A two-stage solution approach for plastic injection machines scheduling problem

Tugba Sarac; Aydin Sipahioglu; Emine Akyol Ozer

doi:10.3934/jimo.2020022

Article Contents

2021, Volume 17, Issue 3: 1289-1314. Doi: 10.3934/jimo.2020022

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A two-stage solution approach for plastic injection machines scheduling problem

1.
Eskisehir Osmangazi University, Department of Industrial Engineering, Meselik Campus, 26040, Eskisehir, Turkey
2.
Eskisehir Technical University, Department of Industrial Engineering, Iki Eylul Campus, 26555, Eskisehir, Turkey

^* Corresponding author: Emine Akyol Ozer
^* Corresponding author: Emine Akyol Ozer

Received: August 2018

Revised: March 2019

Early access: February 2020

Published: May 2021

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Abstract

One of the most common plastic manufacturing methods is injection molding. In injection molding process, scheduling of plastic injection machines is very difficult because of the complex nature of the problem. For example, similar plastic parts should be produced sequentially to prevent long setup times. On the other hand, to produce a plastic part, its mold should be fixed on an injection machine. Machine eligibility restrictions should be considered because a mold can be usually fixed on a subset of the injection machines. Some plastic parts which have same shapes but different colors are used same mold so these parts can only be scheduled simultaneously if their mold has copies, otherwise resource constraints should be considered. In this study, a multi-objective mathematical model is proposed for parallel machine scheduling problem to minimize makespan, total tardiness, and total waiting time. Since NP-hard nature of problem, this paper presents a two-stage mathematical model and a two-stage solution approach. In the first stage of mathematical model, jobs are assigned to the machines and each machine is scheduled separately in the second stage. The integrated model and two-stage mathematical model are scalarized by using goal programming, compromise programming and Lexicographic Weighted Tchebycheff programming methods. To solve large-scale problems in a short time, a two-stage solution approach is also proposed. In the first stage of this approach, jobs are assigned to machines and scheduled by using proposed simulated annealing algorithm. In the second stage of the approach, starting time, completion time and waiting time of the jobs are calculated by using a mathematical model. The performance of the methods is demonstrated on randomly generated test problems.

Keywords:

Mathematics Subject Classification: Primary: 90C29, 90C90; Secondary: 90B35.

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Figure 1. Gantt Chart of Toy Example

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Figure 2. Number of Non-Dominated Solutions

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Table 1. Literature Review

Ref.	$ \alpha $	$ \beta $	$ \gamma $	Solution Methods
[24]	$ FJ_c $	Resource	Number of Tardy Jobs	Hybrid evolutionary algorithm
[8]	$ P_m $	$ r_j $, prec, $ s_{ij} $	$ \sum w_j T_j $	Variable neighborhood search
[7]	$ P_m $	Resource, $ s_{ij} $	Inventory holding, backlogging and setup time	Work center based decomposition approach
[15]	$ P_m $	$ s_{ij} $	Average relative percentage of imbalance	GA and ant colony optimization
[16]	$ P_m $	$ r_j $, $ s_{ij} $	$ C_{max} $, $ \sum T_j $	0-1 MIP model and GA with a fuzzy logic controller
[18]	$ P_m $	$ r_j $, $ s_{ij} $	Maximum tardiness($ T_{max} $)	Greedy Algorithm
[25]	$ P_m $	$ s_{ij} $	$ C_{max} $	A hybrid GA and tabu search approach
[3]	$ P_m $	$ M_j $	$ \sum C_j $	Optimizing Algorithm
[23]	$ P_m $	$ M_j $	$ T_{max} $	A network flow mathematical programming model and a heuristic approach
[17]	$ P_m $	$ M_j $	$ C_{max} $	Simulated annealing algorithm
[11]	$ P_m $	Resource, $ M_j $	$ C_{max} $	Integer programming and constraint programming model
[21]	$ R_m $	Resource $ s_{ij} $	$ \sum C_j $, total amount of resources assigned	Mixed integer programming model
[11]	$ P_m $	Resource, $ M_j $	$ C_{max} $	Integer programming, constraint programming model
[10]	$ P_m $	Resource, $ M_j $	$ C_{max} $	A Combined Integer/Constraint Programming Approach
[9]	$ P_m $	Resource	$ C_{max} $	Integer programming, constraint programming model
[1]	$ R_m $	$ M_j $	$ C_{max} $	An integer programming model, GA
[12]	$ R_m $	$ s_{ij} $	$ C_{max} $	Firefly algorithm
[4]	$ P_m $	$ r_j $, $ s_{ij} $, $ M_j $	Energy consumption and annual income	A multi-start and constructive search algorithm
[2]	$ P_m $	Resource, $ s_{ij} $, $ M_j $	$ \sum w_j C_j $	MIP models and a matheuristic algorithm

