For the following problem, provide the mathematical programming formulation that would find

the optimal solution to it$.$ Clearly, indicate all set$($s$)(6$ points$),$ parameter$($s$)(6$ points$),$ variable$($s$)$

$(6$ points$),$ objective function $(6$ points$),$ and constraint$($s$)(6$ points$).$

You are a door$-$to$-$door vendor of cookies, and you have a list of clients denoted by the set C$.$ Each client i

has a probability $\backslash$alpha it of purchasing your product, which depends on both the time t and the specific client.

The quantity that each client i will buy is denoted by $\backslash$beta i$,$ which is client$-$dependent but not time$-$dependent.

Your objective is to maximize the expected value of sales during your shift.

However, there are several constraints to consider. Your workday lasts $8$ hours, and you can only visit two

clients during each hour. Additionally, you can only visit each client once. The city is divided into two

zones, Zone A and Zone B$.$ If you choose to visit a client in Zone A$,$ you cannot visit a client in Zone B

during the same hour.Furthermore, there is a subset of clients M $$ C who can only be visited during the

afternoon, after t $>4).$ You must ensure that these clients are visited at the appropriate times. Lastly, to

cover transportation costs, the expected sales value from the clients visited during each hour must exceed

a minimum threshold $\backslash$gamma t$.$

$3$