Balancing Risk and Economics for Chemical Supply Chain Optimization under Uncertainty Fengqi You and Ignacio E. Grossmann Dept. of Chemical Engineering, Carnegie Mellon University John M. Wassick The Dow Chemical Company
Introduction Chemical Supply Chain Chemical Supply chain: an integrated network of business units for the supply, production, distribution and consumption of the products. Page 2
Introduction Research Goal Motivation Chemical Supply Chain Planning Costs billions of dollars annually Always under various uncertainties and risks Demand Changes Cost & Prices fluctuations Machine Broken Down Natural Disasters Objective: managing uncertainties and risks in chemical supply chain planning by using optimization techniques Page 3
Case Study Case Study 1 Global Sourcing Project Given Minimum and initial inventory Inventory holding cost and throughput cost Single sourcing and minimum sourcing Transport times of all the transport links Uncertain production reliability and demands Determine Inventory level, transportation and sale amounts ~ 100 facilities ~ 1,000 customers ~ 25,000 shipping links/modes Objective: Minimize Cost Page 4
Case Study Uncertain Parameters of Case Study 1 Production and Demand Uncertainty ~ 12,000 uncertain parameters Normal distribution (central limit theory) Mean from forecasting, variance from historical data Three levels of uncertainties (standard deviations/mean) Production: 2% (1 st month), 5% (2 nd -4 th ), 10% (5 th -12 th ) Demand: 5% (1 st month), 10% (2 nd -4 th ), 20% (5 th -12 th ) Page 5
Case Study Case Study 2 Product Mix Project Given Workcenters, products, and customers Demand lower bound and upper bound Capacity (run time) constraint for workcenters Companion sales for some products Minimum batch size Determine Production levels, transportation and sale amount ~ 1,000 products ~ 100 workcenters ~ global customers Objective: Maximize Profit Page 6
Case Study Uncertain Parameters of Case Study 2 Inaccurate Demand Forecasting Demand upper bound are usually overestimated Deterministic model will push high margin product to UB If demand is not so high, may waste resource and lose profit Forecast Lower bound Forecast upper bound demand Real Lower bound Real upper bound Uncertain UB follows truncated normal distribution Mean = forecast LB, standard deviation = (UB-LB)/3 Deterministic model corresponds to the best scenario ~ 10,000 uncertain parameters LB UB deterministic Page 7
Case Study Challenges of Case Studies Major Challenges Large scale supply chain network Multi-period problem (considers transportation times) A lot of uncertain parameters Uncertain parameters follow continuous probability distribution Approach Stochastic programming combined with Monte Carlo sampling Representing continuous probability distribution with scenarios Reducing the number of scenarios Using Sampling average estimator (SAE) to calculate the confidence interval Page 8
Method Stochastic Programming Stochastic Programming Scenario Planning A scenario is a future possible outcome of the uncertainty To avoid the large number of scenarios, they are generated by using the Monte Carlo sampling combined with statistical methods (SAE) Two-stage Decisions Here-and-now: Decisions (x) are takenbefore uncertainty ω resolute Wait-and-see: Decisions (y ω ) are taken after uncertainty ω resolute as corrective action - recourse x Uncertainty reveal y ω ω= 1 ω= 2 ω= 3 ω= 4 ω= 5 ω= Ω Page 9
Method Stochastic Programming Stochastic Programming for Case Study 1 First stage decisions Here-and-now: decisions for the first month (inventory, shipping, sale) Second stage decisions Wait-and-see: decisions for the remaining 11 months cost of scenario s1 cost of scenario s2 cost of scenario s3 Minimize E [cost] cost of scenario s4 cost of scenario s5 Page 10
Model Formulation Multiperiod Planning Model (Case Study 1) Objective Function: Min: Total Expected Cost Constraints: Mass balance for plants Mass balance for terminals Mass balance for customers Minimum inventory level constraint Single and minimum sourcing constraint Page 11
