Models for Designing Resilient Supply Chain Networks for...

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 September 30, 2008

Snyder,Peng Air Products: Resilient Supply Chains

Outline

1 Introduction

2 The Model

3 Example

4 Improve p-robustness

5 Conclusion


IntroductionThe Model

ExampleImprove p-robustness

Conclusion

Outline

1 Introduction

2 The Model

3 Example


5 Conclusion




Conclusion

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 unmet 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




Conclusion

Disruptions

Suppose disruptions can occur

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

Possible disruptions are described by scenarios




Conclusion

Outline

1 Introduction

2 The Model

3 Example


5 Conclusion




Conclusion

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




Conclusion

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”




Conclusion

Decisions

Need to decide which plants and warehouses to open

Customer locations are fixed

Minimize total cost which is the sum of

Fixed cost (to build/lease facilities)Transportation cost




Conclusion

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)




Conclusion

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, aj0 = 0∀jApplies to non-demand nodes only

ps = desired robustness level in scenario s ∈ S \ 0The standard problem of minimizing the nominal cost withoutconsidering scenario costs can be obtained by setting p =∞

c∗s = the optimal objective value for scenario s ∈ S \ 0




Conclusion

Notation: Bill of Materials(BOM)

Define

αmn: amount of product n that is used to make 1 unit ofproduct m, m, n ∈ N.

We consider the production process in our modelRaw materials are shipped to the plants and made intodifferent products, then the products are shipped to thecustomersDemand of raw materials is calculated from demand ofproducts and BOM




Conclusion

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




Conclusion

Objective Function

Minimize nominal-scenario (no-disruption) cost:

minimize∑j∈V0

fjXj +∑

(i ,j)∈A

N∑n=1

dijnYijn0 (1)




Conclusion

Constraints

Subject to:

∑j∈V0

fjXj +∑

(i ,j)∈A

N∑n=1

dijnYijns ≤ (1 + ps)c∗s ∀s (2)

∑(j ,i)∈A

Yjins −∑

(i ,j)∈A

∑m∈N

αmnYijms = bjn ∀j , s, n (3)

∑(j ,i)∈A

Yjins ≤ (1− ajs)kjnXj ∀j , n, s

(4)

Xj ∈ 0, 1 ∀j (5)

Yijns ≥ 0 ∀i , j , n, s (6)

For convenience we set ps = p ∀s, though the result can beeasily extended to more general cases.




Conclusion

In Words

minimize cost in nominal scenario

subject to cost in scenario s ≤ (1 + p)c∗s ∀sflow balance constraints ∀j , n, sflow can’t exceed capacity, or 0 if disrupted ∀j , n, sXj integer

Yijns non-negative




Conclusion

An Even More Compact Description

minimize c0(X ,Y )

subject to cs(X ,Y ) ≤ (1 + p)c∗s (X ∗,Y ∗) ∀s(X ,Y ) ∈ Ω




Conclusion

Solution Methods

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

Run times generally < 1 hour for problems with

40 suppliers40 plants/warehouses80 customers5 scenarios10 products

For bigger problems, we’ll need customized algorithms

Future research




Conclusion

Outline

1 Introduction

2 The Model

3 Example


5 Conclusion




Conclusion

Numerical Example

10 Suppliers

10 plants

20 customers

7 products(5 raw materials)

5 scenarios

Nominal scenario4 disruption scenarios

Costs

fj = 10,000 on average for plantsdijn = 500 on averagePenalty cost for unmet demand = 10,000

Desired robustness level p = 0.1




Conclusion

Numerical Example

Minimizing Nominal Cost w/o bounds on scenariocost(Method1)

Minimizing worst-case cost (Method2)

Minimizing Nominal Cost with bounds on scenariocost(Method3)




Conclusion

Numerical Example

0

500

1000

1500

2000

2500

3000

Scenario1 Scenario2 Scenario3 Scenario4 Scenario5

Cost (X

$10

00)

Solu/on Comparison

Method1

Method2

Method3

Figure: Cost ComparisionSnyder,Peng Air Products: Resilient Supply Chains



Conclusion

Outline

1 Introduction

2 The Model

3 Example


5 Conclusion




Conclusion

Relative Regret

Suppose that

the optimal objective value for scenario s is c∗s

the best scenario cost for current solution is cs

Define the relative regret Rs for scenario s as

Rs =cs

c∗s− 1 (7)

Rs shows how much worse the current solution is compared tothe optimal solution in that scenario




Conclusion

Finding the Maximum Relative Regret

We can find the maximum relative regret among scenarios for agiven solution (X ,Y ):

0.63

1.55

0.48

0.98

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

Scenario2 Scenario3 Scenario4 Scenario5

Regret

Rela)ve Regret

Figure: Relative RegretsSnyder,Peng Air Products: Resilient Supply Chains



Conclusion

Update the Scenario Upper Bounds

Recall

minimize c0(X ,Y )

subject to cs(X ,Y ) ≤ (1 + p)c∗s (X ,Y ) ∀s(X ,Y ) ∈ Ω




Conclusion

Update the Scenario Upper Bounds

If we reduced p to a little below Rmax , the current solution isno longer feasible

0.63

1.55

0.48

0.98

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

Scenario2 Scenario3 Scenario4 Scenario5

Regret

Rela)ve Regret

p

Figure: Update UpperboundSnyder,Peng Air Products: Resilient Supply Chains



Conclusion

Improve p-robustness

We can find the maximum relative regret Rmax among scenarios fora given solution (X ,Y ):

If we reduced p to a little below Rmax , the current solution isno longer feasible

One can obtain solutions with smaller regret but greaternominal costs

Can we improve the regret without greatly increasing thenominal costs?




Conclusion

Improve p-robustness - Example

Example

10 Suppliers, 10 Plants, 20 Customers10 Scenarios, 10 Productsfj = 10, 000dijn = 250

0

0.5

1

1.5

2

2.5

3

1400000 1405000 1410000 1415000 1420000 1425000 1430000 1435000 1440000

Regret

Nominal Cost

Nominal Cost vs Regret

Figure: Nominal Cost vs. Regret




Conclusion


Example

We can improve robustness without increasing the cost toomuch

0

0.5

1

1.5

2

2.5

3

1400000 1405000 1410000 1415000 1420000 1425000 1430000 1435000 1440000

Regret

Nominal Cost

Nominal Cost vs Regret

67%

0.6%

Figure: Nominal Cost vs. Regret




Conclusion


When p gets smaller, the run time becomes longer.

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3

Time

Regret

Run Time vs Regret




Conclusion

Outline

1 Introduction

2 The Model

3 Example


5 Conclusion




Conclusion

Conclusions

Supply chain network design model that accounts fordisruptions

Minimize nominal cost, subject to bound on cost in eachscenario

Substantial improvements in robustness can be attainedwithout large increases in nominal cost

Next steps:

Empirical studyDesign algorithm




Conclusion

Questions?




Conclusion

Thank you!


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