Questions tagged [big-m]

For questions related to the use of a "big M" (large constant) in a mathematical modeling context, either in the objective function (to initialize the simplex method) or in constraints (to formulate logical constraints, to linearize constraints, and so on).

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Robust way to implement $(x=0) \Rightarrow (y=0)$, with $x$ nonnegative and $y$ binary

I am formulating a MILP in which there is a continuous variable x and a binary variable $y$. In the program formulation there are the following constraints: $Ay\leq x \leq By$ (with $0\leq A\leq B$). ...
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  • 414
3 votes
1 answer
164 views

Disjunctive equality constraints: modelling

I have the following constraint in my model: $$x = 0 \lor \sum_{i=1}^n y_i = 3$$ where $x$ and $y_i$ are all binary variables. How this can be linearized by means of big-M notation? Should I include ...
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  • 33
-2 votes
1 answer
150 views

How to solve this linear programming minimization problem with the BigM method

Given this minimization problem: $$z=100x_1+77x_2+80x_3\\10x_1+7x_2+2x_3 \geq 12\\2x_1+3x_2+4x_3 \geq 3\\x_1+2x_2+x_3 \geq 1\\x_1,x_2,x_3 \geq 0$$ After adding artificial variables: $$z-100x_1-77x_2-...
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4 votes
1 answer
194 views

How do constraints become redundant in a Big-M conjunction?

The following Big-M conjunction appears on page 14:3 of The Path&Cycle Formulation for the Hotspot Problem in Air Traffic Management: \begin{align*} \text{(i)} \quad t_{(g, \; s)} - t_{(f, \; s+...
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1 vote
0 answers
81 views

vehicle routing optimization, Big M method of reformulation of constraints

Please excuse me for the long question, if I dont prrovide this info. my post gets removed! The following optimization problem is called Mixed-Integer Quadratically Constrained Programming (MIQCP) ...
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0 votes
1 answer
421 views

Mutable parameter in Pyomo causes a problem

I am defining a Pyomo model that should have some mutable Big-M parameters whose values should be dynamically assigned (once). However, I am having a problem with a difference equation as described in ...
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  • 1,546
5 votes
3 answers
207 views

Modelling precedence relations

I have two tasks $i$ and $k$ with durations $d_i$ and $d_k$, where $d_i$ and $d_k$ are nonnegative variables. I would like to model that $i$ may precede $k$ or $k$ may precede $i$ and that they may ...
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  • 1,936
1 vote
1 answer
168 views

Non-linear optimization local or global solution

In an NLP, I have a constraint that I would like to formulate in a convex manner preferably without introducing binary variables and/or big M formulations if possible. The actual problem is non-convex ...
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  • 81
2 votes
3 answers
360 views

Linearizing a Max Function in the constraint - not working

I have a minimization function which is in its simplest form looks like below. I am including the index of the variables. ...
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  • 793
7 votes
1 answer
175 views

Formulating two non-negative variables without binary and/or big-M

There are two non-negative integer variables $q$ and $p$, where only one of them can take a positive value. To impose this relation, I write: \begin{align} q &\leq M(1 - y) \tag1 \\ p &\leq M(...
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3 votes
1 answer
143 views

In integer programming what's the difference between using lower upper bound constraints and using a big M constraints?

I've noticed that for integer programming models with binary variables some use upper bound constraints and others use big M constraints in order to have two mutually exclusive choices. I have trouble ...
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1 vote
2 answers
147 views

Modeling the product of two variables

Suppose we have two continuous nonnegative variables $X_{1}$ and $X_{2}$ both bounded by the number $M$ from above. I would like to model the following: If $X_{1} > 0$ then $X_{2} = 0$ If $X_{2} &...
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2 votes
1 answer
375 views

If-Then-Else modeling in MILP using the Big M method

I have trouble finding a solution to the following problem. I have a decision variable $x$. If the value of $x$ is between 0 and a constant $A$, then the binary variable $y_1$ must be equal to 1. If $...
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  • 23
4 votes
2 answers
196 views

