Vehicle scheduling on a tree with release and handling times
- Kyoto Univ. (Japan)
Let T = (V, E, v{sub 0}) be a rooted tree, where V is a set of n vertices, E is a set of edges and v{sub 0} {element_of} V is the root. The travel times d(v{sub i}, v{sub j}) and (v{sub j}, v{sub i}) are associated with each edge (v{sub i}, v{sub j}) {element_of} E, and a job, which is also denoted as v{sub i} is located at each vertex v{sub i}. Each job v{sub i} has release time r(v{sub i}) and handling time h(v{sub i}). The TREE-VSP (Vehicle Scheduling Problem on a tree) asks to find a routing schedule of the vehicle such that it starts from root v{sub 0}. The processing a job v{sub i} cannot be started before its release time t = r(v{sub i}) (hence the vehicle may have to wait if it arrives at v{sub i} too early) and takes h(v{sub i}) time units once its processing has been started (no preemption is allowed). The objective is to find a schedule that minimizes the completion time (i.e., the time to return to v{sub 0} after processing all jobs). We first prove that TREE-VSP is strongly NP-hard. Then we shown that TREE-VSP with depth-first routing constraint can be exactly solved in O({Sigma}{sub j=1}{sup n} log{sub 2} w{sub j}) time, yielding a bound for the 0-1 knapsack problem. The authors show that this relaxation can also be used to generate valid inequalities (cutting planes) for general 0-1 integer programming problem.s The method uses the 0-1 knapsack problems obtained by dropping all but one of the constraints of the 0-1 integer programming problem, but unlike the case of minimal cover inequalities the separation problem for the resulting class of inequalities can be solved in polynomial time (it reduces to the separation problem for a bounded-integer sequential knapsack problem).
- OSTI ID:
- 36149
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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