Abstract
This chapter analyzes the solvability of the strongly \(\mathcal{N}\mathcal{P}\)-hard Capacitated Planned Maintenance Problem (CPMP). The computational complexity of several problem variants is resolved. Finding a feasible solution is already strongly \(\mathcal{N}\mathcal{P}\,\) complete and the CPMP is binary \(\mathcal{N}\mathcal{P}\)-hard for two periods. The CPMP is solvable in time \(O(\min \{n \cdot \frac{\log T} {\sqrt{T}} \cdot \bar{ r}^{max\;T} \cdot 4^{T},n \cdot T^{n+1} \cdot 2^{n}\})\). Therefore, the CPMP is pseudo-polynomially solvable if the number of periods is a constant and strongly polynomially solvable if either the number of maintenance activities is a constant or if the number of periods and the maximal capacity over all periods are constants. Other optimal, strongly polynomial and pseudo-polynomial algorithms to different problem variants are provided. Valid inequalities and polyhedral properties are presented. The relative strength and the computational complexity of 99 lower bounds is evaluated. The lower bounds are derived from Lagrangean relaxation, decomposition and by neglecting constraints completely.
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Notes
- 1.
- 2.
For I tc ′ holds \(\sum _{i\in I_{tc}^{{\prime}}}r_{i} >\bar{ r}_{t}\). Note that the cover I tc ′ is minimal if \(\sum _{j\in I_{tc}^{{\prime}}}r_{j} - r_{i} \leq \bar{ r}_{t}\;\forall i \in I_{tc}^{{\prime}}\) holds. According to (3.53), the extension of a set I is \(I \cup \left \{\left.i \in \left \{1,\ldots,n\right \}\setminus I\right \vert r_{i} \geq \max _{k\in I}r_{k}\right \}\).
- 3.
For I tc ′ ′ holds \(\sum _{i\in I_{tc}^{{\prime\prime}}}r_{i} >\bar{ r}_{t}\) and \(r_{i} + r_{j} >\bar{ r}_{t}\;\forall i,j \in I_{tc}^{{\prime\prime}}\) with i ≠ j. Note that all extended clique inequalities are obtained with Algorithm 3.5. See (O) and (3.53) for the definition of the extension.
- 4.
\(UB \in \mathbb{N}\) always exists because of Remark 2.1. If no upper bound UB can be determined, set UB = T because it is an upper bound on any feasible solution if one exists.
- 5.
A valid value for X iq is obtained with (A) as follows:
$$\displaystyle{ \sum _{t=1\wedge t\notin Q_{q}}^{T}x_{ it} =\sum _{ t=1}^{Q_{q}^{min}-1}x_{ it}+\sum _{t=Q_{q}^{max}+1}^{T}x_{ it} \geq \left \lfloor \frac{Q_{q}^{min} - 1} {\pi _{i}} \right \rfloor +\left \lfloor \frac{T - Q_{q}^{max}} {\pi _{i}} \right \rfloor = X_{iq}. }$$ - 6.
A valid value for Y is: Sort the capacity \((\bar{r}_{1},\ldots,\bar{r}_{T})\) according to non-increasing values and sum up the first UB elements \(\bar{r}_{t_{1}} \geq \ldots \geq \bar{ r}_{t_{UB}}\). Clearly, \(Y =\sum _{ h=1}^{UB}\bar{r}_{t_{h}} \geq \sum _{t=1}^{T}\bar{r}_{t} \cdot y_{t}^{{\ast}}\) holds for an optimal solution (x ∗, y ∗).
- 7.
Disaggregated formulations might have several advantages. The aggregated formulation replaces (V) by the linear combination of (V) that is \(\sum _{i=1}^{n}x_{it} \leq n \cdot y_{t}\;\forall t = 1,\ldots,T\). However, the LP relaxation yields a weaker lower bound than the disaggregated formulation with (V) does (cf. the UFLP (Krarup and Pruzan 1983)). Furthermore, disaggregated formulations might yield integral polytopes (cf. the Standardization Problem (Domschke and Wagner 2005)).
