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EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
Risk Management for Chemical Supply Chain Planning under Uncertainty
Fengqi You and Ignacio E. GrossmannDept. of Chemical Engineering, Carnegie Mellon University
John M. WassickThe Dow Chemical Company
Page 2EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
Chemical Supply chain: an integrated network of business units for the supply, production, distribution and consumption of the products.
Introduction – Chemical Supply Chain
Supply Chain
Page 3EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementMotivationIntroduction
• Objective: managing the risks in supply chain planning
• Chemical Supply Chain PlanningCosts billions of dollars annuallyAlways under various uncertainties and risks
Cost & Prices fluctuationsDemand Changes Natural Disasters Machine Broken Down
Page 4EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
• GivenMinimum and initial inventoryInventory holding cost and throughput costTransport times of all the transport links & modesUncertain customer demands and transport cost
• DetermineTransport amount, inventory and production levels
• Objective: Minimize Cost & Risks
Case StudyIntroduction
Page 5EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementStochastic Programming• Scenario Planning
A scenario is a future possible outcome of the uncertaintyFind a solution perform well for all the scenarios
• Two-stage DecisionsHere-and-now: Decisions (x) are taken before uncertainty ω resoluteWait-and-see: Decisions (yω) are taken after uncertainty ω resolute as“corrective action” - recourse
xyω
Uncertainty reveal
ω= 1ω= 2ω= 3ω= 4ω= 5ω= Ω
Decision-making under Uncertainty
Page 6EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementStochastic 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
Minimize E [cost]
cost of scenario s1
cost of scenario s2
cost of scenario s3
cost of scenario s4
cost of scenario s5
Decision-making under Uncertainty
Page 7EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementObjective Function
Inventory Costs
Throughput Costs
Freight Costs
Demand Unsatisfied
First stage cost Second stage cost
Stochastic Programming Model
Probability of each scenario
Page 8EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementMultiperiod Planning Model (Case Study)
• Objective Function:Min: Total Expected Cost
• Constraints:Mass balance for plantsMass balance for DCsMass balance for customersMinimum inventory level constraintCapacity constraints for plants
Stochastic Programming Model
Page 9EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementResult of Two-stage SP Model
0
0.03
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.27
170 173 176 179 182 185 188 191 194 197 200Cost ($ MM)
Prob
abili
ty
E[Cost] = $182.32MM
Stochastic Programming Model
Page 10EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
• SP model: optimize expected cost (risk-neutral objective)Could not control variance, extreme values, etc.The following distributions have the same E[Cost]=4.1
Risk ManagementRisk Management
• Use risk measures to control the possible outcomeVariance (Mulvey et al., 1995)Upper partial mean (Ahmed and Sahinidis, 1998)Probabilistic financial risk (Barbaro et al., 2002)Downside risk (Eppen et al., 1988)
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10
Cost
P
“Ideal” result
0
0.2
0.4
0.6
0.8
1 2 3 4 5 6 7 8 9 10
Cost
P
High variance
0
0.2
0.4
0.6
0.8
1
1 4 7 10 13 16 19 22 25 28 31 34 37 40
Cost
P
High extreme cost
Page 11EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Cost
Probability ExpectedCost
Desirable Penalty Undesirable Penalty
Objective: Managing the risks by reducing the variance (robust optimization)
Additional Cost
Risk Management
Risk Management using Variance
Page 12EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementRisk Management using Variance
• Goal Programming FormulationNew objective function: Minimize E[Cost] + ρ ·V[Cost]Different ρ can lead to different solution
Expected Cost Variance of all the scenarios
Weighted coefficient
Risk Management
Page 13EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
182
183
184
185
186
187
0 3 6 9 12 15
ρ (1E-5)
Cos
t ($M
M)
0
7
14
21
28
35
Var
ianc
e ($
MM
^2)
Cost ($MM)Variance
Case Study – Robustness vs. CostRisk Management
Page 14EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
0
0.05
0.1
0.15
0.2
0.25
0.3
170 172 174 176 178 180 182 184 186 188 190Cost ($MM)
Prob
abili
ty
E(Cost)=$182.24 MM, ρ=0E(Cost)=$183.14 MM, ρ=1.5E-4
Case Study – Variance ReductionRisk Management
Page 15EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
• First Order Variability indexConvert NLP to LP by replacing two norm to one norm
Risk Management via Variability IndexRisk Management
Page 16EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
• Linearize the absolute value termIntroducing a first order non-negative variability index Δ
Variability IndexRisk Management
Page 17EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementUpper Partial Mean Risk Management
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.50.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Desirable Penalty Undesirable Penalty
• Reduce undesirable penaltyUsing positive variability index Δ by goal programming
(Desirable + undesirable penalty)
(Only undesirable penalty)
Page 18EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementEfficient Frontier – Robustness vs. Cost
178
180
182
184
186
188
190
0 2 4 6 8ρ
Cos
t ($M
M)
0
0.7
1.4
2.1
2.8
3.5
4.2
Var
ianc
e ($
MM
^2)
Cost ($MM)Variance ($MM^2)
Risk Management
Page 19EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementResults – Variability Index Reduction
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
170 172 174 176 178 180 182 184 186 188 190Cost ($MM)
Prob
abili
ty
E(Cost)=$182.24MM, ρ=0E(Cost)=$185.87MM, ρ=2
Risk Management
Page 20EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
0
0.03
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.27
170 173 176 179 182 185 188 191 194 197 200Cost ($ MM)
Prob
abili
ty
Objective: modify the cost distribution in order to satisfy the preferences of the decision maker – manage the probabilistic financial risk
Reduce these probability
OR…
Increase these?
