Constraint Modeling, Feasible Region & Optimal Solutions
Linear Programming is an applied mathematical method used in industrial operations research. Master converting verbal constraints into mathematical inequalities, graphing boundary lines accurately to shade feasible region $R$, and using the objective function line $ax + by = k$ to determine maximum profit or minimum cost.
1. Verbal Phrases into Mathematical Inequalities
| English Phrasing | Bahasa Melayu Phrasing | Mathematical Inequality |
|---|---|---|
| $y$ is at least $k$ / not less than $k$ | $y$ sekurang-kurangnya $k$ / tidak kurang daripada $k$ | $y \ge k$ |
| $y$ is at most $k$ / not more than $k$ | $y$ selebih-lebihnya $k$ / tidak lebih daripada $k$ | $y \le k$ |
| Total of $x$ and $y$ does not exceed $M$ | Jumlah $x$ dan $y$ tidak melebihi $M$ | $x + y \le M$ |
| The number of $y$ exceeds $x$ by at least $d$ | Bilangan $y$ melebihi $x$ sekurang-kurangnya $d$ | $y - x \ge d$ |
| Ratio of $y$ to $x$ is at most $p : q$ | Nisbah $y$ kepada $x$ selebih-lebihnya $p : q$ | $\frac{y}{x} \le \frac{p}{q} \implies qy \le px$ |
| $y$ is at least twice of $x$ | $y$ adalah sekurang-kurangnya dua kali $x$ | $y \ge 2x$ |
2. SPM Examiner Pitfalls in Linear Programming
In SPM Linear Programming, nearly all boundary inequalities are non-strict ($\le$ or $\ge$) and drawn with solid lines. If a strict inequality ($>$ or $<$) is explicitly stated (e.g. "more than"), draw a dashed line. Using solid lines for dashed boundaries loses 1 mark ($K1$).
SPM marking schemes strictly reward showing the objective line $ax + by = k$ on the graph paper! Choose an easy value of $k = \text{LCM}(a, b)$, plot the reference line, and draw a parallel dashed arrow shifting to the furthest vertex in $R$. Do not simply guess points!
If the optimal vertex has non-integer coordinates (e.g. $(14.6, 22.3)$) and the problem involves whole units (cakes, trucks, tables), you MUST test the neighboring integer coordinates inside region $R$. Stating a decimal answer for people or chairs results in an automatic deduction!
3. Full 10-Mark SPM Paper 2 Section C Model Worked Example
(a) Write three linear inequalities, other than $x \ge 0$ and $y \ge 0$, which satisfy all the above constraints. [3 marks]
(b) Using a scale of $2\text{ cm to } 10\text{ units}$ on both axes, construct and shade the feasible region $R$ that satisfies all the constraints. [3 marks]
(c) Using the graph constructed in (b), find:
(i) The maximum number of Model $B$ desks that can be produced if $15$ units of Model $A$ desks are produced. [1 mark]
(ii) The maximum weekly profit earned by the workshop if the profit from each Model $A$ desk is RM 80 and from each Model $B$ desk is RM 60. [3 marks]