Capacitated Concentrator Location Problem Research Articles

Matheuristics with performance guarantee for the unsplit and split delivery capacitated vehicle routing problem

AbstractFor the classical unsplit and split delivery capacitated vehicle routing problems, we carry out a worst‐case analysis for classes of matheuristics and compare their performance on average, on a large set of benchmark instances. The matheuristics are based on the optimal solution of the bin packing problem, the capacitated concentrator location problem, and the unsplit capacitated vehicle routing problem (CVRP). These matheuristics are compared with the classical algorithms having known finite worst‐case performance bound. For the unsplit CVRP, we provide a matheuristic having worst‐case performance bound equal to the one of the classical algorithms, but with an average percent cost increase with respect to the optimal cost equal to 1.13%. For the split delivery case, we provide a matheuristic having worst‐case performance bound 2 and an average percent cost increase with respect to the best‐known cost equal to 0.64%. Moreover, this matheuristic is able to find 22 best‐known solutions, 20 of which are new.

A new extended formulation with valid inequalities for the Capacitated Concentrator Location Problem

We present a new disaggregated formulation of the Capacitated Concentrator Location Problem (CCLP) using the notion of cardinality of terminals assigned to a concentrator. This formulation consists of O(mnn) variables and constraints, where m denotes the number of concentrators and n the number of terminals, respectively. We prove that this extended formulation is stronger than the traditional one. We also present two classes of inequalities exploiting the cardinality effect of the extended formulation. The first class is a generalization of the well-known Cover and (1, k)-Configuration inequalities, which collectively are stronger than the original Cover and (1, k)-Configuration inequalities. The second class, called the 2-Facility Cardinality Matching Inequality, holds for the uncapacitated version of the Concentrator Location Problem and can be lifted to become a strong inequality for CCLP. We solve the LP relaxation of the extended formulation and use separation heuristics to identify and sequentially add the previous valid inequalities to improve the lower bound. This approach is embedded in a branch-and-bound and results in a branch-and-cut approach. We test our solution approach on a large set of benchmark problems. The experimentation shows that we can identify the optimal solution at the root node in most of the problem instances with up to 50 concentrators and 50 terminals. For larger sized test problems with up to 100 concentrators and 1000 terminals, the branch-and-cut procedure using the disaggregated formulation outperforms the branch-and-cut procedure applied to the traditional formulation both in terms of CPU and the required number of branch-and-bound nodes.

A Hybrid Multicast Routing for Large Scale Sensor Networks with Holes

In this article, we present RE $^2$ MR, the first hybrid multicast routing protocol that builds on the strengths of existing topology-based, hierarchical and geographic multicast solutions, while addressing their limitations. In RE $^2$ MR, the multicast path search problem is formulated as the capacitated concentrator location problem (CCLP) which yields the network topology that minimizes the sum of path lengths from the multicast root to multicast members. Furthermore, its trajectory-based lightweight hole detection (THLD) discovers deployment area irregularities (i.e., network holes) that affect its solution and autonomously take them into account to generate updated routing paths, and its Energy-efficient Packet Forwarding (EPF) and Multi-level Facility Computation (MFC) reduce computational and communication overheads. We implement RE $^2$ MR in TinyOS and evaluate it extensively using TOSSIM for relatively large-scale simulations (400 nodes); we also implement RE $^2$ MR on real-hardware and perform experiments on a testbed consisting of 42 TelosB motes. Through the simulations and experiments on real-hardware, we demonstrate that RE $^2$ MR reduces the energy consumption by up to 57 percent and the end-to-end delay by up to 8 percent, when compared with the state-of-the-art multicast routing protocols.

Near‐optimal heuristics and managerial insights for the storage constrained, inbound inventory routing problem

