Question

A retail store stocks two types of customized travel bags, A and B. The store can sell a maximum of 400 type of A bags and a maximum if 300 type B bags per week. However, the store can only store up to 600 bags of both types because of limited storage capacity. The store earnd the profit of 30 per type A bag and a profit of 50 per type B bag. How many of each type of bags should the store keep per week to maximize the profit?

Answer 1

$\displaystyle \text{Let $x_1$ represent bag of type a and $x_2$ represent bag of type b}\\ \text{The linear program in the given problem is }\\ Maximize: 30x_1 +50x_2\\ \text{Subject to: } 400x_1 +300x_2\leq 600\\\ \text{The linear program in its standard form is }\\ Maximize: 30x_1 +50x_2+0x_3\\ \text{Subject to: } 400x_1 +300x_2 +x_3 = 600\\ \text{Next we form our first Tableau from our linear program}\\ \begin{matrix} & & x_1 & x_2 & x_3 \\ & & 30 & 50 & 0 \\\hline x_3 & 400 & 300& 1 & 600 \\\hline & & -30 & -50 & 0 & 0 \\ \end{matrix}\\ \text{Next, we locate the most negative number in the bottom row(-7), and label the column}\\ \text{where it is found the work column , we then form positive ratios by dividing the elements}\\ \text{in the work column by corresponding elements in the last column. Next we label the }\\ \text{the smallest positive ratio, the pivot element. Using elementary row operations we }\\ \text{reduce the pivot element to 1 and other elements in the work column to 0 to obtain}\\ \text{our tableau 2}\\ \begin{matrix} & & x_1 & x_2 & x_3 \\ & & 30 & 50 & 0 \\\hline x_2 & 50 & \frac{4}{3} & 1 & 1 & 600 \\\hline && \frac{110}{3} & 0 &0 &30000\\ \end{matrix}\\ \text{Therefore 50 of $x_2$ should be produced}$