Models for Designing Resilient Supply Chain Networks for Chemicals

Models for Designing ResilientSupply Chain Networks for Chemicals and Gases

Lawrence V. Snyder Peng Peng

Department of Industrial & Systems EngineeringCenter for Value Chain Research

Lehigh UniversityBethlehem, PA

EWO Meeting March 12, 2008

Snyder Air Products: Resilient Supply Chains

Outline

1 Introduction

2 The Model

3 Example


IntroductionThe Model

Example

Outline

1 Introduction

2 The Model

3 Example



Example

Introduction

Supply chain networks are usually designed as though theyfuncion in normal mode all the time

But disruptions are a significant factor

Result in significant cost increases

Increased transportation costsPenalty for missed demands

Objective: Develop model for designing supply chains thatperform well when no disruptions occur but not too badlywhen disruptions occur.

Focus on packaged gases, but model is generic



Example

Introduction

Supply chain networks are usually designed as though theyfuncion in normal mode all the time

But disruptions are a significant factor

Result in significant cost increases

Increased transportation costsPenalty for missed demands

Objective: Develop model for designing supply chains thatperform well when no disruptions occur but not too badlywhen disruptions occur.

Focus on packaged gases, but model is generic



Example

Motivating Example

Plants Warehouses Customers



Example

Decisions

Need to decide which plants and warehouses to open

Customer locations are fixed

Minimize sum of

Fixed cost (to build/lease facilities)Transportation cost

Suppose we optimize ignoring disruptions



Example

Decisions

Need to decide which plants and warehouses to open

Customer locations are fixed

Minimize sum of

Fixed cost (to build/lease facilities)Transportation cost

Suppose we optimize ignoring disruptions



Example

“Optimal” Solution

Cost: $521,163


fig_opt_nominal



Example

Disruptions

Now suppose disruptions can occur

We must design supply chain before we know what facilitieswill be disrupted

Possible disruptions are described by scenarios



Example

Disruptions





fig_scen1Scenario 1Snyder Air Products: Resilient Supply Chains


Example

Disruptions







Example

Disruptions







Example

Solution Performance When Disruptions Occur

Scenario 1: Cost $521,163


fig_soln1_scen1



Example


Scenario 2: Cost $982,241


fig_soln1_scen2

EMERG



Example


Scenario 3: Cost $3,237,107


fig_soln1_scen3

EMERG



Example

Solution Performance: Summary

$0

$500

$1,000

$1,500

$2,000

$2,500

$3,000

$3,500

0 1 2 3

Scenario

Cos

t (x$

1000

)



Example

Outline

1 Introduction

2 The Model

3 Example



Example

Model Objectives

Can we improve performance under the disruption scenariosby choosing better facilities?

“Nominal” cost will increase

But scenario costs will decrease

Two-stage problem:

Stage 1: Choose facility locations(scenario occurs)Stage 2: Assign flows



Example

Model Objectives

Can we improve performance under the disruption scenariosby choosing better facilities?

“Nominal” cost will increase

But scenario costs will decrease

Two-stage problem:

Stage 1: Choose facility locations(scenario occurs)Stage 2: Assign flows



Example

Robust Optimization

This is a robust optimization problem

Prevent performance from being “too bad” in any scenario

Lots of approaches for robust optimization in the literature

Min-max costMin-max regretCVaRetc.

Our approach:

Minimize cost in nominal scenario (no disruptions)Subject to bound on cost in any other scenario“p-robustness”



Example

Robust Optimization

This is a robust optimization problem

Prevent performance from being “too bad” in any scenario

Lots of approaches for robust optimization in the literature

Min-max costMin-max regretCVaRetc.

Our approach:

Minimize cost in nominal scenario (no disruptions)Subject to bound on cost in any other scenario“p-robustness”



Example

Notation: Nodes and Arcs

General network (V ,A)

V = set of nodesA = set of arcsNot necesssarily plants, warehouses, customers

V0 ⊆ V = set of non-demand nodes

N = number of products

bjn = supply of product n at node j ∈ V

> 0 for supply nodes< 0 for demand nodes= 0 for transshipment nodes

kjn = capacity for product n at node j ∈ V0

Depends on type of product and type of storage (bulk, drum,tote)



Example

Notation: Nodes and Arcs

General network (V ,A)

V = set of nodesA = set of arcsNot necesssarily plants, warehouses, customers

V0 ⊆ V = set of non-demand nodes

N = number of products

bjn = supply of product n at node j ∈ V

> 0 for supply nodes< 0 for demand nodes= 0 for transshipment nodes

kjn = capacity for product n at node j ∈ V0

Depends on type of product and type of storage (bulk, drum,tote)