PurposeThe purpose of this paper is to develop efficient heuristics for determining the route design and inventory management of inbound parts which are delivered for manufacturing, assembly, or distribution operations and for which there is limited storage space. The shipment frequencies and quantities are coordinated with the available storage space and the vehicle capacities.Design/methodology/approachTwo heuristics that generate near optimal solutions are proposed. The first heuristic has an iterative routing phase that maximizes the savings realized by grouping suppliers together into routes without considering the storage constraint and then calculates the pickup frequencies in the second phase to accommodate the storage constraint. The second heuristic iteratively executes a routing and a pickup frequency phase that both account for the storage constraint. A lower bound is also developed as a benchmark for the heuristic solutions.FindingsNear optimal solutions can be obtained in a reasonable amount of time by utilizing information about the amount of storage space in the route design process.Practical implicationsThe traditional emphasis on high vehicle utilization in transportation management can lead to inefficient logistics operations by carrying excess inventory or by using longer, less efficient routes. Route formation and pickup quantities at the suppliers are simultaneously considered, as both are important from a logistics standpoint and are interrelated decisions.Originality/valueThe two proposed heuristics dynamically define seed sets such that the solutions to the capacitated concentrator location problem (CCLP) are accurately estimated. This increased accuracy helps in generating near‐optimal solutions in a practical amount of computing time.

Solving the variable size bin packing problem with discretized formulations

In this paper we study the use of a discretized formulation for solving the variable size bin packing problem (VSBPP). The VSBPP is a generalization of the bin packing problem where bins of different capacities (and different costs) are available for packing a set of items. The objective is to pack all the items minimizing the total cost associated with the bins. We start by presenting a straightforward integer programming formulation to the problem and later on, propose a less straightforward formulation obtained by using a so-called discretized model reformulation technique proposed for other problems (see [Gouveia L. A 2n constraint formulation for the capacitated minimal spanning tree problem. Operations Research 1995; 43:130–141; Gouveia L, Saldanha-da-Gama F. On the capacitated concentrator location problem: a reformulation by discretization. Computers and Operations Research 2006; 33:1242–1258]). New valid inequalities suggested by the variables of the discretized model are also proposed to strengthen the original linear relaxation bounds. Computational results (see Section 4) with up to 1000 items show that these valid inequalities not only enhance the linear programming relaxation bound but may also be extremely helpful when using a commercial package for solving optimally VSBPP.

On the capacitated concentrator location problem: a reformulation by discretization

In this paper, we present and discuss a discretized model for the two versions of the capacitated concentrator location problem: a simple version (SCCLP) and a version with modular capacities (MCCLP). We show that the linear programming relaxation of the discretized model is at least as good as the linear programming relaxation of conventional models for the two variations of the problem under study. A technique for deriving valid inequalities from the equations of the discretized model is also given. We will show that this technique provides inequalities that significantly enhance the linear programming bound of the discretized model. Our computational results show the advantage of the new models for obtaining the optimal integer solution for the two versions of the problem.

A GRASP HEURISTIC FOR SOLVING AN EXTENDED CAPACITATED CONCENTRATOR LOCATION PROBLEM

Local Access Networks (LACNs) are often considered the "last mile" over the Internet provided that they are connected to some existing Wide Area Network backbone. In this paper, a revised LACN-WAN interconnection model is presented as an extension to the conventional Capacitated Concentrator Location Problem (CCLP). An efficient heuristic is also developed to seek near-optimal deployment of telecommunication devices (e.g. links, concentrators, etc.) while minimizing the total connection costs for the network interconnection. Our solution algorithm could serve as a decision-making tool in solving problems with a real world size (i.e. 120 nodes and 30 concentrators).

An Efficient Heuristic for Solving an Extended Capacitated Concentrator Location Problem

In this paper, a mathematical model and a solution algorithm are developed for solving an extended capacitated concentrator location problem. Our model extends the conventional formulation by simultaneously addressing the two capacity constraints, total connection ports and maximum data processing rate, on each concentrator to be selected for satisfying the communication demands of the given end-user nodes. Since the problem is NP-complete, an efficient and effective Lagrangian heuristic is developed and tested by solving 100 randomly generated test problems with sizes ranging from 30(nodes)×30(concentrators) to150×30. Altogether 58% of the tested problems are solved optimally with an average solution gap 0.36% from the optimality and average solution times are from a few seconds to one half of a minute.

A multiplier adjustment technique for the capacitated concentrator location problem

We describe a new dual descent method for a pure 0— location problem known as the capacitated concentrator location problem. The multiplier adjustment technique presented is aimed to find an upper bound in a Lagrangean relaxation context permitting both to decrease and to increase multipliers in the course of the search in contrast with methods where that ones are monotonically updated.

A Location Based Heuristic for General Routing Problems

We present a general framework for modeling routing problems based on formulating them as a traditional location problem called the capacitated concentrator location problem. We apply this framework to two classical routing problems: the capacitated vehicle routing problem and the inventory routing problem. In the former case, the heuristic is proven to be asymptotically optimal for any distribution of customer demands and locations. Computational experiments show that the heuristic performs well for both problems and, in most cases, outperforms all published heuristics on a set of standard test problems.

