On lot sizing with capacity acquisition

Jia Guo; Minjiao Zhang; Jia Guo; Minjiao Zhang

doi:10.3934/jimo.2026126

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 7: 3456-3479. doi: 10.3934/jimo.2026126

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Research article

On lot sizing with capacity acquisition

Jia Guo ¹,
Minjiao Zhang ^{2
,
,}

1.
Department of Management, College of Business and Economics, California State University East Bay, Hayward, CA 94542, USA
2.
Department of Economics, Finance, and Quantitative Analysis, Kennesaw State University, Kennesaw, GA, 30144, USA

Received: 17 February 2026 Revised: 29 May 2026 Accepted: 03 June 2026 Published: 26 June 2026
90C11, 90B30

Determining the capacity level is a long-term strategic decision that closely interacts with multi-period lot-sizing choices. We consider a single-echelon lot-sizing problem with capacity acquisition (CALS), where the capacity level, along with production and inventory plans, needs to be determined optimally. In this study, CALS is formulated as a mixed-integer programming problem and we conduct a polyhedral study. We propose two exponential families of valid inequalities, which are shown to be polynomially separable, and we then study the facet conditions for these valid inequalities. Computational studies demonstrate that the proposed valid inequalities significantly strengthen the formulation and improve branch-and-cut performance, outperforming default CPLEX cuts in most test instances through tighter relaxations, reduced runtimes, and improved solution quality.
- capacitated lot sizing,
- capacity acquisition,
- valid inequalities,
- separation,
- facet conditions
Citation: Jia Guo, Minjiao Zhang. On lot sizing with capacity acquisition[J]. Journal of Industrial and Management Optimization, 2026, 22(7): 3456-3479. doi: 10.3934/jimo.2026126

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Abstract

Determining the capacity level is a long-term strategic decision that closely interacts with multi-period lot-sizing choices. We consider a single-echelon lot-sizing problem with capacity acquisition (CALS), where the capacity level, along with production and inventory plans, needs to be determined optimally. In this study, CALS is formulated as a mixed-integer programming problem and we conduct a polyhedral study. We propose two exponential families of valid inequalities, which are shown to be polynomially separable, and we then study the facet conditions for these valid inequalities. Computational studies demonstrate that the proposed valid inequalities significantly strengthen the formulation and improve branch-and-cut performance, outperforming default CPLEX cuts in most test instances through tighter relaxations, reduced runtimes, and improved solution quality.

