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The Team Orienteering Problem with Service Times and Mandatory & Incompatible Nodes
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Abstract
The Team Orienteering Problem with Service Times and Mandatory & Incompatible Nodes (TOP-ST-MIN) is a variant of the classic Team Orienteering Problem (TOP), which includes three novel features that stem from two real-world problems previously studied by the authors. We prove that even finding a feasible solution is NP-complete. Two versions of this variant are considered in our study. For such versions, we proposed two alternative mathematical formulations, a mixed and a compact formulations. Based on the compact formulation, we developed a Cutting-Plane Algorithm (CPA) exploiting five families of valid inequalities. Extensive computational experiments showed that the CPA outperforms CPLEX in solving the new benchmark instances, generated in such a way to evaluate the impact of the three novel features that characterise the problem. The CPA is also competitive for the TOP since it is able to solve almost the same number of instances as the state-of-art algorithms.
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We extend the classical problem setting of the orienteering problem (OP) to incorporate multiple drones that cooperate with a truck to visit a subset of the input nodes. We call this problem the OP with multiple drones (OP-mD). Drones have a limited battery endurance, and thus, they can either move together with the truck at no energy cost for the battery or be launched by the truck onto short flights that must start and end at different customer locations. A drone serves exactly one customer per flight. Moreover, the truck and the drones must wait for each other at the landing locations. A customer prize can be collected at most once, either upon visiting it by the truck or upon serving it by a drone. Similarly to the OP, we maximize the total collected prize under the condition that the truck and the drones return to the depot within a given amount of time. We provide a mixed-integer linear programming formulation for the OP-mD and devise a tailored branch-and-cut algorithm based on a novel decomposition of the problem. We solve instances of the OP-mD with up to 50 nodes within one hour of CPU time with a standard computational setup. Finally, we adapt our framework to solve closely related problems in the literature and compare the resulting computational performance with that of previous studies.
Constructive methods are one of the main families of heuristic approaches to combinatorial optimization problems. They usually start from an empty set and subsequently add elements until a promising feasible solution is found. Their advantages are simplicity in design, analysis and implementation, and an intuitive structure. In general, though not always, they also have a limited computational complexity. An obvious counterpart is offered by destructive methods, which start from the whole ground set in which the solutions are embedded, and subsequently remove elements until they reach a promising feasible solution. This chapter describes the systematic development of a number of constructive and destructive methods for the MaxSum and the MaxMinSum diversity problems and their application, both as stand-alone approaches and to initialize an improvement metaheuristic.
Digital Contact Tracing (DCT) has been proved to be an effective tool to counteract the new SARS-CoV-2 or Covid-19. Despite this widespread effort to adopt the DCT, less attention has been paid to the organisation of the health logistics system that should support the tracing activities. Actually, the DCT poses a challenge to the logistics of the local health system in terms of number of daily tests to be collected and evaluated, especially when the spreading of the virus is soaring. In this paper we introduce a new optimisation problem called the Daily Swab Test Collection (DSTC) problem, that is the daily problem of collecting swab tests at home in such a way to guarantee a timely testing to people notified by the app to be in contact with a positive case. The problem is formulated as a variant of the team orienteering problem. The contributions of this paper are the following: (i) the new optimisation problem DSTC that complements and improves the DCT approach proposed by Ferretti et al. (Science https://doi.org/10.1126/science.abb6936, 2020), (ii) the DSCT formulation as a variant of the TOP and a literature review highlighting that this variant can have useful application in healthcare management, (iii) new realistic benchmark instances for the DSTC based on the city of Turin, (iv) two new efficient and effective hybrid algorithms capable to deal with realistic instances, (v) the managerial insights of our approach with a special regard on the fairness of the solutions. The main finding is that it possible to optimise the underlying logistics system in such a way to guarantee a timely testing to people recognised by the DCT.
The tourist trip design problem (TTDP) is a well-known extension of the orienteering problem, where the objective is to obtain an itinerary of points of interest for a tourist that maximizes his/her level of interest. In several situations, the interest of a point depends on when the point is visited, and the tourist may delay the arrival to a point in order to get a higher interest. In this paper, we present and discuss two variants of the TTDP with time-dependent recommendation factors (TTDP-TDRF), which may or may not take into account waiting times in order to have a better recommendation value. Using a mixed-integer linear programming solver, we provide solutions to 27 real-world instances. Although reasonable at first sight, we observed that including waiting times is not justified: in both cases (allowing or not waiting times) the quality of the solutions is almost the same, and the use of waiting times led to a model with higher solving times. This fact highlights the need to properly evaluate the benefits of making the problem model more complex than is actually needed.
