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
$$\eqalign{ & \,\,\,\,\,\,\,{x_1} \le 4 \cr & \,\,\,\,\,\,\,{x_2} \le 6 \cr & 3{x_1} + 2{x_2} \le 18 \cr & {x_1} \ge 0,\,\,{x_2} \ge 0 \cr} $$

Consider the Linear programme $$(LP)$$
Max $$4x$$ + $$6y$$
Subject to
$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,3x + 2y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,2x + 3y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x,y \ge 0 \cr} $$

The dual for the $$LP$$ is

Consider the Linear programme $$(LP)$$
Max $$4x$$ + $$6y$$
Subject to
$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,3x + 2y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,2x + 3y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x,y \ge 0 \cr} $$

After introducing slack variables $$s$$ and $$t$$, the initial basic feasible solution is represented by the table below (basic variables are $$s=6$$ $$t=6,$$ and the objective function value is $$0$$).

After some simplex iterations, the following table is obtained

From this, one can conclude that