Questions tagged [mixed-integer-programming]

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

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7
votes
1answer
263 views

No, Gurobi, I really do want this variable to be binary

When I mark a variable in a Gurobi MIP model as binary, sometimes Gurobi gives me a solution where that variable has a fractional value other than 0 or 1. How do I constraint a variable to be honest-...
7
votes
1answer
166 views

Strong MIP formulations for a large-scale mixed-integer nonlinear feasibility problem

I'm trying to construct a strong MIP formulation for the following integer nonlinear feasibility problem. Informally: We have a $m \times n$ decision matrix of binary variables Each row of the matrix ...
7
votes
1answer
163 views

Why is there a constant in the objective function of the *best subset selection problem*?

This article presents the following formulation of the best subset selection problem $$\min_{\|\beta\|_0\leq k}\frac{1}{2}\|y-X\beta\|^2_2$$ I wonder where the $1/2$ comes from. Help appreciated.
7
votes
1answer
71 views

Expressing a chain of boolean if-then with logical ANDs using MIP

How to express a chain of boolean If-then as MIP such as: If $(x_{10} \ge b_1$ and $x_{11} \le b_1)$ AND $(x_{20} \ge b_2$ and $x_{21} \le b_2)$... AND... then $y_1 = 1$ else $y_1 = 0$. So basically,...
7
votes
2answers
119 views

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{...
7
votes
2answers
671 views

Finding an Objective Function for Assigning Employees to Sequence Dates

I am using a mixed-integer-program to schedule employees to projects. These projects can have a time window to get completed from a few weeks to a few months. At the moment I am working in a ...
7
votes
1answer
232 views

Finding the optimal, spatially compact set of grid cells

I have a regular grid of cells, maybe square, maybe hexagonal. Each cell has a numeric value associated with it. How can I find a subset of cells that are: a connected, compact set and have an ...
7
votes
1answer
402 views

How to formulate this scheduling problem efficiently?

Let there be $N$ users with individual demands (of some items). Some users can have higher demands while the others can have lower demands. There are exactly $N$ service points. There is a one-to-one ...
7
votes
1answer
247 views

Logical Constraints Modelling using Big-M formulation

I am trying to model some logical constraints in ILOG. Logical constraints could be given such as: Constraint 1 or Constraint 2, Constraint 3 or Constraint 4, Constraint 5 or Constraint 6. The ...
7
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0answers
79 views

Building the Scheurman's Model II constraints for a multi period linear program

Scheurman's paper discusses Model I and model II Formulation to solve harvesting and scheduling problems. It is a specific implementation to solve multi period linear programs. Both models are also ...
7
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0answers
73 views

Modelling a simple ordering problem to have balanced delivery days

Assuming that I should buy 50 items from 25 different vendors with pre-known delivery duration between 2-7 day for each, which day of a week should I place each order so that the delivery days be even ...
6
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5answers
2k views

Algorithms vs LP or MIP

Is there a way of writing an algorithm with if-, while-statements to find an optimal solution without using linear-programming (LP)/MIP? If so, what would the benefits be against the LP/MIP? Is it ...
6
votes
4answers
748 views

Can this be formulated as one inequality

I have two binary variables $x_1$ and $x_2$ and a non-negative continuous variable $y$. In addition, I have the following two parameters $u>q>0$. I would like to formulate the following ...
6
votes
2answers
664 views

How to transform this logical if-then constraint?

Consider the binary variables $x, y, z \in \{0,1\}$. I'd like to formulate the two if-then constraints: $$ x + y \geq 2 \implies z = 0, \tag{1} $$ $$ x + y \leq 1 \implies z = 1. \tag{2} $$ Constraint ...
6
votes
2answers
404 views

Branch and Price Algorithm

Can branch and price be a good solution approach for a routing problem with min-max objective function? For example, minimizing the max length of any vehicle route in a VRP. In the literature, I haven'...
6
votes
2answers
2k views

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 ...
6
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2answers
200 views

Heuristic solution to the graph partitioning problem

I am working on a graph partitioning problem. A static column generation based solution was proposed in How to partition a graph with optimal number of groups? But I need some MILP solver to solve ...
6
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2answers
255 views