References

[1]	M. Florian, M. Klein, Deterministic production planning with concave costs and capacity constraints, Manag. Sci., 18 (1971), 12–20. https://doi.org/10.1287/mnsc.18.1.12 doi: 10.1287/mnsc.18.1.12
[2]	C. P. M. van Hoesel, A. P. M. Wagelmans, An $O(T^3)$ algorithm for the economic lot-sizing problem with constant capacities, Manag. Sci., 42 (1996), 142–150. https://doi.org/10.1287/mnsc.42.1.142 doi: 10.1287/mnsc.42.1.142
[3]	Y. Pochet, L. A. Wolsey, Lot-sizing with constant batches: Formulation and valid inequalities, Math. Oper. Res., 18 (1993), 767–785. https://doi.org/10.1287/moor.18.4.767 doi: 10.1287/moor.18.4.767
[4]	M. Zhao, M. Zhang, Multiechelon lot sizing: New complexities and inequalities, Oper. Res., 68 (2020), 534–551. https://doi.org/10.1287/opre.2019.1867 doi: 10.1287/opre.2019.1867
[5]	M. Florian, J. K. Lenstra, A. H. G. Rinnooy Kan, Deterministic production planning: Algorithms and complexity, Manag. Sci., 26 (1980), 669–679. https://doi.org/10.1287/mnsc.26.7.669 doi: 10.1287/mnsc.26.7.669
[6]	G. R. Bitran, H. H. Yanasse, Computational complexity of the capacitated lot size problem, Manag. Sci., 28 (1982), 1174–1186. https://doi.org/10.1287/mnsc.28.10.1174 doi: 10.1287/mnsc.28.10.1174
[7]	C. P. M. van Hoesel, A. P. M. Wagelmans, Fully polynomial approximation schemes for single-item capacitated economic lot-sizing problems, Math. Oper. Res., 26 (2001), 339–357. https://doi.org/10.1287/moor.26.2.339.10552 doi: 10.1287/moor.26.2.339.10552
[8]	S. Chubanov, M. Y. Kovalyov, E. Pesch, An FPTAS for a single-item capacitated economic lot-sizing problem with monotone cost structure, Math. Program., 106 (2006), 453–466. https://doi.org/10.1007/s10107-005-0641-0 doi: 10.1007/s10107-005-0641-0
[9]	Y. Pochet, Valid inequalities and separation for capacitated economic lot sizing, Oper. Res. Lett., 7 (1988), 109–115. https://doi.org/10.1016/0167-6377(88)90076-4 doi: 10.1016/0167-6377(88)90076-4
[10]	M. Loparic, H. Marchand, L. A. Wolsey, Dynamic knapsack sets and capacitated lot-sizing, Math. Program., 95 (2003), 53–69. https://doi.org/10.1007/s10107-002-0341-y doi: 10.1007/s10107-002-0341-y
[11]	A. Atamtürk, J. C. Muñoz, A study of the lot-sizing polytope, Math. Program., 99 (2004), 443–465. https://doi.org/10.1007/s10107-003-0465-8 doi: 10.1007/s10107-003-0465-8
[12]	L. Deeratanasrikul, S. Mizuno, Multiple-stage multiple-machine capacitated lot-sizing and scheduling with sequence-dependent setup: A case study in the wheel industry, J. Ind. Manag. Optim., 13 (2017), 411–426. https://doi.org/10.3934/jimo.2016024 doi: 10.3934/jimo.2016024
[13]	M. Tang, F. Jing, X. Chao, A dynamic lot sizing model with production-or-outsourcing decision under minimum production quantities, J. Ind. Manag. Optim., 16 (2020), 2389–2406. https://doi.org/10.3934/jimo.2019059 doi: 10.3934/jimo.2019059
[14]	Z. Gao, D. Li, D. Wang, Z. Yu, Lagrange relaxation for the capacitated multi-item lot-sizing problem, Appl. Sci., 14 (2024), 6517. https://doi.org/10.3390/app14156517 doi: 10.3390/app14156517
[15]	J. R. Bradley, B. C. Arntzen, The simultaneous planning of production, capacity, and inventory in seasonal demand environments, Oper. Res., 47 (1999), 795–806. https://doi.org/10.1287/opre.47.6.795 doi: 10.1287/opre.47.6.795
[16]	M. R. Rao, Optimal capacity expansion with inventory, Oper. Res., 24 (1976), 291–300. https://doi.org/10.1287/opre.24.2.291 doi: 10.1287/opre.24.2.291
[17]	S. Chand, T. McClurg, J. Ward, A model for parallel machine replacement with capacity expansion, Eur. J. Oper. Res., 121 (2000), 519–531. https://doi.org/10.1016/S0377-2217(99)00052-1 doi: 10.1016/S0377-2217(99)00052-1
[18]	S. Rajagopalan, J. M. Swaminathan, A coordinated production planning model with capacity expansion and inventory management, Manag. Sci., 47 (2001), 1562–1580. https://doi.org/10.1287/mnsc.47.11.1562.10254 doi: 10.1287/mnsc.47.11.1562.10254
