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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In Markov chains, does $(I-N)^{-1}$ always exist?

Spins-off from these two questions. Under what conditions does $(I-N)^{-1}$ exist? If the entries of an invertible matrix N are between -1 and 1, is its operator norm less than 1? Apparently, in ...
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62 views

The best strategy to increase StackExchange Reputation [closed]

I do not have a lot of background in game theory, but I am curious how would one formally pose the title problem and mathematically describe possible strategies. Are the problems of this type best ...
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1answer
30 views

If the entries of an invertible matrix N are between -1 and 1, is its operator norm less than 1?

For Euclidean norm. If so, why? If not, might $(I-N)^{-1}$ exist some other way? This spins-off from here.
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2answers
59 views

Under what conditions does $(I-N)^{-1}$ exist?

Given an nxn matrix N and $I=I_n$, under what conditions does $(I-N)^{-1}$ exist? On one hand $(I-N)(I + N + N^2 + ...) = (I + N + N^2 + ...) - (N + N^2 + ...) = I?$ On the other hand, $(I-N)(I + N ...
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50 views

Assignment problem with multiple types, capacities and costs

I am trying to solve an optimization problem (variation of assignment problem). I'm stuck with how to represent this problem (as an LP or graph based). If it's formulated as a LP, I'm unsure of how to ...
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9 views

Say optimal solution to the primal is degenerate. Does it hold that optimal solution to dual not unique?

I think it's supposed to be that existence of a degenerate and unique solution of the primal implies multiple solutions to the dual, according to this book (pages 141-145, proof of Theorem 4.5). In ...
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1answer
11 views

How to model a multiobjective problem with a large dataset?

I have a large dataset of businesses (around 5k venues with distance from a predefined point, average price and service quality rating) and I need to create the objective functions to minimize the ...
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1answer
32 views

An LP problem from David G. Luenberger's Linear and Nonlinear Programming book

Could someone help me to solve the following problem? A class of piecewise linear functions can be represented as $f(x) = Maximum (c_{1}^Tx+ d_{1}, c_{2}^Tx, \cdots, c_{p}^Tx + d_{p})$. For such a ...
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42 views

Setting up the Bellman equations for dynamic programming

I have the following question I want to understand. The owner of a chain of three grocery stores has purchased five crates of fresh strawberries. The estimated probability distribution of ...
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18 views

How to model the following banking question?

A group has 200 members. Each member settles up 100,000 S each month. The group runs a lottery every month and pays 10,000,000 $ to one member of the group i.e.,winner. The winner do not settle up ...
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19 views

Not every correlated equilibrium is equivalent to a Nash equilibrium?

This is a statement made under Theorem 3.4.13 on page 84 in the book by Yoav Shoham and Kevin Leyton-Brown, Multiagent Systems. Could someone explain this to a lay-man and also elaborate on why they ...
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10 views

Minimizing handling costs in the one-to-one TSPPD

Currently I'm reading about the Traveling Salesman Problem with Pickups and deliveries. This is similar to the classical TSP but there are $n$ requests and each request has a pickup location and a ...
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2answers
35 views

Prove linear program is unbounded

So I need help on my homework (I feel like a 10 year old). The exercise goes like this: Prove algebraically that the following program is unbounded: Max: $x_1 - x_2$ Constraints: $-2x_1 + x_2 ...
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1answer
66 views

What gambling/board game or real life thing can (surprisingly) be modelled as a linear programming problem?

So I've taken Linear Programming 101. I've read my textbook, took the test and all that, and - besides all the theory, the nice algebraic interpretations, etc - I've encountered a lot of textbook ...
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1answer
28 views

Can a basic variable be the entering variable in Simplex method?

I have got from my teacher that "the entering variable in a maximization problem is the non-basic variable having the most negative coefficient in the Z- row" I think X1,X2 are non basic and ...
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31 views

Simplex Method, which step is wrong?

