Gee, for a while I was Director of Operations Research at FedEx. My office was next to that of FedEx founder, COB, CEO F. Smith.

Poor Smith: One night at about midnight, I was in my office getting in a little violin practice before heading back to my rented room for some sleep (my wife was still in her Ph.D. program at Johns Hopkins in MD). As I left to get some sleep, I saw Smith in his office. He had had to listen to my violin practice of scales, intervals, etc.!

The first good thing I did for Smith was write some software for scheduling the fleet. The BoD was seriously asking if such scheduling could be done at all. One evening R. Frock (he has a book on FedEx) and I used my software and produced a schedule for the full, planned operation. Smith's remark was "Solves the most important problem". Our two BoD representatives of BoD Member General Dynamics said "It's a little tight in a few places, but it's flyable". The BoD was happy, and crucial funding was enabled.

But by the night of the violin practice, I was working on better scheduling. Right, I was encountering the question of P = NP. So, I had chatted with G. Nemhauser about integer linear programming set covering.

Later I went for my Ph.D. in applied math and, in the studies, kept in mind the problems I'd seen at FedEx. So, e.g., model various cases of arrivals as a Poisson process -- from an axiomatic derivation of that process or just from the renewal theorem. Use some stochastic optimal control for some of the operational issues. Have some much more accurate cost estimating of the airplane operations and include those in the integer linear programming, etc.

Gee, for the last 10 years, I didn't know that Amazon was doing so much in such logistics. No way can they be avoiding some optimization problems for their logistics without wasting significant bucks. For a lot they were doing, I'd been there, done that, went to grad school for it, written software for it, etc. Could have helped them!

Scattering warehouses, sort centers, fulfillment centers, ships, airplanes, trucks, bicycles, drones, etc. all around and changing all that ASAP over time has to encounter some severe challenges in applied math and corresponding software, especially having those two keep up with the operational changes.

I know; I know; if assume that the operations will be big enough, then there are some approaches to those discrete optimization problems that use just some continuous techniques that are much easier. In that case, maybe for a while, during the rapid growth period, go ahead and don't try very hard on the optimization, scatter around, and anticipate that, when reach the target size, nearly all those warehouses, sort centers, etc. will be close enough to something optimal.

Then I wonder: What is Wal-Mart doing in response? They have a lot of warehouses, trucks, and stores and a Web site for on-line ordering? Right, they bought Jet.com. But anything else?

Fundamentally I begin to wonder about Amazon and airplanes: For FedEx, sure airplanes are crucial. But for Amazon? I wonder.

Why? What's the difference?

For FedEx, hour by hour, a huge fraction of what FedEx receives and ships is different, literally never done before, that is, could not have been stocked in a warehouse. E.g., FedEx may receive a legal brief, some medical samples, a custom 3D printout, etc. That was the first and last time those particular items were ever shipped. The variety of what FedEx ships is beyond belief -- stand in the sort center in Memphis and watch the stuff go by and have to give up on any simple overview.

For Amazon, all they ship is what's for sale on their Web site. So far, next to none of that stuff is perishable. And, for nearly all of it, the time, hours, days, weeks, from when it is created by the original manufacturer to when it is needed in an Amazon warehouse doesn't much matter.

So, in comparison with FedEx, the variety is much less. So Amazon has an easier time just stocking nearly all their reasonably popular items at each of their local warehouses and delivering to those warehouses by the cheapest way possible, typically 18 wheel trucks.

Okay, maybe Amazon is selling some item, expects to sell only 15 of them in the next month, but has 60 local warehouses. So, can't stock the item at each warehouse. So, for slow moving items, have to stock in some central warehouse and ship from there via airplane. So, have 60 planes, and each day ship from the central warehouse for slow moving items to each of the 60 local warehouses all the slow moving items that warehouse needs.

Right, maybe have fewer than 60 planes, often have one plane serve more than one warehouse, on a given day maybe have some planes make more than one trip from the central warehouse, and each day, given the loads to be moved from the central warehouse to each of the 60 local ones, decide which planes go to which warehouses in what order -- back to integer linear programming set covering again. Okay.

How does that set covering work? Okay, just enumerate all reasonably efficient trips a single airplane might do. If have several airplane types, have to do that separately for each type. Assume that if an airplane stops at a warehouse, then it brings everything that warehouse needs (one warehouse that needs more than one airplane can carry would be a special case, easier in some ways). Then take those candidate airplane trips and pick the ones that move all the loads, within the performance limitations of the airplanes, in the times required and that give the least total operating cost. That last part is the optimization and 0-1 integer linear programming.

But, still, the variety for Amazon is much less than that for FedEx.

Actually, FedEx has long had a warehouse in Memphis. So, shipping customers would use trucks to put their stock in that warehouse, and FedEx would pick from that stock and ship via FedEx planes the orders that had arrived and accumulated from end-user customers. So, in that case, a truck was used for the trip to Memphis and an airplane, for the trip from Memphis.

Airplanes are strange beasts: Generally, if can keep an airplane flying nearly full lots of hours each day, then can have a license to print money. But if have an airplane sit on the ground or, worse, flying with not much load, then have a path to going broke. How much usage FedEx can get from an airplane is well understood, at least by Smith. But I question if Amazon can get enough usage out of airplanes.

Of course, maybe by now, beyond what was in the OP, Amazon has plenty of data to know that they can put a big warehouse there in southern Ohio for their slow moving items and ship those items to local warehouses and distribution centers as orders arrive from the Amazon Web site. Maybe. But, again, for their fast moving items, just stock the local warehouses and distribution centers by 18 wheel trucks -- no expensive airplanes involved.

Gee, in my Ph.D. program, I took courses in optimization, stochastic processes, facility location, multi-objective optimization, etc. from world-class people. I never got a call from an Amazon recruiter! I wasn't that difficult to find: The guy who taught the course in multi-objective optimization and was one of my Ph.D. dissertation advisers was, during a lot of those years of Amazon's growth, President at CMU. The guy who was Chair of the committee that approved my dissertation research was a world-class guy in transportation.

Lesson 1: Go get a Ph.D. for some ways to get approximately optimal solutions to some really complicated NP-complete problems, maybe also with some random elements, and even if a big business has a lot of need for such applied math maybe still won't get an opportunity to apply what studied.

Lesson 2: If want to make money with some applied math and, then, some corresponding computing, then pick own practical problem to solve, do the math to solve the problem (yes, pick the problem so that can use some math to get an advantage in getting a good solution to the problem -- that is, pick the problem and the math together, as a good pair), write the software, and do own startup where are already the CEO. That is, for own future applying math and computing, don't depend on some other CEO. And that's what I'm doing -- my own startup.