In model 1 (Challenge 1), our locating strategy is about to miximize the market share with the least number of new shops opened. That is, only consider the marginal influence of the shops.

In model 2 (Challenge 2), our locating strategy is about to maximize the market share within the given amount of budget, proportional to the k-junction. That is, the cost of opening cost is the only new factor.

Now let's do some math. Assume the cost for the n-th month is C(k). Generally C(k) is increasing due to inflation. Extend C(k) so that the domain the real number. (negative k implies that cost

**projected**in the past), also

*C(k)≦C([k])≦C(k+1)*for any integer k. Therefore it is monotonically increasing. Integrate once on C(k) from zero to infinity, which is diverging. Therefore the cost of running a shop under unlimited time requires infinity cost. Comparing with the cost of opening a new shop, which is a finite cost, the ratio of run cost is far (infinity-ly) more important than the set-up cost. So in case we should take the maximization of profit as the second aim after the maximization of market share. Indeed, what we have to do is build up a model to consider the lattice that supplies. To extend this model, the vector field of supplier and consumer can be built, which will be the ultimate model of this problem.

"We can't open the shop under an unlimited time. Why we have to consider this case?"

This assumption helps us to realize that the running cost is more important than the set-up cost in running a shop, which will be more important than the locating strategy to win the competition.

"If the maximization of profit is concerned, using multiple shops to block the Starbucks will be useless. Then all previous models will be demolished."

Yes this is the main problem of the simplified models.

8)Debug

So far we have finished the analysis of this problem. You may ask that why we finish this problem with just some simple maths, but that is the truth. In fact, our proof about the locating strategy is incomplete, but in the sense of considering every simple case, it works. Combining together, it will works too.

Some advanced tools like matrix and programmiung is highly appreciated in the competition but we tried to finish the problem by simple mathematics, we will be proud of this.

The full solution from different team will be released in 1 month and I'm looking for this and a further analysis.

**End of SIMC report: The Challenge**

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