THE REEL ALLOCATION PROBLEM OF A CORRUGATED PACKAGING COMPANY

Ferreira, Marcelo; Camargo, Victor Claudio Bento; Araujo, Silvio Alexandre de

doi:10.1590/0101-7438.2020.040.00229708

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ARTICLES • Pesqui. Oper. 40 • 2020 • https://doi.org/10.1590/0101-7438.2020.040.00229708 copy

THE REEL ALLOCATION PROBLEM OF A CORRUGATED PACKAGING COMPANY

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

In this paper we address a reel allocation problem that appears in a corrugated cardboard packaging company. Given a production plan which incorporates customer orders and a set of reels in stock, the decision is how to allocate these reels to the corrugator to produce the ordered corrugated cardboard sheets, aiming to minimize the number of reels used, any partial consumption of reels, the unusable leftovers and the number of production stoppages. A mixed-integer mathematical model is proposed and computational tests were performed using instances generated based on real data. The computational results show that the proposed mathematical model is consistent with the reality of the studied company and brings interesting insights that can be used in the decision-making.

Keywords:
mixed-integer programming; mathematical modeling; corrugated packaging industry; reel allocation problem; corrugator

	∑Y	∑T	∑D	∑P	OF
S ¹	11	3	0	0	56.21
S ²	16	2	0	2	1051.30
S ³	13	3	50	0	81.98
S ⁴	14	0	0	0	71.54
S ⁵	10	0	0	0	51.10

	Paper lengths on reels
Stock profile	300m - 749m	750m - 6499m	6500m
Small (S)	90%	5%	5%
Medium (M)	80%	10%	10%
Large (L)	70%	15%	15%

Item analyzed	Totals	Percentages
Total instances analyzed	540	100%
Instances with proof of optimality	266	49.25%
Instances with no proof of optimality	274	50.75%
Instances with unusable reels	36	6.67%
Instances with stoppages	60	11.11%
Instances with all reels totally used	341	63.14%
Instances with one partially used reel	63	11.66%
Instances with two partially used reels	45	8.33%
Instances with three partially used reels	44	8.14%
Instances with four partially used reels	24	4.44%
Instances with five partially used reels	23	4.25%