Question: Maximize $z=2x_1-6x_2$, Subject to $-x_1-x_2-x_3\le-2$ and $2x_1-x_2+x_3\le1$. $x_1,x_2,x_3\ge0$. My work: $-2x_1+6x_2+z=0$ Then ${R_2\over2}\Rightarrow new R_2$, $new ...
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21 views

Consider the maximum flow problem with n nodes and m arcs. You are writing a formulation with f as the maximum flow.

Consider the maximum flow problem with n nodes and m arcs. You are writing a formulation with f as the maximum flow. How many terms the objective function will have?
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1answer
32 views

Primal of Dual of LP problem

Given that the following relation holds: $$\begin{align*} &\textbf{Primal problem} \\ &\max Z = c^Tx \\ &s.t. \\ &Ax \leq b \\ & x \geq 0\end{align*}$$ $\Longrightarrow$ ...
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72 views

A basic question related with the solutions of linear programming problems

I have to select one option from the problem statement given below. Which of the following statements is true in case of linear programming. $1$: An optimal solution exists at extreme points of a ...
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21 views

Is this variant of the Stable Roommate problem NP-hard?

I want to organize $2n$ people ${A, B, C, \dots}$ in pairs. Each people rates every other one with an integer number going from 0 to 10. The ratings may not be reciprocal (i.e., A may rate B a 10, and ...
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107 views

Is optimal solution to dual not unique if optimal solution to the primal is degenerate?

If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
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46 views

Nonlinear non-convex semi-infinite programming with norm equality constraint

In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a ...
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45 views

Assignment problem with serval 'rounds' and additional constraints

Suppose we have an office with $n$ employees and we want to make a planning for the upcoming $T$ weeks. In each week there are $m_t, t\in \{1, 2, \ldots, T\},$ jobs to do. Each employee can give a ...
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1answer
34 views

Removing a max function in the constraints

Can the following problem be transformed into a linear programming problem: Find $x_1,..,x_N$ which maximizes the objective function $$\sum_{i=1}^{N}x_{i}\sum_{j=1}^{n_{i}}c_{ij}$$ subject to the ...
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19 views

Difference between CPM and PERT

What is the difference between Critical Path Method and Program Evaluation and Review Technique?
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77 views

Prove an artificial variable that leaves the basis will never return.

Prove an artificial variable that leaves the basis will never return. Edit: This is for the simplex method (I think). I have no idea how to start this. Anyone know any books with these kinds of ...
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1answer
168 views

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

How do I prove an optimal solution to dual is not unique if an optimal solution to the primal is degenerate? I have no idea how to start this. Anyone know any books with these kinds of questions (and ...
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0answers
113 views

Formulating linear programming treatment plan based on costs, periods, and condition

Note that this question is cross posted from OR-exchange Although we have a software that solves this for us, I'd like to understand the background behind the scenes as well as build a validation ...
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2answers
63 views

Stationary probability in an M/M/$1$ queue with a lazy server

Customers arrive to a single server queue according to a Poisson process with rate $\lambda$. Each customer requires Exponential($\mu$) service time. In the beginning when there are $0$ ...
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1answer
36 views

Duality - linear programming

I have to find a respective dual programme for the given LP $$ \max \ 2 x_1 + 2x_2$$ s.t. $ -x_1 - x_2 \ge -5 \\\phantom{-}x_1,\phantom{,,}x_2 \ge 0$ I got this: $$\min \ 5y_1$$ s.t. $y_1 \ge 2 ...
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58 views

Linear programming of sperner lemma

How can you formulate the 2-D proof of Sperner lemma as a linear programming problem? I know that you have to divide the triangle up into smalled triangles with the original triangle having vertices ...
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28 views

Poisson Distribution Research Question in R

I am working on doing a Poisson distribution based upon the number of potholes and accidents on a given road. The problem I currently have is that I am basing a general linear model off of only the ...
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1answer
30 views

discuss convexity of the following set?

discuss convexity of the following set ? $$M= \{(x,y)∈\Bbb R^2 : x^2+y^2≥a^2 ,x^2+y^2≤b^2 ,x>0,y>0\} $$
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1answer
21 views

