A Lagrangian relaxation approach to multi-period inventory/distribution planning

We consider a multi-period inventory/distribution planning problem (MPIDP) in a one-warehouse multiretailer distribution system where a fleet of heterogeneous vehicles delivers products from a warehouse to several retailers. The objective of the MPIDP is to minimise transportation costs for product delivery and inventory holding costs at retailers over the planning horizon. In this research, the problem is formulated as a mixed integer linear programme and solved by a Lagrangian relaxation approach. A subgradient optimisation method is employed to obtain lower bounds. We develop a Lagrangian heuristic algorithm to find a good feasible solution of the MPIDP. Computational experiments on randomly generated test problems showed that the suggested algorithm gave relatively good solutions in a reasonable amount of computation time.     

A biased random-key genetic algorithm for road congestion minimization

One of the main goals in transportation planning is to achieve solutions for two classical problems, the traffic assignment and toll pricing problems. The traffic assignment problem aims to minimize total travel delay among all travelers. Based on data derived from the first problem, the toll pricing problem determines the set of tolls and corresponding tariffs that would collectively benefit all travelers and would lead to a user equilibrium solution. Obtaining high-quality solutions for this framework is a challenge for large networks. In this paper, we propose an approach to solve the two problems jointly, making use of a biased random-key genetic algorithm for the optimization of transportation network performance by strategically allocating tolls on some of the links of the road network. Since a transportation network may have thousands of intersections and hundreds of road segments, our algorithm takes advantage of mechanisms for speeding up shortest-path algorithms.     

Vector and parallel computing for matrix balancing

Estimating the entries of a large matrix to satisfy a set of internal consistency relations is a problem with several applications in economics, urban and regional planning, transportation, statistics and other areas. It is known as theMatrix Balancing Problem. Matrix balancing applications arising from the estimation of telecommunication or transportation traffic and from multi-regional trade flows give rise to huge optimization problems. In this report, we show that the RAS algorithm can be specialized for vector and parallel computing and used for the solution of very large problems. The algorithm is specialized for vector computations on a CRAY X-MP and is parallelized on an Alliant FX/8. A variant of the algorithm — developed here for its potential parallelism — turns out to be more efficient than the original algorithm even when implemented serially. We use the algorithms to estimate disaggregated input/output tables and a multi-regional trade flow table of the U.S. The larger problem solved has approximately 12 000 constraints and over 370 000 nonlinear variables. This is the first of two papers that aim at the solution of very large matrix balancing problems. Zenios [20] is using the same algorithm for the same models on a massively parallel Connection Machine CM-2.Research partially supported by NSF grants ECS-8718971 and CCR-8811135, and AFOSR grant 89-0145. Computing resources were made available through the ACRF at Argonne National Laboratory and CRAY Research, Inc.     

Synchronization of inventory and transportation under flexible vehicle constraint: A heuristics approach using sliding windows and hierarchical tree structure

This paper investigates the integrated inventory and transportation planning under flexible vehicle constraint. To offer better services at lower prices, more and more companies turn to outsource transportation functions to other professional service providers, namely 3rd party logistics companies. Under these vehicle rental arrangements, the number of vehicles is a decision variable instead of a fixed number, and the transportation cost includes not only the delivery cost but also the cost of vehicle rental that is proportional to the number of vehicles rented in a given planning horizon. In this paper, the problem is formulated as a mixed integer programming problem. A heuristic algorithm is developed, in which sliding windows are applied to approximate the problem by repeatedly solving a series of overlapping short-term subproblems, and a hierarchical tree structure is used to evaluate the closeness of different groups of retailers. Numerical experiments show that a better tradeoff between the inventory cost and transportation cost can be achieved through the proposed heuristic algorithm.     