Disaster management generally includes the post-disaster stage, which consists of the actions taken in response to the disaster damages. These actions include the employment of emergency plans and assigned resources to (i) rescue affected people immediately, (ii) deliver personnel, medical care and equipment to the disaster area, and (iii) aid to prevent the infrastructural and environmental losses. In the response phase, humanitarian logistics directly influence the efficiency of the relief operation. Ambulances routing problem is defined as employing the optimisation tools to manage the flow of ambulances for finding the best ambulance tours to transport the injured to hospitals. Researchers pointed out the importance of equity and fairness in humanitarian relief services: managing the operations of ambulances in the immediate aftermath of a disaster must be done impartially and efficiently to rescue affected people with different priority in accordance with the restrictions. Our research aim is to find the best ambulance tours to transport the patients during a disaster in relief operations while considering fairness and equity to deliver services to patients in balance. The problem is formulated as a new variant of the team orienteering problem with hierarchical objectives to address also the efficiency issue. Due to the limitation of solving the proposed model using a general-purpose solver, we propose a new hybrid algorithm based on a machine learning and neighbourhood search. Based on a new set of realistic benchmark instances, our quantitative analysis proves that our algorithm is capable to largely reduce the solution running time especially when the complexity of the problem increases. Further, a comparison between the fair solution and the system optimum solution is also provided.
Tourism is one of the fastest-growing sectors in the world with a shift from mass tourism to personalized travel. Nevertheless, it generates significant environmental impacts. The current events associated with quarantine measures generated by COVID-19 represent, however, a risk for this sector. It is hence necessary to create strategies that allow efficient decision-making for all echelons and actors for a rapid recovery. Tourists are key actors, which makes necessary to facilitate tourism trip planning according to tourists’ preferences as a complex process. In this paper, we propose a novel model of tourist trip planning for heterogeneous preferences in a tourist group and selection of transport modes, in the first instance, while a second step seeks at minimizing the level of CO2 emissions. A comparison of the two models is made considering the objectives associated with individual tourist benefits and group profit equity, in contrast to the inclusion of the cost of CO2 emissions. A numerical comparison is carried out with a total of 546 data sets. Results illustrate the conflict between those objectives by generating an inverse relationship between the individual and group profit equity of tourists, in addition to individual benefit and emission minimization.
This work is about mission planning in teams of mobile autonomous agents. We consider tasks that are spatially distributed, non atomic, and provide an utility for integral and also partial task completion. Agents are heterogeneous, therefore showing different efficiency when dealing with the tasks. The goal is to define a system-level plan that assigns tasks to agents to maximize mission performance. We define the mission planning problem through a model including multiple sub-problems that are addressed jointly: task selection and allocation, task scheduling, task routing, control of agent proximity over time. The problem is proven to be NP-hard and is formalized as a mixed integer linear program (MILP). Two solution approaches are proposed: one heuristic and one exact method. Both combine a generic MILP solver and a genetic algorithm, resulting in efficient anytime algorithms. To support performance scalability and to allow the effective use of the model when online continual replanning is required, a decentralized and fully distributed architecture is defined top-down from the MILP model. Decentralization drastically reduces computational requirements and shows good scalability at the expenses of only moderate losses in performance. Lastly, we illustrate the application of the mission planning framework in two demonstrators. These implementations show how the framework can be successfully integrated with different platforms, including mobile robots (ground and aerial), wearable computers, and smart-phone devices.
Many tourist applications provide personalized tourist tours which take into account the preferences defined by the user interests and also the constraints derived from location of places to visit, opening hours, time availability, etc. Although user preferences are usually related to the category of the points of interest, a user may also have preferences regarding the travel style, such as the rhythm of the trip, the number of visits to include in the tour or the length of these visits. The goal of this work is to create a customized tourist agenda that includes additional preferences related to the travel style so that the tourist experience in the city can be improved. This problem has been formulated as an extension of the Tourist Travel Design Problem, which is in turn based on the Orienteering Problem. We define several quality metrics in order to measure the user satisfaction with the tourist route generated by different solvers. We also perform extensive experiments in order to evaluate the obtained results.
We have constructed a multi-objective set orienteering problem to model real-world problems more fittingly than the existing models. It attaches (i) a predefined profit associated with each cluster of customers, and (ii) a preset maximum service time associated with each customer of all the clusters. When a customer from cluster visits, it allows the earning of a profit score. Our purpose is primarily to search for a route that (i) on the one hand allows us to service each cluster for maximizing customer satisfaction and (ii) on the other hand allows us to maximize our profits, too. The model assumes that the more time we spend on servicing, the more customer satisfaction it yields. It tries to cover as many clusters as possible within a specified time budget. In this paper, we also consider third-party logistics to allow the flexibility of ending our journey at any cluster of our choice. The proposed model is solved using the nondominated sorting genetic algorithm and the strength Pareto evolutionary algorithm. Here, we also generate the dataset to test the proposed model by using instances from the literature of the generalized traveling salesman problem. Finally, we present a comparative result analysis with the help of some statistical tools and discuss the results.
