global supply chain planning under demand and freight rate

Global Supply Chain Planning under Demand and Freight Rate UncertaintyFengqi YouIgnacio E. Grossmann

Nov. 13, 2007

Sponsored by The Dow Chemical Company (John Wassick)

• Objective:Developing Models and Algorithms for Global Multiproduct Chemical Supply Chains Plannning under Uncertainty

Motivation

• Global Chemical Supply ChainCosts several billion dollars annuallyPlanning under uncertain environment

Introduction

Changes of customer orders

Fluctuations of gas prices

Supply Chain Tactical Planning

Plants Distribution Centers Customers

Introduction

Problem Statement

• GivenMinimum and initial inventoryInventory holding cost and throughput costTransport times of all the transport linksUncertain customer demands and transport cost

• DetermineTransport amount, inventory and production levels

• Objective: Minimize Cost

Problem Statement

Outline

• Stochastic Programming ModelTwo-stage stochastic programming modelSimulation-optimization framework

• Model Extensions for Robustness and RiskRobust optimizationRisk management

• AlgorithmsMulti-cut L-shaped method

Uncertain Parameters

• Demand and Freight Rate UncertaintyNormal distributionForecast value as mean, variances come from historical data

» Demand uncertainty has 3 level variances» Freight rate uncertainty has 2 level variances

Stochastic Programming Model

Scenario Planning• Discretize the probability distribution function

Approximate all the uncertainties to discrete distribution (scenarios)Each scenario represents a possible outcomeGenerate scenarios by Monte Carlo sampling

» Assign each scenario the same probability (i.e. For N sampling, Ps=1/N)» Combine statistical methods for “good” approximation


Decision Stages under Uncertainty• Here-and-now

Decisions (x) are taken before uncertainty ω resolute• Wait-and-see

Decisions (yω) are taken after uncertainty ω resolute as “corrective action” -recourse

xyω

ω= 1ω= 2ω= 3ω= 4ω= 5ω= Ω


Uncertainty reveal

Two-stage Stochastic Programming• 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]


Multi-period Planning Model

• Objective Function:Min: Total Expected Cost

• Constraints:Mass balance for plantsMass balance for DCsMass balance for customersMinimum inventory level constraintCapacity constraints for plants


Second stage cost

Objective Function

Inventory Costs

Throughput Costs

Freight Costs

Demand Unsatisfied

First stage cost


Probability of each scenario

revealed

Rolling Horizon Strategy

uncertain

Inter-facility shipment from the previous time periods considered as pipeline inventoryFacility-customer shipment considers as part of demand realizationInventory level in the previous time period considers as the initial inventory for t =1Consider uncertainty reduction as time period moving forward

Simulation-optimization Framework

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

Robust Optimization

Cost

Probability ExpectedCost

Desirable Penalty Undesirable Penalty

Objective: To find out the optimal solution that yields similar results under the uncertain environment – Robust solution!

Robust Optimization

Cost of robustness

Robust Optimization using Variance

• Goal Programming FormulationNew objective function: Minimize E[Cost] + ρ ·V[Cost]Different ρ can lead to different solution (multi-objective optimization)

Expected Cost Expected Variance of each scenario

Weighted coefficient

Robust Optimization

Robust Optimization via Variability Index

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

Robust Optimization

Variability Index (cont)

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

Robust Optimization

Risk Management (cont)

• Risk: The probability of exceeding certain target cost level Ω

Histogram of Frequencies Cumulative Risk Curve

Ω

Cost

Cumulative Probability = Risk (x, Ω)

Risk Management

Risk Management (cont)

• Risk Management for scenario planningCalculation by

Binary variables Prob

abili

ty

Risk Management

Risk Management Model Formulation

Prob

abili

ty

Risk Management Constraints

Risk Objective

Economic Objective

Multi-objective

Risk Management

Downside Risk• Definition: Positive Profit Deviation

Binary variables are not required, pure LP (MILP -> LP)

Risk Management

Standard L-shaped Method

Scenario sub-problems

Masterproblem

x

1y

Sy

2y

Master problem

Scenario sub-problems

Algorithm: Multi-cut L-shaped Method

3y

Expected Recourse Function

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


Multi-cut L-shaped Method

YesNo

Solve master problem to get a lower bound (LB)

Solve the sub problem to get an upper bound (UB)

UB – LB < Tol ? STOP

Add cut


Conclusion• Current Work

Develop a two-stage stochastic programming approach for global supply chain planning under uncertainty. Simulation studies show that 5.70% cost saving in average can be achievedPresent two robust optimization models and two risk management model. Develop an efficient solution algorithm to solve the large scale stochastic programming problem

• Future WorkMulti-site capacity planning under uncertainty

Questions?

global supply chain planning under demand and freight rate

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