Questions tagged [operations-research]

Operations Research, sometimes known as Management Science or Decision Science, is the discipline of applying appropriate analytical methods to help those who run organisations make better decisions.

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21 views

What are "4 possible cases for primal and dual linear programming problems"?

I am studying for my Linear Programming exam. One of the questions is "4 possible cases for primal and dual linear programming problems"? Neither of my mates knows what it is about. We've ...
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122 views

Expected objective value and median objective value

Suppose we are maximizing profit, where one machine is making four items, composed of product a and b. There is a maximum of A of product a and B of product b. Here is how I have formulated the ...
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Vehicle Routing Problem, Biobjective Integer Linear Program

You manage a fleet of $K$ identical vehicles stationed at a depot. This depot is a node in a complete directed graph whose other nodes are your customers. You wish to use your vehicles to fulfill ...
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45 views

Finding a line with minimum distance among a set of points

Suppose I have a set of points denoted by $a_i$ I want to minimize this model: $$\sum_i (w |(x-a_i)|1(x-a_i)+v|x-a_i|1(a_i-x))$$ where 1(x) denotes an indicator function which takes value $1$ if $x>...
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Check solution: Analyze the optimization problem min $f(x, y) = e^{-x} + e^{-2y}$ over the entire domain of $f$

Analyze the optimization problem min $f(x, y) = e^{-x} + e^{-2y}$ over the entire domain of $f$. Determine if $f$ is convex, concave or neither. If $f$ is neither, what can you say about its possible ...
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Why does this approach fail?

We have the following statement (translated from Spanish to English): A company manufactures skirts, blouses and pants. To do this, use a machine for each type of clothing. The machine for skirts ...
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1answer
47 views

Nonlinear Programming with inequality constraints

I have the following nonlinear optimization problem. $$ \underset {b_1, b_2} \max\, f(b_1, b_2) = N_1(V_1 - b_1)F(b_1;\mu_1,\sigma_1) + N_2(V_2-b_2)F(b_2;\mu_2,\sigma_2) $$ subject to: $$ g_1(b_1, b_2)...
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Modeling contiguity of machine processing in a flow shop environment via a MIP

I'm working on a Mixed-Integer-Programing (MIP) formulation for a flow shop scheduling problem. One of the requirements/wishes is that for each machine $i$, processing should be contiguous, or at ...
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62 views

Solving a linear programming problem using its dual problem.

Solve the following linear programming problem using its dual problem. \begin{align} \min&\quad z = -5x_1 -7 x_2 -12 x_3 + x_4 \\ \mathrm{s.t.}&\quad 2x_1 + 3x_2 +2x_3 +x_4 \leq 38 \\ &\...
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How to create the general formula for minimum volume after using Lagrange's multipliers?

Context: So I was working on finding the relationship between the variables in an isosceles triangular prism a,b(side lengths of rectangle and h (height of triangle). The problem is that I'm trying to ...
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38 views

Optimization Problem: Maximize percentage of dependent sum

I have an optimization problem which I currently solved by brute force but I was wondering if there is a more efficient solution. I wasn't able to reduce the problem to some well known problem in ...
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Minimizing area and maximizing volume, under constraints using Lagrange multipliers

Context I'm in year 12 of school, and for our curriculum, we have to write a 12-page thesis on any mathematical subject. Since I picked maths for Higher Level the maths has to be commensurate for that ...
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Eliminating a free variable in a linear programming problem

Consider the following linear model. \begin{align} \min&\quad z = c^{t}x\\ \mathrm{s.t.}&\quad Ax = b\\ &\quad e^{t} x = 1 \\ &\quad x_{i}\geqslant0 \quad \forall i \in \{1,...,n-1\} \...
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Simplex Method: Why the pivot should be positive?

The pivot shouldn't be negative or 0, but why is this? I'd like to understand the reason. Also why do we choose the smallest number in the ratio column to choose the pivot row?
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How to find the range of the parameters using the basic feasible solutions I have found?

Given the problem: maximize $a_1x_1+a_2x_2$ s.t. $2x_1-x_2\ge 2$ $x_1+x_2\le 4$ $x_1,x_2\ge 0$ a) Given that $a_1=1,a_2=2$, solve the problem graphically. b) Find all the basic feasible ...
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Resources to learn about modeling within the scope of Linear/Integer programming?

