Find a basic feasible solution to the problem: Maximize $\quad \mathrm{x}_{1}+2 \mathrm{x}_{2}+3 \mathrm{x}_{3}+4 \mathrm{x}_{4}$ while satisfying the conditions $$ \begin{aligned} &\mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4}=3 \\ &\mathrm{x}_{1}-\mathrm{x}_{2}+2 \mathrm{x}_{3}+\mathrm{x}_{4}=4 \\ &\mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{x}_{3}-\mathrm{x}_{4}=-1 \end{aligned} $$ \(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4} \geq 0\)

Show through the simplex method and then graphically that the following linear program has no solution. Maximize \(\quad 2 \mathrm{x}+\mathrm{y}\) subject to $$ \begin{aligned} &-x+y \leq 1 \\ &x-2 y \leq 2 \end{aligned} $$ \(\mathrm{x}, \mathrm{y} \geq 0 .\)

According to the Fundamental Theorem of linear programming, if either a linear program or its dual has no feasible point, then the other one has no solution. Illustrate this assertion with an example.

In a manufacturing process, the final product has a requirement that it must weigh exactly 150 pounds. The two raw materials used are \(\mathrm{A}\), with a cost of \(\$ 4\) per unit and \(\mathrm{B}\), with a cost of \(\$ 8\) per unit. At least 14 units of \(\mathrm{B}\) and no more than 20 units of A must be used. Each unit of A weighs 5 pounds; each unit of \(\mathrm{B}\) weighs 10 pounds. How much of each type of raw material should be used for each unit of final product if we wish to minimize cost?

Solve the following linear programming problem: Maximize \(\quad 6 \mathrm{~L}_{1}+11 \mathrm{~L}_{2}\) subject to: \(\quad 2 \mathrm{~L}_{1}+\mathrm{L}_{2} \leq 104\) \(\mathrm{L}_{1}+2 \mathrm{~L}_{2} \leq 76\) and \(L_{1} \geq 0, L_{2} \geq 0\)

A businessman needs 5 cabinets, 12 desks, and 18 shelves cleaned out. He has two part time employees Sue and Janet. Sue can clean one cabinet, three desks and three shelves in one day, while Janet can clean one cabinet, two desks and 6 shelves in one day. Sue is paid \(\$ 25\) a day, and Janet is paid \(\$ 22\) a day. In order to minimize the cost how many days should Sue and Janet be employed?