CHEMICAL REACTION OPTIMIZATION METAHEURISTIC FOR LOCATING SERVICE STATIONS THROUGH THE CAPACITATED <i>P</i>-MEDIAN PROBLEM

Silva, Danilo César Azeredo; Mestria, Mário

doi:10.1590/0101-7438.2018.038.03.0441

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Articles • Pesqui. Oper. 38 (3) • Sep-Dec 2018 • https://doi.org/10.1590/0101-7438.2018.038.03.0441 copy

CHEMICAL REACTION OPTIMIZATION METAHEURISTIC FOR LOCATING SERVICE STATIONS THROUGH THE CAPACITATED P-MEDIAN PROBLEM

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Chemical Reaction Optimization (CRO) is a metaheuristic for solving optimization problems, which mimics the interactions between molecules in a chemical reaction with the purpose of achieving a stable, low-energy state. In the present work, we utilize the CRO metaheuristic to solve, in an efficient manner, the capacitated p-median problem, in order to locate service stations. Results from solving small to medium-sized problems available in the literature, with up to 724 notes and 200 medians, are compared to their optimal or best-known values. Results show that CRO results are comparable, in terms of accuracy and execution time, to many existing successfulmetaheuristics, as well as exact and hybrid methods, having exceeding those in some cases.

Keywords:
Chemical Reaction Optimization capacitated p-median; metaheuristics

CRO’s elementary reactions characteristics
Extent of change	Number of molecules
Large	Unimolecular	Intermolecular
Large	Decomposition	Synthesis
Small	On-Wall ineffective collision	Intermolecular ineffective collision

Instance	Nodes	Medians ( p)	Max Iterations	Max Iterations without Improv.	Optimal Obj.	Min Obj.	Avg. gap (%)	Std Dev.
sjc1	100	10	50	50	17288.99	17393.94	1.34	147.86
sjc1	100	10	100	100	17288.99	17288.99	1.13	137.96
sjc1	100	10	150	150	17288.99	17288.99	0.85	157.41
sjc1	100	10	200	200	17288.99	17288.99	0.66	76.19
sjc1	100	10	250	250	17288.99	17288.99	0.35	75.11
sjc1	100	10	300	300	17288.99	17288.99	0.65	57.63
sjc1	100	10	350	350	17288.99	17288.99	0.48	68.79
sjc1	100	10	400	400	17288.99	17288.99	0.23	59.86
sjc1	100	10	450	450	17288.99	17288.99	0.19	50.43
sjc1	100	10	500	500	17288.99	17288.99	0.19	47.08
sjc2	200	15	50	50	33270.94	33270.94	0.39	87.51
sjc2	200	15	100	100	33270.94	33270.94	0.30	93.38
sjc2	200	15	150	150	33270.94	33270.94	0.25	66.48
sjc2	200	15	200	200	33270.94	33270.94	0.20	54.58
sjc2	200	15	250	250	33270.94	33270.94	0.25	69.45
sjc2	200	15	300	300	33270.94	33270.94	0.16	51.44
sjc2	200	15	350	350	33270.94	33270.94	0.17	49.86
sjc2	200	15	400	400	33270.94	33270.94	0.15	35.87
sjc2	200	15	450	450	33270.94	33270.94	0.07	36.98
sjc2	200	15	500	500	33270.94	33270.94	0.06	34.81
sjc3a	300	25	50	50	45335.16	45480.07	0.88	151.91
sjc3a	300	25	100	100	45335.16	45423.66	0.62	117.61
sjc3a	300	25	150	150	45335.16	45335.16	0.49	107.97
sjc3a	300	25	200	200	45335.16	45335.16	0.47	94.62
sjc3a	300	25	250	250	45335.16	45380.80	0.38	121.29
sjc3a	300	25	300	300	45335.16	45424.09	0.43	79.18
sjc3a	300	25	350	350	45335.16	45421.15	0.35	52.46
sjc3a	300	25	400	400	45335.16	45335.16	0.33	62.38
sjc3a	300	25	450	450	45335.16	45362.39	0.34	64.29
sjc3a	300	25	500	500	45335.16	45413.05	0.32	62.39
sjc3b	300	30	50	50	40635.90	40669.59	0.77	148.01
sjc3b	300	30	100	100	40635.90	40715.97	0.57	81.50
sjc3b	300	30	150	150	40635.90	40689.03	0.51	103.82
sjc3b	300	30	200	200	40635.90	40714.96	0.46	86.73
sjc3b	300	30	250	250	40635.90	40635.90	0.53	100.04
sjc3b	300	30	300	300	40635.90	40664.81	0.41	61.47
sjc3b	300	30	350	350	40635.90	40651.99	0.34	73.22
sjc3b	300	30	400	400	40635.90	40679.59	0.38	66.54
sjc3b	300	30	450	450	40635.90	40635.90	0.31	81.50
sjc3b	300	30	500	500	40635.90	40651.99	0.27	60.81
sjc4a	402	30	50	50	61925.51	62626.99	2.19	338.45
sjc4a	402	30	100	100	61925.51	62527.66	1.90	301.87
sjc4a	402	30	150	150	61925.51	62460.81	1.50	213.19
sjc4a	402	30	200	200	61925.51	62414.62	1.47	260.53
sjc4a	402	30	250	250	61925.51	62351.35	1.22	130.20
sjc4a	402	30	300	300	61925.51	62330.35	1.20	185.05
sjc4a	402	30	350	350	61925.51	62403.76	1.24	215.55
sjc4a	402	30	400	400	61925.51	62240.75	1.06	199.39
sjc4a	402	30	450	450	61925.51	62311.56	1.05	162.92
sjc4a	402	30	500	500	61925.51	62330.35	1.00	126.85
sjc4b	402	40	50	50	52458.00	52495.56	0.58	110.78
sjc4b	402	40	100	100	52458.00	52629.75	0.57	78.04
sjc4b	402	40	150	150	52458.00	52569.75	0.53	74.85
sjc4b	402	40	200	200	52458.00	52568.76	0.55	71.12
sjc4b	402	40	250	250	52458.00	52495.56	0.45	117.54
sjc4b	402	40	300	300	52458.00	52543.17	0.48	87.26
sjc4b	402	40	350	350	52458.00	52565.41	0.47	80.68
sjc4b	402	40	400	400	52458.00	52548.50	0.49	100.83
sjc4b	402	40	450	450	52458.00	52523.13	0.38	77.31
sjc4b	402	40	500	500	52458.00	52472.64	0.37	67.36

