2. Pure Maximisation Problems
• When objective function is of maximisation
• All constraints are ≤ type
• Slack variables are added with 0 coefficient in the objective
function and 1 coefficient in the constraints.
• Optimal solution is when all Cj-Zj ≤ 0
Example
Zmax = 6X1+ 8X2
Subject to :
30 X1+ 20X2 ≤300
5X1+ 10x2 ≤ 110
Where X1, X2 ≥ 0
3. Minimisation problem- Big M Method or Penalty
Method
• When objective function is of minimisation
• All constraints are ≥ type
• Surplus and artificial variables are added with 0 & M coefficient
respectively in the objective function. Surplus variables are
subtracted and artificial variables are added into the constraints
together with one as a coefficient.
• Optimal solution is when all Cj-Zj ≥ 0
Example
Z min= 3X1+ 2.25X2
Subject to :
2X1+ 4X2 ≥ 40
5X1+ 2x2 ≥ 50
Where X1, X2 ≥ 0
4. Mixed Constraints
Constraint
Adjustment in constraint Objective Function
Maximisation
Problem
Minimisation
Problem
≤ Add a Slack Variable 0 0
= Add an Artificial Variable -M M
≥ Subtract a surplus variable and
add an Artificial Variable
0
-M
0
M
5. Example
Z min= 60X1+ 80X2
Subject to :
X1+ X2 = 500
X1 ≤ 400
X2 ≥ 200
Where X1, X2 ≥ 0
6. Example
Z max= 20X1+ 10X2
Subject to :
X1+ X2 = 150
X1 ≤ 40
X2 ≥ 20
Where X1, X2 ≥ 0
7. Special Cases
• Degeneracy: When there is tie for the minimum ratio
for choosing the departed variable.
• Unbounded Problem: When the minimum ratio
column contains negative or infinite the solution is
unbounded.
• Infeasible solution : When an artificial variable is
present as a basic variable.
9. Transportation Problem
Methods for initial basic feasible solution
• NWCM
• LCM
• VAM
Optimality Test
• Stepping Stone Method
• MODI- Modified Distribution Method
10. Special Cases
• Unbalanced Problem
• Degeneracy in Transportation
When m+n-1 not equal to Stone Squares
degeneracy arises
• In order to remove the degeneracy we assign epsilon
to the unoccupied cells.
