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26 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
32 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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27 views
+50

Changing a queueing processes

I am wondering if there are any general results related to how queue behavior changes if one is allowed to repeatedly make one-off behavior changes. Situation Consider a general queueing system ...
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2answers
32 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
41 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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28 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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26 views

Proving equivalence between basic feasible solution and vertex

I stumbled upon this proof of the Bertimas book on Linear optimization and I don't see what the "key ingredient" is that makes it work. Baxic feasible solution $\implies$Vertex Let $x^*$ be a ...
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1answer
12 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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0answers
20 views

Partial dependencies in PERT

Assume that there are 6 activities in a Project: $A$, $B$, $C$, $D$, $E$ and $F$. ...
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1answer
38 views

Network simplex method, leaving and entering variables

Could someone give me a hint on this question, which is a past exam question: Under what circumstances will an entering variable in the network simplex method be the same as the leaving variable? ...
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0answers
19 views

Good MIP formulation of a timetabling problem

I am trying to formulate a university timetabling problem as a mixed-integer program. The choice variables are binary variables of the form $x(c,s,r)$ which is $1$ if a class of course $c$ is held in ...
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1answer
39 views

Probability Density Function and Proof

Given the Probability Density Function: $f(x)=kx(2-x), 0\leq x\leq 1$ Prove that $k=\frac 3 2$ Looks like it should be a Beta Distribution, but all examples of a beta distribution use the format: ...
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27 views

operation research revised simplex

in revised simplex when be compute inverse of basis matrix, it is told in the book that only one column changes(obviously because of entering basic variable)but i could not understand how does it make ...
3
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1answer
50 views

simplex method standard form

i am unable to understand algebraic formulation of simplex method.when we add slack variables, and solve for finding basic feasible solution we put free variables equal to zero. My question is why ...
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0answers
40 views

Sensitivity of coefficients in ODE

I am trying to formulate a mathematical model as part of an op-research problem, and I'm running into a roadblock concerning differential equations of a certain kind; I was hoping to understand if ...
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0answers
115 views

The difference between Dynamic Optimization, Stochastic Programming, Optimal control and Markov Decision Processes

I've seen the following terms thrown around somewhat interchangeably, and I'm confused. What are the distinctions between them, and what are some representative problems that each deals with? ...
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38 views

Double summation of linear function?

I am doing an assignment in operations research and I seem to struggle with some basic arithmetics. I need to formulate an objective function of the form $$\sum^3_{i=1}\sum^8_{j=1}(cij+c'ij)xij$$ ...
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1answer
148 views

Minimizing shipping cost under given constraints

I have a question that has been bugging me for about a day now. A manufacturing company receives orders for engines from two assembly plants. Plant I needs at least 45 engines and Plant II needs at ...
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1answer
367 views

Minimize LPP using graphical method [ operational research ]

Question: Minimize z = 2x + 6y Subject to 2x + y >= 2; 3x + 4y <= 12 x,y >=0 Is min z = 2 the right answer ? if not how do i solve this ?
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1answer
68 views

Application of queueing theory

Jake's Machine Shop contains a grinder for sharpening the machine cutting tools. A decision must now be made on the speed at which to set the grinder. The grinding time required by a machine operator ...
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1answer
55 views

Relation between arg min of two functions

When is $u_F(x) = \underset{u}{\text{argmin}}(F_1(x),\cdots,F_u(x),\cdots,F_U(x))$ $\le$ $\underset{u}{\text{argmin}}(G_1(x),\cdots,G_u(x),\cdots,G_U(x)) = u_G(x)$ where $u \in \{1,2,\cdots,U\}, x \in ...
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0answers
23 views

Highest (lowest) index of positive time-indexed variable

I have a simple problem involving a variable $x_{it}$ representing the amount of a resource allotted to a task $i$ in time $t$. The quantity of the (renewable) resource is constrained at a value $R$ ...
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1answer
29 views

linrar programming [closed]

A fuel manufacturing company wants to mix two fuels (A and B) for its trucks to minimize cost. it needs no fewer than 3,000 litre to run its trucks during the next month. it has a maximum fuel storage ...
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1answer
80 views

How to enforce a constraint that a decision variable can only take 1 of $k$ integer values?

How would you enforce the constraint that $x$, a decision variable, can only take values -3, 7, or 19? I think I probably need to introduce a binary variable here but not sure where to start. Thanks. ...
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1answer
66 views

How to solve a linear program with OR constraints

I have $n$ people. I want assign them to $c$ jobs. A job may be not assigned at all or there must be a minimum and maximum number of people assigned to it. $n$ is about 4000 and $c$ is about 1000. ...
3
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0answers
55 views

Operational Research. (Ressource Management)

I am looking for a solution that i know exists already in the field of "Operational Research"... I Just can't put my finger on the name of the thing. An heuristic to solve a very common and simple ...
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0answers
35 views

Farkas lemma corollary and duality

Which is the application of the Farkas lemma corollary on duality? Let $$ Ax \leq b $$ be a system of inequalities, and let $$ P = \{y^TA = 0, y^Tb \leq -1, y \geq 0\} $$ be a polyhedron. May I say ...
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0answers
59 views

Simply formulated but hard problem on system of linear equations

When does the below system has a solution? $$AX=B\\ X > 0$$, where $A$ is $n\times n$ symmetric positive definite matrix and $X$ is a $n\times 1 $ column vector. Note: (I'm trying to use Farka's ...
3
votes
2answers
185 views

Functions minimized at the median of their arguments

I am doing research on problems of location of a public facility on a network which lead me to the following question. Is there an interesting way to characterize the class of functions $f : ...
0
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1answer
123 views

How to answer this linear algebra question, which is related to operations research?

