WORKSHEET 5
SENSITIVITY ANALYSIS II:
SENSITIVITY ANALYSIS OF RIGHT–HAND SIDE VALUES
Consider the Beaver Creek example:
Max.Z= 40 x1+ 50 x2
s.t.
x1+ 2x2 ≤ 40
4 x1+ 3x2 ≤ 120
x1, x2 ≥ 0
The optimal solution is:
x1 = 24
x2 = 8
Z = $1,360
Let us now consider the effects of changing right-hand side values of the constraints.
Any change in the right-hand side value of a binding constraint will change the
optimal solution. For example, if the amount of labor hours is increased from 40 hours to 41
hours, the new optimal solution is found by solving
x1+ 2x2 = 41 (labor)
4 x1+ 3x2 =120 (mtrl.- clay)
The solution is x1 = 23.4; x2 = 8.8 giving an optimal objective function value of:
40(23.4) + 50(8.8) = $1,376
Figure 1. Effect of increasing the availability of labor hours
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Sensitivity analysis of right-hand side values focuses on two questions:
1. Keeping all other factors the same, how much would the optimal profit change if the
right-hand side of a constraint is changed by one unit?
2. For how many additional or fewer units will this per unit change be valid?
 Answering the First Question: SHADOW PRICES
If the right-hand side of a “≤ ” constraint is increased, the constraint becomes less
restrictive, hence the optimal value of the objective function can only improve (increase for a
maximization problem, decrease for a minimization problem) or stay the same. Conversely, if
the right hand side of a “≥ ” constraint is increased, the constraint becomes more restrictive,
and the optimal value of the objective function can only worsen or stay the same. The change
in the optimal value of the objective function resulting from a one-unit increase in the righthand side of a constraint is called its shadow price.
We have seen that a one unit increase in the amount of labor hours to 41 hours yields a
new objective function value of $1,376. Thus the increase in the value of the objective
function attained by this one additional unit of labor hours is $1,376-$1,360 = $16; that is $16
is the shadow price for labor hours
To determine the shadow price for an extra pound of clay we solve:
x1+ 2x2 = 40
4 x1+ 3x2 =121
Solving for x1 and x2 yields x1 = 24.4; x2 = 7.8 giving an optimal objective function
value of:
40(24.4) + 50(7..8) = $1,366. Hence, the shadow price for an extra pound of clay is
$1,366 - $1,360 = $6.
Figure 2 Effect of Increasing the Availability of Material (Clay)
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The optimal solution is not affected by a change in a right-hand side of a nonbinding
constraint that is less than the slack or surplus for the constraint. Accordingly, because the
optimal solution remains the same, the optimal objective function value does not change.
Hence, we conclude that the shadow price for nonbinding constraints is 0


Shadow Prices
The shadow price for a constraint measures the amount the optimal objective
function value changes per unit increase in the right-hand side value of the constraint
At the optimal solution, either a constraint is satisfied with equality or the
corresponding shadow price is 0.
 Answering the Second Question: RANGES OF FEASIBILITY
If 41 hours (two additonal hours) of labor is available, the optimal solution is found
by solving the following equations:
x1+ 2x2 = 41
4 x1+ 3x2 =120
The solution is x1 = 23.4; x2 = 8.8 giving an optimal objective function value of:
40(23.4) + 50(8.8) = $1,376
This represents an increase of $16 per hour, above the original optimal objective
function value.
In fact, as long as the labor and material (clay) constraints determine the optimal
extreme point, each unit change in the availability of labor hours contributes the shadow price
of $16 to the optimal profit.
The range of feasibility for labor hours gives the limits on the right-hand side value
for labor hours, such that these two constraints continue to determine the optimal point. It is
given this name because, over this range of values, the solution to the current set of binding
constraint equations is a feasible solution.
The figures below illustrate how the range of feasibility for labor hours can be
determined. Notice that by changing the right-hand side of the labor constraint from 40 to
another number, the slope does not change. Thus, by changing the right-hand side from 40,
the boundary of the labor constraint moves paralel to the original constraint, and the optimal
solution continues to be determined by the intersection of these two boundary lines until
they generate an infeasible point.
The focus of sensitivity anaysis for constraint quantity values is to determine the range
over which the constraint quantity values can change without changing the solution
variable mix, specifically including the slack variables.
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Figure 3 Changes to the Feasible Region When Figure 4. Changes to the Feasible Region When
Availability of Labor Hours Increases
Availability of Labor Hours Decreases
In Figure 3, as the availability of labor hours increases, the labor and material (clay)
constraints determine the optimal point until the line passes through the intersection of the
material (clay) boundary and the x2 axis.
This point (x1 = 0 x2 = 40) is determined by solving 4x1+ 3x2 =120 and x1 = 0. At
(0;40), the right-hand side of the labor constraint is:
1(0) + 2(40) = 80.
For values above 80, the intersection of the labor and material (clay) constraints yields
a point outside of the new feasible region. Thus, 80 is the upper limit of the range of
feasibility for labor.
Similarly, in Figure 4, as the availability of labor hours decreases, the labor and
material constraints determine the optimal point until the line passes through the intersection
of the material (clay) boundary and x1 axis at (x1= 30; x2 = 0). As the line passes below this
point, again, the intersection of the labor and material constraints will produce infeasible
points for the new feasible region. At (30;0), the right-hand side value of the labor constraint
is:
1(30) +2(0) = 30. That is, 30 is the lower value of the range of feasibility for labor
hours.
Remember that we have determined the shadow price for an extra pound of material
(clay). Now find the range of feasibility of clay, that is the range of values for which the
shadow price of $6 is valid-?
Note that in the range of feasibility, the values of the variables in the optimal solution
and the objective function value Do Change; it is the shadow price that Do Not Change.
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Range of Feasibility
The range of feasibility for the availability of a resource is the set of right-hand- side
values over which the same set of constraints determines the optimal point.
Within the range of feasibility, the shadow prices remain constant, however, the
optimal solution will change.

Consider the Puck and Pawn Company example:
Max.Z= 2x1+ 4x2
s.t.
41+ 6x2 ≤ 120 (machine A)
2 x1+ 6x2 ≤ 72 (machine B)
x2 ≤ 10 (machine C)
x1, x2 ≥ 0
a) Determine the shadow price for Machine A
b) Detemine the shadow price for Machine B
c) Determine the shadow price for Machine C
d) Determine the ranges of feasibilities of the binding constraints
 Multiple Changes: Analyzing Simultaneous Changes Using the 100%
Rule
A 100% rule, analogous to the 100% rule for objective function coefficients, provides
a sufficient condition for multiple changes that will not affect the shadow prices.
1. For each increase in the right-hand side value, calculate (and express as a percentage)
the ratio of the change in the RHS to the maximum possible increase as determined by
the limits of the range of feasibility.
2. For each decrease in the right-hand side value, calculate (and express as a percentage)
the ratio of the change in the RHS to the maximum possible decrease as determined by
the limits of the range of feasibility.
3. Sum all these percent changes. If the total is less than 100%, the optimal solution will
not change. If the total is greater than or equal to 100%, the optimal solution may
change.
Suppose in the Beaver Creek model, there are 20 additional hours of labor and 30 pounds less
material (clay). By yusing the 100% rule, find whether the current shadow prices are still
valid.
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