ChapterLINEAR PROGRAMMING12.1 Overview12.1.1 An Optimisation ProblemA problem which seeks to maximise or minimisea function is called an optimisation problem. An optimisation problem may involvemaximisation of profit, production etc or minimisation of cost, from availableresources etc.12.1.2 A Linnear Programming Problem (LPP)A linear programming problem deals with the optimisation (maximisation/minimisation) of a linear functionof two variables (say xand y) known asobjective function subject to the conditions that the variables are non-negative andsatisfy a set of linear inequalities (called linear constraints). A linear programmingproblem is aspecial type of optimisation problem.12.1.3 Objective FunctionLinear function Z = ax + by, where aandbareconstants, which has to be maximised or minimised is called a linear objectivefunction.12.1.4 Decision Variables In the objective function Z = ax+ by, xand yare calleddecision variables.12.1.5 ConstraintsThe linear inequalities or restrictions on the variables of an LPPare called constraints. The conditions x ≥0, y ≥0 are called non-negativeconstraints.12.1.6 Feasible RegionThe common region determined by all the constraintsincluding non-negative constraints x ≥ 0, y ≥ 0 of an LPP is called the feasibleregion for the problem.12.1.7 Feasible Solutions Points within and on the boundary of the feasible regionfor an LPP represent feasible solutions.12.1.8 Infeasible Solutions Any Point outside feasible region is called an infeasiblesolution.12.1.9 Optimal (feasible) SolutionAny point in the feasible region that gives theoptimal value (maximum or minimum) of the objective function is called anoptimal solution.12
Following theorems are fundamental in solving LPPs.242 MATHEMATICSLINEAR INEQUALITIES 24212.1.10 Theorem1 Let R be the feasible region (convex polygon) for an LPP andlet Z = ax+ bybe the objective function. When Z has an optimal value (maximumor minimum), where xand yare subject to constraints described by linearinequalities, this optimal value must occur at a corner point (vertex) of the feasibleregion.Theorem2 Let R be the feasible region for a LPP and let Z = ax+ bybe theobjective function. If R is bounded, then the objective function Z has both amaximum and a minimum value on R and each of these occur at a corner point ofR.If the feasible region R is unbounded, then a maximum or a minimumvalue of the objective function may or may not exist. However, if it exits, it mustoccur at acorner point of R.12.1.11 Corner point method for solving a LPP Themethod comprises of the following steps :(1)Find the feasible region of the LPP and determine its corner points (vertices)either by inspection or by solving the two equations of the lines intersectingat that point.