1. Linear Programming Terminology

2. http://www.rajeshtimane.com/2 Contents 1.What is a Mathematical Model? 2.Illustration of LPP: Maximization Case 3.What is Linear Programming Problem (LPP)? 4.Graphical Solution oFeasible Solutions oOptimal Solution 5.Concepts: oWhat is Feasibility? oWhat is an Optimal Solution? oConvex Sets & LPP

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

4. http://www.rajeshtimane.com/4 II. Illustration: Maximize: Z = 3x 1 + 5x 2 Subject to restrictions: x 1 < 4 2x 2 < 12 3x 1 + 2x 2 < 18 Non negativity condition x 1 > 0 x 2 > 0 Objective Function Functional Constraints Non-negativity constraints

5. 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.

6. http://www.rajeshtimane.com/6 IV. Graphical Solution Ex) Maximize: Z = 3x 1 + 5x 2 Subject to restrictions: x 1 < 4 2x 2 < 12 i.e. x 2 < 6 3x 1 + 2x 2 < 18 Non negativity condition x 1, x 2 > 0 Solution: finding coordinates for the constraints (assuming perfect equality), by putting one decision variable equal to zero at a time. Restrictions (Constraints)Co-ordinates x 1 < 4(4, 0) x 2 < 6(0, 6) 3x 1 + 2x 2 < 18(0, 9) & (6, 0)

7. http://www.rajeshtimane.com/7 Restrictions (Constraints)Co-ordinatesNon-negativity Constraint x 1 < 4(4, 0)x 1, > 0 x 2 < 6(0, 6)x 2 > 0 3x 1 + 2x 2 < 18(0, 9) & (6, 0) Feasible Region (Shaded / Points A, B, C, D and E) X2 10 8 6 4 2 0 2 4 6 8 10 X 1 A B C D E

8. 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.

9. http://www.rajeshtimane.com/9 Optimal Solution CornerLimiting ConstraintCo-ordinateMax. Z= 3x 1 + 5x 2 Ax2 = 6(0, 6)30 Bx2 = 6 & 3x 1 + 2x 2 = 18(2, 6)36 Cx1 = 4 & 3x 1 + 2x 2 = 18(4, 3)27 Dx1 = 4(4, 0)12 EOrigin(0, 0)0 From the above table, Z is maximum at point ‘B’ (2, 6) i.e. The Optimal Solution is: X 1 = 2 and X 2 = 6 ANSWER

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

11. 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)”

12. 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’.

13. http://www.rajeshtimane.com/13 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”. A B A B Convex Set Non-convex Set

14. http://www.rajeshtimane.com/14 Contact For any further queries/details be in touch with the author at: http://www.rajeshtimane.com