A comparative study of two key algorithms in multiple objective linear programming

Multiple objective linear programming problems are solved with a variety of algorithms. While these algorithms vary in philosophy and outlook, most of them fall into two broad categories: those that are decision space-based and those that are objective space-based. This paper reports the outcome of a computational investigation of two key representative algorithms, one of each category, namely the parametric simplex algorithm which is a prominent representative of the former and the primal variant of Bensons Outer-approximation algorithm which is a prominent representative of the latter. The paper includes a procedure to compute the most preferred nondominated point which is an important feature in the implementation of these algorithms and their comparison. Computational and comparative results on problem instances ranging from small to medium and large are provided

A Renewed Take on Weighted Sum in Sandwich Algorithms:Modification of the Criterion Space

Sandwich algorithms are commonly used to approximate the Pareto front of a multiobjective (MO) convex problem by enclosing it between an inner and outer approximation. By iteratively improving the approximations, the distance between them is minimized which gives an estimate of how well the Pareto front is approximated. A wellexplainable type of sandwich algorithm is based on weighted sum scalarization (WSS), where the next set of weights is determined by the most promising inner normal of the inner approximation. As these normals can contain negative values, not every optimization will result in finding an efficient point. In order to reduce the number of searches towards the dominated part, we propose an elegant modification of the criterion space which is an advancement on the formulation of Solanki et al. In addition to being well-explainable and easy to integrate within an existing optimization procedure, this modification is theoretically able to obtain all nondominated points of an MO linear programming problem in a finite number of expansions of the inner approximation. Furthermore, we propose two heuristic approaches to determine the distance between the inner and outer approximation that can be used as an alternative for the distance calculation of Solanki et al. These heuristics incorporate the ideas of Solanki et al. And Craft et al. to obtain straightforward and faster methods

A Renewed Take on Weighted Sum in Sandwich Algorithms:Modification of the Criterion Space

Sandwich algorithms are commonly used to approximate the Pareto front of a multiobjective (MO) convex problem by enclosing it between an inner and outer approximation. By iteratively improving the approximations, the distance between them is minimized which gives an estimate of how well the Pareto front is approximated. A wellexplainable type of sandwich algorithm is based on weighted sum scalarization (WSS), where the next set of weights is determined by the most promising inner normal of the inner approximation. As these normals can contain negative values, not every optimization will result in finding an efficient point. In order to reduce the number of searches towards the dominated part, we propose an elegant modification of the criterion space which is an advancement on the formulation of Solanki et al. In addition to being well-explainable and easy to integrate within an existing optimization procedure, this modification is theoretically able to obtain all nondominated points of an MO linear programming problem in a finite number of expansions of the inner approximation. Furthermore, we propose two heuristic approaches to determine the distance between the inner and outer approximation that can be used as an alternative for the distance calculation of Solanki et al. These heuristics incorporate the ideas of Solanki et al. And Craft et al. to obtain straightforward and faster methods

A Comparison of Benson’s Outer Approximation Algorithm with an Extended Version of Multiobjective Simplex Algorithm

The multiple objective simplex algorithm and its variants work in the decision variable space to find the set of all efficient extreme points of multiple objective linear programming (MOLP). Other approaches to the problem find either the entire set of all efficient solutions or a subset of them and also return the corresponding objective values (nondominated points). This paper presents an extension of the multiobjective simplex algorithm (MSA) to generate the set of all nondominated points and no redundant ones. This extended version is compared to Benson’s outer approximation (BOA) algorithm that also computes the set of all nondominated points of the problem. Numerical results on nontrivial MOLP problems show that the total number of nondominated points returned by the extended MSA is the same as that returned by BOA for most of the problems considered

On the simplex, interior-point and objective space approaches to multiobjective linear programming

Most Multiple Objective Linear Programming (MOLP) algorithms working in the decision variable space, are based on the simplex algorithm or interior-point method of Linear Programming. However, objective space based methods are becoming more and more prominent. This paper investigates three algorithms namely the Extended Multiobjective Simplex Algorithm (EMSA), Arbel’s Affine Scaling Interior-point (ASIMOLP) algorithm and Benson’s objective space Outer Approximation (BOA) algorithm. An extensive review of these algorithms is also included. Numerical results on non-trivial MOLP problems show that EMSA and BOA are at par and superior in terms of the quality of a most preferred nondominated point to ASIMOLP. However, ASIMOLP more than holds its own in terms of computing efficiency

Exact And Representative Algorithms For Multi Objective Optimization

In most real-life problems, the decision alternatives are evaluated with multiple conflicting criteria. The entire set of non-dominated solutions for practical problems is impossible to obtain with reasonable computational effort. Decision maker generally needs only a representative set of solutions from the actual Pareto front. First algorithm we present is for efficiently generating a well dispersed non-dominated solution set representative of the Pareto front which can be used for general multi objective optimization problem. The algorithm first partitions the criteria space into grids to generate reference points and then searches for non-dominated solutions in each grid. This grid-based search utilizes achievement scalarization function and guarantees Pareto optimality. The results of our experimental results demonstrate that the proposed method is very competitive with other algorithms in literature when representativeness quality is considered; and advantageous from the computational efficiency point of view. Although generating the whole Pareto front does not seem very practical for many real life cases, sometimes it is required for verification purposes or where DM wants to run his decision making structures on the full set of Pareto solutions. For this purpose we present another novel algorithm. This algorithm attempts to adapt the standard branch and bound approach to the multi objective context by proposing to branch on solution points on objective space. This algorithm is proposed for multi objective integer optimization type of problems. Various properties of branch and bound concept has been investigated and explained within the multi objective optimization context such as fathoming, node selection, heuristics, as well as some multi objective optimization specific concepts like filtering, non-domination probability, running in parallel. Potential of this approach for being used both as a full Pareto generation or an approximation approach has been shown with experimental studies

Operations Research Methods for Optimization in Radiation Oncology

Operations Research has a successful tradition of applying mathematical analysis to a wide range of applications, and problems in Medical Physics have been popular over the last couple of decades. The original application was in the optimal design of the uence map for a radiotherapy treatment, a problem that has continued to receive attention. However, Operations Research has been applied to other clinical problems like patient scheduling, vault design, and image alignment. The overriding theme of this article is to present how techniques in Operations Research apply to clinical problems, which we accomplish in three parts. First, we present the perspective from which an operations researcher addresses a clinical problem. Second, we succinctly introduce the underlying methods that are used to optimize a system, and third, we demonstrate how modern software facilitates problem design. Our discussion is supported by several publications to foster continued study. With numerous clinical, medical, and managerial decisions associated with a clinic, operations research has a promising future at improving how radiotherapy treatments are designed and delivered

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