models for designing resilient supply chain networks for...
TRANSCRIPT
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Outline
1 Introduction
2 The Model
3 Example
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Motivating Example
Plants Warehouses Customers
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
“Optimal” Solution
Cost: $521,163
Plants Warehouses Customers
fig_opt_nominal
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Plants Warehouses Customers
fig_scen1Scenario 1Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Plants Warehouses Customers
fig_scen2Scenario 2Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Plants Warehouses Customers
fig_scen3Scenario 3Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Solution Performance When Disruptions Occur
Scenario 1: Cost $521,163
Plants Warehouses Customers
fig_soln1_scen1
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Solution Performance When Disruptions Occur
Scenario 2: Cost $982,241
Plants Warehouses Customers
fig_soln1_scen2
EMERG
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Solution Performance When Disruptions Occur
Scenario 3: Cost $3,237,107
Plants Warehouses Customers
fig_soln1_scen3
EMERG
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
)
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Outline
1 Introduction
2 The Model
3 Example
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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”
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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”
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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)
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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)
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Objective Function
Minimize nominal-scenario (no-disruption) cost:
minimize∑j∈V0
fjXj +∑
(i ,j)∈A
N∑n=1
dijnYijn0 (1)
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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)
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
An Even More Compact Description
minimize c0(X ,Y )
subject to cs(X ,Y ) ≤ Us ∀s(X ,Y ) ∈ Ω
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Outline
1 Introduction
2 The Model
3 Example
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
New Solution
Cost: $580,758 (vs. $521,163)
Plants Warehouses Customers
fig_soln2_nominal
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Solution Performance When Disruptions Occur
Scenario 1: Cost $580,758 (vs. $521,163)
Plants Warehouses Customers
fig_soln2_scen1
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Solution Performance When Disruptions Occur
Scenario 2: Cost $800,000 (vs. $982,241)
Plants Warehouses Customers
fig_soln2_scen2
EMERG
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Solution Performance When Disruptions Occur
Scenario 3: Cost $800,000 (vs. $3,237,107)
Plants Warehouses Customers
fig_soln2_scen3
EMERG
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Scenario Costs Equal Upper Bounds
Recall:
minimize c0(X ,Y )
subject to cs(X ,Y ) ≤ Us ∀s(X ,Y ) ∈ Ω
Nothing to require optimization in scenarios
Only achieve sufficiently low cost
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Scenario Costs Equal Upper Bounds
Recall:
minimize c0(X ,Y )
subject to cs(X ,Y ) ≤ Us ∀s(X ,Y ) ∈ Ω
Nothing to require optimization in scenarios
Only achieve sufficiently low cost
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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 scenario
Downside: cumbersome
Currently investigating other approaches
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
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
Snyder Air Products: Resilient Supply Chains
IntroductionThe Model
Example
Questions?
Snyder Air Products: Resilient Supply Chains