Sensitivity Analysis
Sensitivity analysis explains, how sensitive the optimal solution is to
changes in various coefficients of the LP model.
Sensitivity analysis can help overcome this skepticism and provide a
better picture of how the solution to a problem will change if different
factors in the model change.
Sensitivity analysis also can help answer a number of practical
managerial questions that might arise about the solution to an LP
problem.
Linear Programming (An Example)
Blue Ridge Hot Tubs manufactures and sells two models of hot tubs: the Aqua-Spa and
the Hydro-Lux. Howie Jones, the owner and manager of the company, needs to decide
how many of each type of hot tub to produce during his next production cycle. Howie
buys prefabricated fiberglass hot tub shells from a local supplier and adds the pump
and tubing to the shells to create his hot tubs. (This supplier has the capacity to deliver
as many hot tub shells as Howie needs.) Howie installs the same type of pump into
both hot tubs. He will have only 200 pumps available during his next production cycle.
From a manufacturing standpoint, the main difference between the two models of hot
tubs is the amount of tubing and labor required. Each Aqua-Spa requires 9 hours of
labor and 12 feet of tubing. Each Hydro-Lux requires 6 hours of labor and 16 feet of
tubing. Howie expects to have 1,566 production labor hours and 2,880 feet of tubing
available during the next production cycle. Howie earns a profit of $350 on each Aqua-
Spa he sells and $300 on each Hydro-Lux he sells. He is confident that he can sell all the
hot tubs he produces. The question is, how many Aqua-Spas and Hydro-Luxes should
Howie produce if he wants to maximize his profits during the next production cycle?
SUMMARY OF LP MODEL (EXAMPLE)
- The complete LP model for Howie’s decision problem can be stated as:
MAX: 350 X1 + 300 X2
Subject to:
Pumps 1 X1 + 1 X2 ≤ 200
Labor 9 X1 + 6 X2 ≤ 1566
Tubing 12 X1 + 16 X2 ≤ 2880
X1 ≥ 0
X2 ≥ 0 - Types of Constraints
- A constraint is binding if it is satisfied as a strict equality in the
- optimal solution; otherwise, it is nonbinding.
Types of Constraints
Types of Solutions
Solutions
Feasible
Solution
Not optimal
Solution
Optimal
Solution
Robust
Solution
Infeasible
Solution
A solution is a set of all the values for the
variables.
Feasible Solution is set of solutions that satisfy all
the equations.
Infeasible solution is a set of solutions that do
not satisfy any one of the equation of LP.
Optimal solution is a solution if and only if it
doesn’t exist a better/best solution.
It is the best possible solution among the available
The Simplex Method
As compared to other problem solving methods, the simplex method
provides us with information about the following:
The range of values the objective function coefficients can assume
without changing the optimal solution
The impact on the optimal objective function value of increases or
decreases in the availability of various constrained resources
The impact on the optimal objective function value of forcing changes
in the values of certain decision variables away from their optimal
values
The impact that changes in constraint coefficients will have on the
optimal solution to the problem,
Slack
The difference between left hand side (LHS) and right hand side (RHS)
in the equation of each constraint,
Binding constraints have Zero slack and non-binding constraints have
some positive value of the slack. A negative slack shows an infeasible
solution.
Slack
- The values in the Slack column indicate that if this solution is
implemented, all the available pumps and labor hours will be used, but
168 feet of tubing will be left over.
Sensitivity Report - This report summarizes information about the variable cells and
constraints for our model. - This information is useful in evaluating how sensitive the optimal
solution is to changes in various coefficients in the model
Shadow Price
- The shadow price for a constraint indicates the amount by which the
objective function value changes given a unit increase in the RHS value
of the constraint, assuming all other coefficients remain constant. - For every one unit increase in resources, will increase in objective
function equal to the shadow price value. - If a shadow price is positive, a unit increase in the RHS value of the
associated constraint results in an increase in the optimal objective
function value. - If a shadow price is negative, a unit increase in the RHS value of the
associated constraint results in a decrease in the optimal objective
function value.
To analyze the effects of decreases in the RHS values, you reverse the sign on the shadow price
Reduced Cost
- The reduced cost means that additional profit for every additional unit we
produce. - The reduced cost for each variable is equal to the per-unit amount the
product contributes to profits minus the per-unit value of the resources it
consumes (where the consumed resources are priced at their shadow prices).
Reduced Cost
- The reduced cost means that additional profit for every additional unit we
produce. - The reduced cost for each variable is equal to the per-unit amount the
product contributes to profits minus the per-unit value of the resources it
consumes (where the consumed resources are priced at their shadow prices).
for Aqua-Spas to any value in
the range from $300 to $450
Similarly, the objective function
coefficient for Hydro Luxes can
assume any value between
$233.33 and $350 without
changing the optimal solution
Degeneracy - The solution to an LP problem sometimes exhibits a mathematical
anomaly known as degeneracy. - An LP is degenerate if in a basic feasible solution, one of the basic
variables takes on a zero value. Degeneracy is a problem in practice,
because it makes the simplex algorithm slower. - The solution to an LP problem is degenerate if the RHS values of any of
the constraints have an allowable increase or allowable decrease of
zero. - Robust Solution
- Robust Solution to an LP problem is a solution in the interior of the
feasible region (rather than on the boundary of the feasible region) that
has a reasonably good objective function value. - Clearly, such a solution will not maximize (or minimize) the objective
function value (except in trivial cases), so it is not an optimal solution in
the traditional sense of the word. - However, a robust solution will generally remain feasible if modest
perturbations or changes occur to the coefficients in the model. - Duality Theory
- Every linear programming model has two forms.
- The first or original form is called the Primal, and the second form is
derived from the primal and is called the Dual. - Since both the primal and the dual are associated with the same
problem, their respective solutions are closely related to each other. - Specifically, the optimal solution values in the primal are equal to the
shadow prices in the dual, and the shadow prices in the primal are
equal to the optimal solution values in the dual. - In addition, the optimal objective function value for the primal is the
same as that for the dual. - Lone Star Company
- Lone Star Company in Little Rock, AR, is developing a new paint that must have a
- brilliance rating of at least 15 degrees, a hue rating of at least 32 degrees, and a clarity
- rating of at least 21 degrees.
- The two ingredients to be mixed to produce the new product are Alpha and Beta,
- which cost $7 per pound and $9 per pound, respectively. Each pound of Alpha
- contributes to 3 degrees of brilliance, 8 degrees of hue, and 7 degrees of clarity
- whereas each pound of Beta contributes 5 degrees of brilliance, 8 degrees of hue, and
- 3 degrees of clarity. Relevant information has been summarized in the following table:
- Currently, only 1.2 pounds of Beta are available. Let A be the amount (in pounds) of
- Alpha and B be the amount (in pounds) of Beta to be mixed to develop the new pain
Currently, only 1.2 pounds of Beta are available. Let A be the amount (in pounds) of
Alpha and B be the amount (in pounds) of Beta to be mixed to develop the new paint.
Lone Star Company (Contd..)
A linear program for determining the quantity of each ingredient to be use to minimize
the total cost is presented below:
(1) Run Solver to solve the LP, summarize the optimal solution as well as the optimal objective
function value, and interpret them. Be sure to show both the Answer Report and the Sensitivity
Report.
(2) Based on the Answer Report, how many constraints are nonbinding at optimality? What are the
respective slacks? What do they mean in this problem and where do they come from?
(3) If a local supplier would like to provide additional Beta at a cost of $2 per pound, should the
offer be accepted based on the Sensitivity Report? Why or why not?
