3 Characteristics of an
Optimization Problem
General description
Decisions that must be
made; represented by
KPiller Illustration
How many of each product to
make next period (two
decision variables
decision variables)
Constraints or
Limited department hours;
restrictions that limit the orders that must be filled;
available decision
minimum hours for labor (5
alternatives
constraints)
A goal or objective that Maximize next period’s total
needs to be maximized profit z
or minimized
Linear Programming (LP)


A mathematical programming problem is
one that seeks to maximize or minimize an
objective function subject to constraints.
If both the objective function and the
constraints are linear, the problem is
referred to as a linear programming
problem.
KPiller Algebraic Formulation
Maximize 5000 X + 4000 Y (profit)
Subject to
10 X + 15 Y <= 150 (dept A hours)
20 X + 10 Y <= 160 (dept B hours)
30 X + 10 Y >= 135 (testing hours)
X >= 1
(current order)
Y >= 3
(current order)
where X, Y are decision variables that represent the production
quantity of E-Supremes and F-Supremes, respectively
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.
3 Steps of Linear Programming

Model Formulation



Model Solution




Spreadsheet Based
Algebraic
Graphical Analysis
Simplex Method (LINDO, CPLEX, etc.)
Excel Solver Add-in
Sensitivity Analysis
Goals For Spreadsheet Design

Communication -

Reliability -

Auditability -

Modifiability -
A spreadsheet's primary business
purpose is that of communicating information to managers.
The output a spreadsheet generates
should be correct and consistent.
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.
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: layout repetitive headings in
same order and in same row/column orientation. Apply
appropriate use of absolute and relative cell references.
Column/row 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.
Solver Modeling Requirements


All components of the optimization problem must be
programmed on the same worksheet. Solver’s
settings are saved with the worksheet.
Solver’s constraint dialog box will not let you enter
formulas. All formulas and calculations must be done
in the worksheet. The constraint dialog box just
compares cells in the current worksheet to determine
feasibility and optimality.
Hours
Required per
E-Supreme
Hours
Required per
F-Supreme
Department A
10
15
25
<= 150
hours
Department B
20
10
30
<= 160
hours
Finishing Dept
30
10
40
>=135
hours
E-Supreme
F-Supreme
Current Orders
1
3
Quantity Made
1
1
Sales Price/Unit
$
20,000
$
24,000
Variable Cost/Unit
$
15,000
$
20,000
Profit per Unit
$
5,000
$
4,000
Total Profit
$
5,000
$
4,000
$
9,000
GRAND TOTAL PROFIT:
Total
Hours
Used
Total Hour
Constraints
Programming the 3 Optimization
Components in Solver
General description
Decision variables
Solver Terminology
Changing Cells
Left Hand Sides (LHS) of Cell Reference in Constraint
Constraints
Dialog box
Right Hand Sides (RHS)
of Constraints
Constraint in Constraint
Objective function
Set Target Cell (click on Max
Dialog box
or Min button)
Types of LP Solutions

Every linear program falls in one of
three categories:

It is infeasible

It has a unique optimal solution or
alternate optimal solutions (different ways
of achieving the same maximum or
minimum objective value)

It has an objective function that can be
increased without bound
“Ideal” Solver Result Message

Solver found a solution. All constraints
and optimality conditions are satisfied
Solver has identified an optimal solution
for the problem you have formulated.
Note that there may be alternative
optimal solutions possible but Solver
will just show you one possible solution.
Examples of Linear
Programming Applications

Production Planning:





several products
multiperiod demand
limited period resources
want minimal production costs or maximum
profitability
Transportation/Distribution Problems:




different routes
limited supply at several sources
demand requirements at various locations
want minimal transportation costs
More Examples of LP
Applications

Investment Planning:




several investment alternatives
risk and capital restrictions
want maximum expected return
Labor Scheduling:




full-time and part-time workforce
multi-period staffing requirements
workforce staffing restrictions
want minimum total labor cost