Notes on the single route lateral transhipment problem

Previous research has analyzed deterministic and stochastic models of lateral transhipments between different retailers in a supply chain. In these models the analysis assumes given fixed transhipment costs and determines under which situations (magnitudes of excess supply and demand at various retailers) the transhipment is profitable. However, in reality, these depend on aspects like the distance between retailers or the transportation mode chosen. In many situations, combining the transhipments may save transportation costs. For instance, one or more vehicle routes may be used to redistribute the inventory of the potential pickup and delivery stations. This can be done in any sequence as long as the vehicle capacity is not violated and there is enough load on the vehicle to satisfy demand. The corresponding problem is an extension of the one-commodity pickup and delivery traveling salesman and the pickup and delivery vehicle routing problem. When ignoring the routing aspect and assuming given fixed costs, transhipment is only profitable if the quantities are higher than a certain threshold. In contrast to that, the selection of visited retailers is dependent on the transportation costs of the tour and therefore the selected retailers are interrelated. Hence the problem also has aspects of a (team) orienteering problem. The main contribution is the discussion of the tour planning aspects for lateral transhipments which may be valuable for in-house planning but also for price negotiations with external contractors. A mixed integer linear program for the single route and single commodity version is presented and an improved LNS framework to heuristically solve the problem is introduced. Furthermore, the effect of very small load capacity on the structure of optimal solutions is discussed.     

A generic auction algorithm for the minimum cost network flow problem

In this paper we broadly generalize the assignment auction algorithm to solve linear minimum cost network flow problems. We introduce a generic algorithm, which contains as special cases a number of known algorithms, including the -relaxation method, and the auction algorithm for assignment and for transportation problems. The generic algorithm can serve as a broadly useful framework for the development and the complexity analysis of specialized auction algorithms that exploit the structure of particular network problems. Using this framework, we develop and analyze two new algorithms, an algorithm for general minimum cost flow problems, called network auction, and an algorithm for thek node-disjoint shortest path problem.     

A solution approach to the inventory routing problem in a three-level distribution system

We consider the infinite horizon inventory routing problem in a three-level distribution system with a vendor, a warehouse and multiple geographically dispersed retailers. In this problem, each retailer faces a demand at a deterministic, retailer-specific rate for a single product. The demand of each retailer is replenished either from the vendor through the warehouse or directly from the vendor. Inventories are kept at both the retailers and the warehouse. The objective is to determine a combined transportation (routing) and inventory strategy minimizing a long-run average system-wide cost while meeting the demand of each retailer without shortage. We present a decomposition solution approach based on a fixed partition policy where the retailers are partitioned into disjoint and collectively exhaustive sets and each set of retailers is served on a separate route. Given a fixed partition, the original problem is decomposed into three sub-problems. Efficient algorithms are developed for the sub-problems by exploring important properties of their optimal solutions. A genetic algorithm is proposed to find a near-optimal fixed partition for the problem. Computational results show the performance of the solution approach.     

部分联合运输策略下的物流车辆路径优化问题研究

论文分析了物流车辆路径优化问题的特点,提出了企业自营物流和第三方物流协同运输的部分联合运输策略。根据客户需求节点的特点进行了节点分类,建立了以车辆调用成本、车辆运输成本、第三方物流运输成本之和最小为目标的整数线性规划模型。根据部分联合运输策略下各类客户需求点运输方式特点,构造了一种新的变维数矩阵编码结构,并对传统算法中概率选择操作方式进行修改,提出了一种新的智能优化算法并与枚举法和遗传算法的运算结果进行了算法性能对比分析。结果显示,本文提出的逆选择操作蚁群算法具有较快的运算速度和较高的稳定性,是求解此类问题的一种有效算法。     

An Algorithm for Solving Capacitated Multicommodity <Emphasis Type="Italic">p</Emphasis>-median Transportation Problems

An algorithm for solving a special capacitated multicommodity p-median transportation problem (CMPMTP), which arises in container terminal management, is presented. There are some algorithms to solve similar kinds of problems. The formulation here is different from the existing modelling of the p-median or some related location problems. We extend the existing work by applying a Lagrangean relaxation to the CMPMTP. In order to obtain a satisfactory solution, a heuristic branch-and-bound algorithm is designed to search for a better solution, if one is possible. A comparison is also made with different algorithms.     

