The two-echelon time-constrained vehicle routing problem in linehaul-delivery systems

Most of the studies address issues relating to the delivery from satellites to customers, which is throughout the end part of the linehaul-delivery system. Differing from the long-term strategic problems including the two-echelon vehicle routing problem (2E-VRP), the two-echelon location routing problem (2E-LRP) and the truck and trailer routing problem (TTRP) which make location decisions in depots or satellites, the paper introduces a short-term tactical problem named the two-echelon time-constrained vehicle routing problem in linehaul-delivery systems (2E-TVRP) that does not involve location decisions. The linehaul level and the delivery level are linked through city distribution centers (CDCs) located on the outskirts of cities. The 2E-TVRP has inter-CDC linehaul on the first level and urban delivery from CDCs to satellites on the second level. Vehicle routes on different levels are interacted by time constraints. A mixed integer nonlinear programming model for the 2E-TVRP is put forward, and a mixed integer linear programming model is used as the benchmark model. The Clarke and Wright savings heuristic algorithm (CW) improved by a local search phase is adopted. The 2E-TVRP formulations and the heuristic algorithm are tested by using 140 randomly-generated instances with up to 10 CDCs and 500 satellites. The computational results indicate that the heuristic can effectively solve various instances of the 2E-TVRP.     

The two-echelon time-constrained vehicle routing problem in linehaul-delivery systems considering carbon dioxide emissions

Multi-echelon distribution strategy is primarily to alleviate the environmental (e.g., energy consumption and emissions) consequence of logistics operations. Differing from the long-term strategic problems (e.g., the two-echelon vehicle routing problem (2E-VRP), the two-echelon location routing problem (2E-LRP) and the truck and trailer routing problem (TTRP)) that make location decisions in depots or satellites, the paper introduces a short-term tactical problem named the two-echelon time-constrained vehicle routing problem in linehaul-delivery systems (2E-TVRP) considering carbon dioxide (CO₂) emissions. The linehaul level and the delivery level are linked through city distribution centers (CDCs). The 2E-TVRP, which takes CO₂ emissions per ton-kilometer as the objective, has inter-CDC linehaul on the 1st level and delivery from CDCs to satellites on the 2nd level. The Clarke and Wright savings heuristic algorithm (CW) improved by a local search phase is put forward. The case study shows the applicability of the model to real-life problems. The results suggest that the vehicle scheduling provided by the 2E-TVRP is promising to reduce the CO₂ emissions per ton-kilometer of the linehaul-delivery system. Adjusting the central depot location or developing the loaded-semitrailer demand among O-D pairs to eliminate empty-running of tractors will contribute to reduce the CO₂ emission factor.     

The impact of depot location,fleet composition and routing on emissions in city logistics

This paper investigates the combined impact of depot location, fleet composition and routing decisions on vehicle emissions in city logistics. We consider a city in which goods need to be delivered from a depot to customers located in nested zones characterized by different speed limits. The objective is to minimize the total depot, vehicle and routing cost, where the latter can be defined with respect to the cost of fuel consumption and CO₂ emissions. A new powerful adaptive large neighborhood search metaheuristic is developed and successfully applied to a large pool of new benchmark instances. Extensive analyses are performed to empirically assess the effect of various problem parameters, such as depot cost and location, customer distribution and heterogeneous vehicles on key performance indicators, including fuel consumption, emissions and operational costs. Several managerial insights are presented.     

An iterative route construction and improvement algorithm for the vehicle routing problem with soft time windows

The solution of routing problems with soft time windows has valuable practical applications. Soft time window solutions are needed when: (a) the number of routes needed for hard time windows exceeds the number of available vehicles, (b) a study of cost-service tradeoffs is required, or (c) the dispatcher has qualitative information regarding the relative importance of hard time-window constraints across customers. This paper proposes a new iterative route construction and improvement algorithm to solve vehicle routing problems with soft time windows. Due to its modular and hierarchical design, the solution algorithm is intuitive and able to accommodate general cost and penalty functions. Experimental results indicate that the average run time performance is of order O(n²). The solution quality and computational time of the new algorithm has been compared against existing results on benchmark problems. The presented algorithm has improved thirty benchmark problem solutions for the vehicle routing problems with soft time windows.     

