Chapter Overview
🧭Your visual roadmap through the chapter
Linear Programming
Inequalities, terminology, graphical method, corner-point & slope methods
LP of Business Problems
Formulating constraints & objective functions from real scenarios
Marginal Analysis
Marginal cost, marginal revenue, marginal profit
An optimization technique that finds the best (maximum or minimum) value of a linear objective function, subject to a set of linear constraints (inequalities).
Product-mix, resource-allocation and cost-minimisation decisions are all classic business problems solved using linear programming's graphical approach.
Translate word problems into inequalities; plot constraints; identify the feasible region; find the optimum using corner points or slopes; spot redundancy/unboundedness.
Key Definitions
📘Source-based terminology, exam-ready
Linear Programming Problem
A problem requiring the optimum (minimum or maximum) value of a function, subject to certain restrictions.
Key word: optimize + restrictions
Objective Function
In an LP problem, the function to be maximized or minimized.
Key word: maximize/minimize
Constraints
The set of restrictions, given by linear inequalities, that the solution must satisfy.
Key word: restrictions
Feasible Region
The area containing all the (usually infinite) combinations of x and y that satisfy every constraint — the feasible solutions.
Key word: all valid combinations
Optimum Solution
The one feasible point, out of infinite feasible points, that gives the maximum or minimum value of the objective function.
Key word: the best point
Positive (Non-Negativity) Constraints
x ≥ 0 and y ≥ 0. These limit the feasible region to the first quadrant and do not need to be plotted — they are the axes themselves.
Key word: x,y ≥ 0
Corner Points
The points where the borders (edges) of the feasible region cross one another.
Key word: where edges meet
Redundant Constraint
A constraint whose removal does not change the feasible region, and so cannot affect the optimal solution.
Key word: removable, no effect
Core Concept
🎯Explained in exam-oriented language
CONCEPT → Linear Programming
MEANING
An "optimization technique" — taking linear inequalities relating to a situation and finding the "best" value obtainable.
PURPOSE
Find the best combination of x and y that maximizes or minimizes an objective function.
APPLICATION
Maximum profit from a product mix limited by resources; minimum cost across alternative inputs.
CONCEPT → Feasible Region & Optimum
MEANING
The shaded area representing all combinations of x, y allowed by the constraints.
PURPOSE
The optimum always lies at a corner point of the feasible region's outer edge.
APPLICATION
Test each corner in the objective function (corner point theorem), or compare slopes.
Key Rules & Principles
📏Every source rule, exam-ready
RULE 1 The 6-step LP approach
1. Define variables 2. Construct inequalities for constraints
3. Plot the constraints on a graph 4. Identify the feasible region
5. Construct the objective function 6. Identify the optimum x, y
RULE 4 Wording → inequality sign
Maximum limit / "cannot exceed" / "no more than" → use ≤
Minimum limit / "at least" / "must supply no less than" → use ≥
A range (e.g. "5 to 7 units") → write as a double inequality (both ≥ and ≤)
RULE 2 Plotting an inequality line
If it has x- and y-intercepts: ignore the inequality sign, treat as an equation, set x=0 to find the y-intercept and y=0 to find the x-intercept, then join the two points
If it passes through the origin (no intercepts): pick any two points that satisfy the equation and join them
RULE 5 Maximizing vs minimizing
Maximizing (e.g. profit): optimum is the corner point furthest from the origin
Minimizing (e.g. cost): optimum is the corner point closest to the origin
RULE 6 Redundancy & unboundedness
A constraint is redundant if removing it leaves the feasible region unchanged
An LP is unbounded if the objective function can improve without limit inside the feasible region — no finite optimal solution exists
If no combination satisfies every constraint together, there is no feasible region / no feasible solution
RULE 3 Shading the correct side
Test the origin (0,0) in the inequality
If it satisfies the inequality → shade the side near the origin
If it does not satisfy the inequality → shade the side away from the origin
Formula Zone
🧮Every formula supported by the source text
OBJECTIVE FUNCTION (GENERAL FORM)
c1, c2 = contribution/profit or cost per unit of x and y. Maximize for profit/contribution; minimize for cost.
CORNER POINT THEOREM
The optimum solution is the corner giving the highest Z (maximize) or lowest Z (minimize).
SLOPE COMPARISON METHOD
The optimum lies at the intersection of the two constraints whose slopes bracket the objective function's slope.
MARGINAL PROFIT
For a linear objective function, marginal profit is constant — it equals the coefficient of that decision variable.
Marginal Analysis — Definitions Recap
Marginal Cost (MC): the additional cost of producing one more unit; the rate of change of cost with output.
Marginal Revenue (MR): the additional revenue from selling one more unit; in a linear objective function, MR is simply the coefficient of that decision variable, and stays constant.
