Workload Equity in Vehicle Routing Problems: A Survey and Analysis

Over the past two decades, equity aspects have been considered in a growing number of models and methods for vehicle routing problems (VRPs). Equity concerns most often relate to fairly allocating workloads and to balancing the utilization of resources, and many practical applications have been reported in the literature. However, there has been only limited discussion about how workload equity should be modeled in VRPs, and various measures for optimizing such objectives have been proposed and implemented without a critical evaluation of their respective merits and consequences. This article addresses this gap with an analysis of classical and alternative equity functions for biobjective VRP models. In our survey, we review and categorize the existing literature on equitable VRPs. In the analysis, we identify a set of axiomatic properties that an ideal equity measure should satisfy, collect six common measures, and point out important connections between their properties and those of the resulting Pareto-optimal solutions. To gauge the extent of these implications, we also conduct a numerical study on small biobjective VRP instances solvable to optimality. Our study reveals two undesirable consequences when optimizing equity with nonmonotonic functions: Pareto-optimal solutions can consist of non-TSP-optimal tours, and even if all tours are TSP optimal, Pareto-optimal solutions can be workload inconsistent, i.e. composed of tours whose workloads are all equal to or longer than those of other Pareto-optimal solutions. We show that the extent of these phenomena should not be underestimated. The results of our biobjective analysis are valid also for weighted sum, constraint-based, or single-objective models. Based on this analysis, we conclude that monotonic equity functions are more appropriate for certain types of VRP models, and suggest promising avenues for further research.Comment: Accepted Manuscrip

Nature of real-world multi-objective vehicle routing with evolutionary algorithms

The Vehicle Routing Problem with Time Windows VRPTW) is an important logistics problem which in the realworld appears to be multi-objective. Most research in this area has been carried out using classic datasets designed for the single-objective case, like the well-known Solomon's problem instances. Some unrealistic assumptions are usually made when using these datasets in the multi-objective case (e.g. assuming that one unit of travel time corresponds to one unit of travel distance). Additionally, there is no common VRPTW multiobjective oriented framework to compare the performance of algorithms because different implementations in the literature tackle different sets of objectives. In this work, we investigate the conflicting (or not) nature of various objectives in the VRPTW and show that some of the classic test instances are not suitable for conducting a proper multi-objective study. The insights of this study have led us to generate some problem instances using d ata from a real-world distribution company. Experiments in these new dataset using a standard evolutionary algorithm NSGA-II) show stronger evidence of multi-objective features. Our contribution focuses on achieving a better understanding about the multi-objective nature of the VRPTW, in particular the conflicting relationships between 5 objectives: number of vehicles, total travel distance, makespan, total waiting time, and total delay time

Immunity-based evolutionary algorithm for optimal global container repositioning in liner shipping

Global container repositioning in liner shipping has always been a challenging problem in container transportation as the global market in maritime logistics is complex and competitive. Supply and demand are dynamic under the ever changing trade imbalance. A useful computation optimization tool to assist shipping liners on decision making and planning to reposition large quantities of empty containers from surplus countries to deficit regions in a cost effective manner is crucial. A novel immunity-based evolutionary algorithm known as immunity-based evolutionary algorithm (IMEA) is developed to solve the multi-objective container repositioning problems in this research. The algorithm adopts the clonal selection and immune suppression theories to attain the Pareto optimal front. The proposed algorithm was verified with benchmarking functions and compared with four optimization algorithms to assess its diversity and spread. The developed algorithm provides a useful means to solve the problem and assist shipping liners in the global container transportation operations in an optimized and cost effective manner. © 2010 The Author(s).published_or_final_versionSpringer Open Choice, 21 Feb 201

Very Fast Non-dominated Sorting

New and very ecient parallel algorithm for the Fast Non-dominated Sorting of Pareto fronts is proposed. By decreasing its computational complexity, the application of the proposed method allows us to increase the speedup of the best up to now Fast and Elitist Multi-objective Genetic Algorithm (NSGA-II) more than two orders of magnitudes. Formal proofs of time complexities of basic as well as improved versions of the procedure are presented. Provided experimental results fully conrm theoretical ndings

Makespan minimizing on multiple travel salesman problem with a learning effect of visiting time

