Vehicle Capacity Constraints Research Articles

A mixed integer linear programming model for dynamic route guidance

One of the major challenges facing ITS (Intelligent Transportation Systems) today is to offer route guidance to vehicular traffic so as to reduce trip time experienced. In a cooperative route guidance system, the problem becomes one of assigning routes to vehicles departing at given times from a set of origins to a set of destinations so as to minimize the average trip time experienced (a so-called system optimal criterion). Since the time to traverse a link will depend upon traffic volume encountered on that link, link times are dynamic. The complex interaction resulting between objective function and constraints makes the dynamic problem significantly more difficult to formulate and solve than the static version. We present a mixed integer linear programming formulation of the problem which is formally derived from a set of traffic flow assumptions. Principal among these is the simplifying assumption that vehicles upon entering a link, assume the speed that traffic would attain were the traffic volume encountered on that link in steady-state. The integer variables correspond to selection of vehicle capacity constraints on the link while the continuous variables correspond to selection of vehicle routes. Implicit within this MILP formulation of the dynamic traffic assignment problem is therefore a decomposition of the problem which results in a conventional capacitated linear programming network flow problem. A small illustrative subnetwork extracted from the city of Sioux Falls is solved to optimality by IBM’s OSL Branch-and-Bound algorithm.

Optimization of Grid Transit System in Heterogeneous Urban Environment

Current analytic models for optimizing urban transit systems tend to sacrifice geographic realism and detail in order to obtain their solutions. The model presented here shows how an optimization approach can be successful without oversimplifying the spatial characteristics and demand patterns of urban areas. This model is designed to optimize a grid transit system in a heterogeneous urban environment whose demand and supply characteristics may vary arbitrarily among adjacent zones. Network characteristics (route and station locations) and operating headways are found that minimize the total cost, including supplier and user costs. Irregular many-to-many demand patterns, zonal variations in route costs, and vehicle capacity constraints are considered in a sequential optimization process.

A heuristic algorithm for the asymmetric capacitated vehicle routing problem

We consider the Asymmetric Capacitated Vehicle Routing Problem (ACVRP[, a particular case of the standard asymmetric Vehicle Routing Problem arising when only the vehicle capacity constraints are imposed. ACVRP is known to be NP-hard and finds practical applications, e.g. in distribution and scheduling. In this paper we describe the extension to ACVRP of the two well-known Clarke-Wright and Fisher-Jaikumar heuristic algorithms. We also propose a new heuristic algorithm for ACVRP that, starting with an initial infeasible solution, determines the final set of vehicle routes through an insertion procedure as well as intea-route and inter-route arc exchanges. The initial infeasible solution is obtained by using the additive bounding procedures for ACVRP described by Fischetti, Toth and Vigo in 1992. Extensive computational results on several classes of randomly generated test problems involving up to 300 customers and on some real instances of distribution problems in urban areas, are presented. The results obtained show that the proposed approach favourably compares with previous algorithms from the literature.

A new exact algorithm for the vehicle routing problem based onq-paths andk-shortest paths relaxations

We consider the basic Vehicle Routing Problem (VRP) in which a fleet ofM identical vehicles stationed at a central depot is to be optimally routed to supply customers with known demands subject only to vehicle capacity constraints. In this paper, we present an exact algorithm for solving the VRP that uses lower bounds obtained from a combination of two relaxations of the original problem which are based on the computation ofq-paths andk-shortest paths. A set of reduction tests derived from the computation of these bounds is applied to reduce the size of the problem and to improve the quality of the bounds. The resulting lower bounds are then embedded into a tree-search procedure to solve the problem optimally. Computational results are presented for a number of problems taken from the literature. The results demonstrate the effectiveness of the proposed method in solving problems involving up to about 50 customers and in providing tight lower bounds for problems up to about 150 customers.

A Matching Based Exact Algorithm for Capacitated Vehicle Routing Problems

A branch and bound algorithm for capacitated vehicle routing is described. Lower bounds are derived by relaxing the subtour elimination and vehicle capacity constraints to yield a perfect b-matching problem. Bounds are strengthened by using Lagrange multipliers to enforce subtour elimination and capacity constraints. Computational results are reported for problems taken from the literature for sizes up to 61 vertices. With the exception of the 48 vertex instance (att48.vrp), the algorithm solves all problems in TSPLIB having 51 or fewer vertices. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

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.