Benders subproblem with product of continuous and discrete variables

I am trying to solve the following problem. The decisions in the problem are $x, y, v, $ and $W$, where $x, y$ are binary and $v, W$ are continuous variables. \begin{equation}\label{eq:3} \begin{...
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4 votes
1 answer
169 views

how to apply Big M to model the logic constraint (if-then-else)

I was hoping to get some help in modelling the following logic as an MIP Constraint c_{m,l}^{RC} is binary decision variable. Simplify it:
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4 votes
2 answers
319 views

Formulating the conditional constraint

I want to develop a model extension of capacitated location problem. The variables are a binary $x_i$ and a continuous $Q_i$. The following condition must be satisfied: if $x_i = 0$, $Q_i$ must be ...
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2 votes
1 answer
125 views

Scenario based approach to value-at-risk optimization using mixed-integer programming

For a discrete set of scenarios, minimising value at risk can be formulated as a mixed integer linear programming problem. If each scenario has equal probability then this can be written as \begin{...
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  • 151
1 vote
0 answers
62 views

Linearize max function in a constraint [duplicate]

I have a constraint as follows: $ \sum_i {r_i} \geq \max \{g_j, B_j\} $ where, $r_i$, $g_j$ are variables and $B_j$ is a parameter. How do I linearize the constraint (I suppose using big-M method)?...
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  • 793
2 votes
1 answer
136 views

How to transform these conditional constraints to linear integer ones in a more efficient way?

The conditional constraints A and B can be transformed to a set of linear integer constraints as follows: A) $\text{if} \ x_1=0 \ \text{then} \ d_1=1 \ \text{else} \ d_1= 0\\ x_1\in {\rm I\!R}^{\geq ...
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  • 284
3 votes
1 answer
307 views

How to fomulate the following conditional constraint in MILP?

How can I formulate the following conditional constraint to a linear constraint using indicator variables? Please note that all variables are continuous and $c \ge 0$ $\text{1: if} \ c=0 \ \& \ ...
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  • 284
0 votes
3 answers
294 views

Using big M values for a constraint

I want to enforce $x_{i,j}=x_{k,j}\implies z_i \neq z_k$ where $k = i-1$ so I used \begin{align}z_k + 1 - (x_{i,j} - x_{k,j})) \leq z_i \leq z_k - 1 - (x_{i,j} - x_{k,j})\quad\text{for each $j$}\end{...
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  • 379
6 votes
2 answers
246 views

How to handle bigM in sub-problem of benders decomposition?

Suppose you want to solve a MIP with Benders decomposition and the binary variables ($y_i$) are fixed in the master problem but these variables are used in the sub-problem with bigM like $x_{ij} \le M....
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  • 2,105
3 votes
2 answers
385 views

Mocking up conditional statements in LP

I would like to know how if condition statements in linear programming can be reformulated using indicator constraints, and hence solved as a mixed integer linear program. Specifically: 1. Is it ...
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  • 151
5 votes
2 answers
352 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 ...
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  • 53
6 votes
2 answers
3k 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 ...
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  • 1,936
5 votes
1 answer
224 views

Converting if else condition to MIP constraints - validation

I have an if else condition as follows: If $g \ge 0$ then $e=1$, else $e=b$. I formulated MIP constraints using big-M as follows where I am setting $\delta=1$ if $g \ge 0$: \begin{alignat}2g &\...
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  • 793
8 votes
1 answer
471 views

if-else condition for the objective variable using big M notation

Let $0\leq \beta\leq 1$ be an objective variable. The size of $\beta$ is $N\!\times\!N$. Now, how can I impose the following? if $\beta_{i,j}>0$ then $\beta_{j,i}=0$ Big M notation can be ...
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  • 83
11 votes
2 answers
1k views

Linear programming: objective function with "buckets"

