Risk Management for Chemical Supply Chain Planning under Uncertaintyegon.cheme.cmu.edu/ewo/docs/EWO_DowMar_2008.pdf · 2016. 2. 8. · EWO Meeting, Mar. 2008, Pittsburgh, PA Risk

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


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


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


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


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


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


Risk ManagementObjective Function

Inventory Costs

Throughput Costs

Freight Costs

Demand Unsatisfied

First stage cost Second stage cost

Stochastic Programming Model

Probability of each scenario


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



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



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


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


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


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


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


Risk Management

• First Order Variability indexConvert NLP to LP by replacing two norm to one norm

Risk Management via Variability IndexRisk Management


Risk Management

• Linearize the absolute value termIntroducing a first order non-negative variability index Δ

Variability IndexRisk Management


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)


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


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


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


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


Risk ManagementRisk Management Model Formulation

Prob

abili

ty

Risk Management Constraints

Risk Objective

Economic Objective

Multi-objective

Risk Management


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


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


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


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


Risk ManagementDownside Risk

• Definition: Positive DeviationBinary variables are not required, pure LP (MILP -> LP)

Risk Management


Risk Management

Prob

abili

ty

Downside Risk Constraints

Risk Objective

Economic Objective

Risk Management

Downside Risk Model Formulation


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


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


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


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


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


Risk ManagementTwo-stage SP Model

Scenario sub-problems

Masterproblem

x

1y

Sy

2y

Master problem

Scenario sub-problems

y



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



Risk ManagementExpected Recourse Function

The expected recourse function Q(x) is convex and piecewise linearEach optimality cut supports Q(x) from below



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



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



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


Risk Management

Questions?

Documents

Risk Management for Chemical Supply Chain Planning under Uncertaintyegon.cheme.cmu.edu/ewo/docs/EWO_DowMar_2008.pdf · 2016. 2. 8. · EWO Meeting, Mar. 2008, Pittsburgh, PA Risk