On FDH efficiency analysis with interval data

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Abstract

Most of DEA models make an assumption that input and output data are exact without any variation. Cooper et al. [Manage. Sci. 45 (1999) 597] introduced imprecise data envelopment analysis (IDEA). When some inputs and outputs are unknown decision variables such as bounded data, ordinal data, and ratio bounded data, the DEA model becomes a non-linear programming problem. In this paper we consider FDH model with imprecise data, first we introduce linear form of FDH model [J. Product. Anal. 16 (2001) 129] then according to [Eur. J. Oper. Res. 140 (2002) 24] we follow efficiency analysis.

Introduction

Data envelopment analysis is a mathematical programming approach to evaluate the relative efficiency of decision making units (DMUs) that use multiple inputs to produce multiple outputs (see [8]). An assumption underlying DEA is that all the data are known exactly. Cooper et al. [6] introduced applications of DEA which data was imprecise. For example, some of data are known only within specified bounds, while other data are known only in terms of ordinal relation.

Cooper et al. [5] investigated IDEA and presented some methods for transforming non-linear model to a linear one. Such problems and their solution technique can be seen in Thompson et al. [12], Charnes et al. [6] and Brockett et al. [3]. In addition Cooper et al. [7] extended the transforming in [5], [6] to the general case. Recently Park [10], [11], Zhu [14], [15], Chen et al. [4] and Despotis et al. [9] investigated IDEA from other perspectives. When we concern with imprecise data, model becomes non-linear and converting the non-linear model to a linear model is important. In this paper we investigate the efficiency analysis in FDH model which the amounts of inputs and outputs are locate within the bounded intervals, then we convert non-linear model to a linear one.

The rest of this paper is organized as follows: In Section 2 we review FDH model. Section 3 is devoted to transforming our problem into a linear model and we discuss on efficiency analysis based on exact data. In Section 4 we present a lower bound and an upper bound for efficiency level of DMUs in the FDH model. In Section 5 we discriminate DMUs according to [9]. An illustrative example is presented in Section 6. Conclusions are given in Section 7.

Section snippets

FDH model and its linear form

Suppose there are n DMUs and the inputs and outputs for DMUj are Xj=(x1j,x2j,…,xmj)t and Yj=(y1j,y2j,…,ysj)t respectively. All Xj∈Rm and Yj∈Rs and Xj>0, Yj>0 for j=1,2,…,n. The input data matrix X and the output data matrix Y can be represented asX=[X1,…,Xj,…,Xn],Y=[Y1,…,Yj,…,Yn],where X is an (m×n) matrix and Y an (s×n) matrix. Let the DMUj to be evaluated on any trial be designated as DMUo where o ranges over 1,2,…,n. First we turn our attention to the FDH model by Tulknes et al. [13]. The

FDH model with interval data

Assume that the inputs and outputs for DMUj, j=1,2,…,n, are not known exactly. Let xij∈[xij,x̄ij] and yrj∈[yrj,ȳrj], where lower and upper bounds are known and positive. According to these assumptions letxij=xij+sij(x̄ij−xij),0⩽sij⩽1,∀i,∀j,yrj=yrj+trj(ȳrj−yrj),0⩽yrj⩽1,∀r,∀j,with these transformations new variables sij and trj enter in our problem. So problem (2.3) is as followsmaxz,s.t.∑mi=1xiovij+(x̄io−xio)siovij=1,∀j,∑sr=1(ȳrj−yro)urj+(ȳrj−yrj)trjurj−(ȳro−yro)trourj−∑mi=1xijvij+(x̄ij−xij

Upper and lower bounds of efficiency values

When we use (3.2) to evaluate the efficiency level of DMUo, inputs and outputs of each DMU are adjusted such that increase efficiency level of DMUo. In fact this model considers the best possible level of inputs and outputs of DMUo and other DMUs in their interval to increase the efficiency level of DMUo. According to (3.2) we can consider upper bound for efficiency index of DMUo as followsmaxz,s.t.∑mi=1xiovij=1,∀j,z⩽∑mi=1xiovio−∑sr=1(ȳro−yro)uro,z⩽∑mi=1x̄ijvij−∑sr=1(ȳrj−ȳro)urj,j≠o,urj,vij

Discrimination and ranking of DMUs

According to the previous section we can discriminate DMUs under FDH model with interval data as followsE1={j∣zj=1},E2={j∣zj<1,z∗j=1},E3={j∣z∗j<1},where j∈{1,2,…,n}. E1 contains whole DMUs which in each level of inputs and outputs are efficient. E2 contains all DMUs which are efficient in the best level of inputs and outputs but there are some combinations of inputs and outputs which units of this set lose their efficiency and according to the differences of upper and lower bounds of efficiency

Numerical example

Consider Table 1. In this table we have six DMUs with two inputs and two outputs. The interval date of inputs and outputs are given below.

Let ε=10−8 we apply , to compute z∗j and zj for DMUj, j=1,…,6. The results are shown in Table 2. Also the inputs and the outputs level for each DMU (when we evaluate DMU1) are represented in Table 2.

We set DMU1 in E1 and it is efficient with any combination of its inputs and outputs. This DMU dominate other DMUs only with its first input,whereas DMU4 which

Conclusion

In this paper we investigated the FDH model with interval data. To deal with these data, we used some transformations to convert the non-linear model to a linear one. To compute the interval that efficiency varies in it we presented two linear models and these models use exact data. Using by these models we obtained lower and upper bounds for efficiency level, also we can use of these models to rank DMUs. Our models are flexible, that is, we can determine the intended level of inputs and

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