Financial Risk ManagementRisk Management
Page 21EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
• Probabilistic Financial RiskProbability of exceeding certain target Ω
Binary variables
Big-M constraints
Ω
Cost
Prob
abili
ty
Cumulative Probability = Risk (x, Ω)
Financial Risk Management ModelRisk Management
Page 22EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementRisk Management Model Formulation
Prob
abili
ty
Risk Management Constraints
Risk Objective
Economic Objective
Multi-objective
Risk Management
Page 23EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
Cost
Risk
Pareto Curve (efficient frontier)
BReduce Risk
D
A
An infinite set of alternative optimal solutions (Pareto curve)
Pareto optimum: Impossible to improve both objective functions simultaneously
C
Reduce Cost
Risk Management
Multi-objective Optimization Model
Page 24EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
Cost
RiskSmallest Risk
Min Optimal Cost
Largest Risk
Max Optimal Cost
Minimize: Cost + ε∙ Risk(ε = 0.001)
Minimize: Risk
Risk Management
ε- constraint Method
Page 25EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementPareto Curve: E[Cost] vs. Risk
180.4
180.6
180.8
181
181.2
181.4
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Risk
E[ C
ost]
($M
M)
Note: Target at $188 MM
Risk Management
Page 26EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
0
0.05
0.1
0.15
0.2
0.25
170 172 174 176 178 180 182 184 186 188 190 192 194Cost ($MM)
Prob
abili
ty
Risk = 0.08 (Min [Cost])Risk = 0.02
Results for Probabilistic Risk ManagementRisk Management
Page 27EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementDownside Risk
• Definition: Positive DeviationBinary variables are not required, pure LP (MILP -> LP)
Risk Management
Page 28EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
Prob
abili
ty
Downside Risk Constraints
Risk Objective
Economic Objective
Risk Management
Downside Risk Model Formulation
Page 29EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementResults for Downside Risk Management
0
0.05
0.1
0.15
0.2
0.25
170 172 174 176 178 180 182 184 186 188 190 192 194Cost ($MM)
Prob
abili
ty
DRisk=36.92 (Min E[Cost])DRsik=5.38
Risk Management
Page 30EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementSimulation Framework
period t-1
Solve Deterministicmodel and execute
decisions for period t
Solve Stochastic model and execute decisions for
period t
Update information on the uncertain parameters(mean and variance)
Update information on the uncertain parameters
(only mean value)
Randomly generate demand and freight rate
period t+1
period tperiod t-1 period t+1
Simulation
Page 31EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementSimulation Flowchart
YesNo
Solve the S/D model and implement the decision for current time period
Update information
t=12 ?
Move to next time period
t = t+1
Randomly generate demand and freight rate information
Next iteration
Calculate the real cost, store data, Set iter = iter+1, t =1
STOPReach iteration limit?
Yes
No
Simulation
Page 32EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementCase Study
6
7
8
9
10
11
12
1 10 19 28 37 46 55 64 73 82 91 100Iterations
Cos
t ($M
M)
Stochastic SolnDeterministic SolnAverage 5.70%
cost saving
Simulation
Page 33EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementProblem Sizes
8,498,429850,27185,4518,910# of Non-zeros
3,701,240370,33837,2483,937# of Variables
1,301,070130,17013,0801,369# of Constraints
1,000 scenarios100 scenarios10 scenarios
Two-stage Stochastic Programming ModelDeterministic ModelToy Problem
40,028,8724,004,697402,26741,899# of Non-zeros
18,149,0771,815,816182,49619,225# of Variables
6,101,280610,37461,2846,373# of Constraints
1,000 scenarios100 scenarios10 scenarios
Two-stage Stochastic Programming ModelDeterministic ModelFull Problem
Note: Problems with red statistical data are not able to be solved by DWS
Algorithm: Multi-cut L-shaped Method
Page 34EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementTwo-stage SP Model
Scenario sub-problems
Masterproblem
x
1y
Sy
2y
Master problem
Scenario sub-problems
y
Algorithm: Multi-cut L-shaped Method
Page 35EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementStandard L-shaped Method
No
Add cut
Solve master problem to get a lower bound (LB)
Solve the subproblem to get an upper bound (UB)
UB – LB < Tol ?Yes
STOP
Algorithm: Multi-cut L-shaped Method
Page 36EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementExpected Recourse Function
The expected recourse function Q(x) is convex and piecewise linearEach optimality cut supports Q(x) from below
Algorithm: Multi-cut L-shaped Method
Page 37EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementMulti-cut L-shaped Method
YesNo
Solve master problem to get a lower bound (LB)
Solve the subproblem to get an upper bound (UB)
UB – LB < Tol ? STOP
Add cut
Algorithm: Multi-cut L-shaped Method
Page 38EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementExample
140
150
160
170
180
190
200
210
220
1 21 41 61 81 101 121 141 161 181Iterations
Cos
t ($M
M)
Standard L-Shaped Upper_boundStandard L-Shaped Lower_boundMulti-cut L-Shaped Upper_boundMulti-cut L-Shaped Lower_bound
Algorithm: Multi-cut L-shaped Method
Page 39EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk ManagementConclusion
• Current WorkDevelop 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 WorkCapacity planning under demand uncertainty
Remarks
Page 40EWO Meeting, Mar. 2008, Pittsburgh, PA
Risk Management
Questions?