def select_team(expected_scores, prices, positions, clubs):
    num_players = len(expected_scores)
    model = pulp.LpProblem("Constrained value maximisation", pulp.LpMaximize)
    decisions = [
        pulp.LpVariable("x{}".format(i), lowBound=0, upBound=1, cat='Integer')
        for i in range(num_players)
    ]
    captain_decisions = [
        pulp.LpVariable("y{}".format(i), lowBound=0, upBound=1, cat='Integer')
        for i in range(num_players)
    ]

    # objective function:
    model += sum((captain_decisions[i] + decisions[i]) * expected_scores[i]
                 for i in range(num_players)), "Objective"

    # cost constraint
    model += sum(decisions[i] * prices[i] for i in range(num_players)) <= 100  # total cost
    model += sum(decisions) == 11  # total team size

    # position constraints
    # 1 goalkeeper
    model += sum(decisions[i] for i in range(num_players) if positions[i] == 1) == 1
    # 3-5 defenders
    model += sum(decisions[i] for i in range(num_players) if positions[i] == 2) >= 3
    model += sum(decisions[i] for i in range(num_players) if positions[i] == 2) <= 5
    # 3-5 midfielders
    model += sum(decisions[i] for i in range(num_players) if positions[i] == 3) >= 3
    model += sum(decisions[i] for i in range(num_players) if positions[i] == 3) <= 5
    # 1-3 attackers
    model += sum(decisions[i] for i in range(num_players) if positions[i] == 4) >= 1
    model += sum(decisions[i] for i in range(num_players) if positions[i] == 4) <= 3

    # club constraint
    for club_id in np.unique(clubs):
        model += sum(decisions[i] for i in range(num_players) if clubs[i] == club_id) <= 3  # max 3 players

    model += sum(captain_decisions) == 1  # 1 captain

    for i in range(num_players):  # captain must also be on team
        model += (decisions[i] - captain_decisions[i]) >= 0

    model.solve()
    return decisions, captain_decisions