Changing a model in to a mixed integer linear programing model

Question

Trying to learn about integer programming in quarantine and I've come across a problem that stumped me. I searched the net but couldn't see anything similar and would appreciate another set of eyes on how to approach it.

Turn the given model in to a binary mixed integer linear programing model:

$\operatorname{Max} z=a(x)+2 b(y)$

s.t $\quad x, y \geq 0$

At minimum two thirds of the given constraints apply:

$$2 x+y \leq 16, \quad x+y \leq 9, \quad x+3 y \leq 12$$

$$a(x)=\begin{cases}10+3 x, & \text{if $0 \leq x \leq 4$}, \\ 14+2 x, &\text{if $x \geq 4$},\end{cases} \quad b(y)=\begin{cases}8+y, &\text{if $0 \leq y \leq 3$} \\ 2+3y, &\text{if $y \geq 3$}\end{cases}$$

It hints to consider making use of multiple $x$ and $y$ variables and I know that if I want to try linearizing the problem I should go with $b(y)$ due to $y$ having a coefficient of $3$ in the third function.

Answer 1

There are many different ways to do this, there are different options here. Here is one, which is definitely not the best, but which is quite natural I guess.

I will only show you how to it for $a(x)$, the method is identical for $b(y)$.

Define a binary variable $\delta$ that takes value $1$ if and only if $x\le 4$:

$$ 4(1-\delta)\le x \le 4 +M(1-\delta) $$ $M$ is an upper bound on $x$. And then define the continuous variable $a$ as follows:

$$ a = (10+3x)\delta + (14+2x)(1-\delta) $$

Now you have non linear terms (in $x\delta$) that you need to linearize, for example like this.

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In practice, before using such transformations blindly, be sure to analyze the nature of the function: is it convex or concave, and are you maximizing or minimizing. As mentioned by Rob Pratt, if you are minimizing (maximizing) a convex (concave) function, a much simpler approach is available. Be sure to check his answer.

For the part where at least $2$ constraints out of $3$ should be satisfied, also please refer to @Rob Pratt's answer, as to my knowledge there is no other way to do it.

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Just for pleasure, I also like this method:

Redefine $x$ in terms of new variables $x_1 \ge 0$ and $x_2 \ge 0$ as follows: \begin{align} x&=x_1+x_2+4(1-\delta)\\ x_1&\le 4 \delta \\ x_2& \le 5 (1-\delta) \end{align}

So $x_1$ is the range on the first interval $[0,4]$, and $x_2$ on the second one $[4,9]$. And so you can rewrite $a(x)$ linearly: $$ a = 3x_1+10\delta +2x_2+22(1-\delta) $$

Answer 2

You can do it with four binary variables. For $i\in\{1,2,3\}$, let binary variable $z_i$ indicate whether constraint $i$ is satisfied, and impose linear constraints \begin{align} 2x+y-16&\le M_1(1-z_1)\\ x+y-9&\le M_2(1-z_2)\\ x+3y-12&\le M_3(1-z_3)\\ z_1+z_2+z_3&\ge 2 \end{align} The original constraints imply upper bounds $x\le 9$ and $y\le 9$, so you can find good values for the $M_i$ constants based on that. For example, take $M_1=2(9)+9-16=11$.

For $a(x)$, the piecewise linear function is concave, so the maximization objective means that you can replace $a(x)$ with a variable $u$ and impose linear constraints \begin{align} u&\le 10+3x\\ u&\le 14+2x \end{align}

For $b(y)$, the piecewise linear function is not concave. Replace $b(y)$ with a variable $v$, introduce a binary variable $z_4$ to indicate which segment is used, and impose linear constraints \begin{align} 0z_4+3(1-z_4)\le y&\le 3z_4+9(1-z_4)\\ 0\le v-(8+y)&\le M_4 z_4\\ 0\le v-(2+3y)&\le M_5(1-z_4) \end{align}