Scheduling family jobs on an unbounded parallel-batch machine to minimize makespan and maximum flow time

Zhichao Geng; Jinjiang Yuan

doi:10.3934/jimo.2018017

Article Contents

2018, Volume 14, Issue 4: 1479-1500. Doi: 10.3934/jimo.2018017

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Scheduling family jobs on an unbounded parallel-batch machine to minimize makespan and maximum flow time

Zhichao Geng and
Jinjiang Yuan^,

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China

* Corresponding author: Jinjiang Yuan
* Corresponding author: Jinjiang Yuan

Received: February 2017

Revised: August 2017

Early access: January 2018

Published: October 2018

The authors are supported by NSFC (11671368), NSFC (11571321), and NSFC (11771406)

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Abstract

This paper investigates the scheduling of family jobs with release dates on an unbounded parallel-batch machine. The involved objective functions are makespan and maximum flow time. It was reported in the literature that the single-criterion problem for minimizing makespan is strongly NP-hard when the number of families is arbitrary, and is polynomially solvable when the number of families is fixed. We first show in this paper that the single-criterion problem for minimizing maximum flow time is also strongly NP-hard when the number of families is arbitrary. We further show that the Pareto optimization problem (also called bicriteria problem) for minimizing makespan and maximum flow time is polynomially solvable when the number of families is fixed, by enumerating all Pareto optimal points in polynomial time. This implies that the single-criterion problem for minimizing maximum flow time is also polynomially solvable when the number of families is fixed.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. The structure of the scheduling problem

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Figure 2. An intuitive interpretation of the proposed algorithm

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Table 1. The definitions of the abbreviations/notations

Abbreviation/Notation	Definition
ERD	earliest release date first (rule)
PoP	Pareto optimal point
PoS	Pareto optimal schedule
ERD-family scheule	a schedule of family jobs defined in the first paragraph in section 3.1
${\mathcal F}_f$	the $f$-th job family
$f$-job	a job which belongs to family ${\mathcal F}_f$
$f$-batch	a p-bath which only includes $f$-jobs
$J_{f, j}$	the $j$-th job in family ${\mathcal F}_f$
$J_{j}$	a general job
$p_{j}~(p_{f, j})$/$r_{j}~(r_{f, j})$	the processing time/the release dates of job $J_{j}~(J_{f, j})$
$n_f$	the number of jobs in family ${\mathcal F}_f$
$(b<n)/(b\geq n)$	the bounded/unbounded capacity of a p-batch
$p(B)$	the processing time of batch $B$
$\pi/\sigma$	a feasible schedule
$C_{j}(\pi)(C_{f, j}(\pi))$	the completion time of job $J_{j}$ ($J_{f, j}$) in $\pi$
$S_{j}(\pi)(S_{f, j}(\pi))$	the starting time of job $J_{j}$ ($J_{f, j}$) in $\pi$
$C_{B}(\pi)$	the completion time of batch $B$ in $\pi$
$S_{B}(\pi)$	the starting time of batch $B$ in $\pi$
$F_{j}(\pi)(F_{f, j}(\pi))$	the flow time of job $J_{j}$ ($J_{f, j}$) in $\pi$ which is $F_{j}(\pi)=C_{j}(\pi)-r_{j}(\pi)$
$C_{\max}(\pi)$	the maximum completion time of all jobs in $\pi$
$F_{\max}(\pi)$	the maximum flow time of all jobs in $\pi$
${\mathcal F}_f^{(i)}$	the set composed by the first $i$ jobs of family ${\mathcal F}_f$, i.e., $\{J_{f, 0}, J_{f, 1}, \cdots, J_{f, i}\}$
$(i_1, \cdots, i_K)$	the instance composed by the job set ${\mathcal F}_1^{(i_1)} \cup \cdots \cup {\mathcal F}_K^{(i_K)}$
$P(i_1, \cdots, i_K)$	the problem $1\|\beta \|^\#(C_{\max}, F_{\max})$ restricted on the instance $(i_1, \cdots, i_K)$
$P_{Y}(i_1, \cdots, i_K)$	the problem $1\|\beta \|C_{\max}: F_{\max}\leq Y$ restricted on the instance $(i_1, \cdots, i_K)$
$P_{Y}^{(f)}(i_1, \cdots, i_K)$	a restricted version of $P_{Y}(i_1, \cdots, i_K)$ with $i_f \geq 1$ for which it is required that feasible schedules end with an $f$-batch
$C_Y^{(f)}(i_1, \cdots, i_K)$	the optimal makespan of problem $P_{Y}^{(f)}(i_1, \cdots, i_K)$
$C_Y^{(f, k_f)}(i_1, \cdots, i_K)$	defined in equation (3)
$D_Y^{(f)}(i_1, \cdots, i_K)$	defined in equation (4)
${\mathcal X}_{Y}(i_1, \cdots, i_K)$	defined in equation (5), the set of family indices attaining the minimum in equation (1)
$\Psi^{(f)}_{Y}(i_1, \cdots, i_K)$	defined in equation (6), the set of the $f$-job indices attaining the minimum in equation (2)
$C_Y(i_1, \cdots, i_K)$	the optimal makespan of problem $P_{Y}(i_1, \cdots, i_K)$
Algorithm DP($Y$)	the proposed dynamic programming algorithm for problem $1\|\beta \|C_{\max}: F_{\max}\leq Y$
Algorithm Family-CF	the proposed algorithm for problem $1\|\beta \|^\#(C_{\max}, F_{\max})$

