Multiobjective Programming

Optimization Problems ◽

Optimal Solution ◽

Objective Functions ◽

Solution Quality ◽

Solution Methods ◽

Feasible Solutions ◽

Optimization problems with multiple criteria measuring solution quality can be modeled as multiobjective programming problems. Because the objective functions are usually in conflict, there is not a single feasible solution that can optimize all objective functions simultaneously. An optimal solution is one that is most preferred by the decision maker (DM) among all feasible solutions. An optimal solution must be nondominated but a multiobjective programming problem may have, possibly infinitely, many nondominated solutions. Therefore, tradeoffs must be made in searching for an optimal solution. Hence, the DM's preference information is elicited and used when a multiobjective programming problem is solved. The model, concepts and definitions of multiobjective programming are presented and solution methods are briefly discussed. Examples are used to demonstrate the concepts and solution methods. Graphics are used in these examples to facilitate understanding.

Duality for a class of nonsmooth multiobjective programming problems using convexificators

Filomat ◽

10.2298/fil1702489j ◽

2017 ◽

Vol 31 (2) ◽

pp. 489-498 ◽

Cited By ~ 1

Author(s):

Anurag Jayswal ◽

Krishna Kummari ◽

Vivek Singh

Keyword(s):

Optimality Conditions ◽

Programming Problem ◽

Optimization Problems ◽

Duality Results ◽

Nonsmooth Multiobjective Programming ◽

Multiobjective Programming Problems ◽

Dual Models

As duality is an important and interesting feature of optimization problems, in this paper, we continue the effort of Long and Huang [X. J. Long, N. J. Huang, Optimality conditions for efficiency on nonsmooth multiobjective programming problems, Taiwanese J. Math., 18 (2014) 687-699] to discuss duality results of two types of dual models for a nonsmooth multiobjective programming problem using convexificators.

Some Approaches for Fuzzy Multiobjective Programming Problems

Journal of Advances in Applied Mathematics ◽

10.22606/jaam.2021.61002 ◽

2021 ◽

Vol 6 (1) ◽

Author(s):

Yves Mangongo Tinda ◽

◽

Justin Dupar Kampempe Busili ◽

Keyword(s):

Optimization Problem ◽

Fuzzy Numbers ◽

Fuzzy Optimization ◽

Objective Functions ◽

Effectiveness And Efficiency ◽

Nearest Interval Approximation ◽

Multiobjective Programming Problems ◽

Interval Approximation

In this paper we discuss two approaches to bring a balance between effectiveness and efficiency while solving a multiobjective programming problem with fuzzy objective functions. To convert the original fuzzy optimization problem into deterministic terms, the first approach makes use of the Nearest Interval Approximation Operator (Approximation approach) for fuzzy numbers and the second one takes advantage of an Embedding Theorem for fuzzy numbers (Equivalence approach). The resulting optimization problem related to the first approach is handled via Karush- Kuhn-Tucker like conditions for Pareto Optimality obtained for the resulting interval optimization problem. A Galerkin like scheme is used to tackle the deterministic counterpart associated to the second approach. Our approaches enable both faithful representation of reality and computational tractability. They are thus in sharp contrast with many existing methods that are either effective or efficient but not both. Numerical examples are also supplemented for the sake of illustration.

An Interval Approximation Approach for a Multiobjective Programming Model with Fuzzy Objective Functions

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488515500373 ◽

2015 ◽

Vol 23 (06) ◽

pp. 845-860 ◽

Cited By ~ 1

Author(s):

M. K. Luhandjula

Keyword(s):

Programming Model ◽

Optimal Solution ◽

Optimization Under Uncertainty ◽

Objective Functions ◽

New Approach ◽

Different Shapes ◽

General Goal ◽

Multiobjective Programming Problems ◽

Interval Approximation

Research in optimization under uncertainty is alive. It assumes different shapes and forms, all concurring to the general goal of designing effective and efficient tools for handling imprecision in an Optimization setting. In this paper we present a new approach for dealing with multiobjective programming problems with fuzzy objective functions. Similar to many approaches in the literature, our approach relies on the deffuzification of involved fuzzy quantities. Our improvement stem from the choice of a deffuzification operator that captures essential features of fuzzy parameters at hand rather than those that yield single values, leading to a loss of many useful information. Two oracles play a pivotal role in the proposed method. The first one returns a near interval approximation to a given fuzzy number. The other one delivers a Pareto Optimal solution of the resulting multiobjective program with interval coefficient. A numerical example is also provided for the sake of illustration.

