In challenge 3, we are required to create a few models considering real factors.

5)Potential Population Density

Recall the assumption that have been made in Challenge 1,2: They go to the closest shop only; continous and even distribution of potential population and a given map of street.

What can you see that assumption different from the reality?

Going to the closest shop implies that the potential customers are in stationery. In the real society, the population forms a vector field (on a 2D/3D coordinate, whatever), where the peference of cumtomers changes from time to time. For example, we won't go to starbucks in the early morning. Therefore a multiplyer t(x|24) is set to be a cycle of population density in different time.

Consider the given potential population density at a given instant. The vector field of the customers can be ignored since the vector field indicates the rate of change in density and indicates the density in the next moment only. At a given moment there're no population flow. Now the question is: it is impossible for us to create a model and a vector field to compare the marginal influence at every instant. How can we create a easy and reasonable model?

Our solutions will be:

i)A simple model: Consider a street, apart from the closest Starbucks and Coffee Bean from a and b units, then the share of Coffee Bean will be a/(a+b) units. The idea comes from that the closer shops will have a higher probability of selling services to the closer customers. The assumption is that the population is flowing, but there will be a higher chance to walk along the closer shops.

Consider the model on a 2D- plane: (0,0)to (4,0), (0,2) to (4,2), (0,0) to (0,2), (2,0) to (2,2), (4,0) to (4,2), total 14 untis. Assume that the Starbucks located at (2,2) while the Coffee Bean is located at (0,0).

It seems that the location of Starbucks will be much better than the Coffee Bean. Actually in the original model we will get the share as 5/15 units at the proportion of 5:9. But it is quite weird as they are close to each other. Then how can we improve the result?

In this model, the new share of Coffee Bean will be 6/14 (6:8) which is more reasonable.

We can see that in this model the strategy of "blocking others" is not effective anymore since two close shop will have similar shraes. The new locating strategy is main locating them in the junction point and near to the centre of map which is more close to reality.

Unfinished and we'll discuss a complicated model considering the population flow tomorrow.

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