Model Formulation Objective Function Expected Cost First stage cost Probability of each scenario Second stage cost Inventory Costs Freight Costs Throughput Costs Demand Unsatisfied Page 12
SP Model Result Result of SP Model for Case Study 1 0.27 0.24 E[Cost] = $182.32±0.36 MM (95% confidence interval) 0.21 0.18 Probability 0.15 0.12 0.09 0.06 0.03 1,000 scenarios 0 170 173 176 179 182 185 188 191 194 197 200 Cost ($ MM) Page 13
Algorithm: Multi-cut L-shaped Method Problem Sizes Case Study 1 Deterministic Model Two-stage Stochastic Programming Model 10 scenarios 100 scenarios 1,000 scenarios # of Constraints 62,187 588,407 5,324,387 52,684,187 # of Cont. Var. 89,014 841,684 7,615,714 75,356,014 # of Disc. Var. 7 7 7 7 Case Study 2 Deterministic Model Two-stage Stochastic Programming Model 10 scenarios 100 scenarios 1,000 scenarios # of Constraints 3,308 13,883 119,633 1,177,133 # of Cont. Var. 3,323 14,625 124,515 1,223,415 # of Disc. Var. 270 270 270 270 Note: Problems with red statistical data are not able to be solved by DWS Page 14
Algorithm: Multi-cut L-shaped Method Multi-cut L-shaped Method Solve master problem to get a lower bound (LB) Add cut Solve the subproblem to get an upper bound (UB) No UB LB < Tol? Yes Page 15
Algorithm: Multi-cut L-shaped Method Example 220 210 200 Standard L-Shaped Upper_bound Standard L-Shaped Lower_bound Multi-cut L-Shaped Upper_bound Multi-cut L-Shaped Lower_bound Cost ($MM) 190 180 170 160 150 Impossible to solve directly takes 5 days by using standard L-shaped only 20 hours with multi-cut version, 12 min if using 100 parallel CPUs and multi-cut version 140 1 21 41 61 81 101 121 141 161 181 Iterations Page 16
Simulation Simulation Framework Solve Stochastic model and execute decisions for period t Update information on the uncertain parameters (mean and variance) period t-1 Randomly generate demand and freight rate period t+1 Solve Deterministic model and execute decisions for period t Update information on the uncertain parameters (only mean value) period t period t-1 period t+1 Page 17
Simulation Rolling Horizon Strategy revealed uncertain Inter-facility shipment from the previous time periods considered as pipeline inventory Facility-customer shipment considers as part of demand realization Inventory level in the previous time period considers as the initial inventory for t =1 Consider uncertainty reduction as time period moving forward Page 18
Simulation Simulation Results for Case Study 12 11 10 Average 5.70±0.03% cost saving Stochastic Soln Deterministic Soln Cost ($MM) 9 8 7 6 1 10 19 28 37 46 55 64 73 82 91 100 Iterations Page 19
Remarks Concluding Remarks Current Progress Develop two-stage stochastic programming models for global supply chain planning under uncertainty. Simulation studies show that 5.70% cost saving can be achieved in average. Present five risk management models. Develop an efficient solution algorithm to solve the industrial size SP problems. Submit a paper to AIChE Journal * recently. * Fengqi You, John M. Wassick, Ignacio E. Grossmann, Risk management for a global supply chain planning under uncertainty: Models and Algorithms, AIChE Journal, Submitted, 2008 Page 20
Multi-site Capacity Planning Capacity Planning w/ Reactor Transformation Train i Train i Site 1 Customer 1 Potential Site 4 Train i Site 2 Customer 2 Potential Site 5 Train i Train i Customer 3 Site 3 Customer 4 Capacity Modifications Potential Site 6 Add a new train (discrete selections) Shut down an existing train Reactor Condenser Distillation Convert a train to produce another product mix Page 21
Multi-site Capacity Planning Problem Statement Capacity modifications Sites Where? What? When? Customers Uncertain demands Product mixes Train Capacities Train network Production levels Costs and prices Max: Net present value Construction times Sourcing amount Planning horizon: 5-10 years Page 22