Integrated location and two-echelon inventory network design under uncertainty

In this paper, we propose a two-stage stochastic model to address the design of an integrated location and two-echelon inventory network under uncertainty. The central issue in this problem is to design and operate an effective and efficient multi-echelon supply chain distribution network and to minimize the expected system-wide cost of warehouse location, the allocation of warehouses to retailers, transportation, and two-echelon inventory over an infinite planning horizon. We structure this problem as a two-stage nonlinear discrete optimization problem. The first stage decides the warehouses to open and the second decides the warehouse-retailer assignments and two-echelon inventory replenishment strategies. Our modeling strategy incorporates various probable scenarios in the integrated multi-echelon supply chain distribution network design to identify solutions that minimize the first stage costs plus the expected second stage costs. The two-echelon inventory cost considerations result in a nonlinear objective which we linearize with an exponential number of variables. We solve the problem using column generation. Our computational study indicates that our approach can solve practical problems of moderate-size with up to twenty warehouse candidate locations, eighty retailers, and ten scenarios efficiently.     

Integration of AI and OR Techniques for Computer-Aided Algorithmic Design in the Vehicle Routing Domain

This paper shows how tools and techniques of artificial intelligence can be successfully integrated into a computer system working in the vehicle routing domain. The aim of this system, called ALTO, is to facilitate the development of routing algorithms for transportation vehicles. In this paper, we describe the general algorithmic framework and the rich interface provided by the system to the expert algorithm designer. We also introduce a methodology for acquiring useful knowledge in the domain, based on examples of successful and unsuccessful problem-solving strategies. With such knowledge, ALTO would then be capable of actively supporting the algorithm designer by suggesting good candidate algorithms for solving new problems.     

A simple greedy algorithm for a class of shuttle transportation problems

Greedy algorithms for combinatorial optimization problems are typically direct and efficient, but hard to prove optimality. The paper presents a special class of transportation problems where a supplier sends goods to a set of customers, returning to the source after each delivery. We show that these problems with different objective functions share a common structural property, and therefore a simple but powerful generic greedy algorithm yields optimal solutions for all of them.     

A two-stage model for a day-ahead paratransit planning problem

We consider a dynamic planning problem for paratransit transportation. The focus is on a decision to take one day ahead: which requests to serve with own vehicles, and which requests to subcontract to taxis? We call this problem the day-ahead paratransit planning problem. The developed model is a non-standard two-stage integer recourse model. Both stages consist of two consecutive optimization problems: the clustering of requests into routes, and the assignment of these routes to vehicles. To solve this model, a genetic algorithm approach is used. Computational results are presented for randomly generated data sets.     

一种新的交通网络设计优化算法

交通网络设计问题是研究如何用定量的方法在已有交通网络上添加或扩容某些路段的问题.文章在回顾交通网络设计问题文献的基础上,提出了基于图论网络优化思想的解决该类问题的一种新思路,给出了启发式算法,并进行了算法复杂性分析,最后通过算例验证了其有效性.     

A genetic algorithm for a single product network design model with lead time and safety stock considerations

We consider a two-stage supply chain with a production facility that replenishes a single product at retailers. The objective is to locate distribution centers in the network such that the sum of facility location, pipeline inventory, and safety stock costs is minimized. We explicitly model the relationship between the flows in the network, lead times, and safety stock levels. We use genetic algorithms to solve the model and compare their performance to that of a Lagrangian heuristic developed in earlier work. A novel chromosome representation that combines binary vectors with random keys provides solutions of similar quality to those from the Lagrangian heuristic. The model is then extended to incorporate arbitrary demand variance at the retailers. This modification destroys the structure upon which the Lagrangian heuristic is based, but is easily incorporated into the genetic algorithm. The genetic algorithm yields significantly better solutions than a greedy heuristic for this modification and has reasonable computational requirements.     

