by Lin Yangchen


System dynamics models were first developed at the Massachusetts Institute of Technology in the 1950s to diagnose emergent phenomena in real-world complex systems that had no obvious explanation.

These mechanistic models transcend statistical correlations to capture causal feedback loops and aspects of human behaviour that are more than the sum of their parts. They can help the postal industry optimize its policies and logistics, from the chief executive trying to formulate a strategy for making snail mail a sustainable business, to the HR manager assessing the potential impact of layoffs, to the operations division tasked with ensuring a steady supply of stamps at the post office. Such models can also help postal historians understand postal dynamics in the past and even simulate alternative scenarios.




Command Centre
the author's experimental model of postal system dynamics in Vensim.


Stamp printing


The stamp resupply rate is given by a modification of Verhulst’s classic model of self-limiting population growth, dN/dt = r(1-N/K), where the Malthusian parameter r denotes the maximal rate of stamp production, N is the instantaneous stock of stamps in the post office and K (Kapazitätsgrenze) is the stamp storage capacity of the post office.

When the stock is very low, it gets replenished slowly because postal authorities are reluctant to put in big orders due to inertia and/or market uncertainty. When the stock is near capacity, the resupply rate levels off again as authorities stop ordering stamps.

If for some reason the number of stamps overshoots stockage capacity, the function will return a negative resupply rate (not shown). Surplus stamps are destroyed or become otherwise incapable of generating revenue. The model is programmed to not refund the post office for the extra stamps ordered.

Consumer appetite

Your appetite for sending a letter depends on whether the postage rate seems reasonable, among other factors. In the model, postage rate influences customer appetite via an inverse logistic function of the form a/(1 + bcx). Parameter a is the maximal potential market demand, or the total weight of mail people will potentially send, b moves the curve left or right, and c controls the direction and gradient of the slope.


Above, Vensim lookup for consumer appetite (output, y-axis) as a function of postage rate (input, x-axis). This is easily adjusted to fit data. But one has to balance the tradeoff between fitting it too closely and fitting it too loosely. Fit it too closely and you capture all the noise together with the signal. Oversimplify the fit and you fail to capture the underlying structure of the system. Either way the model loses predictive power.

When the postage rate tends to 0, people are very happy but the quantity of mail posted converges to an asymptote because there’s a limit to how many letters people can physically send or want to send. When the rate starts creeping up, people don’t react too strongly at first because they still want to send letters and they forgive the post office for struggling with rising costs. But if the situation continues to worsen, consumer appetite goes into a steep dive. When the rate finally reaches stratospheric levels, the curve levels out again. It is programmed to hit bottom when the rate has ballooned to about four times what people perceive as reasonable. The few remaining customers desperately need to send a letter, whatever it takes.

The model exhibits switching behaviour when one progressively increases the postage rate. Profits increase with postage rate until a point where business lost through overcharging customers nullifies any gains from price hikes.

Mail delivery

The total amount of mail sent by customers is affected by consumer appetite and may be constrained by the stock of stamps available at the post office. Some letters will not be delivered on time if mail volume exceeds the workload of the available postmen. Delayed letters are bunched with the new batch of mail for attempted delivery in the next time step.

Philatelic business

Postal authorities issue stamps to mark special occasions, serve as the country’s ambassadors to the world, and make money from philatelists. The model assumes there is no negative feedback between philatelic appetite and the frequency of new stamp issues. No matter how often the post office issues new stamps, philatelists will buy them because of the need for a complete collection.

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The model shown above is only a proof of concept. The structure and level of detail of a working model will depend on the problem at hand. For example, it may well be necessary to explicitly incorporate multiple postage rates differing by destination and mode of transport. The dynamics of multiple post offices might also need scrutiny. Suppose those POs serving remote areas run at a loss, but they could be covered by their busy counterparts in the urban centres.

The model will have to be validated using empirical data and expert knowledge from postal authorities, postal historians, philatelists and all other stakeholders. Finally, sensitivity analysis using stochastic Monte Carlo simulations of the model will infuse the conclusions with statistical rigour.

References


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