Equivalent Single Objective Research Articles

Application of entropy measurement technique in grey based Taguchi method for solution of correlated multiple response optimization problems: A case study in welding

In the present work, an attempt has been made to apply an efficient technique, in order to solve correlated multiple response optimization problems, in the field of submerged arc welding. The traditional grey based Taguchi approach has been extended to tackle correlated multi-objective optimization problems. The Taguchi optimization technique is based on the assumption that the quality indices (i.e. responses) are independent or uncorrelated. But, in practical cases, the assumption may not be valid always. However, the common trend in the solution of multi-objective optimization problems is to initially convert these multi-objectives into an equivalent single objective function. While deriving this equivalent objective function, different priority weights are assigned to different responses, according to their relative importance. But, there is no specific guideline for assigning these response weights. In this context, the present study aims to apply the entropy measurement technique in order to calculate the relative response weights from the analysis of entropy of the entire process. Principal Component Analysis (PCA) has been adopted to eliminate correlation that exists among the responses and to convert correlated responses into uncorrelated and independent quality indices, called principal components. These have been accumulated to calculate the overall grey relational grade, using the theory of grey relational analysis. Finally, the grey based Taguchi method has been used to derive an optimal process environment capable of producing the desired weld quality. The previously mentioned method has been applied to optimize bead geometry parameters of submerged arc bead-on-plate weldment. The paper highlights a detailed methodology of the proposed technique and its effectiveness.

Multiobjective, Multiconstraint Service Restoration of Electric Power Distribution System With Priority Customers

In this paper, a technique based on nondominated sorting genetic algorithm-II (NSGA-II) is presented for solving the service restoration problem in an electric power distribution system. Due to the presence of various conflicting objective functions and constraints, the service restoration task is a multiobjective, multiconstraint optimization problem. In contrast to the conventional genetic-algorithm (GA)-based approach, this approach does not require weighting factors for the conversion of such a multiobjective optimization problem into an equivalent single objective function optimization problem. In this work, various practical distribution system operation issues, such as the presence of priority customers, presence of remotely controlled, as well as manually controlled switches, etc. have also been considered. Based on the simulation results on four different distribution systems, the performance of the NSGA-II-based scheme has been found to be significantly better than that of a conventional GA-based method. Besides, to reduce the software runtime of the NSGA-II algorithm, a faster version of NSGA-II has also been implemented.

Two storage inventory model in a mixed environment

Multi-item inventory models with two storage facility and bulk release pattern are developed with linearly time dependent demand in a finite time horizon under crisp, stochastic and fuzzy-stochastic environments. Here different inventory parameters--holding costs, ordering costs, purchase costs, etc.--are assumed as probabilistic or fuzzy in nature. In particular cases stochastic and crisp models are derived. Models are formulated as profit maximization principle and three different approaches are proposed for solution. In the first approach, fuzzy extension principle is used to find membership function of the objective function and then it's Graded Mean Integration Value (GMIV) for different optimistic levels are taken as equivalent stochastic objectives. Then the stochastic model is transformed to a constraint multi-objective programming problem using Stochastic Non-linear Programming (SNLP) technique. The multi-objective problems are transferred to single objective problems using Interactive Fuzzy Satisfising (IFS) technique. Finally, a Region Reducing Genetic Algorithm (RRGA) based on entropy has been developed and implemented to solve the single objective problems. In the second approach, the above GMIV (which is stochastic in nature) is optimized with some degree of probability and using SNLP technique model is transferred to an equivalent single objective crisp problem and solved using RRGA. In the third approach, objective function is optimized with some degree of possibility/necessity and following this approach model is transformed to an equivalent constrained stochastic programming problem. Then it is transformed to an equivalent single objective crisp problem using SNLP technique and solved via RRGA. The models are illustrated with some numerical examples and some sensitivity analyses have been presented.

Pruned Pareto-optimal sets for the system redundancy allocation problem based on multiple prioritized objectives

In this paper, a new methodology is presented to solve different versions of multi-objective system redundancy allocation problems with prioritized objectives. Multi-objective problems are often solved by modifying them into equivalent single objective problems using pre-defined weights or utility functions. Then, a multi-objective problem is solved similar to a single objective problem returning a single solution. These methods can be problematic because assigning appropriate numerical values (i.e., weights) to an objective function can be challenging for many practitioners. On the other hand, methods such as genetic algorithms and tabu search often yield numerous non-dominated Pareto optimal solutions, which makes the selection of one single best solution very difficult. In this research, a tabu search meta-heuristic approach is used to initially find the entire Pareto-optimal front, and then, Monte-Carlo simulation provides a decision maker with a pruned and prioritized set of Pareto-optimal solutions based on user-defined objective function preferences. The purpose of this study is to create a bridge between Pareto optimality and single solution approaches.

