带次模惩罚和随机需求的设施选址问题

考虑带次模惩罚和随机需求的设施选址问题,目的是开设设施集合的一个子集,把客户连接到开设的设施上并对没有连接的客户进行惩罚,使得开设费用、连接费用、库存费用、管理费用和惩罚费用之和达到最小. 根据该问题的特殊结构,给出原始对偶3-近似算法. 在算法的第一步,构造了一组对偶可行解;在第二步中构造了对应的一组原始整数可行解,这组原始整数可行解给出了最后开设的设施集合和被惩罚的客户集合. 最后,证明了算法在多项式时间内可以完成,并且算法所给的整数解不会超过最优解的3倍.     

A goal programming approach to vehicle routing problems with soft time windows

The classical vehicle routing problem involves designing a set of routes for a fleet of vehicles based at one central depot that is required to serve a number of geographically dispersed customers, while minimizing the total travel distance or the total distribution cost. Each route originates and terminates at the central depot and customers demands are known. In many practical distribution problems, besides a hard time window associated with each customer, defining a time interval in which the customer should be served, managers establish multiple objectives to be considered, like avoiding underutilization of labor and vehicle capacity, while meeting the preferences of customers regarding the time of the day in which they would like to be served (soft time windows). This work investigates the use of goal programming to model these problems. To solve the model, an enumeration-followed-by-optimization approach is proposed which first computes feasible routes and then selects the set of best ones. Computational results show that this approach is adequate for medium-sized delivery problems.     

Heuristics for the multi-depot petrol station replenishment problem with time windows

In the multi-depot petrol station replenishment problem with time windows (MPSRPTW), the delivery of petroleum products stored in a number of different petroleum depots to a set of petrol distribution stations has to be optimized. Each depot has its own fleet of heterogeneous and compartmented tank trucks. Stations specify their demand by indicating the minimum and maximum quantities to be delivered for each ordered product and require the delivery within a predetermined time window. Several inter-related decisions must be made simultaneously in order to solve the problem. For this problem, the set of feasible routes to deliver all the demands, the departure depot for each route, the quantities of each product to be delivered, the assignment of these routes to trucks, the time schedule for each trip, and the loading of the ordered products to different tanks of the trucks used need to be determined. In this paper, we propose a mathematical model that selects, among a set of feasible trips, the subset that allows the delivery of all the demands while maximizing the overall daily net revenue. If this model is provided with all possible feasible trips, it determines the optimal solution for the corresponding MPSRPTW. However, since the number of such trips is often huge, we developed a procedure to generate a restricted set of promising feasible trips. Using this restricted set, the model produces a good but not necessarily optimal solution. Thus the proposed solution process can be seen as a heuristic. We report the results of the extensive numerical tests carried out to assess the performance of the proposed heuristic. In addition, we show that, for the special case of only one depot, the proposed heuristic outperforms a previously published solution method.     

Integer Programming to Minimize Labour Costs

The main purpose of this paper is to demonstrate a real-world application of pure integer programming to find the optimum solution to a labour cost problem. The length of a daily working shift is defined as an integer variable and several shift strategies are analysed to determine the optimum length and shift combinations that satisfy a predicted demand at minimum cost. The state-space model has been used to predict the stochastic behaviour of monthly demands for beer and soft drink. Savings of about 7% of the annual sales have been obtained as a result of implementing the integer programming approach. A numerical example shows that the solution obtained by rounding off the continuous optimal solution does not match with the integer optimal solution. It was also noted that if a rounded-off solution is feasible, then it provides an initial integer solution for the branch-and-bound algorithm that may reduce the computational time.     

Heuristic and lower bound for a stochastic location-routing problem

In this article a stochastic location-routing problem is defined and cast as a two-stage model. In a first stage the set of plants and a family of routes are determined; in a second stage a recourse action is applied to adapt these routes to the actual set of customers to visit, once they are known. A two-phase heuristic is developed. An initial feasible solution is built by solving a sequence of subproblems, and an improvement phase is then applied. A lower bound based on bounding separately different parts of the cost of any feasible solution is also developed. Computational results are reported.     

