I am trying to better understand the origins of Optimal Transport Problems such as Monge's Problem.
For instance, I came across the following references:
https://www.math.ucdavis.edu/~qlxia/Research/monge.pdf , https://www.youtube.com/watch?v=aio1lAE-h_I
In these problems, the objective is to find the "most optimal way to move sand from one location to another". I am trying to better understand where the original problem occurred. For instance, in the very basic set up, it would seem that moving larger amounts of sand would be faster but more difficult (e.g. lifting heavier loads of sand), and moving smaller amounts of sand would be slower but easier. At first glance, there does not seem to be a "real optimal transport order" but rather a proportional tradeoff. It depends more on how quickly and how much weight you are ready to lift.
I can try to think of this problem conceptually. For example, suppose there is a central bakery that produces different types of cakes and it delivers these different cakes to different stores. Each type of cake has a cost to make, and each store has a different demand for different types of cakes, and the cost to deliver cakes depends on the location of the stores relative to the bakery. Thus, when the central bakery has to make multiple trips (e.g. a truck can only carry a certain number of cakes each time) to deliver multiple cakes to multiple bakeries, I can see that there might be "more optimal ways to transport the cakes to different bakeries".
But in the case of Monge's Problem (or Kantorovich), I am trying to better understand the initial problem. I found plenty of references online, but all of these references are far too detailed in terms of the level of theoretical mathematics that they use (e.g. https://arxiv.org/pdf/1701.02826.pdf).
Can anyone please try to explain the objective functions that need to be optimized in Monge's/Kantorovich's Problem, or perhaps recommend references to sources that are less mathematically intensive?
