Integrated lot sizing and production scheduling formulations: an application in a refractory cement industry

Alves, Fernanda de Freitas; Nogueira, Thiago Henrique; Henriques, Rafaella de Souza; Castro, Priscila Vieira de

doi:10.1590/0104-530X2196-15

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Articles • Gest. Prod. 23 (1) • Jan-Mar 2016 • https://doi.org/10.1590/0104-530X2196-15 linkcopy

Integrated lot sizing and production scheduling formulations: an application in a refractory cement industry

Authorship SCIMAGO INSTITUTIONS RANKINGS

This work presents an integrated lot sizing and scheduling problem for a large refractory cement manufacturer. Three mathematical formulations were addressed: two already presented in the literature, and one proposed as an alternative to the existing ones. This study aims to compare these formulations with respect to their performance and applicability as a decision support tool. One of these formulations uses continuous variables, whereas the others are based on time-indexed variables. These mathematical models address the specific concept of how variables and parameters are defined, requiring assumptions and particular settings to suit the real problem. In order to consider the different aspects of the practical situation, several instances were generated from uniform distributions based on real information. Extensive computational tests were run and, based on the results, the formulations were evaluated as a decision support tool and their efficiencies were compared.

Keywords:
Scheduling; Lot sizing; Production planning and control; Mathematical programming formulations

Author	Problem	Solution Method	Objective Function
Manne (1960)	Scheduling	-	Minimize the makespan
Sousa & Wolsey (1992)	Scheduling	Cutting plane/Branch and Bound algorithm	Minimize costs/Maximize profits
Tavares (2002)	Scheduling	-	-
Araujo et al. (2004)	Integrated Lot Sizing and Scheduling	Local Search Algorithm	Minimize inventory costs, backorder and setup costs
Leung (2004)	Scheduling	-	-
Toso & Morabito (2005)	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic	Minimize inventory costs and overtime costs
Carvalho & Santos (2006)	Integrated Lot Sizing and Scheduling	Branch and Bound	Minimize setup costs
Brahimi et al. (2006)	Lot Sizing	-	Minimize production costs, setup costs, inventory holding costs and backlogging costs
Santos (2006)	Integrated Lot Sizing and Scheduling	Branch and Bound	Minimize setup costs
Araujo et al. (2007)	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic/Local Search Algorithm	Minimize a penalty-weighted sum of product backlogs, finished inventories and setup changeovers
Bustamante (2007)	Scheduling	Branch and Bound	Minimize the sum of earliness and tardiness costs
Yamashita & Morabito (2007)	Scheduling	Branch and Bound	Minimize total cost allocation of project resources
Toledo et al. (2007)	Integrated Lot Sizing and Scheduling	Branch and Cut/Linear Relaxation	Minimize production costs, setup costs and inventory holding costs
Allahverdi et al. (2008)	Scheduling	-	Minimize setup costs
Chen & Askin (2009)	Scheduling	Implicit enumeration algorithm	Maximize profits
Ferreira et al. (2009)	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic	Minimize the total sum of product inventory, demand backorder, machine changeover and tank changeover costs
Toso et al. (2009)	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic	Minimize the costs of inventory, overtime and setup costs
Bernardes et al. (2010)	Integrated Lot Sizing and Scheduling	Branch and Cut	Minimize inventory costs, backorder and setup costs
Stadtler (2010)	Integrated Lot Sizing and Scheduling	Branch and Bound	Minimize inventory costs and setup costs
Henriques et al. (2010)	Integrated Lot Sizing and Scheduling	Branch and Bound	Minimize inventory costs
Ferreira et al. (2010)	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic	Minimize the total sum of the inventory, backorder and machine changeover costs
Defalque et al. (2011)	Integrated Lot Sizing and Scheduling	Branch and Cut	Minimize inventory costs, backorder and setup costs
Clark et al. (2011)	Integrated Lot Sizing and Scheduling	-	-
Ramos & Oliveira (2011)	Scheduling	Hybrid evolutionary algorithm	Minimize the sum of earliness and tardiness costs
Shim et al. (2011)	Integrated Lot Sizing and Scheduling	Adapted heuristic	Minimize the sum of setup and inventory holding costs
Stadtler & Sahling (2012)	Integrated Lot Sizing and Scheduling	Relax-and-fix/Relax-and-optimize heuristics	Minimize inventory holding costs, setup costs and penalty costs for backorders
Pinedo (2012)	Scheduling	-	-
Rego (2013)	Scheduling	Multi-objective algorithms	Minimize makespan and total weighted lateness
Molina et al. (2013)	Lot Sizing	Modified Lagrangian Relaxation	Minimize inventory holding costs, backorder, setup costs and transportation costs
Seeanner & Meyr (2013)	Integrated Lot Sizing and Scheduling	Relax-and-fix/LP-and-fix heuristics	Minimize the sum of inventory holding costs, setup costs, production costs, costs for standby, external purchase, overtime and holding of Work-In-Progress stock

Input Data	Distribution Value
Processing Time ( $p_{i}$ ) – Hours	U(1,4)
Setup Time ( $S t_{i j}$ ) – Hours	U(0,2)
Demand ( $d_{i t}$ ) – Units	U(0,5 $α_{1}$ )
Cost ( $H_{i} a n d B_{i}$ ) – R$	U(0, $\bar{j}$ )

