Performance measurement with undesirable outputs is an important subject in production processes. There are papers in which works can be divided into two groups. First group develops production models which undesirable outputs are considered as inputs and in second one undesirable outputs are modeled as outputs. We provide a comparison between these two groups and we axiomatically show that production technology with weak disposability assumption is complete and correct technology. Manuscript profile

The correlated models are introduced in this article regarding revenue efficiency and profit efficiency in dynamic network production systems. The proposed models are not only applicable in measuring efficiency of divisional, periodical and overall efficiencies, but recognizing the exact sources of inefficiency with respect to revenue and profit efficiencies. Two numerical examples, consisting of twoperiods and , are presented to illustrate these proposed models. Manuscript profile

Trade-off reviews the rate of marginal substitution of inputs and outputs onto the efficient frontier. On the other hand, marginal rates of substitution (MRS) are important quantities for analysts and managers. In this paper, by changing the method of Asmild (2004), an algorithm is found that it can be used to calculate the marginal rate of the free disposal hull (FDH) model. Manuscript profile

Regular NDEA models for calculating the efficiency of DMUs with two-stage construction requires the data set to be deterministic. In the real world, however, there are many observations that have stochastic behavior. This paper proposes a network DEA with two-stage structures models by the attention to stochastic data. The stochastic two-stage network DEA models are formulated based on the P-models of chance constrained programming and leader-follower concepts. Manuscript profile

In many situations, the performance and the changes of outputs related to dynamic systems should be estimated while the convexity property is relaxed. Accordingly, first, a dynamic free disposal hull (FDH) model is proposed in this paper to address the efficiency of processes in multiple period of time while the convexity assumption is unsatisfied. Also, two problems, including a mixed integer linear programming and a linear programming model are provided to compute the dynamic FDH model that is a mixed integer non-linear programming problem. Then the changes of multi-period outputs are dealt with for changes of inputs related to several periods using the proposed inverse dynamic FDH model while the efficiency levels are preserved. A case study of gas industry is, moreover, presented to demonstrate the introduced models. The results show the proposed technique is useful to analyze the performance and to estimate outputs in dynamic processes without including convexity. Manuscript profile

In this research, two-stage networks are considered in which there are common resources between both stages. Also, there are links between the first and second stages that are undesirable for the first stage. Two models are rendered, the first is a model for calculating the efficiency of the mentioned two-stage processes under the non-convex technology and the second model is an inverse model for calculating output values through input changes. The changes are such that efficiency values remain constant. Actually, at first, a non-radial network free disposal hull (FDH) model is planned to evaluate the entire and stage efficiencies of two-stage processes with shared resources. The introduced framework is a mixed integer nonlinear programming plan and is computed using a heuristic approach. Then, an inverse two-stage FDH approach is presented to determine outputs related to two stages for perturbing inputs of each stage while the efficiency values remain unchanged. To clarify the proposed models, an application from the literature is used. Manuscript profile