Question

Use simplex method to maximize 𝑓 = 3𝑥 + 5𝑦 + 4𝑧 subject to the conditions 2𝑥 + 3𝑦 ≤ 18 2𝑥 + 5𝑦 ≤ 10 3𝑥 + 2𝑦 + 4𝑧 ≤ 15 and 𝑥, 𝑦, 𝑧 ≥ 0.

Answer 1

One of the theorems of simplex method states that the solution of the linear problem exists at one of the edge points. In our case the region is bounded by 6 planes: $2x+3y=18,2x+5y=10,$ $3x+2y+4z=15$ , $x,y,z=0$. It is possible to find such solution in Maple via the commands:

with(Optimization):

Minimize(-3*x-5*y-4*z,{2*x+3*y<=18,2*x+5*y<=10,3*x+2*y+4*z<=15,x>=0,y>=0,z>=0});

Since it is the minimization problem, the function has the opposite sign.

It is at the intersection of planes $x=0,y=2,z=2.75$. The value is $21$for the original maximization problem. Thus, the solution is at the point $(0,2,2.75)$. The value is $21$.

The manual method of finding solution is the following:

One of conditions from the task formulation is: $3x+2y+4z\leq15$. Thus, $f\leq3y+15$. Using the fact that $x\geq0$ and inequality $2x+5y\leq10$ we get: $y\leq2$. Thus, $f\leq21$. The point that satisfies equality $f=21$ is: $x=0,y=2,z=2.75$. It satisfies equalities $3x+2y+4z=15$, $2x+5y=10$ and $x=0$.