The Team Orienteering Problem (TOP) is an NP-hard routing problem in which a fleet of identical vehicles aims at collecting rewards (prizes) available at given locations, while satisfying restrictions on the travel times. In TOP, each location can be visited by at most one vehicle, and the goal is to maximize the total sum of rewards collected by the vehicles within a given time limit. In this paper, we propose a generalization of TOP, namely the Steiner Team Orienteering Problem (STOP). In STOP, we provide, additionally, a subset of mandatory locations. In this sense, STOP also aims at maximizing the total sum of rewards collected within the time limit, but, now, every mandatory location must be visited. In this work, we propose a new commodity-based formulation for STOP and use it within a cutting-plane scheme. The algorithm benefits from the compactness and strength of the proposed formulation and works by separating three families of inequalities, which consist of some general connectivity constraints, classical lifted cover inequalities based on dual bounds and a class of conflict cuts. To our knowledge, the last class of inequalities is also introduced in this work. A state-of-the-art branch-and-cut algorithm from the literature of TOP is adapted to STOP and used as baseline to evaluate the performance of the cutting-plane. Extensive computational experiments show the competitiveness of the new algorithm while solving several STOP and TOP instances. In particular, it is able to solve, in total, 15 more TOP instances than any other previous exact algorithm and finds eight new optimality certificates. With respect to the new STOP instances introduced in this work, our algorithm solves 30 more instances than the baseline.
Major advances were recently obtained in the exact solution of Vehicle Routing Problems (VRPs). Sophisticated Branch-Cut-and-Price (BCP) algorithms for some of the most classical VRP variants now solve many instances with up to a few hundreds of customers. However, adapting and reimplementing those successful algorithms for other variants can be a very demanding task. This work proposes a BCP solver for a generic model that encompasses a wide class of VRPs. It incorporates the key elements found in the best recent VRP algorithms: ng-path relaxation, rank-1 cuts with limited memory, and route enumeration; all generalized through the new concept of “packing set”. This concept is also used to derive a new branch rule based on accumulated resource consumption and to generalize the Ryan and Foster branch rule. Extensive experiments on several variants show that the generic solver has an excellent overall performance, in many problems being better than the best existing specific algorithms. Even some non-VRPs, like bin packing, vector packing and generalized assignment, can be modeled and effectively solved.
This study focuses on the intrahospital routing of phlebotomists at the University of Iowa Hospitals and Clinics (UIHC). Phlebotomists are responsible for drawing specimens from patients based on doctors' orders. The results of the analysis of these specimens play an important role in determining patient treatment. However, the demand for phlebotomists is likely to outpace supply over the next few years. Therefore, it is important to improve the efficiency of phlebotomists. In this paper, we formulate the phlebotomist intrahospital routing problem as a team orienteering problem with stochastic rewards and service times. The rewards and service times are particularly interesting as they are the result of a queueing process. We present an a priori solution approach and derive a method for efficiently sampling the value of a solution, a value that cannot be determined analytically. Finally, we demonstrate that our proposed approach outperforms the current practice at UIHC.
This paper models and solves a new transportation problem of practical importance; the Consistent Vehicle Routing Problem with Profits. There are two sets of customers, the frequent customers that are mandatory to service and the non-frequent potential customers with known and estimated profits respectively, both having known demands and service requirements over a planning horizon of multiple days. The objective is to determine the vehicle routes that maximize the net profit, while satisfying vehicle capacity, route duration and consistency constraints. A new mathematical model is proposed that captures the profit collecting nature, as well as other features of the problem. For addressing this computationally challenging problem, an Adaptive Tabu Search has been developed, utilizing both short- and long-term memory structures to guide the search process. The proposed metaheuristic algorithm is evaluated on existing, as well as newly generated benchmark problem instances. Our computational experiments demonstrate the effectiveness of our algorithm, as it matches the optimal solutions obtained for small-scale instances and performs well on large-scale instances. Lastly, the trade-off between the acquired profits and consistent customer service is examined and various managerial insights are derived.
Travelling and touring are popular leisure activities enjoyed by millions of tourists around the world. However, the task of travel itinerary recommendation and planning is tedious and challenging for tourists, who are often unfamiliar with the various Points-of-Interest (POIs) in a city. Apart from identifying popular POIs, the tourist needs to construct a travel itinerary comprising a subset of these POIs, and to order these POIs as a sequence of visits that can be completed within his/her available touring time. For a more realistic itinerary, the tourist also has to account for travelling time between POIs and visiting times at individual POIs. Furthermore, this itinerary should incorporate tourist preferences such as desired starting and ending POIs (e.g., POIs that are near the tourist's hotel) and a subset of must-see POIs (e.g., popular POIs that a tourist must visit). We term this the TourMustSee problem, which is based on a variant of the Orienteering problem. Following which, we propose the LP+M algorithm for solving the TourMustSee problem as an Integer Linear Program (ILP). Using a Flickr dataset of POI visits in seven touristic cities, we compare LP+M against various ILP-based baselines, and the results show that LP+M recommends better travel itineraries in terms of POI popularity, total POIs visited, total touring time utilized and must-visit POI(s) inclusion.
We propose a formulation for the pickup and delivery problem with time windows, based on a novel modeling strategy that allows the assignment of vehicles to routes explicitly in two-index flow formulations. It leads to an effective compact formulation that can benefit OR practitioners interested in solving the problem by general-purpose optimization software. Computational experiments indicate that the proposed formulation has interesting features and best overall performance in relation to other compact formulations.
