The relationship between traffic around schools and children’s exposure to pollution is well established. Few studies investigated the extent to which parental decisions to drive or walk to school shape children’s exposure to traffic emissions, as well as the moderating role of traffic-control conditions. In this study, considering different traffic control scenarios, we examined the relationship between traffic emissions resulting from children being driven to school and the pollutant dose experienced by the children walking to school. A simulation-based framework was used to quantify the combined effects of parents’ mode choice (either walk or drive) and traffic control measures (e.g. speed limit, and delay-based actuated signal controls) on traffic related pollutant concentration and the dose experienced by children walking to school. The combined impact assessment showed that the effectiveness of emission reduction strategies around the school was found to vary depending on the school location (high and low background traffic scenario), the proportion of children driven to school, and the traffic control condition. Results revealed the gross inequality of the impacts of high car use, as the dose per child walking to school was significantly higher than that for a child driven to school. For both school locations, reducing school-bound cars and controlling speed were found to result in a decrease in PMx dose experienced by walkers. The results from this study provide key insights into school travel plan management strategies in different school locations.

We review major developments in multi-objective optimization over the past decades. Although mathematical foundations and basic concepts have been established earlier, substantial progress in methods for constructing and identifying preferred solutions started in the late 1950s. We classify these approaches into two broad categories: mathematical programming-based and population-based. The former originated in the late 1950s, and its growth accelerated from the 1970s onward. We differentiate between approaches dealing with problems that operate in a continuous solution space and combinatorial problems where some variables are restricted to integer values. Population-based approaches flourished in the 1990s. Our focus is on evolutionary computation techniques that either aim to discover the entire Pareto front or incorporate the decision maker’s preferences to select the most favorable solution(s) or bias the search toward preferred regions. For all categories, we discuss those approaches that, in our opinion, have made major impacts. We examine current research trends and speculate on future directions in the field.

Optimal Camera Placement (OCP) is the process of finding a subset of cameras that either maximises the coverage, such that the cost of cameras is reduced, or minimises the total cost of cameras, such that coverage constraints are satisfied. By adopting the latter formulation, the OCP problem can be formulated as a Set Covering Problem (SCP), as the concepts of the two problems are inherently similar. Until recently, the literature has not explicitly discussed this similarity. Hence, this paper examines the OCP problem by leveraging the formulation established in prior research. Our focus lies in the practical application, as we implement the model on all instances to derive meaningful insights. Furthermore, we explore techniques from the SCP literature that can be applied to address the OCP problem in future studies. In this study, we address 69 problem instances, utilising a benchmark set generated by other researchers. These instances were employed as part of the GECCO 2021 competition on the optimal camera placement problem and the unicost set covering problem. We provide detailed results, and we conclude with recommendations for future research.

Various dominance structures have been proposed in the multi-objective optimization literature. However, a systematic procedure to understand their effect in determining the resulting optimal set for generic domination principles, besides the standard Pareto-dominance principle, is lacking. In this paper, we analyze and lay out properties of generalized dominance structures which help provide insights for resulting optimal solutions. We introduce the concept of the anti-dominance structure, derived from the chosen dominance structure, to explain how the resulting non-dominated or optimal set can be identified easily compared to using the dominance structure directly. The concept allows a unified explanation of optimal solutions for both single- and multi-objective optimization problems. The anti-dominance structure is applied to analyze respective optimal solutions for most popularly used static and spatially changing dominance structures. The theoretical and deductive results of this study can be utilized to create more meaningful dominance structures for practical problems, understand and identify resulting optimal solutions, and help develop better test problems and algorithms for multi-objective optimization.