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Table 2. Results of Small-scale ($ n $ = 8, $ m $ = 2) Problems

Test No		1-1	2-1	3-1	4-1	5-1	6-1	7-1	8-1
M0-WGP	$ f_1 $	384	273	326	394	346	335	318	280
	$ f_2 $	299	99	296	564	116	295	232	102
	$ f_3 $	4	7	3	0	0	0	0	0
	$ z $	0.1	0.07	0.09	0.24	0.07	0.08	0.11	0.05
	$ t(sec.) $	196	181	299	880	153	175	82	95
M0-CP	$ f_1 $	384	273	343	388	326	383	328	280
	$ f_2 $	299	99	281	506	157	258	222	102
	$ f_3 $	4	7	3	87	14	0	16	0
	$ z $	0.1	0.07	0.09	0.29	0.09	0.08	0.15	0.05
	$ t(sec.) $	322	73	212	522	111	183	85	74
M0-LWT	$ f_1 $	384	273	343	388	326	383	328	280
	$ f_2 $	299	99	281	506	157	258	222	102
	$ f_3 $	4	7	3	87	14	0	16	0
	$ z $	0.1	0.07	0.09	0.29	0.09	0.08	0.15	0.05
	$ t(sec.) $	343	97	323	996	186	194	284	119
M1-M2-WGP	$ f_1 $	423	313	345	433	331	335	310	266
	$ f_2 $	465	130	333	796	291	295	485	141
	$ f_3 $	0	0	0	114	0	0	0	0
	$ z $	0.14	0.09	0.10	0.42	0.11	0.08	0.19	0.05
	$ t(sec.) $	1	1	1	4	1	1	1	1
M1-M2-CP	$ f_1 $	450	320	345	393	380	335	350	351
	$ f_2 $	636	144	333	701	237	295	479	208
	$ f_3 $	0	7	0	167	0	0	0	13
	$ z $	0.19	0.1	0.1	0.42	0.11	0.08	0.19	0.1
	$ t(sec.) $	2	1	0	2	1	1	1	1
M1-M2-LWT	$ f_1 $	450	320	345	393	390	335	350	351
	$ f_2 $	636	144	333	701	237	295	479	208
	$ f_3 $	0	7	0	167	10	0	0	13
	$ z $	0.19	0.1	0.1	0.42	0.12	0.08	0.19	0.1
	$ t(sec.) $	8	3	6	14	4	3	5	3
SA-M3	$ f_1 $	473	320	411	366	402	383	328	295
	$ f_2 $	456	82	349	717	335	258	222	99
	$ f_3 $	90	16	37	112	127	0	16	0
	$ z $	0.19	0.08	0.13	0.37	0.25	0.08	0.15	0.05
	$ t(sec.) $	13	12	13	13	12	12	12	13

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Table 3. Results of Small-scale ($ n $ = 8, $ m $ = 3) Problems

