love & operations research

My February blog post is a bit belated and less technical than it should be.  It has been a busy February for me as I prepare for the arrival of my third child.  I am sure that I will have a few blog posts about love after my little bundle of joy arrives some time during March.

I thoroughly enjoyed Anna Nagurney’s photo essay of why she loves OR as well as Mike Trick’s personal account of finding (and not finding) love via the Secretary problem.  Mike Trick’s blog post reminded me of my father’s criteria for finding a spouse:  he made a list and searched until he met someone who met all of his criteria.  He was chided by many for making a list.  How could anyone meet all of his criteria?  My father is apparently a satisficer not an optimizer.  It turned out that his problem was not over-constrained and infeasible: he met and married my mother, and they will be celebrating their 40th wedding anniversary later this year.

I tell this story about my father because I did not use operations research in choosing a spouse.  Maybe I didn’t need to–my father set such a wonderful example about laying out the right criteria that I think I had a good road map to start with.

I tend to use operations research to run my household so that I can optimally balance work and family life (I will blog about that in the near future).  I have been known to mention critical paths, bottlenecks, and value focused thinking during breakfast conversations about planning a hectic day with my husband.  He doesn’t always appreciate when I try to optimize him, but as a nerd, he always knows that my intentions are good.  And frequently my ideas about scheduling are helpful with budgeting our time on a busy day.  However, my INFORMS blog challenge post about love can be summarized in a single sentence: Do not try to optimize your spouse.

On a somewhat related note, I taught the Secretary problem to my discrete optimization course in February.  As mentioned by Mike Trick, the secretary problem can be used to model finding a spouse (although we’d both agree that it’s not a perfect model for finding love).  The optimal solution to the Secretary problem involves rejecting the first ~exp(-1) candidates (a proportion of the total number of candidates n), meaning that the odds that rejecting the love of your life has probability ~0.368!  However, this is an asymptotic result in terms of the number of love interests.  Most of us mere mortals will end up dating a finite number of people, so we will fare somewhat better (unless we choose a suboptimal strategy).

I required my students to program the Markov decision process model for the Secretary Problem as a homework assignment.  This meant that I had to do the same to be able to go over the solutions.  This homework problem turned out to be really challenging for them, since it was their first time writing a recursion.  I hadn’t written a recursion in awhile, but got my code to work after a few minutes.  Just for kicks, here is the optimal solution value, the optimal threshold, and a simulation over 10,000 replications illustrating the results.

The optimal policy for the secretary problem.  This shows the number of candidates to reject up front.  After that, the optimal policy is to hire the next candidate that is the best seen so far.

The optimal policy for the secretary problem/spouse finding problem. This shows the number of candidates to reject up front. After that, the optimal policy is to hire/marry the next candidate that is the best seen so far (the blue line shows the first candidate that could be hired/married).

The probability of hiring the best secretary/choosing the right spouse as a function of the number of candidates available

The probability of hiring the best secretary/choosing the right spouse as a function of the number of candidates available (n)

The probability of hiring a secretary/finding a spouse as a function of the number of candidates (n) simulated over 10,000 replications

The probability of hiring a secretary/finding a spouse and the probability of hiring the *best* secretary/choosing the right spouse as a function of the number of candidates (n) simulated over 10,000 replications

 

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