At the current basic feasible solution (bfs) $v_0 (v_0 \in \mathbb{R}^5)$, the simplex method yields the following form of a linear programming problem in standard form:

minimize $z = -x_1 - 2x_2$

s.t.

$x_3 = 2 + 2x_1 - x_2$

$x_4 = 7 + x_1 - 2x_2$

$x_5 = 3 - x_1$

$x_1, x_2, x_3, x_4, x_5 \geq 0$

Here the objective function is written as a function of the non-basic variables. If the simplex method moves to the adjacent bfs $v_1 (v_1 \in \mathbb{R}^5)$ that best improves the objective function, which of the following represents the objective function at $v_1$, assuming that the objective function is written in the same manner as above?

In a supplier-retailer supply chain, the demand of each retailer, the capacity of each supplier, and the unit cost in rupees of material supply from each supplier to each retailer are tabulated below. The supply chain manager wishes to minimize the total cost of transportation across the supply chain:

The optimal cost of satisfying the total demand from all retailers is _____ rupees (answer in integer).

For a ball bearing, the fatigue life in millions of revolutions is given by $L = \left(\frac{C}{P}\right)^n$, where $P$ is the constant applied load and $C$ is the basic dynamic load rating. Which one of the following statements is TRUE?

Which one of the following failure theories is the most conservative design approach against fatigue failure?