Routing and capacity assignment in backbone communication networks under time varying traffic conditions

This paper studies the problem of assigning capacities to links in a backbone communication network and determining the routes used by messages for all communicating node pairs in the network under time varying traffic conditions. The best routes are to be chosen from among all possible routes in the network. Tradeoffs between link costs and response time to users are achieved by specifying an upper limit on the average link queueing delay in the network. The goal is to minimize total link fixed and variable costs. The topology of the network and the end-to-end traffic requirements during the different busy-hours are assumed to be known. The problem is formulated as a mathematical programming model. An efficient solution procedure based on a Lagrangian relaxation of the problem is developed. The results of extensive computational experiments across a variety of networks are reported. These results indicate that the solution procedure is effective for a wide range of traffic loads and cost structures.     

Evolutionary Algorithms for the Vehicle Routing Problem with Time Windows

This paper surveys the research on evolutionary algorithms for the Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW can be described as the problem of designing least cost routes from a single depot to a set of geographically scattered points. The routes must be designed in such a way that each point is visited only once by exactly one vehicle within a given time interval. All routes start and end at the depot, and the total demands of all points on one particular route must not exceed the capacity of the vehicle. The main types of evolutionary algorithms for the VRPTW are genetic algorithms and evolution strategies. In addition to describing the basic features of each method, experimental results for the benchmark test problems of Solomon (1987) and Gehring and Homberger (1999) are presented and analyzed.     

A synthesis of assignment and heuristic solutions for vehicle routing with time windows

This paper proposes a three-stage method for the vehicle-routing problem with time window constraints (VRPTW). Using the Hungarian method the optimal customer matching for an assignment approximation of the VRPTW, which is a travel time-based relaxation that partially respects the time windows, is obtained. The assignment matching is transformed into feasible routes of the VRPTW via a simple decoupling heuristic. The best of these routes, in terms of travelling and vehicle waiting times, form part of the final solution, which is completed by the routes provided by heuristic methods applied to the remainder of the customers. The proposed approach is tested on a set of standard literature problems, and improves the results of the heuristic methods with respect to total travel time. Furthermore, it provides useful insights into the effect of employing optimal travel time solutions resulting from the assignment relaxation to derive partial route sets of the VRPTW.     

带时间窗分车运输同时收发车辆路径问题及其启发式算法

本文结合汽车零部件第三方物流的实际背景,提出了带时间窗的可分车运输同时收发车辆路径问题(简称SVRPSPDTW),并给出了问题的数学模型,同时提出两个求解该问题的启发式算法,最后进行了数值试验.由于没有可以利用的算例,本文在Solomn测试基准库的基础上构建了针对新问题的算例.计算结果表明,所有算例计算时间均不超过1秒,且算法1无论是从车辆的使用数还是从车辆行驶的路径总长度上都明显优于算法2,从而说明算法1是寻找SVRPSPDTW问题初始可行解的较为有效的算法.     

On lexicographic optimal solutions in transportation problems

Lexicographic versions of the cost minimizing transportation problem (CMTP) and the time minimizing transportation problem (TMTP) are presented in this paper. In addition to minimizing the quantity sent on the costliest routes in a cost minimizing transportation problem. an attempt is made to minimize the quantity transported on the second-costliest routes. if the shipment on the costliest routes is as small as possible and the quantity shipped on the third-costliest routes, if the shipments on the costliest and the second- costliest routes are as small as possible. and so on. In a lexicographic time minimizing transportation problem one is not only interested in minimizing the transportation cost on the routes of the longest duration but also on the routes of second longest, third-longest duration and so on. For finding lexicographic optimal solutions (LOS) of lexicographic cost minimizing and time minimizing transportation problems a standard cost minimizing transportation problem is formulated whose optimal solution is shown to provide the answer. Some extensions are also discussed     

Equitable Demand Adjustment for Infeasible Transportation Problems

We present a modelling/solution procedure for adjusting demands to obtain an ‘equitably infeasible’ solution for an infeasible transportation problem. The infeasibility may be due to total supply not being equal to total demand, or inadmissible routes (arcs). We show that the problem can be modelled and solved as a pre-emptive, multicriteria, capacitated transportation problem, whose objective is to minimize the maximum deviation between the fractional undersupply to the demand nodes or, equivalently, to minimize the fractional undersupply of the demands. Further, we show that an optimal solution can be obtained by solving either a single-criterion transportation problem (by choosing sufficiently large weights to aggregate the criteria) or a sequence of single-criterion transportation problems.     

