Multiple-stage multiple-machine capacitated lot-sizing and scheduling with sequence-dependent setup: A case study in the wheel industry

Lalida Deeratanasrikul; Shinji Mizuno

doi:10.3934/jimo.2016024

Article Contents

2017, Volume 13, Issue 1: 413-428. Doi: 10.3934/jimo.2016024

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Multiple-stage multiple-machine capacitated lot-sizing and scheduling with sequence-dependent setup: A case study in the wheel industry

Graduate School of Decision Science and Technology, Tokyo Institute of Technology, Tokyo 152-8552, Japan

Received: April 30, 2015

Published: August 2017

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Abstract

This paper studies a real-world problem of simultaneous lot-sizing and scheduling in a capacitated flow shop. The problem combines two significant characteristics in production which are multiple-stage production with heterogeneous multiple machines and sequence-dependent setup time. Setup time does not hold the triangle inequality, thus there may be a setup for a product without actual production. Consequently, a novel mixed integer programming (MIP) formulation is proposed and tested on real data sets of wheel production. Exact approaches cannot find a feasible solution for the model in a reasonable time, so MIP-based heuristics are developed to solve the model more quickly. Test results show that the formulation is able to contain the problem requirements and the heuristics are computationally effective. Moreover, the obtained solution can improve on a real practice at the plant.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Production process flow

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Figure 2. Example of bill of materials from one type of first-stage product

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Figure 3. A disconnected subtour and a main sequence

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Figure 4. A subtour connected to a main sequence at the beginning of period

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Figure 5. Relax and fix heuristic on multi-stage and over the periods

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Figure 6. Comparison of total setup time between the company planning and our model

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Figure 7. Comparison of total inventory level between the company planning and our model

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Figure 8. Comparison of total overtime between the company planning and our model

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Table 1. Average objective values in detailed

q	Setup time (sec)			Inventory level (pieces)			Overtime (sec)
q	W=1000	W=100	W=10	W=1000	W=100	W=10	W=1000	W=100	W=10
20	573,750	511,500	407,850	3,843	11,437	13,906	2,247,857	8,029	7,712
100	529,500	521,100	404,400	10,356	11,557	14,133	17,100	7,713	7,712
200	539,100	521,250	395,280	11,535	11,409	13,680	7,868	7,713	7,712
300	545,250	506,850	398,450	11,443	11,257	13,380	8,245	7,712	7,712
400	559,350	519,300	404,850	11,757	11,579	13,737	7,725	7,712	7,712

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Table 2. Numerical results of small problems

Problem size($N \times M \times T$)	$<$1000		1000—4000		4000—6000
	MIP	Heu.	MIP	Heu.	MIP	Heu.
Avg. Time (sec)	8716	958	35226	1090	81646	1774
Avg. Gap (%)	3.94	5.67	5.44	8.63	6.71	9.22
StDev. Gap	1.81	4.06	1.88	6.19	2.73	3.36

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Table 3. Numerical results of real problems by our heuristics

	Avg. Time(sec)	Avg. LBDev(%)
High variant of products family	8330	18.54
Low variant of products family	2756	1.47

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Table 4. Total objective value between the company solutions and our model solutions

Week	1	2	3	4
Company	1,473,400	1,973,405	2,008,300	15,855,500
Model	1,209,100	1,294,400	1,885,500	11,345,500

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