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