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

Jiayu Shen; Yuanguo Zhu

doi:10.3934/jimo.2018058

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April 2019, 15(2): 577-593. doi: 10.3934/jimo.2018058

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

Received August 2017 Revised January 2018 Published April 2019 Early access April 2018

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: Single machine scheduling, batch delivery, uncertain environment, genetic algorithm, uncertain simulation.

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

Citation: Jiayu Shen, Yuanguo Zhu. An uncertain programming model for single machine scheduling problem with batch delivery. Journal of Industrial and Management Optimization, 2019, 15 (2) : 577-593. doi: 10.3934/jimo.2018058

References:

[1]	E. Cakici, S. Mason and M. Kurz, Multi-objective analysis of an integrated supply chain scheduling problem, International Journal of Production Research, 50 (2012), 2624-2638.
[2]	Y. Chen and Y. Zhu, Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems Journal of Industrial and Management Optimization, to appear. doi: 10.3934/jimo.2017082.
[3]	T. Cheng, V. Gordon and M. Kovalyov, Single machine scheduling with batch deliveries, European Journal of Operational Research, 94 (1996), 227-283.
[4]	Q. Cui and Y. Sheng, Uncertain programming model for solid transportation problem, Information: An International Interdisciplinary Journal, 16 (2013), 1207-1214.
[5]	J. Framinan and P. Gonzalez, On heuristic solutions for the stochastic flow shop scheduling problem, European Journal of Operational Research, 246 (2015), 413-420. doi: 10.1016/j.ejor.2015.05.006.
[6]	B. Fu, Y. Huo and H. Zhao, Coordinated scheduling of production and delivery with production window and delivery capacity constraints, Theoretical Computer Science, 422 (2012), 39-51. doi: 10.1016/j.tcs.2011.11.035.
[7]	Y. Gao, Shortest path problem with uncertain arc lengths, Computers & Mathematics with Applications, 62 (2011), 2591-2600. doi: 10.1016/j.camwa.2011.07.058.
[8]	Y. Gao, L. Yang and S. Li, Uncertain Models on railway transportation planning problem, Applied Mathematical Modelling, 40 (2016), 4921-4934. doi: 10.1016/j.apm.2015.12.016.
[9]	N. Hall, M. Lesaoana and C. Potts, Scheduling with fixed delivery dates, Operations Research, 49 (2001), 134-144. doi: 10.1287/opre.49.1.134.11192.
[10]	N. Hall and C. Potts, Supply chain scheduling: Batching and delivery, Operation research, 51 (2003), 566-584. doi: 10.1287/opre.51.4.566.16106.
[11]	N. Hall and C. Potts, The coordination of scheduling and batch deliveries, Annals of Operations Research, 135 (2005), 41-64. doi: 10.1007/s10479-005-6234-8.
[12]	R. Hallah, Minimizing total earliness and tardiness on a single machine using a hybrid heuristic, Computers & Operations Research, 34 (2007), 3126-3142.
[13]	A. Hamidinia, S. Khakabimamaghani, M. Mazdeh and M. Jafari, A genetic algorithm for minimizing total tardiness/earliness of weighted jobs in a batched delivery system, Computers & Industrial Engineering, 62 (2012), 29-38.