In this paper, an infeasible interior-point technique is proposed to generate the nondominated set of nonlinear multi-objective optimization problems with the help of the direction-based cone method. We derive the proposed method for both convex and nonconvex problems. In order to solve the parametric optimization problems of the cone method, the infeasible interior-point method starts with an initial iterate outside the feasible region, and then gradually reduces the primal and dual infeasibility measures and the objective function value across the iterations with the help of a merit function. Estimates of the reduction of primal and dual infeasibility parameters per iteration are given. The convergence analysis of the method and an estimate of the number of iterations to reach an ϵ-precise solution are also provided. We provide the performance of the proposed methods on a variety of convex and nonconvex multi-objective test problems. Performance comparison between the proposed method and popular existing solvers is provided with respect to two performance measures and the corresponding relative efficiency measures. The reduction of a combined infeasibility measure, as the iterations progress, on the test problems is also shown graphically.

we decided to organise a special issue on theory, computation, and practice of multiobjective optimisation.Since at the two conferences many presentations addressed a variety of different multiobjective optimisation problems, we decided to focus this special issue distinctively on recent developments in multiobjective optimisation falling within the a posteriori paradigm of multiple criteria decision making (MCDM).Motivated by the prevalence of presentations on this topic, our goal was to give the international community an opportunity to publish papers proposing models, methods, and algorithms for multiobjective optimisation and their supporting mathematical theory.In addition, to make the future volume appealing to scientists, engineers, and practitioners, the final call for papers also asked for manuscripts describing important applications of multiobjective optimisation in practice.In total, we received 38 submissions for this issue.Of these submissions, 15 papers were out of scope by addressing other topics in the MCDM area; 9 papers were rejected following reviews; 1 paper was withdrawn by the authors during the review process; and 13 papers were accepted.These 13 papers constitute this special issue.The topics addressed in these papers follow the recent trends observed in the optimisation area in general.The type of optimisation problems addressed ranges from scheduling problems with two objectives to mixed integer linear optimisation problems and nonlinear optimization problems, both with only continuous and with continuous as well as binary variables.Some multiobjective models are specifically bi-or tri-objective while the methods include exact, heuristic, or hybrid algorithms to compute or approximate the Pareto set of these problems.Exact methods are typically used for small-size problems while heuristic or hybrid algorithms are designed for large-scale instances for which they prove to be competitive.The presented applications reflect the type of decision-making situations that are important but challenging and therefore of interest to researchers.

Radiotherapy treatment (RT) irradiates a patient's tumour volume while minimising damage to healthy tissue and surrounding critical organs at risk (OAR). In the conventional RT planning process, the RT planner has to iteratively adjust either the planning objectives (tumour or OAR dose levels) or the weights of the planning objectives until an acceptable plan is obtained that satisfies the minimum requirements. At the end of this iterative process, it remains unknown whether this plan is the best that can be obtained for the patient. The oncologist reviews each plan and decides to either treat using this plan or request further plan development, which may or may not lead to an actual improvement of the reviewed plan. We describe how Data Envelopment Analysis (DEA) is used as a real-time decision support tool to assess quality of RT plans for head and neck cancer patients by applying a knowledge-based comparison of each new plan to a library of previous clinically approved plans. This library allows benchmarking, which gives planners and oncologists a better idea of the relative quality of their plan and its improvement potential, resulting in improved use of resources and better quality treatments for patients. Our DEA-based approach provides a novel way of capturing multiple measures of plan quality as well as anatomical differences between patients in the benchmarking process. We present the developed DEA model and results for a set of benchmark instances. Initial results of integrating DEA-based quality feedback into the RT planning process are presented showing that operations research can contribute significantly to planning quality in this setting. (c) 2021 Elsevier B.V. All rights reserved.

