In order to produce 1000 tons of non-oxidizing steel for engine valves, at least the following units of manganese, chromium and molybdenum, will be needed weekly: 10 units of manganese, 12 units of chromium, and 14 units of molybdenum (1 unit is 10 pounds). These metals are obtainable from dealers in non-ferrous metals, who, to attract markets make them available in cases of three sizes, \(\mathrm{S}, \mathrm{M}\) and \(L\). One S case costs \(\$ 9\) and contains 2 units of manganese, 2 units of chromium and 1 unit of molybdenum. One \(\mathrm{M}\) case costs \(\$ 12\) and contains 2 units of manganese, 3 units of chromium, and 1 unit of molybdenum. One \(L\) case costs \(\$ 15\) and contains 1 unit of manganese, 1 unit of chromium and 5 units of molybdenum. How many cases of each kind should be purchased weekly so that the needed amounts of manganese, chromium and molybdenum are obtained at the smallest possible cost? What is the smallest possible cost?

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 $$ \begin{aligned} &\mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4}=3 \\ &\mathrm{x}_{1}-\mathrm{x}_{2}+2 \mathrm{x}_{3}+\mathrm{x}_{4}=4 \\ &\mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{x}_{3}-\mathrm{x}_{4}=-1 \end{aligned} $$ \(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4} \geq 0\)

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?

Give an example of a problem that is amenable to linear programming methods.

Consider the following problem: Maximize \(\quad \mathrm{P}=5 \mathrm{x}_{1}+8 \mathrm{x}_{2}\) subject to $$ \begin{aligned} &2 \mathrm{x}_{1}+\mathrm{x}_{2} \leq 14 \\ &\mathrm{x}_{1}+3 \mathrm{x}_{2} \leq 12 \\ &\mathrm{x}_{2} \leq 3 \\ &\mathrm{x}_{1} \geq 0, \mathrm{x}_{2} \geq 0 \end{aligned} $$ Suppose that an additional constraint on \(\mathrm{x}_{1}\) and \(\mathrm{x}_{2}\) is imposed: $$ \mathrm{x}_{1}+\mathrm{x}_{2} \leq \mathrm{K} $$ where \(\mathrm{K}\) is some unspecified amount. How does the solution of \((1),(2)\) and (3) change as \(\mathrm{K}\) varies from zero to very large values?

Players \(\mathrm{A}\) and B simultaneously call out either of the numbers 1 and 2 . If their sum is even, \(B\) pays \(A\) that number of dollars, if odd, A pays B. What kind of strategy should both players adopt?