[19]	V. N. Hsu, Dynamic capacity expansion problem with deferred expansion and age-dependent shortage cost, Manuf. Serv. Oper. Manag., 4 (2002), 44–54. https://doi.org/10.1287/msom.4.1.44.294 doi: 10.1287/msom.4.1.44.294
[20]	D. Taş, M. Gendreau, O. Jabali, R. Jans, A capacitated lot sizing problem with stochastic setup times and overtime, Eur. J. Oper. Res., 273 (2019), 146–159. https://doi.org/10.1016/j.ejor.2018.07.032 doi: 10.1016/j.ejor.2018.07.032
[21]	A. Darendeliler, D. Claeys, E. H. Aghezzaf, Integrated condition-based maintenance and multi-item lot-sizing with stochastic demand, J. Ind. Manag. Optim., 19 (2023), 6908–6947. https://doi.org/10.3934/jimo.2022245 doi: 10.3934/jimo.2022245
[22]	S. Ahmed, A. J. King, G. Parija, A multi-stage stochastic integer programming approach for capacity expansion under uncertainty, J. Glob. Optim., 26 (2003), 3–24. https://doi.org/10.1023/A:1023062915106 doi: 10.1023/A:1023062915106
[23]	S. Ahmed, R. Garcia, Dynamic capacity acquisition and assignment under uncertainty, Ann. Oper. Res., 124 (2003), 267–283. https://doi.org/10.1023/B:ANOR.0000004773.66339.df doi: 10.1023/B:ANOR.0000004773.66339.df
[24]	J. Liu, S. Küçükyavuz, Polyhedral study of the static probabilistic lot-sizing problem, Ann. Oper. Res., 261 (2018), 233–254. https://doi.org/10.1007/s10479-017-2641-x doi: 10.1007/s10479-017-2641-x
[25]	A. Agra, A robust model for the lot-sizing problem with uncertain demands, Optim. Lett., 17 (2023), 1279–1293. https://doi.org/10.1007/s11590-023-01994-x doi: 10.1007/s11590-023-01994-x
[26]	Ö. N. Attila, A. Agra, K. Akartunalı, A. Arulselvan, Robust formulations for economic lot-sizing problem with remanufacturing, Eur. J. Oper. Res., 288 (2021), 496–510. https://doi.org/10.1016/j.ejor.2020.06.016 doi: 10.1016/j.ejor.2020.06.016
[27]	M. Schlenkrich, S. N. Parragh, Capacitated multi-item multi-echelon lot sizing with setup carry-over under uncertain demand, Int. J. Prod. Econ., 277 (2024), 109379. https://doi.org/10.1016/j.ijpe.2024.109379 doi: 10.1016/j.ijpe.2024.109379
[28]	A. Özmen, E. Kropat, G. W. Weber, Robust optimization in spline regression models for multi-model regulatory networks under polyhedral uncertainty, Optimization, 66 (2017), 2135–2155. https://doi.org/10.1080/02331934.2016.1209672 doi: 10.1080/02331934.2016.1209672
[29]	E. Savku, G. W. Weber, A stochastic maximum principle for a markov regime-switching jump-diffusion model with delay and an application to finance, J. Optim. Theory Appl., 179 (2018), 696–721. https://doi.org/10.1007/s10957-017-1159-3 doi: 10.1007/s10957-017-1159-3
[30]	E. Savku, G. W. Weber, Stochastic differential games for optimal investment problems in a markov regime-switching jump-diffusion market, Ann. Oper. Res., 312 (2022), 1171–1196. https://doi.org/10.1007/s10479-020-03768-5 doi: 10.1007/s10479-020-03768-5
[31]	G. W. Weber, Ö. Uğur, P. Taylan, A. Tezel, On optimization, dynamics and uncertainty: A tutorial for gene-environment networks, Discrete Appl. Math., 157 (2009), 2494–2513. https://doi.org/10.1016/j.dam.2008.06.030 doi: 10.1016/j.dam.2008.06.030
[32]	E. Kropat, A. Özmen, G. W. Weber, S. Meyer-Nieberg, Ö. Defterli, Fuzzy prediction strategies for gene-environment networks – fuzzy regression analysis for two-modal regulatory systems, RAIRO Oper. Res., 50 (2016), 413–435. https://doi.org/10.1051/ro/2015044 doi: 10.1051/ro/2015044
[33]	A. Atamtürk, D. S. Hochbaum, Capacity acquisition, subcontracting, and lot sizing, Manag. Sci., 47 (2001), 1081–1100. https://doi.org/10.1287/mnsc.47.8.1081.10232 doi: 10.1287/mnsc.47.8.1081.10232
[34]	H. Li, J. Meissner, Capacitated dynamic lot sizing with capacity acquisition, Int. J. Prod. Res., 49 (2011), 4945–4963. https://doi.org/10.1080/00207543.2010.518991 doi: 10.1080/00207543.2010.518991
[35]	I. Barany, T. van Roy, L. A. Wolsey, Uncapacitated lot-sizing: The convex hull of solutions, Math. Program. Stud., 22 (1984), 32–43. https://doi.org/10.1007/BFb0121006 doi: 10.1007/BFb0121006

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