LP Sensitivity Analysis*
The previous courses provided two types of methods for solving LP
problems (i.e. finding an optimal solution and the related optimal value of
the objective function):
1. The graphical method which is applicable to problems with two
variables (and with at most three variables).
2. The simplex methods for solving general LP problems.
Remark. The simplex methods computations are particularly tedious,
repetitive, and boring. Once you have learned how the simplex methods
work (and it is important that you do understand the concepts and the
mathematical foundations of the simplex methods presented in Courses
2&3), computers with carry out the tedious work and you will not need to
solve an LP manually.
*)
This course is mainly based on Chapter 3 in Taha (2007), and Chapters 3
& 4 in Hillier and Liebermann (2005).
This course focuses on LP sensitivity analysis based on the optimal
solution found via both types of methods. With sensitivity analysis we
can ascertain the impact of the uncertainty regarding the parameters’
values on the quality of the optimum solution and related information. In
LP the parameters (input data) of the model can change within certain
limits preserving the optimality of the solution or some other parameters
determined by the optimal solution.
With an optimal solution of an LP problem additional economic
information can be derived, namely the related dual prices of resources
and the reduced (imputed) costs of activities.
 The dual or shadow price of a resource is the technical name of the unit
worth of that resource. The unit worth of a resource is the change in the
optimal objective value per unit change in the availability of a resource.
Knowing the range of a certain resource which keeps the dual price of
that resource unchanged provides support for scientific decision making.
 The reduced cost is defined as
Reduced cost per unit = Cost of consumed resources per unit −
Revenue per unit.
It is used to evaluate the “attractiveness” of a “product” from the
standpoint of optimization which depends on the relative values of the
revenue per unit and the cost of the resources consumed by one unit.
Sensitivity analysis deals with determining:
 the dual prices (corresponding to the optimal solution) and the
conditions regarding the changes in the availability of the
resources that will keep the current dual prices valid;
 the reduced costs (corresponding to the optimal solution) and the
conditions regarding changes in coefficients of the objective
function that will keep the current optimal solution (point)
unchanged.
All software packages provide information for sensitivity analysis. The
material in this course also stresses the interpretation of each iteration in
terms of the solution of the original problem, to make you able to read and
interpret the output reports provided by commercial software. It also
explains how simplex based sensitivity analysis is used to provide important
economic interpretations about the optimum solution, including the dual (or
shadow) prices and the reduced costs.
To explain the basics of sensitivity analysis we start with graphical solution
and then extend to the general LP problem using the simplex tableau results.
 Graphical sensitivity analysis uses the graphical solution to
determine:
(i)
The dual prices (of resources) and their feasibility (conditions
and) ranges (that keep the dual prices unchanged). The current
value of the objective function and the current optimum point
will change owing to the change in the parameters, but their
new values can be easily computing based on the current
optimal solution.
(ii)
The optimality (conditions and) ranges of the unit prices that
keep the optimal solution (point) unchanged. The current value
of the objective function will change owing to the change in its
parameters, but the new value can be easily computing based on
the current optimal solution.
 Algebraic sensitivity analysis uses (the z-row in) the optimal
tableau to determine the dual prices and the reduced costs, and by
recomputing the optimal simplex tableau with the modified righthand side (or the modified coefficients of the objective function) in
terms of deviations from the original data determines:
(i)
(ii)
The feasibility (conditions and) ranges for the resources that
assure that the dual prices associated with the optimal solution
remain valid. The current value of the objective function and
the current optimum point will change owing to the change in
the parameters, but their new values can be easily computing
based on their expression in terms of the current deviations of
the parameters, found in the recomputed optimal simplex
tableau.
The optimality conditions and the optimality ranges for the unit
costs that will keep an optimal solution unchanged. The current
value of the objective function will change owing to the change
in its parameters, but its new value can be easily computing
based on (its row in) the recomputed optimal simplex tableau.
In this course we rewrite the primal basic form of a LP problem
(formula (3) in Courses 2&3) and use its related (primal) tableau. This
approach provides better motivation for the name “basic variables”,
illustrates that the use of the label “−1” in Courses 2&3 is merely a
convention, and is more convenient for the pivot operations and the
scope of sensitivity analysis.
Two cases will be considered for sensitivity analysis:
1. Sensitivity of the optimum solution to changes in the
availability of the resources (right-hand side of the constraints).
2. Sensitivity of the optimum solution to changes in unit profit or
unit cost (coefficients of the objective function).
(I) Graphical Sensitivity Analysis
 Changes in the availability of a resource will move the line
corresponding to its constraint parallel to itself. The dual price for a
certain resource remains valid for changes (increases or decreases) in the
resource that move the line corresponding to its constraint parallel to
itself to any point on the line segment corresponding to the other resource
constraint related to the optimum solution (corner point). The extreme
points of this segment line determine the range of applicability of the
given dual price. We call such a range the feasibility range for (the dual
price corresponding to) a certain resource.