A Branch-and-Bound Algorithm for the Capacitated Vehicle Routing Problem on Directed Graphs

We consider the asymmetric capacitated vehicle routing problem (CVRP), a particular case of the standard asymmetric vehicle routing problem in which only the vehicle capacity constraints are imposed. CVRP is known to be NP-hard and finds practical applications in distribution and scheduling. We describe two new bounding procedures for CVRP, based on the so-called additive approach. Each procedure computes a sequence of nondecreasing lower bounds, obtained by solving different relaxations of CVRP. Effective implementations of the procedures are also outlined which considerably reduce the computational effort. The two procedures are combined into an overall bounding algorithm. A branch-and-bound exact algorithm is then proposed, whose performance is enhanced by means of reduction procedures, dominance criteria, and feasibility checks. Extensive computational results on both real-world and random test problems are presented, showing that the proposed approach favorably compares with previous algorithms from the literature.

Optimal Solution of Vehicle Routing Problems Using Minimum K-Trees

We consider the problem of optimally scheduling a fleet of K vehicles to make deliveries to n customers subject to vehicle capacity constraints. Given a graph with n + 1 nodes, a K-tree is defined to be a set of n + K edges that span the graph. We show that the vehicle routing problem can be modeled as the problem of finding a minimum cost K-tree with two K edges incident on the depot and subject to some side constraints that impose vehicle capacity and the requirement that each customer be visited exactly once. The side constraints are dualized to obtain a Lagrangian problem that provides lower bounds in a branch-and-bound algorithm. This algorithm has produced proven optimal solutions for a number of difficult problems, including a well-known problem with 100 customers and several real problems with 25–71 customers.

Extension of Newell's dispatching policy to the case of sensitive demand

This paper has extended Newell's dispatching policy to the case of sensitive demand which is characterized by a linear mode split model with two major factors, wait time and transit fare. Three objective functions both with and without vehicle capacity constraint are analyzed, including profit maximization, maximization of a combination of net user benefit and operator profit, and maximization of net user benefit subject to a deficit constraint. Closed-form solutions associated with various system parameters are obtained. It is shown that under sensitive demand conditions the optimal dispatching rate is approximately proportional to the square root of the total demand rate, if vehicle size is not binding and it is strictly proportional to the total demand rate, if vehicle size is binding.

A routing and scheduling method in considering trade‐off between the user's and the operator's objectives

A routing and scheduling model is proposed for the one depot, multi‐vehicle, many‐to‐one, and vehicle capacity constraint problem. The model takes both the operator's and the user's objectives into account. The operator's objective is assumed to be the minimization of transportation cost or distance travelled; while the user's objective is represented by the minimum time difference between the desired and the actual times of vehicle arrival. The routing model is based on the classical Clarke‐Wright algorithm; however, node pairs in the model are ranked by an index generated by the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Thus, unlike the traditional Clarke‐Wright model, this model incorporates not only transportation cost but also the desired time of vehicle visit in the route development. Furthermore, different sets of routes can be produced depending upon the degree of preference attached to each objective. A step‐by‐step computation procedure and a set of examples are al...

A Dynamic Programming Solution to the Single Vehicle Many-to-Many Immediate Request Dial-a-Ride Problem

An investigation of the single-vehicle, many-to-many, immediate-request dial-a-ride problem is developed in two parts (I and II). Part I focuses on the “static” case of the problem. In this case, intermediate requests that may appear during the execution of the route are not considered. A generalized objective function is examined, the minimization of a weighted combination of the time to service all customers and of the total degree of “dissatisfaction” experienced by them while waiting for service. This dissatisfaction is assumed to be a linear function of the waiting and riding times of each customer. Vehicle capacity constraints and special priority rules are part of the problem. A Dynamic Programming approach is developed. The algorithm exhibits a computational effort which, although an exponential function of the size of the problem, is asymptotically lower than the corresponding effort of the classical Dynamic Programming algorithm applied to a Traveling Salesman Problem of the same size. Part II extends this approach to solving the equivalent “dynamic” case. In this case, new customer requests are automatically eligible for consideration at the time they occur. The procedure is an open-ended sequence of updates, each following every new customer request. The algorithm optimizes only over known inputs and does not anticipate future customer requests. Indefinite deferment of a customer’s request is prevented by the priority rules introduced in Part I. Examples in both “static” and “dynamic” cases are presented.