Multi-objective Quadratic Programming Research Articles

On Pareto optimality using novel goal programming approach for fully intuitionistic fuzzy multiobjective quadratic problems

Nowadays, intuitionistic fuzzy theory is gaining rapid acceptability as it emphasizes non-membership degree also, in addition to membership degree. In this paper, a simplified novel goal programming method under intuitionistic fuzzy environment (analogous to Angelov’s approach) using both membership/non-membership functions is developed. It is then applied to obtain a Pareto optimal solution to a multiobjective quadratic programming problem (QPP) with all the parameters and variables as intuitionistic fuzzy numbers. The relations between the multiobjective intuitionistic fuzzy QPP and its equivalent crisp problems under various membership/ non-membership functions using goal programming are also established by proving theorems at appropriate places. Moreover, a numerical example and a practical problem in production planning have been projected to illustrate the steps involved in the proposed approach.

Spatial Path Smoothing for Car-like Robots Using Corridor-Based Quadratic Optimization

The global planning module of car-like robots usually plans a coarse spatial path, which may be non-smooth and kinematically infeasible for car-like robots. This study proposes an efficient spatial path smoothing approach, which is capable of optimizing a rough spatial path to be a high quality one. Two novel designs contribute to the proposed approach. One is a direct corridor construction method that provides an optimization region for path optimization. Based on a redefined path representation in the generated corridor, the second one is a core spatial path optimization method, where the optimization problem is formulated as a multi-objective quadratic programming (QP) with corridor and maximum-curvature constraints. Meanwhile, integrating a two-steps strategy into an optimization process yields a good trade-off between efficiency and quality. Experiment results validate that the proposed approach has the ability to online generate a high quality path.

A Non-Archimedean Interior Point Method and Its Application to the Lexicographic Multi-Objective Quadratic Programming

This work presents a generalized implementation of the infeasible primal-dual interior point method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here the non-Archimedean IPM (NA-IPM), is proved to converge in polynomial time to a global optimum and to be able to manage infeasibility and unboundedness transparently, i.e., without considering them as corner cases: by means of a mild embedding (addition of two variables and one constraint), the NA-IPM implicitly and transparently manages their possible presence. Moreover, the new algorithm is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since the NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. To support the theoretical properties of the NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that the NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool, paving the way for its use in real-world problems in the near future.

Fuzzy goal programming approach to solve fully fuzzy multi-objective quadratic programming problem

Fuzzy goal programming approach to solve fully fuzzy multi-objective quadratic programming problem

Solving Bi-Level Multi-Objective Quadratic Programming Problem with Neutrosophic Parameters in Objective Functions and Constraints Using Fuzzy Approach

Solving Bi-Level Multi-Objective Quadratic Programming Problem with Neutrosophic Parameters in Objective Functions and Constraints Using Fuzzy Approach

An algorithm for quadratically constrained multi-objective quadratic fractional programming with pentagonal fuzzy numbers

This study proposes a methodology to obtain an efficient solution for a programming model which is multi-objective quadratic fractional with pentagonal fuzzy numbers as coefficients in all the objective functions and constraints. The proposed approach consists of three stages. In the first stage, defuzzification of the coefficients is carried out using the mean method of α-cut. Then, in the second stage, a crisp multi-objective quadratic fractional programming model (MOQFP) is constructed to obtain a non-fractional model based on an iterative parametric approach. In the final stage, this multiobjective non-fractional model is transformed to obtain a model with a single objective by applying the ε-constraint method. This final model is then solved to get desired solution. Also, an algorithm and flowchart expressing the methodology are given to present a clear picture of the approach. Finally, a numerical example illustrating the complete approach is given.

A Solution Approach for Multi-Level Multi-Objective Quadratic Fractional Programming with Trapezoidal Fuzzy Numbers using Rouben Ranking Function

