Some of my professors had a good way of doing this. Basically all midterms were optional, and the weight of each midterm not taken was added to the final exam.
For instance, in one class there were two midterms, 1 in February, and 1 in March. Each midterm was worth 20% of the final mark, and the final exam was worth 60%. If you didn't write either midterm the final would be worth 100% of your mark.
This worked well because the professor never had to do make-up exams. It was to the advantage of the students to take the midterm exams because in general it helped with final exam preparation, and it allowed good students with difficult midterm schedules not to have their grades pulled down by a single bad mark (e.g. one week I had 5 midterm exams, and 3 major assignments due. If this had been one of my classes I could have chose to only have 4 midterm exams).
If you use this approach, however, the final exam must be cumulative.
Also, I just assumed you students weren't missing final exams. At my undergrad university if a student missed a final exam they had to take it up with the a third party, not the professor.
Ours states that they "have to be offered for legitimate reasons" and list some truly legitimate excuses (e.g. hospitalization), but leave other reasons up to the discretion of the instructor. I've never seen one that actually goes into detail on the format of the exam, when it must be offered, etc. Does yours have one? If so, I'd be curious to see it.
This is my undergrad university's policy:
Exam Policy said:
If you are unable to write your final exam due to illness or another serious reason, you may apply for a deferral and when your application is accepted, you will be permitted to write it during the next deferred exam period. To qualify, you must obtain documentation (such as a doctor's note) explaining your inability to write the exam on the original date, and bring it to your Student Affairs Office as soon as possible after the exam, but no later than January 15 (for Fall courses) or May 15 (Winter and Fall-Winter courses).
If, for serious reasons such as illness or family affliction, you have not written one or more examinations, you may receive the permission of your Faculty Student Affairs Office to defer the examination to the next supplemental examination period, except in the Faculty of Engineering (where students write the examination the next time the course is given); see Deferred Examinations: Faculty of Engineering. You should be aware that the University will only defer examinations for compelling reasons, verified and accepted by the Student Affairs Office. You must provide supporting evidence such as an appropriate medical report, and you must inform the Student Affairs Office as soon as possible to explain why you missed the examination.
If you aren't granted permission to take the deferred exam you automatically fail the course for failing to attend the final exam, regardless of your grade otherwise. This is denoted on your transcript, (You get a non-grade letter that means "You failed to show-up for your final exam.")
If you miss an exam you have to apply to write a deferred exam. All university deferred exams from term 1 are written during the same 1 week period. Deferred exams do not have to be in the same format as the primary exams. It actually sucks because there is basically no way to get information on the format of the deferred exam. I had to defer one exam because I had shingles (provoked by stress). The primary final exam was multiple choice and short answer, the deferred final exam was 2 essay questions.
Also, at my university all formal final exams were three hours long.