Efficient approximation schemes for the maximum lateness minimization on a single machine with a fixed operator or machine non-availability interval

Abstract

In this paper we deal with the single machine scheduling problem with one non-availability interval to minimize the maximum lateness where jobs have positive tails. Two cases are considered. In the first one, the non-availability interval is due to the machine maintenance. In the second case, the non-availability interval is related to the operator who is organizing the execution of jobs on the machine. The contribution of this paper consists in an improved fully polynomial time approximation scheme (FPTAS) for the maintenance non-availability interval case and the elaboration of the first FPTAS for the operator non-availability interval case. The two FPTASs are strongly polynomial.

Appendices

Appendix 1: Proof of Theorem 5

First, we recall the idea of the dynamic programming algorithm which is necessary to explain the proof. Indeed, the problem can be optimally solved by applying the following dynamic programming algorithm APS. This algorithm generates iteratively some sets of states. At every iteration j, a set ${\mathcal {X}}_{j}$ composed of states is generated ($1\le j\le \overline{n}$). Each state $\left[ t,f\right]$ in ${\mathcal {X}}_{j}$ can be associated to a feasible schedule for the first j jobs. Variable t denotes the completion time of the last job scheduled before $T_{1}$ and f is the maximum lateness of the corresponding schedule. This algorithm can be described as follows:

Let UB be an upper bound on the optimal maximum lateness for problem $\left( \mathcal {T}^{\prime \prime }\right)$. If we add the restriction that for every state $\left[ t,f\right]$ the relation $f\le UB$ must hold, then the running time of APS can be bounded by $\overline{n}T_{1}UB$. Indeed, t and f are integers and at each step j, we have to create at most $T_{1}UB$ states to construct ${\mathcal {X}}_{j}$. Moreover, the complexity of APS is proportional to $\sum _{k=1}^{\overline{n}}\left| {\mathcal {X}} _{k}\right|$. In the remainder of the paper, Algorithm APS denotes the version of the dynamic programming algorithm by taking $UB=\mathcal { \varphi }_{JS}\left( \mathcal {T}^{\prime \prime }\right)$.

The main idea of the FPTAS is to remove a special part of the states generated by the algorithm. Therefore, the modified algorithm $APS_{\varepsilon }^{\prime }$ becomes faster and yields an approximate solution instead of the optimal schedule. The approach of modifying the execution of an exact algorithm to design FPTAS, was initially proposed by Ibarra and Kim for solving the knapsack problem (Ibarra and Kim 1975). It is noteworthy that during the last decades numerous combinatorial problems have been addressed by applying such an approach [for instance, see Sahni (1976) and Gens and Levner (1981)]. The worst-case analysis of our FPTAS is based on the comparison of the execution of algorithms APS and $APS_{\varepsilon }^{\prime }$ as described in the following lemma.

Lemma 13

For every state $\left[ t,f\right]$ in ${\mathcal {X}}_{j}$ there exists a state $\left[ t^{\#},f^{\#}\right]$ in ${\mathcal {X}}_{j}^{\#}$ such that:

$$\begin{aligned} t^{\#}\le t\le t^{\#}+j\delta _{2} \end{aligned}$$

(4)

and

$$\begin{aligned} f^{\#}\le f+j\max \{\delta _{1},\delta _{2}\} \end{aligned}$$

(5)

Proof

By induction on j. First, for $j=1$ we have ${\mathcal {X}}_{1}^{\#}={\mathcal {X}}_{1}$. Therefore, the statement is trivial.Now, assume that the statement holds true up to level $j-1$. Consider an arbitrary state $\left[ t,f\right]$ $\in$ ${\mathcal {X}}_{j}$. Algorithm APS introduces this state into ${\mathcal {X}}_{j}$ when job j is added to some feasible state for the first $j-1$ jobs. Let $\left[ t^{\prime },f^{\prime } \right]$ be the above feasible state. Two cases can be distinguished: either $\left[ t,f\right] =\left[ t^{\prime }+p_{j},\max \left\{ f^{\prime },t^{\prime }+p_{j}+q_{j}\right\} \right]$ or $\left[ t,f\right] =\left[ t^{\prime },\max \left\{ f^{\prime },T_{2}+\sum _{i=1}^{j}p_{i}-t^{\prime }+q_{j}\right\} \right]$ must hold. We will prove the statement for level j in the two cases.