The paper proposed a methodology for obtaining a set of efficient solutions for a model which is multi-level multiobjective quadratic with fractional objectives and constraints having trapezoidal fuzzy number (MLMOQFP-TrFN) as coefficients. The model consists of r-levels with several objectives involved to be solved under a set of quadratic constraints. The proposed approach starts with the solution process of the top level and other levels are solved in succession but depending on the solution of the previous levels. The solution process of each level comprises mainly three stages. In the beginning, the Rouben Ranking Function is used to convert the rth-level of fuzzy model into a deterministic or crisp one. After that, the crisp form is reconstructed to get a non-fractional model with the help of an iterative parametric approach. Further, in the last, non-fractional model which is still having multiple objectivesis reconstructed to form a model having only one objective with ɛ -constraint method and is lastly solved by following the solution of (r-1)th- level to get a desired set of efficient solution. Such programming models are very useful in day to day life such as in economic planning, industrial activities, waste management, neural networking, unmanned aerial and underwater vehicle management, agricultural yield improvement, transportation problems with maximizing profits and minimizing wastage of material and cost and so on. An algorithm depicting all the steps of solution approach is also presented to reflect a clear idea for the approach. In addition, a numerical regarding the presentation of complete approach that is studied is given in the end.

Rouben Ranking Function and parametric approach to quadratically constrained multiobjective quadratic fractional programming with trapezoidal fuzzy number coefficients

Rouben Ranking Function and parametric approach to quadratically constrained multiobjective quadratic fractional programming with trapezoidal fuzzy number coefficients

FGP approach to quadratically constrained multi-objective quadratic fractional programming with parametric functions

FGP approach to quadratically constrained multi-objective quadratic fractional programming with parametric functions

Parametric approach to quadratically constrained multi-level multi-objective quadratic fractional programming

The paper proposed a method to study and obtain a set of Pareto optimal solutions or a set of representative solutions to a quadratically constrained multi-level multiobjective quadratic fractional programming problem. This problem involves several objectives to be fulfilled at multi levels under a common set of quadratic constraints. Initially, we used parametric approach to convert the fractional programming model to an equivalent non-fractional programming model by allocating a parametric vector to each fractional objective. Then, $$\varepsilon$$ -constraint method is used to convert this multiobjective programming model into an equivalent model with single objective. The solution of every previous level is followed by the next level in succession to find a solution which is suitable to each level decision maker. An algorithm and numerical example are also presented at the end of the paper to validate the proposed methodology for the Model.

Iterative Parametric Approach for Quadratically Constrained Bi-Level Multiobjective Quadratic Fractional Programming

The paper proposed an iterative parametric approach procedure for solving Bi-level Multiobjective Quadratic Fractional Programming model. The Model is divided into two levels-upper and lower. In the first stage of the approach, a set of pareto optimal solutions of upper Level is obtained by converting the problem into equivalent single non-fractional parametric objective optimization problem by using parametric vector and ε-constraint method. Then for the second stage, the solution of upper level is followed by the lower level decision maker while finding solution with the proposed algorithm to obtain the best preferred solution. A numerical example is solved in the last to validate the feasibility of the approach.

A new approach for solving fuzzy multi-objective quadratic programming of water resource allocation problem

This paper describes an application of fuzzy multi-objective quadratic model with flexible constraints for optimal allocation of limited available water resources among different water-user sectors. Due to the fact that, water resource allocation problem is one of the practical and essential subjects in real world and many of the parameters may be faced by uncertainty. In this paper, we present α - cut approach for transforming fuzzy multi-objective quadratic programming model with flexible constraints into a crisp form. By using this approach a multi-parametric multi-objective programming model corresponding to α and parameters of flexible constraints is obtained. One of the advantages of this model is that the α - cut level is not determined by the decision makers. Actually, this model itself can calculate the α - cut level. In order to achieve a desired Pareto optimal value of multi-parametric multi-objective model, we use goal programming method for illustration of water resource allocation with sensitivity analysis of lower bound of parameters in flexible constraints. To illustrate the efficiency of the proposed approach, we apply it for a real case problem of water resource allocation.

Multiobjective Quadratic Fractional Programming using Iterative Parametric Function

The paper proposed the Model of multiobjective quadratic fractional optimisation problem with a set of quadratic constraints and a methodology for obtaining a set of solutions based on the approach of using iterative parametric functions. Firstly, each fractional objective function is transformed into non-fractional parametric objective function by assigning a vector of parameters to each objective function. In this approach, the Decision Maker(DM) predecides the desired tolerance levels of the objective functions in the form of termination constants. Then, by using ε-constraint method, a set of efficient solutions is obtained and termination conditions are checked for each parametric objective function. Also, a comparative study of the proposed method and fuzzy approach is given to reveal the validity of the method. A numerical for Multiobjective quadratic fractional programming Model (MOQFPM) is given in the end to check the applicability of the approach.

