This is the third blog in a series showcasing the functionality of the Exploratory Modeling workbench. Exploratory modeling entails investigating the way in which uncertainty and/or policy levers map to outcomes. To investigate these mappings, we can either use sampling based strategies (open exploration) or optimization based strategies (directed search) In the first blog, I gave a general overview of the workbench and showed briefly how both investigation strategies can be done. In the second blog, I demonstrated the use of the workbench for open exploration in substantial more detail. In this third blog, I will demonstrate in more detail how to use the workbench for directed search. Like in the previous two blog post, I will use the DPS version of the lake problem.

For optimization, the workbench relies on platypus. You can easily install the latest version of platypus from github using pip

pip install git+https://github.com/Project-Platypus/Platypus.git

By default, the workbench will use epsilon NSGA2, but all the other algorithms available within platypus can be used as well.

Within the workbench, optimization can be used in three ways:

* Search over decision levers for a reference scenario

* Robust search: search over decision levers for a set of scenarios

* worst case discovery: search over uncertainties for a reference policy

The search over decision levers or over uncertainties relies on the specification of the direction for each outcome of interest defined on the model. It is only possible to use `ScalarOutcome`

objects for optimization.

## Search over levers

Directed search is most often used to search over the decision levers in order to find good candidate strategies. This is for example the first step in the Many Objective Robust Decision Making process. This is straightforward to do with the workbench using the optimize method.

from ema_workbench import MultiprocessingEvaluator, ema_logging ema_logging.log_to_stderr(ema_logging.INFO) with MultiprocessingEvaluator(model) as evaluator: results = evaluator.optimize(nfe=10000, searchover='levers', epsilons=[0.1,]*len(model.outcomes), population_size=50)

the result from optimize is a DataFrame with the decision variables and outcomes of interest. The latest version of the workbench comes with a pure python implementation of parallel coordinates plot built on top of matplotlib. It has been designed with the matplotlib and seaborn api in mind. We can use this to quickly visualize the optimization results.

from ema_workbench.analysis import parcoords paraxes = parcoords.ParallelAxes(parcoords.get_limits(results), rot=0) paraxes.plot(results, color=sns.color_palette()[0]) paraxes.invert_axis('max_P') plt.show()

Note how we can flip an axis using the `invert_axis`

method. This eases interpretation of the figure because the ideal solution in this case would be a straight line for the four outcomes of interest at the top of the figure.

### Specifying constraints

In the previous example, we showed the most basic way for using the workbench to perform many-objective optimization. However, the workbench also offers support for constraints and tracking convergence. Constrains are an attribute of the optimization problem, rather than an attribute of the model as in Rhodium. Thus, we can pass a list of constraints to the optimize method. A constraint can be applied to the model input parameters (both uncertainties and levers), and/or outcomes. A constraint is essentially a function that should return the distance from the feasibility threshold. The distance should be 0 if the constraint is met.

As a quick demonstration, let’s add a constraint on the maximum pollution. This constraint applies to the max_P outcome. The example below specifies that the maximum pollution should be below 1.

from ema_workbench import MultiprocessingEvaluator, ema_logging, Constraint ema_logging.log_to_stderr(ema_logging.INFO) constraints = [Constraint("max pollution", outcome_names="max_P", function=lambda x:max(0, x-1))] with MultiprocessingEvaluator(model) as evaluator: results = evaluator.optimize(nfe=1000, searchover='levers', epsilons=[0.1,]*len(model.outcomes), population_size=25, constraints=constraints)

### tracking convergence

To track convergence, we need to specify which metric(s) we want to use and pass these to the optimize method. At present the workbench comes with 3 options: Hyper volume, Epsilon progress, and a class that will write the archive at each iteration to a separate text file enabling later processing. If convergence metrics are specified, optimize will return both the results as well as the convergence information.