Need help using the neighborhood search heuristic

I need help using the following neighborhood search heuristic on the set S that is partitioned into A and B where A = {1, 2} and B = {3, 4}. This set partition is our initial solution x_0, and the ...
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22 views

Need guidance on a Queuing problem

I can't really go into specifics, I'm more just looking for terms that I can research to get on the right track. Classes of model/processes etc. A close analogy to my problem: I need to optimally ...
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2answers
59 views

an interior point of a convex set

How can we prove a point is an interior point of a convex set, considering we don't have all of the extreme points of the given convex set ? or How can we find an interior point of a convex set, ...
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1answer
43 views

Relative Interiors of polyhedra

***Source article: Magnanti, T. L., & Wong, R. T. (1981). Accelerating Benders decomposition: Algorithmic enhancement and model selection criteria. Operations Research, 29(3), 464-484
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1answer
237 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:**$$ x_{i}\ge 0$$ $$-6x_{1}+3x_{2}\ge 9$$ ...
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1answer
33 views

How to distribute groups over activities in rounds

I typed out my problem in a Latex file and I will add an image of it here: If anyone could help me how to solve this problem that would be amazing. Thank in advance. Boris
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20 views

Is “nonanticipating” a measurability property of a function or something more?

I have been reading some operations research papers that throw in the term "nonanticipating" at key points in the exposition, but I can't figure out precisely what they mean. My best guess is that ...
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1answer
28 views

maximization problem with inequalities restriction

I have a function $g(x,y,z)$, and $x+y+z=1, x\geq0,y\geq0,z\geq0$. Now I want to maximize $g$. If I ignore the inequalities, then I can use lagrangian and can solve this thing for maximum. But I am ...
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38 views

Scheduling Algorithm for a multi-server queue problem

I have 4 servers, n customers and m reports. At any time, a customer may request one of m reports. There are only 4 servers which are capable of generating reports. Each server can only process one ...
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1answer
37 views

What is the system equation $f$ in Hamilton equation in $H=g+p^Tf$?

I am studying the Donald Kirk's book Introduction to Dynamic Programming. Suppose some integral $\int g dt$ that must be minimised. Then you are given some constraints. Hamilton equation is $H=g+p^T ...
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1answer
39 views

Euler equation for $\int_0^{\infty}e^{-rt}(x^2+2x+\dot x^2) \ \mathrm dt$? Is $\infty$ in the boundary open or closed?

I am pondering this problem here, the course Mat-2.3148 Dynamic Optimization in Aalto University, i.e. Find the function $x(t)$ such that $\int_0^{\infty}e^{-rt}(x^2+2x+\dot x^2)\ \mathrm dt$ has ...
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144 views

Changing a queueing processes

Situation Consider a general queueing system $\mathscr{S}$, whose customer arrival times are independent, and whose service times are independent; both of these are allowed to have general ...
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40 views

Line that passes between two vectors

I encountered the following in a text book I'm reading and I can't seem to understand why this is true (I'm translating this into English so excuse me if I'm not using the correct english terms): ...
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1answer
69 views

In Courty and Li (2000) “Sequential Screening”, what justifies the last equation in Lemma 3.2?

Regarding the article "Sequential Screening," in Review of Economic Studies, 2000 by Courty and Li: In Lemma 3.2, the last equality states that ...
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133 views

Suppose I have the tableau below for a maximization problem. For the tableau to be optimal what are values for c1, c2, and b?

Suppose I have the tableau below for a maximization problem. For the tableau to be optimal what are values for c1, c2, and b? z x1 x2 x3 x4 x5 x6 RHS 1 c1 c2 0 0 0 0 10 ...
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1answer
16 views

What if objective function $Z$ is also in the constraints?

What if objective function $Z$ is in the constraints? To construct the dual form for this problem? how do I approach to this problem? Maximize $\;\;\;\;\;\;\; z$ subject to $$\;\;\;z - ...
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27 views

Partial dependencies in PERT

Assume that there are 6 activities in a Project: $A$, $B$, $C$, $D$, $E$ and $F$. ...