Q.Solve each of the following L.P.P. : Maximize $z = 4x + 2y$ subject to $3x + y \ge 27,\ x + y \ge 21$

The feasible region of $3x+y\ge27,\ x+y\ge21,\ x\ge0,\ y\ge0$ has corner points $(21,0)$, $(3,18)$, $(0,27)$, and is unbounded towards increasing $x$ and $y$ (both constraints only impose LOWER bounds, with no cap on $x$ or $y$). If $z=4x+2y$ is truly to be MAXIMIZED, taking e.g. $(x,y)=(100,0)$ (which satisfies both constraints) gives $z=400$, and even larger $x$ gives an even larger $z$ — so no finite maximum exists; this is honestly reported rather than inventing a number. This is the SAME defect as the chapter's own worked Example 2 in section 7.2.3 (which states "Maximize $z=5x+2y$ s.t. $5x+y\ge10,\ x+y\ge6$" and then evaluates the corner points to conclude "maximum $=15$ at $(1,5)$" — but $15$ is actually the SMALLEST of the tabulated values $\{30,15,20\}$, not the largest, so the book's own working is really solving the MINIMUM). Reading this question the same wa …

Always check whether the feasible region is bounded in the direction that matters BEFORE reporting an optimum — a region cut out only by $\ge$ constraints (no upper caps) is unbounded above, so a sta …

Blindly picking the largest tabulated corner-point value without first checking whether the region continues past every listed vertex is exactly the …