Chapter 3
Introduction to Optimization Modeling
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Introduction to Optimization

Common elements of all optimization problems
– Decision Variables - the variables whose values the
decision maker is allowed to choose. (Changing cells in
Excel)
– Objective Function - value that is to be optimized –
maximized or minimized (Target cell in Excel)
– Constraints that must be satisfied
– Constraints impose restrictions on the values in the
changing cells. (Subject to constraints in Excel)
– Non-negativity – Decision variables cannot have negative
values (Options, check non-negative in Excel)
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Solutions

A feasible solution is any set of values of the
decision variables that satisfies all of the constraints.

The set of all feasible solutions is called the feasible
region.

An infeasible solution is a solution where at least
one constraint is not satisfied.

The optimal solution is the feasible solution that
optimizes the objective.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Steps
– Model development – decide what the decision variables
are, what the objective is, which constraints are required and
how everything fits together
– Optimize – systematically choose the values of the decision
variables that make the objective as large or small as
possible and cause all of the constraints to be satisfied.
– Sensitivity analysis – How changes to parameters of the
problem will affect the optimal solution
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Example 3.1 – Two Variable Model

Maggie will allow herself no more then 450 calories
and 25 grams of fat in her daily desserts. She
requires at least 120 grams of desserts a day. Each
dessert also has a “taste index”.

What should her daily dessert plan be to stay within
her constraints and maximizes the total taste index of
her dessert?
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Summary of given data
Serving of
snack bar
Serving of
ice cream
Limits
Calories
120
160
450 per day
Fat
5 grams
10 grams
25 grams per
day
Weight
37 grams
65 grams
At least 120
grams/day
Taste per gram
85
95
Taste per serving
37 x 85 =
3145
65 x 95 =
6175
Maximize
total taste
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Formulation

Decision variables
Daily servings of snack bar = Snack Bars
Daily servings of ice cream = Ice Cream

Objective function
– Maximize total taste index
Max Z = 3145 X Snack Bars + 6175 X Ice Cream

Constraints
– Calories consumed per day ≤ 450
– Fat consumed per day ≤ 25 grams
– Total dessert consumed ≥ 120 grams
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Constraints

Calories consumed per day ≤ 450
120 X Snack Bars + 160 X Ice Cream ≤ 450

Fat consumed per day ≤ 25 grams
5 X Snack Bars + 10 X Ice Cream ≤ 25

Total dessert consumed ≥ 120 grams
37 X Snack Bars + 65 X Ice Cream ≥ 120

Non-negativity constraints
X1, X2 ≥ 0
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Complete model
Maximize Z = 3145 X Snack Bars + 6175 X Ice Cream
Subject to:
120 X Snack Bars + 160 X Ice Cream ≤ 450 Calories
5 X Snack Bars + 10 X Ice Cream ≤ 25
Fat
37 X Snack Bars + 65 X Ice Cream ≥ 120 Total wt.
x1,x2 ≥ 0
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Ex. 3.1(cont’d) - Spreadsheet Model

Common elements in all LP spreadsheet models are:
– Inputs – all numeric data given in the statement of the
problem (Blue border)
– Changing cells – the values in these cells can be changed
to optimize the objective (Red border)
– Target(objective) cell – contains the value of the objective
(Double line black boarder)
– Constraints – specified in the Solver dialog box
– Nonnegativity – check an option in a Solver dialog box to
indicate nonnegative changing cells
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Ex. 3.1(cont’d) - Spreadsheet Model

Three stages of the complete solution:
– Model development stage – enter all inputs, trial values for
the changing cells, and formulas relating these in
spreadsheet
– Invoke Solver – designate the objective cell, changing cells,
the constraints and selected options, and tell Solver to find
the optimal solution.
– Sensitivity analysis – see how the optimal solution changes
as the selected inputs vary
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Ex. 3.1(cont’d) - Spreadsheet Model

Solver dialog box for this model.
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Ex. 3.1(cont’d) - Spreadsheet Model

Optimal Solution for the Dessert Model
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Ex. 3.1(cont’d) - Spreadsheet Model

In this solution the calorie and fat constraints have
been met exactly, thus they are binding. The
constraint on grams in nonbinding, the positive
difference in grams is called slack.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc

The SolverTable Add-in allows us to ask sensitivity
questions about any of the input variables.

SolverTable’s can be used in two ways:
– One-way table – single input cell and any number of output
cells
– Two-way table – two input cells and one or more outputs

The results are easily interpreted.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc

For the dessert model, check how sensitive the
optimal dessert plan and total taste index are to
(1) changes in the number of calories
(2) the number of daily dessert calories allowed.

The solution to question (1) can be solved by
selecting the Data/SolverTable menu item and select
a one-way table in the first dialog box. The second
dialog box should be completed as shown on the
next slide.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc

The second question asks us to vary two inputs
simultaneously. This requires a two-way SolverTable.
Select the two-way option in the first SolverTable
dialog box to get the two-way table dialog box.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
3.5 Properties of Linear Models

Linear programming is an important subset of a
larger class of models called mathematical
programming models.

Three important properties that LP models possess
– Proportionality
• If a level of any activity is multiplied by a constant factor, 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.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
– Additivity
• This property implies that the sum of the contributions from the
various activities to a particular constraint equals the total
contribution to that constraint.
– Divisibility
• This property means that both integer and noninteger levels of
the activities are allowed.