Test No		9-1	10-1	11-1	12-1	13-1	14-1	15-1	16-1
M0-WGP	$ f_1 $	296	331	391	239	154	283	183	262
	$ f_2 $	311	325	393	135	191	331	124	117
	$ f_3 $	2	0	0	2	0	0	0	0
	$ z $	0.08	0.13	0.14	0.06	0.06	0.08	0.07	0.04
	$ t(sec.) $	618	2211	1774	337	415	497	439	246
M0-CP	$ f_1 $	296	331	370	198	154	283	143	262
	$ f_2 $	311	325	326	122	191	331	94	117
	$ f_3 $	2	0	70	24	0	0	17	0
	$ z $	0.08	0.13	0.15	0.06	0.06	0.08	0.08	0.04
	$ t(sec.) $	691	1227	1414	401	403	539	252	169
M0-LWT	$ f_1 $	296	331	370	198	154	283	143	262
	$ f_2 $	311	325	326	122	191	331	94	117
	$ f_3 $	2	0	70	24	0	0	17	0
	$ z $	0.08	0.13	0.15	0.06	0.06	0.08	0.08	0.04
	$ t(sec.) $	1045	1231	1980	354	402	876	298	108
M1-M2-WGP	$ f_1 $	341	492	409	286	223	282	243	295
	$ f_2 $	495	870	428	344	431	500	295	342
	$ f_3 $	14	327	155	8	0	0	0	0
	$ z $	0.12	0.58	0.23	0.13	0.14	0.12	0.15	0.09
	$ t(sec.) $	1	2	0	1	1	1	1	1
M1-M2-CP	$ f_1 $	334	411	470	297	230	345	189	530
	$ f_2 $	488	791	567	263	430	600	273	1073
	$ f_3 $	7	274	305	89	0	7	21	0
	$ z $	0.11	0.5	0.34	0.16	0.14	0.15	0.16	0.26
	$ t(sec.) $	1	1	1	1	0	0	0	24
M1-M2-LWT	$ f_1 $	341	411	470	297	230	338	189	295
	$ f_2 $	495	791	567	263	430	600	273	344
	$ f_3 $	14	274	305	89	0	0	21	0
	$ z $	0.12	0.5	0.34	0.16	0.14	0.15	0.16	0.09
	$ t(sec.) $	2	6	3	3	2	2	2	1
SA-M3	$ f_1 $	369	416	469	253	185	336	143	309
	$ f_2 $	317	571	639	128	186	407	94	116
	$ f_3 $	50	282	304	80	44	85	17	0
	$ z $	0.11	0.46	0.36	0.11	0.16	0.18	0.08	0.05
	$ t(sec.) $	12	14	12	13	12	14	14	13

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Table 4. Results of Medium-scale ($ n $ = 40, $ m $ = 2) Problems

Test No		1-1	2-1	3-1	4-1	5-1	6-1	7-1	8-1
M0-GP	$ f_1 $	2365	2135	1928	2115	1574	1805	-	1817
	$ f_2 $	22190	20719	15318	19944	4590	8989	-	10340
	$ f_3 $	396	94	130	124	0	0	-	0
	$ z $	0.27	0.23	0.19	0.28	0.07	0.11	-	0.12
M0-CP	$ f_1 $	2358	2274	1965	1625	1491	1772	1656	1821
	$ f_2 $	18290	17933	15682	14654	4761	5670	8345	9159
	$ f_3 $	280	416	364	301	18	56	0	78
	$ z $	0.23	0.25	0.22	0.22	0.07	0.08	0.10	0.12
M0-LWT	$ f_1 $	5776	8558	7284	2547	2764	5648	2013	4304
	$ f_2 $	21808	33944	26740	56966	7363	16720	19845	14963
	$ f_3 $	577	1458	657	1070	91	240	0	420
	$ z $	0.43	0.74	0.56	0.80	0.16	0.37	0.22	0.33
M1-M2-WGP	$ f_1 $	2077	2022	1779	1771	1447	1789	1474	1463
	$ f_2 $	21191	19565	17681	16175	4779	15204	5719	13613
	$ f_3 $	76	178	4	198	0	0	0	12
	$ z $	0.22	0.23	0.19	0.23	0.06	0.16	0.07	0.14
M1-M2-CP	$ f_1 $	1979	1992	1865	1546	1467	1635	1440	1663
	$ f_2 $	15322	20602	14408	13934	3348	6396	4255	7751
	$ f_3 $	467	128	185	306	0	2	0	186
	$ z $	0.21	0.23	0.18	0.21	0.05	0.08	0.05	0.12
M1-M2-LWT	$ f_1 $	5232	5076	6301	5395	2624	3838	3360	2862
	$ f_2 $	19227	18025	22470	18168	6768	11926	10643	8291
	$ f_3 $	593	489	349	382	83	13	0	41
	$ z $	0.38	0.37	0.45	0.43	0.14	0.22	0.19	0.15
SA-M3	$ f_1 $	1819	1561	1623	1346	1473	1621	1399	1776
	$ f_2 $	8153	5314	3941	4380	2609	1347	2687	3086
	$ f_3 $	479	82	137	290	0	0	0	357
	$ z $	0.14	0.07	0.07	0.1	0.04	0.03	0.04	0.08
	$ t(sec.) $	36	32	38	40	30	31	36	36

