Handling Optimization Under Uncertainty Using Intuitionistic Fuzzy-Logic-Based Expected Value Model

Handling Optimization Under Uncertainty Using Intuitionistic Fuzzy-Logic-Based Expected Value Model

Nagajyothi Virivinti (Indian Institute of Technology Hyderabad, India) and Kishalay Mitra (Indian Institute of Technology Hyderabad, India)
DOI: 10.4018/978-1-5225-2990-3.ch032
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Abstract

Uncertainty in parameters during deterministic optimization studies can have large impact on the outcome of the optimization result. It is pragmatic that these parameters are uncertain as they have direct link with real life scenarios, e.g. fuel price appearing as a parameter in objective function or constraints. However, their variability is ignored while solving the problem in a deterministic optimization framework. While mitigating the above mentioned scenario, it is, therefore, necessary to investigate the development of uncertainty handling techniques for a realistic optimization problem. In this work, we propose intuitionistic fuzzy expected value model (IFEVM), which assumes uncertain parameters as intuitionistic fuzzy variables and derives the solution out of an equivalent transformed deterministic formulation while defining the expected values of the objective functions and constraints. Intuitionistic fuzzy parameters can be regarded as a superset of the conventional fuzzy set where the aspect of non-determinacy of a fuzzy member to a set is additionally taken into account. The proposed IFEVM technique has been applied on two examples: first, with the Binh-korn's multi-objective test function where uncertain parameters are linearly related and next with a real life case study of industrial grinding operation having multiple numbers of non-linearly related uncertain parameters. The technique has been further applied to these case studies considering three different levels of risk scenarios e.g. optimistic, pessimistic and intermediate approaches. The IFEVM technique is fairly generic and advantageous, can be applied to any kind of system for handling uncertainty in parameters.
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Introduction

During the past century, the field of optimization, which is not only a potent methodology of modeling and problem solving but also has found a broad range of applications in industry, engineering, management science and operation research, has been developed extensively and converted into a mature area of research. Optimization refers to the analysis of a problem, in which a single solution (for single objective optimization) or a set of trade-off solutions (for multi-objective optimization), called decision variables, must be chosen from a range of feasible solutions. Solutions are compared for their supremacy based on certain criteria met, known as the objective function, and the feasible solutions are the solutions that satisfy all the constraints defined in the optimization problem. One of the main criticisms of the optimization is, however, that it often produces the solutions that are not robust to uncertainties in the parameters or constants present in the optimization problem. Traditionally, these uncertainties can be handled by over designing or over estimating the parameters or by replacing the uncertain parameters by their nominal values. These methods are either costly or lead to suboptimal, sometime infeasible solutions. Investigations on development of uncertainty handling techniques while solving optimization problems are, therefore, necessitated. Based on the nature of parameters or constants, e.g. deterministic or stochastic, optimization methods can be classified into two categories. First one is the deterministic optimization problem where parameters are deterministic in nature and in this case, the optimization problem can be solved with deterministic or certain values of the parameters. For the second case, the case of optimization under uncertainty (OUU), all or some of the parameters are assumed as non-deterministic or stochastic in nature and the uncertain optimization problem can be solved by first associating a distribution based information e.g. probability with it and next converting it into an equivalent deterministic optimization problem to find the solution for a particular realization of uncertainty. There are several methods available in the literature which performs the above mentioned transformation, namely, two stage stochastic programming (TSSP), chance constrained programming (CCP), fuzzy mathematical programming (FMP), stochastic programming (SP), and expected value model (EVM) to name a few (Liu, 2009 and Sahinidis, 2004).

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