Graphical Solutions

Plot all constraints including nonnegativity
ones

Determine the feasible region. (The feasible
region is the set of feasible solution points)

Identify the optimal solution in the feasible
region using either
– the level curve method
– the extreme point method
Feasible Regions





The feasible region for a two-variable linear programming
problem can be nonexistent, a single point, a line, a polygon, or
an unbounded area.
If a constraint can be removed without affecting the shape of the
feasible region, the constraint is said to be redundant.
An overconstrained LP that does not have one point satisfying
all the constraints simultaneously is said to be infeasible and a
feasible region does not exist.
The feasible region for the KPiller problem is a polygon,
illustrating that an infinite number of feasible solutions exist for
this problem. (See class handout).
Assuming that a feasible region exists, an optimal solution will
always occur at one or more of the extreme points, where an
extreme point is a corner point of the feasible region.
Level Curves

Plot a level curve for a targeted objective function
value
 Level curves are parallel lines that shift away from
the origin as the targeted objective function value
increases
 Slide the level curve in the direction of improvement
(away from the origin if maximizing, toward the origin
if minimizing) until it reaches the last point where it
still touches at least one point in the feasible region
 Solve the two equations which specify the identified
feasible point to determine the optimal solution
Extreme Point Method

Determine all of the
extreme points of
the feasible region:
Extreme Point
–
–
–
–
(3.5, 3)
(6.5, 3)
(4.5, 7)
(1.5, 9)

Calculate the resulting
objective function value for
each point:
Z = 5000*E9 + 4000*F9
–
–
–
–
$29,500
$44,500
$50,500
$43,500
Types of LP Solutions

Any linear program falls in one of three categories:
– is infeasible
– has a unique optimal solution or alternate optimal
solutions
– has an objective function that can be increased
without bound

In the graphical method, if the objective function line
is parallel to a boundary constraint in the direction of
optimization, there are alternate optimal solutions,
with all points on this line segment being optimal.
Solver Result Messages


Solver found a solution. All constraints and optimality
conditions are satisfied: Solver has correctly
identified an optimal solution for the problem you
have formulated. Note that there may be alternative
optimal solutions possible however.
Solver has converged to the current solution. All
constraints are satisfied: You have not selected the
linear programming option in the Solver options.
Thus nonlinear programming is being performed and
this is the best solution Solver has found so far. It is
not guaranteed to be the optimal one however.
Example: Infeasible Problem

Solve graphically for the optimal
solution:
Max
s.t.
z = 2x1 + 6x2
4x1 + 3x2 < 12
2x1 + x2 > 8
x1, x2 > 0
Example: Infeasible Problem

There are no points that satisfy both constraints,
x2
hence this
problem has no feasible region, and no
optimal solution.
2x1 + x2 > 8
8
4x1 + 3x2 < 12
4
3
4
x1
Solver Result Message

Solver could not find a feasible solution
Possible problems if you know the solution
should be feasible are:
– too many constraints in Solver
– one of the constraints may be entered wrong (e.g.
the inequality sign might be going the wrong way)
– you may not have the correct changing cells
(decision variables) specified in Solver
– one or more formulas in the spreadsheet may
have been erased.
Example: Unbounded Problem

Solve graphically for the optimal
solution:
Max
s.t.
z = 3x1 + 4x2
x1 + x2 > 5
3x1 + x2 > 8
x 1, x 2 > 0
Example: Unbounded Problem

The feasible region is unbounded and the objective
function line can be moved parallel to itself without
bound so xthat
z can be increased infinitely.
2
3x1 + x2 > 8
8
5
x1 + x2 > 5
Max 3x1 + 4x2
2.67
5
x1
Solver Result Messages

Set Cell values do not converge:
Your model as formulated is unbounded.
– One or more constraint is missing from the
problem or entered wrong (e.g. an inequality sign
is in the wrong direction).
– Often times the modeler has forgotten to check
the Assume Nonnegativity option in Solver.
Note: A feasible region may be unbounded and yet there may be
optimal solutions. In the case where the objective does not
move in the direction of the unboundedness, you will see the
optimal Solver solution message instead of this message.
Solver Result Messages

The linearity conditions required by this LP
Solver are not satisfied:
Solver’s preliminary tests indicate that your model is
not linear.
– Check for use of functions such as IF, MAX, PV, VLOOKUP
which create linearity problems. Write the algebraic logic
programmed in the objective and constraint cells on paper to
ensure linear relationships.
– Sometimes this message occurs due to poor scaling (see
text section 3.11). If you think your model is linear, try
resolving the model again. Often times Solver can find the
solution the second time. If not, use the Use Automatic
Scaling option in Solver. Solver will attempt to rescale your
data. If that doesn’t work, rescale the data yourself.