Question

Solve the given problem using cutting plane algorithm given that x1 and x3 must be integers

at Exercise 6 under the assumption that only xi and x must be inte gers. he ad

Answer 1

Solution: Given below:

Find solution using integer simplex method (Gomory's cutting plane method) subject to 3x1-6x2 + 9x3 = 9 and x2 >= 0 and xlx3 non-negative integers Solution: Problemis Max Z= x2+ 4x3 subject to 3x1 6x2 + 9x39 andx1,x2, x3 2 0,x, a3 non-negative integers The problem is converted to canonical form by adding slack, suplus and atificial variables as appropiate 1. As the constraint 1 is of type's'we should add slack variable S 2. As the constraint 2 is of type' we should add slack variable S2 After introducing slack variables subject to 3x1-6x2 + 9x3 + S1 3x 22 x3 andxj x2,x3, S S220

Iteration-1 MinRatio CB 1 2 9 3 -6 9 S. Z=0 Negative minimum Zj - Cj is -4 and its column index is 3. So, the entering variable is x3 Minimum ratio is 1 and its row index is 1. So, the leaving basis variable is Si The pivot element is 9. Entering x, Departing-S, Key Element 9 R1(new) = R1 (old) ÷ 9

R2(new) = R2(old)-R1(new) Iteration-2 MinRatio 4 Negative minimumZ- Cjis-and its column index is 2. So, the entering vanable is x- Minimum ratio is 2.25 and its row index is 2. So, the leaving basis variable is S The pivot element is3

Entering x2, Departing -S2, Key Element =- R2(new) = R2(old) 豆 R1(new) = RI (old)--R2(new) Iteration-3 MinRatio 12 4 4 24 49 24 24 Since allZ-C0 Hence, non-integer optimal solution is aived with value ofvariables as

Max Z= To obtain the integer valued solution, we proceed to construct Gomory's fractional cut, with the help ofx3-row as follows: =1x1 + 1x3 +12S1 + 4S2 The fractional cut will become 12 4 Adding this additional constraint at the bottom of optimal simplex table. The new table so obtained is Iteration-1 12 24 Sgl 12 49 Z. 24 24 Z: - C Ratio = -3.5 ↑ -5.5 Sgl,j and Sgl,js 0

Minim um negative.( s--and its row index is 3. So, the leaving basis variable is Sgl Maximum negative ratio is -3.5 and its column index is 4. So, the entering variable is Si The pivot element is -jz- Entering S1Departing Sgl, Key Element R3(new) R3(old)x 12 12 12 1%.

R2(new) = R2(old)-24R3(new) -12 21 Snce all Z-C0 Hence, integer optimal solution is aived with value of variables as 21 The integer optimal solution found after 1-cuts.