Note that the bounds given for the classification are not exact unbreakable bounds. If the system finds better solutions than restricted by the given bounds, those solutions are given to the user. WWW-NIMBUS system assumes that the decision maker always prefers better solutions.

The classification symbols have different meaning depending on the type of the objective function. If the objective function is to be minimized they are:

Symbol | Description |
---|---|

< | Value of the function should be decreased. |

<= | Value of the function should be decreased till an aspiration level (to be specified later). |

== | Value of the function is currently satisfactory. |

>= | Value of the function is allowed to increase till an upper bound (to be specified later). |

> | Value of the function is allowed to change freely. |

For maximized functions, the symbols have reversed interpretation, because the "better" solution lies now to the opposite direction:

Symbol | Description |
---|---|

> | Value of the function should be increased. |

>= | Value of the function should be increased till an aspiration level (to be specified later). |

== | Value of the function is currently satisfactory. |

<= | Value of the function is allowed to decreased till an lower bound (to be specified later). |

< | Value of the function is allowed to change freely. |

**NOTE:**

- For minimized function, less (decreased) is better. For maximized function (increased), more is better.
- There must be at least one objective function in the classes that can give "better" solutions. ("<" or "<=" for minimized functions and ">" or ">=" for maximized).
- There must be at least one objective function in the classes that can give "worse" solutions. (">" or ">=" for minimized functions and "<" or "<=" for maximized).
- The default class for the minimized functions is ">" and for the maximized functions "<".

**Next optimization**-
Choose the global or local optimizer to continue with.
If you do not know when to use a local or a global optimizer,
select the global optimizer every now and then (not at every
iteration.) It requires more computing time but assures
more reliable results.
More information about the optimizers can be found from page NIMBUS Optimizers.

**Maximum number of new solutions to be generated**-
Based on the classification information specified, the system
can form up to four different subproblems. Even though they
use the same information, the results may be different. (The
system shows only results that differ from each other.) If you
do not wish to see different results, set the number of
subproblems used to be equal to one.
The different subproblems are the ones used in the original NIMBUS method (version 2), GUESS method, STOM method and the achievement scalarizing method. For further information, see

**Miettinen, K., Mäkelä, M.M.**,*Synchronous Scalarizing Functions within the Interactive NIMBUS Method for Multiobjective Optimization*, Reports of the Department of Mathematical Information Technology, Series B, Scientific Computing, No. B 9/2002, University of Jyväskylä, Jyväskylä, 2002.**NOTE:**The subproblems based on the GUESS, STOM and achievement scalarizing method can give solutions that break the boundary values. These subproblems are used when generating more than one new solution.

*Pareto optimality:*A criterion vector z* (belonging to the criterion space) is*Pareto optimal*if none of the components of z* can be decreased without increasing at least one of the other components.*Weak Pareto optimality:*A criterion vector z* (belonging to the criterion space) is*weakly Pareto optimal*if there does not exist any other vector for which all the components are better.*ICV=Ideal criterion vector:*The criterion vector is obtained by optimizing each of the objective functions individually. ICV forms the lower boundary of the Pareto optimal set. For maximized functions the ICV forms the upper boundary of the Pareto optimal set.*Current solution:*Current criterion values of the objective functions.

*Nadir point:*Estimated upper bounds of the Pareto optimal set. For maximized functions the estimated lower bounds of the Pareto optimal set.

If this projection fails, the reason can be one of the following: The feasible region is empty, there is something wrong with the problem input or the possible equality constraints have been specified with too small tolerances.

**Another problem**- Define a new problem. In this case, there is no need to
classify the functions.
**Save the current problem**- Save the current problem and the obtained result and
return back to this page.
In this case, there is no need to classify the functions.
**Remove a saved problem**- Remove some of the saved problems.
In this case, there is no need to classify the functions.
**Specify classification parameters if necessary (continue)**- Classify the objective functions and continue by specifying
classification parameters, if they are required.
**Use graphical classification instead**- Classify objective functions by using the graphical
classification form. In this case, there is no need to classify the functions
on this page.
**Go to solution database**- New solutions can be added, saved solutions removed, selected
solutions visualized or new alternatives generated.
**Correct Highest or Lowest values**- Change the values of some components of the ideal criterion
vector or the nadir point. If better estimations are known for
some component, they can be input to the system.
**Show the whole problem**- Description of the whole problem definition. In this case, there
is no need to classify the functions.
**Modify this problem**- Change the problem dimensions. After changing the
dimensions, you must provide for the new function definitions
and constraints.
**Stop**- Stop the solution process after obtaining a satisfying
solution.