• Multi-echelon, networked inventory optimization evolved to dynamic lot sizing with backlogging, enabling coordinated production and stock decisions across series and parallel facilities to minimize combined production and holding costs under backorder assumptions [1], [5], [4], [6], [7].

• Concave cost structures and nonconvex optimization emerged as central in production and inventory planning, with results on backlogging, piecewise concave costs, Leontief substitution, and multi-facility cost minimization guiding minimum-cost schedules [3], [6], [8], [13], [14].

• Policy design for multi-echelon inventory systems emphasized optimal, near-optimal and system-wide policies under constant demand, including dynamic nonstationary policies, continuous-review controls, and policy comparisons across multi-stage networks [9], [5], [19], [4].

• Manufacturing planning under pull-based and material requirements planning paradigms linked kanban, just-in-time and MRP concepts to inventory reduction and improved responsiveness, bridging theory with real-world production environments [11], [15], [16].

• Foundational reviews and surveys established the baseline for inventory theory, detailing the status of mathematical inventory models and informal multi-echelon inventories, shaping later theoretical developments [18], [12].