Bi-objective Set Research Articles

A two-phase method based on the branch and bound algorithm for solving the bi-objective set covering problem

A two-phase method based on the branch and bound algorithm for solving the bi-objective set covering problem

A two-phase method based on the branch and bound algorithm for solving the bi-objective set covering problem

A two-phase method based on the branch and bound algorithm for solving the bi-objective set covering problem

Simultaneous scheduling of replacement and repair of common components in operating systems

In order for a system to stay operational, its components need maintenance. We consider two stakeholders—a system operator and a maintenance workshop—and a contract governing their joint activities. Components in the operating systems that are to be maintained are sent to the maintenance workshop, which should perform all maintenance activities on time in order to satisfy the contract. The maintained components are then sent back to be used in the operating systems. Our modeling of this system-of-systems includes stocks of damaged and repaired components, the workshop scheduling, and the planning of preventive maintenance for the operating systems. Our modeling is based on a mixed-binary linear optimization (MBLP) model of a preventive maintenance scheduling problem with so-called interval costs over a finite and discretized time horizon. We generalize and extend this model with the flow of components through the workshop, including the stocks of spare components. The resulting scheduling model—a mixed-integer optimization (MILP) model—is then utilized to optimize the main contract in a bi-objective setting: maximizing the availability of repaired (or new) components and minimizing the costs of maintaining the operating systems over the time horizon. We analyze the main contract and briefly discuss a turn-around time contract. Our results concern the effect of our modeling on the levels of the stocks of components over time, in particular minimizing the risk for lack of spare components.

A Set Covering Model for a Green Ship Routing and Scheduling Problem with Berth Time-Window Constraints for Use in the Bulk Cargo Industry

This paper presents a set covering model based on route representation to solve the green ship routing and scheduling problem (GSRSP) with berth time-window constraints for multiple bulk ports. A bi-objective set covering model is constructed with features based on the minimization of the total CO2 equivalent emissions and the total travel time subject to a limited number of berths in each port, berthing time windows, and the time window for each job. The solutions are obtained using the ε-constraint method, after which a Pareto frontier is plotted. This problem is motivated by the operations of feeder barges and terminals, where the logistics control tower is used to coordinate the routing and berthing time of its barges. We show that the proposed method outperforms the weighted sum method in terms of the number of Pareto solutions and the value of the hypervolume indicator.

A multiobjective analysis of the potential of scheduling electrical vehicle charging for flattening the duck curve

In this paper, we investigate the demand-flexibility of large-collections of electric vehicles (EVs) by scheduling their demand to flatten the electricity duck curve that emerge as a result of growing solar power production. The scheduling problem is investigated in a bi-objective setting and an additional objective function related to the amount of charge provided to EVs is also analyzed. The first objective is the minimization of the ramp-up requirements of the system. The second objective reflects the quality of service and the potential level of charging station's profit margins. An important characteristics of the proposed model is the effect of total charging capacity on the two objective functions. The analysis is carried out based on a quadratic programming model which is used to calculate the Pareto Front of the two objective functions. This is done through a case study based on real-world data for EV driver behavior, solar generation, and energy consumption. The computational experiments show that there is a high level of competition between these two objectives. Moreover, the effect of different maximal charging capacities on these objectives is observed.

Branch-and-Bound for Bi-objective Integer Programming

In bi-objective integer optimization the optimal result corresponds to a set of nondominated solutions. We propose a generic bi-objective branch-and-bound algorithm that uses a problem-independent branching rule exploiting available integer solutions and takes advantage of integer objective coefficients. The developed algorithm is applied to bi-objective facility location problems and the bi-objective set covering problem, as well as to the bi-objective team orienteering problem with time windows. In the latter case, lower bound sets are computed by means of column generation. Comparison with state-of-the-art exact algorithms shows the effectiveness of the proposed branch-and-bound algorithm.

Using column generation to compute lower bound sets for bi-objective combinatorial optimization problems

We discuss the use of column generation in a bi-objective setting. Just as in single objective combinatorial optimization, the role of column generation in the bi-objective setting is to compute dual bounds (i.e. lower bounds for minimization problems and upper bounds for maximization problems) which can be used to guide the search for efficient solutions or to evaluate the quality of approximate solutions. The general idea used in this paper is to first transform the bi-objective problem into single objective by a scalarization method and then solve the transformed problem several times by varying the necessary parameters. We show that irrespective of the scalarization method used, similar subproblems are solved when applying column generation. For this reason, we investigate possible ways of intelligently searching for columns for these subproblems in order to accelerate the column generation method.

Multi-objective Optimization in Dial-a-ride Public Transportation

In the Dial-a-Ride public transportation systems, each customer requirement is specified in terms of a pickup (origin), of a delivery (destination) and of a time window within it has to be satisfied. The aim is to find a set of routes, each assigned to a vehicle, in order to satisfy the set of requests, under capacity, time windows, precedence and pairing conditions. It is usually assumed that the demand of a request, picked up at its origin, is exactly delivered at its destination (one-to-one service) and that the fleet of the vehicles is based at a single depot. From a modelling point of view, the problem could be addressed as a one-to-one capacitated Pickup and Delivery Problem with Time Windows (PDPTW) and therefore, the mathematical formulation presents, beyond the traditional capacity constraints on the vehicles, also the pairing, the precedence and the time windows conditions. In particular, the pairing conditions guarantee that each couple (pickup, delivery) has to belong to the same route while the precedence constraints impose that each pickup has to be served before the associated delivery. This paper addresses the problem with the aim of optimizing, at the same time, the maximum total ride time and the total waiting time. Then, a bi-objective PDPTW with a constraint on the maximum duration of each route is proposed and solved by a two-step approach. In particular, the first step determines a set of feasible routes by meta-heuristics. These routes are used in second step in a bi-objective set partitioning formulation solved by the epsilon-constraint method to generate efficient solutions. The parameters of the meta-heuristics are properly set by a racing procedure. Computational experiments on some benchmark instances are carried out to assess the performance of the proposed approach.