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Table 2. The jobs in family ${\mathcal F}_1$

${\mathcal F}_1$	$J_{1, 1}$	$J_{1, 2}$	$J_{1, 3}$	$J_{1, 4}$	$J_{1, 5}$	$J_{1, 6}$	$J_{1, 7}$	$J_{1, 8}$	$J_{1, 9}$	$J_{1, 10}$
$r_{1, i}$	0	2	3	4	6	7	9	11	14	17
$p_{1, i}$	2	2	4	5	7	12	3	6	1	3

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Table 3. The jobs in family ${\mathcal F}_2$

${\mathcal F}_1$	$J_{1, 1}$	$J_{1, 2}$	$J_{1, 3}$	$J_{1, 4}$	$J_{1, 5}$	$J_{1, 6}$	$J_{1, 7}$	$J_{1, 8}$	$J_{1, 9}$	$J_{1, 10}$
$r_{1, i}$	1	2	4	6	8	10	11	14	16	19
$p_{1, i}$	2	1	1	4	3	8	10	2	11	9

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Table 4. The PoPs of the instances

jobs	PoPs	jobs	PoPs	jobs	PoPs	jobs	PoPs	jobs	PoPs
${\mathcal J}(1, 1)$	$(4, 3)$	${\mathcal J}(1, 2)$	$(4, 3)$	${\mathcal J}(2, 1)$	$(6, 4)$	${\mathcal J}(2, 2)$	$(6, 4)$	${\mathcal J}(1, 3)$	$(5, 3)$
					$(5, 5)$
${\mathcal J}(2, 3)$	$(5, 4)$	${\mathcal J}(3, 1)$	$(8, 6)$	${\mathcal J}(3, 2)$	$(8, 5)$	${\mathcal J}(3, 3)$	$(9, 5)$	${\mathcal J}(1, 4)$	$(2, 2)$
			$(7, 7)$		$(7, 7)$		$(8, 7)$
${\mathcal J}(2, 4)$	$(4, 2)$	${\mathcal J}(3, 4)$	$(7, 5)$	${\mathcal J}(4, 1)$	$(3, 2)$	${\mathcal J}(4, 2)$	$(4, 2)$	${\mathcal J}(4, 3)$	$(5, 2)$
${\mathcal J}(4, 4)$	$(10, 4)$	${\mathcal J}(1, 5)$	$(2, 2)$	${\mathcal J}(2, 5)$	$(4, 2)$	${\mathcal J}(3, 5)$	$(7, 5)$	${\mathcal J}(4, 5)$	$(13, 5)$
	$(9, 6)$								$(9, 6)$
${\mathcal J}(5, 1)$	$(3, 2)$	${\mathcal J}(5, 2)$	$(4, 2)$	${\mathcal J}(5, 3)$	$(5, 2)$	${\mathcal J}(5, 4)$	$(10, 4)$	${\mathcal J}(5, 5)$	$(13, 5)$
									$(12, 6)$
${\mathcal J}(1, 6)$	$(2, 2)$	${\mathcal J}(2, 6)$	$(4, 2)$	${\mathcal J}(3, 6)$	$(7, 5)$	${\mathcal J}(4, 6)$	$(9, 6)$	${\mathcal J}(5, 6)$	$(13, 10)$
${\mathcal J}(6, 1)$	$(3, 2)$	${\mathcal J}(6, 2)$	$(4, 2)$	${\mathcal J}(6, 3)$	$(5, 2)$	${\mathcal J}(6, 4)$	$(10, 4)$	${\mathcal J}(6, 5)$	$(13, 5)$
									$(12, 6)$
${\mathcal J}(6, 6)$	$(18, 10)$	${\mathcal J}(1, 7)$	$(2, 2)$	${\mathcal J}(2, 7)$	$(4, 2)$	${\mathcal J}(3, 7)$	$(7, 5)$	${\mathcal J}(4, 7)$	$(9, 6)$
${\mathcal J}(5, 7)$	$(13, 10)$	${\mathcal J}(6, 7)$	$(22, 12)$	${\mathcal J}(7, 1)$	$(3, 2)$	${\mathcal J}(7, 2)$	$(4, 2)$	${\mathcal J}(7, 3)$	$(5, 2)$
			$(21, 13)$
			$(19, 15)$
