Applied Problems — Manufacturing, Diet & Allocation · Linear Programming

MATApplied Problems

Applied Problems — Manufacturing, Diet & Allocation

This chapter-wide application article covers real-world LPPs: manufacturing (maximise profit), diet/mixture (minimise cost), and resource-allocation/transport (split a shared resource). The unifying method is to formulate variables and constraints, graph the feasible region, evaluate the objective at every corner, and confirm existence of the optimum, watching especially for bounded vs unbounded regions and multiple optimal solutions.

ISC favours these because they test the full modelling-to-interpretation pipeline.

General LPP form

\text{Optimise } Z = ax + by \quad \text{s.t. } a_i x + b_i y \,(\le \text{ or } \ge)\, c_i,\ \ x,y \ge 0

Z

is profit/cost/benefit; constraints encode resource limits (

\le

) or requirements (

\ge

Corner Point Theorem

Z^{*} = \text{optimum of } \{ax_i + by_i\} \text{ over corner points } (x_i,y_i)

On a convex feasible region the linear objective attains its optimum at a corner point; if equal at two corners, every point of the joining edge is optimal.

Mixed-constraint allocation system

\begin{cases} x + 2y \le 120 \\ x + y \ge 60 \\ x - 2y \ge 0 \end{cases}

A typical allocation/transport region mixing an upper-limit, a minimum-delivery, and a balance constraint, with

x,y\ge0

Existence tests on unbounded regions

ax+by>M \;(\text{no max}), \qquad ax+by<m \;(\text{no min})

M,m

are the best corner values; if the relevant open half-plane meets the feasible region, that optimum fails to exist.

Always start by explicitly defining decision variables with units (e.g. $x=$ number of type-A units, $y=$ tonnes on route $R_1$ ).
Translate each English phrase: "at most" $\Rightarrow \le$ , "at least" $\Rightarrow \ge$ , "exactly" $\Rightarrow =$ .
Bounded region: both the maximum and minimum exist and occur at corners. Unbounded region: an optimum may fail to exist, so always apply the open half-plane test.
When the objective line is parallel to a binding edge (proportional coefficients), there are infinitely many optimal solutions along that edge.
For a $Z=x+y$ style objective on an unbounded $\ge$ region, the minimum exists at a corner but the maximum is typically $+\infty$ (does not exist).
Interpret the answer in context: state how many items to produce / units to mix and the resulting profit or cost.
Decision variables that count objects should be non-negative; some board problems also intend integer values, though corner solutions here are usually integral.
Double-check that each candidate corner satisfies ALL constraints before accepting it (a line intersection may lie outside the feasible region).

Where the marks go

Misclassifying "at most" vs "at least", flipping a $\le$ into a $\ge$ and inverting the whole region.
Declaring a maximum on an unbounded region without the $ax+by>M$ open half-plane test.
Accepting a line-intersection point as a corner without checking it satisfies the remaining constraints.
Missing the multiple-optimal-solutions case when the objective is parallel to an edge (e.g. $Z=4x+2y$ with edge $2x+y=10$ ).

How the board asks it

Applicationmanufacturing maximise-profit formulation
A manufacturer makes two products $A$ and $B$ . Each unit of $A$ needs $1$ hour on machine $M_1$ and $3$ hours on machine $M_2$ , while each unit of $B$ needs $2$ hours on $M_1$ and $1$ hour on $M_2$ . Machine $M_1$ is available for $10$ hours and $M_2$ for $15$ hours per day. If the profit is $\text{₹}20$ per unit of $A$ and $\text{₹}30$ per unit of $B$ , formulate the LPP and find graphically the number of units of each product to be made daily to maximise the profit.
Applicationdiet/mixture minimise-cost formulation
A dietician wishes to mix two foods, $X$ and $Y$ , so that the mixture contains at least $10$ units of vitamin A, at least $12$ units of vitamin B and at least $8$ units of vitamin C. One kg of food $X$ contains $1$ , $2$ , $3$ units and one kg of food $Y$ contains $2$ , $2$ , $1$ units of these vitamins respectively. If food $X$ costs $\text{₹}16$ per kg and food $Y$ costs $\text{₹}20$ per kg, formulate and solve the LPP graphically to minimise the cost of the mixture.
Numericalthe corner point theorem
Solve the following LPP graphically: Maximise $Z = 5x + 3y$ subject to $3x + 5y \le 15$ , $5x + 2y \le 10$ , $x \ge 0$ , $y \ge 0$ . Shade the feasible region, list all its corner points and hence find the maximum value of $Z$ .
Give reasonsexistence tests on unbounded regions
For the LPP Minimise $Z = 3x + 2y$ subject to $x + y \ge 8$ , $3x + 5y \ge 15$ , $x \ge 0$ , $y \ge 0$ , the feasible region is unbounded. State, with reasons, whether the minimum value of $Z$ exists by applying the open half-plane test, and find it if it exists.
Numericalmultiple optimal solutions along a binding edge
Maximise $Z = 4x + 2y$ subject to $2x + y \le 10$ , $x + 3y \le 15$ , $x \ge 0$ , $y \ge 0$ . Show that the objective line is parallel to one edge of the feasible region and hence state the complete set of points at which $Z$ attains its maximum value.
Multiple choicetranslating constraints into inequalities
A factory needs at least $200$ kg of material P and at most $300$ kg of material Q. If $x$ kg of P and $y$ kg of Q are used, which set of constraints is correct? $\;$ (a) $x \le 200,\ y \ge 300$ $\;$ (b) $x \ge 200,\ y \le 300$ $\;$ (c) $x \ge 200,\ y \ge 300$ $\;$ (d) $x \le 200,\ y \le 300$ , with $x \ge 0$ , $y \ge 0$ throughout. Choose the correct option.

Practise this topic

PreviousMinimisation Problems

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.