A BI-OBJECTIVE MULTIPERIOD ONE-DIMENSIONAL CUTTING STOCK PROBLEM

Pierini, Livia Maria; Poldi, Kelly Cristina

doi:10.1590/0101-7438.2022.042.00258432

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ARTICLES • Pesqui. Oper. 42 • 2022 • https://doi.org/10.1590/0101-7438.2022.042.00258432 copy

A BI-OBJECTIVE MULTIPERIOD ONE-DIMENSIONAL CUTTING STOCK PROBLEM

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

In this paper, we investigate the bi-objective multiperiod one-dimensional cutting stock problem that seeks to minimize the cost of production associated with the total length of cut objects (waste) and the inventory costs related to objects and items. A mathematical model is presented and heuristically solved by a column generation method. Computational tests were performed using the Weighted Sum method, the ε-Constraint method and a variation of the Benson method. The Pearson correlation coefficient was calculated in order to investigate the trade-off between the conflicting objectives of the problem. The results confirmed a strong negative correlation between the objective functions of the problem. All the applied scalar methods were able to find multiple efficient solutions for the problem in a reasonable computational time; however, the ε-Constraint and the modified Benson methods performed better.

KEYWORDS:
cutting stock problem; bi-objective optimization; ε-constraint method; weighted sum method; Benson method

Class	Items (N)	Periods (T)	Objects (M)
1	5	8	2
2	10	8	2
3	20	8	2
4	5	12	2
5	10	12	2
6	20	12	2

Instances	Class
Instances	1	2	3	4	5	6
1	-0.99957	-0.99880	-0.99233	-0.99965	-0.99593	-0.99745
2	-0.99656	-0.99828	-0.99206	-0.98776	-0.99450	-0.99981
3	-0.99656	-0.99828	-0.99206	-0.98776	-0.99450	-0.99981
4	-0.99630	-0.99845	-0.99739	-0.99999	-0.99860	-0.98629
5	-0.97490	-0.99703	-0.99985	-0.98151	-0.99972	-0.99961
6	-0.99884	-0.99739	-0.99924	-0.99986	-0.99681	-0.99719
7	-0.99792	-0.99961	-0.99983	-1.00000	-0.99663	-0.99941
8	-0.98504	-0.99420	-1.00000	-0.99541	-0.99987	-0.99811
9	-0.99089	-0.99737	-0.99719	-0.99792	-0.99981	-0.99974
10	-0.99831	-0.99880	-0.99912	-0.98580	-0.99753	-0.99945
Average	-0.99349	-0.99782	-0.99691	-0.99357	-0.99739	-0.99769

Instances	Class
Instances	1	2	3	4	5	6
1	-0.98424	-0.98013	-0.99078	-0.98667	-0.99358	-0.98875
2	-0.99291	-0.98901	-0.96467	-0.99067	-0.99762	-0.99902
3	-0.97251	-0.99540	-0.98907	-0.99175	-0.99436	-0.94412
4	-0.99916	-0.98953	-0.97096	-0.99890	-0.99931	-0.97764
5	-0.96127	-0.98554	-0.99879	-0.98933	-0.99995	-0.98996
6	-0.98191	-0.98475	-0.96764	-0.99859	-0.99542	-0.98148
7	-0.97907	-0.96559	-0.98584	-0.98291	-0.99502	-0.97071
8	-0.96555	-0.97456	-0.97610	-0.98619	-0.96437	-0.95723
9	-0.96744	-0.99378	-0.99707	-0.99006	-0.99998	-0.99842
10	-0.96081	-0.97924	-0.99687	-0.99154	-0.99926	-0.98607
Average	-0.97649	-0.98375	-0.98378	-0.99066	-0.99389	-0.97934

Class	Instances										Average
Class	1	2	3	4	5	6	7	8	9	10	Average
1	4	11	10	11	7	16	14	16	11	16	11.6
2	9	6	18	10	10	9	7	17	14	12	11.2
3	8	12	13	9	9	16	10	18	6	10	11.1
4	11	16	22	11	10	12	12	17	21	16	14.8
5	11	9	10	13	4	6	8	12	5	7	8.5
6	13	10	8	13	13	17	15	13	18	12	13.2
										Final average	11.73

Class	Instances										Average
Class	1	2	3	4	5	6	7	8	9	10	Average
1	50	50	50	50	50	50	50	50	50	50	50.0
2	50	50	50	50	50	50	50	50	50	50	50.0
3	50	50	50	50	50	50	50	50	50	50	50.0
4	50	50	50	50	50	50	50	50	50	50	50.0
5	50	50	50	50	50	50	50	50	50	50	49.9
6	50	50	50	50	50	50	50	50	50	50	50.0
										Final average	49.98

Class	Instances										Average
Class	1	2	3	4	5	6	7	8	9	10	Average
1	36.5	36.6	32.6	40.0	34.4	34.6	33.8	33.3	37.7	36.5	35.4
2	40.8	33.4	43.8	35.6	36.3	36.9	38.2	36.7	35.6	35.6	37.3
3	39.4	41.7	41.1	41.0	36.0	41.1	35.2	43.1	40.2	38.0	39.7
4	56.8	55.8	54.4	53.2	53.8	56.9	56.0	52.1	49.5	56.5	54.5
5	62.8	59.3	55.5	56.9	53.2	58.3	53.6	60.7	52.5	52.9	65.6
6	68.6	63.5	65.8	64.6	69.4	63.6	58.8	72.0	66.9	64.4	65.7
										Final average	48.2

Class	Instances										Average
Class	1	2	3	4	5	6	7	8	9	10	Average
1	35.2	33.1	32.8	34.0	33.0	34.5	33.0	33.6	34.3	37.3	34.1
2	33.7	32.6	51.1	34.9	36.6	35.7	36.9	36.9	35.4	34.9	36.9
3	33.6	36.3	37.5	35.6	33.2	37.5	34.8	39.9	37.2	34.4	36.0
4	52.5	51.7	53.3	59.2	50.7	53.5	52.0	50.8	46.7	52.9	52.2
5	53.2	68.1	64.3	52.9	51.1	52.7	52.1	51.5	47.4	51.5	54.5
6	56.4	64.9	57.3	59.6	60.6	57.0	55.3	54.4	57.1	53.5	57.6
										Final average	45.2

Class	Instances										Average
Class	1	2	3	4	5	6	7	8	9	10	Average
1	37.3	33.3	34.1	34.1	32.9	34.5	33.0	34.0	33.6	36.7	34.1
2	33.9	33.1	34.8	34.8	35.9	36.1	35.6	39.9	36.2	35.2	37.1
3	34.1	37.5	36.7	36.7	34.3	38.1	34.7	41.0	37.3	34.0	36.6
4	51.6	51.8	52.4	52.4	50.7	52.9	51.9	50.5	46.6	53.3	51.6
5	53.2	68.6	53.7	53.7	51.9	53.8	52.0	51.5	47.6	51.7	54.9
6	58.3	64.3	61.5	61.5	62.4	58.1	56.5	56.0	55.6	53.7	58.4
										Final average	45.5

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[1] *Corresponding author