with DEA when the data set has a small number of efficient units. But, when a majority of the DMUs in the sample are efficient, only the modified approach produces consistent ranking with DEA.     

A DEA two-stage decision processes with shared resources

Data envelopment analysis (DEA) is an approach for measuring the relative efficiency of peer decision making units that have multiple inputs and outputs. In most practical applications of DEA presented in the literature, the presented models assume that outputs are produced perfectly (see Charnes et al. Eur J Oper Res 2:429–444, 1978). However, in many real situations, some outputs are imperfect and they need to be repaired. This paper develops a DEA approach for measuring the efficiency of decision processes which can be divided into two interdependent stages, arranged in series. The novelty of the proposed approach is the existence of perfect and imperfect outputs in a two-stage decision process. This application of two-stage process involves shared resources and the paper gives a best split of these shared resources between two stages. The case of Iranian car representatives is presented.     

A multiple criteria ranking method based on game cross-evaluation approach

The application of Data Envelopment Analysis (DEA) as an alternative multiple criteria decision making (MCDM) tool has been gaining more attentions in the literatures. Doyle (Organ. Behav. Hum. Decis. Process. 62(1):87?C100, 1995) presents a method of multi-attribute choice based on an application of DEA. In the first part of his method, the straightforward DEA is considered as an idealized process of self-evaluation in which each alternative weighs the attributes in order to maximize its own score (or desirability) relative to the other alternatives. Then, in the second step, each alternative applies its own DEA-derived best weights to each of the other alternatives (i.e., cross-evaluation), then the average of the cross-evaluations that get placed on an alternative is taken as an index of its overall score. In some cases of multiple criteria decision making, direct or indirect competitions exist among the alternatives, while the factor of competition is usually ignored in most of MCDM settings. This paper proposes an approach to evaluate and rank alternatives in MCDM via an extension of DEA method, namely DEA game cross-efficiency model in Liang, Wu, Cook and Zhu (Oper. Res. 56(5):1278?C1288, 2008b), in which each alternative is viewed as a player who seeks to maximize its own score (or desirability), under the condition that the cross-evaluation scores of each of other alternatives does not deteriorate. The game cross-evaluation score is obtained when the alternative??s own maximized scores are averaged. The obtained game cross-evaluation scores are unique and constitute a Nash equilibrium point. Therefore, the results and rankings based upon game cross-evaluation score analysis are more reliable and will benefit the decision makers.     

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.     

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.     

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.     

Qualitative factors in data envelopment analysis: A fuzzy number approach

Qualitative factors are difficult to mathematically manipulate when calculating the efficiency in data envelopment analysis (DEA). The existing methods of representing the qualitative data by ordinal variables and assigning values to obtain efficiency measures only superficially reflect the precedence relationship of the ordinal data. This paper treats the qualitative data as fuzzy numbers, and uses the DEA multipliers associated with the decision making units (DMUs) being evaluated to construct the membership functions. Based on Zadeh’s extension principle, a pair of two-level mathematical programs is formulated to calculate the α-cuts of the fuzzy efficiencies. Fuzzy efficiencies contain more information for making better decisions. A performance evaluation of the chemistry departments of 52 UK universities is used for illustration. Since the membership functions are constructed from the opinion of the DMUs being evaluated, the results are more representative and persuasive.     

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.     

Allocating costs of a shared server with stochastic service parameters and job class priorities

This paper develops an approach to allocate common costs to two divisions that share a process, where there is a trade-off between the joint investment in the process and the delays that a division’s jobs are expected to experience there. We allow one division’s jobs to have priority over the other division’s jobs. One purpose of allocation is to obtain accuracy in costing of products reflecting their consumption of resources. The second purpose, the incentive issue, is to elicit truthful reports of private information possessed by each division on (i) delay cost parameters or (ii) expected usage, this information being needed for the investment decision. In case (i), we find that when a division’s private information on its delay costs is poor or non-existent, it would prefer to invest at a weakly higher level than its accounting cost information justifies. In other words, a firm that allocates service center costs depending only on accounting measures of delay costs will under-invest in a shared facility. In case (ii), we find that to elicit truthful reporting by divisions requires a cost allocation rule that involves a complex monitoring of various physical parameters broadly related to the pattern of waiting times. This complexity is driven by the fact that one division has priority. However because actual usage provides an ex post estimate of expected usage up to some random error, a penalty scheme based on directly monitoring actual usage can be used to enforce truth telling up to any desired approximation.     

A joint optimisation model for inventory replenishment,product assortment,shelf space and display area allocation decisions

In this paper, we propose an optimisation model to determine the product assortment, inventory replenishment, display area and shelf space allocation decisions that jointly maximize the retailer’s profit under shelf space and backroom storage constraints. The variety of products to be displayed in the retail store, their display locations within the store, their ordering quantities, and the allocated shelf space in each display area are considered as decision variables to be determined by the proposed integrated model. In the model formulation, we include the inventory investment costs, which are proportional to the average inventory, and storage and display costs as components of the inventory costs and make a clear distinction between showroom and backroom inventories. We also consider the effect of the display area location on the item demand. The developed model is a mixed integer non-linear program that we solved using LINGO software. Numerical examples are used to illustrate the developed model.     

