Adjacent vertices on transportation polytopes

To solve linear fixed charge problems with Murty's vertex ranking algorithm, one uses a simplex algorithm and a procedure to determine the vertices adjacent to a given vertex. In solving fixed charge transportation problems, the simplex algorithm simplifies to the stepping-stone algorithm. To find adjacent vertices on transportation polytopes, we present a procedure which is a simplification of a more general procedure for arbitrary polytopes.     

Transshipment along a single road

This article explores the problem of the transshipment of goods between centers that are situated along a single road. Costs are incurred both for goods that are shipped and for goods that remain in their original centers. A procedure is developed that determines a partial optimal assignment by use of a northwest corner rule solution. This partial assignment then is completed by solving a number of smaller tridiagonal transportation problems.     

Integrated scheduling of loading and transportation with tractors and semitrailers separated

Motivated by some practical applications, we study a new integrated loading and transportation scheduling problem. Given a set of jobs, a single crane is available to load jobs, one by one, onto semitrailers with a given capacity. Loaded semitrailers are assigned to tractors for transportation tasks. Subject to limited resources (crane, semitrailers, and tractors), the problem is to determine (1) an assignment of jobs to semitrailers for loading tasks, (2) a sequence for the crane to load jobs onto semitrailers, (3) an assignment of loaded semitrailers to tractors for transportation tasks, and (4) a transportation schedule of assigned tractors such that the completion time of the last transportation task is minimized. We first formulate the problem as a mixed integer linear programming model (MILPM) and prove that the problem is strongly NP‐hard. Then, optimality properties are provided which are useful in establishing an improved MILPM and designing solution algorithms. We develop a constructive heuristic, two LP‐based heuristics, and a recovering beam search heuristic to solve this problem. An improved procedure for solutions by heuristics is also presented. Furthermore, two branch‐and‐bound (B&B) algorithms with two different lower bounds are developed to solve the problem to optimality. Finally, computational experiments using both real data and randomly generated data demonstrate that our heuristics are highly efficient and effective. In terms of computational time and the number of instances solved to optimality in a time limit, the B&B algorithms are better than solving the MILPM. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 416–433, 2015     

The bilinear programming problem

The paper deals with bilinear programming problems and develops a finite algorithm using the “piecewise strategy” for large-scale systems. It consists of systematically generating a sequence of expanding polytopes with the global optimum within each polytope being known. The procedure then stops when the final polytope contains the feasible region.     

The pure fixed charge transportation problem

The pure fixed charge transportation problem (PFCTP) is a variation of the fixed charge transportation problem (FCTP) in which there are only fixed costs to be incurred when a route is opened. We present in this paper a direct search procedure using the LIFO decision rule for branching. This procedure is enhanced by the use of 0–1 knapsack problems which determine bounds on partial solutions. Computational results are presented and discussed.     

The continuous‐time single‐sourcing problem with capacity expansion opportunities

This paper develops a new model for allocating demand from retailers (or customers) to a set of production/storage facilities. A producer manufactures a product in multiple production facilities, and faces demand from a set of retailers. The objective is to decide which of the production facilities should satisfy each retailer's demand, in order minimize total production, inventory holding, and assignment costs (where the latter may include, for instance, variable production costs and transportation costs). Demand occurs continuously in time at a deterministic rate at each retailer, while each production facility faces fixed‐charge production costs and linear holding costs. We first consider an uncapacitated model, which we generalize to allow for production or storage capacities. We then explore situations with capacity expansion opportunities. Our solution approach employs a column generation procedure, as well as greedy and local improvement heuristic approaches. A broad class of randomly generated test problems demonstrates that these heuristics find high quality solutions for this large‐scale cross‐facility planning problem using a modest amount of computation time. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

The bounded interval generalized assignment model

The bounded interval generalized assignment model is a “many-for-one” assignment model. Each task must be assigned to exactly one agent; however, each agent can be assigned multiple tasks as long as the agent resource consumed by performing the assigned tasks falls within a specified interval. The bounded interval generalized assignment model is formulated, and an algorithm for its solution is developed. Algorithms for the bounded interval versions of the semiassignment model and sources-to-uses transportation model are also discussed.     

