is a discipline to apply analytical methods for better decisions. It has many synonyms such as management science, decision science and system science.

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He makes $3$ products for his shop: large bowls, small bowls, and pots

He makes $3$ products for his shop: large bowls, small bowls, and pots. Each large bowl uses $3$ pounds of clay and $6$ fluid ounces of glaze. Each small bowl uses $2$ pounds of clay and $6$ fluid ...
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28 views

Using the simplex method to find the minimum cost

A local food bank puts together complementary gift packages for its donors during their pledge drives. The bank's costs for each package are $\$4$ dollars for the Bronze level package, $\$7$ dollars ...
2
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1answer
35 views

Algorithm for partitioning works to workers

I'm writing a computer program to do work but there's a partitioning problem. In this program, there're workers and works. The main objectif is to give a balanced partition plan, so that works can be ...
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10 views

Determine the operation based on the conditions given below

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
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1answer
4k views

Linear programming vs. Integer programming

I was trying to solve a problem where I want to choose which items to choose where each item has a number b_i associated with it and a reward r_i associated with it. I need to choose items that ...
2
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1answer
22 views

Model cost for a state change in an integer program

I have a problem involving tool selection I am trying to model right now. (I am fairly new to this). I have a series of manufacturing operations I need to perform for $i \in \{1,\dots,n\}$. Each ...
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19 views

Linear programming model to maximize profit [closed]

Operation A B grinding. 1. 2 Turning. 3. 1 Assembling. 6. 3 Testing. 5. 4 Operations in hours for a given time period are ...
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11 views

Simplex Method -why does Gauss Elimination produce a new better solution?

In the simplex method, a tableau is created. An entering variable is generated and then using a criteria the exiting variable is also selected. This gives a pivot column and row and pivot element. The ...
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1answer
40 views

Please show that $f(\beta_0,\beta_1)=\log(1+\operatorname{exp}(-y_1(\beta_0+\beta_1 x_1)))+\log(1+\operatorname{exp}(-y_2(\beta_0+\beta_1 x_2)))$

I would like to show that the following result is indeed true. I am very new with this subject, so I ask for a hint to get me started please. Please show that $f(\beta_0,\beta_1)=\log(1+\...
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0answers
11 views

Proof of the existence of a rational finitely generated cone

Let $P$ be a rational polyhedron and $F$ be the inclusion-wise minimal face. Then we define: $C_F= \left\{c\in \mathbb{R}^n : F \subseteq \left\{x \in P:c^Tx=\max\left\{ c^Ty:y \in P\right\}\right\}\...
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11 views

Percolation theory for Reliability Theory: reference works?

For example, the George Grimmeth's book Percolation covers percolation with Inequalities of Reliability Theory such as the celebrated S-shaped curve on page 38. Please outline books and relevant ...
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29 views

Which Random Graph model on Reliability network when vertices deleted?

The reliability network when edges deleted corresponds to binomial random graph model that is also called Bernoulli Model. The page 3 of Random Graphs book (2000) by Svante Janson, Tomasz Luczak and ...
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7 views

To show total total dual integrality

Let P be the unit cube with, $P=conv\{\left(\begin{array}{c} 1/2 \\ 1/2\\1/2 \end{array}\right),\left(\begin{array}{c} -1/2 \\ 1/2\\1/2 \end{array}\right),\left(\begin{array}{c} 1/2 \\ -1/2\\1/2 \end{...
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2answers
870 views

Introduction into Operations Research

I am a first year graduate student who advisor wants me to learn about operations research and to use stochastic integer programming in my research. He keeps giving me papers to read but they ...
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1answer
79 views

Checking whether a solution to MIP is optimal

Consider a binary integer program \begin{align} \min \quad &\sum _{j \in J}f_j x_j +\sum _{i \in I} c_i y_i \notag \\ \mbox{s.t.} \quad &\sum _{j \in N_i} x_j \ge 1-y_i, \quad \forall i\in I \...
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1answer
26 views

Is convexity the most general dividing line between “easy” and “hard” optimization problems?

Just got started with Boyd's Convex Optimization. It's great stuff and I see how it directly subsumes the all-important linear programming class of models. However, it seems that if a problem is non-...
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1answer
21 views

Can every precedence table be represented as an “activity on arc” activity network

I have the precedence table Activity | depends on A | - B | - C | - D | A,B E | B,C F | A,C And I want to ...
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37 views

If the primal is unbounded, then the dual is infeasible.

