An alternative approach in the estimation of returns to scale under weight restrictions

doi:10.1016/j.amc.2006.11.122

Applied Mathematics and Computation

Volume 189, Issue 1, 1 June 2007, Pages 719-724

https://doi.org/10.1016/j.amc.2006.11.122 Get rights and content

Abstract

This paper discusses the issue of returns to scale (RTS) under weight restrictions in data envelopment analysis (DEA). We first review Tone’s method [K. Tone, On returns to scale under weight restrictions in data envelopment analysis, Journal of Productivity Analysis 16 (2001) 31–47] for estimating returns to scale under weight restrictions. Then a new approach is introduced for this task, based upon the sum of the optimal lambda values in the weighted CCR model. The equivalence of this method and Tone’s method is proved. We then apply the method to a real world data set.

Introduction

Data envelopment analysis (DEA) is a nonparametric technique for measuring and evaluating the relative efficiencies of decision making units (DMUs) with multiple inputs and multiple outputs. Specifically, it determines a set of weights such that the efficiency of a target DMU (DMU_o) relative to the other DMUs is maximized. Following Charnes, Cooper and Rhodes [6], a number of different DEA models have now appeared in the literature (see [7]). During this period of model development, the economic concept of returns to scale (RTS) has also been widely studied within the different frameworks provided by these methods [2], [3], [4], [5], [13], [8]. As we know, the imposition of weight restrictions has been recognized as one of the important factors when applying DEA to actual situations, and several models have been developed for this purpose [11], [12], [15], [9], [10]. Therefore determining the returns to scale status (constant, increasing, or decreasing returns to scale) under weight restrictions is important. And this is the topic to which this paper is devoted. Recently Tone [14] addressed returns to scale under weight restrictions, based upon the sign of $u_{o}^{*}$ in the optimal solution of (DWR_o). In this paper we introduce a new approach to classify the returns to scale under weight restrictions which has computational advantages as compared to the Tone’s method. The rest of this paper structured as follows: Section 2 provides preliminary information that will be used in the succeeding sections. Section 3 presents Tone’s method. Section 4 discusses returns to scale under weight restrictions. Section 5 contains a numerical example. Conclusions are given in Section 6.

Section snippets

Preliminaries

In this paper, we focus on the input-oriented weighted DEA models. Suppose, that we have n DMUs, every DMU_j, (j = 1, 2, … , n) producing the same s outputs in (possibly) different amounts, y_rj (r = 1,2, … , s) using the same m inputs, $x_{ij} (i = 1, 2, \dots, m)$ , also in (possibly) different amounts. All inputs and outputs are assumed to be nonnegative, but at least one input and one output are positive, i.e., $x_{j} = (x_{1 j}, \dots, x_{mj}) ⩾ 0, x_{j} \neq 0$ and $y_{j} = (y_{1 j}, \dots, y_{sj}) ⩾ 0, y_{j} \neq 0$ . We define $X = [x_{1}, \dots, x_{n}]$ as the m × n matrix of inputs and $Y = [y_{1}, \dots$

Tone’s method

Consider the following linear program (multiplier form) weighted BCC model as presented in Tone [14]: $({DWR}_{o}) \max {uy}_{o} - u_{o}$ $\begin{matrix} s.t. {vx}_{o} = 1 \\ uY - vX - 1 u_{o} ⩽ 0 \\ vP ⩽ 0 \end{matrix}$ $uQ ⩽ 0$ $v ⩾ 0, u ⩾ 0,$ where matrices P and Q are associated with weight restrictions. The dual (envelopment form) of the DWR_o model represented in (1) is obtained from the same data which are then used in the following form: $\begin{matrix} ({WR}_{o}) θ_{w}^{BCC *} = \min θ_{w} \\ s.t. X λ - P π ⩽ θ_{w} x_{o} \end{matrix}$ $\begin{matrix} Y λ + Q τ ⩾ y_{o} \\ 1 λ = 1 \\ λ ⩾ 0, π ⩾ 0, τ ⩾ 0, \end{matrix}$ where π and τ are dual variables corresponding to constraints (1.1), (1.2), respectively.

Weighted CCR model and estimation of RTS

Consider the following linear programming problem: $({CWR}_{o}) θ_{w}^{CCR *} = \min θ_{w}$ $\begin{matrix} s.t. X λ - P π ⩽ θ_{w} x_{o} \\ Y λ + Q τ ⩾ y_{o} \\ λ ⩾ 0, π ⩾ 0, τ ⩾ 0 . \end{matrix}$ As can be seen, this model is the same as the “envelopment form” of the DWR_o model in (2) except for the fact that the condition $1 λ = 1$ is omitted. In consequence, the variable u_o, which appears in the “multiplier form” for the DWR_o model in (1), is omitted from the dual (multiplier form) of this CWR_o model. Let $θ_{w}^{CCR *}$ be an optimal value to CWR_o. We define “CWR-efficiency” as

Definition 2 CWR-efficiency

A DMU_o is

Empirical illustration

We will apply our approach to the data set obtained from 14 general hospitals: H1 to H14 (Which is taken from Cooper et al. 2000, chapter 7). Table 1 shows the data of 14 general hospital with two inputs (Doctor and Nurse) and two outputs (Outpatient an Inpatient).

The weight restrictions on inputs and outputs are as follows: $\{\begin{matrix} 0.2 ⩽ \frac{v_{1}}{v_{2}} ⩽ 5, \\ 0.2 ⩽ \frac{u_{1}}{u_{2}} ⩽ 5, \end{matrix}$ where v₁ and v₂ represent the weights for doctor and nurse, respectively. And u₁ and u₂ represent the weights for outpatient and inpatient,

Conclusions

The current study gives an alternative method to classify the returns to scale under weight restrictions for each decision making units. By the addition of weight restrictions, the status of returns to scale (CRS, IRS, and DRS) may suffer a change. Then, it is demonstrated that the region of the most productive scale size (MPSS) will usually be narrowed by this addition. A numerical example is presented to demonstrate the validity of the proposed method. The method, proposed in this work, is

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An alternative approach in the estimation of returns to scale under weight restrictions

Abstract

Introduction

Section snippets

Preliminaries

Tone’s method

Weighted CCR model and estimation of RTS

Empirical illustration

Conclusions

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

International Journal of Management science

Linear Programming and Network Flows

second ed.