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**LECTURE 14 Minimization Two Phase method by Dr. Arshad zaheer**

Simplex Method LECTURE 14 Minimization Two Phase method by Dr. Arshad zaheer

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Two Phase Method

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Illustration Minimize f=4x1+ x2 (Objective Function) Subject to: (Constraints) 3x1+ x2 = 3 4x1+ 3x2 ≥6 x1+ 2x2 ≤4 x1, x2 ≥ 0 (Non-Negativity Constraints)

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**Inequalities Constraints in Equation Form**

Let S1 and S2 be the surplus and slack variables for second and third constraints, respectively. Minimize f =4x1+ x2 (Objective Function) Subject to: (Constraints) 3x1+ x2 = 3 …………. (1) 4x1+ 3x2 – S1 = 6 …………. (2) x1+ 2x2 + S2 = 4 …………. (3) x1, x2, S1 , S2≥ 0

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Initial solution Arbitrary values =# of Variables -- # of Equations = 2 Let x1= 0, x2 = 0 Putting above values in objective function (f =5x1+ 4x2) and equation 1-3, f = 0 0 = 3 (Contradiction) S1 = -6 (Violation) S2 = 4 This situation cannot be called as initial feasible solution because it is not satisfying the condition. In order to make it feasible we need to add Artificial variables In equations having contradictions and violation This is against the basic rules of Mathematics as the 0 cannot be equal to 3. This is against the non negativity constraint that X must be non zero.

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Artificial Variables Let A1 and A2 be the artificial variables for first and second equation respectively. Minimize: f =4x1+ x2 (Objective Function) Subject to: (Constraints) 3x1+ x2 + A1= 3 …………. (4) 4x1+ 3x2 – S1 + A2= 6 …………. (5) x1+ 2x2 + S2 = 4 …………. (6) x1, x2, S1 , S2 , A1 + A2 ≥ 0

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**Solution of Artificial Variable**

Questions which involve artificial variables can not solve straight away. We have to use following two methods to solve it Penalty Method or M-Method (Big M Method) Two Phase Method In this lecture we will solve this problem by

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**Solution by Two Phase Method**

Phase I: In Phase I we introduce artificial objective function Capital “F” as the sum of artificial variables introduced in the constraints. We substitute the values of artificial variables from the constraints and get artificial objective function. We have Two artificial variables so the artificial objective function would be as under; F = A1 + A2

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**Artificial Objective Function**

Use Equation no 4 to find the value of A1 3x1+ x2 + A1= 3 A1 = 3 - 3x1- x2 A2 Use Equation no 5 to find the value of A2 4x1+ 3x2 – S1 + A2= 6 A2= 6 - 4x1- 3x2 + S1

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**x1, x2, S1 , S2 , A1 + A2 ≥0 Putting the values of A1 and A2**

F = ( 3 – 3X1- X2) + (6 – 4X1- 3X2 + S1) =9 - 7x1-4 x2 + S1 Minimize f = 4x1+ x2 F=9 - 7x1-4 x2 + S1 Subject to: 3x1+ x2 + A1 = 3 …………. (4) 4x1+ 3x2 – S1 + A2= 6 …………. (5) x1+ 2x2 + S2 = 4 …………. (6) x1, x2, S1 , S2 , A1 + A2 ≥0

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**Initial Feasible Solution**

Arbitrary values =# of Variables -- # of Equation = 3 Let x1= 0, x2 = 0, S1 = 0 Putting above values in objective functions and equation 4-6, F = 9 f = 0 A1 = 3 A2 = 6 S2 = 4 This solution is called initial feasible solution because it satisfies the all Non negativity constraint and also do not have any contradiction or violation

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Initial Tableau For leaving variable the rule is same as maximization, the variable with minimum ratio;A1 Basics X1 X2 S1 S2 A1 A2 RHS Ratio 3 1 3/3=1(Min) 4 -1 6 6/4=1.5 2 4/1=4 f -4 F 7 9 ↑ In artificial function of minimization we select the maximum positive as the entering variable which is X1

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**Optimal solution for Artificial objective Function**

The solution of artificial objective function is said to be optimal when artificial objective functions coefficients become non-positive or zero

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**Calculation [1 1/3 0 0 1/3 0 1] New Pivot Row = 1 X Old Pivot Row**

