# Questions tagged [integer-programming]

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

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### How to increase the lower bound while solving a MILP model?

I have developed an IP model for a combinatorial problem. The model is faster than the other models in the literature. However, for some instances, the model reaches the optimum solution (which I know ...
• 543
72 views

### PULP: Optimization Assignment of Bicycle production per month

Q1. If bicycles of types A and H are produced, then bicycles of type C can be produced with 20% shorter working hours, while selling profit of bicycles type H can be 20% higher. Q2: If bicycles of ...
1 vote
71 views

### Lexicographic objective to maximize the x-th highest value

Given the following (stylized) IP: \begin{align} \mbox{minimize }& \sum_i c_ix_i&\\ \mbox{s.t. }&\sum_i x_i \leq Q & \\ & f(x_i) \geq L & \forall i \\ & 0 \...
• 3,461
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### Optimizing calls to a separation problem in branch and cut

I have a MIP in which I am able to generate cuts at intermediate relaxation solutions using the context class. These cuts are derived from a separation problem. However, after adding them, the code ...
• 21
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### Restrict the number of non-zero variables to any constant in MILP

I am designing an MILP in which given a set $[n]$ of $n$ agents, we create for each $i \in [n]$ a real variable $x_i$. The variables $x_i$ are between 0 and 1 ($0 \leq x_i < 1$). I would like to ...
180 views

### Unable to find good solution for CVRP with Time Window

I am trying to solve Capacitated VRP with Time Window for 50 demand points. I was trying to optimize this in Gurobi Solver but it doesn't give good answer like optimality gap is 30% or more. Any idea ...
117 views

### Branch & bound: why does the lower bound increase?

For a minimization problem, when running a branch & bound algorithm, I understand that: Every integer feasible solution provides an upper bound on the optimal objective value of the original ...
1 vote
86 views

### How to track the first timestep at which a binary variable becomes 1 in an IP? [duplicate]

I have an MIP where I have a binary variable $y_t$ which is set to 1 or 0 and is indexed by time t. It can be set to 1 at multiple timestamps but it is never continuously 1 for more than single ...
• 33
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### 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,...
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 ...
90 views

### Why this ILP and LP are equivalent?

Let's consider a competition with $n$ questions. Each question has a price $p_i$ and a score $v_i$. To advance to the next round of the competition, we need to accumulate a minimum score of $D$. We ...
1 vote
95 views

### A tool for finding integer solutions to linear systems

I have a system of linear equations $A x = 0$, where $A$ is an integer matrix, and I want to find a non-zero solution, if it exists. In that case, a rational solution exists. Multiplying by the common ...
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 &...
1 vote
67 views

### Connections between Bounds in MIPs

we are currently learning about MIP/MILP minimization at university and have become familiar with the branch-and-bound algorithm. Unfortunately, the relationship between upper bound, lower bound and ...
• 19
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### How to write a If then else constraint with continuous variables

I have a problem under investigation which requires if, elseif and else conditions to implement as a constraint in a mixed integer program. Any leads will be appreciated. Thanks a lot. Let $x_t$, $y_t$...
58 views

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 ...
244 views

### Can we add a certain binary row to a matrix which preserves total unimodularity?

Suppose I have a matrix $A\in \{-1, 0, 1\}^{m\times n}$ which is Totally Unimodular (TU), and a vector $b^T \in \{-1, 0, 1\}^{1\times n}$ which has exactly one entry which is $1$ and exactly one entry ...
97 views

### Small number of constraints, but very large coefficients

I'm looking for advice on solving ILP problems with a relatively small number of constraints and variables, but very large coefficients. I have less than 500 variables and constraints, but my ...
70 views

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### Optimization problem with the Harmonic number

I have an optimization problem: \begin{align*} \text{ minimize } \sum_{i=1}^n H(x_i) \\ \text{ subject to } Ax \geq b, x\geq 0, x\in \mathbb{Z}^n \end{align*} where $H(n)$ is the $n$-th Harmonic ...
1 vote
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'...
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1 vote
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. ...
• 111
1 vote
118 views

### Benchmark problems for Benders Decomposition

We are implementing a scheduling model using Benders Decomposition. Does someone know of any existing implementation of Benders or any repositories that contain continuous or integer problems solved ...
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