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Questions tagged [operations-research]

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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Research articles on Multi-Objective Non-Linear Programming (MONLP)

I'm looking for papers dealing with multi-objective non-linear programming which could help me implement an algorithm to solve my problem. My problem is : Maximize $f(x) = c \cdot x$, while ...
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Vehicle Route assignment with capacity constraint

Problem Background I'm trying to find a solution/model to the following problem: Let's consider a cellular network (mobile network, ie., hexagonal cells) denoted $N$ composed of $|N|$ cells. Each ...
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Linear Programming - Labor Allocation

Looking for guidance on specification for an unknown subset of linear programming. The task at hand: For a firm making staffing allocation decisions, accept exogenous levels of required services (b),...
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Help understanding an example from a paper

I am currently reading a paper titled "Toward Optimal Allocation of Location Dependent Tasks in Crowdsensing" by He, S., Shin, D.H., Zhang, J. and Chen, J (2014). Link for the paper I am studying ...
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Why is the Bellman principle of optimality considered a system of linear equations?

I am currently trying to understand why the Bellman Principle of Optimality is considered a system of linear equations. The Bellman optimality equation, taken from Reinforcement Learning - An ...
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Need help or literature for optimizing problem

This might be long winded, but im struggling to generalize this question. So I'm sorry, but here goes: So I have what I think is/was a linear optimization question. I have a series of vendors and ...
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Why does this happen in the linear program?

Use the BIP brach and cut algorithm to solve the following problem interactively. $$\max \ z=2x_1-x_2+5x_3-3x_4+4x_5\\ s.t. 3x_1-2x_2+7x_3-5x_4+4x_5\le6\\ x_1-x_2+2x_3-4x_4+2x_5\le0 \\ x_j \\ binary$$...
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How is delta value determined in this transshipment problem?

I am trying to understand how this transshipment problem is optimized from step to step. I have the answer on the exercice, but cannot really get it. The main point is clear: We want to transport (in ...
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How to reformulate with Dantzig-Wolfe decomposition technique

I am dealing with the following Binary ILP: \begin{equation*} \label{equation6} minimize \sum_{i=1}^{m}\sum_{j=1}^{n}\sum_{t=0}^{T-p_{ij}}e_{ij}x_{ijt} \end{equation*} subject to \begin{...
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what does small value of some order mean?

I came across the expression that in an asymptotic sense $\lambda$ will be small, it will be of order $\sqrt{\frac{log p }{n}}$. Does it mean $\lambda = O(\sqrt{\frac{log p}{n}})$ or something else?
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Why were the ratios disregarded when forming the model.

I came across this question in a textbook and I dont fully understand why they omitted one of the conditons when they were forming the model for the LPP. A manufacturer produces three models I, II ...
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Stochastic Dynamic Programming - Ross

I am reading Stochastic Dynamic Programming by S. Ross and there are a few things I am having trouble understanding. I was wondering if someone can help. In Example 4.1 - A Gambling Model with ...
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How many iterations are needed when a new constraint is added

Define a non negative linear programming problem as Max $c^T x$ s.t $Ax \leq b$ $x \geq 0$ where $a_{ij}\geq 0, \, b_i \geq 0, \, c\geq 0$ with m rows and n columns of $A$ If at optimality, a ...
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How can I mathematically model and analyze an incremental game like Cookie Clicker?

Recently, I've been interested in the optimization of the infamous incremental game Cookie Clicker. From Wikipedia: The user initially clicks on a big cookie on the screen, earning one cookie per ...
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B&B and simplex algorithm

I'm starting studying OR, I read that when solving PLI problem it's common to use Branch and Bound techinque which "decompose" the problem and solves smaller problems. My question is the following: ...
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Economic Order Quantity with quantity-dependant order price (Operations Research)

In an EOQ inventory model, how can you minimize inventory cost given a Holding Cost (H) and an Ordering Cost (S) that depend solely on the order quantity? For example, if the ordering cost of a ...
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2answers
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What is it called when each element of one vector is greater than each element of another?

