Andrea Raith

A comparison of solution strategies for biobjective shortest path problems

We consider the biobjective shortest path (BSP) problem as the natural extension of the single-objective shortest path problem. BSP problems arise in various applications where networks usually consist of large numbers of nodes and arcs. Since obtaining the set of efficient solutions to a BSP problem is more difficult (i.e. NP-hard and intractable) than solving the corresponding single-objective problem there is a need for fast solution techniques. Our aim is to compare different strategies for solving the BSP problem. We consider a standard label correcting and label setting method, a purely enumerative near shortest path approach, and the two phase method, investigating different approaches to solving problems arising in phases 1 and 2. In particular, we investigate the two phase method with ranking in phase 2. In order to compare the different approaches, we investigate their performance on three different types of networks. We employ grid networks and random networks, as is generally done in the literature. Furthermore, road networks are utilized to compare performance on networks with a structure that is more likely to actually arise in applications.

A two-phase algorithm for the biobjective integer minimum cost flow problem

We present an algorithm to compute a complete set of efficient solutions for the biobjective integer minimum cost flow problem. We use the two phase method, with a parametric network simplex algorithm in phase 1 to compute all non-dominated extreme points. In phase 2, the remaining non-dominated points (non-extreme supported and non-supported) are computed using a k best flow algorithm on single-objective weighted sum problems. We implement the algorithm and report run-times on problem instances generated with a modified version of the NETGEN generator and also for networks with a grid structure.

Tolling Analysis with Bi-objective Traffic Assignment

We present a route choice model for a toll highway network based on bi-objective user equilibrium defined as the condition that traffic arranges itself in such a way that no individual trip maker can improve either their toll or travel time without worsening the other component by unilaterally switching routes. We develop heuristic traffic assignment procedures to find bi-objective user equilibrium solutions. Finally, we demonstrate that bi-objective user equilibrium is not equivalent to a bi-objective optimisation problem by studying a three link example network.

A Dijkstra-like method computing all extreme supported non-dominated solutions of the biobjective shortest path problem

We address the problem of determining all extreme supported solutions of the biobjective shortest path problem. A novel Dijkstra-like method generalizing DijkstraÂ?s algorithm to this biobjective case is proposed. The algorithm runs in O(N(m+nlogn)) time to solve one-to-one and one-to-all biobjective shortest path problems determining all extreme supported non-dominated points in the outcome space and one supported efficient path 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 points in outcome space for the one-to-all biobjective shortest path problem. The memory space required by the algorithm is O(n+m) for the one-to-one problem and O(N+m) for the one-to-all problem. A computational experiment comparing the performance of the proposed methods and state-of-the-art methods is included.

Bi-objective robust optimisation

It is important, in practice, to find robust solutions to optimisation problems. This issue has been the subject of extensive research focusing on single-objective problems. Recently, researchers also acknowledged the need to find robust solutions to multi-objective problems and presented some first results on this topic. In this paper, we deal with bi-objective optimisation problems in which only one objective function is uncertain. The contribution of our paper is three-fold. Firstly, we introduce and analyse four different robustness concepts for bi-objective optimisation problems with one uncertain objective function, and we propose an approach for defining a meaningful robust Pareto front for these types of problems. Secondly, we develop an algorithm for computing robust solutions with respect to these four concepts for the case of discrete optimisation problems. This algorithm works for finite and for polyhedral uncertainty sets using a transformation to a multi-objective (deterministic) optimisation problem and the recently published concept of Pareto robust optimal solutions (Iancu & Trichakis, 2014). Finally, we apply our algorithm to two real-world examples, namely aircraft route guidance and the shipping of hazardous materials, illustrating the four robustness concepts and their solutions in practical applications.

A framework for and empirical study of algorithms for traffic assignment

Traffic congestion is an issue in most cities worldwide. Transportation engineers and urban planners develop various traffic management projects in order to solve this issue. One way to evaluate such projects is traffic assignment (TA). The goal of TA is to predict the behaviour of road users for a given period of time (morning and evening peaks, for example). Once such a model is created, it can be used to analyse the usage of a road network and to predict the impact of implementing a potential project. The most commonly used TA model is known as user equilibrium, which is based on the assumption that all drivers minimise their travel time or generalised cost. In this study, we consider the static deterministic user equilibrium TA model.The constant growth of road networks and the need of highly precise solutions (required for select link analysis, network design, etc.) motivate researchers to propose numerous methods to solve this problem. Our study aims to provide a recommendation on what methods are more suitable depending on available computational resources, time and requirements on the solution. In order to achieve this goal, we implement a flexible software framework that maximises the usage of common code and, hence, ensures comparison of algorithms on common ground. In order to identify similarities and differences of the methods, we analyse groups of algorithms that are based on common principles. In addition, we implement and compare several different methods for solving sub-problems and discuss issues related to accumulated numerical errors that might occur when highly accurate solutions are required.

