The double travelling salesman problem with multiple stacks – Formulation and heuristic solution approaches

This paper introduces the double travelling salesman problem with multiple stacks and presents four different metaheuristic approaches to its solution. The double TSP with multiple stacks is concerned with determining the shortest route performing pickups and deliveries in two separated networks (one for pickups and one for deliveries) using only one container. Repacking is not allowed, instead each item can be positioned in one of several rows in the container, such that each row can be considered a LIFO (last in, first out) stack, but no mutual constraints exist between the rows. Two different neighbourhood structures are developed for the problem and used with each of three local search metaheuristics. Additionally some simpler removal and reinsertion operators are used in a Large neighbourhood search framework. Finally some computational results are given along with lower bounds on the objective value.     

Validating vehicle routing zone construction using Monte Carlo simulation

The primary purpose of this paper is to validate a clustering procedure used to construct contiguous vehicle routing zones (VRZs) in metropolitan regions. Given a set of customers with random demand for pickups and deliveries over the day, the goal of the design problem is to cluster the customers into zones that can be serviced by a single vehicle. Monte Carlo simulation is used to determine the feasibility of the zones with respect to package count and tour time. For each replication, a separate probabilistic traveling salesman problem (TSP) is solved for each zone. For the case where deliveries must precede pickups, a heuristic approach to the TSP is developed and evaluated, also using Monte Carlo simulation. In the testing, performance is measured by overall travel costs and the probability of constraint violations. Gaps in tour length, tour time and tour cost are the measure used when comparing exact and heuristic TSP solutions.     

Lasso solution strategies for the vehicle routing problem with pickups and deliveries

This paper considers the vehicle routing problem with pickups and deliveries (VRPPD) where the same customer may require both a delivery and a pickup. This is the case, for instance, of breweries that deliver beer or mineral water bottles to a set of customers and collect empty bottles from the same customers. It is possible to relax the customary practice of performing a pickup when delivering at a customer, and postpone the pickup until the vehicle has sufficient free capacity. In the case of breweries, these solutions will often consist of routes in which bottles are first delivered until the vehicle is partly unloaded, then both pickup and delivery are performed at the remaining customers, and finally empty bottles are picked up from the first visited customers. These customers are revisited in reverse order, thus giving rise to lasso shaped solutions. Another possibility is to relax the traditional problem even more and allow customers to be visited twice either in two different routes or at different times on the same route, giving rise to a general solution. This article develops a tabu search algorithm capable of producing lasso solutions. A general solution can be reached by first duplicating each customer and generating a Hamiltonian solution on the extended set of customers. Test results show that while general solutions outperform other solution shapes in term of cost, their computation can be time consuming. The best lasso solution generated within a given time limit is generally better than the best general solution produced with the same computing effort.     

Multi-ant colony system (MACS) for a vehicle routing problem with backhauls

The vehicle routing problem with backhaul (VRPB) is an extension of the capacitated vehicle routing problem (CVRP). In VRPB, there are linehaul as well as backhaul customers. The number of vehicles is considered to be fixed and deliveries for linehaul customers must be made before any pickups from backhaul customers. The objective is to design routes for the vehicles so that the total distance traveled is minimized. We use multi-ant colony system (MACS) to solve VRPB which is a combinatorial optimization problem. Ant colony system (ACS) is an algorithmic approach inspired by foraging behavior of real ants. Artificial ants are used to construct a solution by using pheromone information from previously generated solutions. The proposed MACS algorithm uses a new construction rule as well as two multi-route local search schemes. An extensive numerical experiment is performed on benchmark problems available in the literature.     

General solutions to the single vehicle routing problem with pickups and deliveries

The single vehicle routing problem with pickups and deliveries (SVRPPD) is defined on a graph in which pickup and delivery demands are associated with the customer vertices. The problem consists of designing a least cost route for a vehicle of capacity Q. Each customer is allowed to be visited once for a combined pickup and delivery, or twice if these two operations are performed separately. This article proposes a mixed integer linear programming model for the SVRPPD. It introduces the concept of general solution which encompasses known solution shapes such as Hamiltonian, double-path and lasso. Classical construction and improvement heuristics, as well as a tabu search heuristic, are developed and tested over several instances. Computational results show that the best solutions generated by the heuristics are frequently non-Hamiltonian and may contain up to two customers visited twice.     

