# How can preferences be modelled in MILPs?

Suppose a MILP model includes the following constraint:

$$\begin{gather*} B_{1} + B_{2} + B_{3} \le 100 \end{gather*}$$, where $$B_{1},B_{2},B_{3}$$ are nonnegative reals.

Suppose that

$$\begin{gather*} (B_{1},B_{2},B_{3}) = (10,30,60) \\ (B_{1},B_{2},B_{3}) = (20,60,20)\\ (B_{1},B_{2},B_{3}) = (80,10,10) \\ (B_{1},B_{2},B_{3}) = (60,10,10) \\ \cdots \end{gather*}$$

are optimal solution vectors.

Is there a way to tell the solver to prefer one of these solution vectors over the others? For example, is it possible to formulate a constraint that tells the solver to prefer an optimal solution with increasing values (if it exists?)?

Edit

I think I need to clarify the purpose of the approach I am looking for. The purpose is not really to find a "preferred" solution but to help the B&B to get faster to the optimal solution by cutting off solutions that have the same value as the "preferred". So I don't want to determine all alternative solutions and then choose one, I want to get as fast as possible to a solution.

Adding constraints $$B_{1} \le B_{2} \le B_{3} ...$$ would help if I knew that there exists an optimal solution that satisfies these constraints, but I don't know this.

• This might not be the best way, but you could solve it first with no preference instructions provided. Then solve a feasibility problem consisting of the original constraints, plus the constraint that the objective value is at least as good as original, plus constraints $B_i \le B_{I+1}$. If this new problem is feasible, it will return a preferred solution, and if not feasible, it is not possible to get preferred solution without degrading optimal objective value. Apr 28 '20 at 14:48
• If these constraints you mentioned should be added to the model when a new incumbent solution has been found, you would try using user-cut or lazy-cut to define them. Many of the modern solvers have such capability. Is it what you you are looking for? Apr 29 '20 at 0:22

An alternative to @Mark L. Stone's comment would be to define new variables $$\delta_i \in \{0,1\}$$ and add the following constraints: $$B_i \le B_{i+1} + M \delta_i\quad \forall i$$ and then penalize the activation of $$\delta_i$$ in the objective function by adding a term such as $$\sum_{i}\omega_i \delta_i$$. In your example, $$M:=100$$ is suitable.
This way, if there exists a solution with increasing values of $$B_i$$, it will be returned, with $$\delta_i =0$$. Otherwise, the constraint is deactivated, at the expense of a penalty.
Note however that this approach may disrupt your initial objective function, you will have to choose proper weights $$\omega_i$$ for variables $$\delta_i$$.