Unit One:
Standard and Integer Linear Programming
Many real-world situations involve determining levels of activity that will maximize or
minimize some quantity (e.g., maximize profit, minimize cost) while operating under constraints
(e.g., limited resources, contractual promises, quality requirements) that impose restrictions on the
levels of activity.
For example, a company that manufactures several types of ceiling fans may
want to determine a production schedule (i.e., decide how many of each type of fan to produce) for
the next week that will maximize profit without using more labor, machine time, and raw materials
than are available.
As another example, the manager of a nursing home cafeteria may want to
design a meal (i.e., decide on what quantities of various foods to use) of minimum cost while
meeting certain nutritional requirements.
As a third example, a county commission may want to
decide which of several recreational facilities to construct so as to maximize anticipated total daily
usage without exceeding available funds and land acreage.
Many such real-world situations can be
expressed as a standard or integer linear programming model which can then be mathematically
solved to determine the optimal levels of activity given the posed objective and constraints.
Standard and integer linear programming models belong to a general category of constrained
optimization models.
In this handout, LP is an abbreviation for linear programming.
Model Description
Features common to Standard LP and Integer LP models:
•
Have a set of decision variables.
(The decision variables are often designated x
1
, x
2
, x
3
, etc.)
•
The objective is either to maximize or to minimize some quantity Q.
•
Q can be expressed as a linear function of the decision variables, which is called the objective
function.
•
Have a set of constraints.
•
Each constraint can be expressed in one of the following ways:
linear function of the decision variables ≤
a constant
linear function of the decision variables ≥
a constant
linear function of the decision variables =
a constant
•
The decision variables can
not
take on negative values; this is termed the nonnegativity
condition.
note
: a linear function of the decision variables x
1
, x
2
, x
3
, …, x
k
is of the form
c
1
x
1
+ c
2
x
2
+ c
3
x
3
+ … + c
k
x
k
+ c
0
where each c
i
is a constant (real number).
Distinction between Standard LP and Integer LP:
Standard LP:
All the decision variables are continuous and thus can take on fractional values.
Integer LP:
At least one decision variable is restricted to be an integer or is a binary variable (for
which the allowable values are 1, which stands for yes to some activity/condition, and 0, which
stands for no to that activity/condition); the remaining variables (if any) are continuous.
Across all Standard and Integer LP models, then, each decision variable will be one of the following