$9\mathrm{.}2$A CASE STUDY: THE BMZ CO$.$ MAXIMUM FLOW PROBLEM

What a day! First being called into his boss$$s office and then receiving an urgent telephone call from the company president himself. Fortunately, he was able to reassure them that he has the situation under control.

Although his official title is Supply Chain Manager for the BMZ Company, Karl Schmidt often tells his friends that he really is the company$$s crisis manager. One crisis after another. The supplies needed to keep the production lines going haven$$t arrived yet. Or the supplies have arrived but are unusable because they are the wrong size. Or an urgent shipment to a key customer has been delayed. This current crisis is typical. One of the company$$s most important distribution centers$$the one in Los Angeles$$urgently needs an increased flow of shipments from the company.

Karl was chosen for this key position because he is considered a rising young star. Having just received his MBA degree from a top American business school four years ago, he is the youngest member of upper$-$level management in the entire company. His business school training in the latest management science and business analytics techniques has proven invaluable in improving supply chain management throughout the company. The crises still occur, but the frequent chaos of past years has been eliminated.

Karl has a plan for dealing with the current crisis. This will mean calling on prescriptive analytics once again.

Background

The BMZ Company is a European manufacturer of luxury automobiles. Although its cars sell well in all the developed countries, its exports to the United States are particularly important to the company.

BMZ has a well$-$deserved reputation for providing excellent service. One key to maintaining this reputation is having a plentiful supply of automobile replacement parts readily available to the company$$s numerous dealerships and authorized repair shops. These parts are mainly stored in the company$$s distribution centers and then delivered promptly when needed. One of Karl Schmidt$$s top priorities is avoiding shortages at these distribution centers.

The company has several distribution centers in the United States. However, the closest one to the Los Angeles center is over $1,000$ miles away in Seattle, so supplies for the Los Angeles center currently are forwarded from the supply centers in New York and New Orleans that receive their supplies directly from Europe. Since BMZ cars are becoming especially popular in California, it is particularly important to keep the Los Angeles center well supplied. Therefore, the fact that supplies there are currently dwindling is a matter of real concern to BMZ top management$$as Karl learned forcefully today.

Page $380$

Most of the automobile replacement parts are produced at the company$$s main factory in Stuttgart, Germany, along with the production of new cars. It is this factory that has been supplying the Los Angeles center with spare parts after first shipping them to New York and New Orleans. Some of these parts are bulky, and very large numbers of certain parts are needed, so the total volume of the supplies has been relatively massive$$over $300,000$ cubic feet of goods arriving monthly. Now a much larger amount will be needed over the next month to replenish the dwindling inventory.

The Problem

Karl needs to execute a plan quickly for shipping as much as possible from the main factory to the distribution center in Los Angeles over the next month. He already has recognized that this is a maximum flow problem$$a problem of maximizing the flow of replacement parts from the factory to this distribution center.

The problem is to maximize the flow of automobile replacement parts from the factory in Stuttgart, Germany, to the distribution center in Los Angeles.

The factory is producing far more than can be shipped along a single route to this one distribution center. Therefore, the limiting factor on how much can be shipped is the limited capacity of the company$$s distribution network.

This distribution network is depicted in Figure $9\mathrm{.}6,$ where the nodes labeled ST and LA are the factory in Stuttgart and the distribution center in Los Angeles, respectively. There is a rail head at the factory, so shipments first go by rail to one of three European ports: Rotterdam $($node RO$),$ Bordeaux $($node BO$),$ and Lisbon $($node LI$).$ They then go by ship to ports in the United States, either New York $($node NY$)$ or New Orleans $($node NO$).$ Finally, they are shipped by truck from these ports to the distribution center in Los Angeles.

FIGURE $9\mathrm{.}6$ The BMZ Co$.$ distribution network from its main factory in Stuttgart, Germany, to a distribution center in Los Angeles.

The organizations operating these railroads, ships, and trucks are independently owned companies that ship goods for numerous firms. Because of prior commitments to their regular customers, these companies are unable to drastically increase the allocation o