The Pinnacle Company is a U.S.-based manufacturer of furniture and appliances that offshored all of its actual manufacturing operations to Asia about a decade ago. It then set up distribution centers at various locations on the East Coast, near ports where its items were imported on container ships.

In many cases, Pinnacle’s appliances and furniture arrive partially assembled, and the company completes the assembly at its distribution centers before sending the finished products to retailers. For example, appliance motors, electric controls, housings, and furniture pieces might arrive from different Asian manufacturers in separate containers. Recently Pinnacle began exporting its products to various locations in Europe, and demand steadily increased. As a result, the company determined that shipping items to the United States, assembling the products, and then turning around and shipping them to Europe was inefficient and not cost effective. The company now plans to open three new distribution centers near ports in Europe, and it will ship its items from Asian ports to distribution centers at the European ports, offload some of the items for final product assembly, and then ship the partially filled containers on to the U.S. distribution centers. The following table shows the seven possible locations near container ports in Europe, and their container capacity that Pinnacle has identified to construct its proposed three distribution centers; the container shipments from each of its Asian ports; and the container shipping cost from each of the Asian ports to each possible distribution center location.

The following table shows the demand from each of the U.S. ports and the cost for container shipments from each of the possible distribution center locations to each of the U.S. ports:

a. Formulate and solve a linear programming model to determine which three distribution center locations in Europe Pinnacle should select, and the shipments from each of the Asian ports to these selected distribution centers and from the European distribution centers to the U.S. ports.

b. Reformulate the model so that Pinnacle minimizes its shipping costs while selecting its three new distribution centers within a budget of $45 million. What is the difference in this solution, if any?

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Proposed Distribution Center Asian Port F. Rotterdam G. Hamburg H. Antwerp 1. Bremen J. Valencia K. Lisbon L. Le Havre Center Cost $16,725,000 Container $19,351,000 $13,766,000 $15,463,000 $12,542,000 $13,811,000 $22,365,000 shipments A. Hong Kong $3,466 $3,560 $3,125 $3,345 $3,060 $3,120 $3,658 235 B. Shanghai $3,190 $3,020 $3,278 $3,269 $2,987 $2,864 $3,725 170 C. Busan $2,815 $2,700 $2,890 $3,005 $2,465 $2,321 $3,145 165 D. Mumbai $2,412 $2,560 $2,515 $2,875 $2,325 $2,133 $2,758 325 E. Kaoshiung $2,600 $2,800 $2,735 $2,755 $2,473 $2,410 $2,925 405 Capacity 565 485 520 490 310 410 605