A Brief Look at Multi-Criteria Problems: Multi-Threshold Optimization versus Pareto-Optimization

Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality ◽

10.5772/intechopen.91169 ◽

2020 ◽

Author(s):

Nodari Vakhania ◽

Frank Werner

Keyword(s):

Optimization Problems ◽

Optimality Criteria ◽

General Notion ◽

Objective Functions ◽

Objective Criterion ◽

Solution Quality ◽

Pareto Optimal Set ◽

Threshold Optimization ◽

Feasible Solutions

Multi-objective optimization problems are important as they arise in many practical circumstances. In such problems, there is no general notion of optimality, as there are different objective criteria which can be contradictory. In practice, often there is no unique optimality criterion for measuring the solution quality. The latter is rather determined by the value of the solution for each objective criterion. In fact, a practitioner seeks for a solution that has an acceptable value of each of the objective functions and, in practice, there may be different tolerances to the quality of the delivered solution for different objective functions: for some objective criteria, solutions that are far away from an optimal one can be acceptable. Traditional Pareto-optimality approach aims to create all non-dominated feasible solutions in respect to all the optimality criteria. This often requires an inadmissible time. Besides, it is not evident how to choose an appropriate solution from the Pareto-optimal set of feasible solutions, which can be very large. Here we propose a new approach and call it multi-threshold optimization setting that takes into account different requirements for different objective criteria and so is more flexible and can often be solved in a more efficient way.

Saddle point criteria and duality in multiobjective programming via an η-approximation method

The ANZIAM Journal ◽

10.1017/s1446181100009962 ◽

2005 ◽

Vol 47 (2) ◽

pp. 155-172 ◽

Cited By ~ 5

Author(s):

Tadeusz Antczak

Keyword(s):

Saddle Point ◽

Programming Problem ◽

Dual Problem ◽

Lagrange Function ◽

Saddle Points ◽

Duality Theorems ◽

Approximation Approach ◽

AbstractIn this paper, Antczak's η-approximation approach is used to prove the equivalence between optima of multiobjective programming problems and the η-saddle points of the associated η-approximated vector optimisation problems. We introduce an η-Lagrange function for a constructed η-approximated vector optimisation problem and present some modified η-saddle point results. Furthermore, we construct an η-approximated Mond-Weir dual problem associated with the original dual problem of the considered multiobjective programming problem. Using duality theorems between η-approximation vector optimisation problems and their duals (that is, an η-approximated dual problem), various duality theorems are established for the original multiobjective programming problem and its original Mond-Weir dual problem.

Lagrangian Duality for Multiobjective Programming Problems in Lexicographic Order

Abstract and Applied Analysis ◽

10.1155/2013/573408 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6

Author(s):

X. F. Hu ◽

L. N. Wang

Keyword(s):

Saddle Point ◽

Programming Problem ◽

Dual Problem ◽

Lexicographic Order ◽

Existence Results ◽

Saddle Point Theorem ◽

Duality Theorems ◽

This paper deals with a constraint multiobjective programming problem and its dual problem in the lexicographic order. We establish some duality theorems and present several existence results of a Lagrange multiplier and a lexicographic saddle point theorem. Meanwhile, we study the relations between the lexicographic saddle point and the lexicographic solution to a multiobjective programming problem.

