Unit 2
Meaning of Linear programming problem ;
Linear programming can be defined as a technique that is used for optimizing a
linear function in order to reach the best outcome. This linear function or
objective function consists of linear equality and inequality constraints. We
obtain the best outcome by minimizing or maximizing the objective function.
Requirements of LPP
1. Objective Function: Define the objective function that represents the goal of
the problem. The objective function should be linear, which means it can be
expressed as a linear combination of decision variables, and it should be either
maximized or minimized.
2. Decision Variables: Identify the decision variables that determine the solution
to the problem. These variables represent the quantities or actions that can be
adjusted to achieve the objective. Decision variables should be continuous and
have defined bounds.
3. Constraints: Specify the constraints that limit the feasible region of the
problem. Constraints can be expressed as linear equations or inequalities
involving the decision variables. They represent the limitations or requirements
that must be satisfied.
4. Linearity: The objective function and constraints must have linear
relationships with the decision variables. This means that the decision variables
appear in the objective function and constraints with a power of 1 (no
exponents) and without any multiplication or division between them.
5. Non-negativity: In most cases, decision variables are required to be non-
negative, meaning they cannot take on negative values. This restriction is often
imposed to ensure practicality and feasibility of the solutions.
Assumptions of LLP :
1. Proportionality: The relationships between decision variables and the
objective function, as well as the constraints, are assumed to be linearly
proportional. This means that a change in the value of a decision variable results
in a proportional change in the objective function and constraints.
2. Additivity: The total effect of multiple decision variables on the objective
function and constraints is the sum of their individual effects. In other words,
the contributions of decision variables are assumed to be additive.
, 3. Certainty: It is assumed that all the parameters and coefficients in the
objective function and constraints are known with certainty and do not vary
during the optimization process. Uncertainty or variability in the problem data is
not considered in the basic linear programming model.
4. Divisibility: Decision variables are assumed to be continuous and divisible.
This means that fractional values or continuous ranges of values are allowed for
decision variables, rather than being restricted to discrete or integer values.
5. Non-negativity: Decision variables are typically assumed to be non-negative,
meaning they cannot have negative values. This assumption ensures practicality
and feasibility in many real-world situations, where negative quantities may not
make sense.
Applications of LPP
1. Resource Allocation: Linear programming is often used for optimizing the
allocation of limited resources, such as labor, capital, materials, or production
capacities. It helps determine the most efficient utilization of resources to
maximize output or minimize costs.
2. Production Planning and Scheduling: Linear programming can aid in
optimizing production planning and scheduling by determining the optimal
allocation of resources, scheduling of tasks, and sequencing of activities. It
helps in minimizing production costs, meeting demand requirements, and
improving overall operational efficiency.
3. Supply Chain Management: Linear programming is used in supply chain
management to optimize inventory levels, distribution, transportation, and
logistics decisions. It helps in minimizing transportation costs, optimizing
inventory levels, and improving overall supply chain performance.
4. Financial Planning and Portfolio Optimization: Linear programming is
applied in financial planning to optimize investment portfolios based on risk
and return objectives. It helps in determining the optimal allocation of financial
resources among various investment options.
5. Transportation and Logistics: Linear programming is extensively used in
transportation and logistics management. It helps in optimizing routing,
scheduling, and allocation of transportation resources to minimize costs,
maximize efficiency, and meet customer demands.
6. Marketing and Advertising: Linear programming can be employed in
marketing and advertising campaigns to optimize resource allocation, budget
allocation, and media selection. It helps in determining the optimal mix and
Meaning of Linear programming problem ;
Linear programming can be defined as a technique that is used for optimizing a
linear function in order to reach the best outcome. This linear function or
objective function consists of linear equality and inequality constraints. We
obtain the best outcome by minimizing or maximizing the objective function.
Requirements of LPP
1. Objective Function: Define the objective function that represents the goal of
the problem. The objective function should be linear, which means it can be
expressed as a linear combination of decision variables, and it should be either
maximized or minimized.
2. Decision Variables: Identify the decision variables that determine the solution
to the problem. These variables represent the quantities or actions that can be
adjusted to achieve the objective. Decision variables should be continuous and
have defined bounds.
3. Constraints: Specify the constraints that limit the feasible region of the
problem. Constraints can be expressed as linear equations or inequalities
involving the decision variables. They represent the limitations or requirements
that must be satisfied.
4. Linearity: The objective function and constraints must have linear
relationships with the decision variables. This means that the decision variables
appear in the objective function and constraints with a power of 1 (no
exponents) and without any multiplication or division between them.
5. Non-negativity: In most cases, decision variables are required to be non-
negative, meaning they cannot take on negative values. This restriction is often
imposed to ensure practicality and feasibility of the solutions.
Assumptions of LLP :
1. Proportionality: The relationships between decision variables and the
objective function, as well as the constraints, are assumed to be linearly
proportional. This means that a change in the value of a decision variable results
in a proportional change in the objective function and constraints.
2. Additivity: The total effect of multiple decision variables on the objective
function and constraints is the sum of their individual effects. In other words,
the contributions of decision variables are assumed to be additive.
, 3. Certainty: It is assumed that all the parameters and coefficients in the
objective function and constraints are known with certainty and do not vary
during the optimization process. Uncertainty or variability in the problem data is
not considered in the basic linear programming model.
4. Divisibility: Decision variables are assumed to be continuous and divisible.
This means that fractional values or continuous ranges of values are allowed for
decision variables, rather than being restricted to discrete or integer values.
5. Non-negativity: Decision variables are typically assumed to be non-negative,
meaning they cannot have negative values. This assumption ensures practicality
and feasibility in many real-world situations, where negative quantities may not
make sense.
Applications of LPP
1. Resource Allocation: Linear programming is often used for optimizing the
allocation of limited resources, such as labor, capital, materials, or production
capacities. It helps determine the most efficient utilization of resources to
maximize output or minimize costs.
2. Production Planning and Scheduling: Linear programming can aid in
optimizing production planning and scheduling by determining the optimal
allocation of resources, scheduling of tasks, and sequencing of activities. It
helps in minimizing production costs, meeting demand requirements, and
improving overall operational efficiency.
3. Supply Chain Management: Linear programming is used in supply chain
management to optimize inventory levels, distribution, transportation, and
logistics decisions. It helps in minimizing transportation costs, optimizing
inventory levels, and improving overall supply chain performance.
4. Financial Planning and Portfolio Optimization: Linear programming is
applied in financial planning to optimize investment portfolios based on risk
and return objectives. It helps in determining the optimal allocation of financial
resources among various investment options.
5. Transportation and Logistics: Linear programming is extensively used in
transportation and logistics management. It helps in optimizing routing,
scheduling, and allocation of transportation resources to minimize costs,
maximize efficiency, and meet customer demands.
6. Marketing and Advertising: Linear programming can be employed in
marketing and advertising campaigns to optimize resource allocation, budget
allocation, and media selection. It helps in determining the optimal mix and
