Question

Mathematically, how is linear programming applicable? Is it the simplex method?
(a) Yes, the statement is correct.
(b) No, the statement does not imply.
(c) Cannot be Determined
(d) None of These

Answer 1

Hint: In the given question, we are going to take one example to understand the simplex method especially for linear programming. There will be the main equation with maximum or minimum condition given, where certain equations exists/satisfies the main equation in the respective question particularly and then solving it predominantly on the graph (by considering constraint equation by analysing with slope intercept equation for straight line, etc.), we can get the required optimal solution.

Complete step-by-step solution:
Typically, simplex method is a standardized and efficient technique used by drawing and analysing the graphical representation; to solve the linear programming problems which led to a desired optimistic solution for the respective drastic problem/s too which involves the characteristics parameters under the constraint/s motion or solution.
Let’s take an example for more clarification of the problem:-
Q. Solve the following Linear Programming Problem, say, L.P.P.
\[
  {\text{Max: }}z = 11x + 8y \\
  {\text{Sub to: x + y}} \leqslant {\text{6}} \\
  {\text{ x}} \leqslant {\text{4}} \\
  {\text{ y}} \leqslant {\text{6}} \\
  {\text{where, }}x \geqslant 0,y \geqslant 0 \\
 \]
Here, first of all we need to analyse the constraints of the given equation to which it is maximize or minimize (here the equation is maximize),
i.e. Max: $Z=11x+8y$
As a result, we will interpret the given conditions (using the slope-intercept form of straight line i.e. $\dfrac{x}{a}+\dfrac{y}{b}=1$) to which they are subjected with, we get
Step $1$:

Constraints (Equations subjected with…)	Equations in the standard form	$\dfrac{x}{a} + \dfrac{y}{b} = 1$	Intersection points between the axes on the graph
$x + y \leqslant 6$	$x + y = 6$	$\dfrac{x}{6} + \dfrac{y}{6} = 1$	$(6,0)$,$(0,6)$
$x \leqslant 4$	$x = 4$	N.A.	$(4,0)$
$y \leqslant 6$	$y = 6$	N.A.	$(0,6)$

 Step $2$:
Since, from the graph we can say that
$ \Rightarrow $The Feasible region for the respective problem is ‘ABCD’
(Where, ‘$ \leqslant $’ sign assumes that the feasible region exists to the ‘left’ of the respective equation line on the graph where ‘$ \geqslant $’ sign assumes the region at ‘right’ side of the equation line and it does not exists due to unbounded region existing)

As a result, the corners of the feasible region ‘ABCD’ seems that,
D=(0,0); A=(4,0); C=(4,2); D=(0,6)
Step $3$:
Now, in this last step we can get the final optimal or optimum solution for the problem,
So, considering the maximum given equation Max $Z=11x+8y$,
Substituting all the values obtained from the feasible region in the given equation above, we get
(When D=(0,0)),
$
  \therefore z = 11 \times 0 + 8 \times 0 \\
  \therefore z = 0 \\
 $ ... (i)
(When$A = (4,0)$),
$
  \therefore z = 11 \times 4 + 8 \times 0 \\
  \therefore z = 44 \\
 $ ... (ii)
(When $B = (4,2)$),
$
  \therefore z = 11 \times 4 + 8 \times 2 = 44 + 16 \\
  \therefore z = 60 \\
 $ ... (iii)
And,
(When $C = (0,6)$),
$
  \therefore z = 11 \times 0 + 8 \times 6 \\
  \therefore z = 48 \\
 $ ... (iv)
From the above equations (i) , (ii), (iii) and (iv), we get
The maximum value for the given linear problem is $60$ [from (iii)] respectively!
As a result from drawing the respective graph the problem becomes as easy to analyze as the optimal solution,
It seems to be the simplex method than any other system,
Hence, the option (a) is correct.

Note: One must know how to plot the equations on the graph to solve the question. Slope-intercept formula for straight line i.e.$\dfrac{x}{a}+\dfrac{y}{b}=1$ is the condition we analyze the intercepts or the points to which they cut-off. As a result, by putting the given values in the given maximize equation the required answer can be obtained. One must take care of the calculations to be sure of the final answer.