Moshe Dror

SAVINGS BY SPLIT DELIVERY ROUTING

This paper examines a relaxed version of the generic vehicle routing problem. In this version, a delivery to a demand point can be split between any number of vehicles. In spite of this relaxation the problem remains computationally hard. The main contribution of this paper is in demonstrating the potential for cost savings through split deliveries. The solution scheme allowing for split deliveries is compared with a solution in which no split deliveries are allowed on a large set of 540 problems. Significant cost savings are realized both in terms of the total distance and the number of vehicles required. The vehicle routes constructed by our procedure, tend to cover cohesive geographical zones and retain some properties of optimal solutions.

VEHICLE ROUTING WITH STOCHASTIC DEMANDS: PROPERTIES AND SOLUTION FRAMEWORKS

This paper considers the vehicle routing problem with stochastic demands. The objective is to provide an overview of this problem, and to examine a variety of solution methodologies. The concepts and the main issues are reviewed along with some properties of optimal solutions. The existing stochastic mathematical programming formulations are presented and compared and a new formulation is proposed. A new solution framework for the problem using Markovian decision processes is then presented.

Inventory/routing: Reduction from an annual to a short-period problem

The inventory-routing problem is a distribution problem in which each customer maintains a local inventory of a product such as heating oil and consumes a certain amount of that product each day. Given a central supplier, the objective is to minimize the annual delivery costs while attempting to insure that no customer runs out of the commodity at any time. In this article we present a procedure for reducing the long-term version of this problem to a single-period problem, which can be attacked using standard routing algorithms. The reduction procedure involves the definition of single-period costs that reflect long-term costs, the definition of a safety stock level and a specification of the customer subset to be considered during a single period.

Note on the Complexity of the Shortest Path Models for Column Generation in VRPTW

In this note we prove that the relaxation approach in designing the subproblem of pricing out only the feasible routes for the set partition formulation of the VRPTW is justified on complexity grounds. That is, the first dynamic programming model presented in M. Desrochers, J. Desrosiers and M. Solomon 1992, that is able to price out all feasible routes, is NP-hard in the strong sense.

Vehicle routing with split deliveries

Abstract This paper considers a relaxation of the classical vehicle routing problem (VRP), in which split deliveries are allowed. As the classical VRP, this problem is NP-hard, but nonetheless it seems more difficult to solve exactly. It is first formulated as an integer linear program. Several new classes of valid constraints are derived, and a hierarchy between these is established. A constraint relaxation branch and bound algorithm for the problem is then described. Computational results indicate that by using an appropriate combination of constraints, the gap between the lower and upper bounds at the root of the search tree can be reduced considerably. These results also confirm the quality of a previously published heuristic for this problem.

Stochastic vehicle routing with modified savings algorithm

Abstract A stochastic vehicle routing problem (SVRP) differs from the well known vehicle routing problem (VRP) in that the actual customer demand is not known with certainty when the vehicle routes are designed. One aspect that differentiates between these problems is the notion of route failure. Route failure indicates a situation where a vehicle cannot complete all the deliveries on a designed route because its supply is exhausted at some point along the route, before the routes demand is fully satisfied. In this paper, we illustrate the effects of route failure on the expected cost of a route, as well as the impact the direction of a designed route can have on the expected cost. In addition, we present a straight-forward modification of the Clark and Wright savings algorithm to account more fully for all the costs inherent in many real routing problems, where the customers actual demands are uncertain.

A Decomposition Approach to the Inventory Routing Problem with Satellite Facilities

This paper presents a comprehensive decomposition scheme for solving the inventory routing problem in which a central supplier must restock a subset of customers on an intermittent basis. In this setting, the customer demand is not known with certainty and routing decisions taken over the short run might conflict with the long-run goal of minimizing annual operating costs. A unique aspect of the short-run subproblem is the presence of satellite facilities where vehicles can be reloaded and customer deliveries continued until the closing time is reached. Three heuristics have been developed to solve the vehicle routing problem with satellite facilities (randomized Clarke-Wright, GRASP, modified sweep). After the daily tours are derived, a parametric analysis is conducted to investigate the tradeoff between distance and annual costs. This leads to the development of the efficient frontier from which the decision maker is free to choose the most attractive alternative. The proposed procedures are tested on data sets generated from field experience with a national liquid propane distributor.

A computational comparison of algorithms for the inventory routing problem

The inventory routing problem is a distribution problem in which each customer maintains a local inventory of a product such as heating oil and consumes a certain amount of that product each day. Each day a fleet of trucks is dispatched over a set of routes to resupply a subset of the customers. In this paper, we describe and compare algorithms for this problem defined over a short planning period, e.g. one week. These algorithms define the set of customers to be serviced each day and produce routes for a fleet of vehicles to service those customers. Two algorithms are compared in detail, one which first allocates deliveries to days and then solves a vehicle routing problem and a second which treats the multi-day problem as a modified vehicle routing problem. The comparison is based on a set of real data obtained from a propane distribution firm in Pennsylvania. The solutions obtained by both procedures compare quite favorably with those in use by the firm.

Cores of Inventory Centralization Games

Abstract Consider a set of n stores with single-item and single-period demands. Assume an option of centralized ordering and inventory with holding and penalty costs only. In this case, a cooperative inventory “centralization” game “defines” allocations of the cost. We examine the conditions under which such an inventory centralization game has a nonempty core. We prove the existence of nonempty core for demands with symmetric distributions and the existence of nonempty core for joint multivariate normal demand distribution. We establish the equivalency of four different nonempty core conditions for the Newsboy Problem and demonstrate their efficacy for discrete independent and identically distributed (iid) demands. Journal of Economic Literature Classification Numbers: C44, C62, C71.

Mathematical programming formulations for machine scheduling: a survey

Abstract Machine scheduling was and still is a rich and promising field for research with applications in manufacturing, logistics, computer architecture, communications, etc. Combinatorial complexity theory has now classified the great majority of known machine scheduling problems as ‘easy’ or ‘very hard’. However, in most cases, mathematical programming models have not accompanied the algorithmic developments for solving ‘easy’ scheduling problems, nor have they facilitates solutions for ‘hard’ problems. Nevertheless, the analysis of the mathematical programming models for some hard combinatorial problems together with their polyhedral properties has enabled important computational advances for such problems as the TSP. In order to assess the present status and the solution potential of mathematical programming formulations for machine scheduling, we have compiled a systematic, consistent survey of formulations. The discussion has been developed in tandem with the classification of a given problems complexity, since ‘solvability’ (i.e., the status of a problem as P or NP-hard) generally cannot be easily assessed from the formulation itself. A number of excellent survey papers on machine scheduling have appeared over the years (see the reference list), but none of them has been focused on mathematical formulations. This survey is the first one that attempts to compile a large number of mathematical programming formulations for scheduling into a single paper to ease the task of model building and testing scheduling formulations. Both, a newcomer and experienced researcher can use it as a reference point. Ultimately, mathematical programming formulations for scheduling problems might be used as a stepping stone to computational advances for some hard problems.