I'm currently taking a course in OR, and I'm facing some major difficulties trying to formulate my LP/IP problems. I understand most of the topics just fine, but I just get lost trying to formulate ...
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Derivation safety stock formula of an inventory model with probabilistic demand

I am applying the "Probabilistic inventory model with safety stock". There are some assumptions in this model: normally distributed demand, fixed lead time (time between placing the order of ...
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How do I formulate this linear programming problem? (especially the second restriction)

A factory creates different types of oils and mixes them together. There exists two types of vegetarian oils (veg1,veg2) and three types of non-vegetarian oils (oil1,oil2,oil3), the price of each oil (...
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Query about bounded variable simplex method

Are bounded variable simplex and bounded variable primal simplex method same? If not where does these two differ?
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Modeling sequence dependent setup times via a MIP for flow shop scheduling

As part of a Non-Permutation Flowshop Scheduling (NPFS) problem, I would like my MIP model to be able to deal with sequence dependent setup times. That is, for each pair of consecutive jobs, a setup ...
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1answer
64 views

Difference between convex set, closed convex set, polyhedron and polytope?

I'm having a hard time differentiating between a convex set, a closed convex set, a polyhedron and a polytope? I have a good understanding of what a convex set is, but I can't seem to understand was ...
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How to remove non negativity constraint(i.e. make it a free variable)?

I am currently trying to show a polyhedron $P = \{Ax=b, x\ge0\}$ can be written in the form $P = \{Ax\le b, x \in R^n\}$. I know $Ax=b$ can be written as $Ax\le b$ and $-Ax\le -b$, but I am not sure ...
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What is the gradient of plus function?

Let $f:\mathbb{R}^n\to \mathbb{R}$ that $f=\frac{1}{2}\|(Ax-b)_+\|^2$, where $A\in\mathbb{R}^{m\times n}$, $b \in \mathbb{R}^m$ and if $x\in \mathbb{R}^n$: $((x)_+)_i=max\{0,x_i\}$. I think the ...
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91 views

Evaluate $\int_{\mathbb{R}^n} \left(\frac{\lambda}{\lambda^2+|x|^2}\right)^N dx$

As part of my thesis I’m reading the following paper: https://arxiv.org/abs/1612.08225 and I have trouble with the following integral calculation in multiple dimensions. Show that $$\int_{\mathbb{R}...
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Construct linear floorplanning constraints

This question is an extension of a previous question. Right now, what I have are these "cheap" equations. The goal is to have the floorplan allow a circle with diameter, $D$ outside the red ...
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Shortest path to sort a list through item exchanging

What is the shortest path to sort a list through item exchanging? Recently, I met a math problem. I'd like to find an apporach to convert a list to ordered list by item exchanging with corresponding ...
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1answer
26 views

Modeling some constraints

We have two decision variables $x \in \mathbb{Z}^{0+}$ that is the main decision variable and $0 \leqslant y \leqslant 1$ that is an auxiliary decision variable. Now based on the nature of the problem ...
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110 views

Linear constraint

I found this video extremely helpful to determine the floorplanning constraints. The first two constraints (i) make sure the modules are within the feasible floorplan; the subsequent four constraints (...
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vehicle routing optimization, Big M method of reformulation of constraints

Please excuse me for the long question, if I dont prrovide this info. my post gets removed! The following optimization problem is called Mixed-Integer Quadratically Constrained Programming (MIQCP) ...
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Stuck at making a constraint for given LP problem where a machine can make one product or the other.

This is the text for following linear problem: In one factory there is a production machine which is available 170 hours a month. Using this machine it is possible to produce 50 pieces of product A ...
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Reinterpret maximin as LP

I have the following problem For $L,T \in \mathbb{R}^n$ and $G \in \mathbb{R}^{n\times n}$ $$\mathsf{max}_L \;\mathsf{min}_T \sum_{j = 1}^n T_j$$ Subject to \begin{align*}\forall i,j &&T_j \...
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1answer
38 views

Can't understand K-Truss Graph properties

Cross-posted on Operations Research SE. I'm trying to understand K-Truss Graphs which are defined as such The k-truss is a subset of the graph with the same number of vertices, where each edge ...
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26 views

How to proof in the best way possible that this network has a min cut of finite capacity

I am trying to prove that the min cut of the following network is 900. From looking at it I think it is obvious. I think the problem lies in the fact I have some infinite capacities and will need to ...
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Is there an algorithm for project scheduling that is known to be optimal, i.e. minimises #workers needed to complete the project in minimum time?