-The multiple traveling salesman problem (MTSP) involves the assignment and sequencing procedure simultaneously. The assignment of a set of nodes to each visitors and determining the sequence of visiting of nodes for each visitor. Since specific range of process is needed to be carried out in nodes in commercial environment, several factors associated with routing problem are required to be taken into account. This research considers visitors’ skill and category of customers which can affect visiting time of visitors in nodes. With regard to learning-by-doing, visiting time in nodes can be reduced. And different class of customers which are determined based on their potential purchasing of power specifies that required time for nodes can be vary. So, a novel optimization model is presented to formulate MTSP, which attempts to ascertain the optimum routes for salesmen by minimizing the makespan to ensure the balance of workload of visitors. Since this problem is an NP-hard problem, for overcoming the restriction of exact methods for solving practical large-scale instances within acceptable computational times. So, Artificial Immune System (AIS) and the Firefly (FA) metaheuristic algorithm are implemented in this paper and algorithms parameters are calibrated by applying Taguchi technique. The solution methodology is assessed by an array of numerical examples and the overall performances of these metaheuristic methods are evaluated by analyzing their results with the optimum solutions to suggested problems. The results of statistical analysis by considering 95% confidence interval for calculating average relative percentage of deviation (ARPD) reveal that the solutions of proposed AIS algorithm has less variation and Its’ confidence interval of closer than to zero with no overlapping with that of FA. Although both proposed meta-heuristics are effective and efficient in solving small-scale problems, in medium and large scales problems, AIS had a better performance in a shorter average time. Finally, the applicability of the suggested pattern is implemented in a case study in a specific company, namely Kalleh

Two-Stage Multi-Objective Meta-Heuristics for Environmental and Cost-Optimal Energy Refurbishment at District Level

Energy efficiency and environmental performance optimization at the district level are following an upward trend mostly triggered by minimizing the Global Warming Potential (GWP) to 20% by 2020 and 40% by 2030 settled by the European Union (EU) compared with 1990 levels. This paper advances over the state of the art by proposing two novel multi-objective algorithms, named Non-dominated Sorting Genetic Algorithm (NSGA-II) and Multi-Objective Harmony Search (MOHS), aimed at achieving cost-effective energy refurbishment scenarios and allowing at district level the decision-making procedure. This challenge is not trivial since the optimisation process must provide feasible solutions for a simultaneous environmental and economic assessment at district scale taking into consideration highly demanding real-based constraints regarding district and buildings’ specific requirements. Consequently, in this paper, a two-stage optimization methodology is proposed in order to reduce the energy demand and fossil fuel consumption with an affordable investment cost at building level and minimize the total payback time while minimizing the GWP at district level. Aimed at demonstrating the effectiveness of the proposed two-stage multi-objective approaches, this work presents simulation results at two real district case studies in Donostia-San Sebastian (Spain) for which up to a 30% of reduction of GWP at district level is obtained for a Payback Time (PT) of 2–3 years.Part of this work has been developed from results obtained during the H2020 “Optimised Energy Efficient Design Platform for Refurbishment at District Level” (OptEEmAL) project, Grant No. 680676

Green Vehicle Routing Problem with Safety and Social Concerns

Over the two last decades, distribution companies have been aware of the importance of paying attention to the all aspects of a distribution system simultaneously to be successful in the global market. These aspects are the economic, the environmental, the social and the safety aspects. In the Vehicle Routing Problem (VRP) literature, the economic issue has often been used, while the environmental, the safety and the social concerns have been less proportion of studies. The Green vehicle routing problem (GVRP) is one of the recent variants of the VRP, dealing with environmental aspects of distribution systems. In this paper, two developed mixed integer programming models are presented for the GVRP with social and safety concerns. Moreover, a Genetic Algorithm (GA) is developed to deal efficiently with the problem in large size. Different numerical analyses have performed to validate the presented algorithm in comparison to exact solutions and investigate the influence of several key factors like the effect of increasing the cost of safety aspect on route balancing, and customer waiting time. The results confirm that the proposed algorithm performs well and has more social and safety benefits (such as more balanced tours and fewer customers waiting time than the classic GVRP

Matheuristic algorithms for solving multi-objective/stochastic scheduling and routing problems

