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Conceptual understanding of Feasibility, Optimal Solution & Convex Sets.

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- 1. Linear Programming Terminology
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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