Abstract
This paper studies single machine scheduling with a fixed non-availability interval. The processing time of a job is a linear increasing function of its starting time, and each job has a release date. A job is either rejected by paying a penalty cost or accepted and processed on the machine. The objective is to minimize the makespan of the accepted jobs and the total rejection penalties of the rejected jobs. We present a fully polynomial-time approximation scheme for the problem. We also show that the special case without non-availability interval can be solved using the same method with a lower order.
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Appendix
Appendix
This appendix proves two important results which corresponding to (3) (5) and (4) in the proof of Theorem 1, respectively.
Result 1 \(\left| {P_{j+1}^k (x^{*})-P_{j+1}^k (\tilde{x}^{(a_1 a_2 bc)})} \right| \le \delta _1 P_{j+1}^k (x^{*})\) for \(x_{j+1} =k, \,k=1,2.\)
It can be divided into the following four cases, and we show that Result 1 holds for each of them.
Case 1.1 \(r_{j+1} \ge P_j^k (x^{*})\) and \(r_{j+1} \ge P_j^k (x^{(a_1 a_2 bc)})\)
Case 1.2 \(P_j^k (x^{(a_1 a_2 bc)})>r_{j+1} \ge P_j^k (x^{*})\)
Case 1.3 \(P_j^k (x^{*})>r_{j+1} \ge P_j^k (x^{(a_1 a_2 bc)})\)
Case 1.4 \(r_{j+1} <P_j^k (x^{*})\) and \(r_{j+1} <P_j^k (x^{(a_1 a_2 bc)})\)
Based on the above analysis, obviously we have
This completes the proof of Result 1.
Result 2 \(\left| {f_{j+1} (x^{*})-f_{j+1} (\tilde{x}^{(a_1 a_2 bc)})} \right| \le \delta _1 f_{j+1} (x^{*})\), for \(x_{j+1} =1\).
Since \(\left| {f_j (x^{*})-f_j (x^{(a_1 a_2 bc)})} \right| \le \delta f_j (x^{*})\), then it implies the following four cases:
Case 2.1 \(P_j^2 (x^{*})=T_2,\, P_j^2 (x^{(a_1 a_2 bc)})=T_2 \)
Case 2.2 \(P_j^2 (x^{*})>T_2,\, P_j^2 (x^{(a_1 a_2 bc)})>T_2 \)
Case 2.3 \(P_j^2 (x^{*})>T_2,\, P_j^2 (x^{(a_1 a_2 bc)})=T_2 \)
Case 2.4 \(P_j^2 (x^{*})=T_2,\, P_j^2 (x^{(a_1 a_2 bc)})>T_2 \)
In the following we show that Result 2 holds for each of the above four cases. We only consider Case 2.1 and Case 2.3 since Case 2.2 and Case 2.4 can be similarly proved.
For Case 2.1,
For Case 2.3,
If \(P_j^2 (x^{*})+W_j (x^{*})-P_j^1 (x^{(a_1 a_2 bc)})-W_j (x^{(a_1 a_2 bc)})\ge 0\), then
Otherwise,
Either (12) or (13) is hold. Hence, we consider (12) and (13) in the following, respectively.
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If (12) is established, then we have
$$\begin{aligned} \left| {f_{j+1} (x^{*})-f_{j+1} \left( \tilde{x}^{(a_1 a_2 bc)}\right) } \right|&= P_{j+1}^2 (x^{*})-P_{j+1}^1 \left( \tilde{x}^{(a_1 a_2 bc)}\right) +W_{j+1} (x^{*})\nonumber \\&-W_{j+1} \left( \tilde{x}^{(a_1 a_2 bc)}\right) \end{aligned}$$(14)
or
Consider (14), we have
Consider (15), we have
Consider (15), we have
Consequently, for both (12) and (13) in Case 2.3, we can obtain
Based on the analysis above, for each of the four cases we have
This completes the proof of Result 2.
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Li, WX., Zhao, CL. Deteriorating jobs scheduling on a single machine with release dates, rejection and a fixed non-availability interval. J. Appl. Math. Comput. 48, 585–605 (2015). https://doi.org/10.1007/s12190-014-0820-3
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DOI: https://doi.org/10.1007/s12190-014-0820-3