I had a linear programming problem with the following objective function $$f(x) = \sum_{j}x_jq_jp_j - \sum_{i}\left(\sum_{j}x_jq_jC_{ij} \right) c_i$$ Where $q, p, C, c$ are known. This problem was ...
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3 votes
1 answer
58 views

defining Mixed integer linear inequalities for a set of variables

The problem is described as follows: considering $n$ variables which are continuous and bounded such that $$L_i \le x_i \le U_i\quad \forall i=1,2,\dots,n.$$ How can i define a set of mixed integer ...
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  • 135
5 votes
2 answers
386 views

How to model If $A \le B$ then $Y = 1$, otherwise $Y = 0$

Somehow I don't get it right. I would like to model the following conditional: If $A\le B$ then $Y=1$ otherwise $Y=0$ where $A, B$ are reals and $Y$ is binary. I can model as follows: $Y \cdot A \le B$...
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  • 1,936
3 votes
2 answers
692 views

Problem with big-M in CPLEX OPL

I have a model for the no-wait flow shop scheduling problem, that utilizes the linear ordering variables, and there is a constraint with big-M. When I implement the model in ...
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  • 2,034
5 votes
1 answer
624 views

How is Big M calculated?

Because of excessive pollution on the Momiss River, the state of Momiss is going to build pollution control stations. Three sites (1, 2, and 3) are under consideration. Momiss is interested in ...
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9 votes
1 answer
279 views

Introducing a big M variable in given equations

While I do understand the general workings of the Big-M-method I am struggling with the following sample exercise, in which the Big-M-method has to be used to find a first feasible solution: \begin{...
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6 votes
1 answer
574 views

Integer programming example clarification

There is an explanation in my book for an integer programming example, which goes like this: A company is considering manufacturing three types of autos: compact, midsize, and large. The resources ...
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10 votes
1 answer
720 views

Linear programming with if-then-else (big-M)

I am trying to formulate the following in linear programming. \begin{cases}\text{if}\,\,a>b\,\,\text{then}\,\,c=a\\\text{else}\,\,c=b.\end{cases} I tried some things with big $M$, like $$a + my &...
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7 votes
1 answer
287 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 ...
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4 votes
0 answers
65 views

If and then constraint for a special case [duplicate]

I have the following constraint: $$f\geq C_1\left(d-z-E_1-E_2\right)+C_2\left(E_1+E_2\right).$$ Here $f, d,z\in \mathbb{R}_{\geq 0}$ and $y\in \{0,1\}$ are variables and $C_1$, $C_2$, $E_1$, and $E_2$...
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  • 1,229
23 votes
2 answers
3k views

Why is it important to choose big-M carefully and what are the consequences of doing it badly?

The question here discusses the two different use of "big-M method", where one of them is the big-M in logical constraints and linearization in (mixed-)integer programming problems (that's what I'm ...
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  • 5,651
19 votes
5 answers
5k views

When to use indicator constraints versus big-M approaches in solving (mixed-)integer programs

Various optimization modeling languages and solvers allow for both indicator constraints (see for example here, here and here) and traditional binary variable and big-M approaches can be used to model ...
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  • 193
16 votes
1 answer
1k views

What is the "big-M" method? And are there two of them?

I’ve seen the "big-$M$ method" referred to in different ways. What is the "big-$M$ method" and why does it seem to mean two different things?
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13 votes
4 answers
325 views

The effect of choosing big M properly

I have a set of linearized constraints that are modelled using big-Ms. Now, it is, of course, common knowledge to make the value of M and small as possible in order to provide tighter LP relaxations ...
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12 votes
4 answers
1k views

Single reference for Mixed Integer Programming formulations to linearize, handle logical constraints and disjunctive constraints, do Big M, etc?

Is there a single crisp and accessible reference which covers how to generate Mixed Integer Programming formulations to linearize products, handle logical constraints and disjunctive constraints, do ...
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