DEVELOPING SOLUTION METHODOLOGIES FOR DIFFERENT OPTIMIZATION PROBLEMS UNDER UNCERTAINTY WITH APPLICATIONS

Abstract

The thesis is devoted to the mathematical formulation and description of well-performing solution techniques along-with the practical applications of different optimization problems under uncertain conditions. The work includes linear programming problems, quadratic optimization problems, multi-objective transportation problems, and linear fractional programming problems under imprecise environments. Chapter 1 describes the introduction of the whole work. The brief description of each chapter is presented as follows: Chapter 2 deals with the study of intuitionistic fuzzy linear programming problems having mixed constraints and unrestricted decisions variables with all the parameters and variables taken to be triangular intuitionistic fuzzy numbers. A solution algorithm is then developed to solve the problem by employing the accuracy function and (α, β)-cut based ranking technique. A numerical example is demonstrated to clarify the steps of the proposed approach. Additionally, an application of the proposed model to solve a production planning problem is discussed and then solved. Finally, a comparison is drawn between various existing approaches and the proposed algorithm. The objective of Chapter 3 is to solve a class of quadratic programming problems under intuitionistic fuzzy environment. A numerical approach is devised to optimize the problem in which the problem is split into two sub-problems to obtain the lower and upper bound for the objective function value. A duality based methodology is developed to evaluate the upper and lower bounds of the problem. In addition, a modified direct method is also described in alternative to the dual approach. The significance of the modified direct technique is that it considerably reduces the number of variables as well as the constraints resulting into a more efficient algorithm than that of duality approach. Finally, a numerical illustration is exemplified using the proposed method. In Chapter 4, a fully intuitionistic fuzzy multi-objective linear fractional programming problem is discussed. A hybrid solution technique involving the concept of goal programming, fuzzy-based linearization, and a membership function is developed to provide the efficient solution to the problem. The original problem is converted into a crisp linear fractional optimization problem using the weighted goal programming approach. Further, the variable transformation is employed for the under- and the over-deviational variables of the goal programming model to linearize all the fractions involved in the problem. In last, an equivalent deterministic linear optimization model is solved to obtain the solution of the original problem. A numerical example is also illustrated to give a clear understanding of the steps involved in the proposed algorithm. Moreover, a real-life application in the E-education sector is formulated and solved followed by a comparative analysis. Chapter 5 is devoted to study the multiple objective linear fractional pro gramming problems having all the parameters and decision variables as intuitionistic fuzzy numbers. A generalized intuitionistic fuzzy programming and variable transformations based algorithm is proposed to solve the problem. To handle the intuitionistic fuzzy constraints associated with each objective, the linear and exponential membership/non-membership functions are used under normal, optimistic, pessimistic, and mixed approaches. Further, various theorems have also been proved to establish the equivalence between the original problem and its various crisp counterparts. The steps of the solution methodology are elaborated by solving a practical application of the problem in production process of the textile industry. Finally, a comparison is drawn among various approaches. Chapter 6 provides a solution approach to handle the fully interval-valued intuitionistic fuzzy multi-objective balanced transportation problems. The proposed methodology is primarily based on goal programming technique to solve the multiobjective optimization problems. The interval-valued intuitionistic fuzzy constraints related with each objective are tackled using linear, exponential and hyperbolic membership/non-membership functions. A numerical example is also demonstrated to clarify the step-wise procedure of the proposed algorithm. Finally, a comparison is established among different membership functions. In Chapter 7, the notion of LR-type interval-valued intuitionistic fuzzy numbers (IVIFNs) is introduced and the basic arithmetic operations are developed between them. A lexicographic criteria is proposed to rank the LR-type IVIFNs. Then, a linear programming problem having both equality as well as inequality type constraints with all the parameters as LR-type IVIFNs and unrestricted decision variables is investigated. A lexicographic ranking-based method is developed to solve the problem. The theorems have also been proved at relevant places to support the proposed algorithm. Further, a numerical illustration is exhibited using the introduced technique. In addition, a practical application in production planning is formulated and then solved using the developed methodology. The aim of Chapter 8 is to investigate the subtraction and division operations over interval-valued intuitionistic fuzzy values and sets. The complete expressions for the subtraction and division operations over any two arbitrary interval-valued intuitionistic fuzzy values/sets are derived by considering the Hamming distance between them. An equivalent linear programming problem is constructed and further solved to obtain the operations. Additionally, using the developed subtraction and division operations on interval-valued intuitionistic fuzzy values and sets, various fundamental properties and relationships are extensively examined and proved.