MySQL not using multiple column index

Question

I have query with columns in where clause which are part of primary key and have foreign key indexes on all columns.

EXPLAIN SELECT   aggEI.c_id AS companyId, aggEI.ei_uid AS uuid
FROM AGG_EI AS aggEI 
WHERE  aggEI.c_id in (8) and aggEI.tg_id IN (1,2,3,4,5,6,7)
       AND aggEI.dt_id = 20130506

I have also defined multiple columnn index on (c_id,tg_id,dt_id) but EXPLAIN shows that it is using foreign key index on c_id

 1 SIMPLE   aggEI   ref PRIMARY,datedm_id_UNIQUE,agg_ei_comdm_fk_idx,agg_ei_datedm_fk_idx,agg_ei_topgrp_fk_idx,comp_uuid    agg_ei_comdm_fk_idx 8   const   65986   Using where; Using index

agg_ei_comdm_fk_idx is the foreign key index on c_id and comp_uuid is the multiple column index on (c_id,tg_id,dt_id)

Can someone explain why it is happening like this

EDIT: Create table

          'CREATE TABLE `AGG_EI` (
              `dt_id` int(11) NOT NULL,
              `c_id` bigint(20) NOT NULL,
              `tg_id` bigint(20) NOT NULL,
              `ei_uid` varchar(150) NOT NULL
              `ei_name` varchar(150) NOT NULL,
               `rating` double NOT NULL,
              `cnt` double NOT NULL
               PRIMARY KEY (`dt_id`,`c_id`,`tg_id`,`ei_uid`),
                UNIQUE KEY `datedm_id_UNIQUE` (`dt_id`,`c_id`,`ei_uid`,`tg_id`),
               KEY `agg_ei_comdm_fk_idx` (`c_id`),
               KEY `agg_ei_datedm_fk_idx` (`dt_id`),
               KEY `agg_ei_topgrp_fk_idx` (`tg_id`),
               KEY `comp_uuid` (`c_id`,`tg_id`,`dt_id`),
               CONSTRAINT `agg_ei_comdm_fk` FOREIGN KEY (`c_id`) REFERENCES `COMPDM`     (`c_id`) ON DELETE NO ACTION ON UPDATE NO ACTION,
                CONSTRAINT `agg_ei_datedm_fk` FOREIGN KEY (`dt_id`) REFERENCES `DATEDM` (`dt_id`) ON DELETE NO ACTION ON UPDATE NO ACTION,
                CONSTRAINT `agg_ei_topgrp_fk` FOREIGN KEY (`tg_id`) REFERENCES `TOPGRP` (`tg_id`) ON DELETE NO ACTION ON UPDATE NO ACTION
             ) ENGINE=InnoDB DEFAULT CHARSET=utf8'

Answer 1

You can force mysql to use a index like :

EXPLAIN 
SELECT   
    aggEI.c_id AS companyId, 
    aggEI.ei_uid AS uuid
FROM 
    AGG_EI AS aggEI FORCE INDEX(`comp_uuid`)
WHERE  
    aggEI.c_id in (8) 
    AND aggEI.tg_id IN (1,2,3,4,5,6,7)
    AND aggEI.dt_id = 20130506