en Mathematical Modeling and Applications of OR Mathematical Modeling and Applications of OR پژوهشی Original <span style="font-size:12pt"><span new="" roman="" style="font-family:" times="">Extant quantitative models in Resource Allocation and Operational Systems often rely on statistical optimization or behavioral frameworks, frequently failing to model and quantify the deterministic structural conflicts that arise when operational and informational constraints are ignored. This paper addresses a critical analytical gap in Industrial Information Integration: the necessity of a deterministic modeling framework to quantify the systemic risks inherent in resource scarcity and misallocation. We introduce a novel analytical framework, derived from the Generalized Pigeonhole Principle (GPP), to quantify the inevitable consequences of resource imbalance within complex organizational systems. The framework develops a three-tiered conceptual model: (1) the Base Principle (quantifying the inevitable Structural Non-Allocation Rate, (N-M)), (2) the Generalized Principle (modeling Inevitable Operational Congestion, <m:omath><i><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></i></m:omath><span style="font-size:12.0pt"><span style="line-height:115%"><span aptos="" style="font-family:"><span style="position:relative"><span style="top:16.5pt"><img alt="" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAA4CAMAAAARgoJQAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OpDbZgAAZjo6ZjpmZrb/kDoAkGZmkLaQkLbbkNv/tmYAttv/tv//25A62////7Zm/9uQ/9u2//+2///bLdLeDQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAAFiUAABYlAUlSJPAAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAcUlEQVQ4T+2QSQ6AIAxFW0ccUVGZ739N0QiydEPc2N1Lmt/XD5BodI/Y+mwzLLDme3RKZiwiRcZ3ZClGmxyjFEWKJmRams0PcWxVIEUqEcjtMQjEsQbwZhJP4ZtMdz3jPPW0wYqVcORkyi1RfX/sNw0c49oFbtfn0c0AAAAASUVORK5CYII=" style="width:9px; height:37px" ></span></span></span></span></span>_min), and (3) the Weighted Principle (quantifying Qualitative Mismatch, f). We demonstrate the model&#39;s predictive and explanatory power through scenario-based analytical simulations, providing a foundational validation for its applicability across workforce planning and capacity development contexts. The analysis yields testable propositions linking these mathematical inevitabilities directly to systemic consequences (e.g., structural attrition/turnover, intragroup operational conflict). This research provides a new, parsimonious analytical language for Operational Systems Modeling, fundamentally shifting the focus from optimizing individual performance to managing unavoidable structural pressures. The model serves as a robust quantitative decision-support tool for strategic workforce planning and lays the essential groundwork for future dynamic modeling of operational capacity and resource demand.</span></span><br> <sider-trans data-url-id="1111761034"></sider-trans> Industrial Information Integration, Deterministic Modeling, Operational Congestion, Systemic Risk Prediction, Resource Allocation Model. 145 164 http://iors.ir/journal/browse.php?a_code=A-10-6103-1&slc_lang=en&sid=1 fatemeh mohajernia fmohajer12225@gmail.com 00031947532846003163 00031947532846003163 Yes PhD student in Educational Management, Islamic Azad University, Qom Branch, Qom, Iran. Jafar Rahmani 0384459@gmai.com 00031947532846003164 00031947532846003164 No Associate Professor, Department of Politics, Faculty of Islamic Systems, Research Institute of Islamic Culture and Thought, Qom, Iran. Mohsen Mohajernia mohajernia@gmail.com 00031947532846003165 00031947532846003165 No Associate Professor, Department of Politics, Faculty of Islamic Systems, Research Institute of Islamic Culture and Thought, Qom, Iran.