Tour recommendation and itinerary planning are challenging tasks for tourists, due to their need to select points of interest (POI) to visit in unfamiliar cities and to select POIs that align with their interest preferences and trip constraints. We propose an algorithm called PersTour for recommending personalized tours using POI popularity and user interest preferences, which are automatically derived from real-life travel sequences based on geo-tagged photographs. Our tour recommendation problem is modeled using a formulation of the Orienteering problem and considers user trip constraints such as time limits and the need to start and end at specific POIs. In our work, we also reflect levels of user interest based on visit durations and demonstrate how POI visit duration can be personalized using this time-based user interest. Furthermore, we demonstrate how PersTour can be further enhanced by: (i) a weighted updating of user interests based on the recency of their POI visits and (ii) an automatic weighting between POI popularity and user interests based on the tourist’s activity level. Using a Flickr dataset of ten cities, our experiments show the effectiveness of PersTour against various collaborative filtering and greedy-based baselines, in terms of tour popularity, interest, recall, precision and F(Formula presented.)-score. In particular, our results show the merits of using time-based user interest and personalized POI visit durations, compared to the current practice of using frequency-based user interest and average visit durations.
This paper addresses a variant of the Orienteering Problem taking into account mandatory visits and exclusionary constraints (conflicts among nodes). Five mixed integer linear formulations are adapted from the Traveling Salesman Problem literature in order to provide a robust formulation for this problem. The main difference among these formulations lies in the way they deal with the subtour elimination constraints. The performance of the proposed formulations is evaluated over a large set of instances. Computational results reveal that the model that avoids subtours by means of a single-commodity flow formulation allows to solve to optimality more instances than the other formulations, within a time limit of 1 h.
A large class of computational problems involve the determination of properties of graphs, digraphs, integers, arrays of integers, finite families of finite sets, boolean formulas and elements of other countable domains. Through simple encodings from such domains into the set of words over a finite alphabet these problems can be converted into language recognition problems, and we can inquire into their computational complexity. It is reasonable to consider such a problem satisfactorily solved when an algorithm for its solution is found which terminates within a number of steps bounded by a polynomial in the length of the input. We show that a large number of classic unsolved problems of covering, matching, packing, routing, assignment and sequencing are equivalent, in the sense that either each of them possesses a polynomial-bounded algorithm or none of them does.
In the Orienteering Problem (OP), we are given an undirected graph with edge weights and node prizes. The problem calls for a simple cycle whose total edge weight does not exceed a given threshold, while visiting a subset of nodes with maximum total prize. This NP-hard problem arises in routing and scheduling applications. We describe a branch-and-cut algorithm for finding an optimal OP solution. The algorithm is based on several families of valid inequalities. We also introduce a family of cuts, called conditional cuts, which can cut off the optimal OP solution, and propose an effective way to use them within the overall branch-and-cut framework. Exact and heuristic separation algorithms are described, as well as heuristic procedures to produce near-optimal OP solutions. An extensive computational analysis on several classes of both real-world and random instances is reported. The algorithm proved to be able to solve to optimality large-scale instances involving up to 500 nodes, within acceptable computing time. This compares favorably with previous published methods.
In this paper, we introduce a variant of the orienteering problem in which travel and service times are stochastic. If a delivery
commitment is made to a customer and is completed by the end of the day, a reward is received, but if a commitment is made
and not completed, a penalty is incurred. This problem reflects the challenges of a company who, on a given day, may have
more customers than it can serve. In this paper, we discuss special cases of the problem that we can solve exactly and heuristics
for general problem instances. We present computational results for a variety of parameter settings and discuss characteristics
of the solution structure.
New formulations are presented for the Travelling Salesman problem, and their relationship to previous formulations is investigated. The new formulations are extended to include a variety of transportation scheduling problems, such as the Multi-Travelling Salesman problem, the Delivery problem, the School Bus problem and the Dial-a-Bus problem. A Benders decomposition procedure is applied on the new formulations and the resulting computational rocedure is seen to be identical to previous methods for solving the Travelling Salesman problem. Based on the Lagrangean Relaxation method, a new procedure is suggested for generating lagrange multipliers for a subgradient optimization procedure. The effectiveness of the bounds obtained is demonstrated by computational test results. Research supported, in part, by the Office of Naval Research under Contract N00014-75-C-0556.
One of the most crucial issues in the tourism industry that can help attract more tourists is to design optimal travel routes for tourists. The Team Orienteering Problem with Time Windows (TOPTW) is one of the approaches that can solve Tourist Trip Design Problem (TTDP). In this study, TOPTW has been applied to design a multi-objective model to determine the optimal travel routes, considering the accessibility. The present study aims 1) to maximize the profit from visiting the Points of Interest (POIs) to increase the satisfaction level of the travel agency, and 2) to maximize the number of visited POIs to improve the satisfaction level of the tourists. Two concepts of “indirect coverage” and “neighborhood radius” are applied to increase the accessibility of the tourist attractions and POIs. Thus, around each POI, a small radius and a large radius are considered, and if the points are in the radius coverage of the points located in the main routes, they can receive indirect coverage from those points. Also the time-window overlap issue for POIs is vital to determine the possibility of indirect coverage and makes the conceptual model of the research closer to reality. Also, several indicators are proposed for calculating accessibility. A multi-objective genetic algorithm (MOGA) is formulated to solve the studied problem, and the results obtained by the proposed algorithm are compared with those of CPLEX software (for small-scale) and NSGA-II algorithm (for large-scale). The findings indicate that the genetic algorithm outperforms the NSGA-II algorithm. A sensitivity analysis is also performed to evaluate the effects of the proposed model parameters on the optimization process. Finally, the model applicability in practice is examined based on a case study in Tehran, the capital of Iran.
