Types of IP Models
All-integer linear programs
 Mixed integer linear programs (MILP)
 Binary integer linear programs, mixed or
all integer: some or all of the decision
variables can only assume 0 or 1 values
 A LP relaxation of an IP model is the
solution that occurs when the integer
restrictions are relaxed or ignored.

Properties of IP Solutions
The value of the optimal solution to an IP
model yields a value that is the same or
less desirable than the value of the
optimal solution to its LP relaxation.
 In maximization problems, the IP optimal
solution will be lower; in minimization
cases, the IP solution will be higher.

Integer Programming & Solver
To perform Integer Programming, add
constraints in Solver that require specific
decision variables (changing cells) to be
Integer (int) or Binary (bin)
 Solver will perform the Branch and
Bound technique to identify the optimal
solution within a specified tolerance.

Capital Budgeting Problems

A decision maker has several potential
projects or investment alternatives

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Projects or investments require different amounts of
resources (which are limited) and generate different
cash flows for the company
Cash flows are often converted into a NPV as input
into the model
There may be dependency relationships between
some of the projects or investment alternatives
The problem is to identify which set of projects
or investments to select in order to achieve the
maximum possible NPV
Branch and Bound Technique

Solve a LP relaxation of the IP model. For
max problems, the solution is an Upper Bound
for the IP solution. For min problems, the
solution is a Lower Bound for the IP solution.
 If solution is all integer, done. Otherwise, set
the other bound equal to the objective value
that would occur if only the integer variables
identified in the solution were used.
Branch and Bound Technique
(Continued)

Create two different subproblems by adding a
constraint to the problem, forcing a non-integer
value to be a given integer. Solve these
problems one at a time. If a solution is all
integer or if it is outside the bounds of the first
problem, you are done creating more
problems in this direction. If a solution is within
the bounds and still not all integer, create two
more problems by adding another constraint
forcing another variable to be an integer.
Branch and Bound Technique
(Continued II)

When all the necessary sub-problems have
been created and solved, identify the solution
which achieves the objective from all the subproblems which resulted in all integer
solutions.
Solver’s Tolerance Option


Solver allows you to set a tolerance % option to
determine how close the identified solution must be to
the true optimal solution before Solver can stop
searching for a better solution.
By default, the suboptimality tolerance factor is 5%.
This means that the solution found by Solver is within
5% of the optimal solution. It may actually be the
optimal solution but there is no guarantee to that! To
guarantee the optimal solution, you must set the
tolerance % to 0. This will result in much longer
solution times however when you do this.
Example of IP Applications
Capital Budgeting Models
 Municipal Bond Underwriting
 Purchase Ordering Models:

Fixed-Charge or Fixed Cost problems
 Fixed Order Quantities & Quantity
Discounts
 Minimum Order Size Constraints

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Workforce Scheduling Problems
Mutually Exclusive Events

When two alternatives (A and B) cannot
both be picked, force Solver to choose at
most one by including the constraint:
XA + XB <= 1

Similiarly choosing no more than k out of
n events requires adding the constraint:
XA + XB + ……+ Xn <= k
Dependent Decisions

When one alternative (B) cannot be used
unless another alternative is also picked
(A), add the constraint:
XB <= XA
Minimum Purchase Size Constraints

Assume that a particular investment
opportunity that you are interested in
requires a minimum investment of
$25,000. You do not wish to invest more
than $200,000 in any particular
investment. How would you model this
as a constraint for Solver?
Let Yi=0 if you choose not to invest in option i,
1 if you choose to invest in option i.
Let Xi=amount of money you choose to invest in i.
There are two constraints required:
Xi<=200000 Yi (this forces Yi to be 1 if you put
money into this option)
Xi>=25000 Yi (this forces Xi to meet minimum
amount if you put money into this option,
because Yi will be 1)
Investment Problem
You have currently identified 20 stocks as
possible investments. You have
$100,000 available. Their potential
returns have been estimated and you
would like to invest your money so as to
maximize the expected return on your
new portfolio.
Each stock requires a minimum purchase
of $5,000 and you do not want to invest
more than $20,000 in any one particular
stock. You would like to invest in at least
8 different stocks. In order to control your
risk, you do not want to put more than
$60,000 in the the first 10 stocks listed.
To invest in stock 5 you must also invest
some money in stock 16.