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 \(L_{1} \geq 0, L_{2} \geq 0\)

The Brown Company has two warehouses and three retail outlets. Warehouse number one (which will be denoted by \(\mathrm{W}_{1}\) ) has a capacity of 12 units; warehouse number two \(\left(\mathrm{W}_{2}\right)\) holds 8 units. These warehouses must ship the product to the three outlets, denoted by \(\mathrm{O}_{1}, \mathrm{O}_{2}\), and \(\mathrm{O}_{3} \cdot \mathrm{O}_{1}\) requires 8 units. \(\mathrm{O}_{2}\) requires 7 units, and \(\mathrm{O}_{3}\) requires 5 units. Thus, there is a total storage capacity of 20 units, and also a demand for 20 units. The question is, which warehouse should ship how many units to which outlet? (The objective being, of course, to accomplish this at the least possible cost.) Costs of shipping from either warehouse to any of the outlets are known and are summarized in the following table, which also sets forth the warehouse capacities and the needs of the retail outlets: $$ \begin{array}{|c|c|c|c|c|} \hline & \mathrm{O}_{1} & \mathrm{O}_{2} & \mathrm{O}_{3} & \text { Capacity } \\\ \hline \mathrm{W}_{1} \ldots \ldots \ldots & \$ 3.00 & \$ 5.00 & \$ 3.00 & 12 \\\ \hline \mathrm{W}_{2} \ldots \ldots . . & 2.00 & 7.00 & 1.00 & 8 \\ \hline \text { Needs (units) } & 8 & 7 & 5 & \\ \hline \end{array} $$

Give an example of a two-person non-zero-sum game.

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?

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.

Find nonnegative numbers $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}\( which maximize \)3 \mathrm{x}_{1}+\mathrm{x}_{2}+9 \mathrm{x}_{3}-9 \mathrm{x}_{4}$ and satisfy the conditions $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}-5 \mathrm{x}_{4}=4$ \(\mathrm{x}_{1}-\mathrm{x}_{2}+3 \mathrm{x}_{3}+\mathrm{x}_{4}=0\)