The feasible region for a $LPP$ is shown in the following figure. Evaluate $Z = 4x + y$ at each of the corner points of this region. Find the minimum value of $Z$,if it exists.
(N/A) The lines $x + 2y = 4$ and $x + y = 3$ intersect at $(2, 1)$. From the figure,it is an unbounded shaded region with the corner points $A(4, 0)$,$B(2, 1)$,and $C(0, 3)$.
We evaluate $Z = 4x + y$ at these corner points:
The smallest value obtained is $3$ at $(0, 3)$. Since the region is unbounded,we must check if $Z < 3$ has any common points with the feasible region.
Graphing the inequality $4x + y < 3$,we observe that the open half-plane $4x + y < 3$ has no common point with the feasible region. Therefore,the minimum value of $Z$ is $3$ at the point $(0, 3)$.
We evaluate $Z = 4x + y$ at these corner points:
| Corner points | Corresponding value of $Z$ |
| $(4, 0)$ | $Z = 4(4) + 0 = 16$ |
| $(2, 1)$ | $Z = 4(2) + 1 = 9$ |
| $(0, 3)$ | $Z = 4(0) + 3 = 3$ |
The smallest value obtained is $3$ at $(0, 3)$. Since the region is unbounded,we must check if $Z < 3$ has any common points with the feasible region.
Graphing the inequality $4x + y < 3$,we observe that the open half-plane $4x + y < 3$ has no common point with the feasible region. Therefore,the minimum value of $Z$ is $3$ at the point $(0, 3)$.
Explore More
Similar Questions
Determine graphically the minimum value of the objective function
$Z = -50x + 20y$ .....$(1)$
subject to the constraints:
${2x - y \geqslant -5}$ .....$(2)$
${3x + y \geqslant 3}$ .....$(3)$
${2x - 3y \leqslant 12}$ .....$(4)$
${x \geqslant 0, y \geqslant 0}$ .....$(5)$
$Z = -50x + 20y$ .....$(1)$
subject to the constraints:
${2x - y \geqslant -5}$ .....$(2)$
${3x + y \geqslant 3}$ .....$(3)$
${2x - 3y \leqslant 12}$ .....$(4)$
${x \geqslant 0, y \geqslant 0}$ .....$(5)$
Medium
View SolutionThe corner points of the feasible region determined by the system of linear constraints are $(2, 72)$,$(15, 20)$,and $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. The minimum value of $Z$ occurs at:
Maximize and minimize $Z = 3x - 4y$ subject to the constraints $x - 2y \leq 0$,$-3x + y \leq 4$,$x - y \leq 6$,and $x, y \geq 0$.
Difficult
View SolutionThe corner points of the bounded feasible region are $(0,1), (0,7), (2,7), (6,3), (6,0), (1,0)$. For the objective function $Z = 3x - y$:
$(i)$ At which point is $Z$ minimum?
$(ii)$ At which point is $Z$ maximum?
$(iii)$ The maximum value of $Z$ is $\ldots$
$(iv)$ The minimum value of $Z$ is $\ldots$
$(i)$ At which point is $Z$ minimum?
$(ii)$ At which point is $Z$ maximum?
$(iii)$ The maximum value of $Z$ is $\ldots$
$(iv)$ The minimum value of $Z$ is $\ldots$
Difficult
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo