## Thomas W. Knowles and Associates## OPTIMIZATION SPECIALISTS |

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An
Optimization model chooses the value of decision variables that give
the optimal solution. Optimization models have three parts.
The is a function of the decision variables, called the
objective function, that is to be maximized or minimized. Often,
the objective is to either maximize profits or minimize costs, but many
other objectives are possible depending on the decision environment.first partThe is a set of constraints that limit the combinations of
decision variables that can be considered. A constraint function
consists of three parts: a constraint function (dependent upon
the decision variables), a relationship, and a right-hand-side
(RHS). The relationship can be one of <=, >=, and =.
An example would be that the number of people working during a time
period must be >= some specified value. Or the number of hours
used on a machine must be <= the number of hours available.second partDecision variable limitations are the of an optimization
model. These limitations are different from
constraints. The most common of these limitations include: (1)
the values must not be negative, (2) the values are restricted to
integers (General Integer variables), and (3) the values are restricted
to 0 or 1 (Binary Integer variables). An example of the first
type of limitation would be that the number of pounds of a raw material
used couldn’t be negative. An example of the second type of
limitation is that the number of people hired must be an integer.
The third type of limitation is often used to choose between two
discrete alternatives, such as 1 if a distribution center is built and
0 if it is not.third partThe mathematical form of the model affects what software (and associated algorithm) can be used. If all of the functions---objective and constraints---are linear and the variables are continuous, then it is called a linear programming model. If all of the functions are linear but at least one of the decision variables is either a General Integer or a Binary Integer, the model is called an integer linear programming model. If any of the functions is not linear and all the variables are continuous, then the model is called a nonlinear programming model. An integer nonlinear programming model contains at least one nonlinear function and at least one variable with integer restrictions. A crucial issue is whether any of the functions---objective or constraint--- is not linear. A nonlinear function can result in the software achieving a “local” optimal solution that is not a “global” optimal solution. The maximum on the right is a local maximum, i.e., it is larger than the values close to it. However, the maximum on the left is also a local maximum, but it is also the global maximum, i.e., the maximum over all decision variable values. Optimization software for nonlinear programming models finds a local optimal solution. Unfortunately, it cannot guarantee that the local optimal solution it finds is also the global optimal solution. A new global solver option from Lindo Systems, Inc guarantees an optimal solution. However, this extra cost option is frequently not necessary because of the experience we have in linearizing functions that are intrinsically nonlinear. |

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