Multiobjective Nonlinear Programming Problems Research Articles

Non-differentiable multiobjective programming under generalised functions

In this paper, we consider a nonlinear multiobjective programming problem where functions involved are non-differentiable. The use of directional derivative in association with a hypothesis of an invex kind on the set has been of much interest in the recent past. Recently under preinvexity assumptions on fi′(xo;ηi(x,xo)) and g′j (xo;θj(x,xo)), necessary and sufficient optimality conditions for a non-smooth multiobjective optimisation problem under generalised class of dI-V-type I functions have been derived. Working in this direction, we introduce a new class of (φ, dI)-V-type I functions and illustrate through various non-trivial examples that this class is non-empty and extends some known classes introduced in the literature. Also, we obtain sufficient optimality conditions to enable a feasible solution of the primal problem to be its weak efficient/efficient solution. Then through an example, we illustrate the relevance of the sufficient optimality theorem obtained to get the efficient solution of the problem. Further, we formulate Wolfe type and Mond-Weir type multiobjective dual programs and establish various duality theorems under this newly introduced class of functions.

Central Processing Point Deployment for e-Healthcare Services Using Wireless Networks

In this paper, we present an approach to jointly minimize energy consumption of wireless transmitters and harmful interference on sensitive medical devices, and optimize the central controller placement for collecting data from transmitters. In particular, a multiobjective nonconvex mixed-integer nonlinear programming problem is formulated to optimize transmit power of wireless users and central processing point deployment along with guaranteeing minimized electromagnetic interference to comply with the immunity level of medical devices. To solve this problem, an important step is to convexify it, following which, we propose the use of a standard branch-and-bound algorithm to solve it. The solution is implemented in MATLAB using optimization interface Toolbox. We analyze the results numerically and consider different cases within a single floor of a hospital and a few scenarios for a two-storey hospital building. Our results show that the proposed approach offers improvement in throughput and guaranteed immunity for medical devices.

Intuitionistic Fuzzy Goal Programming Technique for Solving Non-Linear Multi-objective Structural Problem

This paper proposes a new multi-objective intuitionistic fuzzy goal programming approach to solve a multi-objective nonlinear programming problem in context of a structural design. Here we describe some basic properties of intuitionistic fuzzy optimization. We have considered a multi-objective structural optimization problem with several mutually conflicting objectives. The design objective is to minimize weight of the structure and minimize the vertical deflection at loading point of a statistically loaded three-bar planar truss subjected to stress constraints on each of the truss members. This approach is used to solve the above structural optimization model based on arithmetic mean and compare with the solution by intuitionistic fuzzy goal programming approach. A numerical solution is given to illustrate our approach.

Fuzzy variable based fuzzy non-linear programming approach for optimization of complex system reliability

All most all systems in the field of science and technology are well complicated by the advent of modernity. Complex system reliability is mainly dependant on the credibility of its components. It is very difficult to evaluate precise value of component's reliability at the beginning stage of reliability evaluation, due to unavailable or inadequate information. In this respect, fuzzy reliability variable can play an important role in complex system reliability evaluation. In this paper, a fuzzy reliability variable based fuzzy non-linear programming problem is formulated to maximize the complex system reliability. Symmetrical trapezoidal fuzzy numbers are used as fuzzy variable for the component's reliability. Since fuzzy variable for optimization is not well defined, we first reduced the problem to an equivalent multi-objective non-linear programming problem. The proposed approach is applied on landing system of Boeing 747D airplane to evaluate the system reliability in fuzzy environment. The proposed approach and the complex system reliability model are well substantiated by providing numerical examples.

A Fuzzy Programming Technique for Solving Multi-objective Structural Problem

This paper proposes a new fuzzy multi-objective optimization approach to solve a multi-objective nonlinear programming problem in context of a structural design. We have been developed a multi-objective structural problem of a planar truss structural model in fuzzy environment. Here, the objectives are (i) to minimize weight of the structure and (ii) to minimize the vertical deflection at loading point. In this model, the design variables are the cross-section of the truss members and the constraints are the stresses in members. This approach is used to solve the structural model under uncertainty based on different operator. A numerical illustration is given to support our approach.