生产可能集为凸包时的相对有效性评价模型

给出了一个评价决策单元相对有效性的新的DEA模型,它所对应的生产可能集被称为凸包形生产可能集,同时讨论了该模型解的存在性,定义了决策单元技术DEA有效和"上投影"的概念,断定一个决策单元的"上投影"相对于原来的决策单元是技术DEA有效的。最后给出一个应用新模型进行设施农业效率评价的例子。     

Data Envelopment Analysis with Preference Structure

It is important to consider the decision making unit (DMU)'s or decision maker's preference over the potential adjustments of various inputs and outputs when data envelopment analysis (DEA) is employed. On the basis of the so-called Russell measure, this paper develops some weighted non-radial CCR models by specifying a proper set of ‘preference weights’ that reflect the relative degree of desirability of the potential adjustments of current input or output levels. These input or output adjustments can be either less or greater than one; that is, the approach enables certain inputs actually to be increased, or certain outputs actually to be decreased. It is shown that the preference structure prescribes fixed weights (virtual multiplier bounds) or regions that invalidate some virtual multipliers and hence it generates preferred (efficient) input and output targets for each DMU. In addition to providing the preferred target, the approach gives a scalar efficiency score for each DMU to secure comparability. It is also shown how specific cases of our approach handle non-controllable factors in DEA and measure allocative and technical efficiency. Finally, the methodology is applied with the industrial performance of 14 open coastal cities and four special economic zones in 1991 in China. As applied here, the DEA/preference structure model refines the original DEA model's result and eliminates apparently efficient DMUs.     

Supplier selection and order allocation problem using a two-phase fuzzy multi-objective linear programming

The aim of this paper is to solve a supplier selection problem under multi-price level and multi-product using interactive two-phase fuzzy multi-objective linear programming (FMOLP) model. The proposed model attempts to simultaneously minimize total purchasing and ordering costs, a number of defective units, and late delivered units ordered from suppliers. The piecewise linear membership functions are applied to represent the decision maker’s fuzzy goals for the supplier selection and order allocation problem, and can be resulted in more flexibility via an interactive decision-making process. To demonstrate effectiveness of the proposed model, results of applying the proposed model are shown by a numerical example. The analytical results show that the proposed approach is effective in uncertain environments and provide a reliable decision tool for integrated multi-objective supplier selection problems.     

Multi-objectives Tabu Search based algorithm for progressive resource allocation

Military course of action planning involves time and space synchronization as well as resource and asset allocation. A mission could be seen as a defined set of logical ordered tasks with time and space constraints. The resources to task rules require that available assets should be allocated to each task. A combination of assets might be required to execute a given task. The couple (task, resources) constitutes an action. This problem is formulated as a multi-objectives scheduling and resource allocation problem. Any solution is assessed based on a number of conflicting and heterogeneous objectives. In fact, military planning officers should keep perfecting the plan based on the Commander’s criteria for success. The scheduling problem and resource allocation problem are considered as NP-Hard Problems [A. Guitouni, B. Urli, J.-M. Martel, Course of action planning: A project based modelling, Working Paper, Faculté des sciences de l’ administration, Université Laval, Québec, 2005]. This paper is concerned with the multi-objectives resource allocation problem. Our objective is to find adequate resource allocation for given courses of action schedule. To optimize this problem, this paper investigates non-exact solution methods, like meta-heuristic methods for finding potential efficient solutions. A progressive resource allocation methodology is proposed based on Tabu Search and multi-objectives concepts. This technique explores the search space so as to find a set of potential efficient solutions without aggregating the objectives into a single objective function. It is guided by the principle of maximizing the usage of any resource before considering a replacement resource. Thus, a given resource is allocated to the maximum number of tasks for a given courses of action schedule. A good allocation is a potential efficient solution. These solutions are retained by applying a combination of a dominance rule and a multi-criteria filtering method. The performance of the proposed Pareto-based approach is compared to two aggregation approaches: weighted-sum and the lexicographic techniques. The result shows that a Pareto-based approach is providing better solutions and allowing more flexibility to the decision-maker.     

基于CAR-DEA方法的环境效率评价研究

现有环境效率评价的DEA方法没有考虑多维偏好约束问题,即不同决策单元对不同期望产出和不期望产出的偏好不同. 以地区为例,不同地区对GDP、废水和废气赋予的权重偏好各不相同. 在这种情况下,由于各决策单元的偏好约束不同,形成多维偏好约束集,在传统DEA模型中容易出现无可行解现象. 针对这一问题,基于CAR-DEA方法,结合保证域理论,提出一种解决多维偏好约束集问题的环境效率评价模型. 采用中国工业系统的环境效率评价实例对提出的方法进行了分析和说明.     

Target setting in the general combined-oriented CCR model using an interactive MOLP method

The traditional data envelopment analysis (DEA) model does not include a decision maker’s (DM) preference structure while measuring relative efficiency, with no or minimal input from the DM. To incorporate DM’s preference information in DEA, various techniques have been proposed. An interesting method to incorporate preference information, without necessary prior judgment, is the use of an interactive decision making technique that encompasses both DEA and multi-objective linear programming (MOLP). In this paper, we will use Zionts-Wallenius (Z-W) method to reflecting the DM’s preferences in the process of assessing efficiency in the general combined-oriented CCR model. A case study will conducted to illustrate how combined-oriented efficiency analysis can be conducted using the MOLP method.