In the context of duality in linear programming, prove that If the primal is unbounded, then the dual is infeasible. My book says that this is a corollary to complementary slackness. What's ...
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60 views

optimal, infeasible, degenerate solutions

Note that $c_i$'s in the $z_j-c_j$ row are not coefficients of the $x_i$'s. I use instead: $r_1, r_2, r_3$. I'm assuming there's a non-negativity constraint. we need to state necessary ...
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3answers
289 views

Variable leaving basis in linear programming - when does it happen?

In the simplex algorithm in linear programming, what are conditions for a variable to leave a basis (not necessarily basis for the/an optimal solution)? I'm supposed to list as many sufficient and ...
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27 views

In graph theory, draw the graph corresponding to the matrix A [closed]

I am studying statistics but decided to have some classes in mathematics. This class is called optimization but apparently, the content is graph theory. This is my first time of taking such class and ...
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1answer
3k views

simplex M-method minimization problem

Solve using the simplex method. Identify the solution of the dual in the final simplex tableau Minimize: $$z=12x_{1}+4x_{2}+2x_{3}$$ **Constraints:**$$ x_{i}\ge 0$$ $$-6x_{1}+3x_{2}\ge 9$$ $$2x_{1}-...
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1answer
446 views

Edge weight function for graph instance of scheduling and allocation problem

I have difficulties developing a proper (non-scalar) edge cost function $c_e$ for my resource scheduling problem, which I mapped into a graph problem. Processes $P_i$ need resources $R_i \in \mathcal{...
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19 views

Factory inspections on a budget

A factory inspector is testing the efficiency of $n$ machines. To pass the inspection, each machine is required to run at or above a certain standard efficiency. The inspector can measure the ...
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27 views

References request: two-queue, one-server model with pre-emptive queue priority and finite buffers

Sorry of the title is a mouthful. I'm developing a queue model with the following characteristics: Two queues: One contains an infinite number of people (Queue A) while the other (Queue B) is ...
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10 views

Is there a known optimal solution for searching an ordered list with non-uniform query cost?

Let $D$ be the set of integers from $1$ to $n$ inclusive for $n \geq 1$, and let $$f(i) = \begin{cases} 0& i \leq k \\ i - k& i > k \end{cases}\,\,\,\forall\, i \in D$$ for some $k \...
2
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1answer
562 views
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11 views

Why does the Dantzig cut require the constraint data to be integral?

Given the following integer linear program, (ILP) $\min c'x$ subject to $Ax \ge b, x \in \mathbb{N}_0$ where all elements of $A$ and $b$ are integral, and assuming its linear-program relaxation (...
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36 views

Properties of polyhedron solving constrained max problem

This is a question for people who don't have trouble to think in more than two dimensions. Don't hesitate to ask clarifying questions! Let us suppose we have $n$ random variables $X_i$ that are iid ...
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1answer
24 views

Algorithm for scheduling event observers

I'm reviewing different algorithms to solve a scheduling problem and was hoping someone with a better breath in the area might help me focus on the right class of algorithms. Basically the problem is ...
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23 views

Formulating a linear transportation problem as a stochastic linear program

[Question provided in picture]http://i.imgur.com/avoARFG.jpg[/img] I am having trouble with part b of this question. For part a, I have the following: let xij = number of units produced by plant i ...
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1answer
26 views

Formulating deterministic and stochastic production models (not solving them) [Beginner's Operations Class]

Question provided in picture This question has been troubling me as I am not used to questions without numbers as it is hard for me to visualise. I also find stochastic problems hard in general. &...
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1answer
26 views

When modeling a multi-objective problem, is there a simple way of choosing to fully minimize one function, then to go on and minimize the second?

I am modelling a problem where I have two objectives. My goal is to fully minimize the first objective function, then choose among the solutions that fully minimized the first objective function to ...
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1answer
25 views

Connection between complementarity problem and optimization problem?

I do not understand the connection between complementarity problems and optimization problems. I have tried to look at other definitions for complementarity problem to see if that would help me with ...
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2answers
26 views

Variational Inequalities - What excatly does the definition say? Why are they useful?

