diff --git a/optuna_dashboard/preferential/_study.py b/optuna_dashboard/preferential/_study.py
index 0f2c7d84..585db64d 100644
--- a/optuna_dashboard/preferential/_study.py
+++ b/optuna_dashboard/preferential/_study.py
@@ -71,7 +71,11 @@ class PreferentialStudy:
         if not isinstance(worse_trials, list):
             worse_trials = [worse_trials]
 
-        report_preferences(self._study._study_id, self._study._storage, [(b.number, w.number) for b in better_trials for w in worse_trials])
+        report_preferences(
+            self._study._study_id,
+            self._study._storage,
+            [(b.number, w.number) for b in better_trials for w in worse_trials],
+        )
 
     def get_preferences(self, *, deepcopy: bool = True) -> list[tuple[FrozenTrial, FrozenTrial]]:
         trials = self._study.get_trials(deepcopy=deepcopy)
@@ -92,8 +96,8 @@ class PreferentialStudy:
         else:
             raise RuntimeError("Unexpected trial type")
         storage.set_trial_system_attr(trial_id, _SYSTEM_ATTR_COMPARISON_READY, True)
-        
-        
+
+
 def get_best_trials(study_id: int, storage: optuna.storages.BaseStorage) -> list[FrozenTrial]:
     ready_trials = [
         t
@@ -107,7 +111,6 @@ def get_best_trials(study_id: int, storage: optuna.storages.BaseStorage) -> list
     preferences = get_preferences(study_id, storage)
     worse_numbers = {worse for _, worse in preferences}
     return [copy.deepcopy(t) for t in ready_trials if t.number not in worse_numbers]
-    
 
 
 def create_study(
diff --git a/optuna_dashboard/preferential/_system_attrs.py b/optuna_dashboard/preferential/_system_attrs.py
index 02e89680..2be921da 100644
--- a/optuna_dashboard/preferential/_system_attrs.py
+++ b/optuna_dashboard/preferential/_system_attrs.py
@@ -3,18 +3,20 @@ from __future__ import annotations
 import uuid
 
 import optuna
+from optuna.storages import BaseStorage
 from optuna.trial import FrozenTrial
 from optuna.trial import TrialState
-from optuna.storages import BaseStorage
+
 from .._storage import get_study_summary
 
+
 _SYSTEM_ATTR_PREFIX_PREFERENCE = "preference:values"
 
 
 def report_preferences(
     study_id: int,
     storage: BaseStorage,
-    preferences: list[tuple[int, int]], # element is number of trail
+    preferences: list[tuple[int, int]],  # element is number of trail
 ) -> None:
     key = _SYSTEM_ATTR_PREFIX_PREFERENCE + str(uuid.uuid4())
     storage.set_study_system_attr(