Help: Classify Functions

Divide the objective functions of the problem at the current solution (in the criterion space) into up to five different classes by using the radio buttons. The classification indicates desired change from the current solution. Each class is described by a symbol and the descriptions of the symbols and the different classes are the following.

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 "<".

Selection:

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.

Selection:

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.

Some terminology

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.

Starting point

The decision vector specified by the user is used as a starting point in the calculation. If this point is not feasible subject to the linear and nonlinear constraint functions, it is projected into the feasible region.

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.

Lowest Value (estim.) and Highest Value (estim.)

The estimated ranges of the objective functions in the Pareto optimal set are provided to support the user in the classification. For minimized functions, ICV (ideal criterion vector) represents the lower bound, and Nadir the upper bound. For maximized functions, ICV represents the upper bound and Nadir the lower bound. The user can compare the components of the current solutions to the ranges. It must be kept in mind that the ranges are only estimations. The maximized functions are marked with blue colour.

If the components of the Lowest Value and Highest Value are equal

If the user knows better estimations to the Lowest Value or the Highest Value, the values can be changed by selecting the appropriate operation below. The correctness of the degree of nonconvexity-values affects the estimations of the Lowest and the Highest Values significantly. These degree values can be modified by changing optimization parameters. To change these optimization parameters, first save the problem and then load it again with the option of modifying it. In this way, the Input Problem-page is achieved. The Change optimization parameters-option is then available.

Available operations:

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.

To return to the previous action, use the 'back to previous page' function of the browser.

nimbus@mit.jyu.fi