A linear programming problem is shown below.
$$\eqalign{ & Maximize\,\,\,\,3x + 7y \cr & Subject\,\,to\,\,\,3x + 7y \le 10 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4x + 6y \le 8 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x,\,\,y \ge 0 \cr} $$

It has ..............

One unit of product $${P_1}$$ requires $$3$$ $$kg$$ of resource $${R_1}$$ and $$1$$ $$kg$$ of resource $${R_2}$$. One unit of product $${P_2}$$ requires $$2$$ $$kg$$ of resource $${R_1}$$ and $$2$$ $$kg$$ of resource $${R_2}$$. The profits per unit by selling product $${P_1}$$ and $${P_2}$$ are Rs. $$2000$$ and Rs. $$3000$$ respectively. The manufacturer has $$90$$ $$kg$$ of resource $${R_1}$$ and $$100$$ $$kg$$ of resource $${R_2}$$.

The unit worth of resource $${R_2}$$. i.e. dual price of resource $${R_2}$$ in Rs. per $$kg$$ is

One unit of product $${P_1}$$ requires $$3$$ $$kg$$ of resource $${R_1}$$ and $$1$$ $$kg$$ of resource $${R_2}$$. One unit of product $${P_2}$$ requires $$2$$ $$kg$$ of resource $${R_1}$$ and $$2$$ $$kg$$ of resource $${R_2}$$. The profits per unit by selling product $${P_1}$$ and $${P_2}$$ are Rs. $$2000$$ and Rs. $$3000$$ respectively. The manufacturer has $$90$$ $$kg$$ of resource $${R_1}$$ and $$100$$ $$kg$$ of resource $${R_2}$$.

The manufacturer can make a maximum profit of Rs.

Consider the following Linear Programming problem $$(LLP)$$

Maximize: $$Z = 3{x_1} + 2{x_2}$$
$$\,\,$$ Subject $$\,\,$$ to
$$\eqalign{ & \,\,\,\,\,\,\,{x_1} \le 4 \cr & \,\,\,\,\,\,\,{x_2} \le 6 \cr & 3{x_1} + 2{x_2} \le 18 \cr & {x_1} \ge 0,\,\,{x_2} \ge 0 \cr} $$