Service restoration in distribution system using non-dominated sorting genetic algorithm

In this paper, a non-dominated sorting genetic algorithm-II (NSGA-II) based approach is presented for service restoration in power distribution systems. In contrast to the conventional GA based methods, the proposed approach does not require weighting factors required for conversion of multi-objective function into an equivalent single objective function. Based on the simulation studies carried out in four different systems, the performance of the proposed scheme has been found to be better than the performance of conventional GA technique based approaches. Moreover, by including the string representing the pre-fault configuration of the distribution system in the initial population, the speed of convergence is enhanced significantly.

An Evolutionary Algorithm for Solving Nonlinear Bilevel Programming Based on a New Constraint-Handling Scheme

In this paper, a special nonlinear bilevel programming problem (nonlinear BLPP) is transformed into an equivalent single objective nonlinear programming problem. To solve the equivalent problem effectively, we first construct a specific optimization problem with two objectives. By solving the specific problem, we can decrease the leader's objective value, identify the quality of any feasible solution from infeasible solutions and the quality of two feasible solutions for the equivalent single objective optimization problem, force the infeasible solutions moving toward the feasible region, and improve the feasible solutions gradually. We then propose a new constraint-handling scheme and a specific-design crossover operator. The new constraint-handling scheme can make the individuals satisfy all linear constraints exactly and the nonlinear constraints approximately. The crossover operator can generate high quality potential offspring. Based on the constraint-handling scheme and the crossover operator, we propose a new evolutionary algorithm and prove its global convergence. A distinguishing feature of the algorithm is that it can be used to handle nonlinear BLPPs with nondifferentiable leader's objective functions. Finally, simulations on 31 benchmark problems, 12 of which have nondifferentiable leader's objective functions, are made and the results demonstrate the effectiveness of the proposed algorithm.

Multi-item stochastic and fuzzy-stochastic inventory models under two restrictions

Multi-item stochastic and fuzzy-stochastic inventory models are formulated under total budgetary and space constraints. Here, the inventory costs are directly proportional to the respective quantities, unit purchase/production cost is inversely related to the demand and replenishment/production rate is assumed to vary directly with demand. Shortages are allowed but fully backlogged. Here, for both models, demand and budgetary resource are assumed to be random. In fuzzy-stochastic model, in addition to the above assumptions, available storage space and total expenditure are imprecise in nature. Impreciseness in the parameters have been expressed with the help of linear membership functions. Assuming random variables to be independent and to follow normal distributions, the models have been formulated as stochastic and fuzzy-stochastic non-linear programming problems. The stochastic problem is first reduced to the equivalent single objective or multiple objectives problems following chance-constraint method. The problem with single objective is solved by a gradient-based technique whereas fuzzy technique is applied to the multi-objective one. In the same way, the fuzzy-stochastic programming problem is first reduced to a corresponding equivalent fuzzy non-linear programming problem and then it is solved by fuzzy non-linear programming (FNLP) following Zimmermann technique. The models are illustrated numerically and the results of different models are compared.

On efficiency and duality theory for a class of multi-objective fractional programming problems with invexity

Multiobjective fractional programming problem involving invex functions is considered. Using parametric approach, two types of duals and appropriate weak and strong duality theorems for invex and quasiinvex functions are established for efficient solutions by considering a sequence of equivalent single objective programming problems.

Using AHP for resource allocation problems

AHP has been used for solving multi-criteria resource allocation problems by converting them into equivalent single objective, maximization-type LP problems. At least two approaches can be identified for such applications. In the first approach, the AHP priorities are used as coefficients in the objective function of the LP format, and in the second approach, the benefit-cost ratios are used as the coefficients. This paper evaluates the two approaches. It is shown here that both the approaches are suitable if the criteria (used in the AHP model) are sought to be maximized. However, they are shown to be inappropriate if the criteria are sought to be minimized. A new, simple approach is suggested for the latter case and an extension has been proposed to tackle problems with mixed criteria. The implications of the study on other preference elicitation methods have also been pointed out.