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In this post, it was asked to describe counter intuitive results in the field of operations research.

Among the answers came up the more-for-less transportation paradox. In essence, this paradox in transportation problems occurs when it is possible to ship more total goods for less total cost, while shipping more or equal amounts from each origin and to each destination.

This paradox has been analysed, for example here, here or here, and typically dual analysis is used to understand the paradox.

My question is : for teaching purposes, what would be the best way to explain this paradox in simple terms (e.g., for someone not familiar with dual analysis or even linear programming) ? What is the intuition behind the paradox ? If dual analysis provides good elements to explain the paradox, how to explain what is going on from a more physical point of view ?

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While it may seem reasonable to think that costs increase when you increase production, the transportation problem only considers the costs of moving things around. As a consequence, increasing production may provide opportunity to change how things are moved around, and that may help save costs.

For problems like this, is it often useful to think about an ideal situation. In the transportation problem, the ideal situation is where your supply is located as close to the demand as possible. That way, the costs of moving stuff around will be the lowest.

Unfortunately, we may have situations where there is a lot of supply at a location where there is not enough demand, and we have to move that supply to other locations. If demand would be increased at the location where there is excess supply, and the supply would be increased at the location where there was excess demand, you can get rid of some transportation costs, because the geographic imbalance of supply and demand is reduced. This is where the paradox comes from.

To give a pretty clear example: if the US produces 1 unit of apples, and Europe produces 2 units of apples, while the US has demand for 2 units of apples and Europe has only demand for 1 unit of apples, it is obvious you need to transport 1 unit of apples over the Atlantic Ocean. Now suppose the US increases it production to 2 units of apples. As a consequence the necessity to transport apples over the Atlantic disappears. The Europeans will be stuck with excess apples, and to overcome this issue they will probably lower the price, increasing demand of apples within Europe to 2 units of apples as well. Thus, by increasing supply and demand, we removed the geographic imbalance between supply and demand (all are 2), saving transportation costs compared to the previous situation where an imbalance existed.

Thus, the important thing to realize is that transportation costs are caused by a locational imbalance between supply and demand. Increasing supply and demand may give you an opportunity to reduce this imbalance and thus reduce transportation costs. The intuition that costs should increase when production is increased is valid, but this only applies to the production costs themselves and not how much we need to move around.

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My question is : for teaching purposes, what would be the best way to explain this paradox in simple terms ... how to explain what is going on from a more physical point of view ?

One of the easiest examples to explain and visualize is the iron ore trains of Sweden. The trains hauled by Iore-class locomotives from Riksgränsen on the national border to the Port of Narvik use only a fifth of the power they regenerate. The regenerated energy is sufficient to power the empty trains back up to the national border. The more ore the train can carry the more electricity that can be generated on the downhill trip. Excess energy from the railway is pumped into the power grid to supply homes and businesses in the region, the railway is a net generator of electricity.

Additional reading:

You are referring to TheSimpliFire's answer, part 3:

"3.The transportation paradox, where transportation can be more costly when the number of demands/supplies is reduced.

References

[3] Charnes, A., Klingman, D. (1971). The more-for-less paradox in the distribution model, Cah. Cent. d’Etud. Rech. Oper. 13:11–22."

That paper is referenced in another paper: "Paradox Algorithm in Application of a Linear Transportation Problem" (Jan 5 2014), by Osuji George A., Opara Jude, Nwobi Anderson C., Onyeze Vitus, and Iheagwara Andrew I.:

Some helpful references mentioned there are:

"... Charnes and Klingman (1971), Szwarc (1973), Adlakha and Kowalski (1998) and Storoy (2007) considered the paradoxical transportation problem. Gupta et al (1993) ... Joshi and Gupta (2010) ...".

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