Abstract
Industrial production systems consist of components that gradually wear off. This chapter introduces the Weighted Uncapacitated Planned Maintenance Problem (WUPMP). The maintenance activities cover a set of periods before they must be executed again. The trade-off results from the cost structure. The strongly \(\mathcal{N}\mathcal{P}\)-hard WUPMP has the single-assignment property and the polytope is quasi-integral. A generalized period covering constraint has an integral polytope. The computational complexity of several problem variants is resolved. The WUPMP is solvable time O(n ⋅ T n+1 ⋅ 2n) and strongly polynomially solvable if the number of maintenance activities is a constant. Other optimal, strongly polynomial algorithms to different problem variants are provided. The WUPMP is a generalization of Uncapacitated Facility Location Problem but a special problem variant of the Uncapacitated Network Design Problem and of the Set Partitioning Problem. Structural insights for the Capacitated Planned Maintenance Problem (CPMP) are provided.
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Notes
- 1.
- 2.
- 3.
See Footnote 10 on page 32 for the UFLP and the UNDP.
- 4.
This holds because of the following well-known argument (Schrijver 1998, pp. 31–32): If the system A ⋅ x = b of rational linear equations has a solution, it has one of a size polynomially bounded by the sizes of A and b. It follows that the solution can be represented by a rational number where the numerator and the denominator have a polynomial number of digits in the input length.
- 5.
The least common denominator of some fractions is the least common multiple of the denominators of these fractions. The least common multiple is calculated in polynomial time with the greatest common divisor, which is calculated in polynomial time by Euclid’s algorithm (Cormen et al. 2001, pp. 856–862; Wolfart 2011, pp. 3–4).
- 6.
Note that the facility-transformation from Sect. 5.3.3.2 is substantially different.
- 7.
Improved results such as O(n) for T ∈ O(1) and O(T ⋅ logT) for n = 1 are found in Kuschel and Bock (2016).
- 8.
Given a directed graph G(V, E) with a node set V, a set E of directed arcs, arc costs \(f_{ij}\;\forall (i,j) \in E\) and two nodes s and t with s, t ∈ V. Find a path of minimal costs in G(V, E) that leads from the node s to t. The Shortest Path Problem (SPP) in graphs without negative cycles is solvable in polynomial time (Ahuja et al. 1993, pp. 121–123, pp. 154–157) but it is \(\mathcal{N}\mathcal{P}\)-complete for general cost structures (Garey and Johnson 1979, p. 213).
- 9.
The binomial coefficient \(\binom{n}{k}\) yields the number of distinct subsets with k elements that can be formed from a set with n elements. It holds that \(\binom{n}{k} = \frac{n!} {k!\cdot (n-k)!}\) if k ≤ n (otherwise, 0) (Mood et al. 1974, p. 529). A special case of the binomial theorem is \(\sum _{k=0}^{n}\binom{n}{k} = 2^{n}\) (Mood et al. 1974, p. 530).
- 10.
The reaching algorithm uses a breadth-first search and is outlined as follows. Given a graph G(V, E) that is cycle-free and let c ab be the arc weights of (a, b) ∈ E. Denote the shortest path from the source node to the node b as SP b . Set all nodes as unlabeled. In an iteration, first select a node a that has the smallest SP a among all non-labeled nodes. This is ensured by topologically ordering the nodes and selecting them in an increasing order of subscripts. Second, examine whether the shortest path to all its successors can be improved. If this is the case because SP a + c ab < SP b holds for a successor b, then update SP b and label the node a. The algorithm terminates when all nodes are labeled and returns shortest paths from the source node to all nodes. The reaching algorithm runs in time O( | V | + | E | ).
- 11.
Addendum: Another pruning criterion reduces the computational effort of the loop to for b = max{a ∗, Pre a +π i } + 1 to \(\min \left \{a +\pi _{i},T + 1\right \}\) do
- 12.
Another argument for the redundancy of \(x_{it} \leq 1\;\forall t\) is presented in Remark 6.1.
- 13.
It is straightforward to show via induction that Lemma 4.15 yields an integer solution if \(c_{it} \in \mathbb{N}\;\forall t\).
- 14.
Algorithm 6.5 requires the optimal slack of (4.43).
- 15.
\(\bar{\pi }_{i} = a \cdot \pi _{1}\) with \(a \in \{ 1,\ldots,\big\lfloor \frac{\pi _{i}} {\pi _{1}} \big\rfloor \}\) satisfies \(\bar{\pi }_{i} \leq \pi _{i}\). Note that \(\bar{\pi }_{1} =\pi _{1}\).
- 16.
The instances were generated as presented in Sect. 7.1 and solved with the Simplex algorithm.
- 17.
See Footnote 4 on page 74.
- 18.
The respective Linear Assignment Problem is polynomially solvable (Papadimitriou and Steiglitz 1998, pp. 247–248).
- 19.
Similar results are known for other combinatorial optimization problems. For the Uncapacitated Lot-Sizing Problem valid inequalities exist that yield a tight formulation such that the polyhedron of the LP relaxation coincides with the convex hull of the MIP (Pochet and Wolsey 1994; Vyve and Wolsey 2006).
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Kuschel, T. (2017). The Weighted Uncapacitated Planned Maintenance Problem. In: Capacitated Planned Maintenance. Lecture Notes in Economics and Mathematical Systems, vol 686. Springer, Cham. https://doi.org/10.1007/978-3-319-40289-5_4
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