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Table 5. Results of Medium-scale ($ n $ = 40, $ m $ = 6) Problems

Test No		9-1	10-1	11-1	12-1	13-1	14-1	15-1	16-1
M0-WGP	$ f_1 $	1539	1896	1515	-	645	709	-	1336
	$ f_2 $	17499	25024	19640	-	3198	4627	-	14430
	$ f_3 $	1092	1396	3426	-	0	3	-	1094
	$ z $	0.27	0.43	0.51	-	0.03	0.05	-	0.29
M0-CP	$ f_1 $	1487	2179	1515	1798	756	784	815	1219
	$ f_2 $	18635	24170	19422	20895	3262	3674	6330	10974
	$ f_3 $	1530	2531	3777	1862	66	7	94	869
	$ z $	0.31	0.55	0.54	0.43	0.04	0.05	0.08	0.23
M0-LWT	$ f_1 $	4093	6059	10466	16743	1002	1947	1726	3677
	$ f_2 $	17052	23743	44256	70611	3124	7355	6522	14022
	$ f_3 $	1434	1912	3808	5674	156	333	126	830
	$ z $	0.38	0.62	1.0	1.0	0.06	0.16	0.12	0.33
M1-M2-WGP	$ f_1 $	1261	1608	1205	1136	598	578	528	633
	$ f_2 $	11303	15041	12840	10871	2202	4029	2336	3878
	$ f_3 $	1350	2117	1138	1645	0	3	2	6
	$ z $	0.23	0.41	0.23	0.29	0.02	0.04	0.02	0.04
M1-M2-CP	$ f_1 $	1278	1674	1217	1069	627	619	531	712
	$ f_2 $	11048	16595	12358	9831	2197	2454	1988	3554
	$ f_3 $	1486	2512	1140	1795	92	50	5	190
	$ z $	0.25	0.47	0.23	0.3	0.04	0.04	0.02	0.06
M1-M2-LWT	$ f_1 $	4806	8617	4168	5018	841	906	727	1253
	$ f_2 $	20346	34350	16584	20148	2316	2578	1798	3606
	$ f_3 $	1711	2793	1427	1619	11	10	56	79
	$ z $	0.46	0.91	0.39	0.51	0.03	0.04	0.04	0.07
SA-M3	$ f_1 $	1249	1807	1047	977	876	651	639	738
	$ f_2 $	9711	14617	6886	5216	2845	1635	1121	1539
	$ f_3 $	1941	2752	1318	1327	315	28	39	457
	$ z $	0.27	0.48	0.20	0.21	0.08	0.03	0.03	0.08
	$ t(sec.) $	231	232	198	235	34	33	37	38

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Table 6. Results of Large-scale ($ n $ = 100, $ m $ = 2) Problems

