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        "input_hash": "d68058d5d1c8ae6db375d0ce91f94116969cb6b348d9980d31a420025eda55f7",
        "preference_sources": [
          "synthetic"
        ],
        "config": {
          "target_size": 3,
          "allow_smaller": true,
          "allow_larger": true,
          "min_size": 2,
          "max_size": 4,
          "social_weight": 0.5
        },
        "assignments": {
          "s01": "climate",
          "s02": "climate",
          "s03": "climate",
          "s04": "housing",
          "s05": "housing",
          "s06": "housing",
          "s07": "transit",
          "s08": "transit",
          "s09": "transit",
          "s10": "food",
          "s11": "food",
          "s12": "food"
        },
        "valid": true,
        "teams": [
          {
            "project_id": "climate",
            "students": [
              "s01",
              "s02",
              "s03"
            ]
          },
          {
            "project_id": "housing",
            "students": [
              "s04",
              "s05",
              "s06"
            ]
          },
          {
            "project_id": "transit",
            "students": [
              "s07",
              "s08",
              "s09"
            ]
          },
          {
            "project_id": "food",
            "students": [
              "s10",
              "s11",
              "s12"
            ]
          }
        ],
        "unused_projects": [
          "library"
        ],
        "students": [
          {
            "student_id": "s01",
            "project_id": "climate",
            "teammates": [
              "s02",
              "s03"
            ],
            "project_rank": 1,
            "project_borda": 4,
            "teammate_borda": 19,
            "project_score": 1.0,
            "social_score": 0.95,
            "combined_score": 0.975,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s02",
            "project_id": "climate",
            "teammates": [
              "s01",
              "s03"
            ],
            "project_rank": 2,
            "project_borda": 3,
            "teammate_borda": 19,
            "project_score": 0.75,
            "social_score": 0.95,
            "combined_score": 0.85,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s03",
            "project_id": "climate",
            "teammates": [
              "s01",
              "s02"
            ],
            "project_rank": null,
            "project_borda": 0,
            "teammate_borda": 19,
            "project_score": 0.0,
            "social_score": 0.95,
            "combined_score": 0.475,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s04",
            "project_id": "housing",
            "teammates": [
              "s05",
              "s06"
            ],
            "project_rank": 1,
            "project_borda": 4,
            "teammate_borda": 19,
            "project_score": 1.0,
            "social_score": 0.95,
            "combined_score": 0.975,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s05",
            "project_id": "housing",
            "teammates": [
              "s04",
              "s06"
            ],
            "project_rank": 2,
            "project_borda": 3,
            "teammate_borda": 19,
            "project_score": 0.75,
            "social_score": 0.95,
            "combined_score": 0.85,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s06",
            "project_id": "housing",
            "teammates": [
              "s04",
              "s05"
            ],
            "project_rank": 3,
            "project_borda": 2,
            "teammate_borda": 19,
            "project_score": 0.5,
            "social_score": 0.95,
            "combined_score": 0.725,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s07",
            "project_id": "transit",
            "teammates": [
              "s08",
              "s09"
            ],
            "project_rank": 1,
            "project_borda": 4,
            "teammate_borda": 19,
            "project_score": 1.0,
            "social_score": 0.95,
            "combined_score": 0.975,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s08",
            "project_id": "transit",
            "teammates": [
              "s07",
              "s09"
            ],
            "project_rank": 2,
            "project_borda": 3,
            "teammate_borda": 19,
            "project_score": 0.75,
            "social_score": 0.95,
            "combined_score": 0.85,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s09",
            "project_id": "transit",
            "teammates": [
              "s07",
              "s08"
            ],
            "project_rank": 2,
            "project_borda": 3,
            "teammate_borda": 19,
            "project_score": 0.75,
            "social_score": 0.95,
            "combined_score": 0.85,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s10",
            "project_id": "food",
            "teammates": [
              "s11",
              "s12"
            ],
            "project_rank": 1,
            "project_borda": 4,
            "teammate_borda": 19,
            "project_score": 1.0,
            "social_score": 0.95,
            "combined_score": 0.975,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s11",
            "project_id": "food",
            "teammates": [
              "s10",
              "s12"
            ],
            "project_rank": 2,
            "project_borda": 3,
            "teammate_borda": 19,
            "project_score": 0.75,
            "social_score": 0.95,
            "combined_score": 0.85,
            "ranked_teammates_received": 2
          },
          {
            "student_id": "s12",
            "project_id": "food",
            "teammates": [
              "s10",
              "s11"
            ],
            "project_rank": 2,
            "project_borda": 3,
            "teammate_borda": 19,
            "project_score": 0.75,
            "social_score": 0.95,
            "combined_score": 0.85,
            "ranked_teammates_received": 2
          }
        ],
        "size_deviation": 0,
        "total_score": 10.199999999999998,
        "statistics": {
          "students": 12,
          "teams": 4,
          "project_first_choices": 4,
          "project_top_three": 11,
          "unranked_project_assignments": 1,
          "students_with_ranked_teammate": 12,
          "minimum_individual_score": 0.475,
          "mean_project_score": 0.75,
          "mean_social_score": 0.9499999999999998
        },
        "solution_status": "optimal",
        "solver": {
          "name": "HiGHS via scipy.optimize.milp",
          "scipy_version": "1.17.1",
          "time_limit_seconds": 60,
          "elapsed_seconds": 0.49925745799555443,
          "variables": 190,
          "constraints": 398,
          "phases": {
            "team_size": {
              "status_code": 0,
              "message": "Optimization terminated successfully. (HiGHS Status 7: Optimal)",
              "optimal": true,
              "objective": 0.0,
              "dual_bound": 0.0,
              "relative_gap": 0.0
            },
            "preferences": {
              "status_code": 0,
              "message": "Optimization terminated successfully. (HiGHS Status 7: Optimal)",
              "optimal": true,
              "objective": -10.200000000000012,
              "dual_bound": -10.200000000000012,
              "relative_gap": 0.0
            }
          }
        },
        "scoring": {
          "borda": "M-rank over the full option universe; singleton gets 1",
          "unlisted": "0 points; retained as unranked, allowed unless explicitly excluded",
          "project_divisor": 4,
          "social_divisor": 20,
          "objective_order": [
            "minimize total absolute team-size deviation",
            "maximize weighted Borda score"
          ],
          "guarantees": "Feasibility checked independently. No stability or strategy-proofness guarantee."
        }
      }
    }
  ],
  "note": "These are selected optimal assignments; ties may change individual memberships across solver versions."
}