INSTANCE			CPLEX 12.6		CRO
ID	Nodes	Medians (p)	Opt	Execution Time (s)	Optimality Achieved (%)	Avg Gap (%)	Avg Obj.	Min Obj.	Std Dev	Avg Exec. Time (s)	Avg Iterations	Avg Decomp	Avg On-Wall Col.	Avg Synth.	Avg Intermol Col.	Max Iter.	Max Iter. w/o Improv	Pop Size
cpmp01	50	5	713	0.53	100.00	0.00	713.00	713	0.00	0.12	1	0	0	0	0	1000	500	10
cpmp02	50	5	740	0.13	100.00	0.00	740.00	740	0.00	0.12	0	0	0	0	0	1000	500	10
cpmp03	50	5	751	0.33	100.00	0.00	751.00	751	0.00	0.14	18	0	16	0	2	1000	500	10
cpmp04	50	5	651	0.25	100.00	0.00	651.00	651	0.00	0.12	0	0	0	0	0	1000	500	10
cpmp05	50	5	664	0.35	100.00	0.00	664.00	664	0.00	0.13	5	0	5	0	0	1000	500	10
cpmp06	50	5	778	0.25	100.00	0.00	778.00	778	0.00	0.15	10	0	9	0	0	1000	500	10
cpmp07	50	5	787	0.80	100.00	0.00	787.00	787	0.00	0.15	1	0	1	0	0	1000	500	10
cpmp08	50	5	820	3.02	16.67	0.21	821.73	820	0.87	1.86	595	17	514	18	43	1000	500	10
cpmp09	50	5	715	0.39	100.00	0.00	715.00	715	0.00	0.16	11	0	9	0	1	1000	500	10
cpmp10	50	5	829	2.70	76.67	0.13	830.10	829	2.55	0.99	319	10	275	7	26	1000	500	10
cpmp11	100	10	1006	3.77	20.00	0.62	1012.20	1006	3.63	4.99	500	12	433	13	40	1000	500	10
cpmp12	100	10	966	3.96	100.00	0.00	966.00	966	0.00	1.09	113	2	99	0	11	1000	500	10
cpmp13	100	10	1026	1.20	100.00	0.00	1026.00	1026	0.00	0.96	89	0	79	0	8	1000	500	10
cpmp14	100	10	982	5.47	6.67	0.13	983.27	982	1.14	7.15	712	18	618	23	46	1000	500	10
cpmp15	100	10	1091	5.52	10.00	0.08	1091.90	1091	0.31	5.10	533	15	460	15	41	1000	500	10
cpmp16	100	10	954	3.41	100.00	0.00	954.00	954	0.00	1.01	100	1	87	0	10	1000	500	10
cpmp17	100	10	1034	5.27	26.67	0.25	1036.63	1034	4.00	5.86	562	13	486	17	41	1000	500	10
cpmp18	100	10	1043	5.50	100.00	0.00	1043.00	1043	0.00	0.65	60	0	53	0	6	1000	500	10
cpmp19	100	10	1031	4.80	86.67	0.03	1031.27	1031	0.74	3.14	297	6	258	6	24	1000	500	10
cpmp20	100	10	1005	369.61	13.33	0.33	1008.27	1005	2.91	18.43	1831	39	1593	53	123	2000	2000	20
Averages				20.86	72.83	0.09				2.62

