The Capacitated Traveling Salesman Problem with Pickups and Deliveries on a Tree

The Capacitated Traveling Salesman Problem with Pickups and Deliveries (CTSP-PD)[1] can be defined on an undirected graph T=(V,E), where V is a set of n vertices and E is a set of edges. A nonnegative weight d(e) is associated with each edge e∈ E to indicate its length. Each vertex is either a pickup point, a delivery point, or a transient point. At each pickup point is a load of unit size that can be shipped to any delivery point which requests a load of unit size. Hence we can use a(v)=1,0,–1 to indicate v to be a pickup, a transient, or a delivery point, and a(v) is referred to as the volume of v. The total volumes for pickups and for deliveries are usually assumed to be balanced, i.e., $\sum_{v\in {\it V}}{\it a}({\it v})=0$, which implies that all loads in pickup points must be shipped to delivery points [1]. Among V, one particular vertex r ∈ V is designated as a depot, at which a vehicle of limited capacity of k ≥ 1 starts and ends. The problem aims to determine a minimum length feasible route that picks up and delivers all loads without violating the vehicle capacity.