Skip to main content

Questions tagged [integer-programming]

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

52 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
15 votes
1 answer
308 views

Integrality gap in bilevel binary linear programming problem

I have a bilevel max-min optimization problem over binary variables, with constraints expressed using linear inequalities. The inner (minimization) problem is $$ \begin{alignat}2 \min\limits_x&\...
abebebebahabe's user avatar
6 votes
0 answers
132 views

Characterization for total dual integrality

A problem I study reduces to whether the polyhedron $P=\{\mathbf{x}\mid A\mathbf{x}=\mathbf{1}, \mathbf{x}\geq0\}$ is integral ($A$ is a matrix with coefficients in $\{0,1\}$). I know that the ...
Surpass2019's user avatar
6 votes
0 answers
86 views

How to communicate number of integer combinations to a user

I'm working on a nifty little feature for our next release, i.e., to print the number of possible integer combinations left during branch and bound. This is really handy for the user because they ...
Nikos Kazazakis's user avatar
6 votes
0 answers
124 views

Help in solving resource allocation optimization problem

I've been pondering on this question for some work optimization, and I need some help in being directed to the right direction. I have multiple customers that require an amount of $X$, $Y$ and $Z$ ...
Sylicas's user avatar
  • 61
5 votes
0 answers
89 views

Bounding the size of the dual solution

Given an primal optimization with bounded feasible set: $\max \{cx: Ax \leq b\}$. The feasible region of the dual is $D = \{y:y^\top A = c^\top, y \geq 0\}$. If the primal feasbile region is a ...
user3680510's user avatar
  • 3,655
5 votes
0 answers
148 views

When is there at least an integral point in a polyhedron?

This problem comes from a problem of economics. Let $x\in [0,1]^n$. $\{x_1,x_2,\ldots,x_n\}$ is partitioned into ${S_1, S_2,\ldots,S_k}$ such that $\sum_{x_i\in S_j}x_i\leq 1$ for each $1\leq j\leq k$....
Surpass2019's user avatar
5 votes
0 answers
180 views

Are there any good models for min-max vehicle routing problem?

I am trying to model a min-max VRP problem with multiple delivery vehicles and I have come up with a model using branch and cut but I do not think it is strong enough as it takes lot of time to ...
Morpheus's user avatar
  • 253
4 votes
0 answers
71 views

Continue on "Is there a known MILP to schedule routes after routes are made"

I have made some progress on my previous question (Is there a known MILP to schedule routes after routes are made). I have derived the sets of the problem, which are: 1) Itineraries of vehicle: $i \in ...
dimboukosis's user avatar
3 votes
0 answers
124 views

Continuous optimization with a Euclidean TSP objective

I am trying to solve a problem of the form $$\min_{x_1,\dots,x_n} f(x_1,\dots,x_n)$$ subject to a constraint that $\mathrm{length}(\mathrm{TSP}(x_1,\dots,x_n))\leq c$, where $x_1,\dots,x_n$ are all ...
Tom Solberg's user avatar
3 votes
0 answers
121 views

Number of Subtour Elimination Constraints for ATSP (DFJ Formulation)

In the DFJ formulation of the symmetric TSP, the subtour elimination constraints are typically written as: $$\sum_{\{i,j\} \in E: \ i \in S, j \notin S} x_{ij} \geq 2, \qquad \forall S \subset V, \; 3 ...
user11589's user avatar
3 votes
0 answers
87 views

Minimize total cost to buy unique products over timespan

I am looking for some guidance on how to better state my problem formally and practically (e.g., relevant Python libraries). Let's assume I have a set of $X$ unique products to buy. I have $Y$ days in ...
M2FKXY's user avatar
  • 31
3 votes
0 answers
111 views

Polynomial Time Solution For a Mixed-Integer Linear Programming Specific Case

Consider the following mixed-integer linear programming (MILP): \begin{equation*} \begin{array}{ll@{}ll} \text{maximize} & 1 & \\ \text{subject to}& x_{i} \geq 0, &i=1 ,\dots, m\\ ...
Samuel Bismuth's user avatar
3 votes
0 answers
186 views

How to linearize a max min objective function?