The dual prices and their feasibility range, determined by using an optimal
solution of an LP problem, allow making economic decisions about that
problem (when variants of changes in resources are considered).
Example 3.6-1 in JOBCO.pdf illustrates:
(1)
(2)
The computation of the dual price and of the
feasibility range of a resource.
How the computed dual prices and their feasibility
range allow making economic decisions about this LP
problem.
 Changes in revenue units (i.e. objective-function coefficients) will
change the slope of z. However, the optimum solution will remain in the
same corner point so long as the line of the objective function lies
between the lines corresponding to the two constraints that define the
optimum point. This means that there is a range for the coefficients of the
objective function that will keep the optimum solution unchanged at the
optimum corner point. If we write z = c1x1 + c2x2 we can obtain the range
for c1/c2 by using the data of the lines that contain the optimum point.
Furthermore, we can determine the optimality range for c1(c2) by fixing
the value for c2(c1). The computation of the optimality range for the unit
costs and how the computed optimality ranges can be used for making
economic decisions on LP problems are also illustrated for the JOBCO
model.
(II) Sensitivity Analysis using the Primal Simplex Method
We start with introducing an equivalent representation for the primal
basic form: we write the objective equation as z − ∑cixi = 0, and the
constraints equations as ∑aijxj + si = bi, where si denotes the
corresponding slack variable.
The primal tableau corresponding to the above basic form will have:
 Its first column is labeled with z; it is followed by columns labeled
with the nonbasic variables (as in the previous representation); then
we place the columns labeled by the basic variables; the last column,
labeled “Solution” (or RHS), contains the constants terms (from the
objective function and the right-hand side of the constraint equations).
 The first row is labeled (in the left side) by the objective function
(referred to as the z-row), and the other rows are labeled (also in the
left side) by the basic variables.
Using this new approach, the simplex algorithm can be reformulated as
follows:
Step 1. Determine a starting basic feasible solution (LP problems in which
all the constraints are (≤) with nonnegative right-hand sides offer a
convenient all-slack starting feasible solution).
Step 2. Select an entering variable using the optimality condition (based on
the z-row).
Optimality condition. The entering variable in a maximization
(minimization) problem is the nonbasic variable having the most negative
(positive) coefficient in the z-row. Ties are broken arbitrarily (or using
Bland’s rule). The optimum is reached at the iteration where all the z-row
coefficients of the nonbasic variables are nonnegative (nonpositive).
STOP if there is no entering variable; the (last) solution is optimal.
Otherwise, go to Step 3.
Step 3. Select a leaving variable using the feasibility condition.
Feasibility condition. For both maximization and minimization
problems, the leaving variable is the basic variable associated with the
smallest nonnegative ratio (with strictly positive denominator). Ties are
broken arbitrarily (or using Bland’s rule).
Step 4. Determine a new basic solution by using the appropriate GaussJordan computations. Go to Step 2.
Gauss-Jordan row operations:
1. Pivot row
a. Replace the (label of the) leaving variable in the Basic column with
the (label of the) entering variable.
b. New pivot row = Current pivot row ÷ Pivot element.
2. All other rows, including z
New row = (Current row) − (pivot column coefficient) × (New pivot
row).
We use the Reddy Mikks model (Example 3.3-1 in ExSimplex.pdf) to
illustrate the new approach.
Changes in the Right-Hand Side
1. Determination of Dual Prices
The z-row in the optimal simplex tableau for a LP problem with ≤
constraints yields directly the dual prices as the coefficients of the slack
variables. (For ≥ constraints, the same idea remains applicable except that
the dual price will assume the opposite sign of that associated with the ≤
constraint. As for the case where the constraint is an equation, the
determination of the dual price from the optimal simplex tableau requires
somewhat involved calculations as will be shown in Course 5.)
2. Determination of the Feasibility Conditions and Ranges
To determine the feasibility ranges in a LP problem with all ≤ constraints,
we introduce in the right-hand sides of the constraints additional variables Di
for changes (positive or negative) in the resources. Then we compute the
optimum simplex tableau for this new problem, and derive the conditions
that will keep the solution feasible – that is the right-hand side of the
optimum tableau remains nonnegative. The individual feasibility ranges
result from changing the resources one at a time.
We illustrates the procedure to obtain the dual prices and the feasibility
(conditions and) ranges (that keep the dual prices unchanged) using the
TOYCO model (Example 3.6-2 in TOYCO1.pdf).