In der Praxis beinhalten Optimierungsprobleme oft unterschiedliche Ziele, welche optimiert werden sollen. Oft ist es nicht möglich die Ziele zu einem einzelnen Ziel zusammenzufassen. Mehrzieloptimierung beschäftigt sich damit, solche Probleme zu lösen. Wie in der Einzieloptimierung muss eine Lösung alle Nebenbedingungen des Problems erfüllen. Im Allgemeinen sind die Ziele konfligierend, sodass es nicht möglich ist eine einzelne Lösung zu finden welche optimal im Sinne aller Ziele ist. Algorithmen zum Lösen von Mehrziel-Optimierungsproblemen, präsentieren dem Entscheider eine Menge von effizienten Alternativen. Effizienz in der Mehrzieloptimierung ist als Pareto-Optimalität ausgedrückt. Eine Lösung eines Optimierungsproblems ist genau dann Pareto-optimal wenn es keine andere zulässige Lösung gibt, welche in allen Zielen mindestens gleich gut wie die betrachtete Lösung ist und besser in mindestens einem Ziel. In dieser Arbeit werden Mehrziel-Optimierungsprobleme aus zwei unterschiedlichen Anwendungsgebieten betrachtet. Das erste Problem, das Multi-objective Project Selection, Scheduling and Staffing with Learning Problem (MPSSSL), entstammt dem Management in forschungsorientierten Organisationen. Die Entscheider in solchen Organisationen stehen vor der Frage welche Projekte sie aus einer Menge von Projektanträgen auswählen sollen, und wie diese Teilmenge von Projekten (ein Projektportfolio) mit den benötigten Ressourcen ausgestattet werden kann (dies beinhaltet die zeitliche und personelle Planung). Aus unterschiedlichen Gründen ist dieses Problem schwer zu lösen, z.B. (i) die Auswahl von Projekten unter Beachtung der beschränkten Ressourcen ist ein Rucksackproblem (und ist damit NP-schwer) (ii) ob ein Projektportfolio zulässig ist oder nicht hängt davon ab ob, man dafür einen Zeitplan erstellen kann und genügend Mitarbeiter zur Verfügung stehen. Da in diesem Problem die Mitarbeiterzuordnung zu den einzelnen Projekten einbezogen wird, muss der Entscheider Ziele unterschiedlicher Art berücksichtigen. Manche Ziele sind ökonomischer Natur, z.B. die Rendite, andere wiederum beziehen sich auf die Kompetenzentwicklung der einzelnen Mitarbeiter. Ziele, die sich auf die Kompetenzentwicklung beziehen, sollen sicherstellen, dass das Unternehmen auch in Zukunft am Markt bestehen kann. Im Allgemeinen können diese unterschiedlichen Ziele nicht zu einem einzigen Ziel zusammengefasst werden. Daher werden Methoden zur Lösung von Mehrziel-Optimierungsproblemen benötigt. Um MPSSSL Probleme zu lösen werden in dieser Arbeit zwei unterschiedliche hybride Algorithmen betrachtet. Beide kombinieren nämlich Metaheuristiken (i) den Nondominated Sorting Genetic (NSGA-II) Algorithmus, und den (ii)~Pareto Ant Colony (P-ACO) Algorithmus, mit einem exakten Algorithmus zum Lösen von Linearen Programmen kombinieren. Unsicherheit ist ein weiterer wichtiger Aspekt der in der Praxis auftaucht. Unterschiedliche Parameter des Problems können unsicher sein (z.B. der aus einem Projekt erzielte Gewinn oder die Zeit bzw. der Aufwand, der benötigt wird, um die einzelnen Vorgänge eines Projekts abzuschließen). Um in diesem Fall das ``beste'' Projektportfolio zu finden, werden Methoden benötigt, welche stochastische Mehrziel-Optimierungsprobleme lösen können. Zur Lösung der stochastischen Erweiterung (SMPSSSL) des MPSSSL Problems zu lösen, präsentieren wir eine Methode, die den zuvor genannten hybriden NSGA-II Algorithmus mit dem Adaptive Pareto Sampling (APS) Algorithmus kombiniert. APS wird verwendet, um das Zusammenspiel von Simulation und Optimierung zu koordinieren. Zur Steigerung der Performance des Simulationsprozesses, verwenden wir Importance Sampling (IS). Das zweite Problem dieser Arbeit, das Bi-Objective Capacitated Vehicle Routing Problem with Route Balancing (CVRPB), kommt aus dem Bereich Logistik. Wenn man eine Menge von Kunden zu beliefern hat, steht man als Entscheider vor der Frage, wie man die Routen für eine fixe Anzahl von Fahrzeugen (mit beschränkter Kapazität) bestimmt, sodass alle Kunden beliefert werden können. Die Routen aller Fahrzeuge starten und enden dabei immer bei einem Depot. Die Einziel-Variante dieses Problems ist als Capacitated Vehicle Routing Problem (CVRP) bekannt, dessen Ziel es ist die Lösung zu finden, die die Gesamtkosten aller Routen minimiert. Dabei tritt jedoch das Problem auf, dass die Routen der optimalen Lösung sehr unterschiedliche Fahrtzeiten haben können. Unter bestimmten Umständen ist dies jedoch nicht erwünscht. Um dieses Problem zu umgehen, betrachten wir in dieser Arbeit eine Variante des (bezeichnet als CVRPB) CVRP, welche als zweite Zielfunktion die Balanziertheit der einzelnen Routen einbezieht. Zur Lösung von CVRPB Problemen verwenden wir die Adaptive Epsilon-Constraint Method in Kombination mit einem Branch-and-Cut Algorithmus und zwei unterschiedlichen Genetischen Algorithmen (GA), (i) einem Einziel-GA und (ii) dem NSGA-II. In dieser Arbeit werden Optimierungsalgorithmen präsentiert, welche es erlauben, Mehrziel- und stochastische Mehrziel-Optimierungsprobleme zu lösen. Unterschiedliche Algorithmen wurden implementiert und basierend auf aktuellen Performance-Maßen verglichen. Experimente