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Tools for Multiobjective Optimizations

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Analyzing Multiobjective Optimizations Pareto Slice Table View Pareto Front Graphs Selected Solution Slice Exporting Multiobjective Solutions Weighted Objective Pareto Slice Multiobjective NBI Output Messages

Analyzing Multiobjective Optimizations

CAGE has additional tools for analyzing optimizations with more than one objective. These are described in the following sections:

• Pareto Slice Table View shows a table of all solutions at one run.

• Pareto Front Graphsshow the successful solutions for the selected run with the current selection highlighted in red. These graphs help you select best solutions for each run.

• Selected Solution Slice for collecting and exporting only the solutions you have decided are optimal at each run.

  You can use the Selected Solution slice, or alternatively you can choose to export all solutions to a data set. You can restrict export to acceptable solutions only. See Exporting Multiobjective Solutions.

• Weighted Objective Pareto Slice shows a weighted sum of the objective values over all runs for each solution.

Check the messages and exit flags for each solution, shown in the Optimization Results table and the Solution Information pane. Multiobjective solutions can have specific exit messages, see Multiobjective NBI Output Messages.

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Pareto Slice Table View

The Pareto Slice table view (click ) is for multiobjective optimization where there is more than one solution at each run. The Pareto Slice shows a table of all solutions at one run; you can scroll through the runs using the arrows or edit box at the top.

To collect best solutions across different runs, you need to use the Select Solution function in the toolbar.

To select a solution for each run:

1. Enable the Selected Solutions view. Select Solution > Selected Solution > Initialize.

   The Create Selected Solution dialog box appears.

   

   The default 1 initializes the first solution for each run as the selected solution. You can edit the solution number here if you want. For example, if you select 4, solution number 4 is initialized as the best solution for every run. When you click OK, the toolbar buttons for the Selected Solution Slice view and Select Solution are enabled.

2. Decide which solution you want to use for the currently selected run. Use these tools to help you:

   • Display the Pareto Front Graphs (click    in the toolbar) which show the available solutions with the current selection highlighted in red.

   • Use the pareto front graphs together with the Objective Slice Graphs to select the best solution for the run. If you have constraints you can also use the Constraint Slice Graphs and Constraint Summary Table to help you decide which solution to choose for each run.

3. When you have decided which solution you want to use for the currently selected run, you can select it as best by clicking Select Solution (   ) in the toolbar. You can also select best solutions in the Solution Slice view, see Solution Slice: Optimization Results Table .

4. Scroll through the runs and select a best solution for each. These selections are collected in the Selected Solutions Slice, where you can use them to fill tables or export to a data set. You can also import them to an optimization. See Selected Solution Slice.

Before you run an NBI optimization you can specify how many solutions you want the optimization to find, using the Set Up and Run Optimization toolbar button.

As in the other table views, you can use the Accept check boxes to choose a selection of rows within the table. In this table view, you can only use this to select solutions within a single run. Each different solution has a check box and colored icon for "Acceptable" status. You can override these selections using the check boxes if you want to choose solutions within a run, for use when exporting to a data set, importing to other optimizations, or for future reference. See Choosing Acceptable Solutions.

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Pareto Front Graphs

The Pareto Front Graphs (click  ) are for multiobjective optimization where there is more than one solution at each run. The Pareto Front graphs show the available solutions for the selected run with the current selection highlighted in red. Click in the tables or graphs to select solutions. The selected solution is displayed in all other graphs (objective and constraint).

Before you run an NBI optimization you can specify how many solutions you want the optimization to find, using the Set Up and Run Optimization toolbar button to access the Optimization Parameters dialog box.

You can use the Pareto Front graphs, shown in the preceding figure, in combination with the table views (Solution Slice and Pareto Slice) and the other plots in the graphs (Objective Slice and Constraint Slice graphs) to help you select best solutions for each run. You can collect these solutions together in the Selected Solution Slice.

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Selected Solution Slice

In a multiobjective optimization, there is more than one possible optimal solution at each run. You can use the Selected Solutions view to collect and export those solutions you have decided are optimal at each run.

Note   You can use the Selected Solution slice, or alternatively you can choose to export all solutions to a data set. See Exporting Multiobjective Solutions

After you enable the Selected Solution view, you can use the plots and table views to help you select best solutions for each run. These solutions are saved in the Selected Solutions view. You can then export your chosen optimization output for each point from the Selected Solutions view to a data set, or use your optimization output to fill tables or import to another optimization.

You cannot select best solutions until you have enabled the Selected Solutions view.

1. Select Solution > Selected Solution > Initialize.

   A dialog box called Create Selected Solution appears. The default 1 initializes the first solution for each run as the selected solution.

2. Edit the solution number in this dialog box if you want. For example if you select 4, solution number 4 is initialized as the best solution for every run. When you click OK, the toolbar buttons for the Selected Solutions view and Select Solution are enabled.

