Sensitivity analysiscontinued…
BSAD 30
Dave Novak
Source: Anderson et al., 2013 Quantitative Methods for Business 12th edition – some slides are directly from J. Loucks © 2013 Cengage Learning
Overview
Special case LP problemsAlternative optimal solutionsInfeasibilityUnbounded problems
Sensitivity analysis continued Non-intuitive shadow prices Example
Feasible region
Recall – using a graphical solution procedure, the optimal solution to an LP is found at the extreme points to the feasible region
The feasible region for a two-variable LP problem (x1, x2) can be nonexistent, a single point, a line, a polygon, or an unbounded area
Feasible region Any linear program falls in one of four
categories:1. has a unique optimal solution2. has alternative optimal solutions3. is infeasible 4. has an objective function that can be
increased without bound (unbounded) A feasible region may be unbounded and
yet still have optimal solutions This is common in minimization problems
and is possible in maximization problems
Alternative optimal solution
In some cases, an LP can have more than one optimal solution – this is referred to alternative optimal solutions There is more than one extreme point that
minimizes or maximizes the OF value
Alternative optimal solution
In the graphical method, if the objective function line is parallel to a boundary constraint in the direction of optimization, there are alternate optimal solutionsAll points on this line segment are optimal
Alternative optimal solution
Consider the following LP
Max 4x1 + 6x2
s.t. x1 < 6
2x1 + 3x2 < 18
x1 + x2 < 7
x1 > 0 and x2 > 0
Alternative optimal solutionexample Boundary constraint 2x1 + 3x2 < 18 (constraint #2)
and OF line 4x1 + 6x2 are parallel. All points on line segment A – B are optimal solutions
x1
x2
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10
Constraint #2: 2x1 + 3x2 < 18
Constraint #3: x1 + x2 < 7
Constraint #1: x1 < 6
OF: Max 4x1 + 6x2A
B
Infeasibility
No solution to the LP problem satisfies all the constraints, including the non-negativity conditions
Graphically, this means a feasible region does not existCauses include
• A formulation error has been made• Management’s expectations are too high• Too many restrictions have been placed on the
problem (i.e. the problem is over-constrained)
Infeasibility
Consider the following LP
Max 2x1 + 6x2
s.t. 4x1 + 3x2 < 12
2x1 + x2 > 8
x1, x2 > 0
Infeasibility
x1
Constraint #1: 4x1 + 3x2 < 12
Constraint #2: 2x1 + x2 > 8
2 4 6 8 10
4
8
2
6
10x2
There are no points that satisfy both constraints, so there is no feasible region (and no feasible solution)
Unbounded
The solution to a maximization LP problem is unbounded if the value of the solution may be made indefinitely large without violating any of the constraintsThis leads to an infinite production scenario
For real problems, this is usually the result of improper formulationQuite likely, a constraint has been
inadvertently omitted
Unbounded
Consider the following LP
Max 4x1 + 5x2
s.t. x1 + x2 > 5
3x1 + x2 > 8
x1, x2 > 0
Unbounded
x1
Constraint #2: 3x1 + x2 > 8
Constraint #1: x1 + x2 > 5
Max 4x1 + 5x
2
6
8
10
2 4 6 8 10
4
2
x2
Standard form of an LP
Involves formulating each constraint as a strict equality and:ADDING a slack variable to ≤ (less than or
equal to) constraintsSUBTRACTING a surplus variable from ≥
(greater than or equal to) constraintsEquality constraints do not have a slack or
surplus variable
Standard form of an LP
Max 5x1 + 6x2
s.t. x1 < 6
2x1 + 3x2 ≥ 5
x1 + x2 < 10
x1 + 5x2 ≥ 2
x1 > 0 and x2 > 0
Standard form of an LP
Sensitivity analysis
Recall that last class we discussed:Changes in objective function coefficientsChanges in RHS valuesHow to interpret shadow prices
Sensitivity analysis involves changing only one coefficient at a time!All other coefficients are held constant
Changes in OF coefficients
Change in OF coefficient does not impact the feasible region or the existing extreme points in any way, although it impacts the value associated with the optimal solution
In a two-decision variable graphical problem, a change in an OF coefficient changes the slope of the OF lineRegardless of how many decision variables
are considered, changes in OF coefficients do not change the feasible region
Changes in OF coefficients
Here we identified the range of optimality The current solution remains optimal for all
changes to OF coefficients within this range• The allowable increase and allowable decrease
