# Chapter 5 – Linear Programming

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

In simple words, linear programming is a branch of mathematics that deals with the methods to maximize or minimize profits. This is called optimizing a problem, and hence linear programming finds importance in the field of telecommunication, commerce, management, science, etc.

To begin with, linear programming is a term made by the combination of two words: ‘linear’, which is used to define the relationship between the various variables that have degree one, and ‘programming’, which is the process of finding the best possible solution to the problem with the given alternatives.

Linear programming is made of linear functions, and these functions are in the form of some inequalities or linear equations. Linear programming is exceptionally efficient in finding the utilization of resources in the most optimum way.

Linear programming takes some assumptions while solving a problem. The assumptions are as follows:

• Qualitative terms should be used to express the number of constraints.
• There should be a linear relationship between the objective and the constraint functions.
• It is necessary to optimize the linear function.

#### What are the Components of Linear Programming?

The linear programming of LP is made up of several components. They are as follows:

• Data
• Objective Functions
• Constraints
• Decision Variables

#### Characteristics of Linear Programming

You might be wondering if linear programming only has components or characteristics as well? To answer your questions, linear programming has five characteristics that are used while solving the problem:

• Linearity: The degree of every variable in linear function must be one, i.e., the relationship between the variables in the function shall be linear.
• Constraints: when it comes to resources in the linear functions, then they must be conveyed in the mathematical form.
• Objective Function: In linear programming, it is necessary to specify the objective function qualitatively.
• Non- negativity: In linear programming, the variable cannot have a negative value and must either be 1 or 0.

#### Linear Programming Applications

You might have come across the situation where you wanted to maximize the potential in the best possible way. For example, consider a company trying to maximize the profit while having limitations of labor and materials. In situations like these, optimization plays an important role, and mathematics can help them reach their end goal. But not just in terms of business, linear programming finds its importance in many other fields.

• Engineering: Linear programming helps solve the problems around designing and manufacturing as it can help optimize the shapes.
• Manufacturing: Linear expressions are used to maximize profits in the manufacturing industry.
• Energy industry: Helps in finding ways to optimize the power systems.
• Transportation: Linear programming helps in optimizing the cost and time to achieve the maximum result.

#### Importance of Linear Programming

Since linear programming is helpful in many different fields, the functional problems can be defined as linear programming problems. It has been noted that linear programming problems such as queries related to multi-commodity flow and network flows have contributed to much research on functional algorithms to find the solutions.

#### Applications of Linear Graphs

Linear graphs can be used in our daily life to represent the relationship between the different quantities. For example, if a school hires more teachers, then the number of students taking admission increases as well and vice-versa.

This relationship between the quantities can be in direct or indirect proportions. The relationship can be represented in a graphical manner called linear graphs.

#### Linear Equations Application

Linear equations can help solve real-world problems efficiently, and for that, it is necessary that the real-life problems are converted to mathematical expressions. The expression should represent the relationship between the variables and all the necessary information.

Certain procedures are involved when converting the situation to a mathematical statement. The steps are as follows:

• To convert the problem to a mathematical statement, the algebraic expression should portray the problem efficiently.
• The unknown values must be assigned variables.
• It is necessary to obtain the data, keywords, and phrases by reading the problem multiple times.
• The information obtained must be organized sequentially.
• With the information obtained, you can frame your equation and solve it using systematic techniques.
• To make sure that the solution is correct, retrace it to the problem statement.

#### Difference between Linear and Non- Linear Equations

 Linear Equations Non- Linear Equations The equation is represented by a straight line in the graph. The equation is represented by a curve in the graph. The degree of variables is 1. The variables can have degrees of 2 or more. The equation for linear equation is: Where, x, y= variablesm= slopec= constant The equation for non- linear equation is Where, x, y= variablesa, b, c= constant

#### FAQs

1. What is the difference between the linear and non-linear equation?

A linear equation is represented by a straight line in the graph, while a non-linear equation is represented by a curve.

2. What are the advantages of linear programming?

Linear programming has various advantages:

• Help get the insight into the real-world problems.
• Helps in solving multi-dimensional problems.
• Helps in finding the best possible solution to a problem with given available options.

3. What are the different types of linear programming?

There are many different types of linear programming, and depending on the situation, you can choose the suitable method:

• Simplex method
• R method
• Graphical method
• Open solver method

Formulas are critical, and one must understand the concept behind them. MSVgo is a learning app that is built on the philosophy that understanding a concept is the core of learning and therefore explains the concepts with examples, animations, or explanatory visualization.

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### High School Physics

• Alternating Current
• Atoms
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• Current Electricity
• Dual nature of Radiation and Matter
• Electric Charges and Fields
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• Force And Laws Of Motion
• Gravitation
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• Magnetism and Matter
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• Motion
• Motion in a plane
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• Moving Charges and Magnetism
• Nuclear Energy
• Nuclei
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• Our Environment
• Paths of Heat
• Physical world
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• Semiconductor Electronics: Materials, Devices and Simple Circuits
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• Sources Of Energy
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• Thermal properties of matter
• Thermodynamics
• Units and Measurement
• Vectors, Scalar Quantities and Elementary Calculus
• Wave Optics
• Waves
• Work, Power and Energy

### High School Chemistry

• Acids, Bases and Salts
• Alcohols, Phenols and Ethers
• Aldehydes, Ketones and Carboxylic Acids
• Aliphatic and Aromatic Hydrocarbons
• Alkyl and Aryl Halides
• Amines
• Analytical Chemistry
• Atomic Structure
• Atoms And Molecules
• Basic concepts of Chemistry
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• Chemical Bonding and Molecular Structures
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• Chemistry in Everyday Life
• Chemistry of p-Block elements
• Chemistry of Transition and Inner Transition
• Classification of Elements
• Coordination Compounds
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• Electrochemistry
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### High School Biology

