An uncertain programming model for single machine scheduling problem with batch delivery

Jiayu Shen; Yuanguo Zhu

doi:10.3934/jimo.2018058

Article Contents

2019, Volume 15, Issue 2: 577-593. Doi: 10.3934/jimo.2018058

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An uncertain programming model for single machine scheduling problem with batch delivery

Jiayu Shen and
Yuanguo Zhu^,

School of Science, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China

* Corresponding author: Yuanguo Zhu
* Corresponding author: Yuanguo Zhu

Received: August 2017

Revised: January 2018

Early access: April 2018

Published: January 2019

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Abstract

A single machine scheduling problem with batch delivery is studied in this paper. The objective is to minimize the total cost which comprises earliness penalties, tardiness penalties, holding and transportation costs. An integer programming model is proposed and two dominance properties are obtained. However, sometimes due to the lack of historical data, the worker evaluates the processing time of a job according to his past experience. A pessimistic value model of the single machine scheduling problem with batch delivery under an uncertain environment is presented. Since the objective function is non-monotonic with respect to uncertain variables, a hybrid algorithm based on uncertain simulation and a g#enetic algorithm (GA) is designed to solve the model. In addition, two dominance properties under the uncertain environment are also obtained. Finally, computational experiments are presented to illustrate the modeling idea and the effectiveness of the algorithm.

Keywords:

Mathematics Subject Classification: Primary: 90B99; Secondary: 90C10.

Citation:

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Figure 1. An example of crossover

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Figure 2. An example of mutation

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Figure 3. The sensitivity of the solution with respect to the confidence level

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Table 1. List of notations

notations	definitions
$i=1, 2, ..., n$	the index of job
$j=1, 2, ...$	the index of position
$b=1, 2, ...$	the index of batch
$l=1, 2, ...$, K	the index of customer
$n_{l}$	the number of jobs of customer $l$, l=1, 2, ..., K
$p_i$	processing time of job $i$, i=1, 2, ..., n
$[d^{s}_{i}, d^{t}_{i}]$	due window of job $i$, i=1, 2, ..., n
$\alpha_i$	unit earliness penalty cost of job $i$, i=1, 2, ..., n
$\beta_i$	unit tardiness penalty cost of job $i$, i=1, 2, ..., n
$h_i$	unit holding cost of job $i$, i=1, 2, ..., n
$D_l$	transportation cost of customer $l$, l=1, 2, ..., K
$C_i$	completion time of job $i$, i=1, 2, ..., n
$T_j$	completion time of the $j$th job on machine}, j=1, 2, ...
$R_i$	delivery date of job $i$, i=1, 2, ..., n
$R^{'}_{lb}$	delivery date of the $b$th batch of customer $l$, l=1, 2, ..., K; b=1, 2, ...

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Table 2. Results for small scale

No	GA			HGA
	min	max	time(s)	min	max	time(s)
1	3079	3361	356	3125	3349	301
2	2953	3415	338	2837	3437	312
3	2556	2889	313	2398	2735	263
4	3582	3764	359	3619	3923	329
5	2046	2358	271	2098	2491	254

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Table 3. Results for mesoscale

No	GA			HGA
	min	max	time(s)	min	max	time(s)
1	8466	8893	1603	8501	8697	1354
2	9320	9671	1754	9455	9612	1340
3	8323	8569	1625	8361	8538	1385
4	8736	9028	1701	8798	9110	1313
5	7954	8077	1590	7839	8154	1397

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Table 4. Results for large scale

No	GA			HGA
	min	max	time(s)	min	max	time(s)
1	21542	22634	8655	21696	21873	6320
2	19556	21391	8367	19374	20065	5966
3	21406	23767	8961	21250	22034	6257
4	21973	23891	9058	21630	22741	6299
5	20592	22378	8537	20063	20097	6035

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Table 5. Average relative percentage errors of GA and HGA

$n$	GA	HGA
$20$	$0.06266$	$0.05713$
$50$	$0.03316$	$0.02685$
$100$	$0.01295$	$0.007372$
$200$	$0.005723$	$0.004993$
Average	$0.028623$	$0.024086$

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