${\mathcal J}(7, 4)$	$(10, 4)$	${\mathcal J}(7, 5)$	$(13, 5)$	${\mathcal J}(7, 6)$	$(18, 10)$	${\mathcal J}(7, 7)$	$(22, 12)$	${\mathcal J}(1, 8)$	$(2, 2)$
			$(12, 6)$				$(21, 13)$
${\mathcal J}(2, 8)$	$(4, 2)$	${\mathcal J}(3, 8)$	$(7, 5)$	${\mathcal J}(4, 8)$	$(9, 6)$	${\mathcal J}(5, 8)$	$(13, 10)$	${\mathcal J}(6, 8)$	$(24, 12)$
									$(23, 13)$
									$(19, 15)$
${\mathcal J}(7, 8)$	$(24, 12)$	${\mathcal J}(8, 1)$	$(3, 2)$	${\mathcal J}(8, 2)$	$(4, 2)$	${\mathcal J}(8, 3)$	$(5, 2)$	${\mathcal J}(8, 4)$	$(10, 4)$
	$(23, 13)$
	$(21, 15)$
${\mathcal J}(8, 5)$	$(13, 5)$	${\mathcal J}(8, 6)$	$(18, 10)$	${\mathcal J}(8, 7)$	$(22, 12)$	${\mathcal J}(8, 8)$	$(24, 12)$	${\mathcal J}(1, 9)$	$(2, 2)$
	$(12, 6)$				$(21, 13)$		$(23, 13)$
${\mathcal J}(2, 9)$	$(4, 2)$	${\mathcal J}(3, 9)$	$(7, 5)$	${\mathcal J}(4, 9)$	$(9, 6)$	${\mathcal J}(5, 9)$	$(13, 10)$	${\mathcal J}(6, 9)$	$(19, 15)$
${\mathcal J}(7, 9)$	$(21, 15)$	${\mathcal J}(8, 9)$	$(25, 16)$	${\mathcal J}(9, 1)$	$(3, 2)$	${\mathcal J}(9, 2)$	$(4, 2)$	${\mathcal J}(9, 3)$	$(5, 2)$
			$(23, 17)$
${\mathcal J}(9, 4)$	$(10, 4)$	${\mathcal J}(9, 5)$	$(13, 5)$	${\mathcal J}(9, 6)$	$(18, 10)$	${\mathcal J}(9, 7)$	$(22, 12)$	${\mathcal J}(9, 8)$	$(24, 12)$
			$(12, 6)$				$(21, 13)$		$(23, 13)$
${\mathcal J}(9, 9)$	$(25, 16)$	${\mathcal J}(1, 10)$	$(2, 2)$	${\mathcal J}(2, 10)$	$(4, 2)$	${\mathcal J}(3, 10)$	$(7, 5)$	${\mathcal J}(4, 10)$	$(9, 6)$
	$(24, 17)$
${\mathcal J}(5, 10)$	$(13, 10)$	${\mathcal J}(6, 10)$	$(19, 15)$	${\mathcal J}(7, 10)$	$(21, 15)$	${\mathcal J}(8, 10)$	$(25, 16)$	${\mathcal J}(9, 10)$	$(25, 16)$
							$(23, 17)$		$(24, 17)$
${\mathcal J}(10, 1)$	$(3, 2)$	${\mathcal J}(10, 2)$	$(4, 2)$	${\mathcal J}(10, 3)$	$(5, 2)$	${\mathcal J}(10, 4)$	$(13, 5)$	${\mathcal J}(10, 5)$	$(18, 10)$
							$(12, 6)$
${\mathcal J}(10, 1)$	$(3, 2)$	${\mathcal J}(10, 2)$	$(4, 2)$	${\mathcal J}(10, 3)$	$(5, 2)$	${\mathcal J}(10, 4)$	$(10, 4)$	${\mathcal J}(10, 5)$	$(13, 5)$
									$(12, 6)$
${\mathcal J}(10, 6)$	$(18, 10)$	${\mathcal J}(10, 7)$	$(22, 12)$	${\mathcal J}(10, 8)$	$(24, 12)$	${\mathcal J}(10, 9)$	$(25, 16)$	${\mathcal J}(10, 10)$	$(25, 16)$
			$(21, 13)$		$(23, 13)$

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