Julien Bramel

A computerized approach to the New York City school bus routing problem

Transportation is an area where operations research has had a great impact on systems by improving service quality and reducing operating costs. In particular, one of the most visible applications is in routing and scheduling school buses. In this paper we investigate various issues related to the development of a computerized system to help route and schedule school buses throughout the five boroughs of New York City. The routing problem presents many challenges because of its magnitude, the vast resources involved and the intricacies one encounters when routing in a completely urban environment. We analyze various aspects of the problem including the generation of routes in the Borough of Manhattan and provide a solution requiring far fewer buses than are currently in use. The computerized system, called CATS, incorporating many of the results obtained in this research, is currently being used to route Special Education students.

Probabilistic Analyses and Practical Algorithms for the Vehicle Routing Problem with Time Windows

In the Vehicle Routing Problem with Time Windows, a set of customers are served by a fleet of vehicles of limited capacity, initially located at a central depot. Each customer provides a period of time in which they require service, which may consist of repair work or loading/unloading the vehicle. The objective is to find tours for the vehicles, such that each customer is served in its time window, the total load on any vehicle is no more than the vehicle capacity, and the total distance traveled is as small as possible. In this paper, we present a characterization of the asymptotic optimal solution value for general distributions of service times, time windows, customer loads and locations. This characterization leads to the development of a new algorithm based on formulating the problem as a stylized location problem. Computational results show that the algorithm is very effective on a set of standard test problems.

On the Effectiveness of Set Covering Formulations for the Vehicle Routing Problem with Time Windows

The Vehicle Routing Problem with Time Windows VRPTW is one of the most important problems in distribution and transportation. A classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem. The method starts by solving its linear programming relaxation, via column generation, and then uses a branch and bound strategy to find an integer solution to the set covering problem: a solution to the VRPTW. An empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely efficient. In this paper we explain this behavior by demonstrating that for any distribution of service times, time windows, customer loads, and locations, the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases.

Approximation algorithms for the capacitated traveling salesman problem with pickups and deliveries

We consider the Capacitated Traveling Salesman Problem with Pickups and Deliv- eries (CTSPPD). This problem is characterized by a set of n pickup points and a set of n delivery points. A single product is available at the pickup points which must be brought to the delivery points. A vehicle of limited capacity is available to perform this task. The problem is to determine the tour the vehicle should follow so that the total distance traveled is minimized, each load at a pickup point is picked up, each delivery point receives its shipment and the vehicle capacity is not violated. We present two polynomial-time approximation algorithms for this problem and analyze their worst-case bounds.

An asymptotic 98.5%-effective lower bound on fixed partition policies for the inventory-routing problem

We consider the Inventory-Routing Problem where n geographically dispersed retailers must be supplied by a central facility. The retailers experience demand for a product at a deterministic rate and incur holding costs for keeping inventory. Distribution is performed by a fleet of capacitated vehicles. The objective is to minimize the average transportation and inventory costs per unit time over the infinite horizon. In this paper, we focus on the set of fixed partition policies. In a fixed partition policy, the retailers are partitioned into disjoint and collectively exhaustive sets. Each set of retailers is served independently of the others and at its optimal replenishment rate. We derive a deterministic (O(n)) lower bound on the cost of the optimal fixed partition policy. A probabilistic analysis of the performance of this bound demonstrates that it is asymptotically 98.5%-effective. That is, as the number of retailers increases, the lower bound is very close to the cost of the optimal fixed partition policy.

Worst-case analyses, linear programming and the bin-packing problem

In this paper we consider the familiar bin-packing problem and its associated set-partitioning formulation. We show that the optimal solution to the bin-packing problem can be no larger than 4/3 ⌈ZLP⌉, whereZLP is the optimal solution value of the linear programming relaxation of the set-partitioning formulation. An example is provided to show that the bound is tight. A by-product of our analysis is a new worst-case bound on the performance of the well studied First Fit Decreasing and Best Fit Decreasing heuristics.

A 53-approximation algorithm for the clustered traveling salesman tour and path problems

We consider the ordered cluster traveling salesman problem (OCTSP). In this problem, a vehicle starting and ending at a given depot must visit a set of n points. The points are partitioned into K,K=

A Probabilistic Analysis of Tour Partitioning Heuristics for the Capacitated Vehicle Routing Problem with Unsplit Demands

In the Capacitated Vehicle Routing Problem with unsplit demands, a customers demand may not be divided over more than one vehicle. We analyze the asymptotic performance of a class of heuristics called route first-cluster second, and in particular, the empirically well-studied, Sweep algorithm and Optimal Partitioning heuristic, for any distribution of the demands and when customers are independent and identically distributed in a given region. We show a strong relationship between this class of heuristics and the familiar Next-Fit bin-packing heuristic.

Vehicle Routing and the Supply Chain

One of the central problems of supply chain management is the coordination of product and material flows between locations. A typical problem involves bringing product located at a central facility to geographically dispersed facilities at minimum cost. For example, a supply of product is located at a plant, warehouse, cross-docking facility or distribution center and must be distributed to customers or retailers. The task is often performed by a fleet of vehicles under direct control of the firm. Depending on the structure of the supply chain, a number of different distribution problems can be defined, e.g., where retailers/customers are external, i.e., not part of the same company, or where retailers are internal to the firm. In this chapter, we present models, analyses and approaches for solving a number of these distribution problems.

Periodic Scheduling with Service Constraints

We consider the problem of servicing a number of objects in a discrete time environment. In each period, we may select an object that will receive a service in the period. Each time an object is serviced, we incur a servicing cost dependent on the time since the objects last service. Problems of this type appear in many contexts, e.g., multiproduct lot-sizing, machine maintenance, and several problems in telecommunications. We assume that at most one object can be serviced in a given period. For the general problem withm objects, which is known to be NP-Hard, we describe properties of an optimal policy, and for the specific case ofm = 2 objects, we determine an optimal policy.