[14]	S. Han, Z. Peng and S. Wang, The maximum flow problem of uncertain network, Information Sciences, 265 (2014), 167-175. doi: 10.1016/j.ins.2013.11.029.
[15]	X. Hao, L. Lin, M. Gen and K. Ohno, Effective estimation of distribution algorithm for stochastic job shop scheduling problem, Procedia Computer Science, 20 (2013), 102-107.
[16]	G. Jiang, An uncertain chance-constrained programming model for empty container allocation, Information: An International Interdisciplinary Journal, 16 (2013), 1119-1124.
[17]	F. Jolai, M. Rabbani, S. Amalnick, A. Dabaghi, M. Dehghan and M. YazdanParas, Genetic algorithm for bi-criteria single machine scheduling problem of minimizing maximum earliness and number of tardy jobs, Applied Mathematics & Computation, 194 (2007), 552-560. doi: 10.1016/j.amc.2007.04.063.
[18]	S. Karimi, Z. Ardalan, B. Naderi and M. Mohammadi, Scheduling flexible job-shops with transportation times: Mathematical models and a hybrid imperialist competitive algorithm, Applied Mathematical Modelling, 47 (2017), 667-682. doi: 10.1016/j.apm.2016.09.022.
[19]	H. Ke, J. Ma and G. Tian, Hybrid multilevel programming with uncertain random parameters, Journal of Intelligent Manufacturing, 28 (2017), 589-596.
[20]	S. Li and J. Peng, A new approach to risk comparison via uncertain measure, Industrial Engineering and Management Systems, 11 (2012), 176-182.
[21]	B. Liu, Uncertainty Theory, Springer-Verlag, Berlin, 2004.
[22]	B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.
[23]	B. Liu and K. Yao, Uncertain multilevel programming: algorithm and applications, Computers & Industrial Engineering, 89 (2015), 235-240.
[24]	B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10.
[25]	A. Mason and E. Anderson, Minimizing flow time on a single-machine with job classes and setup times, Naval Research Logistic, 38 (1991), 333-350. doi: 10.1002/1520-6750(199106)38:3<333::AID-NAV3220380305>3.0.CO;2-0.
[26]	Y. Ning, X. Chen, Z. Wang and X. Li, An uncertain multi-objective programming model for machine scheduling problem, International Journal of Machine Learning and Cybernetics, 8 (2017), 1493-1500.
[27]	Z. Peng and K. Iwamura, A sufficient and necessary condition of uncertainty distribution, Journal of Interdisciplinary Mathematics, 13 (2010), 277-285. doi: 10.1080/09720502.2010.10700701.
[28]	X. Qi, Coordinated logistics scheduling for in-house production and outsourcing, IEEE Transactions on Automation Science and Engineering, 5 (2008), 188-192.
[29]	L. Rong, Two new uncertainty programming models of inventory with uncertain costs, Journal of Information & Computational Science, 8 (2011), 280-288.
[30]	J. Shen and Y. Zhu, Scheduling in a two-stage supply chain with uncertain parameters, Journal of Intelligent and Fuzzy Systems, 30 (2016), 3439-3449.
[31]	A. Singh, J. ValenteMaria and M. Moreira, Hybrid heuristics for the single machine scheduling problem with quadratic earliness and tardiness costs, International Journal of Machine Learning and Cybernetics, 3 (2012), 327-333.
