Analysis of the Best Double Frequency Policy in the Single Link Problem with Discrete Shipping Times

We study an inventory–transportation problem where one product has to be shipped from an origin to a destination by vehicles of given capacity over an infinite time horizon. The product is made available at the origin and consumed at the destination at the same constant rate. The intershipment time must be not lower than a given minimum value. The problem is to decide when to make the shipments and how to load the vehicles to minimize the sum of the transportation and the inventory costs at the origin and at the destination per time unit. We study the case in which the intershipment time is a multiple of the minimum value, i.e., the problem with discrete shipping times. We show that, in this case, the best double frequency policy has a tight performance bound of about 1.1603 with respect to the optimal periodic policy and of about 1.1538 with respect to the best frequency-based policy. Moreover, we show that, from the worst-case point of view, the best double frequency policy is the optimal frequency-based policy.     

Rounding Procedures for the Discrete Version of the Capacitated Economic Order Quantity Problem

The capacitated Economic Order Quantity problem (capacitated EOQ) is a well-known problem where products have to be shipped between two points with a vehicle of given capacity. Each shipment has a fixed cost, independent of the shipped quantity, and an inventory cost is generated in the two points. The problem consists in finding the optimal time between consecutive shipments, which minimizes the total cost. The problem is a capacitated variant of the EOQ problem and has a closed form solution. Since such solution is often irrational, it is often rounded to an integer value. In this paper we investigate the errors which are generated by rounding procedures to integer and powers-of-two values. We show that, although in the worst case a tight general relative error of 2 is generated by all the considered rounding procedures, the procedure which rounds to the best between the lower and upper values (integer or powers-of-two) has a performance of $\frac{1}{2}$ ( $\sqrt 2 + 1/\sqrt 2 $ )≈1.06 on classes of instances of high practical relevance.     

Capacitated transportation problem with bounds on RIM conditions

Motivated by dead-mileage problem assessed in terms of running empty buses from various depots to starting points, we consider a class of the capacitated transportation problems with bounds on total availabilities at sources and total destination requirements. It is often difficult to solve such problems and the present paper establishes their equivalence with a balanced capacitated transportation problem which can be easily solved by existing methods. Sometimes, total flow in transportation problem is also specified by some external decision maker because of budget/political consideration and optimal solution of such problem is of practical interest to the decision maker and has motivated us to discuss such problem. Various situations arising in unbalanced capacitated transportation problems have been discussed in the present paper as a particular case of original problem. In addition, we have discussed paradoxical situation in a balanced capacitated transportation problem and have obtained the paradoxical solution by solving one of the unbalanced problems. Numerical illustrations are included in support of theory.     

The freight consolidation and containerization problem

In today’s global free market, third-party logistics providers (3PLs) are becoming increasingly important. This paper studies a problem faced by a 3PL operating a warehouse in Shanghai, China, under contract with a major manufacturer of children’s clothing based in the United States. At the warehouse, the 3PL receives textile parcel shipments from the suppliers located in China; each shipment is destined for different retail stores located across the United Sates. These shipments must be consolidated and loaded into containers of varying sizes and costs, and then sent along shipping routes to different destination ports. An express company, such as UPS and FedEx, unloads the shipments from the containers at the destination ports and distributes them to their corresponding stores or retailers by parcel delivery. The objective is to find an allocation that minimizes the total container transportation and parcel delivery costs. We formulate the problem into an integer programming model, and also propose a memetic algorithm approach to solve the problem practically. A demonstration of a good solution to this problem was a decisive factor in the awarding of the contract to the 3PL in question.     