I'm currently writing a term paper and while writing some proofs I've used an operator I don't know the name of. Let us assume we have $x_1, x_2 \in \mathcal{R}^n$. Now lets define an operator '$\...
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Adding the sum of absolute values must be equal to 1 as a Constr with a linear solver

I have a big problem with Constr. i was: $$\min \sum \sigma_{x_i}$$ $$s.t \sum x_i = 1, where \space \space0<= x_i<=1$$ and now I need that problem to be: $$\min \sum \sigma_{x_i}$$ $$s.t \...
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1answer
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“Sufficient and necessary” big M constraint

I have been given a problem to translate into a linear model. The relevant part to my question is: we're given $ \{{ 1,..., m}\}$ robots to work on $\{ 1,...,n \} $ products each robot $i$ takes ...
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How to determine the convexity of multiple matrix variables function?

This formula is : $$f(W,V,B) =\|XW-V\|^2_F +\|Y-VB\|^2_F +\operatorname{tr}(V'LV) +2\operatorname{tr}(W'DW),$$ where $X$, $Y$ are constant matrices and $L$ is constant laplace matrix. Suppose $D$ is a ...
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Queueing Theory Help M/M/3

Comtex plc employs three people in its mail room to sort and despatch mail going through its internal mail system. Letters arrive at an average rate of 150 an hour and each employee can deal with 60 ...
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Bus fleet requirement for transporting passengers/baggage between airport terminals

I am trying to determine the optimum number of buses required for loading and unloading of passengers/baggage. The buses perform following tasks: Transport terminating passengers and their carry on ...
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Solve a linear problem using bounded variables method

Consider the following $$\min 3x_1+4x_2\\ s.t. 4x_1+3x_2\ge12 \\ 3x_1+4x_2\le12\\ x_1,x_2\ge0$$ Substitute the first restriction by $x_1\le3$ and solve the LP by bounded variable method. Attempt ...
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Linear program with parameter $t$ as coefficient of basic variable

Consider the following linear problem $$\max tx_1+x_2\\ s.t. 4x_1+3x_2\le12 \\ 3x_1+4x_2\le12\\ x_1,x_2\ge0$$ where the parameter $t$ grows exponentially $t\in[1,\infty).$ Find the sequence of ...
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What is a principal minor of a matrix?

I was going through the book on operation research by Hamdy A.Taha. It referred to principal minor of a hessian matrix. Can someone explain what is meant by a principal minor? Is it different from '...
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linearization Technique_operations research

How can I linearize the following constraint? (X-Y)*(i-p)=<0 where, x and y and ...
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0answers
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Solve transportation problem that includes an M

In this problem I have a problem with the M value.As far as I know its a big number. a) How will I substract the M in the columns and rows, do I simply ignore the M? b) M will never be chosed?
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2answers
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How to formulate as an assignment problem?

How will I formulate this problem as an assignment problem? I really don't know. By definition of assignment problem, all the variables $x_{ij}$ are binary $(0$ or $1)$ and all the supply and ...
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2answers
129 views

Transportation problem into initial simplex tableau

I did (b) . For (a), I got this $$\min 3x_1+2.7x_2+2.9x_3+2.8x_4\\ s.t. x_1+x_2\le 5\\ x_3+x_4\le4\\ x_1+x_3=3\\ x_2+x_4\ge4\\ x_i\ge0$$ The standard form is $$\min 3x_1+2.7x_2+2.9x_3+2.8x_4\\ s.t....
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Obtain an optimal solution for transportation problem

Consider Problem 8.1-1 I did (a) and (b). For (c), should I solve using 1.minimum cost method and then method of multipliers ? or 2.Vogel method and then method of multipliers ? Is there an ...
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1answer
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One redundant equation in linear program?

Consider the general linear programming formulation of the transportation problem (see Table 8.6). Verify that the set of $(m+n)$ functional constraint equations $(m$ supply constraints and $n$ demand ...
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Prove that in finite steps you can know if the solution set of linear programming problem is unbounded [duplicate]

Given is $\max\left\{c^T \cdot x | Ax \leq b, x \geq 0\right\}$ which is a linear programming problem and its solution set $M$. Prove that you can find out in finite steps if $M$ is unbounded. It ...
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LP duality - How is the dual model obtained analytically? Starting from a primal model through mathematical operations and related concepts.

