Plant location

In the network optimization models of Section 12.2.4, we have taken the network structure as given. Hence, the decisions we had to make were tactical or operational, and just linked to flow routing. However, at a more strategic level, we have to make decisions concerning:

• The location (or relocation) of production plants
• The sizing (or the expansion) of production capacities
• The capacity and location planning for distribution centers
• The allocation of retail stores to distribution centers

As far as the last point is concerned, we have considered a purely exogenous demand, which should be met at minimum cost. However, there are problems, such as the location of retail stores, in which the demand is a result of our decisions.

What we describe here is a straightforward extension of the transportation problem, whereby source nodes are just potential locations of plants. We should decide where a plant must be opened, within a set of predefined options, taking into account the related costs. Such decisions, as well as the related variables, are logical in nature: Either we open a plant, or we do not. This is a typical setting in which binary decision variables are used:

When opening a plant, the related costs include a fixed component, linked to the binary decision variables y_i. Finding a good solution calls for trading off the cost of opening a plant against transportation costs. In fact, to minimize the transportation cost, we should open plants close to destinations, but this increases the cost of the network structure. We should also note that there is a timing difference between the two decision levels. We open plants now, and then we transport items over a possibly long time period. Therefore, when considering a fixed charge for opening a plant, we must be careful and make it comparable to transportation costs; we should transform a one-time-only cost into a kind of per-unit-of-time fee, say, a monthly or yearly fee. If we measure demand and transportation costs on some time unit, we must somehow amortize opening costs to make all of them comparable. Doing so, we end up with a fixed charge for operating plant i, denoted by f_i. The classical plant location model, where one item type is considered, has the following form:

Comparing this model against the transportation problem, we see two basic differences:

1. There is an additional term in the objective function (12.54), related to fixed charges.
2. The capacity constraint (12.55) does not include a given capacity, but a capacity depending on our strategic decisions. If a plant is not opened (y_i = 0), there can be no flow going out of the corresponding node.

Since the model includes binary decision variables, it must be solved by mixed-integer programming methods such as branch and bound. Leaving solution issues aside, it is important to realize that the main difference between the two sets of decision variables is not really due to integrality requirements. One set of variables is related to strategic decisions, which are not easy to change on a short timescale; these should be considered as design variables, and are the true output of the model. Another set of variables is related to tactical decisions: Transportation decisions, should the demand pattern change, can be adapted on a short notice, subject to plant capacity constraints; these should be considered as control variables. The flow variables x_ij are not meant for immediate implementation. Rather, their role in a strategic model is to “anticipate” the effects of tactical decisions that will be made later. Therefore, the second term in the objective function (12.54) is actually an anticipation function. Such functions are common in hierarchical optimization models, where we need a link between different decision layers. Such a link will be approximate in nature, but we should mention that in the basic plant location model we are making two gross mistakes:

• On the one hand, we assume deterministic demand, but since the time horizon is relatively long, we should consider an uncertain demand.
• On the other hand, a linear cost structure cannot account for economies of scale which may characterize true transportation costs. Hence, the anticipation function should be nonlinear.

Later, we will see how such limitations can be overcome.