Mixed-integer optimization with bilinear constraint

So I have an optimization problem of the following form: \begin{aligned} \max_{x,y} \quad & \sum_i x_i \\ \text{s.t.} \quad & \sum_i x_iy_i \leq a \\ \quad & x_{\min} \leq x \leq x_{\max} ...
6
votes
2answers
236 views

Forbid transformation of max(x,y) into MILP

The function $\max(x,y)$ can be linearized by making use of additional binary variables. I suppose global optimisers are implemented to perform this transformation automatically. Is there a global ...
6
votes
2answers
215 views

How can this be expressed as a MILP constraint?

I am looking for a constraint to express the following: IF W1 = 0 AND W2 = 0 THEN Y = 1 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 ...
6
votes
1answer
3k views

How to linearize min function as a constraint?

I'm trying to solve an optimization problem including following constraint, and I need to linearize it in a maximization nonlinear programming model. Please help me to reformulate it with mixed ...
6
votes
1answer
254 views

What is the performance improvement when using semi-continuous variables instead of binary + continuous variable pair?

I have a MILP model that solves a master production schedule including capacity decisions. In the model I have a production quantity that should either be 0 or at least the amount that can be produced ...
6
votes
1answer
160 views

TSP subtour elimination by assigning distance traveled

Given a set $S$ which we need to travel following TSP rules. I was wondering if this sub tour elimination method is good enough or not? Let $b_{i,j}$ denote edge from $i$ to $j$ is taken or not and $...
6
votes
3answers
652 views

No solution found from n MIP starts

Currently, I am working on the implementation of a MILP formulation with lazy constraints for an optimization problem, and I am using the MIP Start strategy. I am generating the integer feasible ...
6
votes
1answer
226 views

Interface for Cbc - COIN-OR

I would like to code some IP/MIP models in python and test them with an open-source solver. As of now, I only know the Cbc - COIN-OR open-source solver. I have already tried the or-tools interface, ...
6
votes
1answer
158 views

How to treat a system of bilinear constraints

A model contains constraints of the following form $R(k) \leq X(k) G(k)$ where $X(k)$ binary and $G(k)$, $R(k)$ non-negative variables. The index $k$ runs from $1$ to $50$. I linearise the equations ...
6
votes
2answers
139 views

Optimising the current model

After developing the MIP model I noticed that solver is taking a lot of time to reach the solution. So, how should I approach to optimize the current model? Are there any visualization tools or any ...
6
votes
2answers
71 views

If $t\le0$ then $P=1$, if $t > 0$ then $P =0$ or $P=1$

I am trying to model $t \leq 0.0 \implies P = 1.0$ else $P=1$ or $P=0$ where $0 \leq t \leq H$ is a bounded nonnegative real, and $P$ is binary. I can use the expression $t + \epsilon P \ge \epsilon$ ...
6
votes
1answer
155 views

How to propagate time using linear inequalities?

I have an adjacency matrix $G_{i,j}$ that tells the distance between $i$ to $j$ (between 0 to 1) if there is no edge between $i$ to $j$ I am putting a large integer $100$. This is my previous ...
6
votes
2answers
237 views

A heuristic approach to solve a MILP problem?

I have the following optimization problem which is a MILP. I can solve it with a MILP solver. This one I posted here Is there a heuristic approach to the MILP problem? Since I have an additional but ...
6
votes
1answer
101 views

Upper and lower bounds of a variable equal

I'm working on a MILP (Mixed-Integer Linear Programming) problem with the Java API of Cplex. In order to easily exclude some variables from my problem I thought about setting both their lower and ...
6
votes
1answer
181 views

Guides for strong MILP Formulations

When developing MILPs, often there are different alternatives possible to express a constraint. The question then arises, which of the alternatives is better, meaning, which alternative is expected to ...
6
votes
1answer
109 views

Are there strategies/rules/systematic approaches to derive alternative formulations for an optimization problem?

For a given optimization problem there are often many known math programming formulations. For example for the TSP here is a survey https://link.springer.com/chapter/10.1007/3-540-36626-1_5 of many ...
6
votes
2answers
172 views

What is the definition for an integer-friendly constraint?

