Nondominated Points Research Articles

Optimal platform design using non-dominated sorting genetic algorithm II and technique for order of preference by similarity to ideal solution; application to automotive suspension system

ABSTRACTUnlike conventional approaches where optimization is performed on a unique component of a specific product, optimum design of a set of components for employing in a product family can cause significant reduction in costs. Increasing commonality and performance of the product platform simultaneously is a multi-objective optimization problem (MOP). Several optimization methods are reported to solve these MOPs. However, what is less discussed is how to find the trade-off points among the obtained non-dominated optimum points. This article investigates the optimal design of a product family using non-dominated sorting genetic algorithm II (NSGA-II) and proposes the employment of technique for order of preference by similarity to ideal solution (TOPSIS) method to find the trade-off points among the obtained non-dominated results while compromising all objective functions together. A case study for a family of suspension systems is presented, considering performance and commonality. The results indicate the effectiveness of the proposed method to obtain the trade-off points with the best possible performance while maximizing the common parts.

On nadir points of multiobjective integer programming problems

In this study, we consider the nadir points of multiobjective integer programming problems. We introduce new properties that restrict the possible locations of the nondominated points necessary for computing the nadir points. Based on these properties, we reduce the search space and propose an exact algorithm for finding the nadir point of multiobjective integer programming problems. We present an illustrative example on a three objective knapsack problem. We conduct computational experiments and compare the performances of two recent algorithms and the proposed algorithm.

Finding extreme supported solutions of biobjective network flow problems: An enhanced parametric programming approach

We address the problem of determining a complete set of extreme supported efficient solutions of biobjective minimum cost flow (BMCF) problems. A novel method improving the classical parametric method for this biobjective problem is proposed. The algorithm runs in O(Nn(m +nlogn)) time determining all extreme supported non-dominated points in the outcome space and one extreme supported efficient solution associated with each one of them. Here n is the number of nodes, m is the number of arcs and N is the number of extreme supported non-dominated points in outcome space for the BMCF problem. The memory space required by the algorithm is O(n +m) when the extreme supported efficient solutions are not required to be stored in RAM. Otherwise, the algorithm requires O(N +m) space. Extensive computational experiments comparing the performance of the proposed method and a standard parametric network simplex method are presented.

Weighted sum model with partial preference information: Application to multi-objective optimization

Multi-objective optimization problems often lead to large nondominated sets, as the size of the problem or the number of objectives increases. Generating the whole nondominated set requires significant computation time, while most of the corresponding solutions are irrelevant to the decision maker (DM). Optimizing an aggregation function reduces the computation time and produces one or a very limited number of more focused solutions. This requires, however, the elicitation of precise preference parameters, which is often difficult and partly arbitrary, and might discard solutions of interest. An intermediate approach consists in using partial preference information with an aggregation function. In this work, we present a preference relation based on the weighted sum aggregation, where weights are not precisely defined. We give some properties of this preference relation and define the set of preferred points as the set of nondominated points with respect to this relation. We provide an efficient and generic way of generating this preferred set using any standard multi-objective optimization algorithm. This approach shows competitive performances both on computation time and quality of the generated preferred set.

Integrating column generation in a method to compute a discrete representation of the non-dominated set of multi-objective linear programmes

In this paper we propose the integration of column generation in the revised normal boundary intersection (RNBI) approach to compute a representative set of non-dominated points for multi-objective linear programmes (MOLPs). The RNBI approach solves single objective linear programmes, the RNBI subproblems, to project a set of evenly distributed reference points to the non-dominated set of an MOLP. We solve each RNBI subproblem using column generation, which moves the current point in objective space of the MOLP towards the non-dominated set. Since RNBI subproblems may be infeasible, we attempt to detect this infeasibility early. First, a reference point bounding method is proposed to eliminate reference points that lead to infeasible RNBI subproblems. Furthermore, different initialisation approaches for column generation are implemented, including Farkas pricing. We investigate the quality of the representation obtained. To demonstrate the efficacy of the proposed approach, we apply it to an MOLP arising in radiotherapy treatment design. In contrast to conventional optimisation approaches, treatment design using column generation provides deliverable treatment plans, avoiding a segmentation step which deteriorates treatment quality. As a result total monitor units is considerably reduced. We also note that reference point bounding dramatically reduces the number of RNBI subproblems that need to be solved.