考虑订单合并和货物转运的多式联运路径优化研究

为了获得运输的规模经济效应,本文研究了一种考虑订单合并和货物转运的零担多式联运路径优化问题。首先,以总运输成本为目标函数,以网络中的运输工具容量、可以提供的运输工具最大数量、运输工具服务的关闭时间以及订单时间窗为约束,构建混合整数规划模型,在模型中允许多个订单进行合并运输并考虑运输过程中的转运成本。其次,由于多式联运路径优化问题是典型的NP-hard问题,为了快速求解该模型,开发了一种可以快速为该问题提供近似最优解和下界的列生成启发式算法。最后,生成并测试了大量算例,结果表明所开发的列生成启发式算法可以在较短的时间内提供高质量的近似最优解。文章所构建的模型和开发的列生成启发式算法可以为零担自营多式联运物流企业提供高效的决策支持。     

An Integrated Inventory-Routing System for Multi-item Joint Replenishment with Limited Vehicle Capacity

In this paper, we develop a mathematical programming approach for coordinating inventory and transportation decisions in an inbound commodity collection system. In particular, we consider a system that consists of a set of geographically dispersed suppliers that manufacture one or more non-identical items, and a central warehouse that stocks these items. The warehouse faces a constant and deterministic demand for the items from outside retailers. The items are collected by a fleet of vehicles that are dispatched from the central warehouse. The vehicles are capacitated, and must also satisfy a frequency constraint. Adopting a policy in which each vehicle always collects the same set of items, we formulate the inventory-routing problem of minimizing the long-run average inventory and transportation costs as a set partitioning problem. We employ a column generation approach to determine a lower bound on the total costs, and develop a branch-and-price algorithm that finds the optimal assignment of items to vehicles. We also propose greedy constructive heuristics, and develop a very large-scale neighborhood (VLSN) search algorithm to find near-optimal solutions for the problem. Computational tests are performed on a set of randomly generated problem instances.The work of this author was supported by a scholarship of the Faculty of Engineering of Ubonratchathani University, Ubonratchathani, Thailand., The work of this author was supported in part by the National Science Foundation under Grant No. DMI-0085682.     

A branch-and-price algorithm for the capacitated facility location problem

The capacitated facility location problem (CFLP) is a well-known combinatorial optimization problem with applications in distribution and production planning. It consists in selecting plant sites from a finite set of potential sites and in allocating customer demands in such a way as to minimize operating and transportation costs. A number of solution approaches based on Lagrangean relaxation and subgradient optimization has been proposed for this problem. Subgradient optimization does not provide a primal (fractional) optimal solution to the corresponding master problem. However, in order to compute optimal solutions to large or difficult problem instances by means of a branch-and-bound procedure information about such a primal fractional solution can be advantageous. In this paper, a (stabilized) column generation method is, therefore, employed in order to solve a corresponding master problem exactly. The column generation procedure is then employed within a branch-and-price algorithm for computing optimal solutions to the CFLP. Computational results are reported for a set of larger and difficult problem instances.     

A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts

Lin [T.Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 34 (10) (2010) 3158–3165] recently proposed an EOQ model with imperfect quality and quantity discounts, where the lot-splitting shipments policy is adopted. In this note we first rectify the holding cost terms showed in Lin to obtain a new objective function, then resolve the problem and develop an easy to implement algorithm to find the overall optimal solutions for the model. Besides, we present a new model for items with imperfect quality, where lot-splitting shipments and different holding costs for good and defective items are considered. The closed-form formulas for determining the optimal ordering and shipping policies are derived. Also, the results are examined analytically and numerically to gain more insights of the solutions.     

A continuous approximation procedure for determining inventory distribution schemas within supply chains

This paper presents an integrated inventory distribution optimization model that simultaneously incorporates the issues of location, production, inventory, and transportation within a supply chain. The objective is to determine the optimal number and size of shipments under varying but commonly practiced production and shipping scenarios. A continuous approximation procedure is proposed to determine the optimal number and size of shipments. Three production and shipping scenarios are investigated and closed form expressions for the optimal number of shipments for each scenario are obtained. A numerical example is presented to demonstrate the usefulness of the model.     

The multi-commodity one-to-one pickup-and-delivery traveling salesman problem

This paper concerns a generalization of the traveling salesman problem (TSP) called multi-commodity one-to-one pickup-and-delivery traveling salesman problem (m-PDTSP) in which cities correspond to customers providing or requiring known amounts of m different commodities, and the vehicle has a given upper-limit capacity. Each commodity has exactly one origin and one destination, and the vehicle must visit each customer exactly once. The problem can also be defined as the capacitated version of the classical TSP with precedence constraints. This paper presents two mixed integer linear programming models, and describes a decomposition technique for each model to find the optimal solution. Computational experiments on instances from the literature and randomly generated compare the techniques and show the effectiveness of our implementation.     