Example

Notation: Costs and Disruptions

fj = fixed cost to open node j ∈ V0

dijn = cost to transport one unit of product n on arc (i , j) ∈ A

S = set of scenarios

s = 0 is nominal scenario

ajs = 1 if node j ∈ V0 is disrupted in scenario s

Applies to non-demand nodes only

Us = maximum allowable cost in scenario s ∈ S \ 0



Example

Notation: Costs and Disruptions

fj = fixed cost to open node j ∈ V0

dijn = cost to transport one unit of product n on arc (i , j) ∈ A

S = set of scenarios

s = 0 is nominal scenario

ajs = 1 if node j ∈ V0 is disrupted in scenario s

Applies to non-demand nodes only

Us = maximum allowable cost in scenario s ∈ S \ 0



Example

Notation: Decision Variables

Xj = 1 if node j ∈ V0 is open, 0 otherwise

Yijns = flow of product n on arc (i , j) in scenario s



Example

Objective Function

Minimize nominal-scenario (no-disruption) cost:

minimize∑j∈V0

fjXj +∑

(i ,j)∈A

N∑n=1

dijnYijn0 (1)



Example

Constraints

Subject to:

∑j∈V0

fjXj +∑

(i ,j)∈A

N∑n=1

dijnYijns ≤ Us ∀s (2)

∑(j ,i)∈A

Yjins −∑

(i ,j)∈A

Yijns = bjn ∀j , n, s (3)

∑(j ,i)∈A

Yjins ≤ (1− ajs)kjnXj ∀j , n, s (4)

Xj ∈ 0, 1 ∀j (5)

Yijns ≥ 0 ∀i , j , n, s (6)



Example

In Words

minimize cost in nominal scenario

subject to cost in scenario s ≤ Us ∀sflow balance constraints ∀j , n, s

flow can’t exceed capacity, or 0 if disrupted ∀j , n, s

Xj integer

Yijns non-negative



Example

An Even More Compact Description

minimize c0(X ,Y )

subject to cs(X ,Y ) ≤ Us ∀s(X ,Y ) ∈ Ω



Example

Solution Methods

For now, we solve using an off-the-shelf MIP solver

Run times generally <5 minutes for problems with

40 plants40 warehouses500 customers5 scenarios1 product

For bigger problems, we’ll need customized algorithms

Future research



Example

Outline

1 Introduction

2 The Model

3 Example



Example

Example

Single product

3 plants, each with capacity 280

4 warehouses, each with capacity 280

10 customers, average demand 60

4 scenarios

Nominal scenario3 disruption scenarios

Costs

fj = 120,000 on average for plantsfj = 25,000 on average for warehousesdijn = 500 on averagePenalty cost for unmet demand = 10,000

Scenario upper-bounds = 800,000



Example

Example

Single product

3 plants, each with capacity 280

4 warehouses, each with capacity 280

10 customers, average demand 60

4 scenarios

Nominal scenario3 disruption scenarios

Costs

fj = 120,000 on average for plantsfj = 25,000 on average for warehousesdijn = 500 on averagePenalty cost for unmet demand = 10,000

Scenario upper-bounds = 800,000



Example

New Solution

Cost: $580,758 (vs. $521,163)


fig_soln2_nominal



Example


Scenario 1: Cost $580,758 (vs. $521,163)


fig_soln2_scen1



Example


Scenario 2: Cost $800,000 (vs. $982,241)


fig_soln2_scen2

EMERG



Example


Scenario 3: Cost $800,000 (vs. $3,237,107)


fig_soln2_scen3

EMERG



Example

Solution Performance: Summary

$0

$500

$1,000

$1,500

$2,000

$2,500

$3,000

$3,500

0 1 2 3

Thou

sand

s

Scenario

Cos

t (x$

1000

)

Solution 1Solution 2



Example

Scenario Costs Equal Upper Bounds

Recall:

minimize c0(X ,Y )


Nothing to require optimization in scenarios

Only achieve sufficiently low cost



Example

Scenario Costs Equal Upper Bounds

Recall:

minimize c0(X ,Y )


Nothing to require optimization in scenarios

Only achieve sufficiently low cost



Example

Possible Approaches

Include scenario costs in objective function

minimize c0(X ,Y ) +∑

s

γscs(X ,Y )

Downside: scaling issues

Post-optimize scenarios

Solve model, fix XThen find optimal Y for each scenarioDownside: cumbersome

Currently investigating other approaches



Example

Possible Approaches



s

γscs(X ,Y )







Example

Possible Approaches



s

γscs(X ,Y )



Solve model, fix XThen find optimal Y for each scenario

Downside: cumbersome




Example

Possible Approaches



s

γscs(X ,Y )







Example

Conclusions

Supply chain network design model that accounts fordisruptions

Minimize nominal cost, subject to bound on cost in eachscenario

Next steps:

Resolve sub-optimization in scenariosDesign algorithm



Example

Questions?

[email protected]


Documents

Models for Designing Resilient Supply Chain Networks for Chemicals