[1 1/ / ] New Pivot Row = 1 X Old Pivot Row Pivot No. New Pivot Row = 1 X [ ] 3

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**Calculation New A2 Row = [4 3 -1 0 0 1 6] New S2 Row =[1 2 0 1 0 0 4]**

New Row = Old Row – Pivot Column Coefficient x New Pivot Row New A2 Row = [ ] - (4)[1 1/ /3 0 1] = [0 5/ /3 1 2] New S2 Row =[ ] - (1) [1 1/ /3 0 1] = [0 5/ /3 0 3] New f Row = [ ] - (-4) [1 1/ /3 0 1] = [0 1/ /3 0 4] New F Row = [ ] - (7) [1 1/ /3 0 1] = [0 5/ /3 0 2]

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**Tableau I ↑ Basics X1 X2 S1 S2 A1 A2 RHS Ratio 1 1/3 1÷1/3=3 5/3 -1**

1÷1/3=3 5/3 -1 -4/3 2 2 ÷5/3=1.2 (Min) -1/3 3 3/5 ÷ 3=1.8 f 4/3 4 F -7/3 ↑ Still there is one positive coefficient so we need to make further tableau

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**Calculation [0 1 -3/5 0 -4/5 3/5 6/5] New Pivot Row = 1 X**

[ / /5 3/5 6/5] New Pivot Row = 1 X Old Pivot Row Pivot No. New Pivot Row = 1 X [0 5/ /3 1 2] 5/3

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**Calculation [1 1/3 0 0 1/3 0 1] -(1/3)[0 1 -3/5 0 -4/5 3/5 6/5]**

New Row = Old Row – Pivot Column Coefficient x New Pivot Row New X1 Row = [1 1/ /3 0 1] -(1/3)[0 1 -3/5 0 -4/5 3/5 6/5] = [1 0 1/5 0 3/5 -1/5 3/5] New S2 Row = [0 5/ /3 0 3] -(5/3)[0 1 -3/ /5 3/5 6/5] = [ ]

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Calculation New f Row = [0 1/ /3 0 4] -(1/3)[0 1 -3/5 0 -4/5 3/5 6/5] = [0 0 1/5 0 8/5 -1/5 18/5] New F Row [0 5/ /3 0 2] -(5/3)[0 1 -3/5 0 -4/5 3/5 6/5] = [ ]

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**Tableau II Basics X1 X2 S1 S2 A1 A2 RHS 1 1/5 3/5 -1/5 -3/5 -4/5 6/5**

1/5 3/5 -1/5 -3/5 -4/5 6/5 -1 f 8/5 18/5 F Now there is no positive value in the artificial function so this is the end of Phase I

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**Phase II ↑ Basics X1 X2 S1 S2 RHS Ratio 1 1/5 3/5 3/5÷1/5=3 -3/5 6/5**

In phase two rewrite the previous tableau by dropping the artificial objective function and artificial variables Basics X1 X2 S1 S2 RHS Ratio 1 1/5 3/5 3/5÷1/5=3 -3/5 6/5 1/1=1(Min) f 18/5 ↑

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**Calculation [0 0 1 1 1] New Pivot Row = 1 X Old Pivot Row Pivot No.**

[ ] New Pivot Row = 1 X Old Pivot Row Pivot No. New Pivot Row = 1 X [ ]

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**Calculation [1 0 1/5 0 3/5] -(1/5) [0 0 1 1 1] = [1 0 0 -1/5 2/5]**

New Row = Old Row – Pivot Column Coefficient x New Pivot Row New X1 Row = [1 0 1/ /5] -(1/5) [ ] = [ /5 2/5] New X2 Row = [0 1 -3/ /5] -(-3/5)[ ] = [ /5 9/5]

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Calculation New f Row = [0 0 1/5 0 18/5] -(1/5) [ ] = [ /5 17/5]

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**Tableau Basics X1 X2 S1 S2 RHS 1 -1/5 2/5 3/5 9/5 f 17/5**

-1/5 2/5 3/5 9/5 f 17/5 Now there is no positive value in the objective function so this is the optimal point

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Optimal Solution X1 = 2/5 X2 = 9/5 f = 17/5 Cross checking of maximization point put values of X1 and X2 from above solution into original objective function f=4x1+ x2 =4 (2/5) + (9/5) =17/5

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