Introduction, Requirement of LP, Basic Assumptions, Formulation of LP, General Statement of LP, Solution techniques of LP: Graphical Methods, Analytical Methods: Simplex, Big M and Two Phase, Sensitivity Analysis, Primal and Dual Problems, Economic Interpretation.
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Linear Programming - concept
Linear Programming - concept
Linear Programming (LP) is the most useful optimization technique used for the solution of engineering problems. The term ‘linear’ implies that the objective function and constraints are ‘linear’ functions of ‘nonnegative’ decision variables. Thus, the conditions of LP problems (LPP) are
1. Objective function must be a linear function of decision variables
2. Constraints should be linear function of decision variables
3. All the decision variables must be nonnegative
For example,
Linear Programming - assumptions
Linear Programming - assumptions
Linear programs make the following implicit assumptions.
Proportionality. With linear programs, we assume that the contribution of individual variables in the objective function and constraints is proportional to their value. That is, if we double the value of a variable, we double the contribution of that variable to the objective function and each constraint in which the variable appears. The contribution per unit of the variable is constant. For example, suppose the variable xj is the number of units of product j produced and cj is the cost per unit to produce product j. If doubling the amount of product j produced doubles its cost, the per unit cost is constant and the proportionality assumption is satisfied.
Additivity. Additivity means that the total value of the objective function and each constraint function is obtained by adding up the individual contributions from each variable.
Divisibility. The decision variables are allowed to take on any real numerical values within some range specified by the constraints. That is, the variables are not restricted to integer values. When fractional values do not make a sensible solution, such as the number of flights an airline should have each day between two cities, the problem should be formulated and solved as an integer program.
Certainty. We assume that the parameter values in the model are known with certainty or are at least treated that way. The optimal solution obtained is optimal for the specific problem formulated. If the parameter values are wrong, then the resulting solution is of little value.
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