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Well, you did not define and detail well the problem, hence, I will first write formally the problem definition based on my understanding of what you have written, and then I will propose an Integer Programming formulation. Problem definition Let's first define formally the problem. Let: $t$ be the number of tasks to be serviced; $T = \mathbb{N}_{\leqslant ...


7

Introduce binary variables $z_1$, $z_2$, and $z_3$, and impose linear constraints \begin{align} z_1+z_2 +z_3&= 1 \tag1\\ 1z_1+bz_2+(b+1)z_3 \le y &\le (b-1)z_1+bz_2+Uz_3 \tag2\\ z_2&\le x\tag3 \end{align} Constraints $(1)$ and $(2)$ enforce the three disjoint cases $y<b$, $y=b$, and $y>b$. Constraint $(3)$ enforces $x=0\implies z_2=0$, and $...


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