• Kawsik Azad Pronoy
• Ummul Khayer Fatema

Modern project management relies heavily on rigorous deterministic and stochastic mathematical programming models to optimize resource allocations, schedule timelines, and financial trade-offs. This paper provides an exhaustive mathematical analysis of the Critical Path Method (CPM) and Project Crashing (time-cost trade-off optimization). We establish the graph-theoretic foundations of Activity-on-Node (AON) and Activity-on-Arrow (AOA) networks, presenting formal forward-pass and backward-pass recursive dynamic programming algorithms. We define total float, free float, independent float, and interfering float with exact set-theoretic proofs of their mathematical properties. Subsequently, we construct the Primal and Dual Linear Programming (LP) formulations for baseline network analysis, demonstrating how complementary slackness reveals critical path bottlenecks. Expanding into project crashing, we formulate Primal LP models, Mixed-Integer Linear Programming (MILP) models for discrete crashing decisions, and multi-objective optimization models incorporating time-cost-quality (TCQ) Pareto frontiers. Two extensive numerical benchmarks are solved step-by-step: a 10-activity network demonstrating full float matrix computations, followed by a multi-stage crashing optimization under linear and non-linear cost slopes. Finally, stochastic extensions via Beta distribution integration and sensitivity analysis under parametric budget constraints are derived. This manuscript serves as a definitive mathematical reference for researchers and operations research analysts.

Critical Path Method (CPM), Project Crashing, Linear Programming, Time-Cost Trade-Off, Dual Theory, Integer Programming, Multi-Objective Optimization, Network Optimization. Mathematics Subject Classification (MSC 2020): 90B50, 90C05, 90C11, 90C29, 05C85.

"MATHEMATICAL FORMULATIONS AND OPTIMIZATION OF THE CRITICAL PATH METHOD (CPM) AND PROJECT CRASHING: ALGORITHMIC ADVANCEMENTS, DUAL LP THEORY, AND MULTI-OBJECTIVE TRADE-OFFS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.13, Issue 9, page no.a44-a57, September-2026, Available :http://www.jetir.org/papers/JETIR2609007.pdf

"MATHEMATICAL FORMULATIONS AND OPTIMIZATION OF THE CRITICAL PATH METHOD (CPM) AND PROJECT CRASHING: ALGORITHMIC ADVANCEMENTS, DUAL LP THEORY, AND MULTI-OBJECTIVE TRADE-OFFS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.13, Issue 9, page no. ppa44-a57, September-2026, Available at : http://www.jetir.org/papers/JETIR2609007.pdf