[32]	A. Soukhal, A. Oulamara and P. Martineau, Complexity of flow shop scheduling problems with transportation constraints, European Journal of Operational Research, 161 (2005), 32-41. doi: 10.1016/j.ejor.2003.03.002.
[33]	K. Stecke and X. Zhao, Production and transportation integration for a make-todelivery business mode, Manufacturing & Service Operations Management, 9 (2007), 206-224.
[34]	G. Steiner and R. Zhang, Approximation algorithms for minimizing the total weighted number of late jobs with late deliveries in two-level supply chains, Journal of Scheduling, 12 (2009), 565-574. doi: 10.1007/s10951-009-0109-9.
[35]	G. Wan and B. Yen, Tabu search for single machine scheduling with distinct due windows and weighted earliness/tardiness penalties, European Journal of Operational Research, 142 (2002), 271-281. doi: 10.1016/S0377-2217(01)00302-2.
[36]	X. Wang and T. Cheng, Logistics scheduling to minimize inventory and transport costs, International Journal of Production Economics, 121 (2009), 266-273.
[37]	X. Wang and T. Cheng, Production scheduling with supply and delivery considerations to minimize the makespan, European Journal of Operational Research, 194 (2009), 743-752. doi: 10.1016/j.ejor.2007.12.033.
[38]	X. Wu and X. Zhou, Stochastic scheduling to minimize expected maximum lateness, European Journal of Operational Research, 190 (2008), 103-115. doi: 10.1016/j.ejor.2007.06.015.
[39]	H. Yan, Y. Sun and Y. Zhu, A linear-quadratic control problem of uncertain discretr-time switched systems, Journal of Industrial and Management Optimization, 13 (2017), 267-282.
[40]	X. Yang, Scheduling with generalized batch delivery dates and earliness penalties, IIE Transactions, 32 (2000), 735-741.
[41]	K. Yao, Multi-dimensional uncertain calculus with liu process, Journal of Uncertain Systems, 8 (2014), 244-254.
[42]	Y. Yin, T. Cheng, S. Cheng and C. Wu, Single-machine batch delivery scheduling with an assignable common due date and controllable processing times, Computers & Industrial Engineering, 65 (2013), 652-662.
[43]	Y. Yin, T. Cheng, C. Wu and S. Cheng, Single-machine common due-date scheduling with batch delivery costs and resource-dependent processing times, International Journal of Production Research, 51 (2013), 5083-5099.
[44]	Y. Yin, T. Cheng, C. Hsu and C. Wu, Single-machine batch delivery scheduling with an assignable common due window, Omega, 41 (2013), 216-225.
[45]	Y. Yin, T. Cheng, C. Wu and S. Cheng, Single-machine batch delivery scheduling and common due-date assignment with a rate-modifying activity, International Journal of Production Research, 52 (2014), 5583-5596.
[46]	Y. Yin, Y. Wang, T. Cheng, D. Wang and C. Wu, Two-agent single-machine scheduling to minimize the batch delivery cost, Computers & Industrial Engineering, 92 (2016), 16-30.
[47]	S. Zdrzalka, Approximation algorithms for single-machine sequencing with delivery times and unit batch set-up times, European Journal of Operational Research, 51 (1991), 199-209.
[48]	Y. Zhu, Functions of uncertain variables and uncertain programming, Journal of Uncertain Systems, 6 (2012), 278-288.

