optimal control strategy for relief supply behavior...
TRANSCRIPT
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Optimal Control Strategy for Relief Supply Behavior after a Major Disaster
Riki Kawase
BinN International Research Seminar, Sep. 23, 2019
Doctoral Student, Department of Civil Engineering, Kobe University
Joint work with Dr. Urata and Prof. Iryo
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Background: Humanitarian Logistics2
Humanitarian Logistics is important for minimizing the
damage between the rescue and restoration period after a disaster.
Difficulties
Material flow
Information flow
⋮⋮
⋮
Relief suppliers(RS)
Distribution centers(DCs)
Shelters
■ Node bottleneck on supply network (e.g., DCs don't work)
■ Link bottleneck on material network (e.g., Cannot access)
■ Link bottleneck on information network (e.g., Cannot communicate)
Supply DemandBuffer
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3Chairs lined up to read "Paper, bread, water, SOS“
Transshipment processed inefficiently by human power
Many damaged roads
Kumamoto Earthquake (2016)
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Background: Control Strategy4
Control Strategy in Japan
[Push-mode support (sequence control)] [Pull-mode support (feedback control)]
Demand forecasting Demand feedback
Pre-stock Push Pull
timePast disasters
Number of evacuees and meals supplied after the Kumamoto Earthquake
Meals
Evacuees
■ Long push-mode support caused the gap between supply (meals) and demand (evacuees).
Push Pull
Research Question
When should the control strategy change from push to pull ?
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Background: Humanitarian Logistics5
Material flow
Information flow
⋮⋮
⋮
Relief suppliers(RS)
Distribution centers(DCs)
Shelters
Supply DemandBuffer
■ DCs can adjust gaps by holding inventories on the implicit assumption of supply and information availability.
Upper logistics Lower logistics
⁃ not available
⁃ limited available
⁃ fully available
⁃ Push-mode
⁃ Decentralized Pull-mode
⁃ Centralized Pull-mode
Push
Pull
border
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Purpose and Methodology6
Purpose: Mathematical properties of the optimal control strategy
Research Question: When should the push- be changed to pull-mode ?
Methodology
■ Focus on information availability
■ Mathematically analyze the optimal push-mode (no information) and the optimal pull-mode (decentralized/centralized information).
■ Dynamic optimization approach using the stochastic optimal control theory considering Demand uncertainty.
■ The Bayesian updating process can model two control strategies.
updating interval =
∞ not available
0~∞ (limited available)
0 fully available
■ We drive the sufficient condition to change control strategy from push-mode to pull-mode.
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7
Modeling
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Definition8
𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝑆𝑖𝑗(𝑡): Supply rate from node 𝑖 to node 𝑗 at time 𝑡
𝑆𝑡𝑎𝑡𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝐼𝑖 𝑡 : Inventory level in node 𝑖 at time 𝑡𝐼𝑁 𝑡 : Net inventory level in the shelter at time 𝑡
𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟
𝑟𝑖𝑗: Lead time from node 𝑖 node 𝑗(𝑟13 < 𝑟12 + 𝑟23 = 𝑟123 )
𝑧(𝑡): Standard wiener process
𝐷 𝑡 : Predicted demand rate at time 𝑡 (𝑑𝐷/𝑑𝑡 ≤ 0)
ℎ𝑖′: Inventory cost coefficient
(0 < ℎ1′ < ℎ2
′ < ℎ3′ < 𝑏)
𝑇: The end of time
𝑏: Unsatisfied cost coefficient
𝑐: Handling cost coefficient
Supply Chain Network (SCN)
1
RS
3
Shelter
2
DC
𝑟13𝑆13
𝐼2
𝐼𝑁𝐼1
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𝐷 𝑡 𝑑𝑡~𝑁 𝐷 𝑡 𝑑𝑡, 𝐷𝑆𝐷 𝑡2𝑑𝑡
Information Updating Algorithm9
■ Predicted Demand 𝐷 𝑡 follows the normal distribution.
𝐷(𝑡) 𝑑𝑡 = 𝐷 𝑡 𝑑𝑡 + 𝐷𝑆𝐷 𝑡 𝑑𝑧(𝑡)
■ 𝐷 𝑡 is updated to subjective demand 𝐷𝑙(𝑡) based on information 𝐷 𝑡 , applying the Bayesian Estimation at an interval 𝑘𝑙(𝑘1 ≥ 𝑘2).
1
RS
3
Shelter
2
DC
Push-mode support
𝐷𝐷
𝐷
1
RS
3
Shelter
2
DC
Pull-mode support
𝐷
𝐷2
𝐷
■ The number of updates 𝑛 increases as depot 𝑙 is closer to the shelter
⟹ The Bullwhip Effect (𝑉 𝐷1 𝑡 ≥ 𝑉 𝐷2 𝑡 ∵ 𝑛1 ≤ 𝑛2)
[Dynamics]
[Distribution]
𝑡 = 𝑘1
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Stochastic Optimal Control Problem10
Object Function
[Net inventory holding cost]
[Inventory holding cost]
[Inventory handling cost]
[Outflows at destination 𝑗]
[Cost functions]
State Equation (Inventory Dynamics) Initial Condition
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Stochastic Optimal Control Problem11
Object Function
[Net inventory holding cost]
[Inventory holding cost]
[Inventory handling cost]
[Outflows at destination 𝑗]
[Cost functions]
State Equation (Inventory Dynamics) Initial Condition
▪𝑇𝐶𝑆𝑙 𝑡 : Changes in the inventory level in node 𝑙
⁃ min𝑇𝐶𝑆𝑙 𝑡 = 0 (∴ no transshipment)
⁃ 𝑇𝐶𝑆𝑙 𝑡 ≠ 0 means that the inventory level should change (∴ handling cost).