A MOQFPP Method for Solving Quadratic Fractional Programming under Fuzzy Environment

In this paper, a solution procedure is proposed to solve Fully Fuzzy Quadratic Fractional Programming problem (FFQFPP) where all the variables and parameters are triangular fuzzy numbers. Here, FFQFPP transformed into an equivalent Multi- Objective Quadratic Fractional Programming problem (MOQFPP). Then MOQFPP converted into an equivalent multi objective Quadratic programming problem by using Mathematical programming approach. The proposed solution illustrated through numerical examples

Multi-objective fuzzy probabilistic quadratic programming problem

The aim of the paper is to present a multi-choice multi-objective fuzzy probabilistic quadratic programming problem and its solution methodology. The mathematical programming problem suggested here is difficult to solve directly. Therefore, three major steps are suggested to solve the proposed mathematical programming problem. In first step, fuzzy chance constraint is transformed to its equivalent chance constraint programming problem using α-cut technique. Chance constraint technique is used to obtain a crisp multi-choice multi-objective quadratic programming problem. In the next step, importance is given to handle multi-choice parameter using least square approximation technique. At the end of second step, a multi-objective quadratic mathematical programming is obtained. Finally, goal programming approach is used to solve the transformed multi-objective quadratic mathematical programming. Using existing methodology and software, the final solution of the proposed model is obtained. The proposed method is implemented with a numerical example.

Multi-objective fuzzy probabilistic quadratic programming problem

The aim of the paper is to present a multi-choice multi-objective fuzzy probabilistic quadratic programming problem and its solution methodology. The mathematical programming problem suggested here is difficult to solve directly. Therefore, three major steps are suggested to solve the proposed mathematical programming problem. In first step, fuzzy chance constraint is transformed to its equivalent chance constraint programming problem using α-cut technique. Chance constraint technique is used to obtain a crisp multi-choice multi-objective quadratic programming problem. In the next step, importance is given to handle multi-choice parameter using least square approximation technique. At the end of second step, a multi-objective quadratic mathematical programming is obtained. Finally, goal programming approach is used to solve the transformed multi-objective quadratic mathematical programming. Using existing methodology and software, the final solution of the proposed model is obtained. The proposed method is implemented with a numerical example.

A Fuzzy Goal Programming Approach for Solving Chance Constrained Bi-Level Multi-Objective Quadratic Fractional Programming Problem

A Fuzzy Goal Programming Approach for Solving Chance Constrained Bi-Level Multi-Objective Quadratic Fractional Programming Problem

A Quadratic-Programming Approach to the Design Optimization of Fractional-Slot Concentrated Windings for Surface Permanent-Magnet Machines

Fractional-slot concentrated windings (FSCW) are becoming more and more popular in the design of permanent magnet electric machines. A well-known drawback of their adoption is the occurrence of large magneto-motive force (MMF) harmonics, which produce eddy-current losses in rotor permanent magnets. The use of a multi-layer design, with coils of different phases wound around the same tooth, is a possible countermeasure to mitigate the problem. In this paper, a new general systematic methodology is proposed to optimize the multilayer FSCW design in the form of a multi-objective quadratic programming problem. The maximization of the MMF fundamental and the minimization of total rotor losses are taken as properly weighed objective functions. Constraints are imposed to guarantee the physical feasibility and the electric symmetry of the winding. An application example to a 9-slot 8-pole machine is presented, together with extensive validations by comparison with finite element analysis (FEA) simulations, to prove the effectiveness of the proposed technique.

Multi-objective fuzzy quadratic probabilistic programming problem involving fuzzy Cauchy random variable

In this paper, we have proposed a method for solving a multi-objective quadratic probabilistic programming problem, where the objective functions are multi-objective and quadratic in nature. The right hand side parameters are fuzzy Cauchy distributed independent random variable with location parameter δ and scale parameter β. The proposed mathematical programming problem is solved using two steps. First, the fuzziness is removed by using alpha cut technique and randomness is removed by chance constrained method. In second step, weighting method is used to solve the transformed multi-objective quadratic mathematical programming. This mathematical model is solved by existing methodology or software. A numerical example is presented to illustrate the efficiency and feasibility of the proposed method.

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