Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{ccccc|c} x & y & u & v & P & \text { Constant } \\ \hline 0 & 1 & \frac{5}{7} & -\frac{1}{7} & 0 & \frac{20}{7} \\ 1 & 0 & -\frac{3}{7} & \frac{2}{7} & 0 & \frac{30}{7} \\ \hline 0 & 0 & \frac{13}{7} & \frac{3}{7} & 1 & \frac{220}{7} \end{array} $$

Use the technique developed in this section to solve the minimization problem. $$ \begin{aligned} \text { Minimize } & C=2 x-3 y-4 z \\ \text { subject to } &-x+2 y-z \leq 8 \\ & x-2 y+2 z \leq 10 \\ & 2 x+4 y-3 z \leq 12 \\ & x \geq 0, y \geq 0, z \geq 0 \end{aligned} $$

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{aligned} \text { Minimize } & C=8 x+6 y+4 z \\ \text { subject to } & 2 x+3 y+z \geq 6 \\ & x+2 y-2 z \geq 4 \\ & x+y+2 z \geq 2 \\ x & \geq 0, y \geq 0, z \geq 0 \end{aligned} $$

A veterinarian has been asked to prepare a diet for a group of dogs to be used in a nutrition study at the School of Animal Science. It has been stipulated that each serving should be no larger than \(8 \mathrm{oz}\) and must contain at least 29 units of nutrient \(I\) and 20 units of nutrient II. The vet has decided that the diet may be prepared from two brands of dog food: brand \(\mathrm{A}\) and brand \(\mathrm{B}\). Each ounce of brand A contains 3 units of nutrient \(\mathrm{I}\) and 4 units of nutrient II. Each ounce of brand \(B\) contains 5 units of nutrient I and 2 units of nutrient II. Brand \(A\) costs 3 cents/ounce and brand B costs 4 cents/ounce. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at minimum cost.

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-A vessel has 60 deluxe cabins and 160 standard cabins, whereas a type-B vessel has 80 deluxe cabins and 120 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 360 deluxe and 680 standard cabins for their 15 -day cruise in May. It costs \(\$ 44,000\) to operate a type-A vessel and \(\$ 54,000\) to operate a type-B vessel for that period. How many of each type vessel should be used in order to keep the operating costs to a minimum? What is the minimum cost?

Solve each linear programming problem by the simplex method. $$ \begin{aligned} \text { Maximize } & P=2 x+6 y+6 z \\ \text { subject to } & 2 x+y+3 z \leq 10 \\ & 4 x+y+2 z \leq 56 \\ & 6 x+4 y+3 z \leq 126 \\ & 2 x+y+z \leq 32 \\ & x \geq 0, y \geq 0, z & \geq 0 \end{aligned} $$