Linear Programming

1. Linear Programming Terminology

2. Contents • What is a Mathematical Model? • Illustration of LPP: Maximization Case • What is Linear Programming Problem (LPP)? • Graphical Solution • Feasible Solutions • Optimal Solution • Concepts: • What is Feasibility? • What is an Optimal Solution? • Convex Sets & LPP http://www.rajeshtimane.com/

3. I. What is a Mathematical Model ? F = m a ‘Mathematical Expressions’ • Here m and a are called as ‘Decision Variables’ • F can be called as ‘Objective Functions’ • Now, F can be controlled or restricted by limiting m or a … say m < 50 kg …here, m can be called as a ‘Constraint’ • Similarly if a > o …always, then this condition is called as ‘Non-Negativity Condition’ http://www.rajeshtimane.com/

4. II. Illustration: Maximize: Z = 3x1 + 5x2 Subject to restrictions: x1< 4 2x2 < 12 3x1 + 2x2< 18 Non negativity condition x1> 0 x2> 0 Objective Function Functional Constraints Non-negativity constraints http://www.rajeshtimane.com/

5. III. What is Linear Programming? • The most common application of LP is allocating limited resources among competing activities in a best possible way i.e. the optimal way. • The adjective linear means that all the mathematical functions in this model are required to be linear functions. • The word programming does not refer to computer programming; rather, essentially a synonym for planning. http://www.rajeshtimane.com/

6. IV. Graphical Solution Ex) Maximize: Z = 3x1 + 5x2 Subject to restrictions: x1< 4 2x2 < 12 i.e. x2< 6 3x1 + 2x2< 18 Non negativity condition x1, x2> 0 Solution: finding coordinates for the constraints (assuming perfect equality), by putting one decision variable equal to zero at a time. http://www.rajeshtimane.com/

7. X2 10 8 6 4 2 0 A B C D E 2 4 6 8 10 X1 Feasible Region (Shaded / Points A, B, C, D and E) http://www.rajeshtimane.com/

8. Feasible Solutions • Try co-ordinates of all the corner points of the feasible region. • The point which will lead to most satisfactory objective function will give Optimal Solution. • Note: for co-ordinates at intersection; solve the equations (constraints) of the two lines simultaneously. http://www.rajeshtimane.com/

9. Optimal Solution From the above table, Z is maximum at point ‘B’ (2 , 6) i.e. The Optimal Solution is: X1 = 2 and X2 = 6 ANSWER http://www.rajeshtimane.com/

10. Conceptual Understanding • Feasibility • Optimal Solution • Convex Set http://www.rajeshtimane.com/

11. What is Feasibility ? • Feasibility Region [Dictionary meaning of feasibility is possibility] “The region of acceptable values of the Decision Variables in relation to the given Constraints (and the Non-Negativity Restrictions)” http://www.rajeshtimane.com/

12. What is an Optimal Solution ? • It is the Feasible Solution which Optimizes. i.e. “provides the most beneficial result for the specified objective function”. • Ex: If Objective function is Profit then Optimal Solution is the co-ordinate giving Maximum Value of ‘Z’… While; if objective function is Cost then the optimum solution is the coordinate giving Minimum Value of ‘Z’. http://www.rajeshtimane.com/

13. Convex Set Non-convex Set A B A B Convex Sets and LPP’s “If any two points are selected in the feasibility region and a line drawn through these points lies completely within this region, then this represents a Convex Set”. http://www.rajeshtimane.com/

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