An evolutionary local search for the capacitated vehicle routing problem minimizing fuel consumption under three-dimensional loading constraints

This study introduces a new practical variant of the combined routing and loading problem called the capacitated vehicle routing problem minimizing fuel consumption under three-dimensional loading constraints (3L-FCVRP). It presents a meta-heuristic algorithm for solving the problem. The aim is to design routes for a fleet of homogeneous vehicles that will serve all customers, whose demands are formed by a set of three-dimensional, rectangular, weighted items. Unlike the well-studied capacitated vehicle routing problem with 3D loading constraints (3L-CVRP), the objective of the 3L-FCVRP is to minimize total fuel consumption rather than travel distance. The fuel consumption rate is assumed to be proportionate to the total weight of the vehicle. A route is feasible only if a feasible loading plan to load the demanded items into the vehicle exists and the loading plan must satisfy a set of practical constraints.To solve this problem, the evolutionary local search (ELS) framework incorporating the recombination method is used to explore the solution space, and a new heuristic based on open space is used to examine the feasibility of the solutions. In addition, two special data structures, Trie and Fibonacci heap, are adopted to speed up the procedure. To verify the effectiveness of our approach, we first test the ELS on the 3L-CVRP, which can be seen as a special case of the 3L-FCVRP. The results demonstrate that on average ELS outperforms all of the existing approaches and improves the best-known solutions for most instances. Then, we generate data for 3L-FCVRP and report the detailed results of the ELS for future comparisons.     

Order-first split-second methods for vehicle routing problems: A review

Cluster-first route-second methods like the sweep heuristic (Gillett and Miller, 1974) are well known in vehicle routing. They determine clusters of customers compatible with vehicle capacity and solve a traveling salesman problem for each cluster. The opposite approach, called route-first cluster-second, builds a giant tour covering all customers and splits it into feasible trips. Cited as a curiosity for a long time but lacking numerical evaluation, this technique has nevertheless led to successful metaheuristics for various vehicle routing problems in the last decade. As many implementations consider an ordering of customers instead of building a giant tour, we propose in this paper the more general name of ordering-first split-second methods. This article shows how this approach can be declined for different vehicle routing problems and reviews the associated literature, with more than 70 references.     

Solving the bicriterion routing and scheduling problem for hazardous materials distribution

Hazardous materials routing and scheduling decisions involve the determination of the minimum cost and/or risk routes for servicing the demand of a given set of customers. This paper addresses the bicriterion routing and scheduling problem arising in hazardous materials distribution planning. Under the assumption that the cost and risk attributes of each arc of the underlying transportation network are time-dependent, the proposed routing and scheduling problem pertains to the determination of the non-dominated time-dependent paths for servicing a given and fixed sequence of customers (intermediate stops) within specified time windows. Due to the heavy computational burden for solving this bicriterion problem, an alternative algorithm is proposed that determines the k-shortest time-dependent paths. Moreover an algorithm is provided for solving the bicriterion problem. The proximity of the solutions of the k-shortest time-dependent path problem with the non-dominated solutions is assessed on a set of problems developed by the authors.     

Optimal scheduling of a taxi fleet with mixed electric and gasoline vehicles to service advance reservations

This study addresses the problem of scheduling a fleet of taxis that are appointed to solely service customers with advance reservations. In contrast to previous studies that have dealt with the planning and operations of a taxi fleet with only electric vehicles (EVs), we consider that most taxi companies may have to operate with fleets comprised of both gasoline vehicles (GVs) and plug-in EVs during the transition from GV to (complete) EV taxi fleets. This paper presents an innovative multi-layer taxi-flow time-space network which effectively describes the movements of the taxis in the dimensions of space and time. An optimization model is then developed based on the time-space network to determine an optimal schedule for the taxi fleet. The objective is to minimize the total operating cost of the fleet, with a set of operating constraints for the EVs and GVs included in the model. Given that the model is formulated as an integer multi-commodity network flow problem, which is characterized as NP-hard, we propose two simple but effective decomposition-based heuristics to efficiently solve the problem with practical sizes. Test instances generated based on the data provided by a Taiwan taxi company are solved to evaluate the solution algorithms. The results show that the gaps between the objective values of the heuristic solutions and those of the optimal solutions are less than 3%, and the heuristics require much less time to obtain the good quality solutions. As a result, it is shown that the model, coupled with the algorithms, can be an effective planning tool to assist the company in routing and scheduling its fleet to service reservation customers.     