Instance		CRO		IRMA (α = 2.4)		SATS	SA	TS
ID	Opt.	Best gap (%)	Exec. Time (s)	Best gap. (%)	Exec. Time (s)	Best gap (%)	Best gap (%)	Best gap (%)
cpmp01	713	0.00	0.11	0.00	0.20	0.00	2.94	2.94
cpmp02	740	0.00	0.11	0.00	0.05	0.00	0.00	0.00
cpmp03	751	0.00	0.12	0.00	0.16	0.00	0.00	0.00
cpmp04	651	0.00	0.11	0.00	0.08	0.00	0.00	0.00
cpmp05	664	0.00	0.12	0.00	0.15	0.00	0.00	0.00
cpmp06	778	0.00	0.12	0.00	0.08	0.00	0.00	0.00
cpmp07	787	0.00	0.12	0.00	0.35	0.00	2.28	0.00
cpmp08	820	0.00	0.86	0.00	4.65	0.00	0.00	0.12
cpmp09	715	0.00	0.12	0.00	0.24	0.00	0.00	0.00
cpmp10	829	0.00	0.13	0.00	0.89	0.00	0.00	0.00
cpmp11	1006	0.00	0.63	0.00	1.83	0.00	0.00	0.29
cpmp12	966	0.00	0.2	0.00	1.23	0.00	0.00	0.20
cpmp13	1026	0.00	0.24	0.00	0.43	0.00	0.00	0.00
cpmp14	982	0.00	1.8	0.00	5.15	0.30	0.00	0.30
cpmp15	1091	0.00	3.09	0.00	6.57	0.00	0.00	0.36
cpmp16	954	0.00	0.23	0.00	0.80	0.00	0.00	0.31
cpmp17	1034	0.00	0.32	0.00	2.16	0.48	0.29	0.58
cpmp18	1043	0.00	0.22	0.00	2.26	0.19	0.19	0.19
cpmp19	1031	0.00	0.32	0.00	2.42	0.00	0.09	0.29
cpmp20	1005	0.00	4.61	0.00	64.49	0.00	1.39	0.00
Averages		0.00	0.68	0.00	4.71	0.05	0.36	0.28

INSTANCE				CRO
ID	Nodes	Medians (p)	Opt	Optimality Achieved (%)	Avg Gap (%)	Avg Obj.	Min Obj.	Std Dev	Avg Exec. Time (s)	Avg Iterations	Avg Decomp	Avg On-Wall Col.	Avg Synth.	Avg Intermol Col.	Max Iter.	Max Iter. w/o Improv	Pop Size	α	β
sjc1	100	10	17288.99	36.67	0.49	17373.46	17288.99	72.23	3.08	221	3	193	0	23	300	150	10	10	50000
sjc2	200	15	33270.94	26.67	0.32	33377.71	33270.94	89.16	1.83	55	0	48	0	6	100	50	10	10	50000
sjc3a	300	25	45335.16	3.33	0.27	45457.79	45335.16	50.44	250.95	4136	473	3212	300	150	5000	2500	50	2	500000
sjc3b	300	30	40635.90	3.33	0.26	40740.45	40635.90	49.08	49.89	791	94	611	62	23	1000	500	10	2	500000
sjc4a	402	30	61925.51	0.00	0.82	62434.54	62157.19	84.78	211.64	854	15	748	9	80	1000	500	10	10	50000
sjc4b	402	40	52458.00	0.00	0.76	52854.97	52651.53	99.18	11.60	66	0	58	0	7	100	50	10	10	50000
Averages					0.49				88.17