How can you recognize whether a model satisfies
proportionality and additivity?
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc

Not easy to recognize in a spreadsheet model
because the logic of the model can be embedded in
a series of cell formulas.

Often it is easier to recognize when a model is not
linear. Two 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.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc

Real-life problems are almost never exactly linear.
However, a linear approximation often yields very
useful results.

In terms of Solver, if the model is linear the Assume
Linear Model box must be checked in the Solver
Options dialog box.

Check the Assume Linear Model box even if the
divisibility property is violated.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc

If the Solver returns a message that “the condition for
Assume Linear Model are not satisfied” it
– can indicate a logical error in your formulation.
– can also indicate that Solver erroneously thinks the linearity
conditions are not satisfied.

Try not checking the Assume Linear model box and
see if that works. In any case it always helps to have
a well-scaled model.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
3.6 Infeasibility and Unboundedness

It is possible that there are no feasible solutions to a
model. There are generally two possible reasons for
this:
1. There is a mistake in the model (an input entered
incorrectly) or
2. the problem has been so constrained that there are no
solutions left.

In general, there is no foolproof way to find the
problem when a “no feasible solution” message
appears.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc

A second type of problem is unboundedness.

Unboundedness is that the model can be made as
large as possible. If this occurs it is likely that a wrong
input has been entered or forgotten some constraints.

Infeasibility and unboundedness are quite different. It
is possible for a model to have no feasible solution
but no realistic model can have an unbounded
solution.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Example 3.2 – Product Mix Model

The product mix problem is basically to select the
optimal mix of products to produce to maximize profit.

The Monet company produces four types of picture
frames. The four types differ with respect to size,
shape and materials used.

Each frame requires a certain amount of skilled labor,
metal and glass. They also all have different selling
prices.

Monet can produce in the coming week but they do
not want any inventory at the end of the week.

What should the company do to maximize its profit
for this week?
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Ex. 3.2(cont’d) - Algebraic Model
Maximize
Subject to
6x1 + 2x2 + 4x3 + 3x4 (profit objective)
2x1 + x2 + 3x3 + 2x4  4000 (labor constraint)
4x1 + 2x2 + x3 + 2x4  10,000 (glass constraint)
x1  1000 (frame 1 sales constraints)
x2  2000 (frame 2 sales constraints)
x3  500 (frame 3 sales constraints)
x4  1000 (frame 4 sales constraints)
x1, x2, x3, x4  0 (nonnegativity constraint)
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Ex. 3.2(cont’d) - Spreadsheet Model

To develop the spreadsheet model follow these
steps:
– Inputs - Enter the various inputs in the shaded ranges. Enter
only numbers, not formulas in the input cells.
– Range names – Name the ranges as indicated.
– Changing cells - Enter any four values in the range named
Produced.
– Resources used - Enter the formula
=SUMPRODUCT (B9:E9,Produced) in cell B21
and copy it to the rest of the Used range.
– Revenues, costs, and profits – Enter the formulas to
calculate these values.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc

The optimal solution for the product mix model is
shown on the next slide.

The sensitivity analysis allows us to experiment with
different inputs to this problem. Simply change the
inputs and then rerun Solver.

Use SolverTable to perform a more systematic
sensitivity analysis on one or more input variables.

Additional insight can be gained from Solver’s
sensitivity report.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Example 3.3 – Another Product Mix
Model

Pigskin company must decide how many footballs to
produce each month. It has decided to us a 6-month
planning horizon.

Pigskin wants to determine the production schedule
that minimizes the total production and holding costs.

By modeling this type of problem, one needs to be
very specific about the timing events.

By modifying the timing assumptions in this type of
model, one can get alternative – and equally realistic
– models with very different solutions.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Ex. 3.3(cont’d) - Spreadsheet Model

The difference between this model from the product
mix model is that some of the constraints are built
into the spreadsheet itself by means of the formulas.

The only changing cells are production quantities.

To develop the spreadsheet model:
– Inputs - Enter the inputs in the shaded ranges.
– Name ranges – Name ranges indicated.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Ex. 3.3(cont’d) - Spreadsheet Model
– Production quantities - Enter any values in the range
Produced as the production quantities. As always, you can
enter values that you believe are good, maybe even optimal.
– On-hand inventory - Enter the formula =B4 + B12 in cell
B16. This calculates the first month on-hand inventory after
production. Then enter the “typical” formula =B20 + C12 for
on-hand inventory after production in month 2 in cell C16
and copy it across row 16.
– Ending inventories - Enter the formula =B16 – B18 for
ending inventory in cell B20 and copy it across row 20.
– Production and holding costs - Enter the formula calculate
the monthly holding costs. Finally, calculate the cost totals in
column H by summing with the SUM function.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc
Ex. 3.3(cont’d) - Spreadsheet Model

The optimal solution from Solver.
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Ex. 3.3(cont’d) - Spreadsheet Model

SolverTable can be used to perform a number of
interesting sensitivity analyses.

In multiperiod models, the company has to make
forecasts about the future, such as the level of
demand. The length of the planning horizon is usually
the length of time for which the company can make
reasonably accurate forecasts.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc