Multi-item continuous-review inventory optimization couples integer replenishment policies, item-level service requirements, and shared average-resource limits. Sparse linear programming rounding with resource-exchange descent (LP-SERD) was developed for a finite single-objective planning model with full backordering. Mean inventory and one-time shortage costs were evaluated under an explicit uniform-inventory-position approximation. Service-feasible item policies were enumerated and dominance-pruned to form an exact multiple-choice master for that model. The standard LP rank bound limits fractionality to the independent active resource rows. Common within-item resource ordering guarantees a feasible positive-support corner and an additive support-span cost bound. Two independent descent paths refine this start using single-item moves and directional two-item exchanges. Across 15 instances with 500–2000 items, the pooled median LP-certified gap was 0.000701%, and the maximum was 0.004148%. On 35 instances, LP-SERD had 28 lower-cost, 4 tied, and 3 higher-cost endpoints than repaired maximum-weight rounding followed by single-item descent. At a 60 s solver limit, full MILP obtained lower costs on 14 of 15 instances and closed 3 optimality proofs. A paired correlation experiment did not establish a monotone association between resource-coefficient correlation and exchange gain. All 10 OIL main models had integral LP optima; MAN exposed both small optimization gaps and holdout-service shortfalls. The evidence supports a feasible construction with a finite-model certificate, not a general replacement for tractable exact optimization.
Citation: Qingbo Lu, Shuangcheng Huang, Mengzhen Huang, Daojun Li. Sparse LP rounding with resource-exchange descent for multi-item continuous-review inventory under dual shared-resource constraints[J]. Journal of Industrial and Management Optimization, 2026, 22(12): 6248-6270. doi: 10.3934/jimo.2026210
Multi-item continuous-review inventory optimization couples integer replenishment policies, item-level service requirements, and shared average-resource limits. Sparse linear programming rounding with resource-exchange descent (LP-SERD) was developed for a finite single-objective planning model with full backordering. Mean inventory and one-time shortage costs were evaluated under an explicit uniform-inventory-position approximation. Service-feasible item policies were enumerated and dominance-pruned to form an exact multiple-choice master for that model. The standard LP rank bound limits fractionality to the independent active resource rows. Common within-item resource ordering guarantees a feasible positive-support corner and an additive support-span cost bound. Two independent descent paths refine this start using single-item moves and directional two-item exchanges. Across 15 instances with 500–2000 items, the pooled median LP-certified gap was 0.000701%, and the maximum was 0.004148%. On 35 instances, LP-SERD had 28 lower-cost, 4 tied, and 3 higher-cost endpoints than repaired maximum-weight rounding followed by single-item descent. At a 60 s solver limit, full MILP obtained lower costs on 14 of 15 instances and closed 3 optimality proofs. A paired correlation experiment did not establish a monotone association between resource-coefficient correlation and exchange gain. All 10 OIL main models had integral LP optima; MAN exposed both small optimization gaps and holdout-service shortfalls. The evidence supports a feasible construction with a finite-model certificate, not a general replacement for tractable exact optimization.
| [1] |
B. Ghalebsaz-Jeddi, B. C. Shultes, R. Haji, A multi-product continuous review inventory system with stochastic demand, backorders, and a budget constraint, Eur. J. Oper. Res., 158 (2004), 456–469. https://doi.org/10.1016/S0377-2217(03)00363-1 doi: 10.1016/S0377-2217(03)00363-1
|
| [2] |
K. Das, T. K. Roy, M. Maiti, Multi-item stochastic and fuzzy-stochastic inventory models under two restrictions, Comput. Oper. Res., 31 (2004), 1793–1806. https://doi.org/10.1016/S0305-0548(03)00120-5 doi: 10.1016/S0305-0548(03)00120-5
|
| [3] |
S. Y. Chou, P. C. Julian, K. C. Hung, A note on fuzzy inventory model with storage space and budget constraints, Appl. Math. Model., 33 (2009), 4069–4077. https://doi.org/10.1016/j.apm.2009.02.001 doi: 10.1016/j.apm.2009.02.001
|
| [4] |
X. Zhao, M. Qiu, J. Xie, Q. He, Computing (r, Q) policy for an inventory system with limited sharable resource, Comput. Oper. Res., 39 (2012), 2368–2379. https://doi.org/10.1016/j.cor.2011.12.012 doi: 10.1016/j.cor.2011.12.012
|
| [5] |
Y. Kou, Z. Wan, A new data-driven robust optimization approach to multi-item newsboy problems, J. Ind. Manag. Optim., 19 (2023), 197–223. https://doi.org/10.3934/jimo.2021180 doi: 10.3934/jimo.2021180
|
| [6] |
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
|
| [7] |
I. Saracoglu, S. Topaloglu, T. Keskinturk, A genetic algorithm approach for multi-product multi-period continuous review inventory models, Expert Syst. Appl., 41 (2014), 8189–8202. https://doi.org/10.1016/j.eswa.2014.07.003 doi: 10.1016/j.eswa.2014.07.003
|
| [8] |
A. Azimian, B. Aouni, Multi-item order quantity optimization through stochastic goal programing, Ann. Oper. Res., 346 (2025), 751–779. https://doi.org/10.1007/s10479-024-05903-y doi: 10.1007/s10479-024-05903-y
|
| [9] |
S. Bhalla, E. Alfnes, H. H. Hvolby, F. Sgarbossa, Advances in spare parts classification and forecasting for inventory control: A literature review, IFAC-PapersOnLine, 54 (2021), 982–987. https://doi.org/10.1016/j.ifacol.2021.08.118 doi: 10.1016/j.ifacol.2021.08.118
|
| [10] |
C. B. Haubitz, U. W. Thonemann, How to change a running system—controlling the transition to optimized spare parts inventory policies, Prod. Oper. Manag., 30 (2021), 1386–1405. https://doi.org/10.1111/poms.13327 doi: 10.1111/poms.13327
|
| [11] |
L. M. Johannsmann, E. M. Craparo, T. L. Dieken, A. R. Fügenschuh, B. O. Seitner, Stochastic mixed-integer programming for a spare parts inventory management problem, Comput. Oper. Res., 138 (2022), 105568. https://doi.org/10.1016/j.cor.2021.105568 doi: 10.1016/j.cor.2021.105568
|
| [12] |
V. F. Yu, N. Y. Salsabila, N. Siswanto, P. H. Kuo, A two-stage genetic algorithm for joint coordination of spare parts inventory and planned maintenance under uncertain failures, Appl. Soft Comput., 130 (2022), 109705. https://doi.org/10.1016/j.asoc.2022.109705 doi: 10.1016/j.asoc.2022.109705
|
| [13] |
L. Zhang, Q. Deng, B. Miao, X. Liu, H. Shao, Parallel service mode of production and inventory for spare part inventory optimization, Knowl. Based Syst., 241 (2022), 108282. https://doi.org/10.1016/j.knosys.2022.108282 doi: 10.1016/j.knosys.2022.108282
|
| [14] |
P. Sinha, A. A. Zoltners, The multiple-choice knapsack problem, Oper. Res., 27 (1979), 503–515. https://doi.org/10.1287/opre.27.3.503 doi: 10.1287/opre.27.3.503
|
| [15] |
Y. Chen, J. K. Hao, A reduce and solve approach for the multiple-choice multidimensional knapsack problem, Eur. J. Oper. Res., 239 (2014), 313–322. https://doi.org/10.1016/j.ejor.2014.05.025 doi: 10.1016/j.ejor.2014.05.025
|
| [16] |
L. Lamanna, R. Mansini, R. Zanotti, A two-phase kernel search variant for the multidimensional multiple-choice knapsack problem, Eur. J. Oper. Res., 297 (2022), 53–65. https://doi.org/10.1016/j.ejor.2021.05.007 doi: 10.1016/j.ejor.2021.05.007
|
| [17] |
V. Cacchiani, M. Iori, A. Locatelli, S. Martello, Knapsack problems—an overview of recent advances. Part Ⅱ: Multiple, multidimensional, and quadratic knapsack problems, Comput. Oper. Res., 143 (2022), 105693. https://doi.org/10.1016/j.cor.2021.105693 doi: 10.1016/j.cor.2021.105693
|
| [18] |
T. Szkaliczki, Solution methods for the multiple-choice knapsack problem and their applications, Mathematics, 13 (2025), 1097. https://doi.org/10.3390/math13071097 doi: 10.3390/math13071097
|
| [19] |
M. S. Song, Y. Lu, D. Rando, F. J. Vasko, Statistical analyses of solution methods for the multiple-choice knapsack problem with setups: Implications for OR practitioners, Expert Syst. Appl., 262 (2025), 125622. https://doi.org/10.1016/j.eswa.2024.125622 doi: 10.1016/j.eswa.2024.125622
|
| [20] |
I. Crévits, S. Hanafi, R. Mansi, C. Wilbaut, Iterative semi-continuous relaxation heuristics for the multiple-choice multidimensional knapsack problem, Comput. Oper. Res., 39 (2012), 32–41. https://doi.org/10.1016/j.cor.2010.12.016 doi: 10.1016/j.cor.2010.12.016
|
| [21] |
J. Guo, M. Zhang, On lot sizing with capacity acquisition, J. Ind. Manag. Optim., 22 (2026), 3456–3479. https://doi.org/10.3934/jimo.2026126 doi: 10.3934/jimo.2026126
|
| [22] |
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
|
| [23] |
T. W. Leung, C. K. Chan, M. D. Troutt, A mixed simulated annealing-genetic algorithm approach to the multi-buyer multi-item joint replenishment problem: advantages of meta-heuristics, J. Ind. Manag. Optim., 4 (2008), 53–66. https://doi.org/10.3934/jimo.2008.4.53 doi: 10.3934/jimo.2008.4.53
|
| [24] |
F. Grandoni, R. Zenklusen, Approximation schemes for multi-budgeted independence systems, in Algorithms–ESA 2010, Springer, Berlin, 2010,536–548. https://doi.org/10.1007/978-3-642-15775-2_46 |