During the last decades, research in multi-objective optimisation has seen considerable growth. However, this activity has been focused on linear, non-linear, and combinatorial optimisation with multiple objectives. Multi-objective mixed integer (linear or non-linear) programming has received considerably less attention. In this paper we propose an algorithm to compute a finite set of non-dominated points/efficient solutions of a bi-objective mixed binary optimisation problems for which the sub-problems obtained when fixing the binary variables are convex, and there is a finite set of feasible binary variable vectors. Our method uses bound sets and exploits the convexity property of the sub-problems to find a set of efficient solutions for the main problem. Our algorithm creates and iteratively updates bounds for each vector in the set of feasible binary variable vectors, and uses these bounds to guarantee that a set of exact non-dominated points is generated. For instances where the set of feasible binary variable vectors is too large to generate such provably optimal solutions within a reasonable time, our approach can be used as a matheuristic by heuristically selecting a promising subset of binary variable vectors to explore. This investigation is motivated by the problem of beam angle optimisation arising in radiation therapy planning, which we solve heuristically to provide numerical results.

By rerouting and retiming trains in real-time, the propagation of reactionary delay in complex station areas can be reduced. In this study, we propose a new optimisation model and solution algorithm that can be used to determine the best combination of route and schedule changes to make. Whilst several models have been proposed to tackle this problem, existing models either lack sufficient detail, or cannot be solved to optimality within the stringent time limits associated with a real-time environment. We formulate the problem as a multicommodity flow problem on a time-space graph with a novel representation of the track capacity constraints and a new objective function based on utility maximisation. We present a tailored branch-and-price solution algorithm and test it on a set of new instances based on real data. Our experiments show that most of these instances can be solved to optimality within 20 seconds, and provably near-optimal solutions can be found for the remainder. Keywords— railway optimization, timetable rescheduling, multicommodity flow, branchand-price

Data envelopment analysis is a linear programming-based operations research technique for performance measurement of decision-making units. In this paper, we investigate data envelopment analysis from a multiobjective point of view to compute both the efficient extreme points and the efficient facets of the technology set simultaneously. We introduce a dual multiobjective linear programming formulation of data envelopment analysis in terms of input and output prices and propose a procedure based on objective space algorithms for multiobjective linear programmes to compute the efficient frontier. We show that using our algorithm, the efficient extreme points and facets of the technology set can be computed without solving any optimization problems. We conduct computational experiments to demonstrate that the algorithm can compute the efficient frontier within seconds to a few minutes of computation time for real-world data envelopment analysis instances. For large-scale artificial data sets, our algorithm is faster than computing the efficiency scores of all decision-making units via linear programming.

Most real-world optimization problems are multi-objective by nature, with conflicting and incomparable objectives. Solving a multi-objective optimization problem requires a method that can generate all rational compromises between the objectives. This paper proposes two distinct bound set-based branch-and-cut algorithms for general bi-objective combinatorial optimization problems based on implicit and explicit lower-bound sets. The algorithm based on explicit lower-bound sets computes, for each branching node, a lower-bound set and compares it with an upper-bound set. The other fathoms branching nodes by generating a single point on the lower-bound set for each local nadir point. We outline several approaches for fathoming branching nodes, and we propose an updating scheme for the lower-bound sets that prevents us from solving the bi-objective linear programming relaxation of each branching node. To strengthen the lower-bound sets, we propose a bi-objective cutting-plane algorithm that adjusts the weights of the objective functions such that different parts of the feasible set are strengthened by cutting planes. In addition, we suggest an extension of the branching strategy “Pareto branching.” We prove the effectiveness of the algorithms through extensive computational results.

This book constitutes the refereed proceedings of the 5th International Conference on Evolutionary Multi-Criterion Optimization, EMO 2009, held in Nantes, France in April 2009. The 39 revised full pap

Data envelopment analysis is a linear programming-based operations research technique for performance measurement of decision-making units. In this paper, we investigate data envelopment analysis from a multi-objective point of view to compute both the efficient extreme points and efficient facets of the technology set simultaneously. We introduce a dual multi-objective linear programming formulation of data envelopment analysis in terms of input and output prices and propose a procedure based on objective space algorithms for multi-objective linear programmes to compute the efficient frontier. We show that using our algorithm, the efficient extreme points and facets of the technology set can be computed without solving any optimisation problems. We conduct computational experiments to demonstrate that the algorithm can compute the efficient frontier within seconds to a few minutes of computation time for real world data envelopment analysis instances. For large scale artificial data sets our algorithm is faster than computing the efficiency scores of all decision making units via linear programming.