Let us suppose that I have a $\max \min$ objective function that only depends on one set of variables: $\underset{x}\max \underset{y}\min dy$ Associated with the linear set of constraints and right ...
JKHA's user avatar
  • 679
3 votes
0 answers
92 views

Flexible Job Shop with Preemption

I'm trying to solve a flexible job shop problem variant that has precedence constraints on jobs along with a few other issues. We have a MIP formulation and also a simulated annealing algorithm to ...
Robert Hildebrand's user avatar
3 votes
0 answers
113 views

How to find all covers and minimal covers?

Consider a constraint of type $$c_1x_1+c_2x_2+\cdots+c_nx_n\leq C$$ with $x_i$ binary. We call a cover a subset of the $n$ indices such that the sum of the corresponding coefficients is higher than ...
k88074's user avatar
  • 1,691
3 votes
0 answers
72 views

Theoretical aspect of using extended formulation

If I can show a polyhedron Y is an extended formulation of polyhedron X and every extreme point in Y is integral, does that automatically imply the projection of Y onto the variable space of X gives ...
Octavia's user avatar
  • 31
2 votes
0 answers
34 views

Maximizing value of nodes visited in fixed time

Consider the following three problems. The first is intended to be a simplification of the second that might be amenable to solution methods the second is not amenable to. First problem: Assume we ...
Aldo Leopold's user avatar
2 votes
1 answer
112 views

Rearrange 'x' piles of items into eight possible locations/bins based on item color and length

I have an optimization problem that I believe is a variant of the 'bin packing problem with precedence', but I'm unsure of if that is the correct paradigm to work with and I'm not having a huge amount ...
Jon Knott's user avatar
2 votes
0 answers
77 views

When do two integer linear programs yield the same solution?

This question is cross-posted from math stack exchange An illustrative example Consider an integer linear program $\min -2x_1 + x_2$ subject to $x_1 - x_2 \leq 3$ and $x_1 + x_2 \leq 10$ and integer $...
fool's user avatar
  • 121
2 votes
0 answers
38 views

Recoverable Robustness for an optimization problem

I am relatively new to the concept of recoverable robustness. I am researching the robust version of an optimization problem. I currently have methods to address the problem with perfect knowledge. ...
Pia MiA's user avatar
  • 390
2 votes
0 answers
68 views

Multi Commodity VRP Time Windows Paper

I have been wondering is there any paper that who discuss multi commodity with four index/indices like this formulation below: $$ \begin{gathered} \sum_{k} \sum_{c} F_{j, j, c, k}=1 \quad \forall j \\ ...
overboxed's user avatar
  • 593
2 votes
0 answers
77 views

Good encoding for grid layout problem

How would you encode the problem given by https://oeis.org/A337663 with an off-the-shelf solver? You need to lay out $n$ ones and $2, 3, ..., m$ on an infinite grid, for $m$ as large as possible. For ...
user1502040's user avatar
2 votes
0 answers
60 views

How to construct a restriction for a blending problem that has 2 inputs that are blended, and as result we get 2 outputs?

THE PROBLEM A refinery has 10 million barrels of type A crude and 6 million barrels of Type B crude oil. The refinery has 3 plants to produce Gasoline (it makes a profit of 2 USD / barrel) and Naphtha ...
ergch24's user avatar
  • 67
2 votes
0 answers
277 views

About combinatorial Benders Cuts

I am solving an OR scheduling problem where I assign the patient to (day,OR) tuple in Master Problem. Once the assignment is made, a subproblem can be solved for each (day,OR) tuple independently ...
Amogh Bhosekar's user avatar
2 votes
0 answers
82 views

What if anything do linear relaxations of "nearby" MILP nodes tell us about other MILP nodes

Assume we are given MILP where $y \in (\mathbb{R}^+)^n$, $x_1, x_2 \in \{0, 1\}$ are the integer variables. It is obvious that this problem when solved via branch and bound has a 2 deep b&b-tree. ...
worldsmithhelper's user avatar
2 votes
0 answers
123 views