   

3. After you enable the Selected Solutions view, you can use the table views and the plots in the graphs (Objective Slice, Pareto Front, and Constraint Slice graphs) to help you select best solution for each run.

   1. Click in the Pareto Slice Table View (or Solution Slice) table to select a point to display in the graphs until you can decide which solution you want for a point.

   2. Click Select Solution (   ) in the toolbar to select the current solution as best.

   Repeat steps a and b until you have selected solutions for all points.

These solutions are saved in the Selected Solutions view. This view collects all your selected solutions together in one place. For example, you might want to select solution 7 for the first run, and solution 6 for the second, and so on. You can then use your chosen optimization output for each point to fill tables (see Filling Tables from Optimization Results), or choose the Export to Data Set toolbar and Solution menu option (see Exporting to a Data Set), or use these solutions as starting points in another optimization (see Import from Output).

An example of the Selected Solutions view is shown. It looks similar to the Solution Slice view, except the solution controls at the top are not enabled. You cannot change solution number in this view. The solution chosen as best (in the other views) for the currently selected run is displayed in the grayed-out edit box.

As in the other table views you can use the Accept check boxes to choose a selection of rows within the table. See Choosing Acceptable Solutions.

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Exporting Multiobjective Solutions

You can export multiobjective solutions to a data set. You can restrict export to acceptable solutions only (specified by the Accept check boxes). See Choosing Acceptable Solutions.

Alternatively you can use the Selected Solution slice to collect only the best solution for each run.

1. Select Solution > Export to Data Set or use the toolbar button. The Export to Data Set dialog box appears. For multiobjective optimizations this dialog provides an additional control called Solutions to Export.

2. Use the Solutions to Export drop-down list to select either:

   • All Solutions — this exports every solution from every run. You can choose whether to export acceptable solutions only with the check box Use acceptable solutions only.

   • Selected Solutions — this exports your collected solutions in the Solution Slice.

For more information on exporting optimization results, see Exporting to a Data Set.

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Weighted Objective Pareto Slice

The Weighted Objective Pareto Slice view (click   ) shows a weighted sum Pareto solution. This table shows a weighted sum of the objective values over all runs for each solution. For a single objective optimization there is a single cell, which is the sum of the objective across all runs.

In the following multiobjective example, the value in the Objective1 column in the first row shows the sum of the solution 1 values of the first objective across all runs. The second row shows the sum of solution 2 Objective1 values across all runs, and so on for all ten solutions in this case. This information can be useful, for example, for evaluating total emissions across a drive cycle. The default weights are unity (1) for each run.

You can change the weights; for example, if you need a weighted sum of emissions over a drive cycle, you might want to give a higher weight to the value at idle speed. You can alter weights by clicking Edit Pareto Weights (   ) in the toolbar. The Pareto Weights Editor appears.

In this dialog box, you can select objectives to sum, and select weights for any run by clicking and editing, as shown in the previous example. The same weights are applied to each solution to calculate the weighted sums. Click OK to apply new weights, and the weighted sums are recalculated.

You can also specify weights with a MATLAB vector or any column in the optimization output by selecting the other option buttons. If you select Output column you can also specify which solution; for example, you could choose to use the values of spark from solution 5 at each operating point as weights. Click Table Entry again, and you can then view and edit these new values.

Note   Weights applied in the Weighted Pareto View do not alter the results of your optimization as seen in other views. You can use the weighted sums to investigate your results only. You need to perform a sum optimization if you want to optimize using weighted operating points.

The Accept check box is disabled in this view. The exit flag is the minimum of all of the runs that are summed over, so the Accept status can only go green if all runs are green.

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Multiobjective NBI Output Messages

The NBI algorithm provides exit messages that can be seen in the Optimization output view, in the Solution Information pane, for the currently selected run. Check these messages to check for problems with your optimization.

Shadow solutions are displayed at the start of the solution list and indicated by the prefix "Shadow solution" in the message.

The NBI output messages include the exit flags and the first part of the message returned by fmincon calls. Extra information about the NBI solution is added to this to explain certain situations. The extra NBI messages are shown in the following table.

Exit flag	NBI Message
6	Some shadow solutions do not differ. Remove one of the non-competing objectives. Note   CAGE does not run the subproblems if any pair of shadow solutions are the same (within tolerance). All subproblems will show an exit flag of -8.
0	The solver stopped prematurely in at least one shadow problem and some shadow solutions do not differ. Note   All shadow problems share the exit flag of 0, and CAGE does not run the subproblems (all will show an exit flag of -8)
-7	Solution is dominated by other solutions. Note   Only successful solutions (with an fmincon exit flag >= 0) are used to determine whether a point is dominated.
-8	NBI subproblem was not run because some shadow problems do not differ.
-9	NBI subproblem was not run because a shadow problem failed.

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Viewing Your Optimization Results

Analyzing Point Optimization Output

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