LPP Graphical Method - 10 Redundancy in LPP

#OperationsResearch #Math #Statistics #Linear Programming #GraphicalMethod #Constraint #Maximization #Inequality #FreeLecture #FreeStudy #Solution LPP - Graphical Method Case - Redundancy in LPP Solve the following LPP through the Graphical Method: Maximize: Z = 300X1 + 400X2 Subject to - 5X1 + 4X2 ≤ 200 3X1 + 5X2 ≤ 150 5X1 + 4X2 ≥ 100 8X1 + 4X2 ≥ 80 X1 and X2 ≥ 0 Redundancy in LPP: A redundant constraint is one that doesn't affect the feasible solution region. In an LPP there can be one (or more) constraint such that with its presence or absence there in the LPP the feasible region or ultimately the optimal solution remains the same. Such a redundant constraint can be there in an LPP because of the information available from the real life / managerial problem, which has been formulated as an LPP and now we want to solve. Such information might be different from the other information, either verbally or statistically or in both ways, but ultimately it represents the same relationship between the decision variables and the limiting factor as well, which the other information also represents. At the time of formulation as LPP of this kind of problem we frame constraints from each and every information available and as a result such constraint also becomes the part of LPP. In the graphical solution we can easily see this kind of redundant constraint in the graph and we may prefer to not include it in the determination of the feasible region without affecting the optimal solution. If there are two variables in an LP problem, it can be solved by graphical method. Let the two variables be x1 and x2. The variable x1 is represented on x-axis and x2 on y-axis. Due to non-negativity condition, the variables x1 and x2 can assume positive values and hence if at all the solution exists for the problem, it lies in the first quadrant. The steps used in graphical method are summarized as follows: Step 1: Replace the inequality sign in each constraint by an equal to sign. Step 2: Represent the first constraint equation by a straight line on the graph. Any point on this line satisfies the first constraint equation. If the constraint inequality is ‘less than’ type, then the area (region) below this line satisfies this constraint. If the constraint inequality is of ‘greater than’ type, then the area (region) above the line satisfies this constraint. Step 3: Repeat step 2 for all given constraints. Step 4: Shade the common portion of the graph that satisfies all the constraints simultaneously drawn so far. This shaded area is called the ‘feasible region’ (or solution space) of the given LP problem. Any point inside this region is called feasible solution and provides values of x1 and x2 that satisfy all constraints. Step 5: Find the optimal solution for the given LP problem. The optimal solution may be determined using the Extreme Point method. In this method, the coordinates of each extreme point are substituted in the objective function equation; whichever solution has optimum value of Z (maximum or minimum) is the optimal solution. OR, Operations Research, Math, Mathematics, Linear Programming, LPP, Graphical Method, Maximization, Mixed Constraints, Redundancy, Inequality, Equality, Equation, MBA, MCA, CA, CS, CWA, CMA, CPA, CFA, BBA, BCom, MCom, BTech, MTech, CAIIB, FIII, Graduation, Post Graduation, BSc, MSc, BA, MA, BE, Diploma, Production, Finance, Management, Commerce, Engineering, Grade-11, Grade- 12 - www.prashantpuaar.com