Discrete representation of the non-dominated set for multi-objective optimization problems using kernels

In this paper, we are interested in producing discrete and tractable representations of the set of non-dominated points for multi-objective optimization problems, both in the continuous and discrete cases. These representations must satisfy some conditions of coverage, i.e. providing a good approximation of the non-dominated set, spacing, i.e. without redundancies, and cardinality, i.e. with the smallest possible number of points. This leads us to introduce the new concept of (ε, ε′)-kernels, or ε-kernels when ɛ′=ɛ is possible, which correspond to ε-Pareto sets satisfying an additional condition of ε′-stability. Among these, the kernels of small, or possibly optimal, cardinality are claimed to be good representations of the non-dominated set.We first establish some general properties on ε-kernels. Then, for the bi-objective case, we propose some generic algorithms computing in polynomial time either an ε-kernel of small size or, for a fixed size k, an ε-kernel with a nearly optimal approximation ratio 1+ɛ. For more than two objectives, we show that ε-kernels do not necessarily exist but that (ε, ε′)-kernels with ɛ′≤1+ɛ−1 always exist. Nevertheless, we show that the size of a smallest (ε, ε′)-kernel can be very far from the size of a smallest ε-Pareto set.

A branch-and-bound based heuristic algorithm for convex multi-objective MINLPs

We study convex multi-objective Mixed Integer Non-Linear Programming problems (MINLPs), which are characterized by multiple objective functions and non linearities, features that appear in real-world applications. To derive a good approximated set of non-dominated points for convex multi-objective MINLPs, we propose a heuristic based on a branch-and-bound algorithm. It starts with a set of feasible points, obtained, at the root node of the enumeration tree, by iteratively solving, with an ε-constraint method, a single objective model that incorporates the other objective functions as constraints. Lower bounds are derived by optimally solving Non-Linear Programming problems (NLPs). Each leaf node of the enumeration tree corresponds to a convex multi-objective NLP, which is solved heuristically by varying the weights in a weighted sum approach. In order to improve the obtained points and remove dominated ones, a tailored refinement procedure is designed. Although the proposed method makes no assumptions on the number of objective functions nor on the type of the variables, we test it on bi-objective mixed binary problems. In particular, as a proof-of-concept, we tested the proposed heuristic algorithm on instances of a real-world application concerning power generation, and instances of the convex biobjective Non-Linear Knapsack Problem. We compared the obtained results with those derived by well-known scalarization methods, showing the effectiveness of the proposed method.

A Benson type algorithm for nonconvex multiobjective programming problems

In this paper, an approximation algorithm for solving nonconvex multiobjective programming problems (NCMOPs) is presented. We modify Benson’s method using cones instead of hyperplanes. This algorithm uses an inner approximation and an outer approximation to generate (weakly) efficient solutions and (weakly \(\varepsilon \)-) nondominated points of NCMOPs. Some numerical examples are presented to clarify the proposed algorithm.

A Method for Generating a Well-Distributed Pareto Set in Multiple Objective Mixed Integer Linear Programs Based on the Decision Maker’s Initial Aspiration Level

This paper attempt to generate a representative subset of the Pareto optimal set for multiple objective mixed integer linear programming problem using the weighted L1 norm distance. The procedure presented in this paper is somewhat similar to the one used in the ideal-point methods and its aim is to generate at each iteration the closest-points to the ideal vector corresponding to the decision maker’s initial aspiration level for a new tradeoff parameter. Unlike most of the known algorithms for generating a discrete representation of the Pareto optimal set, the procedure generates at each iteration a nondominated point by solving only one mixed integer linear programming problem. The obtained solution minimizes the weighted L1 norm distance to the ideal vector with respect to the distance between the ideal vector and previously found vectors. More generally, this approach is able to generate all Pareto optimal solutions, where all of the decision variables are restricted to be integer. In order to explain the presented details, several illustrative examples are provided.