Reentry trajectory optimization for hypersonic vehicles using fuzzy satisfactory goal programming method

Constrained reentry trajectory optimization for hypersonic vehicles is a challenging job. In particular, this problem becomes more difficult when several objectives with preemptive priorities are expected for different purposes. In this paper, a fuzzy satisfactory goal programming method is proposed to solve the multi-objective reentry trajectory optimization problem. Firstly, direct collocation approach is used to discretize the reentry trajectory optimal-control problem with nonlinear constraints into nonlinear multiobjective programming problem with preemptive priorities, where attack angles and bank angles at nodes and collocation nodes are selected as control variables. Secondly, the preemptive priorities are transformed into the relaxed order of satisfactory degrees according to the principle that the objective with higher priority has higher satisfactory degree. Then the fuzzy satisfactory goal programming model is proposed. The balance between optimization and priorities is realized by regulating parameter ?, such that the satisfactory reentry trajectory can be acquired. The simulation demonstrates that the proposed method is effective for the multi-objective reentry trajectory optimization of hypersonic vehicles.

The Redundancy Queuing-Location-Allocation Problem: A Novel Approach

Redundancy queuing-location-allocation problems (RQLAPs) involve the economical allocation of facilities, each with a number of servers, to suitable locations with appropriate levels of redundancy or reliability. The goal in RQLAPs is to find the facilities which are both inexpensive and reliable. We take into consideration the congestion of the system by modeling each facility as an M/M/m queuing system and formulate the problem as a multiobjective nonlinear mixed integer programming problem. However, these problems are nondeterministic polynomial-time hard (NP-hard) problems where an exact solution cannot be produced in polynomial time. We propose a novel soft-computing approach based on the vibration theory called vibration damping optimization (VDO) to solve the RQLAP. We develop a multiobjective version of the VDO called multiobjective VDO (MOVDO) based on the fast nondominated sorting and crowding distance concepts in the literature. The performance of the proposed MOVDO is statistically compared with two other commonly used metaheuristic algorithms called the nondominated sorting genetic algorithm and multiobjective simulated annealing. A comparison of the results based on different problem sizes favors the MOVDO method proposed in this study.

Unified Interactive Approach for Solving Multiobjective Nonlinear Programming Problems

Unified Interactive Approach for Solving Multiobjective Nonlinear Programming Problems

Optimality and duality for nonsmooth multiobjective programming using G-type I functions

In this paper, (generalized) G-type I functions are defined for a nonlinear multiobjective programming problem where the functions involved are assumed to be locally Lipschitz. This new class of functions is a generalization of G-invex functions defined in Kang et al. (2012) [20]. Examples are given to show the existence of these functions. G-type Kuhn–Tucker necessary conditions are established for a nondifferentiable multiobjective programming problem (GMP). By using suitable G-type I functions, sufficient optimality conditions are derived for the problem (GMP). Further a Mond–Weir type dual (GMWD) is formulated and using these newly defined functions various duality results are established.

A new decision-making model for solving multi-objective large-scale programming problems with a block angular structure

This paper presents a model based on a novel compromised solution method to solve the multi-objective large-scale nonlinear programming (MOLSNLP) problems with block angular structure. In this method, an aggregating function that is developed from TOPSIS and VIKOR is proposed based on the particular measure of ‘‘closeness” to the ‘‘ideal” solution. The decomposition algorithm is utilized to reduce the large-dimensional objective space into a two-dimensional space. Furthermore, two independent solution methods are proposed to solve each nonlinear sub problem respectively. In the last step of proposed method, Single objective nonlinear programming problem is solved to find the final solution. Finally, to justify the proposed method, an illustrative example is provided. Then, the sensitivity analysis is described.

Compromise Allocation for Combined Ratio Estimates of Population Means of a Multivariate Stratified Population Using Double Sampling in Presence of Non-Response

This paper is an attempt to work out a compromise allocation to construct combined ratio estimates under multivariate double sampling design in presence of non-response when the population mean of the auxiliary variable is unknown. The problem has been formulated as a multi-objective integer non-linear programming problem. Two solution procedures are developed using goal programming and fuzzy programming techniques. A numerical example is also worked out to illustrate the computational details. A comparison of the two methods is also carried out.