- 8.
Some results are improved in Kuschel and Bock (2016).
- 9.
- 10.
A trivial asymptotic upper bound is that the decision problem has to be solved at most O(2T) times. Since \(\frac{2^{T}} {\sqrt{T}} \cdot \log _{a}T \leq 2^{T}\) holds already for \(a \geq e^{\sqrt{4\cdot e^{-2}} } \approx 2.087\), the applied approach has an improved asymptotic running time.
- 11.
The function f(a, b, c) = c a ⋅ b −c⋅ with \(a,b,c \in \mathbb{R}\) has one maximum at \(c^{{\ast}} = \frac{a} {\ln b}\). For \(f(\frac{3} {2}, 4,c) = c^{\frac{3} {2} } \cdot 4^{-c}\) follows c ∗ ≈ 1. 082 and \(f(\frac{3} {2}, 4,c^{{\ast}}) \approx 0.251\). For \(f(\frac{3} {2}, 2,c) = c^{\frac{3} {2} } \cdot 2^{-c}\) holds c ∗ ≈ 2. 164 and \(f(\frac{3} {2}, 2,c^{{\ast}}) \approx 0.710\).
- 12.
Using a word-RAM implementation, the SSP is even solvable in time \(O(\frac{n\cdot \bar{r}_{1}} {\log \bar{r}_{1}} )\) (Kellerer et al. 2004, pp. 76–79).
- 13.
- 14.
In the well-known and so called “stars and bars” principle, n indistinguishable balls are assigned to π 1 different boxes. An assignment is encoded as a binary string comprising n digits with a value of “1” for a ball and π 1 − 1 digits with a value of “0” representing a box. Hence, there are \(\binom{n +\pi _{1} - 1}{n}\) combinations.
- 15.
This assumption is analogous to Cornuejols et al. (1991), who investigate relaxations of the related Capacitated Facility Location Problem.
- 16.
In Z(P)(C), the Lagrangean relaxation is a Shared Fixed-Cost Problem (Rhys 1970).
- 17.
In Z(C)(V), the Lagrangean relaxation can be reformulated with Remark 4.2 to yield n SPPs.
- 18.
This intuitive approach that does not hold in general. One counter example is a production scheduling problem from a tile factory presented in Guignard (2003). A Lagrangean relaxation of the demand constraint leads to a Lagrangean relaxation that decomposes because of the integer linearization property. The subproblems consist of continuous KPs and SPPs. Although all subproblems have the integrality property, the Lagrangean relaxation does not have it and the obtained lower bound is tighter than the LP relaxation.
- 19.
See Footnote 4 on page 82.
- 20.
Note that Theorem 5.22 shows that there is no dominance between Z(C) and Z(P). This result is important for the Lagrangean Hybrid Heuristic that is presented in Sect. 6.3 Some results correspond to similar results for the related Single Source Capacitated Facility Location Problem (SSCFLP) (Klose and Drexl 2005; Klose 2001, pp. 218–222).
- 21.
In this problem, the variable y t is coupled via (C) with the variables x it such that the value of y t decides on an upper bound of all variables x it of the same index t. Specifically, y t is binary. If y t = 0, then x it = 0 for all i. Conversely, y t = 1 and (x 1t , …, x nt ) represents the solution of a subproblem. The optimal objective function value z t of the subproblem zt is the contribution of y t = 1 to the objective function value and the objective function is reformulated with z t ⋅ y t .
- 22.
Using a word-RAM implementation, the tth KP is even solvable in time \(O(\frac{n\cdot m} {\log m} )\) with \(m =\max \{\bar{ r}_{t},z_{t}\}\) (Kellerer et al. 2004, pp. 131–136).
- 23.
