Chapter 3.9, Problem 11P

Explanation of Solution

Formulation of LP:

Let,

${\text{x}}_{\text{i}}$ be the pounds of product sold, for $\text{(i=1,2,3)}$.

${\text{R}}_{\text{1}}$ be the pounds of raw material bought and turned into A and ${\text{R}}_{\text{2}}$ be the pounds of law material bought and turned into B.

Let, ${\text{P}}_{\text{i}}$ be the pounds of product $\text{i}$ processed further, for $\text{(i=1,2,3)}$.

The objective is to maximize chemo’s profit.

$\text{The profit=}\left(\text{cost of product 1 per pound}\right)\left(\text{ponds of product 1 sold}\right)\text{+}\left(\text{cost of product 2 per pound}\right)\left(\text{ponds of product 2 sold}\right)$ $\text{+}\left(\text{cost of product 3 per pound}\right)\left(\text{ponds of product 3 sold}\right)\text{-}\left(\text{cost of raw material per lb}\right)\left(\text{pounds of row material bought and turned into A}\right)$

$\text{-}\left(\text{cost of raw material per lb}\right)\left(\text{pounds of row material bought and turned into B}\right)\text{-}\left(\text{cost per lb}\right)\left(\text{pounds of product A processed further}\right)$ $\text{-}\left(\text{cost per lb}\right)\left(\text{pounds of product B processed further}\right)$

${\text{=10x}}_{\text{1}}{\text{+12x}}_{\text{2}}{\text{+20x}}_{\text{3}}{\text{-5R}}_{\text{1}}{\text{-5R}}_{\text{2}}{\text{-3P}}_{\text{1}}{\text{-2P}}_{\text{2}}$

Thus the objective function is,

Maximize, ${\text{Z=10x}}_{\text{1}}{\text{+12x}}_{\text{2}}{\text{+20x}}_{\text{3}}{\text{-5R}}_{\text{1}}{\text{-5R}}_{\text{2}}{\text{-3P}}_{\text{1}}{\text{-2P}}_{\text{2}}$

Constraint 1:

At most, 1lb of product A can be produced from the raw material purchased.

$\begin{array}{l}\text{(Pounds of raw material bought and turned into A}\hfill \\ \text{-Pounds of pounds of product A proceeded) }\le \text{Pounds of product A}\hfill \end{array}$

$\begin{array}{l}{\text{R}}_{\text{1}}{\text{-P}}_{\text{1}}\le {\text{x}}_{\text{1}}\\ {\text{R}}_{\text{1}}{\text{-P}}_{\text{1}}{\text{-x}}_{\text{1}}\text{=0}\end{array}$

Constraint 2:

At most, 1lb of product A can be produced from the raw material purchased and 0