91Ó°ÊÓ

A marketing manager wishes to maximize the number of people exposed to the company's advertising. He may choose television commercials, which reach 20 million people per commercial, or magazine advertising, which reaches 10 million people per advertisement. Magazine advertisements cost \(\$ 40,000\) each while a television advertisement costs \(\$ 75,000\). The manager has a budget of \(\$ 2,000,000\) and must buy at least 20 magazine advertisements. How many units of each type of advertising should be purchased?

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?

Use the branch and bound method to solve the integer programming problem Maximize \(\quad P=2 x_{1}+3 x_{2}+x_{3}+2 x_{4}\) subject to $$ \begin{aligned} &5 \mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4} \leq 15 \\ &2 \mathrm{x}_{1}+6 \mathrm{x}_{2}+10 \mathrm{x}_{3}+8 \mathrm{x}_{4} \leq 60 \\\ &\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}+\mathrm{x}_{4} \leq 8 \\ &2 \mathrm{x}_{1}+2 \mathrm{x}_{2}+3 \mathrm{x}_{3}+3 \mathrm{x}_{4} \leq 16 \\\ &\mathrm{x}_{1} \leq 3, \mathrm{x}_{2} \leq 7, \mathrm{x}_{3} \leq 5, \mathrm{x}_{4} \leq 5 \end{aligned} $$

How would you solve a game with the following payoff matrix: $$ \begin{array}{|l|l|l|l|} \hline \mathrm{B}: & \mathrm{B}_{1} & \mathrm{~B}_{2} & \mathrm{~B}_{3} \\ \hline \mathrm{A}: & & & \\ \hline \mathrm{A}_{1}: & -1 & 0 & 1 \\ \hline \mathrm{A}_{2}: & 3 & 2 & -1 \\ \hline \mathrm{A}_{3}: & -3 & 1 & 0 \\ \hline \end{array} $$