Given a graph G(V, E) and a positive integer K ≤ | V | . Exists a node cover of size K or less for G(V, E) that is a subset V ′ ⊆ V with | V ′ | ≤ K such that for each edge (a, b) ∈ E at least one node a or b belongs to V ′? VERTEX COVER is strongly \(\mathcal{N}\mathcal{P}\)-complete (Garey and Johnson 1979, p. 190).
- 24.
- 25.
Given a set of facilities with fixed costs and a maximal supply. Given a set of customers with transportation costs and a demand. Serve the customers from the facilities such that the demand of all customers is satisfied and the capacity of each facility is not exceeded. The CFLP is strongly \(\mathcal{N}\mathcal{P}\)-hard (Cornuejols et al. 1991), which is proven with a polynomial reduction with the strongly \(\mathcal{N}\mathcal{P}\)-complete 3-DIMENSIONAL MATCHING (Cornuejols et al. 1991).
- 26.
Given a set M ⊆ W × X × Y with W, X and Y being disjoint and consisting of an equal number of m elements. Does M contain a matching, that is a subset M ′ ⊆ M such that | M ′ | = m and no two elements of M ′ agree in any coordinate? 3-DIMENSIONAL MATCHING is strongly \(\mathcal{N}\mathcal{P}\)-complete (Garey and Johnson 1979, p. 221).
- 27.
The following discussion does not include Z(X)(Y), Z(V)(X)(Y), Z(P)(X)(Y)+(Q), which are LPs, Z(P) because it is equivalent to solving T KPs (see Lemma 5.23), Z(P)/(C) because it is solvable with Z(P) as well as Z(C) and Z(C) because it is a WUPMP with \(c_{it} = 0\;\forall i; t\), which is a particular UFLP (Sect. 4.3).
- 28.
Using the definitions of the UNDP, let an upper bound restrict the total flow of all commodities on an arc (a, b) ∈ E and remove the variables for the fixed costs. Find a flow of minimal cost for all commodities through the graph such that the flow is balanced in every node, the arc capacity is never exceeded and the flow is split (i.e., integral). Using a polynomial reduction with the strongly \(\mathcal{N}\mathcal{P}\)-complete 3-PARTITION, the MCNFP is strongly \(\mathcal{N}\mathcal{P}\)-hard.
- 29.
Using the definitions of the UNDP, let an upper bound restrict the total flow of all commodities on an arc (a, b) ∈ E. Find a flow of minimal cost for all commodities through the graph such that the flow is balanced in every node and the arc capacity is never exceeded. Using a polynomial reduction with the strongly \(\mathcal{N}\mathcal{P}\)-complete STEINER TREE IN GRAPHS (Garey and Johnson 1979, p. 208), the FCNDP is strongly \(\mathcal{N}\mathcal{P}\)-hard (Magnanti and Wong 1984).
- 30.
- 31.
In this transformation, a so called customer (facility) is either a regular or a dummy customer (facility).
- 32.
Given a set of facilities with fixed costs and a maximal supply. Given a set of customers with transportation costs and a demand. Find the cost minimal assignment of customers to facilities such that the demand of all customers is satisfied, the capacity of each facility is not exceeded and each customer is assigned to exactly one facility. A polynomial reduction with the strongly \(\mathcal{N}\mathcal{P}\)-complete 3-PARTITION shows that SSCFLP is strongly \(\mathcal{N}\mathcal{P}\)-hard (Klose and Drexl 2005) with Martello and Toth (1990, p. 8).
- 33.
Cf. Anti-CFLP presented in Klose (2001, pp. 238–239).
- 34.
The Generalized Bin Packing Problem (Hung and Brown 1978; Lewis and Parker 1982) is a special case of the SSCFLP with \(f_{t} = 1\;\forall t\) and \(c_{it} = 0\;\forall i; t\). Strong \(\mathcal{N}\mathcal{P}\)-hardness is shown by a polynomial reduction of the strongly \(\mathcal{N}\mathcal{P}\)-complete BIN PACKING (Garey and Johnson 1979, p. 226).
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Kuschel, T. (2017). Analyzing the Solvability of the Capacitated 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_5
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