{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:42:43Z","timestamp":1760035363080,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T00:00:00Z","timestamp":1751932800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study investigates a stochastic production planning problem with regime-switching parameters, inspired by economic cycles impacting production and inventory costs. The model considers types of goods and employs a Markov chain to capture probabilistic regime transitions, coupled with a multidimensional Brownian motion representing stochastic demand dynamics. The production and inventory cost optimization problem is formulated as a quadratic cost functional, with the solution characterized by a regime-dependent system of elliptic partial differential equations (PDEs). Numerical solutions to the PDE system are computed using a monotone iteration algorithm, enabling quantitative analysis. Sensitivity analysis and model risk evaluation illustrate the effects of regime-dependent volatility, holding costs, and discount factors, revealing the conservative bias of regime-switching models when compared to static alternatives. Practical implications include optimizing production strategies under fluctuating economic conditions and exploring future extensions such as correlated Brownian dynamics, non-quadratic cost functions, and geometric inventory frameworks. In contrast to earlier studies that imposed static or overly simplified regime-switching assumptions, our work presents a fully integrated framework\u2014combining optimal control theory, a regime-dependent system of elliptic PDEs, and comprehensive numerical and sensitivity analyses\u2014to more accurately capture the complex stochastic dynamics of production planning and thereby deliver enhanced, actionable insights for modern manufacturing environments.<\/jats:p>","DOI":"10.3390\/axioms14070524","type":"journal-article","created":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T11:58:07Z","timestamp":1751975887000},"page":"524","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Stochastic Production Planning with Regime-Switching: Sensitivity Analysis, Optimal Control, and Numerical Implementation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7042-9089","authenticated-orcid":false,"given":"Dragos-Patru","family":"Covei","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, No. 6, District 1, 010374 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1137\/0322060","article-title":"Stochastic production planning with production constraints","volume":"22","author":"Bensoussan","year":"1984","journal-title":"SIAM J. Control."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1287\/opre.2013.1181","article-title":"Optimal production management when demand depends on the business cycle","volume":"61","author":"Cadenillas","year":"2013","journal-title":"Oper. Res."},{"key":"ref_3","first-page":"1","article-title":"Optimal production management when there is regime switching and production constraints","volume":"61","author":"Cadenillas","year":"2024","journal-title":"Ann. Oper. Res."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Dong, J., Malikopoulos, A., Djouadi, S.M., and Kuruganti, T. (2016, January 6\u20138). Application of Optimal Production Control theory for Home Energy Management in a Micro Grid. Proceedings of the 2016 American Control Conference (ACC), Boston, MA, USA.","DOI":"10.1109\/ACC.2016.7526148"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1080\/07408170309342346","article-title":"Optimal production control problem in stochastic multiple-product multiple-machine manufacturing systems","volume":"35","author":"Gharbi","year":"2003","journal-title":"IIE Trans."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"106059","DOI":"10.1016\/j.aml.2019.106059","article-title":"An elliptic partial differential equation and its application","volume":"101","author":"Covei","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.3934\/era.2022070","article-title":"On a parabolic partial differential equation and system modeling a production planning problem","volume":"30","author":"Covei","year":"2022","journal-title":"Electron. Arch."},{"key":"ref_8","first-page":"1697","article-title":"Stochastic production planning with regime switching","volume":"19","author":"Canepa","year":"2023","journal-title":"J. Ind. Manag."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1137\/0331056","article-title":"Optimal Control of Switching Diffusions with Application to Flexible Manufacturing Systems","volume":"31","author":"Ghosh","year":"1992","journal-title":"SIAM J. Control Optim."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Borhan, J.R.M., Miah, M.M., Alsharif, F., and Kanan, M. (2024). Abundant Closed-Form Soliton Solutions to the Fractional Stochastic Kraenkel-Manna-Merle System with Bifurcation, Chaotic, Sensitivity, and Modulation Instability Analysis. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8060327"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Hu, C., Bian, J., Zhao, D., He, L., and Dong, F. (2024). Optimal Dynamic Production Planning for Supply Network with Random External and Internal Demands. Mathematics, 12.","DOI":"10.3390\/math12172669"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Covei, D.-P. (2023). Exact Solution for the Production Planning Problem with Several Regimes Switching over an Infinite Horizon Time. Mathematics, 11.","DOI":"10.20944\/preprints202309.0965.v1"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/524\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:06:44Z","timestamp":1760033204000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/524"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,8]]},"references-count":12,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070524"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070524","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,8]]}}}