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# Linear programming graphical method (feasibility)

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Illustration on Graphical Method of Linear Programming.

Conceptual understanding of Feasibility, Optimal Solution & Convex Sets.

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### Linear programming graphical method (feasibility)

1. 1. Linear Programming Terminology
2. 2. What is a Mathematical Model ?F=ma ‘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’ http://www.rajeshtimane.com 2
3. 3. Illustration:Maximize: Z = 3x1 + 5x2 Objective FunctionSubject to restrictions: x1 <4 Functional 2x2 < 12 Constraints 3x1 + 2x2 < 18Non negativity condition x1 >0 Non-negativity x2 >0 constraints http://www.rajeshtimane.com 3
4. 4. What is Linear Programming (LP)? 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 4
5. 5. Graphical SolutionEx) Maximize: Z = 3x1 + 5x2Subject to restrictions: x1 < 4 2x2 < 12 i.e. x2 < 6 3x1 + 2x2 < 18Non negativity condition x1, x2 > 0Solution: finding coordinates for the constraints (assuming perfect equality), by puttingone decision variable equal to zero at a time.Restrictions (Constraints) Co-ordinatesx1 < 4 (4 , 0)x2 < 6 (0 , 6)3x1 + 2x2 < 18 (0 , 9) & (6 , 0) http://www.rajeshtimane.com 5
6. 6. Restrictions (Constraints) Co-ordinates Non-negativity Constraintx1 < 4 (4 , 0) x1, > 0x2 < 6 (0 , 6) x2 > 03x1 + 2x2 < 18 (0 , 9) & (6 , 0) X2 10 8 A B 6 4 C Feasible Region (Shaded / Points A, B, C, D and E) 2 0 D E 2 4 6 8 10 X1 http://www.rajeshtimane.com 6
7. 7. 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 7
8. 8. Optimal SolutionCorner Limiting Constraint Co-ordinate Max. Z= 3x1 + 5x2 A x2 = 6 (0 , 6) 30 B x2 = 6 & 3x1 + 2x2 = 18 (2 , 6) 36 C x1 = 4 & 3x1 + 2x2 = 18 (4 , 3) 27 D x1 = 4 (4 , 0) 12 E Origin (0 , 0) 0From the above table, Z is maximum at point ‘B’ (2 , 6) i.e. TheOptimal Solution is:X1 = 2 andX2 = 6 ANSWER http://www.rajeshtimane.com 8
9. 9. 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 9
10. 10. 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 10
11. 11. 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”. Convex Set Non-convex Set A A B B http://www.rajeshtimane.com 11