A parallel clustering technique for the vehicle routing problem with split deliveries and pickups

In this paper, a parallel clustering technique and route construction heuristic have been developed for the vehicle routing problem (VRP) with split deliveries and pickups. An MILP formulation for determining the exact solution to the problem has also been included. It has been shown through extensive experimentation that the algorithm proposed in this paper statistically produces better results than the only heuristic existing for this class of problems in literature. We also form a basis of comparison between this class of problems and the VRP with simultaneous deliveries and pickups. We note that while heuristics for simultaneous deliveries and pickups cannot be applied in situations where customers' delivery or pickup demands exceed the vehicle capacity, heuristics allowing split deliveries and pickups can, in fact, be applied in every situation, even producing superior results under the combined objective of minimization of the fixed charge and mileage associated with vehicle routes. A guideline as to which heuristic could be used under what parametric conditions and objective functions, has also been provided.     

带集货和配送的多站点VRP优化算法研究

带集货和配送的多站点车辆路线问题(M DVRPPD)是经典VRP的扩展,是多个站点和若干客户既有需求又有供给的VRP问题.研究了该问题的模型并提出了求解该问题的多阶段启发式算法,即先用临界客户的思想把多站点转换为单一站点问题,再使用基于SFC的分组方法来构造初始解,并运用3-opt算法优化回路,之后采用插入算法改善解的可行性,从而得到最终优化解.最后通过实例计算证明了该方法解决M DVRPPD问题的实用可行性和科学有效性.     

Scheduling trucks in local depots for door-to-door delivery services

A scheduling method is suggested for trucks delivering and picking up freight between branch offices and a regional depot in door-to-door delivery services. As the objective functions, different levels of customer service resulting from different timing of deliveries and pickups to/from branch offices are considered as well as the travel cost of trucks. Useful properties of the optimal timing of deliveries and pickups are derived to reduce the size of the search space significantly. Numerical experiments were conducted to evaluate various algorithms to solve the problem.     

The capacitated p-hub median problem with integral constraints: An application to a Chinese air cargo network

Consolidation at hubs in a pure hub-and-spoke network eliminates partial center-to-center direct loads, resulting in savings in transportation costs. In this research, we propose a general capacitated p-hub median model, with economies of scale and integral constraints on the paths. This model requires the selection of a specific p among a set of candidate hubs so that the total cost on the resulting pure capacitated hub-and-spoke network is minimized while simultaneously meeting origin–destination demands, operational capacity and singular path constraints. We explored the problem structure and developed a genetic algorithm using the path for encoding. This algorithm is capable of determining local optimality within less than 0.1% of the Lagrangian relaxation lower bounds on our Chinese air cargo network testing case and has reasonable computational times. The study showed that designating airports with high pickups or deliveries as hubs resulted in a high percentage of origin–destination pairs (ODs) in direct deliveries. Furthermore, the more hubs there are, the higher the direct share and the less likely for double rehandles. Sensitivity analysis on the discount rate showed that the economies of scale on trunk lines of hub-and-spoke networks may have a substantial impact on both the operating costs and the route patterns.     

Exact solutions for the collaborative pickup and delivery problem

In this study we investigate the decision problem of a central authority in pickup and delivery carrier collaborations. Customer requests are to be redistributed among participants, such that the total cost is minimized. We formulate the problem as multi-depot traveling salesman problem with pickups and deliveries. We apply three well-established exact solution approaches and compare their performance in terms of computational time. To avoid unrealistic solutions with unevenly distributed workload, we extend the problem by introducing minimum workload constraints. Our computational results show that, while for the original problem Benders decomposition is the method of choice, for the newly formulated problem this method is clearly dominated by the proposed column generation approach. The obtained results can be used as benchmarks for decentralized mechanisms in collaborative pickup and delivery problems.

     

A tabu search heuristic for a routing problem arising in servicing of offshore oil and gas platforms

This paper introduces a pickup and delivery problem encountered in servicing of offshore oil and gas platforms in the Norwegian Sea. A single vessel must perform pickups and deliveries at several offshore platforms. All delivery demands originate at a supply base and all pickup demands are also destined to the base. The vessel capacity may never be exceeded along its route. In addition, the amount of space available for loading and unloading operations is limited at each platform. The problem, called the Single Vehicle Pickup and Delivery Problem with Capacitated Customers consists of designing a least cost vehicle (vessel) route starting and ending at the depot (base), visiting each customer (platform), and such that there is always sufficient capacity in the vehicle and at the customer location to perform the pickup and delivery operations. This paper describes several construction heuristics as well as a tabu search algorithm. Computational results are presented.     