Instance		CRO		CPLEX		IRMA (α= 2.4)		SS		VNS		CS		Fen-Cplex
ID	Opt.	Best gap (%)	Exec. Time (s)	Best gap. (%)	Exec. Time (s)	Best gap (%)	Exec. Time (s)	Best gap (%)	Exec. Time (s)	Best gap (%)	Exec. Time (s)	Best gap. (%)	Exec. Time (s)	Best gap (%)	Exec. Time (s)
sjc1	17288.99	0.000	1.17	0.000	4.89	0.000	1.90	0.000	60.00	0.000	50.50	0.000	22.72	0.000	37.60
sjc2	33270.94	0.000	0.53	0.000	11.46	0.000	3.25	0.068	600.00	0.000	44.08	0.000	112.81	0.000	127.90
sjc3a	45335.16	0.000	198.93	0.000	62.20	0.000	23.96	0.006	2307.00	0.000	8580.30	0.000	940.75	0.000	495.10
sjc3b	40635.90	0.000	1.30	0.000	16.14	0.000	2.47	0.000	2308.00	0.000	2292.86	0.000	1887.97	0.000	72.20
sjc4a	61925.51	0.374	116.79	0.000	215.60	0.000	56.67	0.000	6109.00	0.000	4221.47	0.005	2885.11	0.000	1209.50
sjc4b	52458.00	0.369	7.08	0.000	35.90	0.000	6.26	0.140	6106.00	0.023	3471.44	0.140	7626.33	0.000	669.70
Averages		0.124	54.30	0.000	57.70	0.000	15.75	0.036	2915.00	0.004	3110.11	0.024	2245.95	0.000	435.33

Instance				IRMA Phase 2		IRMA Phase 3		CRO
ID	Nodes	Med. (p)	BKS	Avg Gap (%)	Avg Exec. Time (s)	Avg Gap (%)	Avg Exec. Time (s)	Avg Gap (%)	Avg Exec. Time (s)	Runs	BKS (%)	Avg Obj.	Min Obj.	Std Dev	Avg Iter.	Avg Dec.	Avg OnWall Col.	Avg Syn	Avg Intermol. Col.	Max Iter.	Max Iter. w/o Impr
lin318_005	318	5	180281.21	0.00	9.15	-	-	0.00	9.76	10	60.00	180281.67	180281.21	0.58	135	3	117	0	14	200	100
lin318_015	318	15	88901.56	0.00	26.35	-	-	0.08	23.88		60.00	88972.76	88901.56	107.09	268	5	234	2	26	300	150
lin318_040	318	40	47988.38	1.01	222.41	0.14	319.46	0.65	97.76	10	0.00	48302.49	48175.96	143.22	483	4	424	8	44	500	250
lin318_070	318	70	32198.64	0.01	127.45	-	-	1.02	728.49	10	0.00	32528.45	32333.03	110.87	3477	35	3063	42	189	4000	2000
lin318_100	318	100	22942.69	2.23	222.65	0.00	364.87	0.97	2075.61	10	0.00	23165.59	23058.23	54.68	3518	28	3109	35	167	4000	2000
ali535_005	535	5	9956.77	0.00	7.08	0.00	45.42	0.00	2.17	10	100.00	9956.77	9956.77	0.00	0	0	0	0	0	4000	2000
ali535_025	535	25	3695.15	0.24	311.36	0.00	544.26	0.07	348.36	10	0.00	3697.90	3696.56	1.18	883	12	777	19	47	1000	500
ali535_050	535	50	2461.41	1.69	377.28	0.00	726.30	1.58	842.99	10	0.00	2500.24	2480.37	14.50	983	7	867	14	55	1000	500
ali535_100	535	100	1438.42	2.61	362.75	0.02	637.64	3.17	1341.98	10	0.00	1483.98	1468.28	11.08	969	4	863	12	43	1000	500
ali535_150	535	150	1032.28	2.54	366.54	0.00	761.31	3.56	3621.14	10	0.00	1069.06	1057.06	9.78	953	3	845	11	47	1000	500
u724_010	724	10	181782.96	0.00	6.64	0.00	59.65	0.03	354.99	10	30.00	181842.37	181777.72	88.99	381	7	330	4	38	500	250
u724_030	724	30	95034.01	0.01	158.05	0.00	300.72	0.14	152.50	10	0.00	95168.34	95106.33	49.40	408	9	354	7	37	500	250
u724_075	724	75	54735.05	0.08	507.56	0.00	546.39	0.96	439.98	10	0.00	55261.34	55074.78	99.18	486	3	434	9	36	500	250
u724_125	724	125	38976.76	0.28	509.01	0.02	643.31	2.52	761.39	10	0	39957.9	39686.84	185.7	981	4	871	12	57	1000	500
u724_200	724	200	28079.97	0.10	508.81	0.11	706.29	2.75	2710.03	10	0	28853.36	28690.35	140.68	976	3	861	11	57	1000	500
Averages				0.72	248.21	0.02	471.30	1.17	900.74

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[1] *Corresponding author.