Dynamic programming approximation algorithms for the capacitated lot-sizing problem

This paper provides a new idea for approximating the inventory cost function to be used in a truncated dynamic program for solving the capacitated lot-sizing problem. The proposed method combines dynamic programming with regression, data fitting, and approximation techniques to estimate the inventory cost function at each stage of the dynamic program. The effectiveness of the proposed method is analyzed on various types of the capacitated lot-sizing problem instances with different cost and capacity characteristics. Computational results show that approximation approaches could significantly decrease the computational time required by the dynamic program and the integer program for solving different types of the capacitated lot-sizing problem instances. Furthermore, in most cases, the proposed approximate dynamic programming approaches can accurately capture the optimal solution of the problem with consistent computational performance over different instances.     

Real-time split-delivery pickup and delivery time window problems with transfers

In this research we present the design and implementation of heuristics for solving split-delivery pickup and delivery time window problems with transfer (SDPDTWP) of shipments between vehicles for both static and real-time data sets. In the SDPDTWP each shipment is constrained with the earliest possible pickup time from the origin and the latest acceptable delivery time to a destination. Split-deliveries occur when two or more vehicles service the same origin or destination. The proposed heuristics were applied to both static and real-time data sets. The heuristics computed a solution, in a few seconds, for a static problem from the literature, achieving an improvement of 60% in distance in comparison to the published solution. In the real-time SDPDTWP problems, requests for pickup and delivery of a package, breakdown of a truck or insertion of a truck can occur after the vehicle has left the origin and is enroute to service the customers. Thirty data sets, each consisting of one to seven real-time customer or truck events, were used to test the efficiency of the heuristics. The heuristics obtained solutions to real-time data sets in under five seconds of CPU time.      

Efficient heuristics for the rectilinear distance capacitated multi-facility Weber problem

In this paper, we consider the capacitated multi-facility Weber problem with rectilinear distance. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the rectilinear distance separating them. We first give a new mixed integer linear programming formulation of the problem by making use of a well-known necessary condition for the optimal facility locations. We then propose new heuristic solution methods based on this formulation. Computational results on benchmark instances indicate that the new methods can provide very good solutions within a reasonable amount of computation time.     

Controlled rounding for tables with subtotals

The problem of rounding in statistical tables to protect confidentiality is an important problem in the area of data publication, especially for official statistics. Controlled rounding involves rounding the table data to a prespecified base while ensuring additivity to totals. Previous research provided a formulation of the controlled rounding problem of a simple two-way table as a transportation problem. This paper extends that work to tables with subtotals by using a capacitated transshipment formulation. It is shown that some forms of tables with subtotals always have a controlled rounding solution. Other table structures cannot be guaranteed such a solution under zero-restrictedness. Initial computational experience suggests that the method is viable for use in practical situations.     

Polyhedral study of the capacitated vehicle routing problem

The capacitated vehicle routing problem (CVRP) considered in this paper occurs when goods must be delivered from a central depot to clients with known demands, usingk vehicles of fixed capacity. Each client must be assigned to exactly one of the vehicles. The set of clients assigned to each vehicle must satisfy the capacity constraint. The goal is to minimize the total distance traveled. When the capacity of the vehicles is large enough, this problem reduces to the famous traveling salesman problem (TSP). A variant of the problem in which each client is visited by at least one vehicle, called the graphical vehicle routing problem (GVRP), is also considered in this paper and used as a relaxation of CVRP. Our approach for CVRP and GVRP is to extend the polyhedral results known for TSP. For example, the subtour elimination constraints can be generalized to facets of both CVRP and GVRP. Interesting classes of facets arise as a generalization of the comb inequalities, depending on whether the depot is in a handle, a tooth, both or neither. We report on the optimal solution of two problem instances by a cutting plane algorithm that only uses inequalities from the above classes.This work was supported in part by NSF grant DDM-8901495.     