Improving the travel experience is a goal of tourist destinations. Tourists demand information and services that help plan and organise the trips adapted to their preferences and resources. Intelligent systems, recommendation systems, and electronic tourist guides can play a decisive role in tourist satisfaction and shaping the offer at the destination. These systems must satisfy the interests of tourists which travel alone or as a group. However, the development of these tools to generate group tourism itineraries is still limited. Group itineraries must also consider individual preferences and the selection of transport modes. The problem associated with the construction of tourist routes is called the Tourist Trip Design Problem. In this work, an extension of the Team Orienteering Problem with Time Windows is developed to model tourism planning. The model considers the construction of group routes, a cost and time limit associated with each participant, the selection of the mode of transport to go from one location to another, and heterogeneous preferences in the group. The model aims to maximise the preferences of group members by considering real and challenging constraints that increase complexity and avoid solutions in polynomial time. This is a novel model, and we generate sets of instances to assess the accuracy of our solution approach. A Greedy Randomised Adaptive Search Procedure is proposed to solve the model. Evaluation criteria based on time, cost, and preferences are used to guide the candidate’s selection in the construction phase. The results provided by our meta-heuristic are compared with those obtained by solving the Mixed Integer Programming formulation with exact solver. Computational results show the effectiveness and efficiency of the proposed approach.
The tourist trip design problem is an extension of the orienteering problem applied to tourism. The problem consists in selecting a subset of locations to visit from among a larger set while maximizing the benefit for the tourist. The benefit is given by the sum of the rewards collected at each location visited. We consider a variant of the problem that deals not only with “typical” constraints such as budget, opening-time hours (i.e., time windows at the locations), and maximum trip duration but also with other practical tourism constraints such as mandatory visits, limits on the number of locations of each type, and the order at which selected locations are visited. To solve this problem, we propose a branch-and-check approach in which the master problem selects a subset of locations, verifying all except time-related constraints, and these locations define candidate solutions to the master problem. For each candidate solution, the sub-problem checks whether a feasible trip can be built using the given locations. To accelerate the branch-and-check approach, we propose and test improvements, including preprocessing to tighten the master-sub problem, valid inequalities generated dynamically to strengthen the master problem, and a local branching and variable neighborhood search to find new feasible solutions. Finally, we report the experimental results and compare the performance of the proposed exact algorithm with that of a mathematical solver.
This paper studies the team orienteering problem, where the arrival time and service time affect the collection of profits. Such interactions result in a nonconcave profit function. This problem integrates the aspect of time scheduling into the routing decision, which can be applied in humanitarian search and rescue operations where the survival rate declines rapidly. Rescue teams are needed to help trapped people in multiple affected sites, whereas the number of people who could be saved depends as well on how long a rescue team spends at each site. Efficient allocation and scheduling of rescue teams is critical to ensure a high survival rate. To solve the problem, we formulate a mixed-integer nonconcave programming model and propose a Benders branch-and-cut algorithm, along with valid inequalities for tightening the upper bound. To solve it more effectively, we introduce a hybrid heuristic that integrates a modified coordinate search (MCS) into an iterated local search. Computational results show that valid inequalities significantly reduce the optimality gap, and the proposed exact method is capable of solving instances where the mixed-integer nonlinear programming solver SCIP fails in finding an optimal solution. In addition, the proposed MCS algorithm is highly efficient compared with other benchmark approaches, whereas the hybrid heuristic is proven to be effective in finding high-quality solutions within short computing times. We also demonstrate the performance of the heuristic with the MCS using instances with up to 100 customers.
Summary of Contribution: Motivated by search and rescue (SAR) operations, we consider a generalization of the well-known team orienteering problem (TOP) to incorporate a nonlinear time-varying profit function in conjunction with routing and scheduling decisions. This paper expands the envelope of operations research and computing in several ways. To address the scalability issue of this highly complex combinatorial problem in an exact manner, we propose a Benders branch-and-cut (BBC) algorithm, which allows us to efficiently deal with the nonconcave component. This BBC algorithm is computationally enhanced through valid inequalities used to strengthen the bounds of the BBC. In addition, we propose a highly efficient hybrid heuristic that integrates a modified coordinate search into an iterated local search. It can quickly produce high-quality solutions to this complex problem. The performance of our solution algorithms is demonstrated through a series of computational experiments.
