Modelling Question

Question

Let $W^C_t$, $W_t$ be binary variables and $p$ an integer variable with $1 \leq p \leq 3$

The variables are related through the following equation:

$$W^C_t = \sum_{\theta=1}^{p} W_{t-\theta}$$

I can linearise this equation by introducing indicator variables $Y_1, Y_2, Y_3$ and requiring:

IF $Y_1=1$ THEN $W^C_t = W_{t-1}$

IF $Y_2=1$ THEN $W^C_t = W_{t-1} + W_{t-2} $

IF $Y_3=1$ THEN $W^C_t = W_{t-1} + W_{t-2} + W_{t-3}$

$Y_1 + Y_2 + Y_3 = 1$

This approach may, depending on the problem, to a large amount of constraints.

The question is, is there a better way to model the equation?

Answer 1

Define set/list as $S=\{1,2,3\}$: basically indices of length N
Then add constraints
$\sum_{j=1}^3 jz_j = p $
$ \sum_j z_j = 1$
Then use
$ z_j \le \sum_{k=1}^j W_{t-k} \ \forall j \in S$
$\sum_{k: (k\gt j)} W_{t-k} \le z_{k} \ \forall j \in S$
$ W_{t}^C = \sum_{\theta=1}^N W_{t-\theta}$