Greedy Hypervolume Subset Selection in Low Dimensions.

Given a nondominated point set [Formula: see text] of size [Formula: see text] and a suitable reference point [Formula: see text], the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size [Formula: see text] that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and/or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of [Formula: see text]. In contrast, the best upper bound available for [Formula: see text] is [Formula: see text]. Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of [Formula: see text] using a greedy strategy. In this article, greedy [Formula: see text]-time algorithms for the HSSP in 2 and 3 dimensions are proposed, matching the complexity of current exact algorithms for the 2-dimensional case, and considerably improving upon recent complexity results for this approximation problem.

Discrete representation of non-dominated sets in multi-objective linear programming

In this paper we address the problem of representing the continuous but non-convex set of non-dominated points of a multi-objective linear programme by a finite subset of such points. We prove that a related decision problem is NP-complete. Moreover, we illustrate the drawbacks of the known global shooting, normal boundary intersection and normal constraint methods concerning the coverage error and uniformity level of the representation by examples. We propose a method which combines the global shooting and normal boundary intersection methods. By doing so, we overcome their limitations, but preserve their advantages. We prove that our method computes a set of evenly distributed non-dominated points for which the coverage error and the uniformity level can be guaranteed. We apply this method to an optimisation problem in radiation therapy and present illustrative results for some clinical cases. Finally, we present numerical results on randomly generated examples.

Efficient computation of the search region in multi-objective optimization

Multi-objective optimization procedures usually proceed by iteratively producing new solutions. For this purpose, a key issue is to determine and efficiently update the search region, which corresponds to the part of the objective space where new nondominated points could lie. In this paper we elaborate a specific neighborhood structure among local upper bounds. Thanks to this structure, the update of the search region with respect to a new point can be performed more efficiently compared to existing approaches. Moreover, the neighborhood structure provides new insight into the search region and the location of nondominated points.

An evolutionary approach to generalized biobjective traveling salesperson problem

We consider the generalized biobjective traveling salesperson problem, where there are a number of nodes to be visited and each node pair is connected by a set of edges. The final route requires finding the order in which the nodes are visited (tours) and finding edges to follow between the consecutive nodes of the tour. We exploit the characteristics of the problem to develop an evolutionary algorithm for generating an approximation of nondominated points. For this, we approximate the efficient tours using approximate representations of the efficient edges between node pairs in the objective function space. We test the algorithm on several randomly-generated problem instances and our experiments show that the evolutionary algorithm approximates the nondominated set well.

The Quadrant Shrinking Method: A simple and efficient algorithm for solving tri-objective integer programs

We present a new variant of the full 2-split algorithm, the Quadrant Shrinking Method (QSM), for finding all nondominated points of a tri-objective integer program. The algorithm is easy to implement and solves at most 3|YN|+1 single-objective integer programs when computing the nondominated frontier, where YN is the set of all nondominated points. A computational study demonstrates the efficacy of QSM.

An interactive approach for biobjective integer programs under quasiconvex preference functions

We develop an interactive algorithm for biobjective integer programs that finds the most preferred solution of a decision maker whose preferences are consistent with a quasiconvex preference function to be minimized. During the algorithm, preference information is elicited from the decision maker. Based on this preference information and the properties of the underlying quasiconvex preference function, the algorithm reduces the search region and converges to the most preferred solution progressively. Finding the most preferred solution requires searching both supported and unsupported nondominated points, where the latter is harder. We develop theory to further restrict the region where unsupported nondominated points may lie. We demonstrate the algorithm on the generalized biobjective traveling salesperson problem where there are multiple efficient edges between node pairs and show its performance on a number of randomly generated instances.

