Linear Programming

"OR is the representation of real-world systems by mathematical models together with the use of quantitative methods (algorithms) for solving such models, with a view to optimizing."We can also define a mathematical model as consisting of:
Decision variables , which are the unknowns to be determined by the solution to the model.
Constraints to represent the physical limitations of the system
An objective function

An optimal solution: to the model is the identification of a set of variable values which are feasible (satisfy all the constraints) and which le ad to the optimal value of the objective function.
An optimization model seeks to find values of the decision variables that optimize (maximize or minimize) an objective function among the set of all values for the decision variables that satisfy the given constraints.
Some notes
The mathematical problem given above has the form
all variables continuous (i.e. can take fractional values)
?
a single objective (maximize or minimize)
?
the objective and constraints are linear i.e. any term is either a constant or a constant multiplied by an unknown
(e.g. 24, 4x, 6y are linear terms but xy or x2 is a non linear term)
Any formulation which satisfies these three conditions is called a linear program
(LP).
We have (implicitly) assumed that it is permissible to work in fractions of days
-
problems where this is not permissible and variables must take integer values will be
dealt with under Integer Programming
(IP).