$***********$ I NEED A SOLUTION PROCESS ON HOW TO SOLVE THIS PROBLEM WITH $$SIMPLEX METHOD$***************$

RMC$,$ Inc., is a small firm that produces a variety of chemical products. In a particular

production process, three raw materials are blended $($mixed together$)$ to produce two prod$-$

ucts: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of $\frac{2}{5}$ ton of

material $1$ and $\frac{3}{5}$ ton of material $3.$ A ton of solvent base is a mixture of $\frac{1}{2}$ ton of material $1,$

$\frac{1}{5}$ ton of material $2,$ and $\frac{3}{10}$ ton of material $3.$ After deducting relevant costs, the profit

contribution is $$40$ for every ton of fuel additive produced and $$30$ for every ton of solvent

base produced.

RMC$\text{'}$s production is constrained by a limited availability of the three raw materials.

For the current production period, RMC has available the following quantities of each raw

material:

Raw Material

Material $1$

Material $2$

Material $3$

Amount Available for Production

$20$ tons

$5$ tons

$21$ tons

Assuming that RMC is interested in maximizing the total profit contribution, the problem

formulation is shown here:

Max $40x_1+30x_2,$

$s.t.$

$,\frac{2}{5}{x}_1+\frac{1}{2}{x}_{2}20$ Material $1$

$,\frac{1}{5}{x}_{2}5$ Material $2$

$,\frac{3}{5}{x}_1+\frac{3}{10}{x}_{2}21$ Material $3$

$,x_1,x_{2}0,$

where

$x_1=$ tons $of$ fuel additive produced

$x_2=$ tons $of$ solvent base produced

Solve the RMC problem using the simplex method. At each iteration, locate the basic

feasible solution found by the simplex method on the graph of the feasible region.