A bi-objective column generation algorithm for the multi-commodity minimum cost flow problem

We present a column generation algorithm for solving the bi-objective multi-commodity minimum cost flow problem. This method is based on the bi-objective simplex method and Dantzig–Wolfe decomposition. The method is initialised by optimising the problem with respect to the first objective, a single objective multi-commodity flow problem, which is solved using Dantzig–Wolfe decomposition. Then, similar to the bi-objective simplex method, our algorithm iteratively moves from one non-dominated extreme point to the next one by finding entering variables with the maximum ratio of improvement of the second objective over deterioration of the first objective. Our method reformulates the problem into a bi-objective master problem over a set of capacity constraints and several single objective linear fractional sub-problems each over a set of network flow conservation constraints. The master problem iteratively updates cost coefficients for the fractional sub-problems. Based on these cost coefficients an optimal solution of each sub-problem is obtained. The solution with the best ratio objective value out of all sub-problems represents the entering variable for the master basis. The algorithm terminates when there is no entering variable which can improve the second objective by deteriorating the first objective. This implies that all non-dominated extreme points of the original problem are obtained. We report on the performance of the algorithm on several directed bi-objective network instances with different characteristics and different numbers of commodities.

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

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 beam angle optimisation (BAO) problem, is solved by finding a beam angle configuration (BAC) that leads to the best dose distribution, according to some objective function. Because there exists a trade-off between the main goals in IMRT (to irradiate the tumour according to some prescription and to avoid surrounding healthy tissue) it makes sense to solve this problem from a multi-objective (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 which 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 to select 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 non-linear programming algorithm. During the second phase a set of non-dominated 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.

Quality assessment for VMAT prostate radiotherapy planning based on data envelopment analysis

The majority of commercial radiotherapy treatment planning systems requires planners to iteratively adjust the plan parameters in order to find a satisfactory plan. This iterative trial-and-error nature of radiotherapy treatment planning results in an inefficient planning process and in order to reduce such inefficiency, plans can be accepted without achieving the best attainable quality. We propose a quality assessment method based on data envelopment analysis (DEA) to address this inefficiency. This method compares a plan of interest to a set of past delivered plans and searches for evidence of potential further improvement. With the assistance of DEA, planners will be able to make informed decisions on whether further planning is required and ensure that a plan is only accepted when the plan quality is close to the best attainable one. We apply the DEA method to 37 prostate plans using two assessment parameters: rectal generalized equivalent uniform dose (gEUD) as the input and D95 (the minimum dose that is received by 95% volume of a structure) of the planning target volume (PTV) as the output. The percentage volume of rectum overlapping PTV is used to account for anatomical variations between patients and is included in the model as a non-discretionary output variable. Five plans that are considered of lesser quality by DEA are re-optimized with the goal to further improve rectal sparing. After re-optimization, all five plans improve in rectal gEUD without clinically considerable deterioration of the PTV D95 value. For the five re-optimized plans, the rectal gEUD is reduced by an average of 1.84 Gray (Gy) with only an average reduction of 0.07 Gy in PTV D95. The results demonstrate that DEA can correctly identify plans with potential improvements in terms of the chosen input and outputs.

Multi-objective minmax robust combinatorial optimization with cardinality-constrained uncertainty

Abstract In this paper, we develop two approaches to find minmax robust efficient solutions for multi-objective combinatorial optimization problems with cardinality-constrained uncertainty. First, we extend an existing algorithm for the single-objective problem to multi-objective optimization. We propose also an enhancement to accelerate the algorithm, even for the single-objective case, and we develop a faster version for special multi-objective instances. Second, we introduce a deterministic multi-objective problem with sum and bottleneck functions, which provides a superset of the robust efficient solutions. Based on this, we develop a label setting algorithm to solve the multi-objective uncertain shortest path problem. We compare both approaches on instances of the multi-objective uncertain shortest path problem originating from hazardous material transportation.