Mixed input and output orientations of Data Envelopment Analysis with Linear Fractional Programming and Least Distance Measures
Statistics, Optimization and Information Computing
Data Envelopment Analysis (DEA) is an optimization technique to evaluate the efficiency of Decision-Making Units (DMU's) together with multiple inputs and multiple outputs on the strength of weighted input and output ratios, where as Linear fractional programming is used to obtain DEA frontier. The efficiency scores of DMU obtained through either input orientation or output orientation DEA model will provide only local optimum solution. However, the mixed orientation of input and output
... and output variables will provide the global optimal solution for getting the efficient DMUs in DEA. This study has proposed the relationships of a mixed orientation of input and output variables using fractional linear programming along with Least-Distance Measure (LDM). Both constant returns to scale (CRS) and variable returns to scale (VRS) are considered for the comparative study. . 327 more publications were due to [22, 24, 23, 27, 14] . The transformation through modification of the feasible set in order to convert the fractional program into a linear program was mentioned by Chanes and Cooper  . A large number of problems in the management science directly or indirectly depend on the fractional programs. A more recent comprehensive survey on DEA studies are found in  . Charnes, Cooper, and Rhodes  introduced a non-parametric optimization technique for evaluates the efficiency of DMU known as DEA which is a special case of FPP. DEA is a very flexible method of comparing the efficiency performance of various decision-making units utilizing the multiple inputs to produce multiple outputs  Initially, DEA maximizing the ratio of virtual output (a linear combination of outputs) by virtual input (a linear combination of inputs)  . The original CCR model was applicable only to technologies characterized by constant returns to scale. The transformations of linear fractional DEA into LPP model was proposed by  .The conventional DEA modeling where estimating efficiency values, only for the specific DMU. However an optimal decision is possible only when there is the full information about slacks variables also. The weakness of conventional DEA-model was improved by Joe Zhu see  . The maximum efficient score is unity in standard DEA-model. If efficient score is unity, then we can conclude that the specific DMU is full efficient. But in case of small number of DMUs, the efficient set can contain almost all DMUs. In such cases, for the further classification of efficient DMUs the super efficiency model is very useful see  . The super efficiency measure is important for ranking of efficient DMU's see  . Then slack based measure is applied in case of super efficiency see  . More recently, DMUs are assumed as the black box process,where the inefficiency of inefficient DMUs can be identified by dividing into stages, for measuring the efficiency as whole, as well as for each stage independently by using conventional DEA methodology. For two stages see  , for three stages see  and for decomposition of efficiency into networking DEA-model see  . DEA has become very popular with more than 40,000 publications in last four decades by over 2,500 authors .Empirical analysis of the performance of universities typically takes the form of estimating cost functions with the focus on economies of size and scope or an analysis of efficiency using data envelopment analysis  .Duality has only established a link between multiplier and envelopment DEA-models  .Efficiency analysis is performed not only to estimate the current level of efficiency but also provided information about how to remove inefficiencies  . Least-Distance Measure (LDM) is a technique which provides the efficiency measure as well as relevant benchmarking information. The LDM define the strongly efficient set first and then calculate the least distance benchmark from the evaluated DMU  . In the aforesaid beck drop, this paper is an attempt to utilize the duality concept in FPP for solving the DEAmodels, and explore the duality in DEA for evaluation of productive efficiencies of an organization is characterized by CRS and VRS. In addition, the attempt has been made to explore the concept of input-oriented and outputoriented models for assessing the productive efficiency by mixed-orientation of inputs and outputs in DEA and find the global optimal solution by using LDM. The DEA-model with mixed-orientation has important practical implications which are discussed in this paper which is structured as follows. Section first introduces linear fractional programming (LFP), transformation of LFP into LPP by exploring Chanes,Cooper-transformation, affine transformations and duality in LFP. Section second explains the development of DEA for evaluation of technical efficiencies of organization which is characterized by constant and variable returns to scales with different orientations. Section third includes BCC DEA model with mixed-orientation. Section four deals with the global optimal solution of DEA-model by using LDM approach. The final section discusses data, results and conclusion. Linear Fractional Programming The linear fractional programming is the optimization technique dealing with the ratio of two linear functions (or a ratio of two linear programming problems) subject to a set of linear inequalities and non-negativity constraints on the variables. In 1956 linear fractional programming was developed by J.R. Isbell and W.H. Marlow, the problem is solved directly beginning with a basic feasible solution and showing the conditions under which the solution can be improved. The technique followed is similar to the simplex method of linear programming problem (LLP).