Solving the Fleet Size and Mix Vehicle Routing Problem with Time Windows via Adaptive Memory Programming

This paper presents a novel Adaptive Memory Programming (AMP) solution approach for the Fleet Size and Mix Vehicle Routing Problem with Time Windows (FSMVRPTW). The FSMVRPTW seeks to design a set of depot returning vehicle routes to service a set of customers with known demands, for a heterogeneous fleet of vehicles with different capacities and fixed costs. Each customer is serviced only once by exactly one vehicle, within fixed time intervals that represent the earliest and latest times during the day that service can take place. The objective is to minimize the total transportation costs, or similarly to determine the optimal fleet composition and dimension following least cost vehicle routes. The proposed method utilizes the basic concept of an AMP solution framework equipped with a probabilistic semi-parallel construction heuristic, a novel solution re-construction mechanism, an innovative Iterated Tabu Search algorithm tuned for intensification local search and frequency-based long term memory structures. Computational experiments on well-known benchmark data sets illustrate the efficiency and effectiveness of the proposed method. Compared to the current state-of-the-art, the proposed method improves the best reported cumulative and mean results over most problem instances with reasonable computational requirements.     

Integrating a heterogeneous fixed fleet and a flexible assignment of destination depots in the waste collection VRP with intermediate facilities

We consider a complex recyclable waste collection problem that extends the class of vehicle routing problems with intermediate facilities by integrating a heterogeneous fixed fleet and a flexible assignment of destination depots. Several additional side constraints, such as a mandated break period contingent on tour start time, multiple vehicle capacities, and site dependencies are also included. This specific problem was inspired by a real-world application and does not appear in the literature. It is modeled as an MILP which is enhanced with several valid inequalities. Due to the rich nature of the problem, state-of-the-art solvers are only able to tackle instances of small to medium size. To solve realistic instances, we propose a multiple neighborhood search heuristic capable of systematically treating all problem features and general enough to respond to the varying characteristics of the case study regions for which it is intended. The results show that the heuristic achieves optimality on small instances, exhibits competitive performance in comparison to state-of-the-art solution methods for special cases of our problem, and leads to important savings in the state of practice. Moreover, it highlights and quantifies the savings from allowing a flexible depot assignment. The data from the state of practice comes from a company in the waste collection industry in Geneva, Switzerland.     

Modeling techniques for periodic vehicle routing problems

This paper presents a continuous approximation model for the period vehicle routing problem with service choice (PVRP-SC). The PVRP-SC is a variant of the period vehicle routing problem in which the visit frequency to nodes is a decision of the model. This variation can result in more efficient vehicle tours and/or greater service benefit to customers. We present a continuous approximation model to facilitate strategic and tactical planning of periodic distribution systems and evaluate the value of service choice. Further, results from the continuous model can provide guidelines for constructing solutions to the discrete PVRP-SC.     

The time-dependent vehicle routing problem with soft time windows and stochastic travel times

This paper studies a vehicle routing problem with time-dependent and stochastic travel times. In our problem setting, customers have soft time windows. A mathematical model is used in which both efficiency for service as well as reliability for customers are taken into account. Depending on whether service times are included or not, we consider two versions of this problem. Two metaheuristics are built: a Tabu Search and an Adaptive Large Neighborhood Search. We carry out our experiments for well-known problem instances and perform comprehensive analyses on the numerical results in terms of the computational time and the solution quality. Experiments confirm that the proposed procedure is effective to obtain very good solutions to be performed in real-life environment.     

Congestion-aware system optimal route choice for shared autonomous vehicles

We study the shared autonomous vehicle (SAV) routing problem while considering congestion. SAVs essentially provide a dial-a-ride service to travelers, but the large number of vehicles involved (tens of thousands of SAVs to replace personal vehicles) results in SAV routing causing significant congestion. We combine the dial-a-ride service constraints with the linear program for system optimal dynamic traffic assignment, resulting in a congestion-aware formulation of the SAV routing problem. Traffic flow is modeled through the link transmission model, an approximate solution to the kinematic wave theory of traffic flow. SAVs interact with travelers at origins and destinations. Due to the large number of vehicles involved, we use a continuous approximation of flow to formulate a linear program. Optimal solutions demonstrate that peak hour demand is likely to have greater waiting and in-vehicle travel times than off-peak demand due to congestion. SAV travel times were only slightly greater than system optimal personal vehicle route choice. In addition, solutions can determine the optimal fleet size to minimize congestion or maximize service.     