Analytical results and algorithms for vehicle routing problems with bin-packing features: Bramel, Julien Dana, Ph.D. Columbia University, 1993. 120 pp. adviser:David Simchi-Levi

Due to the increased importance of the service sector of the economy, there is a need for a better analytical understanding of service and distribution systems. In such systems, servers based at a single warehouse provide delivery, customer pick-up or repair and maintenance services to customers geographically dispersed in a given region. Such distribution systems integrate problems involving inventory and routing issues, as well as vehicle capacity and time constraints. At the core of these distribution problems is the Capacitated Vehicle Routing Problem (CVRP). Here, a fleet of vehicles of limited capacity must be dispatched and routed so that each customer receives a certain load. A thorough analysis of the CVRP entails an understanding of a sub-problem, called the bin-packing problem. Here, a set of items of different sizes must be packed into equal size bins in such a way that the total size assigned to any one bin is at most the capacity of the bin and the number of bins used is as small as possible. We show that as the number of customers increases the capacitated vehicle routing problem can be solved using results from the bin-packing literature. We then look at some of the classical heuristics designed for the CVRP and show that a class of these are provably inefficient. That is, as the number of customers increases, the solutions produced by heuristics in this class tend to be far from the optimal solution. These results are in agreement with the empirical performance reported in the literature for heuristics of this class. We next discuss practical algorithms for the CVRP and the problem of integrating inventory issues with transportation cost. We develop a general framework to solve these problems that involves a traditional location model commonly called the Capacitated Concentrator Location Problem. In the next chapter, we consider a vehicle routing problem where each customer requests service and provides a time window in which service is desired. The objective is to serve each customer in its time window in such a way that the total distance traveled is as small as possible. We show that the problem can be solved using similar techniques to the capacitated vehicle routing problem. Finally, we study the bin-packing problem with a different cost structure. We consider the problem when the cost of a bin is a function of the number of items in the bin. We derive results that bound the relative difference between the heuristics solution and the optimal solution. As a result of this analysis, we also derive some new results for the classical bin-packing problem.

An efficient characterization of some cost allocation solutions associated with capacitated network design problems

We analyze some game-theoretic solution concepts associated with a cost allocation problem arising from the Capacitated Network Design (CND) problem. The problem is formulated as a cost cooperative game in characteristic function form to be referred to as the CND game. We provide an efficient representation of several game-theoretic solution concepts associated with the CND game. In particular, we efficiently characterize the core, and in some cases the nucleolus, the least weighted?-core and a certain "central" point in the least weighted?-core. Our model properly generalizes several previously studied cooperative games. We also employ our model to analyze cost allocation problems associated with several classes of network design problems, which were not previously studied in the literature. Specifically, we efficiently characterize the above cost allocation solutions for cost allocation problems associated with the Capacitated Concentrator Location problem, the Capacitated Minimum Spanning Tree problem, the Capacitated Fixed Cost Spanning Forest problem, and the Capacitated Steiner Tree problem.

An Algorithm for the Design of Multitype Concentrator Networks

In recent years we have witnessed remarkable progress in the development of the topological design of computer communication networks. One of the important aspects of the topological design of computer communication networks is the concentrator location problem. This problem is a complex combinatorial problem that belongs to the difficult class of NP-complete problems where the computation of an optimal solution is still a challenging task. This paper extends the standard capacitated concentrator location problem to a generalization of multitype capacitated concentrator location problems and presents an efficient algorithm based on cross decomposition. This algorithm incorporates the Benders decomposition and Lagrangean relaxation methods into a single framework and exploits the resulting primal and dual structure simultaneously. Computational results are quite satisfactory and encouraging and show this algorithm to be both efficient and effective. The use of cross decomposition as a heuristic algorithm is also discussed.

Efficient algorithms for the capacitated concentrator location problem

Topological design of computer communication networks is a very difficult problem. This problem is generally disaggregated into a number of “simpler” problems. One of these “simpler” problems is the capacitated concentrator location problem. The paper makes use of the Lagrangian relaxation approach to develop optimal and heuristic solution procedures for this problem. Computational experiments show these procedures to be both efficient and effective. The heuristic solution procedure is compared to previously developed algorithms and found to generate significantly better results.