A question about mathematical modeling

[JYJUM]

 

This question is about combine the two contraints to become a mathematical modeling: 

(1) If the number of possible routes is less than 4, probability (a) will be used. All the routes will be considered and the travel time for each route will be calculated. However, if there are more or equal than 4 routes, probability (b) will be used in which 50 percent of the number of routes will be considered and calculated. After determining the number of routes to be considered, step (2) will take place.

Probability (a) P_i=number of routes 

Probability (b) P_i=0.5 * number of routes

 

(2) Let T be the set of possible routes. The routes will be randomly chosen based on probability above. X_(k-1) refers to the last route that is randomly chosen. In the statement, we use Xi≤ Xj to compare their traveling time. The route with the lowest traveling time, Xi, will be the candidate for optimal route.

$$T'=  \left\{  \left( X_0,X_1,...,X_ \left( K-1\right)  \right)| X_i \epsilon T,  \forall 0≤i≤K-1, X_i < X_j,  \forall i < j \right\} $$

 

 

How to combine both (1) and (2) into an appropriate mathematical modeling (equation)?

 

 

Edited October 30, 2015 by JYJUM
equation format