from ema_workbench import MultiprocessingEvaluator, ema_logging from ema_workbench.em_framework.optimization import (HyperVolume, EpsilonProgress, ) from ema_workbench.em_framework.outcomes import Constraint ema_logging.log_to_stderr(ema_logging.INFO) # because of the constraint on pollution, we can specify the # maximum easily convergence_metrics = [HyperVolume(minimum=[0,0,0,0], maximum=[1,1,1,1]), EpsilonProgress()] constraints = [Constraint("max pollution", outcome_names="max_P", function=lambda x:max(0, x-1))] with MultiprocessingEvaluator(model) as evaluator: results_ref1, convergence1 = evaluator.optimize(nfe=25000, searchover='levers', epsilons=[0.05,]*len(model.outcomes), convergence=convergence_metrics, constraints=constraints, population_size=100)

We can visualize the results using parcoords as before, while the convergence information is in a DataFrame making it also easy to plot.

fig, (ax1, ax2) = plt.subplots(ncols=2, sharex=True) ax1.plot(convergence1.epsilon_progress) ax1.set_xlabel('nr. of generations') ax1.set_ylabel('$\epsilon$ progress') ax2.plot(convergence1.hypervolume) ax2.set_ylabel('hypervolume') sns.despine() plt.show()

### Changing the reference scenario

Up till now, we have performed the optimization for an unspecified reference scenario. Since the lake model function takes default values for each of the deeply uncertain factors, these values have been implicitly assumed. It is however possible to explicitly pass a reference scenario that should be used instead. In this way, it is easy to apply the extended MORDM approach suggested by Watson and Kasprzyk (2017).

To see the effects of changing the reference scenario on the values for the decision levers found through the optimization, as well as ensuring a fair comparison with the previous results, we use the same convergence metrics and constraints from the previous optimization. Note that the constraints are in essence only a function, and don’t retain optimization specific state, we can simply reuse them. The convergence metrics, in contrast retain state and we thus need to re-instantiate them.

from ema_workbench import Scenario reference = Scenario('reference', **dict(b=.43, q=3,mean=0.02, stdev=0.004, delta=.94)) convergence_metrics = [HyperVolume(minimum=[0,0,0,0], maximum=[1,1,1,1]), EpsilonProgress()] with MultiprocessingEvaluator(model) as evaluator: results_ref2, convergence2 = evaluator.optimize(nfe=25000, searchover='levers', epsilons=[0.05,]*len(model.outcomes), convergence=convergence_metrics, constraints=constraints, population_size=100, reference=reference)

### comparing results for different reference scenarios

To demonstrate the parcoords plotting functionality in some more detail, let’s combine the results from the optimizations for the two different reference scenarios and visualize them in the same plot. To do this, we need to first figure out the limits across both optimizations. Moreover, to get a better sense of which part of the decision space is being used, let’s set the limits for the decision levers on the basis of their specified ranges instead of inferring the limits from the optimization results.

columns = [lever.name for lever in model.levers] columns += [outcome.name for outcome in model.outcomes] limits = {lever.name: (lever.lower_bound, lever.upper_bound) for lever in model.levers} limits = dict(**limits, **{outcome.name:(0,1) for outcome in model.outcomes}) limits = pd.DataFrame.from_dict(limits) # we resort the limits in the order produced by the optimization limits = limits[columns] paraxes = parcoords.ParallelAxes(limits, rot=0) paraxes.plot(results_ref1, color=sns.color_palette()[0], label='ref1') paraxes.plot(results_ref2, color=sns.color_palette()[1], label='ref2') paraxes.legend() paraxes.invert_axis('max_P') plt.show()

## Robust Search

The workbench also comes with support for many objective robust optimization. In this case, each candidate solution is evaluated over a set of scenarios, and the robustness of the performance over this set is calculated. This requires specifying 2 new pieces of information:

* the robustness metrics

* the scenarios over which to evaluate the candidate solutions

The robustness metrics are simply a collection of `ScalarOutcome`

objects. For each one, we have to specify which model outcome(s) it uses, as well as the actual robustness function. For demonstrative purposes, let’s assume we are use a robustness function using descriptive statistics: we want to maximize the 10th percentile performance for reliability, inertia, and utility, while minimizing the 90th percentile performance for max_P.

We can specify our scenarios in various ways. The simplest would be to pass the number of scenarios to the `robust_optimize`

method. In this case for each generation a new set of scenarios is used. This can create noise and instability in the optimization. A better option is to explicitly generate the scenarios first, and pass these to the method. In this way, the same set of scenarios is used for each generation.