Routing models and solution procedures for regional less-than-truckload operations

Less-Than-Truckload (LTL) carriers are required on a daily basis to solve Intra-Group Line-Haul (IGLH) problems. IGLH problems require the determination of routes to service required pickups and deliveries (i.e., 28-foot trailers) at End-Of-Line (EOL) terminals. The objective is to minimize total costs, given that tractors are able to simultaneously transport two trailers and that all pickups and deliveries must be accomplished. In this paper, an approximate IGLH solution approach is presented. Given pickup and delivery requirements together with relevant distance data, a matching network is constructed in which nodes correspond to sets of pickups and deliveries and links to routes. A minimum weight non-bipartite matching algorithm is solved over this network and the result is an IGLH solution. This solution is improved by again applying a minimum weight matching algorithm, this time to a matching network in which nodes correspond to routes and links to improved routes. Finally, the routes are sequenced so as to achieve balance at each EOL terminal (i.e., empty trailers must be delivered or picked up as necessary to ensure that each EOL terminal has the same number of pickups and deliveries) and to minimize the inventory of empty trailers. The new IGLH solution procedure is tested on randomly generated data and on data provided by a large LTL carrier. Computational tests show that near-optimal solutions are generated rapidly.     

A MIP-based approach to solve the prize-collecting local access network design problem

This paper presents a new combinatorial optimization problem that can be used to model the deployment of broadband telecommunications systems in which optical fiber cables are installed between a central office and a number of end-customers. In this capacitated network design problem the installation of optical fiber cables with sufficient capacity is required to carry the traffic from the central office to the end-customers at minimum cost. In the situation motivating this research the network does not necessarily need to connect all customers (or at least not with the best available technology). Instead, some nodes are potential customers. The aim is to select the customers to be connected to the central server and to choose the cable capacities to establish these connections. The telecom company takes the strategic decision of fixing a percentage of customers that should be served, and aims for minimizing the total cost of the network providing this minimum service. For that reason the underlying problem is called the Prize-Collecting Local Access Network Design problem (PC-LAN).     

Queues where customers of one queue act as servers of the other queue

We consider a system comprised of two connected M/M/?/? type queues, where customers of one queue act as servers for the other queue. One queue, Q ₁, operates as a limited-buffer M/M/1/N?1 system. The other queue, Q ₂, has an unlimited-buffer and receives service from the customers of Q ₁. Such analytic models may represent applications like SETI@home, where idle computers of users are used to process data collected by space radio telescopes. Let L ₁ denote the number of customers in Q ₁. Then, two models are studied, distinguished by their service discipline in Q ₂: In Model 1, Q ₂ operates as an unlimited-buffer, single-server M/M/1/∞ queue with Poisson arrival rate λ ₂ and dynamically changing service rate μ ₂ L ₁. In Model 2, Q ₂ operates as a multi-server M/M/L ₁/∞ queue with varying number of servers, L ₁, each serving at a Poisson rate of μ ₂. We analyze both models and derive the Probability Generating Functions of the system’s steady-state probabilities. We then calculate the mean total number of customers present in each queue. Extreme cases are indicated.     

Achieving sharp deliveries in supply chains through variance pool allocation

Variability reduction and business process synchronization are acknowledged as key to achieving sharp and timely deliveries in supply chain networks. In this paper, we develop an approach that facilitates variability reduction and business process synchronization for supply chains in a cost effective way. The approach developed is founded on an analogy between mechanical design tolerancing and supply chain lead time compression. We first present a motivating example to describe this analogy. Next, we define, using process capability indices, a new index of delivery performance called delivery sharpness which, when used with the classical performance index delivery probability, measures the accuracy as well as the precision with which products are delivered to the customers. Following this, we solve the following specific problem: how do we compute the allowable variability in lead time for individual stages of the supply chain so that specified levels of delivery sharpness and delivery probability are achieved in a cost-effective way? We call this the variance pool allocation (VPA) problem. We suggest an efficient heuristic approach for solving the VPA problem and also show that a variety of important supply chain design problems can be posed as instances of the VPA problem. One such problem, which is addressed in this paper, is the supply chain partner selection problem. We formulate and solve the VPA problem for a plastics industry supply chain and demonstrate how the solution can be used to choose the best mix of supply chain partners.     

