The Branch and Bound Technique

Effroymson and Ray developed a model that involves a procedure using branch and bound methods and linear programming to produce optimal solutions with reasonable computing time. The location system was applied in the Esso Company to several location problems involving 4 plants, 50 warehouses, and 200 customer zones. The producer involves the application of rules for including or excluding warehouse locations, depending in whether or not their competitive savings cover their fixed costs of operation. Linear programming is used at points in the procedure to compute lower bound costs. By following out the branches, computing upper and lower bound costs, warehouse location can be either definitely included in the optimum solution or excluded, leading to the final optimum solution.

Hunt Wesson Foods, Inc., produces several hundred distinguishable commodities at 14 locations and distributes nationally through 12 distribution centers. The company decided to undertake a study of its distribution system, particularly the location of distribution warehouse. Five changes in location were indicated involving the movement of existing distribution centers as well as the opening of new ones. The cost reductions resulting from the study were estimated to be in the low seven figures.

Geoffrion and Graves (1974) formulated the Hunt-Wesson distribution problem in such a way that the multi-commodity linear programming sub-problem decomposes into as many independent classical transportation problems as there are commodities. The resulting problem structure include 17 commodity classes, 14 plants, 45 possible distribution center sites, and 121 customer zones. Thus, three levels of distribution were accounted for – plants, distribution centers or warehouses, and customer zones.

Demand for each commodity in each customer zone was known. Demand was satisfied by shipping via regional distribution centers (warehouses), with each customer zone being assigned exclusively to a warehouse. Upper ad lower limits were set on the annual capacity of each warehouse. Warehouse location sites were selected to minimize total distribution costs, which were composed of fixed plus variable cost components.

The warehouse location study indicates that the cost reductions possible are of great significance. Similar results were obtained in an application for a major manufacturer of hospital supplies with 5 commodity classes, 3 plants, 67 possible warehouse locations, and 127 customer zones.

Markland (1973) applied a computer simulation methodology in evaluating field warehouse location configurations and inventory levels for the Ralston Purina Company.

The basic structure of product flow, is a multilevel, multi-product distribution system involving plant warehouses, field warehouses, wholesalers and finally, retail grocers. Shipment from the five warehouses may go to other plant warehouses to field warehouses or to wholesalers. Inventories are maintained at the plant warehouse, field warehouse, and wholesaler levels. Also, shipments may go from any of the five field warehouses to any of the 29 demand analyses areas representing the wholesale level or to any of the other field warehouses.

Model Structure:

The distribution system was modeled in the basic format of system dynamics as a dynamic feedback control system where product flow is the main control variable. Product flow and inventory level equations were written to represent all flow combinations and inventory levels at plant and field warehouses. Constraints on maximum inventory levels at plant and field warehouses as well as constraints on maximum plant production capacity were established.

The system was simulated for six different field warehouse configurations and for different inventory service levels. The number of field warehouses was varied from zero to the existing five. Thirty-two field warehouse location patterns were tested involving combinations of warehouses, using a procedure of dropping warehouses from the existing pattern. In the example given, Ralston Purina saved $132,000 per year by consolidating field warehouses from five to three. The proposed elimination of intermediate warehouse as a policy would have increased costs by $240,000 per year compared with the optimal policy of using only three field warehouses. In addition to the preceding result, it was found that an 85 percent service level minimized distribution cost.

The three warehouse location methodologies discussed are apparently all quite powerful and capable of dealing with problems of size. The real advantage that the other features such as inventory policy can be included.