• Absorption and Movement of Water in Plants
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• Cell – Unit of Life
• Cell Cycle and Cell Division
• Cell Division and Structure of Chromosomes
• Cell Reproduction
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• Chemical Coordination and Integration
• Circulation
• Control And Coordination
• Crop Improvement
• Digestion and Absorption
• Diversity In Living Organisms
• Ecosystem
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• Excretory Products and their Elimination
• Flowering Plants
• Genes and Chromosomes
• Health and Diseases
• Health and Its Significance
• Heredity And Evolution
• Heredity and Variation
• How Do Organisms Reproduce?
• Human Diseases
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• Human Health and Disease
• Human Population
• Human Reproduction
• Hygiene
• Improvement In Food Resources
• Integumentary System- Skin
• Kingdom Fungi
• Kingdom Monera
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• Life Processes
• Locomotion and Movement
• Microbes in Human Welfare
• Mineral Nutrition
• Molecular Basis of Inheritance
• Morphology of Flowering Plants
• Neural Control And Coordination
• Nutrition in Human Beings
• Organism and Population
• Photosynthesis
• Photosynthesis in Higher Plants
• Plant Growth and Development
• Plant Kingdom
• Pollination and Fertilization
• Pollution; Sources and its effects
• Principles of Inheritance and Variation
• Reproduction and Development in Angiosperms
• Reproduction in Organisms
• Reproductive Health
• Respiration in Human Beings
• Respiration in Plants
• Respiratory System
• Sexual Reproduction in Flowering Plants
• Strategies for Enhancement in Food Production
• Structural Organisation in Animals
• Structural Organisation of the Cell
• The Endocrine System
• The Fundamental Unit Of Life
• The Living World
• The Nervous System and Sense Organs
• Tissues
• Transpiration
• Transport in Plants

### High School Math

• Algebra – Arithmatic Progressions
• Algebra – Complex Numbers and Quadratic Equations
• Algebra – Linear Inequalities
• Algebra – Pair of Linear Equations in Two Variables
• Algebra – Polynomials
• Algebra – Principle of Mathematical Induction
• Algebra – Quadratic Equations
• Binomial Theorem
• Calculus – Applications of Derivatives
• Calculus – Applications of the Integrals
• Calculus – Continuity and Differentiability
• Calculus – Differential Equations
• Calculus – Integrals
• Geometry – Area
• Geometry – Circles
• Geometry – Conic Sections
• Geometry – Constructions
• Geometry – Introduction to Euclid’s Geometry
• Geometry – Three-dimensional Geometry
• Geometry – Lines and Angles
• Geometry – Quadrilaterals
• Geometry – Straight Lines
• Geometry – Triangles
• Linear Programming
• Matrices and Determinants
• Mensuration – Areas
• Mensuration – Surface Areas and Volumes
• Number Systems
• Number Systems – Real Numbers
• Permutations and Combinations
• Probability
• Sequence and Series
• Sets and Functions
• Statistics
• Trignometry – Height and Distance
• Trignometry – Identities
• Trignometry – Introduction

### Middle School Science

• Acids, Bases And Salts
• Air and Its Constituents
• Basic Biology
• Body Movements
• Carbon and Its Compounds
• Cell – Structure And Functions
• Changes Around Us
• Chemical Effects Of Electric Current
• Chemistry in Your Life
• Coal And Petroleum
• Combustion And Flame
• Components Of Food
• Conservation Of Plants And Animals
• Crop Production And Management
• Electric Current And Its Effects
• Electricity And Circuits
• Elements and Compounds
• Fibre To Fabric
• Food production and management
• Force And Pressure
• Forests: Our Lifeline
• Friction
• Fun With Magnets
• Garbage In, Garbage Out
• Getting To Know Plants
• Health and Hygiene
• Heat
• Hydrogen
• Life Processes: Nutrition in Animals and Plants
• Light, Shadows And Reflections
• Materials: Metals And Non-Metals
• Matter and Its States
• Metals and Non-metals
• Micro Organisms: Friend And Foe
• Motion And Measurement Of Distances
• Motion And Time
• Nutrition In Animals
• Nutrition In Plants
• Organization in Living Things
• Our Environment
• Physical And Chemical Changes
• Pollution and conservation
• Pollution Of Air And Water
• Reaching The Age Of Adolescence
• Reproduction In Animals
• Reproduction In Plants
• Respiration In Organisms
• Rocks and Minerals
• Separation Of Substances
• Simple Machines
• Soil
• Some Natural Phenomena
• Sorting Materials Into Groups
• Sound
• Stars And The Solar System
• Structure of Atom
• Synthetic Fibers And Plastics
• The Living Organisms And Their Surroundings
• Transfer of Heat
• Transformation of Substances
• Transportation In Animals And Plants
• Universe
• Waste-water Story
• Water: A Precious Resource
• Weather, Climate And Adaptations Of Animals To Climate
• Winds, Storms And Cyclones

### Middle School Math

• Addition
• Area and Its Boundary
• Boxes and Sketches
• Data Handling
• Fun With Numbers
• Heavy and Light
• How Many
• Long And Short
• Mapping
• Measurement
• Money
• Multiplication and Factors
• Multiply and Divide
• Numbers
• Parts and Wholes
• Pattern Recognition
• Patterns
• Play With Patterns
• Rupees And Paise
• Shapes And Angles
• Shapes And Designs
• Shapes and Space
• Similarity
• Smart Charts
• Squares
• Subtraction
• Tables And Shares
• Tenths and Hundredths
• Time
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