Single vehicle routing problems with a predefined customer sequence, compartmentalized load and stochastic demands

We consider the problem of finding the optimal routing of a single vehicle that delivers K different products to N customers according to a particular customer order. The demands of the customers for each product are assumed to be random variables with known distributions. Each product type is stored in its dedicated compartment in the vehicle. Using a suitable dynamic programming algorithm we find the policy that satisfies the demands of the customers with the minimum total expected cost. We also prove that this policy has a specific threshold-type structure. Furthermore, we investigate a corresponding infinite-time horizon problem in which the service of the customers does not stop when the last customer has been serviced but it continues indefinitely with the same customer order. It is assumed that the demands of the customers at different tours have the same distributions. It is shown that the discounted-cost optimal policy and the average-cost optimal policy have the same threshold-type structure as the optimal policy in the original problem. The theoretical results are illustrated by numerical examples.     

New neighborhood structures for the Double Traveling Salesman Problem with Multiple Stacks

The changing requirements in transportation and logistics have recently induced the appearance of new vehicle routing problems that include complex constraints as precedence or loading constraints. One of these problems that have appeared during the last few years is the Double Traveling Salesman Problem with Multiple Stacks (DTSPMS), a vehicle routing problem in which some pickups and deliveries must be performed in two independent networks, verifying some precedence and loading constraints imposed on the vehicle. In this paper, four new neighborhood structures for the DTSPMS based on reinsertion and permutation of orders to modify both the routes and the loading planning of the solutions are introduced and described in detail. They can be used in combination with any metaheuristic using local search as a subprocedure, guiding the search to unexplored zones of the solution space. Some computational results obtained using all proposed neighborhood structures are presented, providing good quality solutions for real sized instances.      

An efficient mixed integer programming model for pairing containers in inland transportation based on the assignment of orders

The inland transportation takes a significant portion of the total cost that arises from intermodal transportation. In addition, there are many parties (shipping lines, haulage companies, customers) who share this operation as well as many restrictions that increase the complexity of this problem and make it NP-hard. Therefore, it is important to create an efficient strategy to manage this process in a way to ensure all parties are satisfied. This paper investigates the pairing of containers/orders in drayage transportation from the perspective of delivering paired containers on 40-ft truck and/or individual containers on 20-ft truck, between a single port and a list of customer locations. An assignment mixed integer linear programming model is formulated, which solves the problem of how to combine orders in delivery to save the total transportation cost when orders with both single and multiple destinations exist. In opposition to the traditional models relying on the vehicle routing problem with simultaneous pickups and deliveries and time windows formulation, this model falls into the assignment problem category which is more efficient to solve on large size instances. Another merit for the proposed model is that it can be implemented on different variants of the container drayage problem: import only, import–inland and import–inland–export. Results show that in all cases the pairing of containers yields less cost compared to the individual delivery and decreases empty tours. The proposed model can be solved to optimality efficiently (within half hour) for over 300 orders.     

Dynamic scheduling of a GI/GI/1+GI queue with multiple customer classes

We consider a dynamic control problem for a GI/GI/1+GI queue with multiclass customers. The customer classes are distinguished by their interarrival time, service time, and abandonment time distributions. There is a cost c _k >0 for every class k∈{1,2,…,N} customer that abandons the queue before receiving service. The objective is to minimize average cost by dynamically choosing which customer class the server should next serve each time the server becomes available (and there are waiting customers from at least two classes). It is not possible to solve this control problem exactly, and so we formulate an approximating Brownian control problem. The Brownian control problem incorporates the entire abandonment distribution of each customer class. We solve the Brownian control problem under the assumption that the abandonment distribution for each customer class has an increasing failure rate. We then interpret the solution to the Brownian control problem as a control for the original dynamic scheduling problem. Finally, we perform a simulation study to demonstrate the effectiveness of our proposed control.     

The gravity p-median model

In this paper we propose a new model for the p-median problem. In the standard p-median problem it is assumed that each demand point is served by the closest facility. In many situations (for example, when demand points are communities of customers and each customer makes his own selection of the facility) demand is divided among the facilities. Each customer selects a facility which is not necessarily the closest one. In the gravity p-median problem it is assumed that customers divide their patronage among the facilities with the probability that a customer patronizes a facility being proportional to the attractiveness of that facility and to a decreasing utility function of the distance to the facility.