Indicator function for integer variable with inequality constraint

I have $n$ integer variables $\vec{x}$ with the following integer programming problem. $$ COST = \sum^{n-1}_{i = 0} a_i x_i + \sum^{n-1}_{j=0} b_j I(x_j > 0) $$ Here, $a_i, b_j \in \mathbb{R}_+$ ...
Omar Shehab's user avatar
2 votes
0 answers
122 views

Condition for an integer program and its linear relaxation to have the same value

Let $A$ be a $(0,1)$-matrix where no row or column is a zero vector, and consider the following optimization programs \begin{align}(1):\min&\quad y\cdot1\\\text{s.t.}&\quad yA\ge w\\&\quad ...
James Alex's user avatar
2 votes
0 answers
176 views

Operation hours optimization for circular schedule

Here is my problem. A store has X = 15 electical devices with the ability to work non-stop, fully charged, up to 8 hours. Their battery charge lasts 2 hours and the operating hours of the store differ ...
Psyndrom Ventura's user avatar
1 vote
0 answers
108 views

What is the suitable optimization method for this case?

What is the best optimization method to solve a large-scale problem (about 300 thousand variables)? The problem is nonlinear, nonconvex, involves only binary variables, and is unconstrained. The ...
Hussein Sharadga's user avatar
1 vote
0 answers
110 views

Applications of Knapsack and Cutting Stock in Pure Math

I'm giving a seminar to PhD students in pure math, and one of the things I'd like to do is show that more applied optimization can also make its way into pure Mathematics. As for classical problems, I'...
J. Dionisio's user avatar
1 vote
1 answer
123 views

Maximizing sum of probabilities with variable distributions

Suppose $\\{X_i\\}$ are binary decision variables and $\\{A_j\\}$ are Skellam random variables with $(\mu_1, \mu_2) = (\sum_i b_{i} X_i, c_j)$. Here, $b_i, c_j \in \mathbb{R}^{\geq 0}$ are constants. ...
Jacob's user avatar
  • 111
1 vote
0 answers
264 views

Deriving a valid inequality

Given a set of facilities $I$ and days $J$, each facility $i \in I$ has a capacity of $C_i$, and a set of days $J$ where in each day $j \in J$ there's a total demand of $q_j$ that can be satisfied by ...
CHE's user avatar
  • 113
1 vote
0 answers
88 views

Optimize cherry picking runs

I am trying to optimize a cherry picking procedure on 96-well microplates. The plates are 12X8 (12 columns, 8 rows). We pass a command file that has many lines like this to a robot: ...
Ryan's user avatar
  • 111
1 vote
1 answer
137 views

Constraints to avoid disjointed solutions in a MIP

Given an directed graph $G= (N,E)$, where $N$ is the set of nodes and $E$ is the set of all edges, each associated with a direction. $G$ is a connected graph but not necessarily a complete graph. A ...
CHE's user avatar
  • 113
1 vote
0 answers
139 views

Assembly Line Balancing---(Minimizing Cycle Time)

A factory has a four workstations assembly line which produces a bluetooth speaker. This production requires twelve assembly operations, respecting some precedence constraints. Table 4 indicates the ...
user11048's user avatar
1 vote
0 answers
59 views

How to avoid complementarity constraints in continuous nonlinear program?

In my two-stage continuous NLP problem, I have a constraint in second stage: $X_{g,k}$ = $X_{g,0} + a_{g} d_{g} $, if $X_{g,k} \in [X_g^u,X_g^l]$ $X_{g,k} = X_g^u$, if $X_{g,k} \geq X_g^u$ $X_{g,k} ...
Ghulam Mohy-ud-din's user avatar
1 vote
0 answers
93 views

Bipartite matching

If I have two matrix $$A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} $$ and $$B = \begin{bmatrix} 3 & 4 \\ 5 & 3 \end{bmatrix} $$ We have to make a matching between $A$ and $B$ ...
Ishaan's user avatar
  • 139
1 vote
0 answers
58 views

Product allocation to vendor according to their demand

Company X has 3 types of products and due to the limited availability of raw materials, the production of products are also limited. They have partnered with Store A for them to sell their products. ...
sonyeoja's user avatar
1 vote
0 answers
60 views

Building blocks for unimodular matrices

I read Chapter 19.4 of Schrijver(1986) and get to know that every totally unimodular matrix can be produced by taking operations on network matrices and two certain matrices. I find that some paper ...
Surpass2019's user avatar
1 vote
0 answers
137 views

How can I set the solution of gurobi to be a multiple of 10 instead of all integers?