If we want to specify a constraint, this can easily be done. Note however, that in case of robust optimization, the constrains will apply to the robustness metrics instead of the model outcomes. They can of course still apply to the decision variables as well.

import functools from ema_workbench import Constraint, MultiprocessingEvaluator from ema_workbench import Constraint, ema_logging from ema_workbench.em_framework.optimization import (HyperVolume, EpsilonProgress) from ema_workbench.em_framework.samplers import sample_uncertainties ema_logging.log_to_stderr(ema_logging.INFO) percentile10 = functools.partial(np.percentile, q=10) percentile90 = functools.partial(np.percentile, q=90) MAXIMIZE = ScalarOutcome.MAXIMIZE MINIMIZE = ScalarOutcome.MINIMIZE robustnes_functions = [ScalarOutcome('90th percentile max_p', kind=MINIMIZE, variable_name='max_P', function=percentile90), ScalarOutcome('10th percentile reliability', kind=MAXIMIZE, variable_name='reliability', function=percentile10), ScalarOutcome('10th percentile inertia', kind=MAXIMIZE, variable_name='inertia', function=percentile10), ScalarOutcome('10th percentile utility', kind=MAXIMIZE, variable_name='utility', function=percentile10)] def constraint(x): return max(0, percentile90(x)-10) constraints = [Constraint("max pollution", outcome_names=['90th percentile max_p'], function=constraint)] convergence_metrics = [HyperVolume(minimum=[0,0,0,0], maximum=[10,1,1,1]), EpsilonProgress()] n_scenarios = 10 scenarios = sample_uncertainties(model, n_scenarios) nfe = 10000 with MultiprocessingEvaluator(model) as evaluator: robust_results, convergence = evaluator.robust_optimize(robustnes_functions, scenarios, nfe=nfe, constraints=constraints, epsilons=[0.05,]*len(robustnes_functions), convergence=convergence_metrics,)

fig, (ax1, ax2) = plt.subplots(ncols=2) ax1.plot(convergence.epsilon_progress.values) ax1.set_xlabel('nr. of generations') ax1.set_ylabel('$\epsilon$ progress') ax2.plot(convergence.hypervolume) ax2.set_ylabel('hypervolume') sns.despine() plt.show()

paraxes = parcoords.ParallelAxes(parcoords.get_limits(robust_results), rot=45) paraxes.plot(robust_results) paraxes.invert_axis('90th percentile max_p') plt.show()

## Search over uncertainties: worst case discovery

Up till now, we have focused on optimizing over the decision levers. The workbench however can also be used for worst case discovery (Halim et al, 2016). In essence, the only change is to specify that we want to search over uncertainties instead of over levers. Constraints and convergence works just as in the previous examples.

Reusing the foregoing, however, we should change the direction of optimization of the outcomes. We are no longer interested in finding the best possible outcomes, but instead we want to find the worst possible outcomes.

# change outcomes so direction is undesirable minimize = ScalarOutcome.MINIMIZE maximize = ScalarOutcome.MAXIMIZE for outcome in model.outcomes: if outcome.kind == minimize: outcome.kind = maximize else: outcome.kind = minimize

We can reuse the reference keyword argument to perform worst case discovery for one of the policies found before. So, below we select solution number 9 from the pareto approximate set. We can turn this into a dict and instantiate a Policy objecti.

from ema_workbench import Policy policy = Policy('9', **{k:v for k, v in results_ref1.loc[9].items() if k in model.levers}) with MultiprocessingEvaluator(model) as evaluator: results = evaluator.optimize(nfe=1000, searchover='uncertainties', epsilons=[0.1,]*len(model.outcomes), reference=policy)

Visualizing the results is straightforward using parcoords.

paraxes = parcoords.ParallelAxes(parcoords.get_limits(results), rot=0) paraxes.plot(results) paraxes.invert_axis('max_P') plt.show()

## Closing remarks

This blog showcased the functionality of the workbench for applying search based approaches to exploratory modelling. We specifically looked at the use of many-objective optimization for searching over the levers or uncertainties, as well as the use of many-objective robust optimization. This completes the overview of the functionality available in the workbench. In the next blog, I will put it all together to show how the workbench can be used to perform Many Objective Robust Decision Making.