APPROXIMATION SCHEMES FOR SCHEDULING JOBS WITH CHAIN PRECEDENCE CONSTRAINTS

We study the problem of scheduling jobs with precedence constraints and present two polynomial time approximation schemes for it. The first one is for the case when the machines are identical and the precedence graph partitions the jobs into groups of fixed size. As an interesting particular case, this yields a polynomial time approximation scheme for scheduling jobs with chain or tree precedence constraints on identical machines, when each chain or tree is of constant size. The second approximation scheme is for scheduling jobs with arbitrarily long chain precedence constraints on uniformly related machines when the number of machines is fixed.

A Study on Optimal Inspection Strategies of Reliability Systems with Parallel-Chain Precedence Constraints under Failure States

Testability is a new and important technology as reliability and maintainability. A lot of literature has discussed the optimal problems of testability in recent years. In Chiu’s paper, a new kind of precedence constraints, parallel-chain precedence constraints, was presented, which is a common constraint in many products and systems. He discussed the optimal inspection strategies to determine the states (good/failed) of some typical reliability systems with parallel-chain precedence constraints. But in reality we encounter more problems with fault-testing models, and how to locate all the failed components with minimal cost is our interest. Suppose that a coherent system has n independent components, each of which has a known prior probability of fault. The true state of each component can be obtained by inspecting it with some cost. This paper discusses the optimal inspection strategies of the coherent system with parallel-chain precedence constraints when the system fails, which can minimize the total expected inspection cost. If the system is known to have failed, there are two situations: one is that the number of the failed components in the system is unknown, which is a common phenomenon; another is that the number of the failed components is known. In addition, the coherent systems concerned in this paper are all typical reliability systems, such as series, parallel, parallel-series, series-parallel and k -out-of- n systems, and all of them are subject to parallel-chain precedence constraints.