SOME INVESTIGATIONS IN THE AREA OF OPTIMIZATION AND IMPLICATION IN UNCERTAIN ENVIRONMENT

Abstract

In real-life problems such as incorporate or in industry, decision making is a continuous process. The experts and the decision-makers (DMs), usually, have to su er with uncertainty as well as with hesitation, due to the complexity of the situations. The main reasons behind these complexities are lack of good communications with all involved persons, error in data, understanding of markets, unawareness of customers, etc. So, the prediction of the parameters is a complex and challenging task. The classical methods encounter great di culty in dealing with uncertainty and complexity involved in such situations. In general, the parameters of an optimization problem are considered as crisp numbers. These crisp values are determined from past occurrences which are very uncertain since the systems environment keep on changing. Therefore, some degree of uncertainty exists in such a determination. This led to the development of the fuzzy set (FS) theory by Zadeh [194]. In order to handle the insu cient information, the fuzzy approach is used to model the problem and evaluate the optimal solution. FS theory has been shown to be a useful tool to handle the situations in which the data are imprecise by attributing a degree to which a certain object belongs to a set. An FS is a generalization of an ordinary set in that it allows the degree of membership for each element to range over the unit interval [0, 1]. Thus, the membership function of an FS maps each element of the universe of discourse to its range space, which, in most cases, is assumed to be the unit interval. Using an FS approach, quantities are represented by FSs. The membership functions represent the uncertainties involved in various parameters of the problem. During the last decades, FS theory played an important role in modeling uncertain and optimization problems. Zimmermann [203] showed that the solutions of fuzzy linear programming problems (FLPPs) are always e cient. Since the FS theory came into existence, many extensions of FSs also appeared over time , e.g., L-FSs proposed by Goguen [80], interval-valued fuzzy set (IVFS) proposed by Gorzalczany [81] represents the degree of membership of an element by an interval rather than exact numerical value, intuitionistic fuzzy set (IFS) proposed by Atanassov [11] etc. One among these extensions is IFS which is playing an important role in decision making under i ii uncertainty and gained popularity in recent years. It helps more adequately to represent situations where DMs abstain from expressing their assessments. In this way, IFSs provide a richer tool to grasp impression and ambiguity than the conventional FSs. These characteristics of IFSs led to the extension of optimization methods in an intuitionistic fuzzy environment (IFE). An application of IFSs to optimization problems is introduced by Angelov [9]. His technique is based on maximizing the degree of membership, minimizing the degree of non-membership and the crisp model is formulated using the IF aggregation operator. In decision making, one chooses the best alternative from the given set of feasible alternatives. There exist several processes in literature but there are mainly four stages required to choose the best alternative: (i) Evaluate the set of feasible alternatives from the given information. (ii) Determine the weight vector corresponding to alternatives or attributes which depend on DM. (iii) Aggregate alternatives by taking the weight vector given by DM. (iv) Rank the alternatives in the order of preferences and select the best one. During last decades, IFS theory played an important role in modeling uncertain and vague systems, received much attention from the researchers and meaningful results were obtained in the eld of decision-making problems [138], pattern recognition [54, 143] to name a few. There are several information measures in IFE, such as divergence measures, similarity measures, dissimilarity measures, and distance measures. They model uncertain and vague information. The inclusion between two IFSs can be measured by the concept of inclusion measure [79] and the commonality between two IFSs can be measured by the concept of similarity measure [95]. In fuzzy logic, the fuzzy implication is equally important from both the theoretical and practical points of view. From the theoretical point of view, the development of algebra is done and their properties are studied. From the practical point of view, the fuzzy implication is used to study approximate reasoning and network problems, etc. (see [19, 106]). One among the several extensions of FS is the IVFS. It has become very popular from both the theoretical and practical aspects. It has become one of the most important operators in logic [174]. The