{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T10:43:02Z","timestamp":1777286582691,"version":"3.51.4"},"reference-count":80,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,22]],"date-time":"2023-08-22T00:00:00Z","timestamp":1692662400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Minimizing a company\u2019s operational risk by optimizing the performance of the manufacturing and distribution supply chain is a complex task that involves multiple elements, each with their own supply line constraints. Traditional approaches to optimization often assume determinism as the underlying principle. However, this paper, adopting an entropy approach, emphasizes the significance of subjective and objective uncertainty in achieving optimized decisions by incorporating stochastic fluctuations into the supply chain structure. Stochasticity, representing randomness, quantifies the level of uncertainty or risk involved. In this study, we focus on a processing production plant as a model for a chain of operations and supply chain actions. We consider the stochastically varying production and transportation costs from the site to the plant, as well as from the plant to the customer base. Through stochastic optimization, we demonstrate that the plant producer can benefit from improved financial outcomes by setting higher sale prices while simultaneously lowering optimized production costs. This can be accomplished by selectively choosing producers whose production cost probability density function follows a Pareto distribution. Notably, a lower Pareto exponent yields better supply chain cost optimization predictions. Alternatively, a Gaussian stochastic fluctuation may be proposed as a more suitable choice when trading off optimization and simplicity. Although this may result in slightly less optimal performance, it offers advantages in terms of ease of implementation and computational efficiency.<\/jats:p>","DOI":"10.3390\/e25091245","type":"journal-article","created":{"date-parts":[[2023,8,22]],"date-time":"2023-08-22T08:58:54Z","timestamp":1692694734000},"page":"1245","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Stochastically Optimized Two-Echelon Supply Chain Model: An Entropy Approach for Operational Risk Assessment"],"prefix":"10.3390","volume":"25","author":[{"given":"Konstantinos","family":"Petridis","sequence":"first","affiliation":[{"name":"Department of Business Administration, University of Macedonia, 54006 Thessaloniki, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Prasanta Kumar","family":"Dey","sequence":"additional","affiliation":[{"name":"Aston Business School, Aston University, Birmingham B4 7ET, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5499-6008","authenticated-orcid":false,"given":"Amit K.","family":"Chattopadhyay","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics & Data Science, College of Engineering and Physical Sciences, Aston University, Aston Triangle, Birmingham B4 7ET, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2239-3744","authenticated-orcid":false,"given":"Paraskevi","family":"Boufounou","sequence":"additional","affiliation":[{"name":"Department of Economics, National and Kapodistrian University of Athens, 10559 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3345-495X","authenticated-orcid":false,"given":"Kanellos","family":"Toudas","sequence":"additional","affiliation":[{"name":"Department of Agribusiness and Supply Chain Management, Agricultural University of Athens, 75, 11855 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0378-3939","authenticated-orcid":false,"given":"Chrisovalantis","family":"Malesios","sequence":"additional","affiliation":[{"name":"Department of Agricultural Economics and Rural Development, Agricultural University of Athens, 75, 11855 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,22]]},"reference":[{"key":"ref_1","unstructured":"Gruen, T.W., Corsten, D.S., and Bharadwaj, S. 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