THEORETICAL INSIGHTS AND A NEW CLASS OF VALID INEQUALITIES FOR THE TEMPORAL BIN PACKING PROBLEM WITH FIRE-UPS

Martinovic, John; Strasdat, Nico

doi:10.1590/0101-7438.2023.043.00283503

John Martinovic ^**Corresponding author

Institute of Numerical Mathematics, Technische Universität Dresden, 01062, Dresden, Germany - E-mail: john.martinovic@tu-dresden.de

https://orcid.org/0000-0002-5428-9351

Nico Strasdat

Institute of Numerical Mathematics, Technische Universität Dresden, 01062, Dresden, Germany - E-mail: nico.strasdat@tu-dresden.de

https://orcid.org/0000-0003-0953-3367

*Corresponding author

Conflicts of interest: The authors declare that they do not have any conflicts of interest.

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Figure 1
An assignment of eight items to one bin of size C=7, following the basic idea from (Dell’Amico et al., 2020DELL’AMICO M, FURINI F & IORI M. 2020. A branch-and-price algorithm for the temporal bin packing problem. Computers & Operations Research, 114: 104825., Fig. 2). The blue item i=8 with [s _i , e _i )=[2, 5) and c _i =2 is not placed in a contiguous manner. Hence, this configuration would not be feasible for classical 2D packing problems.

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Figure 2
Illustration of instance E

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Figure 3
The summed activity Y _t: =Σ_k∈K y _tk over time in an optimal solution of E. The left picture shows the development of Y _t for Model 1, whereas the right picture visualizes the situation after having modified the model by using w _tk and for all t∈T. In both cases, the red regions define some lower and upper bounds for Y _t that are derived from , , and of M1. Even if this is not clearly visible in the illustrations, please note that the value of Y ₄ is zero in both scenarios.

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Table 1
An overview of the (average) LP bounds obtained from different variants of M1-type formulations.

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Table 2
An overview of the (average) LP bounds obtained from different variants of M1-type formulations for very large instances.

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Table 3
A brief numerical comparison for two versions of M1**, with and without (R0). The column ’Diff’ measures the percentage difference between the new and the old value.

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Table 4
The average LP bound obtained from M1** and M1**+(R0). By ’Gap’, here we mean the relative discrepancy between the two LP values (transformed into a percentage scale). The averages are built based on 20 instances each.

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Table 5
Average solution times and number of instances solved to optimality (in brackets) for M1** and M1**+(R0) where r∈{0, 0.25}. Moreover, the average exit gap (UB-LB)/LB given by the solver at the end of the time limit is reported. The averages are built based on 20 instances each.

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Table 6
Average solution times and number of instances solved to optimality (in brackets) for M1** and M1**+(R0) where r∈{0.5, 0.75}. Moreover, the average exit gap (UB-LB)/LB given by the solver at the end of the time limit is reported. The averages are built based on 20 instances each.

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Table B1
An overall comparison of M1** and M1**+(R0). We use boldface to indicate the better formulation (with respect to instances solved to proven optimality, breaking ties by the smaller required computation time).

