Question 2: How will changes in the objective function’s coefficients change the
optimal solution?
In the previous question, we examined how changing the constants in the constraints
changed the optimal solution to a standard maximization problem. Such modification led
to a different feasible region and changes to the optimal solution.
In this question, we examine how changes in the coefficients of the objective function
change the optimal solution. If one of the coefficients of the objective function increases
or decreases, the feasible region remains the same. Since the feasible region is defined
by the constraints and not the objective function, there are no changes to the corner
points. A change in one of the coefficients changes the slope of the isoline, the line that
corresponds to setting the objective function equal to a constant.
Let’s recall the problem we have been analyzing:
Maximize P  70 x  120 y
subject to
x  y  35
x  3 y  75
2 x  y  60
x  0, y  0
Figure 10 - The feasible region for the standard maximization
problem on the left.
The objective function takes on the optimal value P  3450 at the corner point (15, 20).
The isoprofit line 70 x  120 y  3450 just touches the feasible region at that point verifying
this assertion.
Now let’s try increasing and decreasing the coefficient on x in the objective function.
Figure 11 – This graph shows the feasible region with the isoprofit
line at the optimal solution (gray dashed line) and the isoprofit line
for the optimal solution in the modified problem (purple dashed
line). The coefficient on x has decreased from 70 to 40.
As the coefficient on x is decreased, the isoprofit line for the optimal solution gets less
steep. At a value of 40, the isoprofit line passes through (0, 25) and (15, 20). At this
point, the isoprofit line forms the border of the feasible region there so all points on the
line between (0, 25) and (15, 20) yield the same maximum profit. If the coefficient is
decreased to a value less than 40, (15, 20) is no longer part of the optimal solution.
Figure 12 -This graph shows the feasible region with the isoprofit
line at the optimal solution (gray dashed line) and the isoprofit
line for the optimal solution in the modified problem (purple
dashed line). The coefficient on x has increased from 70 to 120.
As the coefficient on x is increased, the isoprofit line for the optimal solution gets
steeper. At a value of 120, the isoprofit line passes through (0, 25) and (25, 10). The
isoprofit line forms the border of the feasible region there, so all points on the line
between (0, 25) and (25, 10) yield the same maximum profit. If the coefficient is
increased to a value greater than 120, (15, 20) is no longer part of the optimal solution.
As long as the coefficient on x in the objective function falls between 40 to 120, the
optimal solution includes  x, y   15, 20  . If the coefficient is equal to 40 or 120, any
ordered pair on a line connecting 15, 20  to one of the adjacent corner points is optimal.
However, if the coefficient on x falls outside of 40 to 120 the optimal solution moves to a
new corner point.
Recall that the coefficient on x in the objective function is the profit for each frame bag. If
the profit per frame bag were to increase by up to $50 (to $120) or decrease by up to
$30 (to $40), the optimal solution would not change. This solution, 15 frame bags and 20
panniers, is fairly insensitive to changes in the coefficient on x. Even though the location
of the optimal solution does not change over this range, the profit does not change.
Notice that we are only varying one coefficient at a time. Varying both coefficients
simultaneously is beyond the scope of this presentation.
Example 3
Find the Range of Values For Which a Corner Point
Remains Optimal
The coefficient on y in the objective function for the linear programming
problem
Maximize P  70 x  120 y
subject to
x  y  35
x  3 y  75
2 x  y  60
x  0, y  0
can be varied to find a range of values over which 15, 20  is the
optimal solution. Over what range of coefficient values will 15, 20 
remain optimal?
Solution Let’s examine the graphical solution for this linear
programming problem.
From the earlier discussion on varying the coefficient on x, we observed
that changing the coefficient causes the slope of the isoprofit to change.
If the slope of the isoprofit line increases so that it coincides with the
constraint x  y  35 , the optimal solution is all points on the line
connecting 15, 20  and  25,10  . If the slope of the isoprofit line
decreases so that it coincides with the constraint x  3 y  75 , the optimal
solution is all point on the line connecting 15, 20  and  0, 25  .
Figure 13 – In a, the coefficient on y in the objective function has been increased from 120 to 210. In b,
the coefficient on y has been decreased from 120 to 70.
As long as the coefficient on y falls in the range 70 to 210, the solution
15, 20 
is optimal. Outside of this range, 15, 20  is no longer optimal. If
we can determine the coefficient values that cause the isoprofit line to
fall on the binding constraint borders, we know the range of values over
which the solution remains optimal.
In Figure 13a, the constraint border x  3 y  35 and isoprofit line
70 x  210 y  5250 coincide. Notice that on each line, the coefficient on y
is three times as large as the coefficient on x. Because of this, both
lines have the same slope. We can always use this information to find
the values of any missing constraints.
For instance, to find the coefficient on y in the objective function that
makes its slope the same as the slope on x  y  35 , we can observe
that the coefficients must be the same. Therefore, the coefficients in the
objective function must be the same. Since we have fixed the coefficient
on x and are only varying the coefficient on y, we conclude that isoprofit
lines for P  70 x  70 y are parallel to the constraint border x  y  35 . In
particular, the isoprofit line 70 x  70 y  2450 is exactly the same line.
Putting this together, we observe that the coefficient on y can range
from 70 to 210 and the corner point at 15, 20  is optimal.