# Tagged Questions

Questions on optimization constrained to integer variables.

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### How to linearize this constraint a summation of a product of a integer with a binary

I have to linearize the following constraint, $$\sum_{i \in V_C} \sum_{j \in V} \sum_{k \in K} y_{ik} \cdot x_{ijk\ell} \leq I_\ell \qquad \forall \ell \in V_D$$ where $y$ is a integer variable ...
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### Operations Resarch Optimal Scheduling

Consider the following problem: A car manufacturing company needs to transport car frames, which are $10$ cubic units each, and wheels, which are $2$ cubic units each, across the Atlantic ocean. ...
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### minimising quadratic function subject to integer solutions

I would appreciate if one could help me to solve this problem. I have a bivariate quadratic function: $$f(a_1,a_2)=(1-u_1^2)a_1^2 +(1-u_2^2)a_2^2 -2u_1u_2a_1a_2$$ where $u_1^2+u_2^2=1$ and $a_1$ ...
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### Solving an integer linear programming problem without a graph

I am new to linear prorgramming and so far I have been solving LP problems with the help of a graph solution. However, when there are more than 2 variables obviously I can't plot them on the graph. ...
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### Chocolatier sampler boxes problem: applying goal programming and mixed-integer programing to optimally compromise goals.

QUESTION: A boutique chocolatier is planning to make a number of sampler boxes, each containing $36$ chocolates. (Therefore the total number of chocolates should be divisible by $36$.) The ...
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### Help with Math software (macaulay 2)

I just started working with Macaulay 2 and need some help. I need to find the number of solutions of a system of equations. I am having difficulty imputing this into the software so please be specific ...
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### Binary depending on the sign of another variable

I'm writing a mixed integer linear problem, where I have an indicator function in the objective function counting the instances of negative values of a decision variable. I thought of defining a ...
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### Book recommendation on integer programming ? (in order to solve a set cover problem)

I'm trying to solve a set cover problem. To put it shortly, my problem is about covering a $N \times M$ grid, by using various rectangles which have associated cost depending on their shape and ...
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### How do I convert a constraint with a product of two integer variables to a linear constraint?

I have a constraint of the form: $$\theta \leq a_1x_1 + a_2x_2 + a_3x_1x_2$$ where, $x_1$ and $x_2$ are integer variables with ranges $x_1 \in \{0, m\}$ and $x_2 \in \{0, n\}$. I would want to ...
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### Linear Integer Optimzation Problem (scheduling problem)

Does any of you know how to get this done?
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### How to find smallest integer which is greater than N positive primes

I know this can't be computed exactly, but I just need a rough estimate. I know one can compute a rough estimate of the number of primes less than N using the famous formula: ...
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### Escaping from a point in linear programming

Is there a trick for explaining the following constraint as a set of linear (in)equalities? $$\sum_{i=1}^n|x_i-a_i|>0,$$ where $a_i$'s are real constants.
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### Determining information in minimum trials (combinatorics problem)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the ...
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### One solution of a diophantine system

How to find one solution of $Ax = b$, where $A$ is a $(m, n)$ matrix and $x$ a vector of size $(n, 1)$. $A$, $x$ and $b$ are matrices of integers entries. How to check whether is a solution exists?
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### Determine largest possible position for integers

I'm sorry if I'm not using the proper terminology but here's my question. When two numbers are multiplied, the position of the largest number can only be at the position of the sum of their operands. ...
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### Is there any methods to solve for integer solution of a quadratic equation like $ax^2 + bx + c = 0$

Is there any method to solve for integer solution of a quadratic equation like following: $$ax^2 + bx + c = 0$$ where $a, b, c \in \mathbb{Z}$ If not is it possible for the Special case: ? x^2 -x ...