1st case: $\left[ t,f\right] =\left[ t^{\prime }+p_{j},\max \left\{ f^{\prime },t^{\prime }+p_{j}+q_{j}\right\} \right]$ Since $\left[ t^{\prime },f^{\prime }\right] \in {\mathcal {X}}_{j-1}$, there exists $\left[ t^{\prime \#},f^{\prime \#}\right] \in {\mathcal {X}}_{j-1}^{\#}$ such that $t^{\prime \#}\le t^{\prime }\le t^{\prime \#}+\left( j-1\right) \delta _{2}$ and $f^{\prime \#}\le f^{\prime }+\left( j-1\right) \max \{\delta _{1},\delta _{2}\}$. Consequently, the state $\left[ t^{\prime \#}+p_{j},\max \left\{ f^{\prime \#},t^{\prime \#}+p_{j}+q_{j}\right\} \right]$ is generated by Algorithm $APS_{\varepsilon }^{\prime }$ at iteration j. However it may be removed when reducing the state subset. Let $\left[ \lambda ,\mu \right]$ be the state in ${\mathcal {X}}_{j}^{\#}$ that is in the same box as $\left[ t^{\prime \#}+p_{j}, \max \left\{ f^{\prime \#},t^{\prime \#}+p_{j}+q_{j}\right\} \right]$. Hence, we have:

$$\begin{aligned} \lambda \le t^{\prime \#}+p_{j}\le t^{\prime }+p_{j}=t \end{aligned}$$

(6)

Moreover,

$$\begin{aligned} \lambda +\delta _{2}\ge t^{\prime \#}+p_{j}\ge t^{\prime }-\left( j-1\right) \delta _{2}+p_{j}=t-\left( j-1\right) \delta _{2} \end{aligned}$$

which implies

$$\begin{aligned} t\le \lambda +j\delta _{2} \end{aligned}$$

(7)

Finally,

$$\begin{aligned} \mu&\le \max \left\{ f^{\prime \#},t^{\prime \#}+p_{j}+q_{j}\right\} +\delta _{1} \nonumber \\&\le \max \left\{ f^{\prime }+\left( j-1\right) \max \{\delta _{1},\delta _{2}\},t^{\prime }+p_{j}+q_{j}\right\} +\delta _{1} \nonumber \\&\le \max \left\{ f^{\prime },t^{\prime }+p_{j}+q_{j}\right\} +\left( j-1\right) \max \{\delta _{1},\delta _{2}\}+\delta _{1} \nonumber \\&<f+j\max \{\delta _{1},\delta _{2}\}. \end{aligned}$$

(8)

Consequently, the statement holds for level j in this case.

2nd case: $\left[ t,f\right] =\left[ t^{\prime },\max \left\{ f^{\prime },T_{2}+\sum _{i=1}^{j}p_{i}-t^{\prime }+q_{j}\right\} \right]$Since $\left[ t^{\prime },f^{\prime }\right] \in {\mathcal {X}}_{j-1}$, there exists $\left[ t^{\prime \#},f^{\prime \#}\right] \in {\mathcal {X}}_{j-1}^{\#}$ such that $t^{\prime \#}\le t^{\prime }\le t^{\prime \#}+\left( j-1\right) \delta _{2}$ and $f^{\prime \#}\le f^{\prime }+\left( j-1\right) \max \{\delta _{1},\delta _{2}\}$. Consequently, the state $\left[ t^{\prime \#},\max \left\{ f^{\prime \#},T_{2}+\sum _{i=1}^{j}p_{i}-t^{\prime \#}+q_{j}\right\} \right]$ is generated by algorithm $APS_{\varepsilon }^{\prime }$ at iteration j. However it may be removed when reducing the state subset. Let $\left[ \lambda ^{\prime },\mu ^{\prime }\right]$ be the state in ${\mathcal {X}}_{j}^{\#}$ that is in the same box as $[t^{\prime \#},\max \{f^{\prime \#},T_{2}+\ \sum _{i=1}^{j}p_{i}$ $-t^{\prime \#}+q_{j}\}]$. Hence, we have:

$$\begin{aligned} \lambda ^{\prime }\le t^{\prime \#}\le t^{\prime }=t \end{aligned}$$

(9)

Moreover,

$$\begin{aligned} \lambda ^{\prime }+\delta _{2}\ge t^{\prime \#}\ge t^{\prime }-\left( j-1\right) \delta _{2}=t-\left( j-1\right) \delta _{2} \end{aligned}$$

which implies

$$\begin{aligned} t\le \lambda +j\delta _{2} \end{aligned}$$

(10)

and

$$\begin{aligned} \mu ^{\prime }&\le \max \left\{ f^{\prime \#},T_{2}+\sum _{i=1}^{j}p_{i}-t^{\prime \#}+q_{j}\right\} +\delta _{1} \end{aligned}$$