I am having issues understanding the definition of variational inequalities. We have the following definition: Given a set $X \subset \mathcal{R}^n$ and a mapping $F: X \rightarrow \mathcal{R}^n$ a ...
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13 views

Reduced cost in linear programming maximization sensitivity analysis?

My sensitivity report of maximization problem shows negative reduced cost although my optimal values of variables are not zero. So, what does it mean by the negative values of reduced cost? Here is ...
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2answers
31 views

Is Reliability Component a vertex?

The term component has a distinct definition in graph theory from vertex while the terms components and vertices can be mostly the same in Realiability Engineering, my intuition. So how is the term ...
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19 views

Origins of Operations Research and original meanings to different terms? [closed]

I am confused by Reliability Engineering to the extent that sometimes the terms used are graph-theoretical: this aspired to be researched here and here. In comparison, terms are sometimes more slack, ...
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0answers
16 views

Area of Operations Research on graph theory and reliability engineering? [closed]

I am confused by the jargon in Operations Research (OR) when it is the same as in Graph theory such as component but it can mean just a vertex. So I am confused to the extent that reliability ...
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100 views

Are any tools or techniques available to solve the “placement of safety points” problem?

Definition 0. Given a metric space $X$ and subsets $H$ and $S$ thereof, define: $$d(H,S) = \sup_{h \in H} \inf_{s \in S}d(h,s)$$ (This as an asymmetric version of the Hausdorff distance.) Here's ...
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2answers
40 views

Trying to sell the most batches of animals using linear programming

I'm trying to sell the most batches of animals... Let's say I have 200 dogs, 100 cats, and 100 ferrets. ...
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2answers
227 views

Convert a piecewise linear non-convex function into a linear optimisation problem.

Update: Problem and solution found here (p. 17, 61), although my prof's solution (formulation) is different. Convert $$\min z = f(x)$$ where $$f(x) = \left\{\begin{matrix} 1-x, &...
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2answers
34 views

Shortest path in a graph with weighted edges and vertices

I am considering a problem of route planning (from a source $s$ to sink $t$) in a undirected graph with weighted edges and vertices. The goal is to find a shortest path between the source $s$ and the ...
1
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1answer
15 views

Tree-width of a quadratic pseudo-Boolean function

A pseudo-Boolean function $f : \mathbb{B}^n \mapsto \mathbb{R}$ is of the following form. $$ f \left(x_1, \ldots, x_n\right) = \sum_{S\subseteq V} c_S \prod_{j \in S} x_j $$ Here $c_S \in \mathbb{R}$,...
3
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1answer
31 views

Transforming a $0$-$1$ knapsack problem into the standard form

I have the following $0$-$1$knapsack problem: $$\begin{align*} &\mathrm{Max} : \quad z= 3x_1 -4x_2+5x_3+7x_4-6x_5+x_6\\ & \mathrm{subject\ to}: -2x_1 +x_2 +10x_3 +3x_4 -5x_5+12x_6 \leq 4 \...
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40 views

Transform an assignment problem to use the Hungarian algorithm

I have this assignement problem: There are three machines to perform four tasks. The costs of assigning to a machine each of the tasks are given by the following matrix: $$\begin{pmatrix} ...
3
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1answer
889 views

Prove optimal solution to dual is not unique if optimal solution to the primal is degenerate and unique.

How do I prove an optimal solution to dual is not unique if an optimal solution to the primal is degenerate and unique? What I tried: Let the primal be $$\max z=cx$$ subject to $$Ax \...
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1answer
57 views

Setting up an LP problem on producing linear board in jumbo reels

I have to set up a linear programming problem corresponding to the following scenario: What I tried: I think we have 8 templates for 1 $68 \times l$ reel (or whatever): $22,22,22$ (66) $20,...
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0answers
9 views

Correct definition of the co-occurrence graph of a pseudo-Boolean function

In section 4.6 of Pseudo-Boolean Optimization, Boros and Hammer have defined the co-occurrence graph of a pseudo-Boolean function as follows. If a pseudo-Boolean function $f : \mathbb{B}^n \mapsto ...
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3answers
57 views

Mixed Integer Linear Programming Conditional Constraints

I have a set of variables: $x_1,x_2,x_3,x_4$ $x_1$ is a binary integer variable while the rest are real numbers all between 0 and 1 I want a constraint such that: if $x_2+x_3+x_4$>0 then $x_1=1$ ...