Characterization of facility assignment costs for a location-inventory model under truckload distribution

We consider a two-stage distribution system, where the first stage consists of potential distribution centres (DCs) and the second stage consists of geographically dispersed existing retailers. Our goal is to determine the set of open DCs and assignment of open DCs to retailers simultaneously with inventory decisions of retailers. In addition to the DC-specific fixed facility location costs, we explicitly model the inventory replenishment and holding costs at the retailers and truckload transportation costs between the DCs and the retailers. The transportation costs are subject to truck/cargo capacity, leading to an integrated location-inventory problem with explicit cargo costs. We develop a mixed-integer nonlinear model and analyse its structural properties leading to exact expressions for the so-called implied facility assignment costs and imputed per-unit per-mile transportation costs. These expressions analytically demonstrate the interplay between strategic location and tactical inventory/transportation decisions in terms of resulting operational costs. Although both the theory and practice of integrated logistics have recognized the fact that strategic and tactical decisions are interrelated, to the best of our knowledge, our paper is the first to offer closed-form results demonstrating the relationship explicitly. We propose an efficient solution approach utilizing the implied facility assignment costs and we demonstrate that significant savings are realizable when the inventory decisions and cargo costs are modelled explicitly for facility location purposes.     

A Stochastic Bottleneck Transportation Problem

In this paper, a stochastic bottleneck transportation problem, which aims at minimizing the transportation time target subject to a chance constraint, is formulated and an algorithm based on a parametric programming approach is developed to solve it. Further, assuming the transportation costs to be deterministic, a trade-off analysis between the transportation time target and the total cost is given. In addition, methods are developed which give the whole spectrum of optimal solutions to the problems mentioned above. The algorithms are illustrated by numerical examples. The computational complexity of the algorithms is also discussed.     

Stowage planning in maritime container transportation

We consider a stowage-planning problem of arranging containers on a container ship in the maritime transportation system. Since containers are accessible only from the top of the stack, temporary unloading and reloading of containers, called shifting, is unavoidable if a container required to be unloaded at the current port is stacked under containers to be unloaded at later ports on the route of the ship. The objective of the stowage planning problem is to minimize the time required for shifting and crane movements on a tour of a container ship while maintaining the stability of the ship. For the problem, we develop a heuristic solution method in which the problem is divided into two subproblems, one for assigning container groups into the holds and one for determining a loading pattern of containers assigned to each hold. The former subproblem is solved by a greedy heuristic based on the transportation simplex method, while the latter is solved by a tree search method. These two subproblems are solved iteratively using information obtained from solutions of each other. To see the performance of the suggested algorithm, computational tests are performed on problem instances generated based on information obtained from an ocean container liner. Results show that the suggested algorithm works better than existing algorithms.     

A parallel interior point algorithm for linear programming on a network of transputers

Interior Point algorithms have become a very successful tool for solving large-scale linear programming problems. The Dual Affine algorithm is one of the Interior Point algorithms implemented in the computer program OB1. It is a good candidate for implementation on a parallel computer because it is very computing-intensive. A parallel Dual Affine algorithm is presented which is suitable for a parallel computer with a distributed memory. The algorithm obtains its speedup from parallel sparse linear algebra computations such as Cholesky factorisation, matrix multiplication, and triangular system solving, which form the bulk of the computing work. Efficient algorithms based on the grid distribution of matrices are presented for each of these computations. The algorithm is implemented in occam 2 on a square mesh of transputers. The resulting parallel program is connected to the sequentialFortran 77 program OB1, which performs the preprocessing and the postprocessing. Experimental results on a mesh of 400 transputers are given for a test set of seven realistic planning and scheduling problems from Shell and seven problems from the NETLIB LP collection; the results show a speedup of 88 for the largest problem.     

Approximation algorithms for <Emphasis Type="Italic">k</Emphasis>-echelon extensions of the one warehouse multi-retailer problem

In this paper, we consider k-echelon extensions of the deterministic one warehouse multi-retailer problem. We give constant factor approximation algorithms for some of these extensions when k is fixed. We focus first on the case without backorders and we give a \((2k-1)\)-approximation algorithm under general assumptions on the evolution of the holding costs as products move toward the final customers. We then improve this result to a k-approximation when the holding costs are monotonically non-increasing or non-decreasing (which is a natural situation in practice). Finally we address problems with backorders: we give a 3-approximation for the one-warehouse multi-retailer problem with backlog and a k-approximation algorithm for the k-level Joint Replenishment Problem with backlog (a variant where inventory can only be kept at the final retailers). Ours results are the first constant approximation algorithms for those problems. In addition, we demonstrate the potential of our approach on a practical case. Our preliminary experiments show that the average optimality gap is around 15%.