# Questions tagged [mixed-integer-programming]

For questions about mathematical optimization problems involving both continuous and binary or general integer variables.

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### How to determine whether the solution is integer feasible in a CPLEX heuristic callback?

I am working with a MILP formulation of a routing problem using Concert CPLEX 12.10. And I am implementing a greedy heuristic that uses the variables fractional values to attempt to construct an ...
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1answer
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### What kind of scheduling problem is this?

I am trying to develop algorithm to solve following basic version of the problem. 1) I would like to know what this problem is called in literature so that I can look it up 2) What are efficient ...
1answer
846 views

### Non-constant expressions cannot be multiplied

I am trying to solve a valued n-queens problem, in which queens in black squares worth double of those in white squares. I solved it in AMPL just fine, but I would like to try that in Python using a ...
0answers
63 views

### Complexity of determining whether a LP or MIP is infeasible

What is the best complexity for the worst case scenario and the algorithm associated with it to determine if a linear programming (LP) is infeasible ? Further, what if we consider a mixed integer ...
3answers
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### Open source MILP solver for quick “good enough” solution

I have a problem that I have already posted elsewhere in OR.stack, but the question is focused around a large binary MILP (about 1 million decision variables). Ultimately, I am more time constrained ...
1answer
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### Are valid inequalities worth the effort given modern solvers?

In Laurence Wolsey's Integer Programming, he presents a well-known procedure for deriving valid inequalities (VI) suitable for integer and mixed integer linear problems (see Section 8.3, and also ...
1answer
170 views

### How to write quantity discount constraint?

In a simple transportation problem like below image, how to model a constraint for shipment discount e.g. if quantity transported from any origin to any destination is more than 100 unit then the ...
1answer
205 views

### Nested set in OPL/CPLEX

I'm trying to solve a scheduling problem using OPL/CPLEX. In the model, there are nine tasks in which each task has its specific precedence. the precedence relationship is as follows: ...
1answer
94 views

### Is working with equations in an MILP more efficient than working with inequaltities?

I have the following model \begin{align}\max&\quad B_{1} + B_{2}\\\text{s.t.}&\quad 0 \leq B_{1} \leq c_{1}\cdot Y_{1} \\&\quad 0 \leq B_{2} \leq c_{2}\cdot Y_{2} \\&\quad \end{align} ...
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3answers
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0answers
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### Adding redundant inequality to a model

The following relationship is part of my optimization model ($W_k$ denotes a binary variable) $$W_{1} \le \cdots \le W_{k-1} \le W_{k} \le W_{k+1} \le \cdots \le W_{k^{\max}}$$ My question is, do ...
2answers
169 views

### Modeling Linear Program to decide if an inequality is facet

Suppose you have a set of points $v_1,\ldots,v_n$, which are the vertices of the polytope $P=\operatorname{conv}\{v_1,\ldots,v_n\}$ and a linear inequality $a^\top v \leq b$. What would be a linear ...
1answer
246 views

### Advanced Heuristics for MIP vehicle routing problem in PuLP

I am currently trying to speed up an MIP. An approach I was considering was to implement a cut callback heuristic with PuLP (one which rounds relaxed integer variables greater than .9 to 1). ...
2answers
239 views

### Linearizing objective function with variables inside an indicator function

I am working on a problem in which I am trying to maximize the average of a variable only for the data that meet a certain condition with a constraint on the number of data that meet this condition. I ...
1answer
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### How to linearize difference of absolutes?

How to linearize difference of absolutes? $$|a|\ge k|b|$$ where $a,b$ are variables and $k$ is a constant.
1answer
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### How to design the objective function of an MILP problem

In an MILP problem, I wonder if it is better to use an objective function that involves as many of the variables of the problem as possible as opposed to an objective function with only a few ...
1answer
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### How to write a LP/MIP to mark the items in a time-series list after a fixed delta?

I have a time series list like this. ...
2answers
150 views

### How to exploit known solution in MILP

I have an MILP model to which I get an integer feasible solution as a result of a heuristic search. In this particular example, the initial solution turns out to be the optimal solution, which I prove ...
1answer
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### Benders subproblem feasible region dependent upon solution master problem

Suppose I want to solve a naturally MINLP problem of the following form: $$\min_{x,y} \{c'x + y \mid Ax \leq b, Dx + Ey \leq f, G(x)y\leq g, x \in \mathbb{Z}, y \in \mathbb{R}^+\}$$ Here $G(x)$ ...
1answer
158 views

### Find the cheapest shopping cart

Given a list of items to buy on an e-commerce, generate the cheapest possible cart (which obviously includes all the items). The items can be purchased from many sellers at different prices. Sellers ...
0answers
61 views

### Integer variable optimization - decreasing execution time

I have a variable declared as follows in AMPL: ...
2answers
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### IF X = 0 THEN Y = 1, IF X > 0 THEN Y => 0

I'm trying to model the following IF $tS = 0$ THEN $Y = 1$, IF $tS \gt 0$ THEN $Y \ge 0$ $tS$ is a positive real number and $Y$ is binary. I tried the following: $tS - \epsilon \ge -M Y$ but ...
2answers
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### Why do we need to measure the difficulty of mixed-integer programming problems?

I'm doing a project about the estimation of the difficulty of mixed-integer programming problems. The MIP instances are from MIPLIB 2017. And there are three categories of MIPs provided by MIPLIB 2017,...
2answers
101 views

### How can I model this binary logic?

I am looking for a constraint to express the following: IF W1 = 0 AND W2 = 0 THEN Y = 0 IF W1 = 0 AND W2 = 1 THEN Y = 1 IF W1 = 1 AND W2 = 0 THEN Y = 0 IF W1 = 1 AND W2 = 1 THEN Y <= 1 ...