Questions tagged [mixed-integer-programming]

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

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1answer
64 views

How to solve this clustering problem with heuristic or meta-heuristic approach?

I have clustering problem with servers and users. This is different to the one posted in https://math.stackexchange.com/questions/4088441/what-will-be-an-efficient-joint-clustering-solution-to-this-...
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1answer
79 views

Is it known that any boolean function is equivalent, in some sense, to a system of linear inequalities?

I cannot believe that below is a "fresh result" in mathematical programming as my rather intensive literature search did not bring references to any similar assertions. I will appreciate any ...
4
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4answers
387 views

How to visit a subset of network nodes in a single trip?

I have a connected network where I want to visit a set of destinations which may require visiting intermediate nodes as well because there may be no direct edge between source and destination nodes. I ...
2
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2answers
74 views

MILP modelling on minimal disturbance of right-hand-side to make a linear system infeasible

I try to model the following problem: given $z\in\{0,1\}^m$ and a linear system $Ax\le b(z), x\in\Bbb R^n, A\in\Bbb R^{d\times n}, b(z)\in\Bbb R^{d}$, where $b(z)$ means that some entries of $b$ are ...
2
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1answer
82 views

Electricity market clearing price using fixed-MIP formulation?

Dual information of electricity markets clearing problem is required to calculate the marginal clearing price. As most electricity market problems are based on MIP (and dual information of MIP is not ...
5
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1answer
86 views

How can I strengthen a family of constraints in the presence of a clique constraint?

Suppose $x_i$ are binary variables, $y_j$ are arbitrary variables, $a_j$ and $b$ are constants, and I have the following linear constraints: \begin{align} x_i + \sum_j a_j y_j &\le b &&\...
3
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1answer
51 views

Name for subclass of ILP without any inequality constraints (including constraints on x)

In "Myths and Counterexamples of Mathematical Programming" myth "IP Myth 21" says: The problem of finding $x\in \mathbb{Z}$ such that $Ax=b$, where $A\in\mathbb{Z}^{m\times n}$ ...
2
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1answer
91 views

How to deal with log0 in optimization problem

I am adding some constraints to my model described in my previous post, where a discontinuous piecewise-quadratic functions is the objective to be minimized in cvx. Here I have an additional terms, ...
3
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2answers
142 views

How to reformulate a discontinuous piecewise-quadratic functions

I am trying to develop a model, solving an optimization problem which has the following objective function: variable p(i); minimize sum(cost) subject to p>=0 ...
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1answer
38 views

How to interpret no-overlap constraints with rotation as a mixed integer programming

Suppose, we want to locate some given facilities $\left \{ (i,j) \ |\ (i,j) \in \text[{1,\cdots, N}]\right \}$ in a specific area. Each facility has a predefined dimension with a length $l_{i}$ and ...
3
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1answer
35 views

Obtaining linear relaxation objective value from MILP model coded in Pyomo

I would like to seek some advice on modeling the following: I am currently using Pyomo to generate my MILP model in Pyomo. It seems that it is not possible to cast the integer and binary variables to ...
5
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1answer
150 views

How to treat a system of bilinear constraints

A model contains constraints of the following form $R(k) \leq X(k) G(k)$ where $X(k)$ binary and $G(k)$, $R(k)$ non-negative variables. The index $k$ runs from $1$ to $50$. I linearise the equations ...
2
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1answer
51 views

Constraints that set values to binary variables depending on other binaries

I am trying to write a mathematical problem that involves some conditions based on binary variables. More specifically, I have a set of three binary variables $d_1$, $d_2$, $d_3$ and depending on ...
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6answers
876 views

Where can I find documentation on good practices for efficient formulations of a problem?

I am sort of new to mathematical optimization and have to build some fairly complicated models for my thesis. I was wondering where I could find literature to help me develop more efficient versions ...
2
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2answers
81 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 ...
4
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1answer
174 views

Assignment problem with batching costs

I am studying an assignment problem with batching costs, and I would like to know if there is a standard name or algorithm for this problem. I know this problem can be formulated as mixed-integer ...
2
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1answer
59 views

How does the RCPSP's precedence constraint work?

