2.3.2 Yes/No Decisions

xi j R, δi Z i I, j J

Notice this use of the quantifier ‘ ’, but here it is indices which are being quantified over the domains of their index sets. Such a notation is common in certain modelling languages as discussed in Chapter 3. The use of the Greek letter ‘δ’ for 0–1 variables is common in contrast to the Roman alphabet for continuous variables.

It is necessary to model this problem as a (mixed) IP because of the discrete nature of the decision to build, or not build, a warehouse at each location. The building of a fraction of a warehouse (resulting in an LP model) is not an option. The fi are fixed costs which will either be incurred in their entirety, or not at all. These are in contrast to the ci j which are variable costs which can be incurred in continuously varying fractions depending on the continuously varying values of the variables xi j .

In (2.89) we have represented the variables δi as general integer variables which must lie in the range 0–1, so restricting them to the two values 0 and 1. It is more usual to specify them directly as binary variables δi {0, 1}.

This problem is an instance of the fixed charge problem, which is further discussed in Chapter 3. Note that it is not necessary to explicitly model the condition that if a warehouse is opened (and therefore a fixed cost incurred) one will send a positive quantity out of it. This must happen by virtue of optimality for to open it and not send any quantity from it would be non-optimal. Hence this condition is guaranteed by the objective function.

2.3.4 Logical Decisions

Once we have 0–1 variables we can make logical statements about the corresponding decisions. Example 2.13 illustrates this.

Example 2.13 Model the following extra conditions in the facility location problem.

i. At most n warehouses can be built where n |I |.

ii. If any warehouses are built at i P then no warehouses can be built at k Q but at least three warehouses must be built at l L.

52 2 Integer Programming

Many variants and extensions of the facility location problem can also be modelled. The problem typifies a whole class of problems involving discrete decisions which have to be made if other activities are to be carried out.

Example 2.14 A blending problem has been formulated as an LP with variables xi representing the proportions of ingredients i. In addition it is stipulated that

i.If an ingredient i is included it must be at a proportion of at least αi .

ii.At most n ingredients can be used.

iii.If any ingredient i I is used then at least one ingredient j J must also be used.

We introduce 0–1 variables δi representing the presence of ingredient i and the following constraints:

If xi > 0 then (2.93) forces δi = 1. Then δi = 1 forces xi αi by (2.94). Condition (ii) is similar to that in Example 2.13 and is modelled as

Again the above types of constraints may also arise in other contexts than blending.

It is sometimes natural to model certain conditions as products of 0–1 variables. Of course such a product is itself a 0–1 quantity which can be represented as a 0–1 variable with a logical dependence on the original variables. Example 2.15 illustrates this.

This type of model arises in a number of contexts. One of the most important is crew scheduling (particularly for airlines) where each subset corresponds to a roster which, if used, must be assigned to a possible crew. The objective is to minimise the number of crews necessary to cover all the flights. Each flight gives rise to a constraint. The model also arises in computer circuit design and logic simplification. This latter application is discussed in Chapter 4.

It is worth pointing out that, in practice such models can have a huge number of variables. However, they are often amenable to simplification by preprocessing. For example δ5 is redundant since δ4 covers everything that δ5 does and more at equal objective ‘cost’. The fifth constraint can then be removed as it is implied by the second. This reduces the model to

δ1, δ2, δ3, δ4 {0, 1}

Exercise 2.6.17 is to solve this model by branch-and-bound.

Set-covering models look deceptively easy to solve but can be very difficult, usually on account of their size. They do have important mathematical properties. In particular there will always be an optimal integer solution which corresponds to a vertex of the polytope for the LP relaxation (but normally not the vertex associated with the LP optimum). The Chvatal´ rank for the class of set-covering models is not bounded.

Closely related are set packing models where the objective is to ‘pack’ as many subsets into a given set, as possible, with no overlap. A particularly important case arises where the set of elements can be represented as the nodes of a graph. The subsets are pairs of nodes which are joined by an edge, The packing problem becomes that of finding the maximum ‘pairing’ of nodes where each node has a unique ‘mate’ (if it has one at all). This is known as the matching problem. Formulated as an IP it can be shown to be of Chvatal´ rank 1. A special case of the matching problem is to ‘bipartite’ graphs. Here there are two sets of nodes and matchings can only be between nodes of different sets. This problem is usually known as the assignment problem. An optimal solution to the LP relaxation is integer. The Chvatal´ rank is therefore 0. A set partitioning model is that of partitioning the original elements into a collection of the given subsets. Packing and partitioning models can be converted