(11)

$$\begin{aligned}&\le \max \left\{ f^{\prime }+\left( j-1\right) \max \{\delta _{1},\delta _{2}\},T_{2}+\sum _{i=1}^{j}p_{i}-t^{\prime }+\left( j-1\right) \delta _{2}+q_{j}\right\} +\delta _{1} \end{aligned}$$

(12)

$$\begin{aligned}&\le \max \left\{ f^{\prime },T_{2}+\sum _{i=1}^{j}p_{i}-t^{\prime }+q_{j}\right\} +\left( j-1\right) \max \{\delta _{1},\delta _{2}\}+\delta _{1} \end{aligned}$$

(13)

$$\begin{aligned}&\le f+j\max \{\delta _{1},\delta _{2}\}. \end{aligned}$$

(14)

In conclusion, the statement holds also for level k in the second case, and this completes our inductive proof. $\square$

Now, we give the proof of Eq. (2) in Theorem 5. By definition, the optimal solution can be associated to a state $\left[ t^{*},f^{*}\right]$ in ${\mathcal {X}}_{\overline{n}}$. From Lemma 13, there exists a state $\left[ t^{\#},f^{\#}\right]$ in ${\mathcal {X}}_{\overline{n}}^{\#}$ such that:

$$\begin{aligned} f^{\#}&\le f^{*}+\overline{n}\max \{\delta _{1},\delta _{2}\} \nonumber \\&=f^{*}+\overline{n}\max \left\{ \frac{\mathcal {\varphi }_{JS}\left( {\mathcal {I}} ^{\prime \prime }\right) }{\omega _{1}},\frac{T_{1}}{\omega _{2}}\right\} \nonumber \\&=f^{*}+n\max \{\frac{\mathcal {\varphi }_{JS}\left( {\mathcal {I}} ^{\prime \prime }\right) }{\left\lceil \frac{2\overline{n}}{\varepsilon }\right\rceil }, \frac{T_{1}}{\left\lceil \frac{\overline{n}}{\varepsilon }\right\rceil }\} \nonumber \\&\le f^{*}+\max \left\{ \varepsilon \frac{\mathcal {\varphi }_{JS}\left( \mathcal { I}^{\prime \prime }\right) }{2},\varepsilon T_{1}\right\} =\left( 1+\varepsilon \right) \mathcal {\varphi }^{*}\left( {\mathcal {I}} ^{\prime \prime }\right) . \end{aligned}$$

(15)

Since $\mathcal {\varphi }_{APS_{\varepsilon }^{\prime }}\left( {\mathcal {I}} ^{\prime \prime }\right) \le f^{\#}$, we conclude that Equation (5) holds.

Appendix 2: Proof of Lemma 6

The first step consists in applying heuristic JS, which can be implemented in $O\left( \overline{n}\ln \overline{n}\right)$ time. In the second step, algorithm $APS_{\varepsilon }^{\prime }$ generates the state sets ${\mathcal {X}} _{j}^{\#}$ ($j\in \left\{ 1,2,\ldots ,\overline{n}\right\}$). Since $\left| {\mathcal {X}}_{j}^{\#}\right| \le \left( \omega _{1}+1\right) \left( \omega _{2}+1\right)$, we deduce that

$$\begin{aligned} \sum _{j=1}^{\overline{n}}\left| {\mathcal {X}}_{j}^{\#}\right|&\le \overline{n}\left( \omega _{1}+1\right) \left( \omega _{2}+1\right) =\overline{n}\left( \left\lceil \frac{\overline{n}}{\varepsilon } \right\rceil +1\right) \left( \left\lceil \frac{2\overline{n}}{\varepsilon } \right\rceil +1\right) \nonumber \\&\le \overline{n}\left( \frac{\overline{n}}{\varepsilon }+2\right) \left( \frac{2\overline{n}}{\varepsilon }+2\right) . \end{aligned}$$

(16)

Note that algorithm $APS_{\varepsilon }^{\prime }$ generates ${\mathcal {X}} _{j}^{\#}$ by associating every new created state to its corresponding box if and only if such a state has a smaller value of t (in this case, the last state associated to this box will be removed). Otherwise, the new created state will be immediately removed. This allows us to generate ${\mathcal {X}}_{j}^{\#}$ in $O\left( \omega _{1}\omega _{2}\right)$ time. Hence, our method can be implemented in $O\left( \overline{n}\ln \overline{n}+ \overline{n}^{3}/\varepsilon ^{2}\right)$ time and this completes the proof.