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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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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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 ...
214 views

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 ...
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 $x>... 0answers 30 views Check solution: Analyze the optimization problem min$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$is convex, concave or neither. If$fis neither, what can you say about its possible ... 2answers 49 views 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 ... 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)... 1answer 67 views 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 ... 1answer 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 \\ &\... 0answers 17 views 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 ... 0answers 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 ... 0answers 59 views 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 ... 1answer 18 views 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\} \... 0answers 19 views 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? 1answer 25 views 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 ... 1answer 20 views 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 ... 0answers 7 views 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 ... 1answer 49 views 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 (... 1answer 22 views Query about bounded variable simplex method Are bounded variable simplex and bounded variable primal simplex method same? If not where does these two differ? 1answer 35 views 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 ... 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 ... 0answers 25 views 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 ... 2answers 75 views 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 ... 1answer 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}... 0answers 31 views 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$outside the red ... 0answers 28 views Shortest path to sort a list through item exchanging What is the shortest path to sort a list through item exchanging? Recently, I met a math problem. I'd like to find an apporach to convert a list to ordered list by item exchanging with corresponding ... 1answer 26 views Modeling some constraints We have two decision variables$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 ... 2answers 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 (... 0answers 24 views 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) ... 1answer 39 views 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 ... 1answer 38 views 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 \... 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 ... 1answer 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 ... 1answer 30 views 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$... 1answer 61 views Finding unknown in an optimal simplex tableau I have a problem about this simplex problem for my Operations Research class. The following tableau belongs to the optimal solution of a Linear Programming Problem. Calculate the value of objective ... 1answer 65 views Maximizing$\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$f^*(n)$... 2answers 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 ... 0answers 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 ... 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 ... 0answers 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: ... 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,... 0answers 13 views 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 ... 1answer 36 views 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 ... 1answer 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: ... 0answers 44 views 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 ... 2answers 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 } ... 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 ... 0answers 49 views 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 ... 0answers 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$...
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 ...