Displaying posts sorting by a custom criterion

Save the order in a Custom Field, then modify the main query to order on that custom field via pre_get_posts. Example for posts page:

function wpa_custom_order( $query ) {
    if ( $query->is_home() && $query->is_main_query() ) {
        $query->set( 'meta_key', 'your_order_key' );
        $query->set( 'orderby', 'meta_value' );
    }
}
add_action( 'pre_get_posts', 'wpa_custom_order' );