The planning of long-duration missions for resource-constrained agents is a longstanding challenge not only in field robotics research, but also in the fields of vehicle routing and logistics. To efficiently plan missions over a larger time scale than an agent's resource capacity natively allows, it is necessary to exploit the availability of recharging stations, where an agent can replenish its spent resources at the cost of time. This paper presents the Orienteering Problem with Replenishment, a novel extension of the orienteering problem, which can fully utilise the existence of recharging stations by not only revisiting such stations, but also intelligently managing the amount of time spent recharging at each stop. By leveraging the above mechanisms for overcoming the resource limitations of the platform, the proposed problem formulation allows for the planning of longer term missions. A suite of computational tests are performed to assess the proposed formulation, and its performance is analysed with respect to the impact of intelligent utilisation of the recharge points on reward acquisition. Finally, limitations of the proposed problem formulation are discussed, and avenues for future work are presented.
Mission planning for Autonomous Marine Vehicles (AMVs) is non-trivial because significant uncertainty is present when profiling the operating environment, especially for underwater missions. Mission complexity is compounded for each vehicle added to the mission. In practice, fleet operations are formulated as separate temporal problems by the operator and solved using a temporal planner. This paper proposes a planning method that uses energy as the base planning resource instead of time. Unlike temporal planners, energy planners account for physical loads endured by the vehicles. The extent of uncertainty in the vehicle loads is clarified by using the vehicle dynamics model and Monte Carlo simulation on the model parameters. The planning method is a multistage procedure to decompose operator specified task, obstacle, and vehicle data into an energy formulation of the Team Orienteering Problem (TOP) which is then solved using Discrete Strengthened PSO (DStPSO). The DStPSO algorithm has been modified to include a selective swarm size decay method that allows for larger initial swarm sizes to promote early exploration and preserves a percentage of the best performing particles on each iteration to save computational resources. The planner produces near-optimal routes containing feasible trajectories for individual vehicles that maximise tasks completed according to individual vehicle energy constraints. A case-study mission for long-term, large-scale, underwater inspection of a wind turbine array was converted into input data to evaluate the planner. Energy planning presents the opportunity for vehicles to actively monitor the feasibility of their individual plan against their current energy consumption, allowing for advanced reasoning and fault handling to occur in situ without operator assistance.
We study a new variant of the Team Orienteering Problem (TOP) where precedence constraints are introduced. Each customer has a set of tasks that have to be accomplished according to a predefined order by an heterogeneous fleet of vehicles. If a customer is selected, then all the tasks have to be completed by possibly different vehicles. To tackle the problem, we propose an enhancement of the Kernel Search (KS) framework that makes use of different sorting strategies and compare its performance to a Branch-and-Cut algorithm embedding the dynamic separations of different valid inequalities and the use of a simplified KS as primal heuristic. The Branch-and-Cut strongly improves the performance of Gurobi when used to solve the compact problem formulation, whereas the variant of KS comes up to be an extremely effective approach also as primal heuristic embedded into a MIP solver. New benchmark instances and corresponding best known values are provided. Both solution approaches have also been tested on instances of the special case TOP providing extremely good results.
This tutorial introduces readers to several variants of routing problems with profits. In these routing problems each node has a certain profit, and not all nodes need to be visited. Since the orienteering problem (OP) is by far the most frequently studied problem in this category of routing problems, the book mainly focuses on the OP. In turn, other problems are presented as variants of the OP, focusing on the similarities and differences. The goal of the OP is to determine a subset of nodes to visit and in which order, so that the total collected profit is maximized and a given time budget is not exceeded.
The book provides a comprehensive review of variants of the OP, such as the team OP, the team OP with time windows, the profitable tour problem, and the prize-collecting travelling salesperson problem. In addition, it presents mathematical models and techniques for solving these OP variants and discusses their complexity. Several simple examples and benchmark instances, together with their best-known results, are also included. Finally, the book reviews the latest applications of these problems in the fields of logistics, tourism and others.
Urban tourism is a worldwide form of tourism and is one of the most important social and economic impetus for urban development. The urban tourism market has been increasingly dominated by the demand for personalized experiences. Accordingly, this study aims to design personalized itineraries with hotel selection for multi-day urban tourists. A two-level heuristic approach is proposed, which embeds genetic algorithm, variable neighborhood search, and differential evolution algorithm into the structure of memetic algorithm. A case study in Xiamen, a coastal city in Southeast China, is carried out to evaluate the performance of our approach. Results of paired sample t-tests show that our proposed approach is remarkably superior to existing methods. In addition, compared with previous methods, our approach can design more reasonable and personalized itineraries for tourists.
A substantial transformation has occurred in tourist behavior in the postmodern tourism era, where the tourism market is dominated by the demand for tailored experiences. Therefore, the design of personalized tourist routes plays a fundamental role in improving tourists’ travel experiences and the success of tourist attractions. This study proposes a hybrid heuristic algorithm based on random simulation (RS-H²A) to design a personalized day tour route in a time-dependent stochastic environment. To evaluate the performance of this algorithm, we conducted a case study at Jiuzhai Valley in Sichuan, China. The results of paired sample t-tests indicated that the proposed algorithm performed significantly better than existing methods. Furthermore, our proposed approach had the ability to design more realistic and personalized routes for tourists than previous methods. We designed an experiment to further explore how uncertain environments affect tourists with different levels of risk awareness.