Heuristic approaches for solving large-scale bus transit vehicle scheduling problem with route time constraints

This paper presents new models for multiple depot vehicle scheduling problem (MDVS) and multiple depot vehicle scheduling problem with route time constraints (MDVSRTC). The route time constraints are added to the MDVS problem to account for the real world operational restrictions such as fuel consumption. Compared to existing formulations, this formulation decreases the size of the problem by about 40% without eliminating any feasible solution. It also presents an exact and two heuristic solution procedures for solving the MDVSRTC problem. Although these methods can be used to solve medium size problems in reasonable time, real world applications in large cities require that the MDVSRTC problem size be reduced. Two techniques are proposed to decrease the size of the real world problems. For real-world application, the problem of bus transit vehicle scheduling at the mass transit administration (MTA) in Baltimore is studied. The final results of model implementation are compared to the MTA's schedules in January 1998. The comparison indicates that, the proposed model improves upon the MTA schedules in all respects. The improvements are 7.9% in the number of vehicles, 4.66% in the operational time and 5.77% in the total cost.     

A decomposition-based iterative optimization algorithm for traveling salesman problem with drone

This study investigates a new delivery problem that has emerged after the attempts of several e-commerce and logistics firms to deploy drones in their operations to increase efficiency and reduce delivery times. In this problem, a delivery truck that carries a drone on its roof serves customers in coordination with a drone. The drone is considered to complement the truck due to its cost-efficiency and ability to access difficult terrains and to travel without exposure to congestion. This study presents an iterative algorithm that is based on a decomposition approach to minimize delivery completion time. In the first stage of the proposed methodology, the truck route and the customers assigned to the drone are determined. In the second stage, a mixed-integer linear programming model is solved to optimize the drone route by fixing the routing and the assignment decisions that are made in the first stage. Beginning with the shortest truck route, the assignment and the routing decisions are iteratively improved. The solution times of our algorithm are compared with the solution times of the state-of-the-art formulations that are solved by CPLEX. The results demonstrate that our algorithm yields shorter solution times for the instances that we generated with the specified parameters. An optimization-based heuristic algorithm, which obtains solutions for medium-sized instances, is developed by reducing the feasible search area.     

An integrated modelling approach for the bicriterion vehicle routing and scheduling problem with environmental considerations

The consideration of pollution in routing decisions gives rise to a new routing framework where measures of the environmental implications are traded off with business performance measures. To address this type of routing decisions, we formulate and solve a bi-objective time, load and path-dependent vehicle routing problem with time windows (BTL-VRPTW). The proposed formulation incorporates a travel time model representing realistically time varying traffic conditions. A key feature of the problem under consideration is the need to address simultaneously routing and path finding decisions. To cope with the computational burden arising from this property of the problem we propose a network reduction approach. Computational tests on the effect of the network reduction approach on determining non-dominated solutions are reported. A generic solution framework is proposed to address the BTL-VRPTW. The proposed framework combines any technique that creates capacity-feasible routes with a routing and scheduling method that aims to convert the identified routes to problem solutions. We show that transforming a set of routes to BTL-VRPTW solutions is equivalent to solving a bi-objective time dependent shortest path problem on a specially structured graph. We propose a backward label setting technique to solve the emerging problem that takes advantage of the special structure of the graph. The proposed generic solution framework is implemented by integrating the routing and scheduling method into an Ant Colony System algorithm. The accuracy of the proposed algorithm was assessed on the basis of its capability to determine minimum travel time and fuel consumption solutions. Although the computational results are encouraging, there is ample room for future research in algorithmic advances on addressing the proposed problem.     

Physical distribution from a warehouse: Vehicle coverage and inventory levels

This paper studies the costs involved in distributing items from a warehouse or depot to randomly scattered customers on a day-to-day basis. Two trade-offs are explored simultaneously. The first one arises because by accumulating large inventories at the depot it is possible to build more efficient distribution tours. This trade-off has already been explored for both distribution of goods (Burns et al., 1983) and passengers (Daganzo et al., 1977; Hendrickson, 1978). Another tradeoff, which involves the length of individual vehicle tours (Clarens and Hurdle, 1975), balances the inventory inside the vehicles against the transportation cost. Banks et al. (1982) have considered both of these tradeoffs simultaneously in the context of passenger transportation, but used a somewhat unrealistic model for vehicle routing. This paper is similar to the latter reference but uses a different routing strategy. It also illustrates how the nature of the objects carried (cheap goods, expensive goods, people, etc.) affects the optimal configuration of the distribution system and the overall distribution costs. Usually there is an optimum partitioning of the service area into districts and an optimum dispatching frequency in each district. The results can vary tremendously, depending on factors such as: the inventory carrying cost per item per unit time, the transportation costs, the demand per unit area and unit time, the average distance from the depot, the average vehicle speed and the time per stop.As an illustration of the ideas, a hypothetical limousine service from an airport is analyzed. The example is used to demonstrate how dramatically the optimal system configuration depends on the nature of the items carried.     