arithmetic operators in IVFS theory [55] and one can nd theoretical articles concerned with di erent classes of interval-valued logical connectives, like, interval-valued fuzzy negations [26], interval-valued t-norms [56, 174], interval-valued fuzzy uninorms [57], interval-valued fuzzy implications [5, 28, 111]. IF t-norms and t-conorms are noted in [59]. The expression, construction, classi cation and several properties with applications of intuitionistic and interval-valued fuzzy implications are given in [33] and [45]. IFIs [33, 45] and IF relations [146] are studied. The main objectives of the thesis are as follows: iii (i) Modeling and analysis of optimization problems in IFE and development of algorithms for solving such problems. (ii) Analysis of duality theory in IFE. (iii) Analysis and development of algorithms for selecting the best alternative from the given set of feasible alternatives in decision-making problems in IFE. (iv) Algebraic analysis of implication operators in IFE and their uses for solving distributive equations and Boolean-like laws. The thesis is organized into eight chapters. The chapter-wise summary of the thesis is as follows: Chapter 1 is introductory in nature. In this chapter, basic de nitions of FS and IFS, various types of fuzzy and IF numbers, and their mathematical operations are introduced. A ranking function is introduced. Ranking function transforms a fuzzy or IF number into an equivalent real number. Also, basic de nitions and axioms of implication operator, negation, t-norm, t-conorm in fuzzy and IF environments are introduced. It also presents a brief review of the research work done in the eld of fuzzy and IF optimizations and implications. In Chapter 2, the product of unrestricted LR-type IFNs is proposed. Then with the help of the proposed product, an algorithm is proposed to nd the optimal solutions of unrestricted LR-type IFLPPs. A test example is given to support the proposed method and investigated the applicability of existing approaches. In Chapter 3, we introduce a pair of primal-dual LPPs in IFE and prove duality results by using an aspiration level approach in which membership and non-membership functions are taken in the form of reference functions. Since the fuzzy environment and IFE cause the duality gap, we propose to investigate the impact of membership function governed by reference function on duality gap. Also, the duality gap obtained by the approach has been compared with the duality gap obtained by existing approaches. In Chapter 4, the formulation of the multi-objective optimization problem (MOOP), accuracy index and value function in IFE are introduced. For resolving the mutual con icting nature of objectives in MOOP in IFE, we introduce the membership and non-membership functions governed by reference function which do not depend on the upper and lower levels of acceptability. An e cient algorithm is developed for solving MOOP in IFE from di erent viewpoints, viz., optimistic, pessimistic and mixed. The optimal solution obtained by the proposed approach is compared with the solutions obtained by existing approaches. In Chapter 5, the information measures are introduced in IFE to measure the uncertainty and hesitancy. We introduce and study the continuity of considered measures. Next, we prove iv some results that can be used to generate measures for FSs as well as for IFSs and we also prove some approaches to construct point measures from set measures in IFE. We de ne weight set for one and many preference orders of alternatives and investigate the properties based on these ordering. Based on the weight set, we develop the model for nding the uncertain weights corresponding to attribute. Also, we develop the model to nd attribute weights in a certain environment by using attribute weights in an uncertain environment. An algorithm is developed for choosing the best alternative according to the preference orders of alternatives. In Chapter 6, a new type of implication on L, known as the residual implication, is derived from powers of continuous t-norm T and satis es certain properties of residual implications by imposing some extra conditions. Moreover, some additional important properties are studied and analyzed. The solutions of Boolean-like laws in IIT are obtained. In Chapter 7, a new class of IFIs known as (fI; !)-implications is introduced which is a generalized form of Yager's f-implications in IFE. Basic properties of these implications are discussed in detail. The distributive equations II(T (u; v);w) = S(II(u;w); II(v;w)) and II(u; T1(v;w)) = T2(II(u; v); II(u;w)) over t-representable t-norms and t-conorms generated from nilpotent and strict t-norms in IFE are discussed. Finally, in Chapter 8, conclusions are drawn based on the present study and future research work is suggested in this direction.