Applied Mathematics · Class 12 Commerce
Ch 10Linear Programming Problem — Class 12 Applied Mathematics, concept-first.
This concept map shows how the chapter fits together. A linear programming problem is built from four ingredients — decision variables, an objective function, linear constraints and non-negativity conditions. Common types of LPP include manufacturing, diet, transportation and assignment problems.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Linear Programming Formulation
Imagine you run a small factory that makes two products: chairs and tables. Each chair gives you ₹200 profit, each table gives you ₹300 profit. You have limited wood (240 units) and limited labour hours (100 hours).
Most relevant Q&A
- A furniture manufacture makes two products: chairs and tables. Processing of these products is done on two machines A and B. A chair require…Free
- A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry shop, and then sent to the machin…Free
- A firm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. In view of the need to ensure certain nutrients…Preview
- There is a factory located at each of the two places P and Q. From these locations, a certain commodity is delivered to each of the three de…Preview
- A company has two groups of inspectors, namely, group A and B, who are assigned to do a quality inspection work. It is required that at leas…Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Concept Map
This concept map shows how the chapter fits together. A linear programming problem is built from four ingredients — decision variables, an objective function, linear constraints and non-negativity con…
Introduction
Linear programming is a powerful mathematical tool for making the best possible decision when resources are limited.
Linear Programming Problem
Linear programming is a mathematical technique for finding the best possible outcome — maximum profit or minimum cost — under a given set of constraints.
Mathematical Formulation of a Linear Programming Problem
Every real-world optimization problem — whether it’s a factory deciding how many units of two products to make, or a farmer choosing how much land to allocate to different crops — can be translated in…
Types of Linear Programming Problems
Linear programming problems in the real world rarely come with a neat, ready-made mathematical model. Instead, they appear in different flavours depending on what you are trying to do — maximise profi…
+−Worked Examplesi5 questions
- Example 1A furniture manufacture makes two products: chairs and tables. Processing of these products is done on two machines A and B. A chair require…Free
- Example 2A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry shop, and then sent to the machin…Free
- Example 3A firm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. In view of the need to ensure certain nutrients…Preview
- Example 4There is a factory located at each of the two places P and Q. From these locations, a certain commodity is delivered to each of the three de…Preview
- Example 5A company has two groups of inspectors, namely, group A and B, who are assigned to do a quality inspection work. It is required that at leas…Preview
Solving a Linear Programming Problem
A linear programming problem is not solved by guesswork — it is solved by systematically exploring the feasible region.
Graphical Method of Solving Linear Programming Problem
The graphical method is the most intuitive way to solve a linear programming problem when you have only two decision variables.
Corner-Point Method
The corner-point method is the most direct way to solve a linear programming problem when the feasible region is bounded.
+−Worked Examplesi6 questions
- Example 6Solve the following Linear Programming Problem graphically. Maximize $Z = 5x + 3y$ Subject to constraints: $3x + 5y \le 15$ $5x + 2y \le 10$…Free
- Example 7Solve the following Linear Programming Problem graphically. Maximize $Z = 2x + 4y$ Subject to constraints: $x + 2y \le 5$ $x + y \le 4$ $x \…Free
- Example 8Minimise $Z = x + 2y$ Subject to constraints: $x \geq 0$ and $y \geq 0$ $2x + y \geq 3$ $x + 2y \geq 6$ Show that minimum $Z$ has more than…Preview
- Example 9Solve the following Linear Programming Problem Graphically. Maximize $Z = 6x + y$ Subject to constraints: $2x + y \geq 3$ $y - x \geq 0$ $x…Preview
- Example 10Maximize $Z = x + y$ Subject to constraints: $x, y \geq 0$ $x - y \leq -1$ $x \geq y$Preview
- Example 11Minimize $Z = 3x + 5y$ Subject to constraints: $x, y \geq 0$ $x + 3y - 3 \geq 0$ $x + y - 2 \geq 0$Preview
Iso-Profit/Iso-Cost Method
The iso-profit (or iso-cost) method is a graphical way to locate the optimal solution when the feasible region is already drawn.
+−Worked Examplesi2 questions
- Example 12Solve the following Linear Programming Problem Graphically. Maximize $Z = 15x + 10y$ Subject to $4x + 6y \leq 360$ $3x \leq 180$ $5y \leq 20…Free
- Example 13Solve the following Linear Programming Problem Graphically. Minimize $Z = 18x + 10y$ Subject to $4x + y \geq 20$ $2x + 3y \geq 30$ and $x \g…Preview
Unit Summary
This section gathers together everything the chapter has covered on the linear programming problem (LPP) — one of the core topics of the CBSE Class 12 Applied Mathematics syllabus.
Check Your Progress
+−Check Your Progress9 questions
- Q1To maintain his health, a person must fulfill certain minimum daily requirements for several kinds of nutrients. Assuming that there are onl…Free
- Q2Vitamins A and B are found in two different foods F1 and F2. One unit of food F1 contains two units of Vitamin A and 3 units of Vitamin B, o…Free
- Q3A brick manufacturer has two depots, A and B with stocks of 30,000 and 20,000 bricks respectively. He receives orders from three builders P,…Free
- Q4A Cooperative Society of farmers has 50 hectares of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated…Preview
- Q5A company has two grades of inspectors, I and II to undertake quality control inspection. At least 1,500 pieces must be inspected in an 8-ho…Preview
- Q6Solve the following Linear Programming Problem graphically: **i.** Maximize $Z = 3x + 2y$ Subject to the constraints: $-2x + y \leq 1$ $x \l…Preview
- Q7Solve the following Linear Programming Problem graphically by using Iso-cost method: Minimize $Z = 4x - 2y$ Subject to the constraints: $x +…Preview
- Q8Solve the following Linear Programming Problem graphically by using Iso-cost method: Maximize $Z = 3x + 9y$ Subject to the constraints: $x +…Preview
- Q9Solve the following Linear Programming Problem graphically by using Iso-cost method: Maximize $Z = 3x + 2y$ Subject to the constraints: $-2x…Preview