Articles of operations research

Arbitrage opportunity

Given odds $o_i$ for $i=1,2,\ldots,n$ and the possibility to bet the amount $b_i\in \mathbb{R}$ on each event such that if event $i$ occurs you receive $b_io_i$ and if it doesn’t you recieve $-b_i$. I am trying to find out the condition for arbitrage. My immediate thoughts are that $1/o_i$ represents probability, and since these events […]

Example about the Reduced cost in the Big-M method?

I want to gather examples about the reduced cost in different cases, now for the Big-M method. I hope this makes the methods more accesible. So How does the Big-M method work with the below? $$\min x_1-2x_2+x_4$$ $$\text{ s.t. } -x_1+x_2=1$$ $$x_2-2x_3+3x_4=10$$ $$x_1+x_3+4x_4=4$$ $$x_1,x_2,x_3,x_4\geq 0$$

Convert a piecewise linear non-convex function into a linear optimisation problem.

Update: Problem and solution found here (p. 17, 61), although my prof’s solution (formulation) is different. Convert $$\min z = f(x)$$ where $$f(x) = \left\{\begin{matrix} 1-x, & 0 \le x < 1\\ x-1, & 1 \le x < 2\\ \frac{x}{2}, & 2 \le x \le 3 \end{matrix}\right.$$ s.t. $$x \ge 0$$ into a linear integer […]

Operations research book to start with

for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear optimization, convex optimization and quadratic programming, dynamic programming, multicriterial optimizations (did I miss something?) Thanks, Lucian

How to solve this operation research problem using dual simplex method?

Maximize $$ z = 2x_1 -x_2 +x_3$$ Subject to constraints $$2x_1 + 3x_2 -5x_3 \ge 4$$ $$-x_1 +9x_2 -x_3 \ge 3$$ $$4x_1 +6x_2 +3x_3 \le 8$$ And $x_1, x_2, x_3 \ge 0$ I managed to solve this through simplex method(by 2 stage method) but I was asked solve it using dual simplex method, I found […]

Arbitrage sports betting

Player A vs Player B. Bookie 1 offers 1.36 odds on player A winning. Bookie 2 offers 5.5 on player B winning. We have $1000 in total to bet. How would you place your bets such that profit is maximized? I have been told that this can be solved using linear programming, but I don’t […]

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? Thank you for your help.

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 + N^2 + …) = \lim_{m \to \infty} (I-N) […]

Jeep problem variant: cross the desert with as much fuel as possible

I’m dealing with the following variant of the well-known Jeep problem: A 1000 mile wide desert needs to be crossed in a Jeep. The mileage is one mile / gallon and the Jeep can transport up to 1000 gallons of gas at any time. Fuel may be dropped off at any location in the desert […]

Is Reliability Component a vertex?

The term component has a distinct definition in graph theory from vertex while the terms components and vertices can be mostly the same in Realiability Engineering, my intuition. So how is the term component Operations Research (OR) such as Reliability Engineering usually defined? Is the reliability component a vertex as defined in graph theory?