For example, the solution for gurobi has two solutions, as follows: [10,20,50,70] [55,79,30,80] I only want to output solutions that contain only multiples of 10. The sample example as follow: ...
Zying's user avatar
  • 57
1 vote
0 answers
48 views

Unifying constraint matrices in sparse situations

$\DeclareMathOperator\Set{Set}$ Let $Set=\{x\in\mathbb Z^{n}:\exists y\in\mathbb Z^m\text{ satisfying } A[x,y]'\leq b\}$ where $A$ has $r=km$ rows and $k=O(1)$. I am trying to write $$ Set=\{x\in\...
User2021's user avatar
1 vote
1 answer
113 views

ILP program to find a centrosymmetric Hadamard matrix

A question in mathoverflow asks if there exists a centrosymmetric Hadamard matrix of order 36. An $n \times n$ matrix $A = (a_{i,j})$ is centrosymmetric if: $$a_{i,j} = a_{n-i+1, n-j+1}, \space i=1,\...
Fabius Wiesner's user avatar
0 votes
0 answers
33 views

Shipment Allocation using Pulp

I am trying to create a model where each shipment must be allocated to a route to minimize the freight cost. I am a complete newbie at LP Optimization. Explanation of dataset: Every destination (D1,...
Jayit Ghosh's user avatar
0 votes
1 answer
63 views

Minimize Expenses For Workers

My goal is to minimize the labor expenses. Say we have 3 types of workers: $x_1$ = Permanent Driver, rate = 693.875/day $x_2$ = Reliever Drivers rate = 435/day $x_3$ = Crews rate = 400/day There are 6 ...
Siazam's user avatar
  • 1
0 votes
0 answers
71 views

Improving the lower bound

Good afternoon. I have a very difficult to a MIP model. I have tried several different strategies to reduce the gap. I am using Gurobi and in this case, I already have an incumbent solution. I've set &...
Angelo Aliano Filho's user avatar
0 votes
0 answers
58 views

Shopping Basket Deal Optimization

I am looking for guidance on a solution to the problem of picking the best special offers that can be applied to a given basket of items. In the system, a special offer has N collections of qualifying ...
Robert Snipe's user avatar
0 votes
0 answers
34 views

Stationarity conditions for IPs

Let's consider the following (MQ)IP: $\min x^T Q x$ s.t. $g(x) \geqslant 0$ $x_i \in \mathbb{Z}$ $i \in I$ By ignoring the integrality constraints we end up with the QP: $\min x^T Q x$ s.t. $g(x) \...
Matheus Diógenes Andrade's user avatar
0 votes
0 answers
62 views

How do I solve this non-linear optimisation problem based on simulations?

I have an optimisation problem that is essentially a knapsack problem with a non-linear objective. I have an input dataframe that contains a row for each item, each item has columns defining its mean ...
will's user avatar
  • 31
0 votes
0 answers
61 views

Gamma uncertainty in the RHS of a constraint

I am new to the concept of robust optimization.I am trying to formulate the robust variation of a Binary Integer Program. Suppose we have a constraint of the form $\sum{x_{i,j}} \geq b_j$ for $ i \in ...
Pia MiA's user avatar
  • 390
0 votes
0 answers
65 views

Decision whether to serve the current table now or skip it

Can the following problem be formulated as a LP/IP problem, where an agent (waiter) that has just arrived at a certain table has to decide whether to visit that table now, or skip the current table ...
gbullins's user avatar