Tabu Search heuristics for the Vehicle Routing Problem with Time Windows

This paper surveys the research on the Tabu Search heuristics for the Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW can be described as the problem of designing least cost routes for a fleet of vehicles from one depot to a set of geographically scattered points. The routes must be designed in such a way that each point is visited only once by exactly one vehicle within a given time interval; all routes start and end at the depot, and the total demands of all points on one particular route must not exceed the capacity of the vehicle. In addition to describing basic features of each method, experimental results for Solomon’s benchmark test problems are presented and analyzed. This work was partially supported by the Emil Aaltonen Foundation, Liikesivistysrahasto Foundation, the Canadian Natural Science and Engineering Research Council and the TOP program funded by the Research Council of Norway. This support is gratefully acknowledged.     

A survey of recent research on location-routing problems

The design of distribution systems raises hard combinatorial optimization problems. For instance, facility location problems must be solved at the strategic decision level to place factories and warehouses, while vehicle routes must be built at the tactical or operational levels to supply customers. In fact, location and routing decisions are interdependent and studies have shown that the overall system cost may be excessive if they are tackled separately. The location-routing problem (LRP) integrates the two kinds of decisions. Given a set of potential depots with opening costs, a fleet of identical vehicles and a set of customers with known demands, the classical LRP consists in opening a subset of depots, assigning customers to them and determining vehicle routes, to minimize a total cost including the cost of open depots, the fixed costs of vehicles used, and the total cost of the routes. Since the last comprehensive survey on the LRP, published by Nagy and Salhi (2007), the number of articles devoted to this problem has grown quickly, calling a review of new research works. This paper analyzes the recent literature (72 articles) on the standard LRP and new extensions such as several distribution echelons, multiple objectives or uncertain data. Results of state-of-the-art metaheuristics are also compared on standard sets of instances for the classical LRP, the two-echelon LRP and the truck and trailer problem.     

A genetic-algorithm-based approach to the two-echelon capacitated vehicle routing problem with stochastic demands in logistics service

This paper addresses the two-echelon capacitated vehicle routing problem (2E-CVRP) with stochastic demands (2E-CVRPSD) in city logistics. A stochastic program with recourse is used to describe the problem. This program aims to minimize the sum of the travel cost and the expected cost of recourse actions resulting from potential route failures. In a two-echelon distribution system, split deliveries are allowed at the first level but not at the second level, thereby increasing the difficulty of calculating the expected failure cost. Three types of routes with or without split deliveries are identified. Different methods are devised or adapted from the literature to compute the failure cost. A genetic-algorithm-based (GA) approach is proposed to solve the 2E-CVRPSD. A simple encoding and decoding scheme, a modified route copy crossover operator, and a satellite-selection-based mutation operator are devised in this approach. The numerical results show that for all instances, the expected cost of the best 2E-CVRPSD solution found by the proposed approach is not greater than that of the best-known 2E-CVRP solution with an average relative gap of 2.57%. Therefore, the GA-based approach can find high-quality solutions for the 2E-CVRPSD.     

A multi-shift vehicle routing problem with windows and cycle times

In many applications of the vehicle routing problem with time windows (VRPTW), goods must be picked up within desired time frames. In addition, they have some limitations on their arrival time to the central depot. In this paper, we present a new version of VRPTW that minimizes the total cycle time of the goods. In order to meet the time windows and also minimize the cycle time, the courier’s schedule is allowed to vary. An algorithm, named VeRSA, is proposed to solve this problem. VeRSA employs concepts of scheduling theorems and algorithms to determine feasible routes and schedules of the available couriers. We prove a theoretical lower bound that provides a useful bound on the optimality gap. We also conduct a set of numerical experiments. VeRSA obtains a feasible solution faster than solving the MIP. The optimality gap using our proposed lower bound is smaller than the gap found with the standard LP relaxation.     

