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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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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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10 views

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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25 views

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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2answers
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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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18 views

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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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
119 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
111 views

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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25 views

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
14 views

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
38 views

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
28 views

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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Writing a model (just need a check)

3 different products are produced by a company. Each product should be processed in each 3 machines. The time for processing each product in each machine is shown in the info table below and the ...
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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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Find the final tableu knowing the optimal solution

Consider the following linear program $$\max z=5x_1+2x_2+3x_3\\ s.t. x_1+5x_2+2x_3\le b_1\\ x_1-5x_2-6x_3\le b_2 \\ x_1,x_1,x_3\ge0$$ If the optimal solution is reached at $x_1=30,x_5=10,$ write ...
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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
32 views

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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2answers
217 views

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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41 views

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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52 views

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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0answers
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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 ...
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how to understand a manufacturing process w/ uncertain delays & rerouting?

Can anyone tell me what branch of mathematics I need to study in order to better understand the factors that govern the completion times in a multi-step process, in which the thing that's traveling ...
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help: eoq model with backlogging with different demand rates

Consider the following modification of the EOQ model with backlogging. When the inventory is positive, demand arrives at rate lamda1; when the inventory is negative, demand arrives at rate lamda2. ...
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NP-hardness proof of a model with a convex objective

Let $T=(V,E)$ denote a tree. Each node $j \in V$ in the tree has a known attribute $c_j$. From T, construct a bi-directional graph $G' = (V, E')$ where $E' = \{(j,k), (k,j)| (j,k) \in E\}$. Simply, ...
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1answer
39 views

Non linear optimization, KKT

max: $10x_1-2x_1^2-x_1^3+8x_2-x_2^2$ s.t. $x_1+x_2≤2$ $x_1≥0$ $x_2≥0$ I'm supposed to write down the KKT conditions, show that (-1,-1) is not optimal and to find the solution to this problem. ...
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2answers
39 views

Does there exist any Linear Programming model where every Basic Feasible Solution is degenerate?

I need to know if such a Linear Programming model possible where each basic feasible solution shows degeneracy. If it is not possible, then why.
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Big M on convex optimization

Given the following convex optimization problem \begin{equation} \begin{array}{rl} F(x):= \min_{x\in\mathbb{R}^n}\ &x^{T}Ax+b^{T}x\\ \text{s.t.}\ & |x| \leq M \quad (i.e., |x_1| \leq M, |x_2|...
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62 views

LP model that will minimize total cost for the company

Can someone help me figure out the deciding variables and constraints for this problem? A company has signed contracts to deliver 30 units of their product in June, 15 units in July and 15 units in ...
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2answers
72 views

Total number of variables for a general maximum flow varaible

Consider the maximum flow problem with n nodes and m arcs. You are writing a formulation with f as the maximum flow. The total number of variables is _____ ? I have been asked this question for a ...
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1answer
91 views

How to maximize the infinity norm over a convex region?

Consider the following optimization model $$\begin{array}{ll} \text{maximize} & \displaystyle\max_{i \in S} |x_{i}|\\ \text{subject to} & Q(x)+ \displaystyle\sum_{i \in S}|x_{i}| \leq m\end{...
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1answer
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optimization loss due to misperceived probability

Suppose $a$ is chosen to maximize the expected value of $u(a,x)$ under a probability measure of $x$. Image the true distribution is $P(x)$, but the optimization may be conducted under a misperceived ...
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Graph Clustering - Capacitated VRP on a MultiDiGraph

I'm working on the problem of the CVRP on a Multi Directed non-complete graph that has been extracted from OpenStreetMaps using OSMnx. In the extracted graph I have also 'flagged' several delivery ...
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3answers
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Linear Programming optimization with multiple optimal solutions

I am trying to solve the following optimization problem using linear programming (deterministic operations research). According to the book, there are multiple optimal solutions, I don't understand ...
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Practical production scheduling at big factory

At this big factory: $3$ production lines:Line$_1$,Line$_2$,Line$_3$; $10$ products: $p_1,p_2,p_3,p_4,p_5,p_6,p_7,p_8,p_9,p_{10}$; Each production line can produce each of the products, but each ...
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1answer
62 views

How to formulate this optimization problem mathematically?

Suppose we have a discrete function $f(x,y,z),g_1(x,y),g_2(y,z)$ in which $x,y,z\in \{1,...N\}$. I want to find several $\{(x_1,y_1,z_1),...,(x_K,y_K,z_K)\}$ triples such that $g_1(x_i,y_i)$ ranks in ...
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1answer
71 views

Min cost flow change in objective function by changing flow of some arc

I solving a problem that is formulated as a min cost flow problem. After finding the optimal solution, I would like to determine how the objective function will change if I increase/decrease flow of ...
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1answer
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Is my LP model correct?

A Digital Image Printing (DIP) company makes two types of printers: industrial and home printers. The company makes P400 profit from each industrial and a P200 profit from each piece of home printer. ...
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Determine is a solution is basic in linear system.

We know that the maximum number of basic solution of a linear system is given by nCm where n is number of unknowns and m is the rank of coefficient matrix and also of augmented matrix. By setting (n-m)...
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3answers
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Need to create $3-4$ different box sizes and to minimize material waste for a set of $n$ objects that need to fit into these boxes

Excuse the non-mathematical way I've phrased the question. I have the following problem: I have $N$ square paper documents with side lengths between $150$mm and $860$mm. I know each document side's ...