Optimal Intersection Operation with Median U-Turn

The median U-turn intersection is a potential method for relieving traffic congestion at intersections. Median U-turn intersections can be classified into six types, according to the prohibited traffic movements. Although much is known about the technology of the median U-turn, efforts have focused on its geometric design, safety analysis, and evaluation. In this paper, a lane-based optimization model is proposed for the integrated design of median U-turn intersection types, lane markings, distances between the median crossovers and the main intersection, and signal timings. The capacity maximization problem is considered, and a set of constraints is determined that ensures the feasibility and safety of the optimization results. The optimization is formulated as a multi-objective mixed-integer nonlinear programming problem. It is solved by transformation to a series of mixed-integer linear programming models. The latter can be solved by the standard branch-and-bound technique. Extensive numerical analysis and a case study have shown the effectiveness of the proposed method. In addition, the method can assist transportation professionals in the proper selection of U-turn types and in the design of intersections with median U-turns.

OPTIMIZATION OF ENERGY SAVING FOR WIRELESS SENSOR NETWORKS

Wireless sensor network (WSN) has been widely used in national defense, military and security, environmental monitoring and many other fields. It s of great significance to design WSN which consumes little energy but supports service quality flexibly and effectively. To further improve the performance of WSN, this article, focusing on aspects of low energy consumption and long life cycle of the WSN and taking energy limits and coverage requirements in WSN into account, attributes the coverage control problem of WSN to a multi-objective nonlinear programing proble. A mathematical model for multi-objective nonlinear programming problem is established and a virtual force optimization algorithm based on energy-saving is presented in this article. Sensor nodes and target coverage areas of virtual force operations and nodes of virtual force operations are carried out at the same time for energy saving so as to realize the optimization of coverage control technology. In terms of network routing optimization, taking QoS requirements, topology structure of the network and energy consumption into consideration, this article puts forward the hierarchical ecological niche ACA to optimize routing technology to analyze energy and realize simultion. Finally, advantages of the new optimization strategy on the enhancement of the network lifetime are illustrated.

Fuzzy Goal Programming Approach in Selective Maintenance Reliability Model

800x600 In the present paper, we have considered the allocation problem of repairable components for a parallel-series system as a multi-objective optimization problem and have discussed two different models. In first model the reliability of subsystems are considered as different objectives. In second model the cost and time spent on repairing the components are considered as two different objectives. These two models is formulated as multi-objective Nonlinear Programming Problem (MONLPP) and a Fuzzy goal programming method is used to work out the compromise allocation in multi-objective selective maintenance reliability model in which we define the membership functions of each objective function and then transform membership functions into equivalent linear membership functions by first order Taylor series and finally by forming a fuzzy goal programming model obtain a desired compromise allocation of maintenance components. A numerical example is also worked out to illustrate the computational details of the method.  Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4

An Optimal Chance Constraint Multivariate Stratified Sampling Design Using Auxiliary Information

When we are dealing with multivariate problem then we need an allocation which is optimal for all the characteristics in some sense because the individual optimum allocations usually differ widely unless the characteristics are highly correlated. So an allocation called “Compromise allocation” is to be worked out suggested by Cochran. When auxiliary information is also available, it is customary to use it to increase the precision of the estimates. Moreover, for practical implementation of an allocation, we need integer values of the sample sizes. In the present paper the problem is to determine the integer optimum compromise allocation when the population means of various characteristics are of interest and auxiliary information is available for the separate and combined ratio and regression estimates. This paper considers the optimum compromise allocation in multivariate stratified sampling with non-linear objective function and probabilistic non-linear cost constraint. The probabilistic non-linear cost constraint is converted into equivalent deterministic one by using Chance Constrained programming. The formulated multi-objective nonlinear programming problem is solved by Fuzzy Goal programming approach and Chebyshev approximation. Numerical illustration is also given to show the practical utility of the approaches.

