SIMC 2010 Report (2): Challenge 2

The aim of tis problem is to maximize the market share within a given budget relating to the position of the shops. Note that the cost is not related to the marginal influence, but the n-junction road,
3)Basic Strategy
Of course we should we most of the budget.
Proof: setting up a new shop shop increase market, so putting more is better.
Another important strategy is that we should bulid more shops in the 2-junction road.
Proof: it's the matter of blocking others and obtaining more "weak" controlling area.
Why obtaining more "weak" area is better? A basic frame is that the area of "strong controlling" is almost the same for every shops. It means that if you bulid a shop in their "strong controlling" area, you will get a little share. Then the "weak" area will be relatively bigger. Now when the number of shops is equal, the brand who controls more "weak" area wins. When three shops nearby forms a close triangle, the area within these three shops is somehow likely a "strong controlled" area so that a large area can be obtained.
comparing the 4-2, buliding two with 2 units of cost is obviously better than buliding one shop in the 4-junction road (even two besides the junction must be better). But how about the 3-2 comparason? It seems that the case of 3-2 is much more complicated since there're too many junction to be drawn and the stright road is undetermined. However we simplify the case for putting two at the end of the junction and the last one in the end of one of the junction. Then the share is also miximized.
Therefore our conclusion about the strategy will be maximized when we bulid as much at 2-junction as possible.
4)Application
Now we go the the core idea of the problem. How can we apply the strategy?
our solution is putting them in front of Starbucks to block them, and we get 27.5/47. however we can see that the optimal solution will be 28/47 where two blocks starbucks below and one above the 4-junction road.
Now we realize that the solution aimed to rule the whole street in the bottom. It seems that it's still the same target: to block Starbucks and acquire a large area. But what's the difference?
The difference is that we are trying to block Starbucks in their "strong" controlling area, and to obtain the whole street in the center. In their solution they tried to obtian the weak area.
Note that when we block starbucks, we can only block one of their way, but they still control the other ways. But acquiring the whole "weak" area is different from that. It will be impossible for Starbucks to go inside the "controlling triangle" such that we can surely win the competition.
Locating near the 40junction road aims at acquiring more market share from different road, then they will have the higher chance the control a "weaker" area. In our solution,
Recall the two strategy : near to the junction road and vlocking others, they have different advantage so how to balance them and maximize the share? That will be the key idea of our report.
Anyway I think this problem is just a further application of Q1, but not very hard if we have solved Q1 in a good way (or many infinite times of trial and of course error?)