Wyndor with variations
Every LP problem will fall into one of the following
categories:
1. The problem has one (or more) optimal solution(s)
2. The problem has no optimal solution
because
a) The problem is unbounded
b) The problem is inconsistent
(infeasible)
Example (No Feasible Solution)
Maximize Z = 3x1 + 5x2
subject to
x1 ≥ 5
x2 ≥ 4
3x1 + 2x2 ≤ 18
and
x1 ≥ 0, x2 ≥ 0.
x2
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
10
x1
Example (Unbounded Solution)
Maximize Z = 5x1 + 12x2
subject to
x1 ≤ 5
2x1 –x2 ≤ 2
and
x1 ≥ 0, x2 ≥ 0.
x2
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
10
x1
If we find an optimal solution we
either have a
• Normal solution
• Multiple optima or
• Degenerate solution
Wyndor Glass normal solution
Wyndor Glass Co. Product-Mix Problem
Unit Profit
Doors
$300
Windows
$500
Hours
Hours Used Per Unit Produced Used
Plant 1
1
0
2
Plant 2
0
2
0
Plant 3
3
2
0
Units Produced
Doors
2
Windows
6
<=
<=
<=
Hours
Available
4
12
18
Total Profit
$3 600
GLP optimal solution
GLP sensitivity report
SYNOPSIS OF THE SOLUTION OUTPUT
• Optimal values for the
- decision variables
- slack/surplus variables
- objective function
• Shadow prices and allowable range of RHS changes
• Allowable increase/decrease of objective
coefficients
• Reduced costs
Multiple optima – graphical solution
Wyndor – Multiple optima
Wyndor, Degenerate solution
– graphical representation
Wyndor – degenerate solution
•
The meaning of "Adjustable cells"
(upper part of the sensitivity report)
•
Nondegenerate solution
•
Allowable increase/decrease tells us how much we can increase/reduce a given coefficient in the
objective function witkout changing the optimal solution (everything else fixed)
•
Whenever a coefficient is changing less than allowable the optimal solution will not change
•
If the coefficient of a variable is increased with the exact upper allowable amount there will be an
alternative optimal solution with, for a max(min) model, higher (lower) value of the variable
•
If the coeff. is reduced with the exact maximum allowable amount there will be an alternative optimal
solution with, for a max (min) model, lower (higher) value of the variable
•
Signs we can use to point out multiple optima: Allowable increase/decrease for the objective function
coeff.. = 0 (i.e. any change of the coefficient will lead to another unique solution)
•
Degenerate solution
•
The signs of multiple optima are not valid
•
The objective funtion coefficient must change with at least, and possibly a lot more than "allowable
increase/decrease" to give a new optimal solution
Sensitivity analysis contd.
• Meaning of ”Constraints” (lower part)
• Shadow price: The change in OV (objective function value)
when the RHS (right hand side) of that constraint is increased
by one unit, everyhing else held constant
• Allowable increase/decrease ranges give us the region for
which the shadow prices are valid
• Signs of a degenerate solution: Some of the shadow prices will
have a 0 allowable increase or decrease (we only get effects on
OV of one-sided changes in RHS for these)
Shortcomings of Solver’s Sensitivity Report
1. The report gives sensitivity information for
perturbations of parameters only in the
immediate neighborhood of the solution,
and only for changing one parameter at a
time.
2. The report gives sensitivity only for effects
upon the OV.
3. The report gives no sensitivity information
for changes in the model’s technical
coefficients.