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n	$\bar{s}$	d_i	c_i	M1	M1+(R0)	M1*	M1*+(R0)	M1**	M1**+(R0)
50	50	d_S	c_L	18.0	19.2	19.6	19.6	19.6	19.6
			c_H	22.2	24.4	25.6	25.6	25.6	25.6
		d_L	c_L	26.8	29.4	30.0	30.0	30.0	30.0
			c_H	40.0	43.8	46.0	46.0	46.0	46.0
	60	d_S	c_L	15.4	16.6	16.8	16.8	16.8	16.8
			c_H	19.6	22.0	23.6	23.6	23.6	23.6
		d_L	c_L	27.0	28.8	29.6	29.6	29.6	29.6
			c_H	35.2	39.6	41.6	41.6	41.6	41.6
Average				25.5	28.0	29.1	29.1	29.1	29.1
100	100	d_S	c_L	18.8	21.8	22.4	22.4	22.4	22.4
			c_H	25.2	30.8	33.6	33.6	33.6	33.6
		d_L	c_L	31.0	34.2	34.4	34.4	34.4	34.4
			c_H	41.6	47.4	49.2	49.2	49.4	49.4
	120	d_S	c_L	17.2	19.6	20.0	20.0	20.0	20.0
			c_H	22.0	25.6	27.2	27.4	27.4	28.4
		d_L	c_L	25.0	30.0	30.8	30.8	30.8	30.8
			c_H	37.8	42.8	44.4	44.4	44.4	44.4
Average				27.3	31.5	32.8	32.8	32.8	32.9
150	150	d_S	c_L	17.8	21.6	21.6	21.6	21.6	21.6
			c_H	29.2	32.0	34.8	36.0	34.8	36.8
		d_L	c_L	33.0	38.4	38.8	38.8	38.8	38.8
			c_H	43.8	50.8	52.8	52.8	52.8	52.8
	180	d_S	c_L	15.2	19.2	19.6	19.6	19.6	19.6
			c_H	21.8	26.4	28.0	31.2	28.0	32.4
		d_L	c_L	26.6	31.0	31.6	31.6	31.6	31.6
			c_H	37.6	44.6	47.6	47.6	47.6	47.6
Average				28.1	33.0	34.4	34.9	34.4	35.1
200	200	d_S	c_L	18.8	23.6	24.4	24.4	24.4	24.4
			c_H	25.2	30.0	31.6	32.4	31.6	33.2
		d_L	c_L	32.4	37.4	38.0	38.0	38.0	38.0
			c_H	43.4	51.4	53.6	53.6	53.6	53.6
	240	d_S	c_L	16.0	20.6	21.2	21.2	21.6	21.6
			c_H	23.8	28.2	30.6	34.6	30.6	37.0
		d_L	c_L	27.6	31.8	32.4	32.4	32.4	32.4
			c_H	36.6	43.4	46.0	46.2	46.0	46.2
Average				28.0	33.3	34.7	35.3	34.8	35.8
Total: Average				27.2	31.5	32.7	33.0	32.8	33.2

n	$\bar{s}$	d_i	c_i	M1	M1+(R0)	M1*	M1*+(R0)	M1**	M1**+(R0)
500	500	d_S	c_L	19.6	24.0	24.8	25.4	24.8	26.0
			c_H	28.2	34.0	37.0	47.0	37.6	49.2
		d_L	c_L	32.6	40.0	40.8	40.8	40.8	41.0
			c_H	44.4	55.0	59.2	59.2	59.2	59.4
	600	d_S	c_L	16.6	21.8	22.4	23.0	22.4	25.0
			c_H	22.8	28.0	30.8	46.6	32.6	49.8
		d_L	c_L	27.8	35.2	35.6	35.6	35.6	35.6
			c_H	39.4	48.0	50.0	50.8	50.0	51.0
Average				28.9	35.8	37.6	41.0	37.9	42.1
1000	1000	d_S	c_L	20.2	24.4	24.8	26.6	25.0	28.2
			c_H	30.6	37.0	45.4	72.6	47.4	77.6
		d_L	c_L	32.4	39.0	39.6	39.6	39.6	39.6
			c_H	48.0	58.6	63.2	64.6	63.2	65.2
	1200	d_S	c_L	17.0	21.4	22.0	26.0	22.0	29.2
			c_H	26.8	31.8	41.2	71.2	44.8	76.4
		d_L	c_L	30.6	35.0	36.0	36.0	36.0	36.0
			c_H	40.2	48.4	52.0	52.0	52.0	53.4
Average				30.7	37.0	40.5	48.6	41.2	50.7

indicator	units	M1**	M1**+(R0)	Diff
n _var	10³	6.1	6.6	+8.2%
n _con	10³	12.6	14.1	+11.9%
n _nz	10³	110.0	114.0	+2.7%
z _LP	-	32.8	33.2	+1.2%
Exit gap	%	3.4	3.0	-11.8%
opt	-	104	106	+1.9%
t	s	705.5	703.7	-0.3%

n	r=0.00		r=0.25		r=0.50		r=0.75
n	M1**	M1**+(R0)	M1**	M1**+(R0)	M1**	M1**+(R0)	M1**	M1**+(R0)
100	25.9	26.6	25.9	26.4	25.9	26.1	25.9	26.0
150	25.9	27.9	25.9	27.9	26.0	27.6	26.2	27.3
200	27.0	30.3	27.0	29.6	27.0	29.2	27.3	28.8
Aver.	26.2	28.3	26.3	27.9	26.3	27.6	26.5	27.3
Gap	8.0%		6.0%		4.6%		3.0%

