# Questions tagged [mixed-integer-programming]

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

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178 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 ...
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### A discrete location problem

I have a discrete optimization problem in which there are some predetermined nodes in the set A. a vehicle must visit them as the traveling salesman problem. there are some other nodes in the set B ...
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### 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, $a_i, b_j \in \mathbb{R}_+$ ...
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### Issue in solving a large scale MIQP problem

I am solving a large scale MIQP optimisation problem at each step of a model predictive control problem. The problem description is as below. \begin{align} \min_{u} \quad (x_{k}&-x_\text{ref})^{T}...
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### Portfolio optimization with indicator function constraint in CVXPY

I have the following portfolio optimization problem that I want to solve using CVXPY: \begin{align}\min_w&\quad w^\top\Pi\\\text{s.t.}&\quad\sum_{i=1}^nw_i=1\\&\quad w^\top\Sigma w\le\...
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### 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 ...
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### Issues modeling portfolio optimization with rebalancing in gurobipy

I want to solve the following portfolio optimization problem by means of the Python API of Gurobi: \begin{align}\max_w&\quad w^\top\Pi\\\text{s.t.}&\quad w^\top\Sigma w=\sigma^{\rm target}\\&...
37 views

### Interger programming using gray encoding

Could anyone suggest me a tool or library which takes an integer programming problem written in DOCPLEX or CVXPY as input and outputs the equivalent problem using Gray binary encoding? I am happy to ...
131 views

### Formulate a problem as Mixed Linear Programming problem

I need to formulate the following problem as a Mixed Integer Linear Programming problem A farmer needs to establish a 17-year business plan where he will decide when to sell or buy a new truck. The ...
211 views

### Does it make sense to use a MILP solver for a scheduling problem but without the obligation to schedule all the tasks?

I know that MILP solvers are bad with scheduling problems. However, if we are allowed to keep unscheduled some tasks (i.e a solution with 0 scheduled tasks is a feasible solution but we add the ...
338 views

### How can I identify the reason that makes a MILP model hard for solvers such as CPLEX?

I'm solving a MILP model whose native lower bound (via linear relaxation) is very poor. We could provide a lower bound by providing a given value (derived based on the problem itself). I know that ...
744 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 ...
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### distance specific constraint

I have some points with determined coordinates $(a_i,b_i)$. A vehicle can move between these points based on rectangular distance. In more detail, we consider that the path between points is an ...
253 views

### How to model y = floor(x)?

I went through the answers to this question: Modeling floor function exactly, but I still do not get how to model y = floor(x). Is that question answered and I just ...
91 views

### How this problem can be defined as MultiObjective optimisation

I need to optimize the end-to-end latency of a multi-component application. Assuming that the application has 10 components, component 1-5 is hosted by device 1, and device 2 is hosting the other 5 ...
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### Benders Decomposition Problem

$$r_{m_h,s}(n)=\frac B{m_hb_\ell s}\log_2(1+\gamma_{m_h,s}(n))$$ How to deal with multiple subproblems in Benders decomposition when the original objective function is in product form of an integer ...
108 views

### How to write distance specific constraint?

Suppose there are a few plants (p) and few customers (c). The supply (Sp), distance (Dpc), cost (COSTpc) and demand (DEMANDc) between them is given. I have a constraint that 90% of total demand of all ...
227 views

### How to linearize the Min function while letting the binary variable to be fixed for x1==x2 as well?

As discussed here, the min function, i.e $X = \min\{x_1,x_2\}$, can be linearized as follows: \begin{align} X & \le x_1 \\ X & \le x_2 \\ X & \ge x_1 - ...
118 views

### Is my formulation correct and how to formulate this IF-THEN constraint?

I have system with $N_U$ users and $N_T$ transmitters. Multiple transmitters can transmit to a single users and one transmitter can transmit to many users, i.e., two sets of transmitters serving two ...
98 views

### Formulating these logical constraint in an ILP

I have these two constraints : $z \leq My$ $t \leq M'y$ where $z$ and $t$ are two integer variables $z, t\geq 0$, $y$ is a binary variable, and $M$, $M'$ are two big numbers. So basically these ...
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### How to express this logical constraint for an ILP?

I am trying to write an ILP for a problem but I have this logical constraint and I'm stuck. In my model I have: two binary variables: $x$ and $y$ One Integer variable: $z$ The logical constraint I am ...
119 views

### A Question on A tutorial on column generation and branch-and-price for vehicle routing problems by Dominique Feillet

I am reading A tutorial on column generation and branch-and-price for vehicle routing problems by Dominique Feillet to learn the column generation approach, but I have a problem. in section 3.3 ...
213 views

### Column Generation algorithm for vehicle routing problem

I want to solve a VRP with a column generation algorithm. The objective of the problem is makespan minimization. In more detail, I want to minimize the arrival time of the last vehicle in the depot. I ...
217 views

### Best way to solve an allocation problem

I have the following problem: I have products with different attributes (price, weight, category) and I have a list of clients. Every client has an "affinity value" with every product, the ...
173 views

### Optimized Production Planning & Scheduling

I want to create an optimized schedule and production to meet demand of four different buyers. I am starting with Pulp but not able to account for continuous production variable and discrete dispatch ...
82 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 ...
178 views

### Mixed integer programming vs Constraint programming

I see that certain studies are comparing MIP and constraint programming (CP) performances. And generally, they claim that CP outperforms MIP. But I believe that it is not completely true. Because ...
110 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{...
158 views

### Column generation when intractable variables appear in the objective function

Is it possible to implement a column generation for a problem that the variables in the "complicating" constraint appear in the objective function? Suppose the MIP is: \begin{align} z = \...
67 views

### Finding Optimal Route using different Paths

I have a list of paths e.g. path 1 takes you from point A to B. A person needs to complete 5 of such paths. Route1 = path1 + ...
111 views

### Is a convex or MILP (without big-M) formulation possible for this problem

Assume we are given a directed acyclic graph (DAG) $G(V, A)$, where $|V| = n, |A| = m$, and the graph contains a source node $\mathbf{s}$ (i.e. every node in $V \backslash \mathbf{s}$ is connected by ...
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### Output of binary variable greater than one

I am working on a shipping demurrage problem that uses a binary variable to denote the date a specific vessel can be loaded (I have been kindly helped by Wesley on OR before with this). I am confident ...
138 views

### Limit number of switches in employee scheduling problem

Here is a scheduling problem I need to solve. Given the demand for 2 positions in 1 week with 3 shifts per position, I need to allocate the employees accordingly with some extra operational ...
100 views

### Pyomo: Help setting up my model

I have been working with a model for the past couple of weeks, but I am not sure what am I doing wrong as every time Pyomo returns that is infeasible or ...