On alternative mixed integer programming formulations and LP-based heuristics for lot-sizing with setup times

We address the multi-item, capacitated lot-sizing problem (CLSP) encountered in environments where demand is dynamic and to be met on time. Items compete for a limited capacity resource, which requires a setup for each lot of items to be produced causing unproductive time but no direct costs. The problem belongs to a class of problems that are difficult to solve. Even the feasibility problem becomes combinatorial when setup times are considered. This difficulty in reaching optimality and the practical relevance of CLSP make it important to design and analyse heuristics to find good solutions that can be implemented in practice. We consider certain mixed integer programming formulations of the problem and develop heuristics including a curtailed branch and bound, for rounding the setup variables in the LP solution of the tighter formulations. We report our computational results for a class of instances taken from literature.     

Solving the capacitated location-routing problem

This is a summary of the main results presented in the author’s Ph.D thesis, available at http://prodhonc.free.fr/homepage. This thesis, written in French, was supervised by Christian Prins and Roberto Wolfler-Calvo, and defended on 16 October 2006 at the Université de Technologie de Troyes. Several new approaches are proposed to solve the capacitated location-routing problem (CLRP): heuristic, cooperative and exact methods. Their performances are tested on various kinds of instances with capacitated vehicles and capacitated or uncapacitated depots.      

Planning and consolidating shipments from a warehouse

A typical warehouse or distribution centre ships material to various customer locations across the country, using various modes of transportation. Each mode has different constraints on size of shipment, different cost structures and different transportation times. Typically, for a given warehouse there are certain customer locations that receive frequent shipments of material. It is often possible, therefore, for the warehouse to consolidate different orders for the same customer location into a single shipment. The transportation mode and the day of shipment must be chosen such that the consolidated shipment meets the size constraints and arrives within an agreed-upon ‘delivery window’. In preparing a warehouse distribution plan, a planner seeks to achieve transportation economies of scale (by consolidating two or more orders into fewer shipments) while levelling the workload on warehouse resources and ensuring that material arrives at a customer location during the acceptable delivery window.The problem of deciding what shipments to make daily can be formulated as a set partitioning problem with side constraints. This paper describes a heuristic solution approach for this problem. Computational experiments using actual warehouse select activity indicate that, for moderate-size problems, the heuristic produces solutions with transportation costs that are within a few percent of optimal. Larger problems found in practice are generally too large to be solved by optimal algorithms; the heuristic easily handles such problems. The heuristic has been integrated into the transportation planning system of a leading distributor of telecommunications products.     

Degeneracy in fixed cost transportation problems

A perturbation method is introduced which transforms any fixed cost transportation problem F into a degeneracy-free equivalent F′. If a basic optimal solution to F′ is known, an optimal solution to F can be obtained by means of simple rounding.     

Bike sharing systems: Solving the static rebalancing problem

This paper deals with a new problem that is a generalization of the many to many pickup and delivery problem and which is motivated by operating self-service bike sharing systems. There is only one commodity, initially distributed among the vertices of a graph, and a capacitated single vehicle aims to redistribute the commodity in order to reach a target distribution. Each vertex can be visited several times and also can be used as a buffer in which the commodity is stored for a later visit. This problem is NP-hard, since it contains several NP-hard problems as special cases (the TSP being maybe the most obvious one). Even finding a tractable exact formulation remains problematic.This paper presents efficient algorithms for solving instances of reasonable size, and contains several theoretical results related to these algorithms. A branch-and-cut algorithm is proposed for solving a relaxation of the problem. An upper bound of the optimal solution of the problem is obtained by a tabu search, which is based on some theoretical properties of the solution, once fixed the sequence of the visited vertices. The possibility of using the information provided by the relaxation receives a special attention, both from a theoretical and a practical point of view. It is proven that to build a feasible solution of the problem by using the one obtained by the relaxation is an NP-hard problem. Nevertheless, a tabu search initialized with the optimal solution of the relaxation often shows that it is the optimal one.The algorithms have been tested on a set of instances coming from the literature, proving their effectiveness.