In [1] the authors define the RCPSP (resource-constrained project scheduling problem) as follows: minimize $$ \sum_{t} t x_{n t} $$ subject to $$ \begin{array}{c} \sum_{t} x_{j t}=1, \quad j \in J, \\ ...
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1answer
124 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 ...
3
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1answer
157 views

Linearizing a quadratic function with more variables or not in Gurobi?

Suppose I want to set the price $0 \le p_t \le p_{max} $ and based on the price, demand is determined $D_t(p_t)=a-bp_t$. Inventory level at each time is denoted by $I_t$ and it is determined by $I_t= ...
2
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3answers
120 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. ...
4
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2answers
191 views

Mixed-integer optimization with bilinear constraint

So I have an optimization problem of the following form: \begin{aligned} \max_{x,y} \quad & \sum_i x_i \\ \text{s.t.} \quad & \sum_i x_iy_i \leq a \\ \quad & x_{\min} \leq x \leq x_{\max} ...
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1answer
226 views

How to mathematically formulate the optimization problem?

I have a system with $S$ service points. There are also $U$ users in the system. We have $$U>S>G$$ One group can have maximum $M$ service points, but there is no restrictions on the number of ...
3
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1answer
61 views

Clustering problem involving multidepots and customers requiring commodities located exclusively in an specific depot

I'm trying to solve a clustering problem that's similiar to a VRP Pickup and Delivery problem with multiple depots and customers. Each customer demands a commodity that is exclusively found on one ...
2
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1answer
66 views

Impose binary constraint on integer matrix with CVXPY

So I have the following matrix: \begin{equation} P_{i,j}= \begin{bmatrix} x_0 & x_1 & x_2 \\ y_0 & y_1 & y_2 \\ z_0 & z_1 & z_2 \end{bmatrix} \end{equation} where ...
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0answers
65 views

Strange Result from GurobiPy

I am trying to understand why GurobiPy gives me a strange result for a simple linear programming model coded as below? Why is the optimality gap is 0%? Please let me know if you spot any errors in the ...
6
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2answers
157 views

Index of element in MILP vector decision variable that equals 1

Consider a decision variable in a MILP constrained: $$\sum_i p_i = 1$$ $$p_i\ \in \{0, 1\}$$ Obviously one element in $p$ is 1 and all others are 0. How can I set a decision variable to the index i of ...
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1answer
187 views

Linearize x different of y in ILP

I am surprised I couldn't find an already written answer for my question in the internet. I want to linearize $x$ different of $y$ for two nonegative integer decision variables. I am not considering ...
3
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1answer
149 views

Oscillations with (online) mixed-integer optimization problem

I have the following mixed-integer optimization problem: \begin{aligned} \max_{x,y} \quad & \sum_i x_i - \|wx\|_2 \\ \text{s.t.} \quad & \sum_i x_i \leq A \\ \quad & x \leq x_{\max} y \\ ...
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1answer
83 views

Mixed Integer Programming - How to model the dependency of two variables in an objective function

I have two variables $a$ and $b$, in which $a$ is the amount of goods and $b$ is the amount of boxes of the given sizes. So $b$ (box size + number) is dependent on a (goods quantity). If $a$ is ...
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0answers
77 views

Mixed Integer Programming - Model Formulation for A Resource Allocation Problem

There are a number of orders, which needs to be shipped. For each order, there may be 1 to 3 route options. The problem here is to find out the best allocation (combination) of orders among these ...
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1answer
119 views

MILP constrained by the minimum number of satisfied constraints

I have an MILP where we have $$ t_k = \sum_i P_i\cdot C_{ik} : P_i\ \in \{0,1\}, C_{ik} \in I^+ $$ and our model is constrained by the number of times $t_k$ is bigger than a certain value $T_k$. $$ \...
2
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1answer
43 views

How to define a stationary point of the MINLP problem?

As we all know, KKT point and stationary point are well defined when the optimization variables are continuous in the problem. Now, I want to know whether there exist some special points except for ...
4
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3answers
125 views

Optimization formulation has slow performance

I am formulating and solving a scheduling problem. The problem consists of scheduling items on a single machine, where the only time element is the transition from item $i$ to item $j$. For example, ...
3
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0answers
40 views

How to formulate a storage component?