show all references

References:

[1]	E. Cakici, S. Mason and M. Kurz, Multi-objective analysis of an integrated supply chain scheduling problem, International Journal of Production Research, 50 (2012), 2624-2638.
[2]	Y. Chen and Y. Zhu, Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems Journal of Industrial and Management Optimization, to appear. doi: 10.3934/jimo.2017082.
[3]	T. Cheng, V. Gordon and M. Kovalyov, Single machine scheduling with batch deliveries, European Journal of Operational Research, 94 (1996), 227-283.
[4]	Q. Cui and Y. Sheng, Uncertain programming model for solid transportation problem, Information: An International Interdisciplinary Journal, 16 (2013), 1207-1214.
[5]	J. Framinan and P. Gonzalez, On heuristic solutions for the stochastic flow shop scheduling problem, European Journal of Operational Research, 246 (2015), 413-420. doi: 10.1016/j.ejor.2015.05.006.
[6]	B. Fu, Y. Huo and H. Zhao, Coordinated scheduling of production and delivery with production window and delivery capacity constraints, Theoretical Computer Science, 422 (2012), 39-51. doi: 10.1016/j.tcs.2011.11.035.
[7]	Y. Gao, Shortest path problem with uncertain arc lengths, Computers & Mathematics with Applications, 62 (2011), 2591-2600. doi: 10.1016/j.camwa.2011.07.058.
[8]	Y. Gao, L. Yang and S. Li, Uncertain Models on railway transportation planning problem, Applied Mathematical Modelling, 40 (2016), 4921-4934. doi: 10.1016/j.apm.2015.12.016.
[9]	N. Hall, M. Lesaoana and C. Potts, Scheduling with fixed delivery dates, Operations Research, 49 (2001), 134-144. doi: 10.1287/opre.49.1.134.11192.
[10]	N. Hall and C. Potts, Supply chain scheduling: Batching and delivery, Operation research, 51 (2003), 566-584. doi: 10.1287/opre.51.4.566.16106.
[11]	N. Hall and C. Potts, The coordination of scheduling and batch deliveries, Annals of Operations Research, 135 (2005), 41-64. doi: 10.1007/s10479-005-6234-8.
[12]	R. Hallah, Minimizing total earliness and tardiness on a single machine using a hybrid heuristic, Computers & Operations Research, 34 (2007), 3126-3142.
[13]	A. Hamidinia, S. Khakabimamaghani, M. Mazdeh and M. Jafari, A genetic algorithm for minimizing total tardiness/earliness of weighted jobs in a batched delivery system, Computers & Industrial Engineering, 62 (2012), 29-38.
[14]	S. Han, Z. Peng and S. Wang, The maximum flow problem of uncertain network, Information Sciences, 265 (2014), 167-175. doi: 10.1016/j.ins.2013.11.029.
[15]	X. Hao, L. Lin, M. Gen and K. Ohno, Effective estimation of distribution algorithm for stochastic job shop scheduling problem, Procedia Computer Science, 20 (2013), 102-107.
[16]	G. Jiang, An uncertain chance-constrained programming model for empty container allocation, Information: An International Interdisciplinary Journal, 16 (2013), 1119-1124.
[17]	F. Jolai, M. Rabbani, S. Amalnick, A. Dabaghi, M. Dehghan and M. YazdanParas, Genetic algorithm for bi-criteria single machine scheduling problem of minimizing maximum earliness and number of tardy jobs, Applied Mathematics & Computation, 194 (2007), 552-560. doi: 10.1016/j.amc.2007.04.063.
[18]	S. Karimi, Z. Ardalan, B. Naderi and M. Mohammadi, Scheduling flexible job-shops with transportation times: Mathematical models and a hybrid imperialist competitive algorithm, Applied Mathematical Modelling, 47 (2017), 667-682. doi: 10.1016/j.apm.2016.09.022.
[19]	H. Ke, J. Ma and G. Tian, Hybrid multilevel programming with uncertain random parameters, Journal of Intelligent Manufacturing, 28 (2017), 589-596.
[20]	S. Li and J. Peng, A new approach to risk comparison via uncertain measure, Industrial Engineering and Management Systems, 11 (2012), 176-182.
[21]	B. Liu, Uncertainty Theory, Springer-Verlag, Berlin, 2004.
[22]	B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.
[23]	B. Liu and K. Yao, Uncertain multilevel programming: algorithm and applications, Computers & Industrial Engineering, 89 (2015), 235-240.
[24]	B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10.
[25]	A. Mason and E. Anderson, Minimizing flow time on a single-machine with job classes and setup times, Naval Research Logistic, 38 (1991), 333-350. doi: 10.1002/1520-6750(199106)38:3<333::AID-NAV3220380305>3.0.CO;2-0.
[26]	Y. Ning, X. Chen, Z. Wang and X. Li, An uncertain multi-objective programming model for machine scheduling problem, International Journal of Machine Learning and Cybernetics, 8 (2017), 1493-1500.
[27]	Z. Peng and K. Iwamura, A sufficient and necessary condition of uncertainty distribution, Journal of Interdisciplinary Mathematics, 13 (2010), 277-285. doi: 10.1080/09720502.2010.10700701.
[28]	X. Qi, Coordinated logistics scheduling for in-house production and outsourcing, IEEE Transactions on Automation Science and Engineering, 5 (2008), 188-192.
