Risk Management for Chemical Supply Chain Planning under Uncertainty

Transcription

1 for Chemical Supply Chain Planning under Uncertainty Fengqi You and Ignacio E. Grossmann Dept. of Chemical Engineering, Carnegie Mellon University John M. Wassick The Dow Chemical Company

2 Introduction Chemical Supply Chain Supply Chain Chemical Supply chain: an integrated network of business units for the supply, production, distribution and consumption of the products. Page 2

3 Introduction 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 the risks in supply chain planning Page 3

4 Introduction Case Study Given Minimum and initial inventory Inventory holding cost and throughput cost Transport times of all the transport links & modes Uncertain customer demands and transport cost Determine Transport amount, inventory and production levels Objective: Minimize Cost & Risks Page 4

5 Decision-making under Uncertainty Stochastic Programming Scenario Planning A scenario is a future possible outcome of the uncertainty Find a solution perform well for all the scenarios 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 5

6 Decision-making under Uncertainty Stochastic Programming for Case Study First stage decisions Here-and-now: decisions for the first month (production, inventory, shipping) 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 6

7 Stochastic Programming Model Objective Function First stage cost Probability of each scenario Second stage cost Inventory Costs Freight Costs Throughput Costs Demand Unsatisfied Page 7

8 Stochastic Programming Model Multiperiod Planning Model (Case Study) Objective Function: Min: Total Expected Cost Constraints: Mass balance for plants Mass balance for DCs Mass balance for customers Minimum inventory level constraint Capacity constraints for plants Page 8

9 Stochastic Programming Model Result of Two-stage SP Model 0.27 E[Cost] = $182.32MM Probability Cost ($ MM) Page 9

10 SP model: optimize expected cost (risk-neutral objective) Could not control variance, extreme values, etc. The following distributions have the same E[Cost]= P 0.4 P P Cost High variance Cost Ideal result Cost High extreme cost Use risk measures to control the possible outcome Variance (Mulvey et al., 1995) Upper partial mean (Ahmed and Sahinidis, 1998) Probabilistic financial risk (Barbaro et al., 2002) Downside risk (Eppen et al., 1988) Page 10

11 using Variance Probability 0.9 Additional Cost Expected Cost Desirable Penalty Undesirable Penalty Cost Objective: Managing the risks by reducing the variance (robust optimization) Page 11

12 using Variance Goal Programming Formulation New objective function: Minimize E[Cost] + ρ V[Cost] Different ρ can lead to different solution Expected Cost Variance of all the scenarios Weighted coefficient Page 12

13 Case Study Robustness vs. Cost Cost ($MM) Variance ($MM^2) ρ (1E-5) Cost ($MM) Variance 7 0 Page 13

14 Case Study Variance Reduction E(Cost)=$ MM, ρ=0 E(Cost)=$ MM, ρ=1.5e Probability Cost ($MM) Page 14

15 via Variability Index First Order Variability index Convert NLP to LP by replacing two norm to one norm Page 15

16 Variability Index Linearize the absolute value term Introducing a first order non-negative variability index Δ Page 16

17 Upper Partial Mean Reduce undesirable penalty Using positive variability index Δ by goal programming (Desirable + undesirable penalty) Desirable Penalty Undesirable Penalty (Only undesirable penalty) Page 17

18 Efficient Frontier Robustness vs. Cost Cost ($MM) Variance ($MM^2) Cost ($MM) Variance ($MM^2) ρ Page 18

19 Results Variability Index Reduction E(Cost)=$182.24MM, ρ=0 E(Cost)=$185.87MM, ρ=2 0.5 Probability Cost ($MM) Page 19

20 Financial OR Increase these? Probability Reduce these probability Cost ($ MM) Objective: modify the cost distribution in order to satisfy the preferences of the decision maker manage the probabilistic financial risk Page 20

21 Financial Model Probabilistic Financial Risk Probability of exceeding certain target Ω Binary variables Ω Cumulative Probability = Risk (x, Ω) Big-M constraints Probability Cost Page 21

22 Model Formulation Risk Objective Economic Objective Multi-objective Constraints Probability Page 22

23 Multi-objective Optimization Model An infinite set of alternative optimal solutions (Pareto curve) Pareto optimum: Impossible to improve both objective functions simultaneously Cost B Reduce Risk A D C Reduce Cost Pareto Curve (efficient frontier) Risk Page 23

24 ε- constraint Method Cost Max Optimal Cost Minimize: Cost + ε Risk (ε = 0.001) Min Optimal Cost Minimize: Risk Smallest Risk Largest Risk Risk Page 24

25 Pareto Curve: E[Cost] vs. Risk E [Cost ] ($MM) Risk Note: Target at $188 MM Page 25

26 Results for Probabilistic Risk = 0.08 (Min [Cost]) Risk = 0.02 Probability Cost ($MM) Page 26

27 Downside Risk Definition: Positive Deviation Binary variables are not required, pure LP (MILP -> LP) Page 27

28 Downside Risk Model Formulation Risk Objective Economic Objective Downside Risk Constraints Probability Page 28

29 Results for Downside 0.25 DRisk=36.92 (Min E[Cost]) DRsik= Probability Cost ($MM) Page 29

30 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 30

31 Simulation Simulation Flowchart Calculate the real cost, store data, Set iter = iter+1, t =1 Solve the S/D model and implement the decision for current time period Next iteration Move to next time period Randomly generate demand and freight rate information t = t+1 Update information No No t=12? Yes Reach iteration limit? Yes STOP Page 31

32 Simulation Case Study Average 5.70% cost saving Stochastic Soln Deterministic Soln Cost ($MM) Iterations Page 32

33 Algorithm: Multi-cut L-shaped Method Problem Sizes Toy Problem Deterministic Model Two-stage Stochastic Programming Model 10 scenarios 100 scenarios 1,000 scenarios # of Constraints 1,369 13, ,170 1,301,070 # of Variables 3,937 37, ,338 3,701,240 # of Non-zeros 8,910 85, ,271 8,498,429 Full Problem Deterministic Model Two-stage Stochastic Programming Model 10 scenarios 100 scenarios 1,000 scenarios # of Constraints 6,373 61, ,374 6,101,280 # of Variables 19, ,496 1,815,816 18,149,077 # of Non-zeros 41, ,267 4,004,697 40,028,872 Note: Problems with red statistical data are not able to be solved by DWS Page 33

34 y Algorithm: Multi-cut L-shaped Method Two-stage SP Model Master problem Scenario subproblems y 1 Master problem x y 2 y S Scenario sub-problems Page 34

35 Algorithm: Multi-cut L-shaped Method Standard 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 STOP Page 35

36 Algorithm: Multi-cut L-shaped Method Expected Recourse Function The expected recourse function Q(x) is convex and piecewise linear Each optimality cut supports Q(x) from below Page 36

37 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 STOP Page 37

38 Algorithm: Multi-cut L-shaped Method Example Standard L-Shaped Upper_bound Standard L-Shaped Lower_bound Multi-cut L-Shaped Upper_bound Multi-cut L-Shaped Lower_bound Cost ($MM) Iterations Page 38

39 Remarks Conclusion Current Work Develop a two-stage stochastic programming model for global supply chain planning under uncertainty. Simulation studies show that 5.70% cost saving can be achieved in average Present four risk management model. Develop an efficient solution algorithm to solve the large scale stochastic programming problem Future Work Capacity planning under demand uncertainty Page 39

40 Questions? Page 40