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Besides @RobPratt's answer, the first condition would be (for simplicity I omitted indices $i$ and $j$ and continued with only two $y$ variables: $$ x \implies (y_1 \oplus y_2) $$ $$ \lnot x \lor (y_1 …

I assumed that your variables are $X_{w}$, $X_{s}$, $X_{e}$, and $p_{t}$. Then as @RobPratt suggested, we can use two separate disjuncts. $$ W_t X_w+S_t X_s - 0.1 X_e \geq m(1-z_t) \quad (1)$$ $$ p_t …

If the binary variable $\delta_{i,j}^d$ is defined to linearize, Your first clause can be converted as: $$Iff \quad ((x_i \geq N) \implies \delta_{i,j}^d = 1)) \quad \implies ((\delta_{i,j}^d = 1) \im …

Another way would be by introducing the new binary auxiliary variables $z_{i, j, a}$ and $w_a$ and imply the following constraints: $$ z_{i,j,a} = 1 \iff\left(\sum_{b} x_{i,j,a,b} \geq 1\right) \quad …

Why not try to calculate the overall required operators at each shift or any specific period of time, weekly/monthly, as a predefined parameter, in the best case, for each of $N$ parts, and incorporat …

By updating the question, the mentioned conditional inequalities can be linearized by introducing new indicator variables, $z_i$, for each part and coupling those to make linearization. …

I know the logical constraints can be linearized by either the logical representation of whose relation, (for pure binary variables e.g. CNF/DNF) or for general form by using Big-M formulation. As I h …

One possible way to linearize such a constraint would be by dividing this expression into four parts and then linearizing each of which separately. $$ Iff \quad (w=1) \rightarrow (x \rightarrow y) \qu …

I want to linearize the following disjunctive form. $$\left[\begin{gathered}w_{1}\\x \geq a\end{gathered}\right] \vee \left[\begin{gathered}w_{2}\\x \geq b\end{gathered}\right]$$ where $w_1$ and $w_2 …

written by CPLEX in the following form: c1: (y1==1 && x>=2) || (y2==1 && x>=3); c2: (y3==1 && x<=8) || (y4==1 && x==2.5); or c1: (y1==1 || y2==1); c2: (y3==1 || y4==1); and adding appropriate linearization …

lor \delta$$ $$ (\sum_{i=1}^n x_{i} \leq 1) \lor ((\sum_{i=1}^n x_{i} \geq 3) \lor \delta$$ By introducing the auxiliary variables, the last line would be: $$ z_{1} + z_{2} + \delta \geq 1 $$ Now, the linearization …

It is not uncommon that with different objective functions, there are different complexity that comes with the specific problem. For example, in the scheduling theory, it is often of interest to deter …

I would like to know about the linearization of the $(If, Then)$ constraints as follows: $$\begin{array}{l} \text { If: } \\ 15 \leqslant x \leqslant 25 \\ \text { then: } \quad y=\color{blue}{a} x+\ … I knew that we can use specific linearization to do this. I was wondering if, is there another way to formulate this problem efficiently? …

Nowadays, mathematical programming solvers have been frequently used to solve lots of practical/academic problems. Many of these might be interpreted as a MIP or MINLP to represent a specific problem …

Suppose that, the following non-linear objective arises (MAX/MIN): \begin{equation} \max\frac{\sum\limits_{j} a_{j} x_{j}}{\sum\limits_{j} b_{j} x_{j}} \end{equation} 1) Replace the expression $\dfra …