Chapter 3
Introduction to
Optimization Modeling
CBTL212
Managerial Problem Solving
3.1 INTRODUCTION
• Spreadsheet optimization is one of the most
powerful and flexible methods of quantitative
analysis.
• The specific type of optimization discussed
here is linear programming (LP).
3.2 INTRODUCTION TO
OPTIMIZATION
• All optimization problems:
1. Decision variables (Changing Cells)
– the variables whose values the decision maker
can choose.
– directly or indirectly, the values of these variables
determine such outputs as total cost, revenue,
and profit
– variables an organization must know to function
properly
3.2 INTRODUCTION TO
OPTIMIZATION cont’
2. Objective function (Target Cells)
– value that is to be optimized—maximized or
minimized
3. Constraints
– must be satisfied
– usually physical, logical, or economic restrictions
that depend on the nature of the problem
3.2 INTRODUCTION TO
OPTIMIZATION cont’
Excel refers to the:
• decision variables as the changing cells
– cells must contain numbers that are allowed to
change freely;
– they are not allowed to contain formulas
• objective as the target cell
– There can be only one target cell—which could
contain profit, total cost, total distance traveled
– must be related through formulas to the changing
cells
3.2 INTRODUCTION TO
OPTIMIZATION cont’
3.2 INTRODUCTION TO
OPTIMIZATION cont’
• Constraints can come in a variety of forms.
• One very common form is nonnegativity.
• Nonnegativity constraints state that changing
cells must have nonnegative (zero or positive)
values.
• Nonnegativity constraints are usually included
for physical reasons.
• For example, it is impossible to produce a
negative number of automobiles.
3.2 INTRODUCTION TO
OPTIMIZATION cont’
• Two steps are involved in solving an
optimization problem.
1. Model Development
– what are the variables, objective and constraints,
and how everything fits together?
– Spread sheet model - relate all variables with
appropriate cell formulas
– model contains formulas for relating the
changing cells to the target cell and that it
contains formulas for operationalizing the
constraints
3.2 INTRODUCTION TO
OPTIMIZATION cont’
2. Optimize
– systematically choose the values of the decision
variables that make the objective as large (for
maximization) or small (for minimization) as
possible and satisfy all the constraints
3.2 INTRODUCTION TO
OPTIMIZATION cont’
• LINEAR PROGRAMMING INTRO NOTES
• WE HAVE SOLVER in Excel!!!!!!!!!!!!!!!
3.2 INTRODUCTION TO
OPTIMIZATION cont’
• All we need to do is develop the model and
then tell Solver the target cell, the changing
cells, the constraints, and the type of model
(linear, integer, or nonlinear).
• Solver then goes to work, finding the best
feasible solution with the most suitable
algorithm.
3.2 INTRODUCTION TO
OPTIMIZATION cont’
• Sensitivity analysis.
– what-if questions
– What if the unit costs increased by 5%?
– What if forecasted demands were 10% lower?
– What if resource availabilities could be increased
by 20%?
– What effects would such changes have on the
optimal solution?
3.5 PROPERTIES OF LINEAR MODELS
• LP models possess three important properties
that distinguish them from general
mathematical programming models:
– proportionality,
– additivity, and
– divisibility.
3.5 PROPERTIES OF LINEAR MODELS
cont’
Proportionality
– if the level of any activity is multiplied by a
constant factor, then the contribution of this
activity to the objective, or to any of the
constraints in which the activity is involved, is
multiplied by the same factor. Or
– A change in a variable results in a proportionate
change in that variable’s contribution to the value
of the function
3.5 PROPERTIES OF LINEAR MODELS
cont’
Additivity
– the sum of the contributions from the various
activities to a particular constraint equals the total
contribution to that constraint
– Implies to the objective, i.e., the value of the
objective is the sum of the contributions from the
various activities or
– The function value is the sum of the contributions
of each term
3.5 PROPERTIES OF LINEAR MODELS
cont’
Divisibility
• both integer and noninteger levels of the
activities are allowed or
• The decision variable can be divided into noninterger values, taking on fractional values.
Interger programming techniques can be used
if the divisibility assumption does not hold
3.5 PROPERTIES OF LINEAR MODELS
cont’
If we want the levels of some activities to be integer
values, there are two possible approaches:
– (1) We can solve the LP model without integer
constraints, and if the solution turns out to have
noninteger values, we can attempt to round them to
integer values; or
– (2) we can explicitly constrain certain changing cells to
contain integer values. The latter approach, however,
takes us into the realm of integer programming, which
is discussed in Chapter 6.
3.5 PROPERTIES OF LINEAR MODELS
cont’
Discussion of the linear properties
• when a model is not linear?
• Two particular situations that lead to
nonlinear models are when :
– (1) there are products or quotients of expressions
involving changing cells and
– (2) there are nonlinear functions, such as squares,
square roots, or logarithms, of changing cells.
3.5 PROPERTIES OF LINEAR MODELS
cont’
• In Excel’s Solver, if the model is linear—in
particular if it satisfies the proportionality and
additivity properties—then you should check
the Assume Linear Model box in the Solver
Options dialog box.
3.6 INFEASIBILITY AND
UNBOUNDEDNESS
• Two things can go wrong when you invoke
Solver.
• Both of these can indicate that there is a
mistake in the model
3.6 INFEASIBILITY AND
UNBOUNDEDNESS cont’
A. Infeasibility
• it’s possible that there are no feasible
solutions to the model.
• There are generally two reasons for this:
1. There is a mistake in the model (an input was
entered incorrectly, such as a >= instead of a
<= ), or
2. the problem has been so constrained that no
solutions are left
3.6 INFEASIBILITY AND
UNBOUNDEDNESS cont’
B. Unboundedness
• the model has been formulated in such a way
that the objective is unbounded—that is, it
can be increased (or decreased for
minimization problems) without bound.
• If this occurs, we have probably entered a
wrong input or forgotten some constraints