In case of one dimensional heat conduction in a medium with constant properties, $$T$$ is the temperature at position $$x,$$ at time $$t.$$ Then $${{\partial T} \over {\partial t}}$$ is proportional to

A company has two factories $${S_1},$$ $${S_2}$$ and two warehouses $${D_1},$$ $${D_2}$$ . the supplies from $${S_1}$$ and $${S_2}$$ are $$50$$ and $$40$$ units respectively. Warehouse $${D_1},$$ requires a minimum of $$20$$ units and a maximum of $$40$$ units. Warehouse $${D_2},$$ requires a minimum of $$20$$ units and, over and above, it can take as much as can be supplied. A balanced transport-ation problem is to be formulated for the above situation. The number of supply points, the number of demand points, and the total supply (or total demand) in the balanced transportation problem respectively are

Consider a linear programming problem with two variables and two constraints. The objective function is: Maximize $${x_1} + {x_2}.$$ The corner points of the feasible region are $$(0,0), (0,2), (2,0)$$ and $$(4/3, 4/3).$$

Let $${y_1}$$ and $${y_2}$$ be the decision variables of the dual and $${v_1}$$ and $${v_2}$$ be the slack variables of the dual of the given linear programming problem. The optimum dual variables are

Consider a linear programming problem with two variables and two constraints. The objective function is: Maximize $${x_1} + {x_2}.$$ The corner points of the feasible region are $$(0,0), (0,2), (2,0)$$ and $$(4/3, 4/3).$$

If an additional constraint $${x_1} + {x_2} \le 5$$ is added, the optimal solution is