THE TRANSPORTATION AND ASSIGNMENT PROBLEMS:THE TRANSPORTATION PROBLEM

■ THE TRANSPORTATION PROBLEM

Prototype Example

One of the main products of the P & T COMPANY is canned peas. The peas are pre- pared at three canneries (near Bellingham, Washington; Eugene, Oregon; and Albert Lea, Minnesota) and then shipped by truck to four distributing warehouses in the western United States (Sacramento, California; Salt Lake City, Utah; Rapid City, South Dakota; and Albuquerque, New Mexico), as shown in Fig. 9.1. Because the shipping costs are a major expense, management is initiating a study to reduce them as much as possible. For the upcoming season, an estimate has been made of the output from each cannery, and each warehouse has been allocated a certain amount from the total supply of peas. This information (in units of truckloads), along with the shipping cost per truckload for each cannery-warehouse combination, is given in Table 9.2. Thus, there are a total of 300 truckloads to be shipped. The problem now is to determine which plan for assigning these shipments to the various cannery-warehouse combinations would minimize the total ship- ping cost.

By ignoring the geographical layout of the canneries and warehouses, we can pro- vide a network representation of this problem in a simple way by lining up all the can- neries in one column on the left and all the warehouses in one column on the right. This representation is shown in Fig. 9.2. The arrows show the possible routes for the truck- loads, where the number next to each arrow is the shipping cost per truckload for that route. A square bracket next to each location gives the number of truckloads to be shipped out of that location (so that the allocation into each warehouse is given as a negative number).

The problem depicted in Fig. 9.2 is actually a linear programming problem of the transportation problem type. To formulate the model, let Z denote total shipping cost, and let xij (i = 1, 2, 3; j = 1, 2, 3, 4) be the number of truckloads to be shipped from cannery

An Application Vignette

Procter & Gamble (P&G) is the world’s largest and most profitable consumer products company. It makes and markets hundreds of brands of consumer goods worldwide and had over $83 billion in sales in 2012. Fortune magazine ranked the company at 5th place in its “World’s Most Admired Companies” list in 2011.

The company has grown continuously over its long history tracing back to the 1830s. To maintain and ac- celerate that growth, a major OR study was undertaken to strengthen P&G’s global effectiveness. Prior to the study, the company’s supply chain consisted of hundreds of suppliers, over 50 product categories, over 60 plants, 15 distribution centers, and over 1,000 customer zones. However, as the company moved toward global brands, management realized that it needed to consolidate plants to reduce manufacturing expenses, improve speed to market, and reduce capital investment. Therefore, the study focused on redesigning the company’s production and distribution system for its North American operations. The result was a reduction in the number of North American plants by almost 20 percent, saving over $200 million in pretax costs per year.

A major part of the study revolved around formulat- ing and solving transportation problems for individual product categories. For each option regarding the plants to keep open, and so forth, solving the corresponding transportation problem for a product category showed what the distribution cost would be for shipping the prod- uct category from those plants to the distribution centers and customer zones.

Source: J. D. Camm, T. E. Chorman, F. A. Dill, J. R. Evans,

D. J. Sweeney, and G. W. Wegryn: “Blending OR/MS, Judg- ment, and GIS: Restructuring P & G’s Supply Chain,” Inter- faces, 27(1): 128–142, Jan.–Feb. 1997. (A link to this article is provided on our website, www.mhhe.com/hillier.)