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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Outline
1 Introduction
2 The Model
3 Example
4 Improve p-robustness
5 Conclusion
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Disruptions
Suppose disruptions can occur
We must design supply chain before we know what facilitieswill be disrupted
Possible disruptions are described by scenarios
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Outline
1 Introduction
2 The Model
3 Example
4 Improve p-robustness
5 Conclusion
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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”
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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”
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove 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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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)
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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)
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Objective Function
Minimize nominal-scenario (no-disruption) cost:
minimize∑j∈V0
fjXj +∑
(i ,j)∈A
N∑n=1
dijnYijn0 (1)
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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.
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Outline
1 Introduction
2 The Model
3 Example
4 Improve p-robustness
5 Conclusion
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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)
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Outline
1 Introduction
2 The Model
3 Example
4 Improve p-robustness
5 Conclusion
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
IntroductionThe Model
ExampleImprove p-robustness
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?
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Improve p-robustness - Example
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Improve p-robustness - Example
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Outline
1 Introduction
2 The Model
3 Example
4 Improve p-robustness
5 Conclusion
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
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
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Questions?
Snyder,Peng Air Products: Resilient Supply Chains
IntroductionThe Model
ExampleImprove p-robustness
Conclusion
Thank you!
Snyder,Peng Air Products: Resilient Supply Chains