The above results can be summarized as follows:
__________________________________________
1. The dual prices remain valid so long as the changes Di, i= 1, …, m, in
the right-hand sides of the constraints satisfy all the feasibility
conditions when the changes are simultaneous or fall within the
feasibility ranges when the changes are made individually.
2. For other situations where the dual prices are not valid because the
simultaneous feasibility conditions are not satisfied or because the
individual feasibility ranges are violated, the recourse is to either resolve the problem with the new values of Di or apply the post-optimal
analysis presented in Course 5.
_______________________________________________
Available LP packages usually present this information as standard output.
Practically, none provide the case of simultaneous conditions, presumably
because its display is cumbersome, particularly for large LP problems.
Changes in the Objective Function
1. Determination of Reduced Costs
The reduced costs can be read in the optimal simplex tableau (z-equation
coefficients).
With the given definition of the reduced cost we can now see that an
unprofitable variable can be made profitable in two ways:
1. By increasing the unit revenue.
2. By decreasing the unit cost of consumed resources.
In most real-life situations, the price per unit may not be a viable option
because its value is dictated by market conditions. The real option then is to
reduce the consumption of resources, perhaps by making the production
process more efficient.
2. Determination of the Optimality Conditions and Ranges
To determine the conditions that will keep an optimal solution unchanged (in
a maximization problem), called the optimality conditions, we introduce
additional variables di for the changes (increases or decreases) in unit
revenues of the coefficients in the objective function. Then we generate the
simplex tableaus using the same sequence of entering and leaving variables
as in the original model. The new optimal tableau is exactly the same as the
original optimal tableau except that the reduced costs (z-equation
coefficients) have changed. This means that changes in the objectivefunction coefficients can affect the optimality of the problem only.
A convenient way for computing the new reduced costs is to add a new top
row and a new leftmost column to the optimal tableau:
 the entries in the top row are the changes dj associated with each
variable;
 the entries in the leftmost column are 1 in the z-row and the associated
dj in the row of each basic variable (keep in mind that dj = 0 for the
slack variables).
Now, to compute the new reduced cost for each variable (or the value of z),
multiply the elements of its column by the corresponding elements in the
leftmost column, add them up, and subtract the top-row element from the
sum.
Note that the application of these computations to the basic variables will
always produce a zero reduced cost, a proven theoretical result.
The optimality conditions corresponding to nonbasic variables (in a
maximization problem) are obtained by the nonnegativity of the
corresponding new reduced costs (z-equation coefficients). These conditions
should be satisfied simultaneously to maintain the optimality of the current
optimum.
Remark. The only difference in the minimization case is that the reduced
costs (z-equation coefficients) must be ≤ 0 to maintain optimality.
The general optimality conditions can be used to determine the individual
optimality ranges for di by one at a time changes.
We illustrate the above procedure by using the TOYCO model
(TOYCO2.pdf).
The above results can be summarized as follows:
___________________________________________
1. The optimal values of the variables remain unchanged so long as
the changes dj, j = 1, …, n, in the objective function coefficients
satisfy all the optimality conditions when the changes are
simultaneous or fall within the optimality ranges when a change
is made individually.
2. For further situations where the simultaneous optimality
conditions are not satisfied or the individual ranges are violated,
the recourse is to either resolve the problem with the new values
of dj or apply the post-optimal analysis presented in Course 5.
___________________________________________
(III) Sensitivity Analysis with Solver
Click Solve in Solver parameters dialogue box.
↓
The new dialogue box Solver results will give you the opportunity to
request further details about the solution, including the important
sensitivity analysis report (click Sensitivity).
↓
The report will be stored in a separate Excel sheet, as shown by the
choices on the bottom of the screen. You can then click Sensitivity
Report 1 to view the results.
The output includes:
1. The reduced cost.
2. The shadow (dual) price.
3. The optimality (feasibility) ranges for the changes dj and Di.
To illustrate the Excel Solver sensitivity analysis we use the report for
the TOYCO model (in SA_Solver.pdf)
Solution for Homework Week 1 (SpecialCases.pdf)
1.
2.
3.
4.
Degeneracy (Problem 1)
Alternative optima (Problem 2)
Unbounded solutions (Problem 3)
Nonexisting (or infeasible) solutions (Problem 4).
Degeneracy
In the application of the feasibility condition of the simplex method, a tie
for the minimum ratio may occur and can be broken arbitrarily. When
this happens, at least one basic variable will be zero in the next iteration
and the new solution is said to be degenerate. In the graphical solution of
Problem 1 three lines pass through the optimum point x1 = 0, x2 =2.
Because this is a two-dimensional problem, the point is overdetermined
and one of the constraints (x1 + 4x2 ≤ 8) is redundant.
