[ Home | Support | Sales | Solvers | Documentation | Model Libraries | Search | Contact Us ]

bilinear.gms : Convexification of bilinear term binary times x

The model demonstrates various forumlations to represent bilinear
product terms of one continuous and one binary variable.

A set of 60 products i is produced on a set of machine with a given
total capacity. Some machine are special in the sense that if a
product is produced on one of them, cleaning treatment costs apply
caused by a set of cleaning treatment machines t.

A binary variable, delta(i), indicates that product i is produced on
one of the special machines. The model is simplified regarding the
machine-product relations.

Here we mimic a larger roduction problem, and just require that

     E1.. sum(iE, delta(iE)) =e= 2 ;
     E2.. sum(iO, delta(iO)) =e= 5 ;

which represents the fact that it cannot be avoided to use the special
machine and, thus, to have some cleaning treatment.

If product i is produced on a special machine, then the amount, y(i),
of the by-product is given by the recipe constraint y(i)=0.164*p(i),
where the non-negative variable p(i) is the amount produced on special
machines. For each product there is a specific yield of YS(i) $/ton.
The by-product is burnt and leads to an energy yield of YB(i) $/ton,
where YB(i)<YS(i). The by-product also passes the treatment plant.

The production is limited by the production capacity C, where x(i),
100+i <= x(i) <= XUB, is the amount of product i produced.

The amount produced on special machines is p(i)=x(i)*delta(i).

We compare the non-convex MINLP formulation to equivalent linear
forms of p(i)=x(i)*delta(i) using big-M, convex hull, and indicator
forumlations.  Moreover, a new SOS-1 formulation is presented which is
described in:

• Kallrath, J, Combined Strategic Design and Operative Planning in the Process Industry, 2009. Submitted to Computers \& Chemical Engineering