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

learn more… | top users | synonyms

-2
votes
0answers
20 views

MSc research topics in Global Optimization [on hold]

Currently I am a masters student in Applied Mathematics. I am interested in Global Optimization, and I was looking for a research topic around it. I was told by my Professor (advisor) to bring a ...
0
votes
0answers
16 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: $$d(H,S) = \sup_{h \in H} \inf_{s \in S}d(h,s)$$ Here's some extremely dodgy intuition. Imagine $S$ is a set of ...
1
vote
0answers
17 views

Linear programming: choosing entering variable

maximize 10π‘₯1 + 12π‘₯2 +12π‘₯3 subject to π‘₯1 + 2π‘₯2 + 2π‘₯3 + π‘₯4= 20 2π‘₯1 + π‘₯2 + 2π‘₯3+π‘₯5= 20 2π‘₯1 + 2π‘₯2 + π‘₯3 +π‘₯6= 20 π‘₯1, … , π‘₯6 β‰₯ 0 This is my first step for simplex tableau x1 x2 ...
0
votes
1answer
50 views

Max $z = x_1(1-x_2)x_3$ s.t. $x_1 - x_2 + x_3 \le 1$

Using dynamic programming, Maximise $$z = x_1(1-x_2)x_3$$ subject to $$x_1 - x_2 + x_3 \le 1$$ $$x_1, x_2, x_3 \ge 0$$ Here's the outline of my solution 1. How is it? Let ...
0
votes
2answers
22 views

Simulation methods and generating random variables

Twenty aircraft are sent to bomb a target that is rectangular in shape. It has dimensions 150m by 50m. Each aircraft makes a bombing run along the horizontal x axis and drops one bomb. The point ...
0
votes
0answers
20 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. What I tried: The short version is that unbounded primal means a column ...
2
votes
1answer
93 views

Convert a piecewise linear function into a linear optimisation problem.

Consider $$f(x) = \left\{\begin{matrix} 1-x, & 0 \le x < 1\\ x-1, & 1 \le x < 2\\ \frac{x}{2}, & 2 \le x \le 3 \end{matrix}\right.$$ where $x \ge 0$. Convert $$\min z = ...
0
votes
1answer
31 views

Dependence of the derivative of a pseudo-Boolean function on its variables

I am going through Pseudo-Boolean optimization by Boros et al. In the section 2, the paper introduces the idea of derivative and residual of a peudo-Boolean function. It is claimed that both ...
0
votes
0answers
22 views

Travelling Salesman Variation

Is there a name for this variation and a recommended algorithm for solving this problem: You have a large boat with many leaks on it. As soon as you patch a leak, it resets, and slowly begins ...
1
vote
1answer
18 views

Is Graph with multiple-inputs and multiple-outputs called MIMO?

MIMO (systems with multiple-inputs and multiple-outputs) is a term in engineering areas and applied mathematics such as process-control and wireless communication. Suppose you have a directed graph ...
0
votes
0answers
32 views

Weights in goal programming

I'm not quite convinced about assigning weights in goal programming. Here is an example formulation problem. What I tried: Let $x_j$ be the number of minutes for ad $j = R, T$ We want to ...
2
votes
0answers
50 views

How to model task scheduling with constraints

I am trying to model a task scheduling with constraints, in order to understand which model or algorithm is better suited to compute an optimal solution. Suppose I have $n$ jobs with CPU loads ...
0
votes
0answers
15 views

Linear programming exercise verification

I am working on this exercise (translation mine): A Motel provides a 24 hour service and needs a minimum number of workers depending on the time slot: ...
1
vote
0answers
70 views

What programs or websites solve linear integer or goal programming problems?

I don't think I can use Excel. My solver doesn't work so I can't even use Excel for regular linear programming. Something like this but for integer or goal programming. This seems to allow integer ...
0
votes
0answers
17 views

How to find extreme directions?

objective:min $βˆ’3x_1βˆ’2x_2βˆ’x_3$ The set is : $X=\lbrace (x_1,x_2,x_3):2x_1+x_2-x_3\le2; x_1,x_2,x_3\ge0 \rbrace$ Attempt: $2d_1+d_2-d_3\le0$ (a) $d_1+d_2+d_3=1$ and $d_1,d_2,d_3\ge0$ Since from ...
0
votes
0answers
5 views

Parallel series systems defined in OR? Isomorphism to SP-graphs in graph theory?