This study considers a navigation method for finding the most preferable radiotherapy plan from a discrete set using planner-defined clinical criteria. The method is based on repeatedly solving an optimization model to identify a plan that best satisfies the aspiration values set by the planner. During navigation, the planner iteratively adjusts the aspiration values to match the preference information learned from previous plans until the most preferable plan is identified. The use of soft constraints to model aspiration values enables navigation amongst a discrete set and allows the planner to freely specify the aspiration values without producing an infeasible model. We demonstrate the use of the model by applying it to a prostate cancer case. This illustrates that improvements in optimization criteria do not necessarily lead to improvements in clinical criteria. Hence, the method obviates the need to simultaneously monitor both optimization and clinical criteria in current navigation systems. Instead, the direct use of clinical criteria for navigation aids the planner to quickly identify the most preferable plan.

Due to inherent trade-offs between tumour control and sparing of organs at risk, optimisation problems arising in intensity modulated radiation therapy planning are naturally modelled as multi-objective optimisation problems. Nevertheless, the vast majority of studies in the literature consider single objective approaches to these problems. The beam angle optimisation problem, that we address ion this paper, is one of these problems. It attempts to identify “good” beam angle configurations that allow the delivery of efficient treatment plans. In this paper two bi-objective local search algorithms are developed for the bi-objective beam angle optimisation problem, namely Pareto local search (PLS) and a variation of PLS we call adaptive PLS (aPLS). Both algorithms are able to find a set of (approximately) efficient beam angle configurations. While the PLS algorithm aims to find a set of efficient BACs by performing a very focused search over a specific region of the objective space, the aPLS algorithm aims to produce a set of efficient BACs that are well-distributed over the objective space. We test both algorithms on two prostate cancer cases and compare them to our previously proposed single objective local search algorithm.

Selecting 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 thisMOproblem 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.

In this paper, we propose a user equilibrium model considering the 3 most important factors influencing route choice behaviour in a road network, namely, travel time, travel time reliability, and monetary cost. We further develop the time surplus maximisation bi-objective user equilibrium model and incorporate the concept of travel time budget to model how users might react to uncertainty induced by day-to-day variability in travel time caused by traffic incidents. This results in a three-objective user equilibrium model, which has a possibly infinite set of equilibrium flows. To compute equilibrium flows, we introduce time budget surplus defined as the maximum travel time a user is willing to spend minus the actual time budget required for a desired level of travel time reliability. At equilibrium, for each origin-destination pair, all individuals are travelling on the path with the highest time budget surplus value among all the efficient paths between this origin-destination pair. This becomes a time budget surplus maximisation three-objective user equilibrium model (TBSmaxTUE). We show that the TBSmaxTUE model is a special case of three-objective user equilibrium considering minimisation of expected travel time, travel time variance, and toll (monetary cost) as objectives. We illustrate the model and our results on a small network.

Optimization over the efficient set of a multi-objective optimization problem is a mathematical model for the problem of selecting a most preferred solution that arises in multiple criteria decision-making to account for trade-offs between objectives within the set of efficient solutions. In this paper, we consider a particular case of this problem, namely that of optimizing a linear function over the image of the efficient set in objective space of a convex multi-objective optimization problem. We present both primal and dual algorithms for this task. The algorithms are based on recent algorithms for solving convex multi-objective optimization problems in objective space with suitable modifications to exploit specific properties of the problem of optimization over the efficient set. We first present the algorithms for the case that the underlying problem is a multi-objective linear programme. We then extend them to be able to solve problems with an underlying convex multi-objective optimization problem. We compare the new algorithms with several state of the art algorithms from the literature on a set of randomly generated instances to demonstrate that they are considerably faster than the competitors.