Exact and meta-heuristic approach for a general heterogeneous dial-a-ride problem with multiple depots

Dial-a-ride problems are concerned with the design of efficient vehicle routes for transporting individual persons from specific origin to specific destination locations. In real-life this operational planning problem is often complicated by several factors. Users may have special requirements (e.g. to be transported in a wheelchair) while service providers operate a heterogeneous fleet of vehicles from multiple depots in their service area. In this paper, a general dial-a-ride problem in which these three real-life aspects may simultaneously be taken into account is introduced: the Multi-Depot Heterogeneous Dial-A-Ride Problem (MD-H-DARP). Both a three- and two-index formulation are discussed. A branch-and-cut algorithm for the standard dial-a-ride problem is adapted to exactly solve small problem instances of the MD-H-DARP. To be able to solve larger problem instances, a new deterministic annealing meta-heuristic is proposed. Extensive numerical experiments are presented on different sets of benchmark instances for the homogeneous and the heterogeneous single depot dial-a-ride problem. Instances for the MD-H-DARP are introduced as well. The branch-and-cut algorithm provides considerably better results than an existing algorithm which uses a less compact formulation. All seven previously unsolved benchmark instances for the heterogeneous dial-a-ride problem could be solved to optimality within a matter of seconds. While computation times of the exact algorithm increase drastically with problem size, the proposed meta-heuristic algorithm provides near-optimal solutions within limited computation time for all instances. Several best known solutions for unsolved instances are improved and the algorithm clearly outperforms current state-of-the-art heuristics for the homogeneous and heterogeneous dial-a-ride problem, both in terms of solution quality and computation time.     

The load-dependent vehicle routing problem and its pick-up and delivery extension

The present paper examines a Vehicle Routing Problem (VRP) of major practical importance which is referred to as the Load-Dependent VRP (LDVRP). LDVRP is applicable for transportation activities where the weight of the transported cargo accounts for a significant part of the vehicle gross weight. Contrary to the basic VRP which calls for the minimization of the distance travelled, the LDVRP objective is aimed at minimizing the total product of the distance travelled and the gross weight carried along this distance. Thus, it is capable of producing sensible routing plans which take into account the variation of the cargo weight along the vehicle trips. The LDVRP objective is closely related to the total energy requirements of the vehicle fleet, making it a credible alternative when the environmental aspects of transportation activities are examined and optimized. A novel LDVRP extension which considers simultaneous pick-up and delivery service is introduced, formulated and solved for the first time. To deal with large-scale instances of the examined problems, we propose a local-search algorithm. Towards an efficient implementation, the local-search algorithm employs a computational scheme which calculates the complex weighted-distance objective changes in constant time. Solution results are presented for both problems on a variety of well-known test cases demonstrating the effectiveness of the proposed solution approach. The structure of the obtained LDVRP and VRP solutions is compared in pursuit of interesting conclusions on the relative suitability of the two routing models, when the decision maker must deal with the weighted distance objective. In addition, results of a branch-and-cut procedure for small-scale instances of the LDVRP with simultaneous pick-ups and deliveries are reported. Finally, extensive computational experiments have been performed to explore the managerial implications of three key problem characteristics, namely the deviation of customer demands, the cargo to tare weight ratio, as well as the size of the available vehicle fleet.     

A multiple type bike repositioning problem

This paper investigates a new static bicycle repositioning problem in which multiple types of bikes are considered. Some types of bikes that are in short supply at a station can be substituted by other types, whereas some types of bikes can occupy the spaces of other types in the vehicle during repositioning. These activities provide two new strategies, substitution and occupancy, which are examined in this paper. The problem is formulated as a mixed-integer linear programming problem to minimize the total cost, which consists of the route travel cost, penalties due to unmet demand, and penalties associated with the substitution and occupancy strategies. A combined hybrid genetic algorithm is proposed to solve this problem. This solution algorithm consists of (i) a modified version of a hybrid genetic search with adaptive diversity control to determine routing decisions and (ii) a proposed greedy heuristic to determine the loading and unloading instructions at each visited station and the substitution and occupancy strategies. The results show that the proposed method can provide high-quality solutions with short computing times. Using small examples, this paper also reveals problem properties and repositioning strategies in bike sharing systems with multiple types of bikes.