Pitam Singh

A recent survey on computational techniques for solving singularly perturbed boundary value problems

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of ideas and methods of singular perturbation theory. In continuation of a survey performed earlier, this paper limits its coverage to some standard numerical methods developed by numerous researchers between 2000 and 2005. A summary of the results of some recent methods is presented and this leads to conclusions and recommendations regarding methods to use on singular perturbation problems. Because of space constraints, we considered one-dimensional singularly perturbed boundary value problems only.

A fuzzy based branch and bound approach for multi-objective linear fractional (MOLF) optimization problems

Abstract In the present study, a new fuzzy based branch-bound approach is attempted for solving multi-objective linear fractional (MOLF) optimization problems. The original MOLF optimization problem is converted into equivalent fuzzy multi-objective linear fractional (FMOLF) optimization problem. Then branch and bound techniques is applied on FMOLF optimization problem. The feasible space of FMOLF optimization problem is bounded by triangular simplex space. The weak duality theorem is used to generate the bound for each partition and feasibility conditions are applied to neglect one of the partition in each step. After finite number of steps, a fuzzy efficient (Pareto-optimal) solution is obtained for FMOLF optimization problem which is also efficient (Pareto-optimal) solution of the original MOLF optimization problem. Some theoretical validations are also established for the proposed approach on FMOLF optimization problem. For the efficiency of proposed approach, it has been performed on two numerical applications. The method is coded in Matlab (2016). The results are compared with earlier reported methods.

Weakly fuzzy efficient conditions for multiobjective fractional programming problem

It is the purpose of this article to establish some weakly fuzzy efficient conditions for multiobjective linear fractional programming problem MOFP under generalized convexity. Osuna et al., [27] derived some results regarding the optimality and duality properties for multiobjective fractional programming in classical sense. In this article, an equivalent fuzzy multiobjective fractional programming problem is formulated by using fuzzy goal and tolerance limits for each objective functions. Some weakly efficient criteria for MOFP are established with fuzzy membership functions parallel to Osuna et al., [27].

Fuzzy efficient and pareto — Optimal solution for multiobjective linear plus linear fractional programming problem

Many practical optimization problems usually have several conflicting objectives. In those multiobjective optimization, no solution optimizing all the objective functions simultaneously exists in general. Instead, pareto — optimal solutions, which are efficient in terms of all objective functions, are introduced. In general we have many optimal solutions. Therefore we need to decide a final solutions among pareto — optimal solutions taking in to account the balance among objective functions. In this paper we find fuzzy efficient and pareto — optimal solution to the multiobjective linear plus linear fractional programming problem and show that, in the case that, when any goal is fully achieved, then fuzzy efficient solution may or may not be pareto — optimal solution and therefore we propose a procedure to obtain fuzzy efficient solution which is pareto — optimal also and review some results. In the proposed approach each objective function is transformed into linear functions by using Taylors theorem. Then the MOLLFP is changed into equivalent multiobjective linear programming problem (MOLP) and then find fuzzy efficient and pareto — optimal solution in finite number of steps. Efficiency of proposed method is verified by numerical examples. To explore the potential use of the proposed method, three numerical examples are solved. AMS 2000 Subject Classification: 90C29, 90C32

Duality-based branch–bound computational algorithm for sum-of-linear-fractional multi-objective optimization problem

Optimizing the sum-of-fractional functions under the bounded feasible space is a very difficult optimization problem in the research area of nonlinear optimization. All the existing solution methods in the literature are developed to find the solution of single-objective sum-of-fractional optimization problems only. Sum-of-fractional multi-objective optimization problem is not attempted to solve much by the researchers even when the fractional functions are linear. In the present article, a duality-based branch and bound computational algorithm is proposed to find a global efficient (non-dominated) solution for the sum-of-linear-fractional multi-objective optimization (SOLF-MOP) problem. Charnes–Cooper transformation technique is applied to convert the original problem into non-fractional optimization problem, and equivalence is shown between the original SOLF-MOP and non-fractional MOP. After that, weighted sum method is applied to transform MOP into a single-objective problem. The Lagrange weak duality theorem is used to develop the proposed algorithm. This algorithm is programmed in MATLAB (2016b), and three numerical illustrations are done for the systematic implementation. The non-dominance of obtained solutions is shown by comparison with the existing algorithm and by taking some feasible solution points from the feasible space in the neighborhood of obtained global efficient solution. This shows the superiority of the developed method.

Multimodal data modeling for efficiency assessment of social priority based urban bus route transportation system using GIS and data envelopment analysis

Multimodal data modeling is fast growing area of research. It may used to combine the information from the different sources. The research interest in multimodal data modeling is increasingly attracting the attention in the field of transportation planning. In this study, multi-modal data is used to assess and design a socially efficient public transport bus route plan for the Allahabad city of Uttar Pradesh state, India. Data envelopment analysis (DEA) method is used for the efficiency assessment of existing 24 public transport bus routes by taking access point to locations of social facilities like as hospitals, shopping malls, colleges, coaching centers, schools, banks and the population, near to the particular route. Geographical Information System (GIS) technology is used for multimodal data modeling to design new more socially efficient routes for the existing roads of the city. DEAP Solver software is used for the evaluation of efficiency and rank for social priority routes and route number 15 and 24 are relative efficient route among the existing 24 routes. Finally, the social efficiency of existing public bus transport routes and newly designed routes are compared. We suggested ways to improve the performance of bus routes based on the social perspectives using multimodal data.

Fuzzy efficient and Pareto-optimal solution for multi-objective linear fractional programming problems

Many practical optimisation problems usually have several conflicting objectives. In these multi-objective optimisation problems, solution optimising all the objective functions simultaneously does not exist, in general. Instead, Pareto-optimal solutions, which are efficient in terms of all objective functions, are introduced. Nevertheless, many optimal solutions exist. A final solution among Pareto-optimal solutions is to be selected based on the balance among objective functions. In this paper, we find fuzzy efficient and Pareto-optimal solution to the multi-objective linear fractional programming problem (MOLFP). It has shown that when any fuzzy goal is fully achieved, the fuzzy efficient solution may or may not be Pareto-optimal. Therefore, a procedure is proposed to obtain fuzzy efficient solution which is also Pareto-optimal. The efficiency of proposed method is verified by numerical examples and a practical application in production planning.