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Piecewise Linear Approximation and Branch and Refine Algorithm for PX Tank Sizing under Uncertainty Problem Fengqi You Ignacio E Grossmann Jose M Pinto EWO Meeting March 10 2010 Vehicle Routing Tank Sizing Problem Tank Sizing Tank installation upgrade downgrade Several available discrete tank sizes Safety stock optimization for uncertainty Vehicle Routing Several available truck sizes Routing and timing decisions Integration Capture the effects of customer synergies and truck availability Integration requires to solve extended routing problem for long term e g years Integrated MILP model is very large 4000 3000000 of Possible Routes 3500 2500000 of Binary Variables in the MILP 3000 2000000 2500 1500000 2000 1500 1000000 1000 500000 500 0 of 0 1 Variables in the MILP Tradeoff operating cost vs capital cost of Possible Routes 0 1 2 3 of Customers 4 5 2 Complexity 4 customer case Departure 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 256 possible route 230 448 binary variables in the MILP model for integrating tank sizing and vehicle routing Return 3 Modeling Challenges How to effectively integrate tank sizing with vehicle routing Continuous approximation CA approach Cont Approximation Model safety stock optimization obtain tank sizing decisions Fix tank sizes tradeoff capital and operating cost in the strategic level Customer Clustering Select the 1st clustering solution Detailed Routing Model obtain vehicle routing decisions reduce most integer variables with some nonlinearities Next cluster Termination How to optimize the safety stocks for demand uncertainty Employ stochastic inventory model Integrate safety stock optimization with tank sizing How to model the uncertainty of adding losing customers Two stage stochastic programming A network structure for each scenario in each year x Uncertainty reveal y 1 2 3 4 5 4 Optimal Safety Stock Level Service Level Lead time T Safety Stock 5 Cyclic Inventory Routing Key Assumption each customer is replenished in a cyclic way with interval

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