| [25] |
V. Beiranvand, W. Hare, Y. Lucet, Best practices for comparing optimization algorithms, Optim. Eng., 18 (2017), 815–848. https://doi.org/10.1007/s11081-017-9366-1 doi: 10.1007/s11081-017-9366-1
|
| [26] |
P. Escalona, D. Araya, E. Simpson, M. Ramirez, R. Stegmaier, On the shortage control in a continuous review (Q, r) inventory policy using αL service-level, RAIRO Oper. Res., 55 (2021), 2785–2806. https://doi.org/10.1051/ro/2021125 doi: 10.1051/ro/2021125
|
| [27] |
H. Zhang, A note on the convexity of service-level measures of the (r, q) system, Manage. Sci., 44 (1998), 431–432. https://doi.org/10.1287/mnsc.44.3.431 doi: 10.1287/mnsc.44.3.431
|
| [28] |
E. Porras Musalem, R. Dekker, An inventory control system for spare parts at a refinery: An empirical comparison of different re-order point methods, Eur. J. Oper. Res., 184 (2008), 101–132. https://doi.org/10.1016/j.ejor.2006.11.008 doi: 10.1016/j.ejor.2006.11.008
|
| [29] |
D. D. Haan, Benchmarking spare part demand forecasting methods, Erasmus University Rotterdam, 2021. |
Figures(7) / Tables(3)
Qingbo Lu, Shuangcheng Huang, Mengzhen Huang, Daojun Li. Sparse LP rounding with resource-exchange descent for multi-item continuous-review inventory under dual shared-resource constraints[J]. Journal of Industrial and Management Optimization, 2026, 22(12): 6248-6270. doi: 10.3934/jimo.2026210
LP-SERD computational procedure. Complete MILP is an independent reference route. The support-rounded policy is integer and feasible before either descent path begins; the lower-cost endpoint is retained
Scale-dependent solution quality and workflow cost. (a) Median and maximum LP-certified gaps, with five instances per size. (b) Median phase-summed workflow estimates. Labels on full-MILP points give proofs closed within the configured tolerance out of five; the nominal solver limit is 60 s. Both vertical axes are logarithmic
Finite-budget full-MILP results for 15 instances with 500–2000 items. (a) Feasible-incumbent availability and proof closure. (b) LP-certified gaps of available incumbents. (c) Actual phase-summed workflow times. Short horizontal segments denote medians. Missing incumbents have no objective value; objective populations therefore differ across limits. Numerical categories are nominal solver limits, while "Approx. matched" uses the LP-SERD workflow target. Each category is an independent solver invocation, not a point along one search trajectory. LP-SERD is shown at its own termination without a proof-closure claim. Panels (b) and (c) have logarithmic vertical axes
Boundaries of feasible support rounding, independent of the inventory evaluator. (a) Two-policy supports at five order-violation levels, 30 instances per level; error bars are 95% Wilson intervals. (b) Crossed supported resources A = (0, 2), B = (2, 0) with unit limits and equal LP weights: neither support corner is feasible, although the excluded C = (1, 1) is feasible (costs 1, 1, and 2, respectively). (c) Three supported policies with weights (0.25, 0.50, 0.25); B uses no more of either resource than the support average (1.75, 1.75), without being a common resource minimum
Resource-correlation sensitivity of the LP certificate and exchange gain. (a) Median certificates and 95% bootstrap intervals from 10 cases per target correlation, with maxima. (b) All 40 paired exchange-versus-single-path gains. The Spearman association is −0.115; the seed-block bootstrap 95% interval is [−0.398, 0.180], and the two-sided blocked permutation p-value is 0.4932. No fitted trend is imposed
OIL optimization and temporal marginal-service assessment. (a) Holdout-service pass proportions on nested item sets, not independent replenishment cycles. (b) Policies retained after dominance pruning as a percentage of the full finite policy domain. (c) Phase-summed workflow estimates on a logarithmic scale. The empirical windows retain calendar continuity; pack size, equivalent storage, and unobserved costs follow the assumptions in Section 5.1
MAN optimization and temporal marginal-service assessment. (a) Retained policies as a percentage of the original quantity–reorder-point domain; analytical reorder-point reduction precedes dominance pruning. (b) LP-certified gaps, retaining zero values on a linear scale. (c) Holdout-service pass proportions for nested item sets. The quantity increment is one, MOQ is a lower bound, and the zero shortage-penalty assumption fixes reorder points at their service-feasible minima
DownLoad: