In the referenced table, there must be an index where the referenced columns are listed as the first columns in the same order.
Refer: http://dev.mysql.com/doc/refman/5.6/en/innodb-foreign-key-constraints.html
So try:
CREATE TABLE `project` (
`id` int(11) NOT NULL AUTO_INCREMENT,
`project_id` varchar(60) NOT NULL,
`project_name` varchar(500) NOT NULL,
`cons_bal` int(11) NOT NULL,
`non_cons_bal` int(11) NOT NULL,
`budget_head` varchar(100) NOT NULL,
PRIMARY KEY (`id`),
KEY `project_id` (`project_id`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8
CREATE TABLE `project_map` (
`id` int(11) NOT NULL,
`project_id` varchar(60) NOT NULL,
`head` int(11) NOT NULL,
PRIMARY KEY (`id`),
KEY `project_id` (`project_id`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8
ALTER TABLE `project_map` ADD CONSTRAINT
`p_map_fk001` FOREIGN KEY (`project_id`)
REFERENCES project(`project_id`);