Definition of linear programming problem model decision variable, objective function, constraints and method of LPP
DEFINITION OF LINEAR PROGRAMMING
DECISION VARIABLE, OBJECTIVE FUNCTION,
CONSTRAINTS AND METHOD OF LPP
Presented by Sunny Mervyne Baa
LINEAR AND PROGRAMMING’.
• The world linear stand for indicating the relationships between different variables of degree
one whereas another word programming means planning and refers to the process of
selecting best course of action from various alternatives.
• Thus, linear programming is a mathematical technique for allocating limited resources is
• In the words of William M. Fox, “Linear programming is a planning technique that permits
some objective function to be minimized or maximized within the framework of given
• Linear Programming (LP) is a mathematical modelling technique useful for allocation of
limited resources such as material, machines etc to several competing activities such as
projects, services etc.
• By George B. Dantzig
• Decision variables describe the quantities that the decision makers would like to
• hey are the unknowns of a mathematical programming model.
• Typically we will determine their optimum values with an optimization method.
• In a general model, decision variables are given algebraic designations such as .
• The number of decision variables is n, and is the name of the jth variable. In a
specific situation, it is often convenient to use other names such as or or .
• There must be clearly defined objective which can be stated in quantitative way.
• In business problems the objective is generally profit maximization or cost
• The objective function evaluates some quantitative criterion of immediate
importance such as cost, profit, utility, or yield. The general linear objective
function can be written as coefficient of the jth decision variable. The criterion
selected can be either maximized or minimized.
• A constraint is an inequality or equality defining limitations on decisions.
Constraints arise from a variety of sources such as limited resources, contractual
obligations, or physical laws. In general, an LP is said to have m linear constraints
that can be stated as
• One of the three relations shown in the large brackets must be chosen for each
constraint. The number is called a "technological coefficient," and the number is
called the "right-side" value of the ith constraint. Strict inequalities (<, >, and ) are
not permitted. When formulating a model, it is good practice to give a name to
each constraint that reflects its purpose.
• The business problems involving two variables can be easily solved by drawing the graph for various
constraints. Following are the steps in graphical solution of linear programming problem (LPP):
• 1. Formulate LPP by writing the objective function (generally maximize profit) and the constraints.
• 2. Constraints are changed into equalities.
• 3. Plot the constraints on the graph.
• 4. Identify the feasible region and ascertain their coordinates.
• 5. Test which point is most profitable