Capacity Expansion Model¶

Definition¶

The course's capacity expansion model uses a binary purchase variable $y$ to switch additional capacity on or off. If existing capacity is 100 and a new machine supplies 50 more, the restriction is

x_1+x_2\le100+50y,\qquad y\in\{0,1\}.

Here $x_1,x_2$ are the production quantities that share this capacity, and $y=1$ means the additional machine is purchased.

Interpretation¶

A capacity expansion model lets the optimization decide whether to invest in extra capacity instead of fixing that choice in advance. One binary variable $y$ records the investment, and the capacity constraint $x_1+x_2\le100+50y$ changes with it.

Two parallel lines x1 + x2 = 100 and x1 + x2 = 150 with the region below each shaded; point A lies below the first, while point B lies between them and needs the extra capacity.

The two lines are the two possible capacity limits. With $y=0$ only the blue triangle below $x_1+x_2=100$ is allowed; with $y=1$ the orange band up to $x_1+x_2=150$ is added. The point $A=(50,40)$ uses $90$ units and fits either way, while $B=(70,50)$ uses $120$ units and is allowed only when the machine is bought.

The right-hand side $100+50y$ is linear in $y$ , so the model stays a mixed integer program: the quantities $x_1,x_2$ may be continuous while $y$ is binary. The constraint allows the extra $50$ units when $y=1$ but does not force their use. Without a cost attached to $y$ , choosing $y=1$ is therefore never worse, and nothing in the model discourages buying the machine. The investment cost has to appear in the objective, for example as a term $-Fy$ in a profit maximization with purchase cost $F$ , or in a budget constraint.

The integrality of $y$ is essential. In the LP relaxation, $y=0.4$ would be allowed and would give capacity $100+50\cdot0.4=120$ $120$ , as if part of a machine could be bought. The same device, a right-hand side switched by a binary variable, is also used in either-or constraints.

Key Results¶

With $y=0$ $0$ , the capacity limit is 100.
With $y=1$ $1$ , the capacity limit is 150.
Any specified acquisition cost must also be represented in the economic objective or budget; the isolated course example only demonstrates the capacity restriction.

Examples¶

The illustrated point $A=(50,40)$ uses capacity 90 and is allowed with $y=0$ $0$ ; $B=(70,50)$ uses 120 and requires $y=1$ $1$ .

A plan with total production 120 requires $y=1$ $1$ .
A plan with total production 90 satisfies the capacity restriction with $y=0$ $0$ .

Prerequisites¶

Sources¶

OR Modelling · Week 3

Symbols: element of

Last updated: 20 October 2026 03:02