Graphical Sensitivity Analysis
Sensitivity Analysis
Source: Introduction to Management Science 10 e, Anderson Sweeney Williams
Example 1
Max s.t. 5x1 + 7x2 x1 < 6 2x1 + 3x2 < 19 x1 + x2 < 8 x1, x2 > 0
x2
8 7 6 5 4 3 2 1
x1 + x2 < 8
Max 5x1 + 7x2 x1 < 6 Optimal: x1 = 5, x2 = 3, z = 46 2x1 + 3x2 < 19 x1
1
2
3
4
5
6
7
8
9
10
x2
8 7 6 5 4 3 2 1
5 5
Feasible Region
1 1
1 2 3 4
4 4 3 3 2 2
5 6 7 8 9 10
x1
Example 1
• Range of Optimality for c1 The slope of the objective function line is -c1/c2. The slope of the first binding constraint, x1 + x2 = 8, is -1 and the slope of the second binding constraint, x1 + 3x2 = 19, is -2/3. Find the range of values for c1 (with c2 staying 7) such that the objective function line slope lies between that of the two binding constraints: -1 < -c1/7 < -2/3 Multiplying through by -7 (and reversing the inequalities): 14/3 < c1 < 7
Example 1
• Range of Optimality for c2 Find the range of values for c2 ( with c1 staying 5) such that the objective function line slope lies between that of the two binding constraints: -1 < -5/c2 < -2/3 Multiplying by -1: Inverting, Multiplying by 5: 1 > 5/c2 > 2/3 1 < 5 < c2/5 < 3/2 c2 < 15/2
Right-Hand Sides
• The improvement in the value of the optimal solution per unit increase in the right-hand side is called the dual price. • The range of feasibility is the range over which the dual price is applicable. • As the RHS increases, other constraints will become binding and limit the change in the value of the objective function.
Dual Price
• Graphically, a dual price is determined by adding +1 to the right hand side value in question and then resolving for the optimal solution in terms of the same two binding constraints. • The dual price is equal to the difference in the values of the objective functions between the new and original problems. • The dual price for a nonbinding constraint is 0. • A negative dual price indicates that the objective function will
Source: Introduction to Management Science 10 e, Anderson Sweeney Williams
Example 1
Max s.t. 5x1 + 7x2 x1 < 6 2x1 + 3x2 < 19 x1 + x2 < 8 x1, x2 > 0
x2
8 7 6 5 4 3 2 1
x1 + x2 < 8
Max 5x1 + 7x2 x1 < 6 Optimal: x1 = 5, x2 = 3, z = 46 2x1 + 3x2 < 19 x1
1
2
3
4
5
6
7
8
9
10
x2
8 7 6 5 4 3 2 1
5 5
Feasible Region
1 1
1 2 3 4
4 4 3 3 2 2
5 6 7 8 9 10
x1
Example 1
• Range of Optimality for c1 The slope of the objective function line is -c1/c2. The slope of the first binding constraint, x1 + x2 = 8, is -1 and the slope of the second binding constraint, x1 + 3x2 = 19, is -2/3. Find the range of values for c1 (with c2 staying 7) such that the objective function line slope lies between that of the two binding constraints: -1 < -c1/7 < -2/3 Multiplying through by -7 (and reversing the inequalities): 14/3 < c1 < 7
Example 1
• Range of Optimality for c2 Find the range of values for c2 ( with c1 staying 5) such that the objective function line slope lies between that of the two binding constraints: -1 < -5/c2 < -2/3 Multiplying by -1: Inverting, Multiplying by 5: 1 > 5/c2 > 2/3 1 < 5 < c2/5 < 3/2 c2 < 15/2
Right-Hand Sides
• The improvement in the value of the optimal solution per unit increase in the right-hand side is called the dual price. • The range of feasibility is the range over which the dual price is applicable. • As the RHS increases, other constraints will become binding and limit the change in the value of the objective function.
Dual Price
• Graphically, a dual price is determined by adding +1 to the right hand side value in question and then resolving for the optimal solution in terms of the same two binding constraints. • The dual price is equal to the difference in the values of the objective functions between the new and original problems. • The dual price for a nonbinding constraint is 0. • A negative dual price indicates that the objective function will