I have a LP model: $$ Max\sum_{i=1}^n c_{j}x_{j}$$ S.t. $$\sum_{i=1}^n a_{i,j}x_{j} \le b_{i}$$ $$ x_{j} \ge 0$$ $$ i=1 \cdots m$$ $$ j=1 \cdots n$$ and I get this (Dual model): $$ ...
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Operation research: two-phases simplex method

I'm preparing just now the operation research exam, but I'm blocked on a topic: the simplex method (on the tableau) with "two-phases" method. Briefly, I've to optimize a problem which has a surplus ...
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1answer
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Probability in Process Control Limit Charts

I am currently taking an Operations Managment class where we are discussing control limits for various processes/tasks. For example, we discuss a machine that produces memory cards of a specific width ...
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In the transportation problem why is it necessary to make m+n-1 allocations?

I understand that if the system has less than m+n-1 allocations then we have 'degeneracy' but why exactly is that a bad thing? And what is the real world effect of degeneracy? Apologies if this ...
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Are we allowed to simplify the goal constraints in goal programming?

I know that we can't simplify the resource constraints while solving goal programming problems. But are we allowed to simplify the goal constraints?
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1answer
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Prove that Dual linear program does not have finite optimal solution

Consider the following $\displaystyle \max z=x+2y\\s.t.-x+y\le-2\\4x+y\le4\\ x,y\ge0$ Find the dual program and prove graphically that D has no finite optimal solution. Solution The dual is given ...
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1answer
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Trying to find the inverse of $B$ knowing the optimal solution

Can one know how is $B^{-1}$ and $\left(\matrix{ b_1 \\b_2}\right)$ defined knowing that $c_BB^{-1}b=150$ and $B^{-1}b=B^{-1}$ $\left(\matrix{b_1\\ b_2 }\right)=\left(\matrix{30 \\ 10}\right)$ ? We ...
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Lagrangian of non-separable multiple variables

I am stuck on how to work out the lagragian of the problem below, the quadratic variables are making me very lost. min $4x_1^2-(x_2+1)^2$ subject to $(x_1-x_2)^2-9 \leq 0$ My best guest currently is ...
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1answer
68 views

Find the final tableau knowing the optimal solution

Consider the following linear program $$\displaystyle \max z=5x_1+2x_2+3x_3\\ s.t. x_1+5x_2+2x_3\le b_1\\ x_1-5x_2-6x_3\le b_2 \\ x_1,x_1,x_3\ge0$$ If the optimal solution is reached at $x_1=30,...
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Mathematical Modeling Approach-The Recruitment Of New Air Hostesses (2)

The Alpha Airlines passenger service director is trying to decide how many new ones flight attendants to recruit and train in the next six months. Requirements in time flights are: The problem ...
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1answer
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Positioning items to maximize separation subject to constraints

Say we want to place n items on the real line. Let us denote the position of item i by $p_i$. We have interval constraints on the position $p_i$, i.e. we are given $l_i, r_i$ such that $l_i \le p_i \...
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Modelling a linear minimization program

Attempt: Let $x_l=$ number of barrel of light crude oil. $x_h=$ number of barrel of heavy crude oil. Then we should have as objective function $z=20x_l+15x_h$ And as conditions s.t. $.4x_l+....
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Transportation problem path finding

I'm trying to understand the transportation problem and implement a solution for it based on this material: http://web.tecnico.ulisboa.pt/mcasquilho/compute/_linpro/TaylorB_module_b.pdf The first ...
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Constructing an extreme direction from a simplex tableau indicating unboundedness

Context: In a question, we are asked to show that a problem does not have a finite optimal solution, then told to construct an extreme direction of the feasible region using the final tableau. The ...
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1answer
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Question about alternative optimal extreme points?

Could someone please check if what I've done is correct? And how could I answer b. ? Thank you. Consider the following $$\max 2x+3y\\ s.t.\ \ x+y\le2\\ 4x+6y\le9\\ x,y\ge0$$ a. Sketch the ...
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A linear program with unbounded optimal solution

Consider the following $$\max 3x_1+x_2\\ s.t. -x_1+2x_2\le0\\ x_2\le4$$ a) Sketch the feasible region b)Verify that the problem has an unbounded optimal solution value Attempt: a) (Here the ...
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Operation Research penalty function and KKT

I have a problem as this. Wish someone could help me! Thanks a lot!
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Find number of reserved machines so that cost be minimum

In a company we have $20$ similar machines . Number of damaged machines has binomial distribution with $p = 0.3$. In other words, $$\mathbb{P}[X = k] = { 20 \choose k}0.3^k 0.7^{20-k}.$$ In order to ...