Integer Programming extends linear programming by requiring some or all decision variables to take integer values. This is essential for problems involving discrete decisions such as: number of facilities to open, assignment of tasks, selection of projects, or routing decisions. IP problems are NP-hard and significantly more difficult to solve than LP. Types include: pure integer programming (all variables integer), mixed-integer programming (MIP - some variables integer), and binary integer programming (variables restricted to 0 or 1).

Interior Point Methods solve linear and nonlinear programs by traversing the interior of the feasible region rather than following boundary edges. For LP, the barrier (primal-dual) method adds logarithmic barrier function μΣln(xᵢ) to prevent boundary violations, solving a sequence of barrier problems as μ → 0. The method maintains strict feasibility and complementarity, iterating using Newton steps on KKT conditions. Advantages over simplex: polynomial worst-case complexity (Karmarkar's algorithm), better performance on large sparse problems, warm-start capability. Extensions include: nonlinear programming (IPOPT), semidefinite programming, and conic optimization. Interior point methods revolutionized large-scale optimization and are implemented in all major solvers (CPLEX, Gurobi, Mosek).

Inventory Management involves determining optimal ordering and stocking policies to balance holding costs, ordering costs, and stockout costs. Classic models include: (1) Economic Order Quantity (EOQ) - determines optimal order size Q* = √(2DS/H) where D is demand rate, S is ordering cost, H is holding cost; (2) Economic Production Quantity (EPQ) - extends EOQ for production scenarios; (3) Newsvendor Model - single-period stochastic demand; (4) (Q,r) policy - order Q units when inventory drops to reorder point r; (5) (s,S) policy - order up to S when inventory falls below s. Advanced topics: multi-echelon inventory, stochastic inventory models, and inventory optimization under uncertainty.