一种面向多制式路径规划的网络模型

随着综合交通运输与导航应用技术的发展,综合利用多种交通方式实施路径规划的应用需求日益迫切,传统的单一制式路径规划系统与服务正在向多制式的方向发展.然而,传统的路径规划模型与算法难以直接应用在多制式的条件下.针对多制式路径规划的特点,构建合理的数据模型是解决该问题的基础.为此,提出了制式切换点的概念,并以此为基础建立了包含多制式图集合、顶点属性表和切换点矩阵三个主要部分的多制式网络模型,并在真实的城市道路网数据集上进行了多制式网络构建实验.该工作能够从理论和实践两个方面为多制式路径规划算法的设计提供基础性的技术支持.     

Solving the multiperiod assignment problem with start-up costs using dual ascent

In this article we consider a multiperiod assignment problem where the assignment cost of assigning job i to machine j varies from one time period to the next. A start-up cost is incurred whenever the job processed by a machine in the current time period is different from the one processed in the previous time period. This problem is modeled as an integer programming problem for which a dual ascent approximate procedure is developed. Our computational results show that our procedure outperforms the more common Lagrangian-relaxation-based subgradient procedure by a significant margin. It is also found to be faster than MPSX/370 by many orders of magnitude. © 1993 John Wiley & Sons, Inc.     

A versatile scheme for ranking the extreme points of an assignment polytope

A cutting plane scheme embedded in an implicit enumeration framework is proposed for ranking the extreme points of linear assignment problems. This method is capable of ranking any desired number of extreme points at each possible objective function value. The technique overcomes storage difficulties by being able to perform the ranking at any particular objective function value independently of other objective values. Computational experience on some test problems is provided.     

Heuristics for the multi‐resource generalized assignment problem

The well‐known generalized assignment problem (GAP) involves the identification of a minimum‐cost assignment of tasks to agents when each agent is constrained by a resource in limited supply. The multi‐resource generalized assignment problem (MRGAP) is the generalization of the GAP in which there are a number of different potentially constraining resources associated with each agent. This paper explores heuristic procedures for the MRGAP. We first define a three‐phase heuristic which seeks to construct a feasible solution to MRGAP and then systematically attempts to improve the solution. We then propose a modification of the heuristic for the MRGAP defined previously by Gavish and Pirkul. The third procedure is a hybrid heuristic that combines the first two heuristics, thus capturing their relative strengths. We discuss extensive computational experience with the heuristics. The hybrid procedure is seen to be extremely effective in solving MRGAPs, generating feasible solutions to more than 99% of the test problems and consistently producing near‐optimal solutions. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 468–483, 2001     

An exact analysis of a joint production‐inventory problem in two‐echelon inventory systems

We consider a two‐echelon inventory system with a manufacturer operating from a warehouse supplying multiple distribution centers (DCs) that satisfy the demand originating from multiple sources. The manufacturer has a finite production capacity and production times are stochastic. Demand from each source follows an independent Poisson process. We assume that the transportation times between the warehouse and DCs may be positive which may require keeping inventory at both the warehouse and DCs. Inventory in both echelons is managed using the base‐stock policy. Each demand source can procure the product from one or more DCs, each incurring a different fulfilment cost. The objective is to determine the optimal base‐stock levels at the warehouse and DCs as well as the assignment of the demand sources to the DCs so that the sum of inventory holding, backlog, and transportation costs is minimized. We obtain a simple equation for finding the optimal base‐stock level at each DC and an upper bound for the optimal base‐stock level at the warehouse. We demonstrate several managerial insights including that the demand from each source is optimally fulfilled entirely from a single distribution center, and as the system's utilization approaches 1, the optimal base‐stock level increases in the transportation time at a rate equal to the demand rate arriving at the DC. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Time minimizing flows in directed networks

An important class of network flow problems is that class for which the objective is to minimize the cost of the most expensive unit of flow while obtaining a desired total flow through the network. Two special cases of this problem have been solved, namely, the bottleneck assignment problem and time-minimizing transportation problem. This paper addresses the more general case which we shall refer to as the time-minimizing network flow problem. Associated with each arc is an arc capacity (static) and a transferral time. The objective is to find a maximal flow for which the length (in time) of the longest path carrying flow is minimized. The character of the problem is discussed and a solution algorithm is presented.     

