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 […]

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? $$\min x_1-2x_2+x_4$$ $$\text{ s.t. } -x_1+x_2=1$$ $$x_2-2x_3+3x_4=10$$ $$x_1+x_3+4x_4=4$$ $$x_1,x_2,x_3,x_4\geq 0$$

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 […]

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

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 […]

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 […]

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.

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) […]

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 […]

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?

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