---
title: "Basic requirements of linear programming"
subject: "Project Engineering"
chapter: 3
key: "semester_7/project_engineering:linear-programming-requirements"
source: https://www.bctengineer.com/q/semester_7/project_engineering/linear-programming-requirements
site: BCT Engineering
---

# Basic requirements of linear programming

Project Engineering · Chapter 3: Project Planning and Scheduling

- Asked in: 2070 Chaitra (old course), 6 marks
- Video: [Intro to Linear Programming](https://www.youtube.com/watch?v=K7TL5NMlKIk) — Dr. Trefor Bazett
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/project_engineering/linear-programming-requirements
- Whole chapter: https://www.bctengineer.com/md/semester_7/project_engineering/chapter-3

## Question

Explain the basic requirements of linear programming.

## Short answer

Linear programming requires a linear objective function to maximize or minimize, decision variables, linear resource constraints, non-negativity, alternative feasible actions, and proportionality/additivity. Example: maximize 40x+30y subject to 2x+y<=100, x+y<=80, x,y>=0, giving maximum Rs 2600 at x=20,y=60.

## Answer

**Linear programming (LP)** is a mathematical technique for allocating limited resources (money, labour, machines, material) among competing activities in the best way, so as to maximise profit or minimise cost.

#### Basic requirements

1. **Objective function:** a clearly stated aim, to be maximised (profit) or minimised (cost, time), expressed as a linear function, e.g. $Z = c_1x_1 + c_2x_2$.
2. **Decision variables:** unknown quantities ($x_1, x_2, ...$) whose values are to be found; they are inter-related.
3. **Constraints:** limits on resources, expressed as linear inequalities or equations, e.g. $a_1x_1 + a_2x_2 \le b$.
4. **Non-negativity:** $x_j \ge 0$, as negative production is meaningless.
5. **Alternative courses of action:** more than one feasible solution must exist, from which the best is chosen.
6. **Linearity, proportionality and additivity:** objective and constraints are linear; total use of a resource is the sum of the use of each activity.
7. **Divisibility and certainty:** variables can take fractional values and all coefficients are known and constant.

#### Example

A contractor makes two jobs $x$ and $y$ with profit Rs 40 and Rs 30 per unit. Labour: $2x + y \le 100$; machine hours: $x + y \le 80$; $x, y \ge 0$.

$$
\text{Maximise } Z = 40x + 30y
$$

Corner points of the feasible region: $(0,0)$: $Z = 0$; $(50,0)$: $Z = 2000$; $(20,60)$: $Z = 2600$; $(0,80)$: $Z = 2400$.

**Answer: maximum $Z = $ Rs 2600 at $x = 20$, $y = 60$.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What must the objective function of an LP problem be?
   A. A non-linear function of cost
   B. A set of resource limits
   C. A list of alternatives
   D. A linear function to be maximised or minimised
   Answer: D. LP needs a clear linear aim, such as maximise profit Z = c1x1 + c2x2 or minimise cost. (Key point: Objective function)

2. Why is non-negativity (x >= 0) required?
   A. Negative production has no meaning
   B. To make constraints equalities
   C. To ensure many solutions exist
   D. To make the objective linear
   Answer: A. Decision variables such as quantities produced cannot be negative. (Key point: Non-negativity)

3. What does additivity mean in LP?
   A. Variables must be whole numbers
   B. Total use of a resource is the sum of the use by each activity
   C. Coefficients vary with output
   D. Only one solution exists
   Answer: B. Together with proportionality, additivity keeps objective and constraints linear. (Key point: Additivity)

4. Maximise Z = 40x + 30y with 2x + y <= 100, x + y <= 80. Which corner point gives the maximum?
   A. (50, 0) with Z = 2000
   B. (0, 80) with Z = 2400
   C. (20, 60) with Z = 2600
   D. (0, 0) with Z = 0
   Answer: C. Evaluate Z at all corners; (20, 60) gives 800 + 1800 = 2600, the highest. (Key point: Corner point solution)

## Related in Project Engineering

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