I have the following question: Let $A$ be an $(m \times n)$-matrix and $b$ a vector in $\mathbb{R}^m$. The system of inequalities $Ax \leq b$ has a solution $x \geq 0$, if and only if $yb \geq 0$ for ...
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0answers
50 views

Travelling Salesman on Subset of Points

I'd like to solve the travelling salesman problem, except that the salesman only needs to travel to a subset of the locations. Each location has exactly one client, and each client has a "type". For ...
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0answers
35 views

Finding the dual of a linear program

I have an exam next week and I would like to make sure I am doing this problem correctly and I would also appreciate if somebody could explain to me the purpose of duality? What is the ultimate goal ...
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1answer
1k views

Linear Programming Inventory Problem

I'm still trying to get used to the nature of these problems and I'd appreciate some further explanation. ...
2
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1answer
821 views

Linear Programming: Three variable graphical solution

A small bank offers three type of loans: housing loans at $8.50$% interest, education loans at $13.75$% interest rates, and loans to senior citizens at $12.25$% interest. Further, it needs to ...
3
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2answers
804 views

Financial Linear Programming Problem

I'm very new at linear programming and I'm trying to figure out a way to approach this problem below: ...
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0answers
608 views

Graphically solving a Linear Programming Problem?

I was given the following linear programming problem and have been asked to find all optimal solutions graphically. I am quite new to the subject, so please forgive my naivety. ...
4
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1answer
100 views

Mathematical formulation in operations research

Does anyone know how I would enforce the following constraints using a mathematical formulation? Any help or feedback is appreciated. a) If person A is given project 1, then person D must be given ...
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0answers
22 views

Complexity of Earlist Avaible Due Date for Scheduling Problem 1|ri, pi=1|Lmax

Let us consider the scheduling problem 1|ri,pi=1|Lmax (basically, this means there is one machine on which we have to schedule n jobs (all with identical procssing time 1) in such a way that the ...
0
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1answer
49 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 ...
0
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1answer
69 views

How would I answer this question? (Reworded)

George wants your help to work out how many of each type he should stock in order to maximise his profit. There are three types of Snackboxes: A, B and C. A and C both cost 5 to produce, and B cost 7 ...
0
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3answers
71 views

Summation notation [duplicate]

Does anyone have any suggestions as to how I would be able to formulate this problem using summation notation for those of you who are familiar with it? Hermione has been busy packing her bag with ...
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2answers
87 views

Operations research - summation notation [duplicate]

Outline: Hermione has been thinking about the imminent return of the Dark Lord, so she has been busy packing her bag with all the items required for her survival. Because she has so many different ...
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1answer
58 views

Integral Polyhedra: Integer on each face

The general topic is unimodular matrices and integral polyhedra. I am really new to this field and I am studying for an exam in an advanced operations research course. In this case we are always ...
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1answer
147 views

Operations research - linear programming help appreciated.

George Weasley, the owner of Weasleys' Wizard Wheezes, recently found that his Skiving Snackboxes have become extremely popular amongst the students at Hogwarts School of Witchcraft and Wizardry, and ...
0
votes
1answer
169 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 ...
2
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1answer
69 views

Software for Binary Integer Linear Programs

I am aware that there is good software out there to solve integer linear programs (ILPs). However, is there (preferably free or low cost) software I could use to solve large binary integer linear ...
0
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1answer
107 views

Partial linear relaxation yields an integer solution

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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0answers
60 views

Infinite loop in column generation algortihm

I have to program the following: Input: I have k commodities that have to go from place i to j with a certain demand There are n nodes and the cost for traveling one piece between the nodes i and j ...
2
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1answer
303 views

How to solve this LP problem as a Dynamic Programming problem?

The standard form LP problem is $$\min -3x_1-7x_2-10x_3 \text{ s.t. }$$ $$x_3\leq 2$$ $$40x_3+40x_2+20x_1\leq 180$$ $$x_1,x_2,x_3\geq 0$$ My last lecture covered the Bellman equation ...
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1answer
418 views

Cost-to-go form of Dynamic Programming algorithm?

My lecture of Mat-2.3148 (Finnish) defines dynamic-programming-algorithm so that$J_N(x_N)=g_N(x_N)$ and $J_k(x_k)=\min_{u_k}\left\{g_k(x_k,u_k)+J_{k+1}(f_k(x_k,u_k))\right\}$ where the state ...