| The problem of how to determine a fair evaluation of the efficiency of a decision making unit (DMU), such as a bank or hospital, relative to its peers, has been the focus of an enormous volume of research over many decades. One important methodology developed more than three decades ago, Data Envelopment Analysis (DEA), is an effective non-parametric tool for evaluating the relative efficiencies of decision making units when multiple inputs and outputs are present. It has been widely used in the evaluation and improvement of those DMUs. Literally thousands of journal articles and books have been devoted to this topic.One particular area of recent research is the measurement of efficiency when the DMU appears in the form of multiple stages. Many important real world problems assume this form; supply chains are an important application area. Two-stage network DEA is an important subset of multistage DEA research, and a substantial volume of research has focused on this. Even within this restricted class of multistage processes, there are different forms of the two-stage situation. One of these, the closed serial process, is one where all outputs from the first stage are intermediate in the sense that their sole function is to act as the input set to the second stage. Nothing leaves the system at this point. An important problem is how to decompose the efficiency of the multistage DMU into efficiency measures for the individual components or stages. Some of the important research in this area applies game theory principles to determine an aggregate or overall efficiency measure for the DMU as some form of average of the efficiency measures for the individual stages. Both geometric mean and arithmetic mean approaches have been developed.Another important research thrust in recent years has involved the evaluation of efficiency when DMUs are non-homogeneous. This phenomenon can take many forms such as different output mixes or different input mixes from one DMU to another. Generally, the presence of such non-homogeneity is addressed by viewing the DMU as a business unit consisting of a set on mutually exclusive subunits. This being the case, the modelling of efficiency implies splitting inputs across those subunits.The chapters in this thesis involve problems that concern either the issue of non-homogeneous DMUs or network/multistage problems.The main contributions of this thesis are as follows:(1)A study of the Efficiency of Economic Ecosystems in 31 Regions of China. A balance between environmental regulation and economic prosperity has become a major issue of concern to attain a sustainable society in China. This study proposes the application of DEA for measuring the efficiencies of the ecological systems in various regions of that country. A two-stage network DEA model is presented. The proposed approach differs from most of the previous ecological systems models in that we view it in a two stage setting; the first stage models the ecological system itself, while the second stage models water recycling as a feedback process, and the treatment of other undesirable outputs coming from the first stage and entering as inputs to stage two (decontamination system). There, polluting gases and water are split into two parts; one part is treated, while the other is discharged. The model considers two major desirable outputs from the first stage, namely the Population and Gross Region Product by expenditure (GRP), as well as undesirable variables such as consumed water, and pollutants in the form of nitrogen oxide, sulfur dioxide and soot, etc. At the same time, these undesirable outputs from the first stage are inputs to the second stage. As well, recycled water is fed back into stage 1. Thus, intermediate variables such as consumed water and waste gas emission simultaneously play dual roles of both outputs and inputs in the ecological system.(2)A study on Modelling Efficiency in the Presence of Multiple Partial Input to Output Processes. In the original DEA model, it is assumed that in a multiple input, multiple output setting, all members of the input bundle affect the entire output bundle. There are many situations, however, where this assumption does not hold.. In a manufacturing setting, for example, packaging resources (inputs) only influence the production of those products that require packaging. This is referred to as "partial input to output interactions", where the DEA model is based on the view of a DMU as a business unit consisting of a set of independent subunits. In that situation, the overall efficiency of the DMU can be defined as a weighted average of the efficiencies of those subunits. The current paper presents an extension to that methodology to allow for efficiency measurement in situations where there exist multiple procedures or processes for generating certain output bundles. The proposed model is then applied to the problem of evaluating the efficiencies of a set of steel fabrication plants.(3)A study involving DEA Models for Non-Homogeneous DMUs with Different Input Configurations. This work is somewhat related to the previous paper in that partial input to output impacts are involved. In earlier research by other authors the issue of non-homogeneity on the output side was investigated. Specifically, not all DMUs possess the same output profile. That research examined a set of steel fabrication plants where not all plants produced the same set of products/outputs. In the current research we investigate non-homogeneity on the input side. Such can occur in manufacturing plants, for example, when the output bundle can be produced using different mixes of machines, robots and laborers. Thus, we can have an input configuration existing in one DMU that is different from the configuration in another DMU. As a practical application of this phenomenon, we examine the measurement of efficiencies of a set of provinces in China. There, all provinces have the same common set of outputs, but on the input side this commonality is missing. While all provinces have water, capital investment and natural resources, the latter of these (natural resources) takes several different forms; coal, natural gas and petroleum. However, not all provinces have the same mix of these resources. This means that one cannot directly apply the conventional DEA methodology. We develop herein a DEA type of methodology to evaluate this non-homogeneous setting. This evaluation provides important insights into not only the overall performance of each province, but as well provides measures of the efficiency of the various configurations of the three natural resources.(4)A study on Measuring Efficiency with Products, By-Products and Parent-Offspring Relations:A Conditional Two-Stage DEA Model. This study is somewhat related to paper 2 above in that there are aspects involving partial input to output impacts as well as multiple processes. Missing from that earlier work is consideration of the presence of outputs in the form of by-products, giving rise to a parent-offspring phenomenon. One of the modelling complications there is that the parent assumes two different roles; as an input affecting the offspring, while at the same time being the dominant output. Another complication is that in the presence of multiple processes, by-products often arise out of only a subset of those processes. We develop a DEA-type of methodology to handle partial input to output impacts in the presence of by-products.
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