		OF	T	B	FU	PU	NU	LNU	S	GAP
01	SSSS	152.73	547.50	29.80	29.70	0.10	0.10	0.30	0.00	0.41%
02	SSSM	274.41	186.55	53.70	53.70	0.00	0.00	0.00	0.00	0.03%
03	SSSL	8702.17	7.32	85.90	85.90	0.00	0.00	0.00	17.20	0.00%
04	SSMS	97.67	830.45	16.90	14.30	2.60	0.00	0.00	0.00	1.52%
05	SSMM	205.91	191.18	40.20	40.10	0.10	0.10	1.10	0.00	0.25%
06	SSML	411.87	11.48	80.60	80.60	0.00	0.00	0.00	0.00	0.00%
07	SSLS	116.35	1462.04	20.30	17.40	2.90	0.00	0.00	0.00	2.97%
08	SSLM	158.48	1800.00	29.80	28.50	1.30	0.50	10.90	0.00	5.93%
09	SSLL	388.30	373.48	75.90	75.80	0.10	0.10	0.40	0.00	0.13%
10	SMSS	296.38	190.40	58.00	58.00	0.00	0.00	0.00	0.00	0.05%
11	SMSM	549.33	372.87	107.50	107.50	0.00	0.00	0.00	0.00	0.02%
12	SMSL	19612.76	11.40	171.50	171.50	0.00	0.00	0.00	39.00	0.00%
13	SMMS	173.47	1800.00	32.40	30.60	1.80	0.30	1.50	0.00	5.73%
14	SMMM	392.96	195.72	76.90	76.90	0.00	0.00	0.00	0.00	0.08%
15	SMML	818.11	199.60	160.10	160.10	0.00	0.00	0.00	0.00	0.00%
16	SMLS	211.13	1632.34	39.70	37.80	1.90	0.00	0.00	0.00	3.85%
17	SMLM	305.56	1452.71	59.20	58.50	0.70	0.00	0.00	0.00	2.15%
18	SMLL	772.63	217.67	151.20	151.20	0.00	0.00	0.00	0.00	0.01%
19	SLSS	436.91	200.11	85.50	85.50	0.00	0.00	0.00	0.00	0.01%
20	SLSM	817.60	28.38	160.00	160.00	0.00	0.00	0.00	0.00	0.00%
21	SLSL	33741.62	562.03	257.00	257.00	0.00	0.00	0.00	67.50	0.07%
22	SLMS	250.28	1800.00	48.00	46.90	1.10	0.10	4.30	0.00	2.65%
23	SLMM	602.98	31.45	118.00	118.00	0.00	0.00	0.00	0.00	0.00%
24	SLML	1225.89	416.55	239.90	239.90	0.00	0.00	0.00	0.00	0.01%
25	SLLS	305.25	1534.72	58.80	57.70	1.10	0.00	0.00	0.00	1.89%
26	SLLM	450.46	1800.00	87.30	86.30	1.00	0.10	0.20	0.00	1.16%
27	SLLL	1152.82	213.84	225.60	225.60	0.00	0.00	0.00	0.00	0.00%
28	DSSS	152.64	1439.39	29.70	29.50	0.20	0.10	0.10	0.00	2.34%
29	DSSM	272.87	303.31	53.40	53.40	0.00	0.00	0.00	0.00	0.10%
30	DSSL	5434.22	144.88	85.60	85.50	0.10	0.10	0.10	10.40	0.00%
31	DSMS	110.84	1800.00	17.60	12.90	4.70	0.10	9.20	0.00	14.00%
32	DSMM	204.19	1252.65	39.70	39.40	0.30	0.30	0.40	0.00	1.35%
33	DSML	413.33	529.90	80.80	80.70	0.10	0.10	0.10	0.00	0.25%
34	DSLS	134.17	1800.00	22.00	17.00	5.00	0.00	0.00	0.00	14.35%
35	DSLM	170.53	1645.78	31.30	28.90	2.40	0.40	3.00	0.00	10.78%
36	DSLL	391.67	1172.98	76.30	75.90	0.40	0.40	0.70	0.00	0.71%
37	DMSS	293.75	1357.39	57.40	57.30	0.10	0.00	0.00	0.00	1.53%
38	DMSM	535.53	1134.51	104.80	104.80	0.00	0.00	0.00	0.00	0.52%
39	DMSL	21248.21	515.41	171.90	171.90	0.00	0.00	0.00	42.40	0.09%
40	DMMS	185.92	1800.00	33.30	29.70	3.60	0.10	1.90	0.00	10.87%
41	DMMM	406.68	1312.54	79.50	79.40	0.10	0.00	0.00	0.00	0.94%
42	DMML	822.71	809.41	161.00	161.00	0.00	0.00	0.00	0.00	0.16%
43	DMLS	220.71	1800.00	39.70	35.60	4.10	0.10	0.10	0.00	8.38%
44	DMLM	313.25	1800.00	58.90	56.10	2.80	0.40	1.90	0.00	5.87%
45	DMLL	773.08	965.35	151.20	151.10	0.10	0.10	0.20	0.00	0.29%
46	DLSS	451.14	936.51	88.00	88.00	0.10	0.00	0.00	0.00	0.66%
47	DLSM	818.11	1629.38	160.10	160.10	0.00	0.00	0.00	0.00	0.40%
48	DLSL	33888.81	972.64	257.60	257.60	0.00	0.00	0.00	67.80	0.14%
49	DLMS	261.28	1800.00	48.10	44.60	3.50	0.10	5.30	0.00	7.44%
50	DLMM	613.56	1614.38	119.90	119.70	0.20	0.00	0.00	0.00	0.94%
51	DLML	1227.86	1356.66	240.20	240.10	0.10	0.10	0.10	0.00	0.12%
52	DLLS	319.52	1672.56	60.40	57.90	2.50	0.00	0.00	0.00	4.09%
53	DLLM	462.86	1800.00	89.00	87.00	2.00	0.20	0.70	0.00	3.59%
54	DLLL	1155.74	1637.42	226.00	226.00	0.00	0.10	0.10	0.00	0.32%

	OF	T	B	FU	PU	NU	LNU	S	GAP
Single Wall	2689.78	669.86	95.17	94.63	0.54	0.05	0.69	4.58	1.07%
Double Wall	2640.12	1297.78	95.68	94.46	1.21	0.10	0.89	4.47	3.34%

	OF	T	B	FU	PU	NU	LNU	S	GAP
Small Stock	988.46	861.62	48.31	47.18	1.13	0.13	1.46	1.53	3.06%
Medium Stock	2662.90	976.99	95.23	94.39	0.84	0.06	0.31	4.52	2.25%
Large Stock	4343.48	1112.84	142.74	142.07	0.67	0.04	0.59	7.52	1.31%

	OF	T	B	FU	PU	NU	LNU	S	GAP
Short Profile	7093.29	585.55	112.09	112.06	0.03	0.02	0.03	13.57	0.35%
Medium Profile	468.08	987.59	90.73	89.72	1.01	0.07	1.33	0.00	2.57%
Large Profile	433.47	1378.30	83.46	81.87	1.59	0.13	1.01	0.00	3.69%

	OF	T	B	FU	PU	NU	LNU	S	GAP
Small Schedule	231.67	1357.78	43.66	41.69	1.96	0.06	1.26	0.00	4.60%
Medium Schedule	419.74	1031.55	81.61	80.99	0.62	0.11	1.01	0.00	1.89%
Large Schedule	7343.43	562.11	161.02	160.96	0.06	0.06	0.09	13.57	0.13%

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[1] *Corresponding author