Calculation Roadmap
🗺️General path for any LP question
1· Read Question
Identify the decision variables and limits
2· Build Constraints
Translate wording into inequalities
3· Plot & Shade
Find intercepts, shade the feasible region
4· Find Corners
Solve intersecting lines simultaneously
5· Optimize & Check
Test corners in objective function, state answer
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Formulating an LP Problem
- 1Define a variable for each product/outcome to be optimized
- 2Write one constraint per limiting resource, using ≤ for maximums, ≥ for minimums
- 3Add non-negativity constraints x,y ≥ 0
- 4Write the objective function to maximize or minimize
METHOD C · Corner Point Theorem
- 1Identify every corner point of the feasible region
- 2Solve pairs of intersecting constraint lines simultaneously for each corner's (x,y)
- 3Substitute each corner's (x,y) into the objective function
- 4Select the corner giving the best (max/min) result
METHOD B · Graphical Method (Plotting)
- 1For each constraint, set x=0 to find the y-intercept, and y=0 to find the x-intercept
- 2Plot and join the two intercepts for each line
- 3Test the origin in each inequality to decide which side to shade
- 4The overlapping shaded area is the feasible region
METHOD D · Slope Comparison Method
- 1Rearrange the objective function and each constraint into y = ... + (slope)x
- 2Rank the slopes in order
- 3Find the two constraints whose slopes bracket the objective function's slope
- 4Solve those two constraints simultaneously for the optimum (x,y)
METHOD E · Testing for Redundancy / Unboundedness / No Feasible Region
- 1Plot all constraints and identify the feasible region
- 2Remove one constraint at a time — if the region is unchanged, it is redundant
- 3If the feasible region is open-ended in the direction of optimization, the LP is unbounded
- 4If no region satisfies every constraint together, there is no feasible solution
Different Cases / Scenarios
🔀Recognise the wording, pick the right route
IF wording says "cannot exceed" / "no more than" / maximum
→ USE ≤ in the constraint
IF wording says "at least" / "no less than" / minimum
→ USE ≥ in the constraint
IF a range is given (e.g. "5 to 7 units")
→ Write two inequalities (≥ lower, ≤ upper)
IF the objective is to maximize (e.g. profit)
→ Optimum is the corner furthest from the origin
IF the objective is to minimize (e.g. cost)
→ Optimum is the corner closest to the origin
IF removing a constraint leaves the feasible region unchanged
→ That constraint is redundant
IF the objective function can improve without limit in the feasible region
→ The LP is unbounded — no finite solution
IF no combination satisfies every constraint at once
→ There is no feasible region / solution
Comparison Tables
⚖️Don't confuse these!
| Corner Point Theorem | Slope Comparison Method | |
|---|---|---|
| Approach | Test every corner's (x,y) directly in the objective function | Compare the objective function's slope with each constraint's slope |
| When useful | Few corner points; straightforward substitution | Quickly narrows down which two constraints bound the optimum |
| Result | Directly gives the best objective function value | Identifies which two lines to solve simultaneously for the optimum |
| Redundancy | Unboundedness | |
|---|---|---|
| Definition | A constraint that does not affect the feasible region | No limit to improving the objective value within the feasible region |
| Effect on feasible region | No change | Infinite extension in the direction of optimization |
| Implication | Can be removed; simplifies the model | No finite optimal solution exists — model needs revising |
| Constraint wording | Sign to use | Example |
|---|---|---|
| Maximum limit | ≤ | Maximum sales demand of 5,000 units → x ≤ 5,000 |
| Minimum limit | ≥ | Supply at least 2,000 units → x ≥ 2,000 |
| Non-negativity | ≥ 0 | Cannot make a negative number of products → x,y ≥ 0 |
Graph Interpretation
📈Reading a linear programming graph
Reading the feasible region
The shaded (often double- or triple-shaded) area is the feasible region. Any (x,y) inside or on its boundary satisfies every constraint. The straight-line boundary of each constraint is its outer limit (maximum constraints) or inner limit (minimum constraints).
Reading corner points
Corner points are where boundary lines of the feasible region cross. The optimum solution always lies at one of these corners — never in the middle of the shaded area.
Reading the objective function line
Since the objective function's value (C) does not affect its slope, it can be plotted at any convenient value. Moving this line outward (maximizing) or inward (minimizing), parallel to itself, shows which corner it last touches — that corner is the optimum.
No feasible region
If, after plotting and shading, the constraints leave no overlapping shaded area, the problem has no feasible solution — there is no combination of x and y that satisfies every constraint at once.
Worked Calculation Logic
✍️Method students must reproduce in the exam
GIVEN → FORMULA → SUBSTITUTION → CALCULATION → ANSWER
A company makes Product X (contribution $8) and Product Y (contribution $12). Direct labour: 3x+2y≤20,000; Machine time: x+2.5y≤12,000; sales demand x≤6,000, y≤4,000. Maximize C=8x+12y.
GIVEN: Objective C=8x+12y; constraints as above; corner points found by solving intersecting lines simultaneously.
FORMULA: Corner point theorem — substitute each corner's (x,y) into C=8x+12y
SUBSTITUTION / CALCULATION: Solve each pair of intersecting constraints for their corner (x,y), then compute C at each corner
FINAL ANSWER: Compare all corner values of C — the corner giving the highest C is the optimal product mix
Exam Question Patterns
📝Formulation — write constraints & objective function from a word problem
Graph reading — match a shaded region to its set of inequalities (or vice versa)
Corner point calculation — solve simultaneous equations for a corner
Optimization — maximize profit/contribution or minimize cost
Redundancy / unboundedness / no feasible solution identification
Marginal cost/revenue/profit calculation questions
Exam Traps
⚠️⚠ WATCH OUT
- ✗ ≤ vs ≥ confusion — "no more than" is ≤, "at least" is ≥; reversing them flips the feasible region.
- ✗ Forgetting non-negativity (x,y≥0) — without it, the feasible region can wrongly extend into other quadrants.
- ✗ Shading the wrong side — always test the origin (or another point, if the line passes through the origin).
- ✗ Maximize/minimize mix-up — picking the nearest corner for a maximization problem, or vice versa.
- ✗ Treating a redundant constraint as binding — check whether removing it actually changes the feasible region.
Quick Decision Tree
🌳Chapter 4 in 60 Seconds
⚡6 steps: variables→constraints→plot→feasible region→objective→optimum
Max limit: ≤ Min limit: ≥
Shade test: plug in origin (0,0)
Corner point theorem: test each corner in Z
Slope method: objective slope sits between two constraint slopes Max → furthest corner; Min → nearest corner
Redundant: removable, no effect on region Marginal Profit = MR − MC
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