The series parallel graph definition is inductive with respect to series operation and parallel operation in graph theory. In comparison to series parallel systems in OR (Operations Research) and ...
1
vote
1answer
54 views

Find the optimal solution without going through the ERO's

All I got is that $$12y_1 + 7y_2 + 10y_3 = 2(0) + 4(10.4) + 3(0) + 1(0.4)$$ and $y_2 = 0$ because $x_6$ is in basis. How do I find $y_1$ and $y_3$ without going through the simplex method? I ...
0
votes
0answers
45 views

What values make the solutions in the optimal? infeasible? degenerate? etc

Note that $c_i$'s in the $z_j-c_j$ row are not coefficients of the $x_i$'s. We can use instead $r_1, r_2, r_3$ (r for row). I'm assuming there's a non-negativity constraint. we need to state ...
0
votes
1answer
37 views

Convert non-linear into linear

What I tried: Let $$u_1 = x_1^3$$ $$u_2 = x_2 x_3$$ $$u_3 = x_3^3$$ Then we have $z = u_1 + u_2 + u _3$ s.t. $1 \le u_3 \le 343$ $u_3^{1/3}$ should be integer $u_1 \in \{0, 1\}$ $u_2 \in ...
7
votes
0answers
100 views

When might some a variable leave the basis?

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 ...
0
votes
0answers
25 views

Need optimal tableaus be unique assuming unique solution?

If so, why? If not, do they differ by some ERO/s? That is, they are row equivalent? This is the problem (taken from Chapter 2 here): My classmate gave an optimal tableau that is different ...
0
votes
1answer
45 views

Interpreting optimal tableau in manufacturing

What I tried: 3a1 Make $c_3$ greater than the current $z_3$ so any value greater than 20/3? 3a2 I just do EROs to make the $x_3$ column into $[0, 0, 1]^T$ ? 3b $c_1 \ge 10$? idk 3c ...
0
votes
0answers
16 views

RINS: start - incumbent and relaxation solution have all different values

RINS: Relaxation induced neighborhood search; introduced by Danna et.al in 2010 this paper. I wanted to design an example how the RINS algorithm works to make sure that I have it completly understood ...
1
vote
0answers
36 views

Strict inequalities in an LP problem

So I have to formulate an LP problem out of this scenario converting 3 variables into just 2 variables in order to use the graphical method. I guess I do that by using the demand constraint: I ...
0
votes
0answers
26 views

Use graphical analysis to solve a parameterised LP problem

For $c < 0$, we have no feasible solutions and hence no optimal solutions. For $c=0$, our only feasible solution is $z=0$ obtained by $(0,0)$. For $c > 0$, well... I graphed the ...
1
vote
1answer
32 views

Assigning 2 Tasks to each Agent w/ Hungarian Algorithm?

Suppose I have 4 agents and 8 tasks and I would like to assign each agent 2 tasks each. Is there a way to use the Hungarian Algorithm to solve this problem? I worked it out with 2 agents and 4 tasks ...
2
votes
1answer
26 views

Check that a Nash equilibrium point is given by $\left(0,\frac {1} {2}, \frac {1} {2}\right)$ $\left(0,\frac {1} {2}, \frac {1} {2}\right)$

Given the game matrix \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 0 \\ 1 & 0 & 2 \end{bmatrix} I already see a Nash equilibrium in pure strategies, which is $a_{11}$, ...
0
votes
1answer
71 views

LP problem involving producing assemblies

I have to construct an LP problem based on the ff scenario that might be similar to a scenario in another question (in the sense that I felt the need to use $mod$): The productivities are ...
2
votes
1answer
52 views

How does one use the 'input/hr' column in the table below in setting up the problem?

I have to set up a linear programming problem corresponding to the following scenario: If my understanding of the problem is correct, I use $mod$: Let $i$ be $A$ or $B$. Let $x$ be amount ...
0
votes
0answers
21 views

Find a Nash Equilibrium point in mixed strategies using the simplex algorithm

Here is the game matrix A. First I note that $C_4<C_1$ and $C_4 <C_3$ So I can remove $C_1 $ and $C_3$ since they are dominated strategies. Then considering only $C_2,C_4$, I note that $R_3$ ...
0
votes
0answers
27 views

Expected value of Max/Min Function whose gradient is $P(X<x)$

Consider the function $$f(x) = E[min(D,x)]$$ where $D$ is some random variable, with $x$ constant. $$\nabla f(x) = P(D\geq x) $$ Is there a function $g(x)$ such that $$\nabla g(x) = P(D\leq x) ...
-1
votes
1answer
29 views

mutivariable unconstrained optimization using gradient search procedure [closed]

Multi-variable unconstrained optimization problem: Maximize the function, $$f(x)=2xy+2y-x^2-2y^2$$ using the gradient search procedure.
0
votes
1answer
50 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) ...
1
vote
1answer
51 views

Are my constraints in this LP problem correct? Any redundant?