Evolutionary algorithm using surrogate models for solving bilevel multiobjective programming problems

PLoS ONE ◽

10.1371/journal.pone.0243926 ◽

2020 ◽

Vol 15 (12) ◽

pp. e0243926

Author(s):

Yuhui Liu ◽

Hecheng Li ◽

Hong Li

Keyword(s):

Evolutionary Algorithm ◽

Programming Problem ◽

Computational Cost ◽

Hierarchical Structures ◽

Surrogate Models ◽

Weighted Sum Method ◽

Np Hard Problem ◽

A bilevel programming problem with multiple objectives at the leader’s and/or follower’s levels, known as a bilevel multiobjective programming problem (BMPP), is extraordinarily hard as this problem accumulates the computational complexity of both hierarchical structures and multiobjective optimisation. As a strongly NP-hard problem, the BMPP incurs a significant computational cost in obtaining non-dominated solutions at both levels, and few studies have addressed this issue. In this study, an evolutionary algorithm is developed using surrogate optimisation models to solve such problems. First, a dynamic weighted sum method is adopted to address the follower’s multiple objective cases, in which the follower’s problem is categorised into several single-objective ones. Next, for each the leader’s variable values, the optimal solutions to the transformed follower’s programs can be approximated by adaptively improved surrogate models instead of solving the follower’s problems. Finally, these techniques are embedded in MOEA/D, by which the leader’s non-dominated solutions can be obtained. In addition, a heuristic crossover operator is designed using gradient information in the evolutionary procedure. The proposed algorithm is executed on some computational examples including linear and nonlinear cases, and the simulation results demonstrate the efficiency of the approach.

The Application of Stochastic Order to Stochastic Multiobjective Programming Problems

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.505-506.524 ◽

2014 ◽

Vol 505-506 ◽

pp. 524-527

Author(s):

Ming Fa Zheng ◽

Qi Hang He ◽

Zu Tong Wang ◽

Dong Qing Su

Keyword(s):

Programming Problem ◽

Theoretical Basis ◽

Efficient Solution ◽

Stochastic Order ◽

Stochastic Approach ◽

Stochastic Problems ◽

Pareto Efficient ◽

This paper is devoted to the application of stochastic order to the with stochastic multiobjective programming problem. A new method, called stochastic approach, is originally presented based on stochastic order. The partial Pareto efficient solution is defined first, and then several types of stochastic order from the viewpoint of practical problems are proposed. The results obtained can provide theoretical basis for dealing with the stochastic problems in field of civil engineering and transportation.

Second-order optimality conditions for efficiency in $$C^{1,1}$$-smooth quasiconvex multiobjective programming problem

Computational and Applied Mathematics ◽

10.1007/s40314-021-01625-0 ◽

2021 ◽

Vol 40 (6) ◽

Author(s):

Tran Van Su ◽

Dinh Dieu Hang

Keyword(s):

Optimality Conditions ◽

Programming Problem ◽

Multiobjective Programming Problem

Second Order ◽

Second Order Optimality Conditions ◽

An evolutionary-based optimization algorithm for truss sizing design

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/7476 ◽

2016 ◽

Vol 38 (4) ◽

pp. 307-317

Author(s):

Pham Hoang Anh

Keyword(s):

Local Search ◽

Objective Function ◽

Optimization Algorithm ◽

Optimization Problems ◽

Optimal Solution ◽

The Other ◽

Truss Structures ◽

Benchmark Problems ◽

Discrete Variables ◽

Objective Functions

In this paper, the optimal sizing of truss structures is solved using a novel evolutionary-based optimization algorithm. The efficiency of the proposed method lies in the combination of global search and local search, in which the global move is applied for a set of random solutions whereas the local move is performed on the other solutions in the search population. Three truss sizing benchmark problems with discrete variables are used to examine the performance of the proposed algorithm. Objective functions of the optimization problems are minimum weights of the whole truss structures and constraints are stress in members and displacement at nodes. Here, the constraints and objective function are treated separately so that both function and constraint evaluations can be saved. The results show that the new algorithm can find optimal solution effectively and it is competitive with some recent metaheuristic algorithms in terms of number of structural analyses required.