Multiobjective programming under (φ, d)-V-type I univexity

In this paper, we have introduced a new class of (φ, d)-V-type I univex, quasi (φ, d)-V-type I univex, pseudo (φ, d)-V-type I univex, quasi-pseudo (φ, d)-V-type I univex and pseudo-quasi (φ, d)-V-type I univex functions in case of nonlinear multiobjective programming problem where functions involved are nondifferentiable and illustrated through non-trivial examples that this class extends some known classes in literature. Various Karush-Kuhn-Tucker type sufficient optimality conditions are obtained under this newly introduced class of functions. Also, for mixed type multiobjective dual program, we have established weak, strong, converse and strict converse duality results in order to relate the efficient and weak efficient solutions of primal and dual problem.

On the Problem of Compromise Allocation in Multi-Response Stratified Sample Surveys

In stratified sample surveys, the problem of determining the optimum allocation is well known due to articles published in 1923 by Tschuprow and in 1934 by Neyman. The articles suggest the optimum sample sizes to be selected from each stratum for which sampling variance of the estimator is minimum for fixed total cost of the survey or the cost is minimum for a fixed precision of the estimator. If in a sample survey more than one characteristic is to be measured on each selected unit of the sample, that is, the survey is a multi-response survey, then the problem of determining the optimum sample sizes to various strata becomes more complex because of the non-availability of a single optimality criterion that suits all the characteristics. Many authors discussed compromise criterion that provides a compromise allocation, which is optimum for all characteristics, at least in some sense. Almost all of these authors worked out the compromise allocation by minimizing some function of the sampling variances of the estimators under a single cost constraint. A serious objection to this approach is that the variances are not unit free so that minimizing any function of variances may not be an appropriate objective to obtain a compromise allocation. This fact suggests the use of coefficient of variations instead of variances. In the present article, the problem of compromise allocation is formulated as a multi-objective non-linear programming problem. By linearizing the non-linear objective functions at their individual optima, the problem is approximated to an integer linear programming problem. Goal programming technique is then used to obtain a solution to the approximated problem.

A fuzzy goal programming approach in stochastic multivariate stratified sample surveys

This paper deals with fuzzy goal programming (FGP) approach to stochastic multivariate stratified sampling with non linear objective function and probabilistic non linear cost constraint which is formulated as a multiobjective non linear programming problem (MONLPP). In the model formulation of the problem, we first determine the individual best solution of the objective functions subject to the system constraints and construct the non linear membership functions of each objective. The non linear membership functions are then transformed into equivalent linear membership functions by first order Taylor series at the individual best solution point. Fuzzy goal programming approach is then used to achieve maximum degree of each of the membership goals by minimizing negative deviational variables and finally obtain the compromise allocation. A numerical example is presented to illustrate the computational procedure of the proposed approach.

Design of Lightweight Hybrid Materials for the Cab Body-In-White of a Commercial Vehicle Based on Multi-Objective Optimization

In this paper, the lightweight hybrid materials cab BIW of a commercial vehicle is investigated for material selection. In order to improve the dynamic performance and reduce total mass, the lightweight materials are employed in the BIW. Nine groups of sensitive components are selected to apply the lightweight materials. The design problem is formulated as a multi-objective nonlinear programming problem and the modified non-dominated sorting genetic algorithm (NSGA-II) is used to solve the problem. The surrogate model based on radial basis function (RBF) is adopted in the optimization. The research results show that the usage of steel-aluminum hybrid materials can reduce the total mass and increase the first torsion modal frequency of the cab BIW.

Multi-objective optimization for optimum allocation in multivariate stratified sampling with quadratic cost

In stratified sampling, usually the cost function is taken as a linear function of sample sizes n h . In many practical situations, the linear cost function does not approximate the actual cost incurred adequately. For example, when the cost of travelling between the units selected in the sample within a stratum is significant, instead of a linear cost function, a cost function that is quadratic in √n h will be a more close approximation to the actual cost. In this paper, the problem is formulated as multi-objective nonlinear integer programming problem with quadratic cost under three different situations, i.e. complete, partial or null information about the population. A numerical example is also presented to illustrate the computational details.