使用相关子查询建立最后日期

drop table if exists t;
    create table t(drug_refill_id int auto_increment primary key,drug_inventory_id int,drug_entity_id int,start_date date);

    insert into t (drug_inventory_id,drug_entity_id, start_date) values
    (1,1,'2018-01-01'),(1,3,'2018-01-02'),(2,4,'2018-01-01'),(1,5,'2018-01-01'),(1,5,'2018-02-01');

    select * 
    from t 
    where start_date = (select max(start_date) from t t1 where t1.drug_entity_id = t.drug_entity_id) and
            drug_inventory_id = 1;

+----------------+-------------------+----------------+------------+
| drug_refill_id | drug_inventory_id | drug_entity_id | start_date |
+----------------+-------------------+----------------+------------+
|              1 |                 1 |              1 | 2018-01-01 |
|              2 |                 1 |              3 | 2018-01-02 |
|              5 |                 1 |              5 | 2018-02-01 |
+----------------+-------------------+----------------+------------+
3 rows in set (0.00 sec)

顺便说一句，如果你希望答案更接近你的数据，那么在你的问题中包括文本数据，而不是图像。