Multi-directional local search

This paper introduces multi-directional local search, a metaheuristic for multi-objective optimization. We first motivate the method and present an algorithmic framework for it. We then apply it to several known multi-objective problems such as the multi-objective multi-dimensional knapsack problem, the bi-objective set packing problem and the bi-objective orienteering problem. Experimental results show that our method systematically provides solution sets of comparable quality with state-of-the-art methods applied to benchmark instances of these problems, within reasonable CPU effort. We conclude that the proposed algorithmic framework is a viable option when solving multi-objective optimization problems.

Evolutionary, constructive and hybrid procedures for the bi-objective set packing problem

Abstract The bi-objective set packing problem is a multi-objective combinatorial optimization problem similar to the well-known set covering/partitioning problems. To our knowledge and surprise, this problem has not yet been studied whereas several applications have been reported. Unfortunately, solving the problem exactly in a reasonable time using a generic solver is only possible for small instances. We designed three alternative procedures for approximating solutions to this problem. The first is derived from the original ‘Strength Pareto Evolutionary Algorithm’, which is a population-based metaheuristic. The second is an adaptation of the ‘Greedy Randomized Adaptative Search Procedure’, which is a constructive metaheuristic. As underlined in the overview of the literature summarized here, almost all the recent, effective procedures designed for approximating optimal solutions to multi-objective combinatorial optimization problems are based on a blend of techniques, called hybrid metaheuristics. Thus, the third alternative, which is the primary subject of this paper, is an original hybridization of the previous two metaheuristics. The algorithmic aspects, which differ from the original definition of these metaheuristics, are described, so that our results can be reproduced. The performance of our procedures is reported and the computational results for 120 numerical instances are discussed.

A comparison of stochastic programming and bi-objective optimisation approaches to robust airline crew scheduling

A prominent problem in airline crew scheduling is the pairings or Tour-of-Duty planning problem. The objective is to determine a set of pairings (or Tours-of-Duty) for a crew group to minimise the planned cost of operating a schedule of flights. However, due to unforeseen events the performance in operation can differ considerably from planning, sometimes causing significant additional recovery costs. In recent years there has been a growing interest in robust crew scheduling. Here, the aim is to find solutions that are "cheap" in terms of planned cost as well as being robust, meaning that they are less likely to be disrupted in case of delays. Taking the stochastic nature of delays into account, Yen and Birge (Transp Sci 40:3---14, 2006) formulate the problem as a two-stage stochastic integer programme and develop an algorithm to solve this problem. Based on the contradictory nature of the goals, Ehrgott and Ryan (J Multi-Criteria Decis Anal 11:139---150, 2002) formulate a bi-objective set partitioning model and employ elastic constraint scalarisation to enable the solution by set partitioning algorithms commercially used in crew scheduling software. In this study, we compare the two solution approaches. We improve the algorithm of Yen and Birge (Transp Sci 40:3---14, 2006) and implement both methods with a commercial crew scheduling software. The results of both methods are compared with respect to characteristics of robust solutions, such as the number of aircraft changes for crew. We also conduct experiments to simulate the performance of the obtained solutions. All experiments are performed using actual schedule data from Air New Zealand.

A Multiobjective Branch-and-Bound Framework: Application to the Biobjective Spanning Tree Problem

This paper focuses on a multiobjective derivation of branch-and-bound procedures. Such a procedure aims to provide the set of Pareto-optimal solutions of a multiobjective combinatorial optimization problem. Unlike previous works on this issue, the bounding is performed here via a set of points rather than a single ideal point. The main idea is that a node in the search tree can be discarded if one can define a separating hypersurface in the objective space between the set of feasible solutions in the subtree and the set of points corresponding to potential Pareto-optimal solutions. Numerical experiments on the biobjective spanning tree problem are provided that show the efficiency of the approach in a biobjective setting.

Two-phase method and Lagrangian relaxation to solve the Bi-Objective Set Covering Problem

This paper deals with the Bi-Objective Set Covering Problem, which is a generalization of the well-known Set Covering Problem. The proposed approach is a two-phase heuristic method which has the particularity to be a constructive method using the primal-dual Lagrangian relaxation to solve single objective Set Covering problems. The results show that this algorithm finds several potentially supported and unsupported solutions. A comparison with an exact method (up to a medium size), shows that many Pareto-optimal solutions are retrieved and that the other solutions are well spread and close to the optimal ones. Moreover, the method developed compares favorably with the Pareto Memetic Algorithm proposed by Jaszkiewicz.

A Comparative Study of Multiple-Objective Metaheuristics on the Bi-Objective Set Covering Problem and the Pareto Memetic Algorithm

The paper describes a comparative study of multiple-objective metaheuristics on the bi-objective set covering problem. Ten representative methods based on genetic algorithms, multiple start local search, hybrid genetic algorithms and simulated annealing are evaluated in the computational experiment. Nine of the methods are well known from the literature. The paper introduces also a new hybrid genetic algorithm called Pareto memetic algorithm. The results of the experiment indicate very good performance of hybrid genetic algorithms, however, no algorithm was able to outperform all other methods on all instances. Furthermore, the results indicate that the performance of multiple-objective metaheuristics may differ radically even if the methods are based on the same single objective algorithm and use exactly the same problem-specific operators.