Reallocating multiple inputs and outputs of units to improve overall performance

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Abstract

All decision-making units (DMUs) in the private or public sector are provided with a set of inputs of different values by their governing decision maker (GDM), and are required to generate a set of outputs. The GDM is able to reallocate the inputs/outputs among the DMUs to estimate the maximum absolute decision making efficiency of the sector. Serial models are presented to manage the interaction between two decision-making levels, GDM and DMUs, to provide the reallocated targets of inputs/outputs for DMUs in the next operating period. The 25 branches of a commercial bank in Taiwan are used as an illustration.

Introduction

A set of performance indices is used to measure the efficiency of a group of decision-making units (DMUs) in the private or public sector. These DMUs operate under their governing decision maker (GDM), who has the power to allocate the resources and set targets for the individual DMUs. The relative efficiency of each DMU or the efficiency of the GDM may be evaluated to determine optimal practices with the available data of each DMU in the indices. Available literature measures the ‘relative decision-making efficiency’ of each DMU, for example, by using the data on all of the DMUs in the sector as a reference set. The conventional data envelopment analysis (DEA) would obtain a set of favorable weights from the indices and associate those with a target for improved efficiency to reduce the values of the inputs and increase values of the outputs [1], [2], [3]. The set of weights for each DMU represents the best course of measurement, among a collection of possible alternatives, en route to selecting the optimal approach. In this capacity, the set of weights serves to indicate ex post facto evaluations of the relative importance among the indices.

Centralized resource allocation models may also be used to obtain the set of weights from the indices for the GDM. Resource allocation problems arise when the GDM, which possesses authority, seeks to reallocate the inputs and outputs among the DMUs to maximize the ‘absolute decision-making efficiency’ of the sector. Our use of the terms ‘DMU’ and ‘GDM’ help emphasize our interest in the decision making by GDM and DMUs on different levels.

Thanassoulis and Dyson [4] combined goal programming (GP) and DEA to obtain the maximal interests of each DMU. Athanassopoulos [5] suggested another goal programming model based on DEA, in which the central decision maker, the GDM, considers the goal of the whole organization when determinging global targets and the maximal contribution of each DMU. In a later study, Athanassopoulos [6] proposes another non-linear programming model that includes the restriction of the weights in the model.

Golany et al. [7] proposed three models based on an additive DEA model [8]. They proposed suggestions concerning the allocation of resources in each DMU after considering the costs and benefits of the input/output. In addition, there are five stages related to the allocation of the resources. This model does not consider output targets, but only maps out the input resources of DMUs. Gloany and Tamir [9] suggested an output-oriented model (maximum output) that considers input and output targets and resource allocation simultaneously. However, this model discusses a single output: each output index must be weighed subjectively before analyzing mutiple output indices.

Beasley [10] utilizes the method of cross-efficiencies to propose a non-linear programming model that aims to maximize the average efficiency of DMUs, and also discusses the fixed allocation of costs and resource allocation of the inputs. Korhonen and Syrjänen [11] suggest a multi-objective linear programming model (MOLP) to perform the resource allocation. Fang and Zhang [12] propose a bicriteria DEA-based model that the GDM can search to find the preferred resource allocation solution, by exploring trade-offs between the total efficiency of the organization and the equity among the individual DMUs, according to the preference of the GDM. Golany [13] and Golany and Tamir [9] emphasized that resource reallocation is an important approach for improving overall performance.

Similar to the conventional radial-based DEA, the radial-based centralized resource allocation model is considered either input-oriented or output-oriented, depending on whether it is concerned with minimum consumption or maximum total output production, respectively. The model proposed by Lozano and Villa [14] can be considered a special case, with the common weights restrictions under the radial-based model. Lozano and Villa [15] also suggest three models, which discuss resource allocation when the number of DMUs decreases and the output remains unchanged. The first model addresses whether the DMUs should be deleted or retained for maximal efficiency. In this model, only the DMUs with high efficiency are selected. The second model addresses the number of the DMUs that should be reserved and resources that should be reallocated for maximum efficiency. The final model looks for the resource reallocation that minimizes the number of DMUs and maximizes the overall efficiency. Lozano et al. [16] propose a serial model that corresponds to three objectives that are pursued lexicographically to address the problem of emission permits. Asmild et al. [17] reconsider the centralized model proposed by Lozano and Villa [14] and suggest modifying it to consider only adjustments to previously inefficient DMUs, to stabilize the original efficient frontier.