A new method for optimizing a linear function over the efficient set of a multiobjective integer program

We present a new algorithm for optimizing a linear function over the set of efficient solutions of a multiobjective integer program (MOIP). The algorithm’s success relies on the efficiency of a new algorithm for enumerating the nondominated points of a MOIP, which is the result of employing a novel criterion space decomposition scheme which (1) limits the number of subspaces that are created, and (2) limits the number of sets of disjunctive constraints required to define the single-objective IP that searches a subspace for a nondominated point. An extensive computational study shows that the efficacy of the algorithm. Finally, we show that the algorithm can be easily modified to efficiently compute the nadir point of a multiobjective integer program.

A matheuristic approach to solve the multiobjective beam angle optimization problem in intensity‐modulated radiation therapy

AbstractSelecting a suitable set of beam angles is an important but difficult task in intensity‐modulated radiation therapy (IMRT) for cancer treatment. From a single objective point of view, this problem, known as the beam angle optimization (BAO) problem, is solved by finding a beam angle configuration (BAC) that leads to the best dose distribution, according to some objective function. As there exists a trade‐off between the main goals in IMRT (to irradiate the tumor according to some prescription and to avoid surrounding healthy tissue), it makes sense to solve this problem from a multiobjective (MO) point of view. When doing so, a solution of the BAO problem is no longer a single BAC, but instead a set of BACs that lead to a set of dose distributions that, depending on both dose prescription and physician preferences, can be selected as the preferred treatment. We solve this MO problem using a two‐phase strategy. During the first phase, a deterministic local search algorithm is used for selecting a set of locally optimal BACs, according to a single‐objective function. During this search, an optimal dose distribution for each BAC, with respect to the single‐objective function, is calculated using an exact nonlinear programming algorithm. During the second phase, a set of nondominated points is generated for each promising locally optimal BAC and a dominance analysis among them is performed. The output of the procedure is a set of (approximately) efficient BACs that lead to good dose distributions. To demonstrate the viability of the method, the two‐phase strategy is applied to a prostate case.

A Method for Constrained Multiobjective Optimization Based on SQP Techniques

We propose a method for constrained and unconstrained nonlinear multiobjective optimization problems that is based on an SQP-type approach. The proposed algorithm maintains a list of nondominated points that is improved both for spread along the Pareto front and optimality by solving single-objective constrained optimization problems. These single-objective problems are derived as SQP problems based on the given nondominated points. Under appropriate differentiability assumptions we discuss convergence to local optimal Pareto points. We provide numerical results for a set of unconstrained and constrained multiobjective optimization problems in the form of performance and data profiles, where several performance metrics are used. The numerical results confirm the superiority of the proposed algorithm against a state-of-the-art multiobjective solver and a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort necessary to compute the approximation.

Norm–based approximation in E-[0,1] convex multi-objective programming

This paper addresses the problem of capturing nondominated points on non-convex pareto frontier, which are encountered in E-[0,1] convex multi-objective programming problems. We use E-[0,1] map to transfer non-convex pareto frontier to convex pareto frontier, then An algorithm to find a piecewise linear approximation of the nondominated set of convex pareto frontier are applied. Finally, the inverse map of E-[0,1] is used to get the nondominated set of non-convex pareto frontier.

Using Dynamic Programming Technique for Solving a Class of Bi-Criteria Multistage Transportation Problem

This paper presents a solution algorithm to solve one class of bicriteria multistage transportation problem. This class represents bi-criteria multistage transportation problem without any restrictions on the intermediate stages. A way to find the non-dominated extreme points in the criteria space is developed. This method involves a parametric search in the criteria space. The presented algorithm is based also on fruitful application of the methods for treating transportation problems and the dynamic programming technique to obtain the shortest route for transportation networks and methods for solving bicriteria linear programming. The solution algorithm as well as an illustrative example are also included in this paper.