Irregular Types of LP Models Dr. Sania Bhatti Irregular Types of LP Models • Theorem: Every linear program either: 1. is infeasible, 2. is unbounded, 3. has a unique optimal solution value (single optimal point or multiple optima) Infeasible LP Model • Definition A linear program is infeasible if it has no feasible solutions, i.e. the feasible region is empty. • Example: Max 8X1 + 5X2 s.t. 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 X1 - X2 ≤ 350 X1 ≥ 800 X1, X2 ≥ 0 X2 1000 900 Models With No Solutions Infeasibility . Max s.t. 800 700 600 8X1 + 5X2 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 X1 - X2 ≤ 350 X1 ≥ 800 X1, X2 ≥ 0 No points in common. No points satisfy all constraints simultaneously. 500 400 No Solutions! Problem is 300 INFEASIBLE. 200 100 100 200 300 400 500 600 700 800 X1 Infeasibility • A problem is infeasible when there are no solutions that satisfy all the constraints. • Infeasibility can occur from – Input Error – Misformulation – Simply an inconsistent set of constraints Models With An “Unbounded” Solution X2 1000 Max s.t. 900 800 8X1 + 5X2 X1 - X2 ≤ 350 X1 ≥ 200 X2 ≥ 300 700 600 Unbounded Feasible Region 500 Can increase indefinitely 400 300 200 100 Unbounded Solution 100 200 300 400 500 600 700 800 X1 X2 1000 Models With An Unbounded Feasible Region – Optimal Solution Min s.t. 900 800 8X1 + 5X2 X1 - X2 ≤ 350 X1 ≥ 200 X2 ≥ 200 700 600 Unbounded Feasible Region 500 400 300 200 OPTIMAL POINT 100 100 200 300 400 500 600 700 800 X1 Unboundedness • An unbounded feasible region extends to infinity in some direction. • If the problem is unbounded, the feasible region must be unbounded. • If the feasible region is unbounded, the problem may or may not be unbounded. • An unbounded solution means you left out some constraints – you cannot make an “infinite” profit. Multiple Optimal Solutions • When an objective function line is parallel to a constraint the problem can have multiple optimal solutions. • A problem can have multiple optima(alternative optima) but a single optimal value of Z. Example: MAX 8X1 + 4X2 s.t. 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 1X1 - 1X2 ≤ 350 X1, X2 ≥ 0 Multiple Optimal Solutions X2 1000 MAX 8X1 + 4X2 s.t. 2X1 + 1X2 ≤ 1000 3X1 + 4X2 ≤ 2400 1X1 - 1X2 ≤ 350 All points on the X1, X2 ≥ 0 boundary between 900 800 700 600 Optimal Extreme Point optimal extreme points are also optimal 500 Optimal Extreme Point 400 300 200 100 100 200 300 400 500 600 700 800 X1 Class Exercise • Solve graphically the following LP models and categorize them as Infeasible, unbounded or a problem with single/multiple optima. 1. Max z = 2x1 + 6x2 s.t. 4x1 + 3x2 < 12 2x1 + x2 > 8 x1, x2 > 0 Class Exercise 2. Max z = 3x1 + 4x2 s.t. x1 + x2 > 5 3x1 + x2 > 8 x1, x2 > 0 Class Exercise: Solution • There are no points that satisfy both constraints, hence this problem has no x2 feasible region, and no optimal solution. 2x1 + x2 > 8 8 4x1 + 3x2 < 12 4 x1 3 4 Class Exercise: Solution • The feasible region is unbounded and the objective function line can be moved parallel to itself without bound so that z can be x2 increased infinitely. 3x1 + x2 > 8 8 5 x1 + x2 > 5 Max 3x1 + 4x2 2.67 5 x1