LOT SIZING AND CUTTING STOCK PROBLEMS IN A PAPER PRODUCTION PROCESS

Pierini, Livia Maria; Poldi, Kelly Cristina

doi:10.1590/0101-7438.2021.041s1.00235094

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ARTICLES • Pesqui. Oper. 41 (spe) • 2021 • https://doi.org/10.1590/0101-7438.2021.041s1.00235094 copy

LOT SIZING AND CUTTING STOCK PROBLEMS IN A PAPER PRODUCTION PROCESS

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

In many manufacturing industries, the production process involves the production of objects and the cutting of such objects into smaller pieces in order to meet a specific demand. In the optimization of these processes, one can identify the lot sizing and the cutting stock problems. In the literature, these problems are mostly separately dealt. However, treating these two problems in an integrated approach can decrease overall costs. In this research, we deal with the coupled lot sizing and cutting stock problem. It is proposed a mathematical formulation for the production and cutting of paper based on a real case, which considers setup costs and limited machine capacity in the production process of the objects. For the solution of the proposed model, we used a column generation approach and a relax-and-fix heuristic. Computational tests were carried out in order to analyze the methodology used in the resolution of the model. The results showed to be competitive in a reasonable computational time.

Keywords:
cutting stock problem; lot sizing problem; integrated; Relax-and-Fix heuristic; column generation; paper industry

Parameter	Value
Weight of produced reel	$b_{m} = L_{m} ρ$
Production cost of reel	$c_{m t} \in [\begin{matrix} 0.015 & 0.025 \end{matrix}] b_{m}$
Setup cost	$s_{m t} \in [\begin{matrix} 0.03 & 0.05 \end{matrix}] c_{m t}$
Wasted paper in setting up the machine	$f_{m} \in [\begin{matrix} 0.01 & 0.05 \end{matrix}] b_{m}$
Inventory cost for reel	$h_{t} \in [\begin{matrix} 0.0000075 & 0.0000125 \end{matrix}]$
Inventory cost for final items	$σ_{i t} = 0.5 \cdot h_{t}$
Trim loss cost in the cutting process	$c p_{t} = \frac{\sum_{m = 1}^{M} c_{m t}}{M} \cdot 10$
Items length	$l_{i} \in [\begin{matrix} 0.1 & 0.3 \end{matrix}] \cdot \frac{\sum_{m = 1}^{M} L_{m}}{M}$
Demand of final items	$d_{i t} \in [\begin{matrix} 0 & 300 \end{matrix}]$ . If $d_{i t} \leq 50$ , then $d_{i t} = 0$
Capacity of the production machine	$C_{m t} = \frac{b_{m}}{\sum_{m = 1}^{M} b_{m}} \cdot α^{\frac{\sum_{t = 1}^{T} \sum_{m = 1}^{M} (\frac{D_{t}}{M} + f m)}{T}}$

Class	*T/N/C*	Iteration	Generated Col.	GC Time
1	08/05/NC	5.7	12.0	2.6
2	08/10/NC	11.2	43.6	2.3
3	08/20/NC	19.2	95.9	4.4
4	10/05/NC	5.6	12.6	1.4
5	10/10/NC	12.3	43.2	3.2
6	10/20/NC	18.8	86.9	5.8
7	12/05/NC	5.5	14.7	1.3
8	12/10/NC	11.8	48.0	3.7
9	12/20/NC	18.0	101.2	6.5
10	08/05/TC	4.8	9.8	0.9
11	08/10/TC	12.4	42.4	2.7
12	08/20/TC	20.5	78.3	5.3
13	10/05/TC	5.2	13.4	1.0
14	10/10/TC	11.8	39.8	2.9
15	10/20/TC	17.3	82.1	5.6
16	12/05/TC	5.1	12.4	1.2
17	12/10/TC	12.6	40.4	3.6
18	12/20/TC	18.0	88.2	7.0
	Average	12.0	48.1	3.4

Class	*T/N/C*	RF Gap (%)	CPLEX Gap (%)
1	08/05/NC	0.09	0.10
2	08/10/NC	0.07	0.11
3	08/20/NC	0.05	0.10
4	10/05/NC	0.09	0.09
5	10/10/NC	0.05	0.09
6	10/20/NC	0.03	0.09
7	12/05/NC	0.13	0.14
8	12/10/NC	0.05	0.08
9	12/20/NC	0.04	0.11
10	08/05/TC	0.16	0.09
11	08/10/TC	0.09	0.12
12	08/20/TC	0.03	0.10
13	10/05/TC	0.07	0.08
14	10/10/TC	0.09	0.13
15	10/20/TC	0.04	0.10
16	12/05/TC	0.09	0.10
17	12/10/TC	0.05	0.10
18	12/20/TC	0.04	0.10
	Average	0.07	0.10

Class	*T/N/C*	RF Time	CPLEX Time	RF Total	CPLEX Total
1	08/05/NC	0.3	0.1	2.9	2.6
2	08/10/NC	0.3	0.4	2.6	5.1
3	08/20/NC	0.8	28.1	5.2	35.4
4	10/05/NC	0.7	0.1	2.2	1.3
5	10/10/NC	25.0	0.4	28.3	3.8
6	10/20/NC	1.2	111.1	7.0	118.5
7	12/05/NC	0.4	0.1	1.7	1.6
8	12/10/NC	0.5	0.5	4.2	4.3
9	12/20/NC	1.4	105.9	7.9	114.8
10	08/05/TC	0.5	0.0	1.4	0.9
11	08/10/TC	0.3	0.3	3.0	3.7
12	08/20/TC	0.9	65.0	6.2	72.4
13	10/05/TC	0.3	0.1	1.3	1.3
14	10/10/TC	0.4	0.5	3.3	4.1
15	10/20/TC	1.2	25.6	6.8	33.4
16	12/05/TC	0.4	0.1	1.5	1.7
17	12/10/TC	0.5	0.5	4.1	5.0
18	12/20/TC	1.4	23.4	8.4	33.5
	Average	2.0	20.1	5.4	24.7

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