The Orienteering Problem with Mandatory Visits and Exclusionary Constraints (OPMVEC) is to visit a set of mandatory nodes (locations) and some optional nodes, while respecting the compatibility constraint between nodes and the maximum total time budget constraint. It is a variation of the classic orienteering problem that originates from a number of real-life applications. We present a highly effective memetic algorithm (MA) for OPMVEC combining: (i) a dedicated tabu search procedure that considers both feasible and infeasible solutions by constraint relaxation, (ii) a backbone-based crossover, and (iii) a randomized mutation procedure to prevent from premature convergence. Experiments on six classes of 340 benchmark instances from the literature demonstrate highly competitive performance of MA – it reports improved results for 104 instances compared with the existing heuristic approach, while finding matching best-known results for the remaining cases. Additionally, MA can be used to produce a starting point for an exact solver (e.g., CPLEX), leading to an increased number of problem instances that are solved to optimality. We further investigate the contribution of the key algorithmic elements to the success of the proposed approach.
This study investigates the team orienteering problem with time windows and mandatory visits (TOPTW-MV), a new variant of the well-known team orienteering problem with time windows. In TOPTW-MV, some customers are important customers that must be visited. The other customers are called optional customers. Each customer carries a positive score. The goal is to determine a given number of paths to maximize the total score collected at visited nodes, while observing side constraints such as mandatory visits and time window constraints.
We constructed a mathematical programming model and designed a multi-start simulated annealing (MSA) heuristic for TOPTW-MV. Computational study showed that MSA outperforms Gurobi on solving small-scale benchmark instances. Among the 72 small TOPTW-MV instances, MSA obtained better solutions than Gurobi for 13 instances and the same solutions as those obtained by Gurobi for the remaining instances. Moreover, the average computational time of MSA is shorter than that of Gurobi. In addition, computational study based on 168 TOPTW-MV benchmark instances adapted from existing TOPTW benchmark instances indicated that MSA significantly improves the performance of basic simulated annealing heuristic and outperforms the artificial bee colony algorithm on solving large TOPTW-MV instances.
During a trip planning, tourists gather information from different sources, select and rank the places to visit according to their personal interests, and try to devise daily tours among them. This paper addresses the complex selection and touring problem and proposes a “filter-first, tour-second” framework for generating personalized tour recommendations for tourists based on information from social media and other online data sources. Collaborative filtering is applied to identify a subset of optional points of interest that maximize the potential satisfaction, while there are some preselected mandatory points that the tourists must visit. Next, the underlying orienteering problem is solved via an Iterated Tabu Search algorithm. The goal is to generate tours that contain all mandatory points and maximize the total score collected from the optional points visited daily, taking into account different day availabilities and opening hours, limitations on the tour lengths, budgets and other restrictions. Computational experiments on benchmark datasets indicate that the proposed touring algorithm is very competitive. Furthermore, the proposed framework has been evaluated on data collected from Foursquare. The results show the practical utility and the temporal efficacy of the recommended tours.
The Team Orienteering Problem (TOP) aims at maximizing the total amount of profit collected by a fleet of vehicles while not exceeding a predefined travel time limit on each vehicle. In the last years, several exact methods based on different mathematical formulations were proposed. In this paper, we present a new two-index formulation with a polynomial number of variables and constraints. This compact formulation, reinforced by connectivity constraints, was solved by means of a branch-and-cut algorithm. The total number of instances solved to optimality is 327 out of 387 benchmark instances, 26 more than any previous method. Moreover, 24 not previously solved instances were closed to optimality.
This paper addresses a variant of the Orienteering Problem in which some constraints related to mandatory visits and incompatibilities among nodes are taken into account. A hybrid algorithm based on a reactive GRASP and a general VNS is proposed. Computational experiments over a large set of instances show the efficiency of the algorithm. Additionally, we also validate the performance of this algorithm on some instances taken from the literature of the traditional Orienteering Problem.
This article introduces, models, and solves a generalization of the orienteering problem, called the the orienteering problem with variable profits (OPVP). The OPVP is defined on a complete undirected graph G = (V,E), with a depot at vertex 0. Every vertex i∈V \{0} has a profit pi to be collected, and an associated collection parameter αi∈[0, 1]. The vehicle may make a number of “passes,” collecting 100αi percent of the remaining profit at each pass. In an alternative model, the vehicle may spend a continuous amount of time at every vertex, collecting a percentage of the profit given by a function of the time spent. The objective is to determine a maximal profit tour for the vehicle, starting and ending at the depot, and not exceeding a travel time limit.
It is shown that a certain tour of 49 cities, one in each of the 48 states and Washington, D.C., has the shortest road distance.
Operations Research, ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.
This paper explores new approaches to the symmetric traveling-salesman problem in which 1-trees, which are a slight variant of spanning trees, play an essential role. A 1-tree is a tree together with an additional vertex connected to the tree by two edges. We observe that (i) a tour is precisely a 1-tree in which each vertex has degree 2, (ii) a minimum 1-tree is easy to compute, and (iii) the transformation on “intercity distances” cij → Cij + πi + πj leaves the traveling-salesman problem invariant but changes the minimum 1-tree. Using these observations, we define an infinite family of lower bounds w(π) on C*, the cost of an optimum tour. We show that maxπw(π) = C* precisely when a certain well-known linear program has an optimal solution in integers. We give a column-generation method and an ascent method for computing maxπw(π), and construct a branch-and-bound method in which the lower bounds w(π) control the search for an optimum tour.
