This work is partially supported by the National Science Foundation of China with grant numbers 717110111013. The theorems can be used in a pre-processing step to increase the density of the precedence networks.Ī DP algorithm has been proposed for solving the Acknowledgments We have proved three precedence theorems that can identify dominant precedence constraints, which are respected by at least one optimal solution. We have developed a linear formulation for the problem via Dantzig-Wolfe decomposition of a compact nonlinear formulation. ![]() In this article, we have studied the sequential testing problem for n:n systems. (2013), which was generated by the random network generator RanGen (Demeulemeester, Vanhoucke, & Herroelen, 2003), and which is available online 1 Conclusions For solving the linear programs, we use CPLEX version 12.6.3. A memory limit of 8GB and a time limit of two hours are applied. RLF is solved iteratively, and at each iteration, the pricing problem determines whether a new variable exists that if added to the restricted master problem, will Experimental setupĪll the computational experiments are performed on a Dell Latitude E7250 laptop with an Intel Core i5-5200U (2.20GHz) processor. It is called restricted (RLF) when it includes only a subset of its variables (columns). Here, the linear relaxation of ILF, denoted by LF, is the master problem. The algorithm consists of a master and a pricing problem. This technique enables us to solve significantly larger instances (by Column generationĬolumn generation (CG) (Martin, 2012) is a technique for solving linear programs with a large number of variables. We use an improved version of the memory management technique in Rostami et al. (2013), but tuned specifically towards the n:n problem. The recursion of DPn (Section 4.1) is close to the recursion in Coolen et al. In the remainder of the text, we will refer to this algorithm as DPn. ![]() In this section, we propose a DP algorithm for solving the sequential testing problem for n:n systems. The condition is intuitive: for a series DP algorithm We know that when E = ∅, a sequence L = ( l 1, l 2, …, l n ) is optimal (Ünlüyurt, 2004) if c l 1 1 − p l 1 ≤ c l 2 1 − p l 2 ≤ ⋯ ≤ c l n 1 − p l n. Note that since every feasible solution satisfies the constraints in E, we have E⊆ D. ![]() Precedence theorems may help us identify some of the constraints in D. ![]() Given an instance G( N, E), let D be the set of all dominant constraints. A test can be performed only if all its predecessors are done, where j ∈ N is said to Precedence theoremsĪ precedence constraint ( i, j) ∈ N × N is said to be dominant iff there is at least one optimal solution with i before j. The precedence network of an instance of the problem can be visualized using an acyclic directed graph G( N, E), where the nodes represent the tests, and there is an arc from node i to j iff ( i, j) ∈ E. The set E is a partial order on N, representing the precedence constraints. We consider a set N of n tests (or inspections) with pre-specified testing costs c i ≥ 0, i ∈ N, and success probabilities p i ∈ . With these challenges in mind for preventive and corrective maintenance, the importance of sequential testing was highlighted by the U.S. The components need to be tested in a specific order to detect the state of the machine with minimum cost or time. The state of the machine (working or failing) depends on the state of its components. Section snippets Introduction and related workĪ multi-component machine, e.g., a missile radar set, that is inactive except during a state of military alert, undergoes periodic checkups.
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