D

x2

C

B

A

x1

Fig. 2.9 Geometric representation of non-terminating branch-and-bound tree

2.2.2 The Convex Hull of an IP

In Fig. 2.10 we draw the feasible region of the LP relaxation of Example 2.7 (with the same lettering as in Fig. 2.7).

We also mark the points with integer coordinates inside this region. These are the feasible solutions to the IP in Example 2.7. Although the feasible solutions are disconnected, if we are using LP to help us solve the IP, we can surround these integer points by the smallest convex region containing them. This is known as the convex hull of feasible integer points. It is marked by the dotted lines in the figure. It is obvious that its boundaries must be straight lines in this two-dimensional case. In higher dimensions they will be hyperplanes. They could therefore be represented by linear constraints. In this example these are

x2

I

If we were to use objective (2.41) and replace (or append) the constraints (2.42)–(2.45) by those above then the IP in Example 2.7 could be solved as a (much easier) LP. The optimal solution would be given by vertex I in Fig. 2.10.

The boundaries of the convex hull are known as facets and the constraints giving rise to them are known as facet-defining constraints.

In practice they may be very difficult to calculate. However, it may be possible to calculate some of them or approximate them by constraints which cut off some of the feasible points of the LP relaxation feasible region, but are not facets. All such constraints are known as cutting planes. Appending them to a model, when known, could be expected to reduce the size of the tree search in branch-and-bound. However, another problem, which frequently arises, is that there may be an astronomic number of facets. This is the case with, for example, all known IP formulations of the famous travelling salesman problem (TSP) (which is discussed in Sect. 2.3). In practice one may only add them (or other cutting planes) iteratively when they are known to cut off the current fractional solution.

Formulations that lead to constraints representing the convex hull are known as ‘sharp’ and those which are closer to the convex hull than others are known as ‘tighter’.

Such procedures can be implemented as another method of solving IP models, i.e. solve the LP relaxation, add cutting plane(s) to remove the fractional solution

and keep repeating. Alternatively this approach can be implemented in a branch- and-bound tree (known as branch-and-cut) by applying it to the IPs corresponding to intermediate nodes in the solution tree.

If a model is a mixed IP then one is only restricted to integer coordinates in some directions. If, for example, we were only to restrict x1 to take integer values in Example 2.7 the feasible solutions would be represented by the vertical lines in Fig. 2.11. The convex hull would be as represented by the dotted lines, and the optimal solution lies at vertex H.

Fig. 2.11 Convex hull of a mixed integer programme

The facet-defining constraints for this example are then

The facet-defining constraints of an IP can be regarded as the strongest bounds on certain linear expressions which can be inferred from the constraints of an IP. This is analogous to the way in which LP can be regarded as an inference problem as mentioned in Sect. 1.1. Hence both types of problem could be regarded as inference problems. For example suppose we were to choose the objective

Maximise

subject to the constraints of Example 2.7. The answer would clearly be x2 = 3 as a result of the facet 2, i.e. the constraints imply x2 3 but nothing stronger.

It should be obvious that it is not possible to derive the constraint x2 3 by an argument involving only multipliers on the constraints, leading to a dual model. The best that can be obtained by such an argument is x2 4 16 . 4 16 is the optimal objective value to the LP relaxation with objective (2.72). Viewed as inference problems finding the strongest implied constraints of an IP is therefore more difficult than simply adding constraints in certain multiples (the LP dual values). The difference between the optimal LP objective value and the optimal IP objective value is sometimes known as the duality gap for an IP because the LP optimal value is the best that can be obtained by dual multipliers. It is an indication of the difficulty of an IP. Clearly it is the difference between the objective value for the LP relaxation at the top of a branch-and-bound tree and the objective value at the node giving the optimal IP solution. Therefore it is some (albeit crude) measure of the likely size of the tree.

There is, however, a remarkable procedure for finding the strongest implied constraints (the facet-defining constraints) of a pure IP. It consists of the repeated application of the following operations. (For simplicity we will consider the case of a maximisation model with ‘ ’ constraints.)

i.Adding constraints together in suitable non-negative multiples.

ii.Dividing through by the greatest common divisor of the left-hand-side coefficients and rounding down the right-hand-side coefficient.

We illustrate this by reference to the constraints of Example 2.7.

Dividing constraints (2.43) and (2.44) each through by 2 we obtain, respectively

These imply, since the left-hand expressions are integer,

δ3

(0,0,1) A

(0,1,1/2) B

δ2

F (0,1,0)

D

(1,0,1/2)

C (0,0,0)O (1,1,0)

E (1,0,0)

δ1

Fig. 2.12 The LP relaxation of a 0–1 IP model

Constraint (2.80) is represented by the plane ABCD. It can be seen that the polytope of the LP relaxation has vertices O,A,B,C,D,E and F. Hence optimising an objective subject to (2.80) and (2.81) (ignoring the integrality requirement) could lead to one of the solutions associated with the fractional vertices at B or D.

The convex hull of feasible 0–1 integer solutions is shown in Fig. 2.13.

The convex hull of the feasible integer solutions is defined by the constraints

together with (2.81).

It can be seen that all the vertices have 0–1 coordinates.

In general the feasible region associated with both the LP relaxation and the convex hull of a 0–1 IP model is a multidimensional hypercube with some of the corners sliced off.

δ3

(0,0,1)

δ2

(0,1,0)

(1,0,0)

δ1

Fig. 2.13 The convex hull of a 0–1 IP model

2.3 The Use of 0–1 Variables

As has already been stated the majority of integer variables in practical IP models are restricted to the two values 0 and 1. If there are general integer variables then they would be restricted to a small integer range in almost any sensible IP model. Variables that, in reality, should be integer (e.g. number of cars manufactured or people of a certain category), which could take large values, would be treated as continuous and their solution values in the optimal solution rounded to give sensible answers.

2.3.1 Expressing General Integer Variables as 0–1 Variables

It should be pointed out that general (bounded) integer variables can always be expressed as a combination of 0–1 variables. We demonstrate by the following example.

Example 2.11 Express the integer variable x (0 x U ) using 0–1 variables.

The most compact way of doing this is a ‘binary’ expansion