mtts 02163We consider a common variant of the vehicle routing problem in which a vehicle fleet delivers products stored at a central depot to satisfy customer orders. Each vehicle has a fixed capacity, and each order uses a fixed portion of vehicle capacity. The routing decision involves determining which of the demands will be satisfied by each vehicle and what route each vehicle will follow in servicing its assigned demand in order to minimize total delivery cost. We present a heuristic for this problem in which an assignment of customers to vehicles is obtained by solving a generalized assignment problem with an objective function that approximates delivery cost. This heuristic has many attractive features. It has outperformed the best existing heuristics on a sample of standard test problems. It will always find a feasible solution if one exists, something no other existing heuristic can guarantee. It can be easily adapted to accommodate many additional problem complexities. By parametrically varying the number of vehicles in the fleet, our method can be used t o optimally solve the problem of finding the minimum size fleet that can feasibly service the specified demand.

We describe a branch and bound algorithm for the generalized assignment problem in which bounds are obtained from a Lagrangian relaxation with the multipliers set by a heuristic adjustment method. The algorithm was tested on a large sample of small random problems and a number of large problems derived from a vehicle routing application. Computation times were reasonable in all cases and the branch and bound trees generated had nearly two orders of magnitude fewer nodes than for competing algorithms. Although comparison of running times on different machines is difficult, the multiplier adjustment method appears to be about one order of magnitude faster than the best previously existing algorithms for this problem.generalized assignment problem, branch and bound, Lagrangean relaxation

For Air Products and Chemicals, Inc., inventory management of industrial gases at customer locations is integrated with vehicle scheduling and dispatching. Their advanced decision support system includes on-line data entry functions, customer usage forecasting, a time/distance network with a shortest path algorithm to compute intercustomer travel times and distances, a mathematical optimization module to produce daily delivery schedules, and an interactive schedule change interface. The optimization module uses a sophisticated Lagrangian relaxation algorithm to solve mixed integer programs with up to 800,000 variables and 200,000 constraints to near optimality. The system, first implemented in October, 1981, has been saving between 6% to 10% of operating costs.

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