Pachkova [18] considers the restrictions on reallocation. For example, access to resources can be restricted, or the resources can be extremely expensive, especially in the short run, so that moving production between individual DMUs becomes impossible. The organization may thus be unable to achieve full efficiency due to the existing limits on reallocation. The approach is a trade-off between the maximum allowed reallocation cost and the highest level of efficiency that the organization can achieve.

However, the efficiency of the radial-based model is not able to consider the slacks of inputs and outputs. For example, the efficiency score that is estimated by a radial-based model might be achieved with positive slacks. Liu and Tsai [19] propose a slacks-based centralized resource allocation model. By incorporating this model, the problem of a missing slack can be solved. Hosseinzadeh Lotfi et al. [20] proposed an enhanced Russell model that can be expressed as a non-radial centralized resource allocation.

Liu and Tsai [19] proposed [CSBM-CW] model which is used to maximize the aggregate efficiency score of the GDM. The two decision-making levels, the GDM and the DMUs under the GDM, would interpret the primal and dual solutions in different ways. The primal solution provides a set of reallocated values of inputs and outputs to those DMUs as targets to improve the performance of the GDM. Each DMU would then strive to achieve its deadline targets in the indices during the next operation period. By contrast, the dual solution is a set of common weights of inputs and outputs that is applied to all DMUs. The set of common weights indicates the relative importance among the inputs and outputs, regarding the performance of the GDM in the current period. Therefore, during the next period, DMUs are supposed to meet all their targets but may expend more effort on the indices with higher weights. Finally, several indices would have values beyond the targets. At the end of the next period, the set of common weights is used to measure the performance of DMUs in the following period. The GDM would then re-evaluate the aggregate score for the next period, and set new targets for DMUs in the following period.

We consider that certain inputs and outputs are uncontrollable, and their values cannot be altered because they owe their influence to certain congenital or acquired causes. For example, the total square footage of floor space in a bank is one of the performance indices used to assess a bank branch. However, it can be difficult to find another suitable location to achieve the desired square footage of floor space because a change in location directly influences other factors, such as sales. We thus introduce the general resource (re)allocation model [CSBM-G]. Therefore, the [CSBM-CW] model is a special case of the [CSBM-G] model, in which all input and output values can be altered.

The [CSBM-G] model provides a set of common weights for controllable inputs and outputs, and a favorable weight for each uncontrollable input or output. Furthermore, side constraints may be added to the [CSBM-G] model to limit the ranges of alteration in the desired inputs and outputs.

The remainder of the paper is arranged as follows. In the next section, we demonstrate our serial slacks-based centralized resource reallocation model, and discuss ways to reallocate the input resources to achieve optimal performance. We also discuss the restrictions affecting resource reallocation, as decision makers may set restrictions to each index in DMUs, to meet practical needs. In Section 3, the case of a commercial bank is analyzed. Lastly, Section 4 presents a discussion of other resource allocation models, and suggests follow-up studies.

Section snippets

Slacks-based centralized resource allocation models

We demonstrate serial slacks-based centralized resource allocation models that employ the idea of a radial-based centralized resource allocation model [14], and a slacks-based measure (SBM) [21]. The models are described in the following subsections.

Resource allocation problems of a commercial bank

In the case of the commercial bank, the district manager controls resource adjustments and reallocation in the branches. The four input indices are the number of employees, the operating costs (tens of thousands of dollars/year), the rental costs (monthly), and the number of ATMs, denoted as x1, x2, x3, and x4, respectively. The five output indices are the business transactions in a branch (monthly), the amount of money drawn from ATMs (monthly), the amount of savings, the amount of credit

Conclusion and discussion

The [CSBM-CW] and the [CSBM-G] models are introduced to solve the resource (re)allocation problems by maximizing the aggregated efficiency score of the GDM. The solutions are associated with reallocated values of inputs and outputs for those DMUs in the next operation period. The general model [CSBM-G] can handle uncontrollable inputs and outputs for the practical problems.

The values for each input and output index can be unified and still retain the same differentiations among the DMUs, as the

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