集货送货一体化的物流配送车辆路线问题的标号算法

本文结合实际情况,对具有时间窗约束的集货送货一体化的车辆路线问题进行了研究,针对该问题的特点,采用修正的多属性标号算法对该问题进行求解,并通过C 编程语言实现了该算法,最后用一个示例表明本文的算法是有效的.     

Selection of Capacity Expansion Projects

The problem of determining a project selection schedule and a production-distribution-inventory schedule for each of a number of plants so as to meet the demands of multiregional markets at minimum discounted total cost during a discrete finite planning horizon is considered. We include the possibility of using inventory and/or imports to delay the expansion decision at each producing region in a transportation network. Through a problem reduction algorithm, the Lagrangean relaxation problem strengthened by the addition of a surrogate constraint becomes a 0–1 mixed integer knapsack problem. Its optimal solution, given a set of Lagrangean multipliers, can be obtained by solving at most two generally smaller 0–1 pure integer knapsack problems. The bound is usually very tight. At each iteration of the subgradient method, we generate a primal feasible solution from the Lagrangean solution. The computational results indicate that the procedure is effective in solving large problems to within acceptable error tolerances.     

Topological design of wide area communication networks

This paper develops a model for designing a backbone network. It assumes the location of the backbone nodes, the traffic between the backbone nodes and the link capacities are given. It determines the links to be included in the design and the routes used by the origin destination pairs. The objective is to obtain the least cost design where the system costs consist of connection costs and queueing costs. The connection costs depend on link capacity and queueing costs are incurred by users due to the limited capacity of links. The Lagrangian relaxation embedded in a subgradient optimization procedure is used to obtain lower bounds on the optimal solution of the problem. A heuristic based on the Lagrangian relaxation is developed to generate feasible solutions.     

An integrated approach for earthwork allocation,sequencing and routing

Planning techniques for large scale earthworks have been considered in this article. To improve these activities a “block theoretic” approach was developed that provides an integrated solution consisting of an allocation of cuts to fills and a sequence of cuts and fills over time. It considers the constantly changing terrain by computing haulage routes dynamically. Consequently more realistic haulage costs are used in the decision making process. A digraph is utilised to describe the terrain surface which has been partitioned into uniform grids. It reflects the true state of the terrain, and is altered after each cut and fill. A shortest path algorithm is successively applied to calculate the cost of each haul, and these costs are summed over the entire sequence, to provide a total cost of haulage. To solve this integrated optimisation problem a variety of solution techniques were applied, including constructive algorithms, meta-heuristics and parallel programming. The extensive numerical investigations have successfully shown the applicability of our approach to real sized earthwork problems.     

Vehicle routing in the 1-skip collection problem

In this paper, we consider a real problem, which we call the 1-skip collection problem, where a fleet of vehicles must collect a number of skips situated in different locations and transport them to one among different plants chosen on the basis of the kind of waste contained in the skip. Each vehicle has a capacity of one skip and it starts and ends its tour at the depot. Each time a vehicle collects a skip, it has to go to a plant and empty it. A number of constraints are imposed, which involve time windows for the customers and the plants, shift-time, different kinds of skips, number of drivers available to carry out the service and priorities assigned to the customers who have to be served. The objective is to minimize the total cost of the service given by the fixed cost of the drivers engaged to carry out the service, the cost of the extra time and the penalty cost paid if a customer is not served. A heuristic algorithm to solve the real problem is presented. The algorithm first constructs a feasible solution by means of the nearest-neighbour algorithm. Then, if it finds a feasible solution, it improves it. The computational results show that the solution of the algorithm is much better than the solution applied by the firm that carries out the service since it serves a higher number of skips with a smaller number of drivers.