Finding equivalent transportation formulations for constrained transportation problems

This paper describes a procedure for determining if constrained transportation problems (i.e., transportation problems with additional linear constraints) can be transformed into equivalent pure transportation problems by a linear transformation involving the node constraints and the extra constraints. Our results extend procedures for problems in which the extra constraints consist of bounding certain partial sums of variables.     

Resource allocation for transportation

A linear programming formulation is described that will permit the optimal assignment of transportation resources (vehicles) and movement requirements to a network consisting of a set of designated origins, ports of embarkation, enroute bases, ports of debarkation, and destinations to achieve a prescribed schedule of arrivals.     

虚拟场景中的碰撞检测算法

综述了当前出现的虚拟场景中的各种碰撞检测算法。重点对包围盒层次法中的轴向包围盒法(AABB)、方向包围盒法(OBB)、离散方向多面体法(K-DOPs)的检测原理和检测效率进行了分析比较。得出的主要结论是:包围盒层次法应用极为广泛,检测效率取决于对包围盒的选取。K-DOPs是介于AABBs和凸包之间的包围盒,只要合理地选取平行平面对的个数和方向,就可以在碰撞检测的简单性和包裹物体的紧密性之间取得较好的折衷。     

Coordinated replenishments from multiple suppliers with price discounts

In this study we present an integer programming model for determining an optimal inbound consolidation strategy for a purchasing manager who receives items from several suppliers. The model considers multiple suppliers with limited capacity, transportation economies, and quantity discounts. We propose an integrated branch and bound procedure for solving the model. This procedure, applied to a Lagrangean dual at every node of the search tree, combines the subgradient method with a primal heuristic that interact to change the Lagrangean multipliers and tighten the upper and lower bounds. An enhancement to the branch and bound procedure is developed using surrogate constraints, which is found to be beneficial for solving large problems. We report computational results for a variety of problems, with as many as 70,200 variables and 3665 constraints. Computational testing indicates that our procedure is significantly faster than the general purpose integer programming code OSL. A regression analysis is performed to determine the most significant parameters of our model. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 579–598, 1998     

Transportation type problems with quantity discounts

It is known to be real that the per unit transportation cost from a specific supply source to a given demand sink is dependent on the quantity shipped, so that there exist finite intervals for quantities where price breaks are offered to customers. Thus, such a quantity discount results in a nonconvex, piecewise linear functional. In this paper, an algorithm is provided to solve this problem. This algorithm, with minor modifications, is shown to encompass the “incremental” quantity discount and the “fixed charge” transportation problems as well. It is based upon a branch-and-bound solution procedure. The branches lead to ordinary transportation problems, the results of which are obtained by utilizing the “cost operator” for one branch and “rim operator” for another branch. Suitable illustrations and extensions are also provided.     

Optimal empty vehicle redistribution for hub‐and‐spoke transportation systems

This article considers the empty vehicle redistribution problem in a hub‐and‐spoke transportation system, with random demands and stochastic transportation times. An event‐driven model is formulated, which yields the implicit optimal control policy. Based on the analytical results for two‐depot systems, a dynamic decomposition procedure is presented which produces a near‐optimal policy with linear computational complexity in terms of the number of spokes. The resulting policy has the same asymptotic behavior as that of the optimal policy. It is found that the threshold‐type control policy is not usually optimal in such systems. The results are illustrated through small‐scale numerical examples. Through simulation the robustness of the dynamic decomposition policy is tested using a variety of scenarios: more spokes, more vehicles, different combinations of distribution types for the empty vehicle travel times and loaded vehicle arrivals. This shows that the dynamic decomposition policy is significantly better than a heuristics policy in all scenarios and appears to be robust to the assumptions of the distribution types. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Solving quadratic assignment problems with rectangular distances and integer programming

The problem considered involves the assignment of n facilities to n specified locations. Each facility has a given nonnegative flow from each of the other facilities. The objective is to minimize the sum of transportation costs. Assume these n locations are given as points on a two-dimensional plane and transportation costs are proportional to weighted rectangular distances. Then the problem is formulated as a binary mixed integer program. The number of integer variables (all binary) involved equals the number of facilities squared. Without increasing the number of integer variables, the formulation is extended to include “site costs” Computational results of the formulation are presented.