From the theoretical standpoint degeneracy has two implications:
 The phenomenon of cycling or circling.
 The solution may be temporarily degenerate. Note that two
consecutive iterations (though differing in the basic-nonbasic
categorization of the variables) yield identical values for all the
variables and the objective value. The computation should not be
stopped at the iteration where degeneracy first appears.
From the practical point of view, degeneracy reveals that the model
has at least one redundant constraint. In practice, the mere knowledge that
some resources are superfluous can be valuable during the implementation
of the solution. The information may also lead to discovering irregularities
in the construction of the model. Unfortunately, there are no efficient
computational techniques for identifying the redundant constraints directly
from the tableau.
Alternative Optima
When the objective function is “parallel” to a non-redundant binding
constraint (i.e. a constraint that is satisfied as an equation at the optimal
solution), the objective function can assume the same optimal value at more
than on solution point, thus giving rise to alternative optima. Problem 2
shows that there is an infinite number of optimal solutions (the lines z = 2x1
+ x2 and that corresponding to the constraint x1 + 2x2 ≤ 5 are parallel); it also
demonstrates the practical significance of encountering such solutions.
In practice, alternative optima are useful because we can choose from many
solutions without experiencing deterioration in the objective value.
Unbounded Solution
In some LP models, the values of the variables may be increased indefinitely
without violating any of the constraints – meaning that the solution space is
unbounded in at least one variable. As a result, the objective value may
increase (maximization case) or decrease (in the minimization case)
indefinitely. Unboundedness points to the possibility that the model is
poorly constructed. The most likely irregularity in such models is that one or
more non-redundant constraints have not been accounted for, and the
parameters (constants) of some constraints may not have been estimated
correctly. In the simplex method unboundedness can be recognized when
there is no leaving variable (all the denominators of the ratios of the
feasibility conditions are negative or zero). In the graphical solution of
Problem 3 the solution space is unbounded in the direction of x2, and the
value of z can be increased indefinitely.
Infeasible Solution
LP models with inconsistent constraints have no feasible solution. We notice
that this situation can never occur if all the constraints are of the type ≤ with
nonnegative right-hand sides because the slack variables provide a feasible
solution. For other types of constraints we use artificial variables. Although
the artificial variables are penalized in the objective function to force them
to zero at the optimum, this can occur only if the model has a feasible space.
Otherwise, at least one artificial variable will be positive in the final
(optimum) iteration. From the practical point of view, an infeasible space
points to the possibility that the model is not correctly formulated. The
graphical solution of Problem 4 illustrates such a situation.
More about the Artificial Starting Solution
The procedure for “ill-behaved” LPs with (=) and (≥) constrained is to
use artificial variables that play the role of slacks at the first
iteration, and then dispose of them legitimately at a later iteration.
Two closely related methods are introduced here: the M-method and
two-phase method.
M-Method
The M-method starts with the LP equation form. In any equation i
which does not have a slack variable (or a variable that can play the
role of a slack) we introduce an artificial variable Ri. Then, a starting
basic solution is obtained (similar to the convenient all-slack basic
solution). However, because the artificial variables are not part of the
original LP model, they are assigned a very high penalty in the
objective function, thus forcing them (eventually) to equal 0 in the
optimum solution. This will always be the case if the problem has a
feasible solution.
Penalty rule for artificial variables. Given M, a sufficiently large
positive value (mathematically, M → ∞), the artificial variable
objective coefficient equals −M in maximization problems and M in
minimization problems.
From the standpoint of solving the problem on the computer, M must
assume a numeric value. What value of M should we use? The
answer depends on the data of the original LP. M should be large
relative to the actual objective coefficients of the model, but should
not be too large because potential severe round-off errors can result
when very large values are manipulated with much smaller values
(that may impair the accuracy of the simplex calculations). Chosen M,
before proceeding with the simplex method computations, we need to
make the z-row consistent with the rest of the tableau by substituting
out the artificial variables in the z-row using the appropriate constraint
equations.
The application of the M-Method is illustrated using Example 3.4-1 in
M-Method.pdf.
Two-Phase Method
Phase I. Put the problem in equation form, and add the necessary
artificial variables to the constraints (exactly as in the M-method) to
secure a starting basic solution. Next, find a basic solution of the
resulting equations that, regardless of whether the LP is maximization
or minimization, always minimizes the sum of the artificial variables.
If the minimum value of the sum is positive, the LP problem has no
feasible solution, which ends the process (recall that a positive
artificial variable signifies that an original constraint is not satisfied).
Otherwise, proceed to Phase II.
Phase II. Use the feasible solution from Phase I as a starting basic
feasible solution of the original problem.
We illustrate the
Two_Phases.pdf.
Two-phases
method
using
Example
3.4-2
in