Questions tagged [mixed-integer-programming]

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

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2
votes
1answer
116 views

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 ...
3
votes
0answers
48 views

Linearisation using SOS2

I am trying to linearise the following expresssion. $C(k) = B(k) e^{-d(k)}, B(k) \ge 0 , d(k) \ge 0 $ I am trying to do this by using SOS2 sets. I set $X(k) = e^{-d(k)}$ and I get $C(k) = B(k) X(...
6
votes
2answers
165 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 ...
11
votes
2answers
672 views

Can presolve reductions change the value of the linear programming relaxation?

For integer programs solvers (like Gurobi, Cplex, ...) report the value of the linear programming relaxation for integer programs, i.e. ...
25
votes
4answers
685 views

Stochastic programming MIP solvers

I am aware that Benders Decomposition is readily available in CPLEX and in SCIP; but are there any (free) solvers that provide off the shelf stochastic programming MIP algorithms or a nice to work ...
22
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9answers
488 views

Reference for column generation applications

When talking about column generation algorithms, the main example is the cutting stock problem. I'm aware that variations of vehicle routing problem (VRP) can be solved using a column generation ...
1
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0answers
71 views

Ways to improve lower bounds while solving MIPs

What are the ways to improve lower bounds while solving a minimization problem (MILP)?
1
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2answers
305 views

Pulp: slack variable to identify & measure extent infeasible in supply problem

I am currently modelling a supply problem that attempts to optimise a rail schedule which moves products from a production plant, to a warehouse to satisfy sales. The model is working fine (thanks in ...
3
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0answers
52 views

What are the efficient ways to model this scheduling problem and ways to improve running time

There are $N - \{1 \ldots N\}$ jobs, each with processing time $p_j$, to be scheduled on $M - \{1 \ldots M\}$ machines over span of $D - \{1 \ldots D\}$ days and while working $T - \{1 \ldots E \ldots ...
3
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1answer
104 views

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 ...
5
votes
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 ...
5
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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 ...
8
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3answers
1k views

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 ...
3
votes
1answer
191 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 \ \& \ ...
3
votes
1answer
130 views

Modeling the sum of binary variables

Suppose $x_{1},x_{2}, \cdots ,x_{n}$ are binaries. I would like to model the following: IF $x_{1} + x_{2}+ \cdots +x_{n} \ge 2$ THEN $x_{1} + x_{2} = 2$ IF $x_{1} + x_{2}+ \cdots +x_{n} \ge 3$ ...
1
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3answers
212 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{...
5
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3answers
206 views

Problem with binary decision variable constraints in VRP

I would like to create non-linear violation costs in my VRP. I already created my whole VRP with time windows in which I have these decision variable: ...
1
vote
1answer
36 views

Enforcing discontinuous range for a variable

I want to enforce the following ranges for a set of variables, where the ranges for each variable are discontinuous. For example, I have two sets of ranges for each variable, \begin{align}\sf LB_1 &...
2
votes
1answer
168 views

Does this problem fall into any common problem definition…Knapsack maybe?

I am struggling to find a representative problem formulation for this optimization challenge. I have implemented a MILP in Matlab, but the run time is taking more then a day. My goal is to see if it ...
1
vote
1answer
58 views

Derived variables of when a decision variable appears?

I am dealing with a multi-travelling passenger problem. $x_{i,j}$ is a binary variable that allocate a passenger $i$ to a vehicle $j$, every vehicle can carry only $n_{pv}$ passenger where $i \in \{1,...
21
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3answers
2k views

Are valid inequalities worth the effort given modern solvers?

In Laurence Wolsey's Integer Programming[1], 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 ...
4
votes
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 ...
2
votes
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: ...
4
votes
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} ...
7
votes
2answers
118 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{...
17
votes
1answer
1k views

The difference between max-min and min-max

I am solving two-stage optimization problems in the form of $$\max_{x \in X}\min_{y \in Y} f(x,y),$$ where $f(x,y)$ is the solution of a mixed integer linear program (MIP). As the constraints of the ...
12
votes
2answers
328 views

“Out of the box” Mixed Integer Programming Heuristics

I currently have a complex Mixed Integer Program (a sort of vehicle routing problem variant, with multiple vehicle types and without assigned pickup -> delivery routes, among other complications) ...
4
votes
2answers
294 views

What are the most useful evaluation metrics when comparing the performance of different model formulations

Suppose you are writing a paper about a certain new problem class. You have certain problem instances of different size (real-world as well as random) given. You developed different integer ...
5
votes
3answers
450 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 ...
7
votes
2answers
201 views

Is the iteration-limited Simplex dual solution of a MIP node useful?

Idea Sometimes I encounter problems where Simplex spends many iterations for final convergence to the optimal objective value. Let's suppose, this happens when solving branch and bound-tree nodes as ...
7
votes
2answers
260 views

Do Benders cuts exclude current solutions?

I am wondering if optimality cuts in Benders algorithm exclude the possibility to have the same solutions and as a result, have the same optimality cut? I don't know why it is not possible to have the ...
5
votes
1answer
69 views

Help with a MILP formulation for service scheduling

I'm trying to formulate a MILP for scheduling service jobs for multiple devices. Let's assume that each device $i$ has a life $\ell_i$ and that I have $n$ total service visits to allocate at times $\{...
4
votes
3answers
179 views

Formulate relationship between four binary variables

I have four binary variables $x_{h}$, $x_{h'}$, $y_h$ and $y_{h'}$. I need to have the following relationships satisfied between the variables: 1- If $y_h = 1$ and $y_{h'} = 1$, then exactly one of $...
3
votes
2answers
301 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 ...
5
votes
1answer
424 views

Where can I find resources to learn mathematical modelling for real life operation research problems like combinatorial optimization?

I find it hard to form math models for real life operations research problems, how can I learn this? Any books, tutorials available?
2
votes
1answer
132 views

How to formulate logical or constraint in cplex?

How to write the following constraint in cplex? the cost for full truck load is $700 and the cost of fractional truck load is \$800 *fraction up to $700
4
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0answers
69 views

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 ...
8
votes
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 ...
6
votes
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). ...
5
votes
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 ...
5
votes
1answer
88 views

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.
3
votes
1answer
84 views

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 ...
10
votes
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 ...
12
votes
1answer
98 views

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 $G(x)$ ...
5
votes
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 ...
3
votes
0answers
61 views

Integer variable optimization - decreasing execution time

I have a variable declared as follows in AMPL: ...
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 ...
14
votes
2answers
1k views

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,...
-4
votes
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 ...

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