In this paper, the selective travelling salesperson problem with stochastic service times, travel times, and travel costs (SSTSP) is addressed. In the SSTSP, service times, travel times and travel costs are known a priori only probabilistically. A non-negative value of reward for providing service is associated with each customer and there is a pre-specified limit on the duration of the solution tour. It is assumed that not all potential customers can be visited within this tour duration limit, even under the best circumstances. And, thus, a subset of customers must be selected. The objective of the SSTSP is to design an a priori tour that visits each chosen customer once such that the total profit (total reward collected by servicing customers minus travel costs) is maximized and the probability that the total actual tour duration exceeds a given threshold is no larger than a chosen probability value. We formulate the SSTSP as a chance-constrained stochastic program and propose both exact and heuristic approaches for solving it. Computational experiments indicate that the exact algorithm is able to solve small- and moderate-size problems to optimality and the heuristic can provide near-optimal solutions in significantly reduced computing time.Journal of the Operational Research Society (2005) 56, 439452. doi:10.1057/palgrave.jors.2601831 Published online 27 October 2004
The problem of maximizing diversity deals with selecting a set of elements from some larger collection such that the selected elements exhibit the greatest variety of characteristics. A new model is proposed in which the concept of diversity is quantifiable and measurable. A quadratic zero-one model is formulated for diversity maximization. Based upon the formulation, it is shown that the maximum diversity problem is NP-hard. Two equivalent linear integer programs are then presented that offer progressively greater computational efficiency. Another formulation is also introduced which involves a different diversity objective. An example is given to illustrate how additional considerations can be incorporated into the maximum diversity model.
In the team orienteering problem, start and end points are specified along with other locations which have associated scores. Given a fixed amount of time for each of the M members of the team, the goal is to determine M paths from the start point to the end point through a subset of locations in order to maximize the total score. In this paper, a fast and effective heuristic is presented and tested on 353 problems ranging in size from 21 to 102 points. The computational results are presented in detail.
This report describes an implementation of the Lin-Kernighan heuristic, one of the most successful methods for generating optimal or nearoptimal solutions for the symmetric traveling salesman problem. Computational tests show that the implementation is highly effective. It has found optimal solutions for all solved problem instances we have been able to obtain, including a 7397-city problem (the largest nontrivial problem instance solved to optimality today). Furthermore, the algorithm has improved the best known solutions for a series of large-scale problems with unknown optima, among these an 85900-city problem.
In 1970 Held and Karp introduced the Lagrangean approach to the symmetric traveling salesman problem. We use this 1-tree relaxation in a new branch and bound algorithm. It differs from other algorithms not only in the branching scheme, but also in the ascent method to calculate the 1-tree bounds. urthermore we determine heuristic solutions throughout the computations to provide upperbounds. We present computational results for both a depth-first and a breadth-first version of our algorithm. On the average our results on a number of Euclidean problems from the literature are obtained in about 60% less 1-trees than the best known algorithm based on the 1-tree relaxation. For random table problems (up to 100 cities) the average results are also satisfactory.
The multiple tour maximum collection problem (MTMCP) consists of determining the m optimal time constrained tours which visit a subset of weighted nodes in a graph such that the total weight collected from the subset of nodes is maximized. In this paper, a heuristic for the MTMCP is developed. The results of computational testing reveal that the heuristic has three very attractive features. First, the heuristic will always produce a feasible solution if one exists. Second, the heuristic produces very good solutions. In most cases where the exact solution is known, it produces the optimal solution. And finally, the heuristic has the ability to handle large problems in a very short amount of computation time.
During the last decade, a number of challenging applications in logistics, tourism and other fields were modelled as orienteering problems (OP). In the orienteering problem, a set of vertices is given, each with a score. The goal is to determine a path, limited in length, that visits some vertices and maximises the sum of the collected scores. In this paper, the literature about the orienteering problem and its applications is reviewed. The OP is formally described and many relevant variants are presented. All published exact solution approaches and (meta) heuristics are discussed and compared. Interesting open research questions concerning the OP conclude this paper.
After [15], [31], [19], [8], [25], [5], minimum cut/maximum flow algorithms on graphs emerged as an increasingly useful tool for exact or approximate energy minimization in low-level vision. The combinatorial optimization literature provides many min-cut/max-flow algorithms with different polynomial time complexity. Their practical efficiency, however, has to date been studied mainly outside the scope of computer vision. The goal of this paper is to provide an experimental comparison of the efficiency of min-cut/max flow algorithms for applications in vision. We compare the running times of several standard algorithms, as well as a new algorithm that we have recently developed. The algorithms we study include both Goldberg-Tarjan style "push-relabel" methods and algorithms based on Ford-Fulkerson style "augmenting paths." We benchmark these algorithms on a number of typical graphs in the contexts of image restoration, stereo, and segmentation. In many cases, our new algorithm works several times faster than any of the other methods, making near real-time performance possible. An implementation of our max-flow/min-cut algorithm is available upon request for research purposes.