$(10$ points$)$ Consider a different version of the Traveling Salesman Problem in which the salesman collects a prize $w_k$ in every city $k$ that he visits and pays a penalty $C_l$ to every city $l$ that he fails to visit. Suppose that the cost of traveling from city $i$ to city $j$ is $t_{ij}$ and there is a lower bound $W$ on the amount of prize to be collected. Formulate the problem of finding a tour that minimizes the sum of travel costs and penalties of the salesman.