provide the range of values each OF coefficient can take on where the current optimal solution remains optimal
Outside of this range, the slope of the OF line changes enough that a different extreme point in the feasible region becomes the new optimal solution
Changes in RHS constraint coefficients Change in RHS coefficients are a bit more
tricky to interpret because changing the RHS of a constraint can impact both the feasible region and change the extreme points for the problem
In a two-decision variable graphical problem, a change in RHS coefficient can increase or decrease the area of the feasible region
Changes in RHS constraint coefficients Here we identified the range of feasibility
The allowable increase and allowable decrease associated with the shadow price for each constraint provides the range of values that each RHS coefficient can take on where the:• Current binding constraints remain binding AND• Interpretation of the shadow price is valid
Changes in RHS constraint coefficients Outside of the range of feasibility
the interpretation of the shadow price does not hold, AND
a different set of constraints become binding
When the RHS of a constraint changes outside of the range of feasibility, you must re-formulate and re-solve the LP
Changes in the LHS constraint coefficients Classical sensitivity analysis provides no
information about changes resulting from a change in the LHS coefficients in a constraint
We must change the coefficient and rerun the model to learn the impact the change will have on the solution
Non-intuitive shadow prices
Constraints with variables naturally on both the left-hand (LHS) and right-hand (RHS) sides often lead to shadow prices that have a non-intuitive explanation
Example of non-intuitive shadow price Recall from last class that Olympic Bike is
introducing two new lightweight bicycle frames, the Deluxe (x1) and the Professional (x2) , to be made from special aluminum and steel alloys
The objective was to maximize total profit, subject to limits on the availability of aluminum and steel
Olympic bike example
Consider the problem from Lecture 16
Max 10x1 + 15x2
s.t. 2x1 + 4x2 < 100
3x1 + 2x2 < 80
x1 > 0 and x2 > 0
ObjectiveFunction
“Regular”Constraints
Non-negativity Constraints
Olympic bike example
Let’s introduce an additional constraint The number of Deluxe frames produced (x1)
must be greater than or equal to the number of Professional frames produced (x2)
Olympic bike example
Max 10x1 + 15x2
s.t. 2x1 + 4x2 < 100
3x1 + 2x2 < 80
x 1 > x2
x1 > 0 and x2 > 0
ObjectiveFunction
“Regular”Constraints
Non-negativity Constraints
Olympic answer report
Objective Cell (Max)Cell Name Original Value Final Value
$B$4 Objective Function (Maximize Profit) 0 400
Variable CellsCell Name Original Value Final Value Integer
$B$1 X1 (# of Deluxe frames) 0 16 Contin$B$2 X2 (# of Professional frames) 0 16 Contin
ConstraintsCell Name Cell Value Formula Status Slack
$B$8 1) Constraint#1 (materials (aluminum) constraint) LHS 96 $B$8<=$C$8 Not Binding 4$B$9 2) Constraint#2 (materials (steel) constraint) LHS 80 $B$9<=$C$9 Binding 0$B$10 3) Constraint #3 (production constraint) LHS 16 $B$10>=$C$10 Binding 0
Olympic sensitivity report
Variable CellsFinal Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease$B$1 X1 (# of Deluxe frames) 16 0 10 12.5 25$B$2 X2 (# of Professional frames) 16 0 15 1E+30 8.333333333
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$B$8 1) Constraint#1 (materials (aluminum) constraint) LHS 96 0 100 1E+30 4$B$9 2) Constraint#2 (materials (steel) constraint) LHS 80 5 80 3.333333333 80$B$10 3) Constraint #3 (production constraint) LHS 16 -5 0 26.66666667 2.5
Olympic bike example
Interpret the shadow prices of:Constraint #1
Olympic bike example
Interpret the shadow prices of:Constraint #2
Olympic bike example
Interpret the shadow prices of:Constraint #3
Olympic bike example
Interpret the range of optimality for OF coefficient:c1
Olympic bike example
Interpret the range of optimality for OF coefficient:c2
Reduced costs
The reduced cost of a variable is typically the shadow price of the corresponding non-negativity constraintSo most, variables have a reduced cost = 0x1, x2, etc. ≥ 0
We will discuss a situation when the reduced costs ≠ 0 in a four decision variable problem before the final exam
Summary
Special case LP problemsAlternative optimal solutionsInfeasibilityUnbounded problems
Sensitivity analysis continued Non-intuitive shadow prices Example