Say we have a finite number of activities $\{A_i\}_{i=1}^n$ that each take $\{d_i\}_{i=1}^n$ units of time to complete with $d_i>0$ for all $i$. Some activities are dependent on others, ex. $A_5$ ...
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61 views

Finding unknown in an optimal simplex tableau

I have a problem about this simplex problem for my Operations Research class. The following tableau belongs to the optimal solution of a Linear Programming Problem. Calculate the value of objective ...
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1answer
65 views

Maximizing $\sum_{n\in[m]}|f(n)-n|$ where $f(n)$ is a permuation of $[m]$

Given $m\in\Bbb Z_{\ge1},[m]=\{1,2,...,m\}$ and $f(n)$ an arbitrary bijection from $[m]\to[m]$, I am interested in finding the maximum value of$$I_f=\sum_{n\in[m]}|f(n)-n|$$and the permutation $f^*(n)$...
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50 views

Sufficient graph conditions for VRP

I'm trying to create a program to solve basic Vehicle Routing Problem and, to test it, I would like to write a function to generate valid graphs. I found in this course (p. 11) that a graph must be ...
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29 views

Optimal ordering policy in inventory model

Determine the optimal ordering policy in the case of a single period inventory model with no setup cost instantaneous stock replenishment and amount demanded is a continuous random variable. In this ...
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2answers
98 views

Non-linear optimization problem using Lagrange's method/K.K.T. conditions

We are given the following problem: $$\text{minimize } 2x_1^2 + x_2^2 + 3x_3^2 \text{ subject to } x_1+x_2+x_3=10, x_1\le5, \text{ and } x_1,x_2,x_3\ge0$$ I understand that I have to check all ...
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33 views

Stepping stone algorithm in Operational Research

I'm trying to solve a transportation cost problem using simplex tableu and the stepping stone algorithm. I've done one iteration using the minimum cost method and this is the tableu I have obtained: ...
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1answer
46 views

Trying to derive dual formula for linear programming

I have the following basic linear optimization problem. Let $A \in \mathbb{R}^{n,n},c\in \mathbb{R}^n$, then solve $$\inf_{x \in \mathbb{R}^n} c^Tx,\ \text{ s.t. }\begin{cases}A^Tx=c \\x_i \geq 0, i=1,...
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Explanation of proof Matrix operation

I don't fully understand the process of the following preposition, Could someone explain what is happening more in detail? A step by step guide if possible :) Let u, v ∈ $C^N$ be vectors with u ≤ s ...
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Sensitivity analysis adding a new variable lpp

Some friends and I are having trouble with this sensitivity analysis in a LPP. https://imgur.com/a/HgBi8Fj We got the first question right with a python code, but we're having trouble with the second ...
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85 views

Why does the AMPL Solver (Minos) display a wrong solution (equal to zero)?

I am using AMPL to solve a non-linear program and, although I know the answer is (x, y) = (6, 4), the solver returns (x, y) = (0, 0) and I really cannot understand why. This is my attempt: ...
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Where can I find good practices on building models for optimization?

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 ...
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64 views

How can we express an integer linear program as a dynamic program? (Operations Research)

We are given the following integer linear program: \begin{equation} \begin{array}{c} \max x_{1}+2 x_{2}+2 x_{3}+3 x_{4} \\ \text { such that } 2 x_{1}+3 x_{2}+x_{3}+2 x_{4} \leq 4, \\ \text { where } ...
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1answer
18 views

Interpret input quantity matrix multiplied by the transpose of a price matrix

I have two $77 \times 8$ matrices where rows represent firm ids: input quantity matrix, $X$, where columns represent quantity types (e.g., grain, seed, and chemical). price matrix, $W$, where ...
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Why VAM returns better bfs than least-cost method?

I came across this question quite a few times while studying operation research. But could not find any satisfactory answer. I understand why both methods provides better bfs compared to North-West ...
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36 views

Modeling a piece-wise objective function for a linear program

I am attempting to design a linear program where I optimize the amount of money I make by selling good $A$. Selling units of $A$ below a threshold $t$ results in an income of $a$ dollars per unit of $...
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Deposits in the context of linear programming

Suppose that $x$ is amount invested in the low-risk term deposit and $y$ is amount invested in the balanced deposit, and $z$ is amount invested in the high-risk deposit. To minimize risk, you decide ...

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