formulate but do not solve the following exercise as a linear - programming problem.\na company manufactures…

formulate but do not solve the following exercise as a linear - programming problem.\na company manufactures x units of product a, y units of product b, and z units of product c. each product is processed in three departments: i, ii, and iii. the total available labor - hours per week for departments i, ii, and iii are 920, 1200, 830, respectively. the time requirements (in hours per unit) and profit per unit for each product are as follows.\n\n| | product a | product b | product c |\n|--|--|--|--|\n| dept. i | 2 | 1 | 2 |\n| dept. ii | 3 | 1 | 2 |\n| dept. iii | 2 | 2 | 1 |\n| profit | $18 | $12 | $14 |\n\nmaximize $p = 18x + 12y + 14z$ subject to the constraints\ndepartment i $2x+1y + 2zgeq920$\nhow many units of each product should the company produce in order to maximize its profit, p in dollars? department ii\ndepartment iii\n$xgeq0$\n$ygeq0$\n$zgeq0$\nresources\nread it\nshow my work (optional)

formulate but do not solve the following exercise as a linear - programming problem.\na company manufactures x units of product a, y units of product b, and z units of product c. each product is processed in three departments: i, ii, and iii. the total available labor - hours per week for departments i, ii, and iii are 920, 1200, 830, respectively. the time requirements (in hours per unit) and profit per unit for each product are as follows.\n\n| | product a | product b | product c |\n|--|--|--|--|\n| dept. i | 2 | 1 | 2 |\n| dept. ii | 3 | 1 | 2 |\n| dept. iii | 2 | 2 | 1 |\n| profit | $18 | $12 | $14 |\n\nmaximize $p = 18x + 12y + 14z$ subject to the constraints\ndepartment i $2x+1y + 2zgeq920$\nhow many units of each product should the company produce in order to maximize its profit, p in dollars? department ii\ndepartment iii\n$xgeq0$\n$ygeq0$\n$zgeq0$\nresources\nread it\nshow my work (optional)

Answer

Explanation:

Step1: Determine the profit - function

The profit per unit of product A is $18$, of product B is $12$ and of product C is $14$. So the profit function to be maximized is $P = 18x+12y + 14z$.

Step2: Set up the constraint for Department I

Product A requires 2 hours per unit, product B requires 1 hour per unit and product C requires 2 hours per unit in Department I. The total available labor - hours in Department I is 920. So the constraint is $2x + y+2z\leq920$ (it should be less - than or equal to since we can't use more hours than available).

Step3: Set up the constraint for Department II

Product A requires 3 hours per unit, product B requires 1 hour per unit and product C requires 2 hours per unit in Department II. The total available labor - hours in Department II is 1200. So the constraint is $3x + y+2z\leq1200$.

Step4: Set up the constraint for Department III

Product A requires 2 hours per unit, product B requires 2 hours per unit and product C requires 1 hour per unit in Department III. The total available labor - hours in Department III is 830. So the constraint is $2x+2y + z\leq830$.

Step5: Add non - negativity constraints

The number of units $x,y,z$ cannot be negative. So $x\geq0,y\geq0,z\geq0$.

Answer:

Maximize $P=18x + 12y+14z$ subject to the constraints: $2x + y+2z\leq920$ $3x + y+2z\leq1200$ $2x+2y + z\leq830$ $x\geq0,y\geq0,z\geq0$