A benchmark test problem toolkit for multi-objective path optimization

Due to the complexity of multi-objective optimization problems (MOOPs) in general, it is crucial to test MOOP methods on some benchmark test problems. Many benchmark test problem toolkits have been developed for continuous parameter/numerical optimization, but fewer toolkits reported for discrete combinational optimization. This paper reports a benchmark test problem toolkit especially for multi-objective path optimization problem (MOPOP), which is a typical category of discrete combinational optimization. With the reported toolkit, the complete Pareto front of a generated test problem of MOPOP can be deduced and found out manually, and the problem scale and complexity are controllable and adjustable. Many methods for discrete combinational MOOPs often only output a partial or approximated Pareto front. With the reported benchmark test problem toolkit for MOPOP, we can now precisely tell how many true Pareto points are missed by a partial Pareto front, or how large the gap is between an approximated Pareto front and the complete one

Two-phase strategies for the bi-objective minimum spanning tree problem

This paper presents a new two-phase algorithm for the bi-objective minimum spanning tree (BMST) prob-lem. In the first phase, it computes the extreme supported efficient solutions resorting to both mathematicalprogramming and algorithmic approaches, while the second phase is devoted to obtaining the remaining ef-ficient solutions (non-extreme supported and non-supported). This latter phase is based on a new recursiveprocedure capable of generating all the spanning trees of a connected graph through edge interchanges basedon increasing evaluation of non-zero reduced costs of associated weighted linear programs. Such a procedureexploits a common property of a wider class of problems to which the minimum spanning tree (MST) prob-lem belongs, that is the spanning tree structure of its basic feasible solutions. Computational experimentsare conducted on different families of graphs and with different types of cost. These results show that thisnew two-phase algorithm is correct, very easy to implement and it allows one to extract conclusions on thedifficulty of finding the entire set of Pareto solutions of the BMST problem depending on the graph topologyand the possible correlation of the edge cost

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

Network Interdiction under Uncertainty

We consider variants to one of the most common network interdiction formulations: the shortest path interdiction problem. This problem involves leader and a follower playing a zero-sum game over a directed network. The leader interdicts a set of arcs, and arc costs increase each time they are interdicted. The follower observes the leader\u27s actions and selects a shortest path in response. The leader\u27s optimal interdiction strategy maximizes the follower\u27s minimum-cost path. Our first variant allows the follower to improve the network after the interdiction by lowering the costs of some arcs, and the leader is uncertain regarding the follower\u27s cardinality budget restricting the arc improvements. We propose a multiobjective approach for this problem, with each objective corresponding to a different possible improvement budget value. To this end, we also present the modified augmented weighted Tchebychev norm, which can be used to generate a complete efficient set of solutions to a discrete multi-objective optimization problem, and which tends to scale better than competing methods as the number of objectives grows. In our second variant, the leader selects a policy of randomized interdiction actions, and the follower uses the probability of where interdictions are deployed on the network to select a path having the minimum expected cost. We show that this continuous non-convex problem becomes strongly NP-hard when the cost functions are convex or when they are concave. After formally describing each variant, we present various algorithms for solving them, and we examine the efficacy of all our algorithms on test beds of randomly generated instances

Ordered Weighted Average optimization in Multiobjective Spanning Tree Problem

Rework adversely impacts the performance of building projects. In this study, data were analyzed from 788 construction incidents in 40 Spanish building projects to determine the effects of project and managerial characteristics on rework costs. Finally, regression analysis was used to understand the relationships among contributing factors and to develop a model for rework prediction. Interestingly, the rework prediction model showed that only the original contract value (OCV) and the project location in relation to the company’s headquarters contributed to the regression model. Project type, type of organization, type of contract, and original contract duration (OCD), which represents the magnitude and complexity of a project, were represented by the OCV. This model for rework prediction based on original project conditions enables strategies to be put in place prior to the start of construction, to minimize uncertainties, to reduce impacts on project cost and schedule, and, thus, to improve productivity.Peer ReviewedPostprint (author's final draft

Biobjective Optimization over the Efficient Set Methodology for Pareto Set Reduction in Multiobjective Decision Making: Theory and Application

A large number of available solutions to choose from poses a significant challenge for multiple criteria decision making. This research develops a methodology that reduces the set of efficient solutions under consideration. This dissertation is composed of three major parts: (i) the formalization of a theoretical framework; (ii) the development of a solution approach; and (iii) a case study application of the methodology. In the first part, the problem is posed as a multiobjective optimization over the efficient set and considers secondary robustness criteria when the exact values of decision variables are subjected to uncertainties during implementation. The contributions are centered at the modeling of uncertainty directly affecting decision variables, the use of robustness to provide additional trade-off analysis, the study of theoretical bounds on the measures of robustness, and properties to ensure that fewer solutions are identified. In the second part, the problem is reformulated as a biobjective mixed binary program and the secondary criteria are generalized to any convenient linear functions. A solution approach is devised in which an auxiliary mixed binary program searches for unsupported Pareto outcomes and a novel linear programming filtering excludes any dominated solutions in the space of the secondary criteria. Experiments show that the algorithm tends to run faster than existing approaches for mixed binary programs. The algorithm enables dealing with continuous Pareto sets, avoiding discretization procedures common to the related literature. In the last part, the methodology is applied in a case study regarding the electricity generation capacity expansion problem in Texas. While water and energy are interconnected issues, to the best of our knowledge, this is the first study to consider both water and cost objectives. Experiments illustrate how the methodology can facilitate decision making and be used to answer strategic questions pertaining to the trade-off among different generation technologies, power plant locations, and the effect of uncertainty. A simulation shows that robust solutions tend to maintain feasibility and stability of objective values when power plant design capacity values are perturbed