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

# Short note: Linear programming

Project Engineering · Chapter 3: Project Planning and Scheduling

- Asked in: 2068 Baisakh (old course), 8 marks; 2067 Asar (old course), 4 marks; 2065 Shrawan (old course), 4 marks (asked 3 times)
- Video: [Linear Programming (Optimization) 2 Examples Minimize & Maximize](https://www.youtube.com/watch?v=Y7e7DCsDUMY) — Mario's Math Tutoring
- 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-note
- Whole chapter: https://www.bctengineer.com/md/semester_7/project_engineering/chapter-3

## Question

Write a short note on linear programming.

## Short answer

Linear programming optimizes a linear objective function subject to linear resource constraints. For a profit maximization problem with constraints 2x+y≤100, x+y≤80, x≤40, the corner-point method yields maximum profit $2600 at x=20, y=60.

## Answer

**Linear programming (LP)** is a mathematical technique to find the best (maximum or minimum) value of a linear objective function, subject to linear constraints on limited resources. In project work it is used for optimum allocation of resources, time-cost trade-off (crashing) and product mix.

#### Components
- **Decision variables:** quantities to be decided ($x$, $y$).
- **Objective function:** to maximise profit or minimise cost.
- **Constraints:** limits on resources such as labour, machine hours, material.
- **Non-negativity:** $x, y \ge 0$.

#### Methods
Graphical method (two variables), simplex method (many variables), and software (Excel Solver).

#### Example
A contractor makes two products, $x$ and $y$, with profit \$40 and \$30 per unit. Maximise $Z = 40x + 30y$ subject to $2x + y \le 100$, $x + y \le 80$, $x \le 40$, $x, y \ge 0$.

Corner points of the feasible region and values of $Z$:

| Point $(x,y)$ | $Z = 40x+30y$ |
|---|---|
| (0, 0) | 0 |
| (40, 0) | 1600 |
| (40, 20) | 2200 |
| (20, 60) | 2600 |
| (0, 80) | 2400 |

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

#### Assumptions and limits
Linearity, certainty of data, divisibility, and a single objective. Real project data are often uncertain.

## Check yourself

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

1. LP optimises:
   A. A nonlinear function
   B. A linear objective subject to linear constraints
   C. Only minimisation without constraints
   D. Probabilities
   Answer: B. LP maximises or minimises a linear objective under linear constraints. (Key point: Definition)

2. Which is NOT a component of LP?
   A. Decision variables
   B. Objective function
   C. Constraints
   D. Float
   Answer: D. Float belongs to scheduling. (Key point: Components)

3. In the example, max Z occurs at:
   A. (40, 20)
   B. (0, 80)
   C. (20, 60)
   D. (40, 0)
   Answer: C. Z = 40(20)+30(60) = 2600, the highest corner value. (Key point: Corner point)

4. LP assumes:
   A. Linearity, certainty, divisibility and single objective
   B. Random data
   C. Many objectives
   D. Integer only
   Answer: A. Real project data are often uncertain. (Key point: Assumptions)

## Related in Project Engineering

- [Basic requirements of linear programming](https://www.bctengineer.com/question.md?key=semester_7%2Fproject_engineering%3Alinear-programming-requirements) — Project Engineering, chapter 3

## Related in other subjects

- [Linear programming problem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alinear-programming-problem) — Mathematics III, chapter 5