n	r=0.00		r=0.25
n	M1**	M1**+(R0)	M1**	M1**+(R0)
100	870.9 (11)	774.3 (12)	831.2 (12)	467.5 (18)
150	976.9 (10)	1013.5 (9)	902.4 (11)	868.6 (11)
200	1201.6 (7)	1042.7 (10)	1192.7 (7)	915.0 (10)
Total	1016.5 (28)	943.5 (31)	975.4 (30)	750.4 (39)
Exit Gap	14.7%	11.1%	6.1%	3.8%

n	r=0.50		r=0.75
n	M1**	M1**+(R0)	M1**	M1**+(R0)
100	288.1 (18)	91.0 (20)	142.4 (19)	39.7 (20)
150	773.2 (12)	662.7 (14)	612.0 (16)	396.5 (17)
200	1081.6 (9)	910.7 (10)	918.9 (10)	809.8 (13)
Total	714.3 (39)	554.8 (44)	557.7 (45)	415.3 (50)
Exit Gap	6.1%	3.8%	8.2%	5.7%

Y_{t} : = \{\begin{cases} m i n \{Σ_{i \in I_{t}} c_{i}, m a x \{Y_{t - 1}, \frac{Σ_{i \in I_{t}} c_{i}}{C}\}\}, i f t \notin T_{S}, \\ m i n \{Σ_{i \in I_{t}} c_{i}, m a x \{1, Y_{t - 1}, \frac{Σ_{i \in I_{t}} c_{i}}{C}\}\}, i f t \in T_{S,} \end{cases} (t \in T),

Y_{t} : = \{\begin{cases} m a x \{Y_{t - 1}, \frac{Σ_{i \in I_{t}} c_{i}}{C}\}, i f t \notin T_{S}, \\ m a x \{1, Y_{t - 1}, \frac{Σ_{i \in I_{t}} c_{i}}{C}\}, i f t \in T_{S,} \end{cases}

n	$\bar{s}$	d_i	c_i	M1**		M1**+(R0)
n	$\bar{s}$	d_i	c_i	t	opt	t	opt
50	50	d_S	c_L	2.3	(5)	4.0	(5)
			c_H	2.5	(5)	1.3	(5)
		d_L	c_L	360.3	(4)	360.8	(4)
			c_H	0.4	(5)	0.5	(5)
	60	d_S	c_L	1.7	(5)	1.2	(5)
			c_H	1.8	(5)	1.7	(5)
		d_L	c_L	1.6	(5)	11.2	(5)
			c_H	1.3	(5)	0.6	(5)
Average (Sum)				46.5	(39)	47.7	(39)
100	100	d_S	c_L	1.1	(5)	3.6	(5)
			c_H	120.3	(5)	78.6	(5)
		d_L	c_L	1442.7	(1)	1449.8	(1)
			c_H	1090.1	(2)	1092.9	(2)
	120	d_S	c_L	69.2	(5)	85.3	(5)
			c_H	474.8	(4)	83.6	(5)
		d_L	c_L	433.2	(4)	685.9	(4)
			c_H	628.1	(4)	546.9	(4)
Average (Sum)				532.4	(30)	503.3	(31)
150	150	d_S	c_L	49.8	(5)	85.7	(5)
			c_H	1722.5	(1)	1464.1	(1)
		d_L	c_L	1182.8	(2)	1462.2	(1)
			c_H	1485.7	(2)	1372.0	(2)
	180	d_S	c_L	15.1	(5)	37.2	(5)
			c_H	939.8	(3)	853.9	(3)
		d_L	c_L	1408.5	(2)	1198.4	(3)
			c_H	1509.8	(1)	1494.0	(1)
Average (Sum)				1039.2	(21)	995.9	(21)
200	200	d_S	c_L	44.4	(5)	99.2	(5)
			c_H	1800.0	(0)	1800.0	(0)
		d_L	c_L	1494.2	(1)	1800.0	(0)
			c_H	1496.1	(1)	1624.6	(1)
	240	d_S	c_L	83.6	(5)	123.1	(5)
			c_H	1800.0	(0)	1693.9	(2)
		d_L	c_L	1110.7	(2)	1201.9	(2)
			c_H	1800.0	(0)	1800.0	(0)
Average (Sum)				1203.6	(14)	1267.8	(15)
Total: Average (Sum)				705.5	(104)	703.7	(106)

[1] *Corresponding author

Brasil

Brasil

THEORETICAL INSIGHTS AND A NEW CLASS OF VALID INEQUALITIES FOR THE TEMPORAL BIN PACKING PROBLEM WITH FIRE-UPS

ABSTRACT