I have to set up an LP problem based on the situation below: What I tried: Let $b_i$ denote sacks bought at month i (i=1,2,3) Let $s_i$ denote sacks sold at month i (i=2,3,4) We want to ...
0
votes
2answers
71 views

How do I go about splitting up 1 LP problem into 2?

I have to set up a linear programming problem corresponding to the following scenario: From Chapter 2 here.
0
votes
1answer
43 views

LP problem: Does ratio of capacity refer to volume? Weight?

I have to set up an LP problem based on this situation below: What I tried: Let $x_{i,j}$ denote amount of loot i in hold j for i = 1,2,3 corresponding to materials, gold and spice for j = ...
2
votes
1answer
32 views

Graphs with weighted edges and vertices

I am considering a route planning problem, which I try to model with a graph. I understand that 1. to find a shortest path in a graph, we need to know the weights on the edges. 2. as some places are ...
1
vote
0answers
32 views

Primal/Dual Simplex methods clarification

I have several questions regarding these methods. Primal Simplex Method Does the pivot element always have to be a positive entry in the table? Does the RHS always have to be positive in the pivot ...
0
votes
0answers
42 views

What is a Hungarian forest: definition

I have a doubt in the definition of the Hungarian forest. This is from the book Matching theory by Lovasz. Let $G$ be a bipartite graph with partite sets $A,B$ and let $M$ be a matching of $G$. Let ...
0
votes
1answer
49 views

How to configure simplex method to start from a specific point

If I have a linear programming problem e.g. $$\max 2x_1 + x_2$$ with these constraints $$x_1-2x_2 \leq 14$$ $$2x_1-x_2\leq 10$$ $$x_1-x_2Β \leq 3$$ And I want to solve the problem starting from a ...
0
votes
1answer
41 views

Canonical form simplex method

In 2-phases simplex method what kind of operations must be done to get the canonical form tableau? In this step(phase 2 of 2-phases method) after the remotion of artificial variables columns of ...
0
votes
0answers
50 views

Do corner points optimise a linear function over a bounded convex region?

This proof says if $Z_P \ne Z_Q$... ...then $Z$ is maximised (or minimised, I guess) at one of the $\color{red}{\text{endpoints}}$ -- of what exactly? $\overline{PQ}$? So the maximum value of ...
-1
votes
1answer
18 views

Optimization Problem Maximize $z= 60x_1+20x_2$

Restate the absolute value constraint as a combination of two linear constraints: I know how to find the optimal solution (std form, canonical form, simplex algorithm ...etc) I don't know how to put ...
0
votes
1answer
45 views

Polytopes defined by $x_i >=0, Ax = b$ are generic ? (Understanding simplex method)

Consider polytopes in $R^n$ defined by $x_i >= 0, Ax = b$, for $b > 0$. Assume $A$ is of full rank $r$ and $Ax=b$ has solutions. The following properties seems to be correct. I would be ...
0
votes
1answer
129 views

Normalized objective function in optimization problem

I have fairly standard linear optimization model with two objectives \begin{align*} \text{max}\, (f_1 &= 4x_1+5 x_2\,,\,f_2 = 1x_1 + 0x_2 ) \\ \text{subject to}& \\ 1x_1 + 1x_2 ...
4
votes
1answer
54 views

Exercise 2.27 from Bazaraa (LP)

Consider the system $Ax=b$ where $A=[a_1,a_2,...,a_n]$ is an $m \times n$ matrix of rank $m$. Let $x$ be any solution of this system. Starting with $x$, construct a basic solution. There is a hint ...
0
votes
1answer
40 views

Linear Programming Free Variables

I am using a book called Introduction to Operations Research. I'm not sure how to deal with free variables that are not constrained i.e. they could be positive or negative. I understand how any ...
0
votes
0answers
19 views

selecting N integers with constraints

I need to write a program which takes 4 inputs as follows N = The number of integers to be generated ($10 <$ N $< 10000$) Start = The minimum value of the integers ($100 <$ Start) End ...
1
vote
1answer
63 views

MILP optimization constraint formulation

I'm trying to find a sensible way to add constraint for my optimization problem. Lets assume we have binary decision variables $x_i\in\{0,1\}$ and two constraints \begin{align*} \sum\limits_{i=1}^n ...
0
votes
1answer
45 views

Travelling salesman problem as an integer linear program

So the travelling salesman problem is a problem wherein a salesman has to travel through all cities in a way that the total travelling distance is minimal. You can rewrite this as an integer linear ...