Let’s say I have a drink for several customers, and I know their demand. Besides, I also have a storage tank for the drink during times when demand exceeds supply. The storage tank's size is not yet ...
3
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1answer
46 views

How to assign task, resources to trips

In a scheduling problem I want to assign the resources to tasks, each task has earliest start date, latest start date and duration. Also each resource has fixed number of available hours in a day. The ...
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0answers
74 views

MILP formulation for “if (a>=b) then c=1, 0 otherwise”

I need to build a MILP (Mixed integer linear programming) constraint form this if-else statement. In my formulation a and b are two continuous variables and c is boolean. if (a >= b) then c = 1, 0 ...
0
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1answer
104 views

Assignment problem with variable tasks to be done

I'm dealing with a kind of assignment problem, in which I have a set of tasks $t$ to be executed by machines $w$, but these tasks depend on the variatns $v$ of components $m$ being selected, which is ...
0
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1answer
79 views

How to linearise this nonlinear constraint?

I have a constraint in the form $\sum_{n=1}^{N}x_{m,n}\omega_{m,n}\ge (t_u-1)\beta_u, \forall u, u=1,2,\cdots, U$ where $x_{m,n}$ is binary variable $t_u$ and $\beta_u$ are continuous optimization ...
1
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1answer
87 views

How to transform this problem with logarithmic objective function into an approximated convex optimization problem?

I have an objective function as follows $\underset{x_{m,n}}{\max}\hspace{1mm}\hspace{1mm}\sum_{m=1}^{M}\log_2\left(\frac{\sum_{n=1}^{N}(1-x_{m,n})\omega_{m,n}+z}{\sum_{n=1}^{N}x_{m,n}\omega_{m,n}}\...
1
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1answer
88 views

How can I linearise this nonlinear proportional relation constraint?

My optimisation problem has a constraint in the form \begin{equation} \begin{array}{*{35}{l}} \text{}\hspace{16.5mm}\text{ C4:} \hspace{2mm}\sum_{u=1}^U d_{u,1}L_{u}:\sum_{u=1}^U d_{u,2}L_{u}:\cdots:\...
5
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3answers
1k views

How does a solver generally know whether a solution is optimal?

I was wondering how does the solver for a MILP determine whether a solution is optimal. I am having a hard time to believe that the solver actually tries all solutions, since in some cases I have over ...
0
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1answer
43 views

How can we understand which solution approach is suitable for our mathematical problem?

I'm looking for a solution approach for my MIP model, but I couldn't find any specific books about this issue whose solution approach is suitable for my model. There are a lot of exact, heuristic or ...
3
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2answers
148 views

A relaxed version of job shop scheduling

I am working on a formulation for a problem that seems similar to the bin packing problem. My problem variables include items that are to be placed in bins, special events that are conditionally ...
1
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1answer
110 views

Why some decision variables don't get values in Cplex?

I use this code in the cplex and don't know why some decision variables don't get value, I attach my code below. I don't know my model is wrong or my code? I haven't error in the code but have some ...
5
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2answers
620 views

How to transform this logical if-then constraint?

Consider the binary variables $x, y, z \in \{0,1\}$. I'd like to formulate the two if-then constraints: $$ x + y \geq 2 \implies z = 0, \tag{1} $$ $$ x + y \leq 1 \implies z = 1. \tag{2} $$ Constraint ...
3
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0answers
56 views

Solving a nonlinear model with constraints of exponential functions and continuous variable multiplications

I have a nonlinearly-constrained model and wonder if a nonlinear solver like Ipopt or Knitro can solve the problem. Briefly, my objective function is linear. I have the following variables with their ...
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1answer
99 views

is this a mistake or not in this tutorial?

Please consider https://stoprog.org/what-stochastic-programming and look at "A SIMPLE INTEGER RECOURSE MODEL" at "Stochastic Integer Programming" section. Should not $b_i$ be $...
3
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0answers
32 views

PuLP Python: How to linearize an inequality involving an integer variable

I am working on a Copper payables problem where the objective function is to maximise the sum of copper payable over a time period, T. The total amount of payable tonnes i.e. what the customer will ...
5
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0answers
54 views

Reference request — fishery yield optimization

I'm looking for references to do a review of research on managing fisheries in industry. I've seen adaptions of population growth models which include some harvesting constant or function and was ...
1
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1answer
62 views

How define variable in CPLEX and What is diffrence between decision variables and variable in CPLEX

I want to code a problem by CPLEX, in this problem I have variables and decision variables, how define them? In this picture you can see the variables: which we have: I use these codes for variables:...

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