Quantative Modelling Part 2.1
BRING RULER TO EXAM BECAUSE YOU NEED TO DRAW (slide 19 e.g.)
Linear programming:
objectives of business decisions frequently involve maximizing profit or minimizing costs
linear programming uses linear algebraic relationships to represent a firm’s decisions, given
by a business objective, and resource constraints
Steps in application:
1. Identify problem as solvable by linear programming
2. Formulate mathematical model of the unstructured problem
3. Solve the model
4. Implementation
Model components:
Decision variables = mathematical symbols representing levels of activity of a firm. It defines
the decision that the company would like to make (number of production)
Objective function = a linear mathematical relationship describing an objective of the firm. It
is about maximising something or minimising something.
Constraints = requirements or restrictions placed on the firm by the operating environment,
stated in linear relationships of the decision variables
Parameters = numerical coefficients and constants used in the objective functions and
constrains.
Summary of Model Formulation Steps:
- Step 1: Clearly define the decision variables
- Step 2: Construct the objective function
- Step 3: Formulate the constraints
LP Model Formulation:
,Feasible solutions:
feasible solution: does not violate any of the constraints:
Infeasible solution: violates at least one of the constraints
Graphical solution of LP models:
◼ Graphical solution is limited to linear programming models containing only two
decision variables (can be used with three variables but only with great difficulty)
◼ Graphical methods provide visualisation of how a solution for a linear programming
problem is obtained.
To find the optimal solution by using graphical solution:
1. Find feasible region
2. Draw objective function line
3. Find the moving direction of the objective function line
4. Shift the objective function and find the optimum point
,
, Feasible region = area common to both constraints
Binding and Nonbinding activities
◼ Once the optimal solution to an LP has been found, it is useful to classify each
constraint as being a binding constraint or a nonbinding constraint
Binding constraint = if the left-hand side and the right-hand side of the constraint are equal
when the optimal values of the decision variables are substituted into constraints.
Nonbinding constraint = if the left-hand side and the right-hand side of the constraint are
unequal when the optimal values of the decision variables are substituted into the constraint
Slack variables:
◼ Standard form requires that all constraints be in the form of equations (equalities)
A slack variable is (weak inequality) to convert it to an equation
(=)
A slack variable typically represents an unused resource
And it contributes nothing to the objective function value
Linear programming standard form
BRING RULER TO EXAM BECAUSE YOU NEED TO DRAW (slide 19 e.g.)
Linear programming:
objectives of business decisions frequently involve maximizing profit or minimizing costs
linear programming uses linear algebraic relationships to represent a firm’s decisions, given
by a business objective, and resource constraints
Steps in application:
1. Identify problem as solvable by linear programming
2. Formulate mathematical model of the unstructured problem
3. Solve the model
4. Implementation
Model components:
Decision variables = mathematical symbols representing levels of activity of a firm. It defines
the decision that the company would like to make (number of production)
Objective function = a linear mathematical relationship describing an objective of the firm. It
is about maximising something or minimising something.
Constraints = requirements or restrictions placed on the firm by the operating environment,
stated in linear relationships of the decision variables
Parameters = numerical coefficients and constants used in the objective functions and
constrains.
Summary of Model Formulation Steps:
- Step 1: Clearly define the decision variables
- Step 2: Construct the objective function
- Step 3: Formulate the constraints
LP Model Formulation:
,Feasible solutions:
feasible solution: does not violate any of the constraints:
Infeasible solution: violates at least one of the constraints
Graphical solution of LP models:
◼ Graphical solution is limited to linear programming models containing only two
decision variables (can be used with three variables but only with great difficulty)
◼ Graphical methods provide visualisation of how a solution for a linear programming
problem is obtained.
To find the optimal solution by using graphical solution:
1. Find feasible region
2. Draw objective function line
3. Find the moving direction of the objective function line
4. Shift the objective function and find the optimum point
,
, Feasible region = area common to both constraints
Binding and Nonbinding activities
◼ Once the optimal solution to an LP has been found, it is useful to classify each
constraint as being a binding constraint or a nonbinding constraint
Binding constraint = if the left-hand side and the right-hand side of the constraint are equal
when the optimal values of the decision variables are substituted into constraints.
Nonbinding constraint = if the left-hand side and the right-hand side of the constraint are
unequal when the optimal values of the decision variables are substituted into the constraint
Slack variables:
◼ Standard form requires that all constraints be in the form of equations (equalities)
A slack variable is (weak inequality) to convert it to an equation
(=)
A slack variable typically represents an unused resource
And it contributes nothing to the objective function value
Linear programming standard form