Test No		1-1	2-1	3-1	4-1	5-1	6-1	7-1	8-1
M0-WGP	$ f_1 $	-	-	-	-	-	-	-	-
	$ f_2 $	-	-	-	-	-	-	-	-
	$ f_3 $	-	-	-	-	-	-	-	-
	$ z $	-	-	-	-	-	-	-	-
M0-CP	$ f_1 $	-	-	-	-	-	-	-	-
	$ f_2 $	-	-	-	-	-	-	-	-
	$ f_3 $	-	-	-	-	-	-	-	-
	$ z $	-	-	-	-	-	-	-	-
M0-LWT	$ f_1 $	5499	7943	-	-	5224	5683	-	-
	$ f_2 $	371810	394667	-	-	394518	388008	-	-
	$ f_3 $	1399	5206	-	-	942	1450	-	-
	$ z $	0.71	0.89	-	-	0.72	0.76	-	-
M1-M2-WGP	$ f_1 $	5737	8535	6578	6779	5073	5495	5340	5873
	$ f_2 $	133487	275567	218122	226738	114247	154587	1222567	151816
	$ f_3 $	1928	6108	3476	3618	310	1371	64	1643
	$ z $	0.35	0.75	0.55	0.53	0.24	0.39	0.23	0.4
M1-M2-CP	$ f_1 $	-	-	-	-	5196	-	-	6133
	$ f_2 $	-	-	-	-	122797	-	-	148961
	$ f_3 $	-	-	-	-	847	-	-	1784
	$ z $	-	-	-	-	0.29	-	-	0.41
M1-M2-LWT	$ f_1 $	4975	7435	5715	-	-	5068	5098	6142
	$ f_2 $	371810	394667	381471	-	-	388008	405443	393012
	$ f_3 $	797	3158	1725	-	-	999	336	1841
	$ z $	0.67	0.80	0.72	-	-	0.72	0.66	0.79
SA-M3	$ f_1 $	3536	4105	3452	3849	3124	3530	4285	4252
	$ f_2 $	18409	24746	26997	23597	15347	20652	9298	26957
	$ f_3 $	192	158	241	86	0	173	0	108
	$ z $	0.05	0.07	0.07	0.06	0.04	0.06	0.04	0.07
	$ t(sec.) $	66	70	72	82	64	65	74	68

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Table 7. Results of Large-scale ($ n $ = 100, $ m $ = 8) Problems

Test No		9-1	10-1	11-1	12-1	13-1	14-1	15-1	16-1
M0-WGP	$ f_1 $	-	-	-	-	-	-	-	-
	$ f_2 $	-	-	-	-	-	-	-	-
	$ f_3 $	-	-	-	-	-	-	-	-
	$ z $	-	-	-	-	-	-	-	-
M0-CP	$ f_1 $	-	-	-	-	-	-	-	-
	$ f_2 $	-	-	-	-	-	-	-	-
	$ f_3 $	-	-	-	-	-	-	-	-
	$ z $	-	-	-	-	-	-	-	-
M0-LWT	$ f_1 $	-	-	-	-	-	-	-	-
	$ f_2 $	-	-	-	-	-	-	-	-
	$ f_3 $	-	-	-	-	-	-	-	-
	$ z $	-	-	-	-	-	-	-	-
M1-M2-WGP	$ f_1 $	1865	4259	1637	4430	1048	578	1093	1744
	$ f_2 $	49222	144672	38766	156241	15984	4029	17236	399726
	$ f_3 $	3562	17320	2274	22013	1	3	0	2472
	$ z $	0.24	0.95	0.15	1.0	0.03	0.04	0.03	0.25
M1-M2-CP	$ f_1 $	1862	4431	2063	4710	1415	2107	1291	2050
	$ f_2 $	56691	158564	59000	186431	25063	41366	21170	42777
	$ f_3 $	3674	19700	4637	22460	1388	1930	125	1954
	$ z $	0.25	1.0	0.27	1.0	0.12	0.20	0.04	0.22
M1-M2-LWT	$ f_1 $	11668	-	13751	-	2141	16990	2203	14119
	$ f_2 $	123224	-	142194	-	15990	178534	17515	140421
	$ f_3 $	3611	-	4665	-	0	3370	71	2534
	$ z $	0.48	-	0.54	-	0.04	0.66	0.05	0.57
SA-M3	$ f_1 $	1617	4096	1443	3695	1113	1746	1095	1350
	$ f_2 $	25013	109062	18602	93653	3156	11361	5052	9681
	$ f_3 $	2138	15728	2054	13899	0	1535	88	878
	$ z $	0.14	0.84	0.11	0.70	0.01	0.13	0.02	0.09
	$ t(sec.) $	90	276	275	267	72	173	69	277

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