[29]	L. Rong, Two new uncertainty programming models of inventory with uncertain costs, Journal of Information & Computational Science, 8 (2011), 280-288.
[30]	J. Shen and Y. Zhu, Scheduling in a two-stage supply chain with uncertain parameters, Journal of Intelligent and Fuzzy Systems, 30 (2016), 3439-3449.
[31]	A. Singh, J. ValenteMaria and M. Moreira, Hybrid heuristics for the single machine scheduling problem with quadratic earliness and tardiness costs, International Journal of Machine Learning and Cybernetics, 3 (2012), 327-333.
[32]	A. Soukhal, A. Oulamara and P. Martineau, Complexity of flow shop scheduling problems with transportation constraints, European Journal of Operational Research, 161 (2005), 32-41. doi: 10.1016/j.ejor.2003.03.002.
[33]	K. Stecke and X. Zhao, Production and transportation integration for a make-todelivery business mode, Manufacturing & Service Operations Management, 9 (2007), 206-224.
[34]	G. Steiner and R. Zhang, Approximation algorithms for minimizing the total weighted number of late jobs with late deliveries in two-level supply chains, Journal of Scheduling, 12 (2009), 565-574. doi: 10.1007/s10951-009-0109-9.
[35]	G. Wan and B. Yen, Tabu search for single machine scheduling with distinct due windows and weighted earliness/tardiness penalties, European Journal of Operational Research, 142 (2002), 271-281. doi: 10.1016/S0377-2217(01)00302-2.
[36]	X. Wang and T. Cheng, Logistics scheduling to minimize inventory and transport costs, International Journal of Production Economics, 121 (2009), 266-273.
[37]	X. Wang and T. Cheng, Production scheduling with supply and delivery considerations to minimize the makespan, European Journal of Operational Research, 194 (2009), 743-752. doi: 10.1016/j.ejor.2007.12.033.
[38]	X. Wu and X. Zhou, Stochastic scheduling to minimize expected maximum lateness, European Journal of Operational Research, 190 (2008), 103-115. doi: 10.1016/j.ejor.2007.06.015.
[39]	H. Yan, Y. Sun and Y. Zhu, A linear-quadratic control problem of uncertain discretr-time switched systems, Journal of Industrial and Management Optimization, 13 (2017), 267-282.
[40]	X. Yang, Scheduling with generalized batch delivery dates and earliness penalties, IIE Transactions, 32 (2000), 735-741.
[41]	K. Yao, Multi-dimensional uncertain calculus with liu process, Journal of Uncertain Systems, 8 (2014), 244-254.
[42]	Y. Yin, T. Cheng, S. Cheng and C. Wu, Single-machine batch delivery scheduling with an assignable common due date and controllable processing times, Computers & Industrial Engineering, 65 (2013), 652-662.
[43]	Y. Yin, T. Cheng, C. Wu and S. Cheng, Single-machine common due-date scheduling with batch delivery costs and resource-dependent processing times, International Journal of Production Research, 51 (2013), 5083-5099.
[44]	Y. Yin, T. Cheng, C. Hsu and C. Wu, Single-machine batch delivery scheduling with an assignable common due window, Omega, 41 (2013), 216-225.
[45]	Y. Yin, T. Cheng, C. Wu and S. Cheng, Single-machine batch delivery scheduling and common due-date assignment with a rate-modifying activity, International Journal of Production Research, 52 (2014), 5583-5596.
[46]	Y. Yin, Y. Wang, T. Cheng, D. Wang and C. Wu, Two-agent single-machine scheduling to minimize the batch delivery cost, Computers & Industrial Engineering, 92 (2016), 16-30.
[47]	S. Zdrzalka, Approximation algorithms for single-machine sequencing with delivery times and unit batch set-up times, European Journal of Operational Research, 51 (1991), 199-209.
[48]	Y. Zhu, Functions of uncertain variables and uncertain programming, Journal of Uncertain Systems, 6 (2012), 278-288.

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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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, ...

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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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

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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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

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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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

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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$n$	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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2020 Impact Factor: 1.801

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2020 Impact Factor: 1.801

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2020 Impact Factor: 1.801

American Institute of Mathematical Sciences

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

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

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