3 Components for a Spreadsheet Linear
Programming Problem



There is one cell which can be identified as the
Target or Set Cell, the single objective of the
problem that is to be maximized or minimized.
There is at least one cell that is limited to
assume values over specified ranges. These
cells are referred to as the Constraint Cells of
the problem.
There is at least one Changing or Variable Cell
(decision variable). The set of changing cells
must influence the target cell and all of the
constraint cells. The changing cells should not
have formulas entered in the cells.
3 Steps of Linear Programming
Model Formulation
• Spreadsheet Based
• Algebraic
Model Solution
• Graphical Analysis (for 2 decision variable
problems)
• Simplex Method
• Excel Solver
Sensitivity Analysis
Product Mix Algebraic Formulation
Maximize 5000 E9 + 4000 F9
Subject to
10 E9 + 15 F9 <= 150 (dept A hours)
20 E9 + 10 F9 <= 160 (dept B hours)
30 E9 + 10 F9 >= 135 (testing hours)
E9 >= 1
(current order)
F9 >= 3
(current order)
where E9,F9 represent the quantity to
produce next period of each model
Important LP Terminology

Linear functions are functions in which each variable
appears in a separate term raised to the first power
and multiplied by a constant, which may be 0.

A feasible solution is an assignment of values to the
decision variables that satisfies all the problem's
constraints.

The objective does not affect the feasibility of the
problem.

An optimal solution is a feasible solution that results
in the largest possible objective function value, z,
when maximizing or smallest z when minimizing
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 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.

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.

If a constraint can be removed without affecting the shape of the
feasible region, the constraint is said to be redundant.

A linear program which is overconstrained so that no point
satisfies all the constraints is said to be infeasible.
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

Extreme Point
–
–
–
–
(3.5, 3)
(6.5, 3)
(4.5, 7)
(1.5, 9)

¶=5000E9 + 4000F9
–
–
–
–
$29,500
$44,500
$50,500
$43,500
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:
– too many constraints
– 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
– one or more formulas in the spreadsheet may
have been erased.
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: 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. This is common in minimization problems and
is possible in maximization problems. In these cases, you will
see the optimal Solver solution message.
Solver Result Messages

The conditions for Assume Linear Model are
not satisfied: Solver’s preliminary tests
indicate that your model is not linear.

This may be the case. However sometimes this
message occurs due to poor scaling (e.g. some #s
are in % and others in millions). 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 solve the problem, you can try rescaling the
data yourself.
Solver Modeling Requirements


All components of the optimization problem must be
on the same worksheet. Solver’s settings are saved
with the sheet.
Solver’s constraint dialog box will not let you enter
formulas. All formulas and calculations must be done
on the worksheet. The constraint dialog box just
compares cells to determine feasibility.
Goals For Spreadsheet Design

Communication - A spreadsheet's primary business
purpose is that of communicating information to managers.

Reliability - The output a spreadsheet generates
should be correct and consistent.

Auditability - A manager should be able to retrace the
steps followed to generate the different outputs from the
model in order to understand the model and verify results.

Modifiability - A well-designed spreadsheet should be
easy to change or enhance in order to meet dynamic user
requirements.
Spreadsheet Design Guidelines

Organize the data, then build the model around the data.

Do not embed numeric constants in formulas!

Things which are logically related should be physically related.

Use formulas that can be copied.

Column/rows totals should be close to the columns/rows being
totaled.

The English-reading eye scans left to right, top to bottom.

Use color, shading, borders and protection to distinguish
changeable parameters from other model elements.

Use text boxes and cell notes to document various elements of
the model.
Network Of Routes
120
Amsterdam (500)
62
61
Antwerp (700)
130
41
Leipzig (400)
Nancy(900)
40
100
110
102.5
90
Liege (200)
122
Le Havre (800)
42
Tilburg (500)
The Transportation Tableau
To
From
Leipzig
Nancy
Liege
Tilburg
62
120
130
41
61
40
100
110
102.5
90
122
42
Amsterdam
Antwerp
Le Havre
Demand
400
900
200
500
Supply
500
700
800