Linear Programming Terminology

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

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’

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

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.

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)

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) X X 1 A B C D E

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.

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

Conceptual Understanding  Feasibility  Optimal Solution  Convex Set

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)”

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’.

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

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