I have seen some papers claiming that their proposed model is integer-friendly. I would like to get more information about what type of constraints we can call integer-friendly. Probably, it can be ...
6
votes
1answer
383 views

Creating a Decision Variable in Python-MIP with Dictionaries

I have already seen in the Python-MIP documentation on how to implement a variable with multiple indices. In each example I read, this is done with the use of lists and integers as indices. This is ...
6
votes
1answer
247 views

Multi-period linear dynamic programming with differing in-period dependencies and changes

I’m not sure if I’m wording this right but in a nutshell, my problem is: I’m modelling potential actions a boat owner can do to their boat. Let’s say he wants to know over the 50 year lifespan of the ...
6
votes
1answer
286 views

Can GLPK be used to solve an optimal team selection problem?

My Problem I am quite new to optimisation, so any advice is appreciated. I am currently trying to solve a problem as follows: Given a pool of people, we want to create n teams such to find the optimal ...
6
votes
1answer
195 views

Does the weighted sum approach find all pareto-optimal solutions in MILP

I use the weighted sum approach for a multiobjective optimization problem that is formulated as a MILP. This means that the objective function is linear. I read quite often that the weighted sum ...
6
votes
1answer
214 views

Convert summation of min functions into linear constraints for optimization

I have the following optimization problem: $$ \mbox{maximize } j^{*} \mbox{ subject to:} \sum_{j^{*}\leq j\leq J} \min({\bf A}_j,{\bf B}_j) \geq \lambda, \lambda \in \mathbb{R} \mbox{ and } {\bf A}_j,{...
6
votes
1answer
279 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). ...
6
votes
1answer
154 views

Dual of a model to obtain reduced costs

I have the following model which I am going to solve with column generation. Basically it is a VRP problem with pick-up and delivery in which couriers are assigned to a set of shifts based on their ...
6
votes
0answers
120 views

Water quality component optimization

I have an optimization problem that I'm attempting to tackle. As you can see in the image below, there's a graph network through which water flows. I've drawn out the problem in the image to explain ...
6
votes
0answers
50 views

What are useful plots/statistics/metrics when analyzing the solution sensitivity in multi-objective optimization?

Consider an optimization problem with $n>3$ objectives. For handling this there exists often two approaches: a) some weighting of the objectives, b) fix an order of objectives and then optimize ...
6
votes
0answers
82 views

Benefits of removing slack variables during presolve

I was reading Tobias Achterberg's thesis, and on page 138 he mentions the following presolving technique for linear equations (I'm slightly paraphrasing Example 10.2): Consider the equation $4x_1+...
6
votes
0answers
72 views

Data Formulation for Mixed-Integer-Programming Models

Until now, I have used the Gurobi, CPLEX and OR-Tools (GCO) interface to formulate mixed-integer-programming models. Recently, I have discovered MiniZinc and want to utilize it to formulate big ...
6
votes
0answers
317 views

Cplex is stuck after root node relaxation solution

I am solving an MIP through Benders decomposition (coded both generic and legacy callback versions), by employing Java with Cplex 12.9. For some of the instances, Cplex is stuck for two hours (time ...
5
votes
3answers
451 views

How can I represent null or dashes in a cost matrix or incidence matrix in CPLEX?

In the image below, the cost matrix of customers and supplier has several dashes which indicates the impossibility of certain suppliers with certain customers. How can I represent these dashes in ...
5
votes
3answers
2k views

How to represent an integer variable via binary variables?

Suppose we have a model with $N$ integer variables, i.e. $x \in \mathbb{Z}^{N}$ with $L \leq x \leq U$. How can we represent the integer variables via binary variables? Or in other words: how can we ...
5
votes
3answers
1k views

How does a solver generally know whether a solution is optimal?

I was wondering how does the solver for a MILP determine whether a solution is optimal. I am having a hard time to believe that the solver actually tries all solutions, since in some cases I have over ...
5
votes
4answers
448 views

How to visit a subset of network nodes in a single trip?

I have a connected network where I want to visit a set of destinations which may require visiting intermediate nodes as well because there may be no direct edge between source and destination nodes. I ...

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