haben gezeigt, dass die entwickelten Methoden gut geeignet sind, die betrachteten Optimierungsprobleme zu lösen. Die hybriden Algorithmen, welche Metaheuristiken mit exakten Methoden kombinieren, waren entweder ausschlaggebend um das Problem zu lösen (im Fall des Project Portfolio Selection Problems) oder konnten die Performance des Lösungsprozesses signifikant verbessern (im Fall des Vehicle Routing Problems).In practice decision problems often include different goals which can hardly be aggregated to a single objective for different reasons. In the field of multi-objective optimization several objective functions are considered. As in single objective optimization a solution has to satisfy all constraints of the problem. In general the goals are conflicting and there will be no solution, that is optimal for all objectives. Algorithms for multi-objective optimization problems provide the decision maker a set of efficient solutions, among which she or he can choose the most suitable alternative. In multi-objective optimization efficiency of a solution is expressed as Pareto-optimality. Pareto-optimality of a solution is defined as the property that no other solution exists that is better than the proposed one in at least one objective and at least equally good in all criteria. The first application that is considered in this thesis, the Multi-objective Project Selection, Scheduling and Staffing with Learning problem (MPSSSL) arises from the field of management in research-centered organizations. Given a set of project proposals the decision makers have to select the ``best'' subset of projects (a project portfolio) and set these up properly (schedule them and provide the necessary resources). This problem is hard to solve for different reasons: (i) selecting a subset of projects considering limited resources is a knapsack-type problem that is known to be NP-hard, and (ii) to determine the feasibility of a given portfolio, the projects have to be scheduled and staff must be assigned to them. As in this problem the assignment of workers is influenced by the decision which portfolio should be selected, the decision maker has to consider goals of different nature. Some objectives are related to economic goals (e.g. return of investment), others are related to the competence development of the workers. Competence oriented goals are motivated by the fact that competencies determine the attainment and sustainability of strategic positions in market competition. In general the objectives cannot be combined to a single objective, therefore methods for solving multi-objective optimization problems are used. To solve the problem we use two different hybrid algorithms that combine metaheuristic algorithms, (i) the Nondominated Sorting Genetic Algorithm (NSGA-II), and (ii) Pareto Ant Colony (P-ACO) algorithm with a linear programming solver as a subordinate. In practice, uncertainty is another typically encountered aspect. Different parameters of the problem can be uncertain (e.g. benefits of a project, or the time and effort required to perform the single activities required by a project). To determine the ``best'' portfolio, methods are needed that are able to handle uncertainty in optimization. To solve the stochastic extension (SMPSSSL) of the MPSSSL problem we present an algorithm that combines the aforementioned NSGA-II algorithm with the Adaptive Pareto Sampling (APS) algorithm. APS is used to handle the interplay between multi-objective optimization and simulation. The performance of the simulation process is increased by using importance sampling (IS). The second problem, the Bi-objective Capacitated Vehicle Routing Problem with Route Balancing (CVRPB) arises from the field of vehicle routing. Given a set of customers, the decision makers have to construct routes for a fixed number of vehicles, each starting and ending at the same depot, such that the demands of all customers can be fulfilled, and the capacity constraints of each vehicle are not violated. The traditional objective of this problem (known as the Capacitated Vehicle Routing Problem (CVRP)) is minimizing the total costs of all routes. A problem that may arise by this approach is that the resulting routes can be very unbalanced (in the sense of drivers workload). To overcome this problem a second objective function that measures the balance of the routes of a solution is introduced. In this work, we use the Adaptive Epsilon-Constraint Method in combination with a branch-and-cut algorithm and two genetic algorithms (i) a single-objective GA and (ii) the multi-objective NSGA-II, to solve the considered problem. Prototypes of different algorithms to solve the problems are developed and their performance is assessed by using state of the art performance measures. The computational experiments show that the developed solution procedures will be well suited to solve the considered optimization problems. The hybrid algorithms combining metaheuristic and exact optimization methods, turned out to be crucial to solve the problem (application to project portfolio selection) or to improve the performance of the solution procedure (application to vehicle routing)