		MBILP-CH						MBILP-DH						MBILP-DHP
Class	Product Family	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$
1	2	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.8	100.0%	0.0%	7.5	100.0%
1	3	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	2.4	100.0%	0.0%	51.6	100.0%
1	4	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	5.2	100.0%	0.0%	109.2	100.0%
1	5	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	9.0	100.0%	0.0%	315.4	100.0%
1	6	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	15.2	100.0%	-	-	0.0%
1	7	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	20.9	100.0%	-	-	0.0%
1	10	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	-	-	0.0%	-	-	0.0%
1	15	99.9%	0.1	100.0%	100.0%	3600.0	0.0%	84.8%	0.1	100.0%	40.0%	1212.9	67.0%	-	-	0.0%	-	-	0.0%
1	20	99.9%	0.1	100.0%	100.0%	3600.0	0.0%	57.6%	0.2	100.0%	26.0%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
Average		99.9%	0.0	100.0%	22.2%	800.1	78.0%	15.8%	0.1	100.0%	7.3%	534.8	85.0%	0.0%	8.9	67.0%	0.0%	120.9	44.0%
Standard Deviation		0.0%	0.0	0.0	44.1%	1587.4	0.4	32.1%	0.0	0.0	15.0	1217.4	0.3	0.0%	7.8	0.5	0.0%	136.2	0.5
Class	Product Family	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$
2	2	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.0	100.0%	0.0%	0.8	100.0%	0.0%	16.7	100.0%
2	3	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	2.4	100.0%	5.0%	41.1	100.0%
2	4	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	5.1	100.0%	0.0%	297.0	100.0%
2	5	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	9.0	100.0%	0.0%	552.7	100.0%
2	6	33.3%	0.1	100.0%	0.0%	0.1	100.0%	33.3%	0.1	100.0%	0.0%	0.1	100.0%	33.3%	14.2	100.0%	-	-	0.0%
2	7	33.3%	0.1	100.0%	0.0%	0.3	100.0%	33.3%	0.1	100.0%	0.0%	0.7	100.0%	33.3%	20.3	100.0%	-	-	0.0%
2	10	99.9%	0.1	100.0%	78.7%	3600.0	0.0%	41.5%	0.1	100.0%	7.7%	2026.0	67.0%	-	-	0.0%	-	-	0.0%
2	15	99.9%	0.1	100.0%	99.3%	3600.0	0.0%	12.6%	0.1	100.0%	4.3%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
2	20	99.9%	0.1	100.0%	100.0%	3600.0	0.0%	7.9%	0.1	100.0%	3.0%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
Average		40.7%	0.0	100.0%	30.9%	1200.1	67.0%	14.3%	0.1	100.0%	1.7%	1025.2	74.0%	11.1%	8.6	67.0%	1.3%	226.9	44.0%
Standard Deviation		46.4%	0.0	0.0	46.7%	1799.9	0.5	17.1%	0.0	0.0	2.8%	1603.3	0.4	17.2%	7.5	0.5	2.5%	251.5	0.5
Class	Product Family	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$
3	2	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.8	100.0%	0.0%	8.0	100.0%
3	3	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	2.4	100.0%	0.0%	75.8	100.0%
3	4	33.2%	0.1	100.0%	0.0%	0.5	100.0%	2.3%	0.1	100.0%	0.0%	3.4	100.0%	2.3%	5.0	100.0%	33.3%	353.6	67.0%
3	5	75.6%	0.0	100.0%	0.0%	3.3	100.0%	39.4%	0.1	100.0%	0.0%	131.0	100.0%	39.4%	9.1	100.0%	32.0%	3547.5	33.0%
3	6	99.9%	0.0	100.0%	0.0%	6.2	100.0%	18.0%	0.1	100.0%	0.0%	2.5	100.0%	18.0%	14.7	100.0%	-	-	0.0%
3	7	99.8%	0.1	100.0%	0.0%	51.7	100.0%	6.6%	0.1	100.0%	0.0%	82.9	100.0%	6.6%	21.1	100.0%	-	-	0.0%
3	10	99.9%	0.1	100.0%	9.7%	3261.8	33.0%	4.2%	0.2	100.0%	0.7%	2876.3	33.0%	-	-	0.0%	-	-	0.0%
3	15	99.8%	0.1	100.0%	39.3%	3600.0	0.0%	3.0%	0.1	100.0%	0.7%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
3	20	99.8%	0.1	100.0%	53.3%	3600.0	0.0%	1.6%	0.1	100.0%	0.7%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
Average		67.6%	0.0	100.0%	11.4%	1169.3	70.0%	8.3%	0.1	100.0%	0.2%	1144.0	70.0%	11.0%	8.8	67.0%	16.3%	996.2	33.0%
Standard Deviation		44.1%	0.0	0.0	20.4%	1741.3	0.5	12.9%	0.0	0.0	0.3%	1674.7	0.5	15.4%	7.8	0.5	18.9%	1707.4	0.4
Model Average		69.4%	0.0	100.0%	21.5%	1056.5	72.0%	12.8%	0.1	100.0%	3.1%	901.4	77.0%	7.4%	8.8	67.0%	5.9%	448.0	41.0%
Model Standard Deviation		26.2%	0.0	0.0	14.5%	109.8	0.0	10.1%	0.0	0.0	7.8%	246.0	0.1	9.5%	0.2	0.0	10.2%	875.8	0.0

Formulations	Binary Variables	Constraints
MBILP-CH	O( $\bar{j}$ ²)	O( $\bar{j}$ ²)
MBILP-DH	O( $\bar{j}$ ³)	O( $\bar{j}$ ³)
MBILP-DHP	O( $\bar{j} \bar{s}$ )	O( $\bar{j}$ ^{2 $\bar{s}$ 2})

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E-mail: gp@dep.ufscar.br

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