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Previously in IEMS 310… • • • • • • Notation of optimization problems Linear Programs Sensitivity Analysis Duality Piecewise linear functions Assignment Problems Logistics • Hwk 2 due Friday • Yue prefers – “hard copy so I can mark on it,” – “and email me the excel file in case I need to look through.” • Please fill out survey Yue’s Office Hours Options: • Fri 3-5, (current) hwk due at 5 1. Thurs 9-11, hwk due Fri in class 2. Thurs 1:30-3:30, hwk due Fri in class Economies of Scale • Related to piecewise linear functions • costs are concave – decreasing derivative cost – decreasing marginal costs • returns/profits are convex quantity profit quantity • Diseconomies of scale are the opposite Hwk 1 Problem B • Design optimization – price, features, shape, packaging • • • • • Civil: price with safety constraint ME: air conditioning budgeting websites: load, reliability Chem Eng: reaction throughput Agenda • Network Flow problem – transportation – staffing • Sequential Decision Processes – production problem The Network Flow Problem • data: c 1 x1 L1,U1 – network structure – bounds on flow through each link – cost/profit per unit flow on each link • decision variables: flow on links • constraints: – bounds on flow through links – conservation of flow • max profit / min cost x2 x3 x5 x4 x6 x7 x8 x9 L1 ≤ x1 ≤ U1 x1 - x2 - x5 = 0 min c1x1 + c2x2 + … Network Flow • Recognize the structure – includes assignment problem jobs machines 2 1 3 7 3 2 5 0.5 jobs machines 1.1 • LP • If the flow bounds (Li,Ui) are integer, then solution will be integer 4.5 Transportation Problem ≤ 2500 Plants (supply) Depots (demand) =2000 ≤ 4000 =3000 ≤ 3500 =2500 =1500