⁃ min𝑇𝐶𝑆𝑙 𝑡 means supply constraints for node 𝑙𝑗𝑆𝑖𝑗 𝑡 − 𝑟𝑖𝑗 𝑇𝑆𝑙𝑗 𝑡
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12
- Optimal Push-mode Support -
Mathematical Analysis
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Assumption13
Cost functions𝑓𝐼 𝑥 = 𝑓𝐵 𝑥 = 𝑓𝑆 𝑥 = 𝑥𝛼 , 𝛼 > 1
Assumption
1. Let 𝑡 = 𝑇 be the time when demand becomes 0, 𝐷𝑙 𝑇 = 0.
2. Demand decreases constantly over time, 𝑑𝐷𝑙 𝑡 /𝑑𝑡 = 𝐷𝑙 < 0.
3. The inventory holding cost at the shelter is twice that at the RS, ℎ𝐼𝑁′ = 1/2ℎ1
′ .
4. The DC is not ready after a disaster, 𝑆23 𝑡 = 0 ∀𝑡 ∈ 0, 𝑟12 .
The following mathematical properties will be proved:
Lemma 1. There is no need to pre-store at DC and to add stock
after a disaster, 𝐼∗ 𝑡 = 0.
Lemma 2. "Direct Supply" is effective, 𝐸 𝑆12∗ 𝑡 < 𝐸 𝑆13
∗ 𝑡 .
Theorem 1. DC is unnecessary for sequence control strategy.
Theorem 2. Maintaining inventory at the shelter is the optimal
when the penalty cost is sufficiently high, 𝐼𝑁∗ 𝑡 > 0 𝑖𝑓 𝑏 → ∞.
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Optimal Control : 𝑰 𝒕 14
Optimal Inventory
■ 𝐼 𝑡 asymptotically approaches 0 from initial value 𝐼 (0) > 0.
→ Solve 𝐼∗(0) = argmin𝐼 0 𝑉∗
> 0, therefore 𝐼∗ 0 = 0
𝑉∗ = 𝑉(𝐼∗, 𝐼𝑁∗, 𝑆∗)
Pre-stock
There is no need to pre-store at DC and to add stock, 𝐼∗ 𝑡 = 0.
Lemma 1
■ Positive inventory level
■ Decreasing function
■ The limit value is 0
*
> 𝟎
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Optimal Control : 𝑺𝟏𝒋 𝒕15
Optimal supply rate from RSS13∗ 𝑡S12
∗ 𝑡
Lemma 2
"Direct Supply" is effective, 𝐸 𝑆12∗ 𝑡 < 𝐸 𝑆13
∗ 𝑡 .
𝑡 ∈ 𝑟123 − 𝑟13, 𝑇 − 𝑟123
*
*𝑟13 < 𝑟123
Differentiate 𝐸 𝑆 𝑡 as follows:
We obtain,
< 0
When 𝑑𝐸 𝑆 𝑡 /𝑑𝑡 < 0, we obtain 𝐸 𝑆12∗ 𝑡 < 𝐸 𝑆13
∗ 𝑡 (∵ 𝑟13 < 𝑟123)
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Optimal Control : 𝑺𝟏𝒋 𝒕16
Optimal supply rate from RSS13∗ 𝑡S12
∗ 𝑡
Lemma 2
"Direct Supply" is effective, 𝐸 𝑆12∗ 𝑡 < 𝐸 𝑆13
∗ 𝑡 .
𝑡 ∈ 𝑟123 − 𝑟13, 𝑇 − 𝑟123
*
*𝑟13 < 𝑟123
Differentiate 𝐸 𝑆 𝑡 as follows:
We obtain,
< 0
When 𝑑𝐸 𝑆 𝑡 /𝑑𝑡 < 0, we obtain 𝐸 𝑆12∗ 𝑡 < 𝐸 𝑆13
∗ 𝑡 (∵ 𝑟13 < 𝑟123)
Lemma 2
"Direct Supply" is effective, 𝐸 𝑆12∗ 𝑡 < 𝐸 𝑆13
∗ 𝑡 .
There is no need to pre-store at DC and to add stock, 𝐼∗ 𝑡 = 0.
Lemma 1
DC is unnecessary for sequence control strategy.
Theorem 1
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[Dynamics]
[Optimal]
Optimal Control : 𝑰𝑵 𝒕 17
Optimal net inventory
0
𝐼𝑁 𝑡
𝑡 𝑡 + Δ𝑡
■ Dynamics of 𝐼𝑁 𝑡 is
𝐼𝑁 𝑡
𝑡 𝑡 + Δ𝑡
0
𝑏 → ∞
Maintaining inventory at the shelter is
the optimal when the penalty cost is
sufficiently high, 𝐼𝑁∗ 𝑡 > 0 𝑖𝑓 𝑏 → ∞.
Theorem 2
■ Assume 𝑏 → ∞
■ Long-term expected value is 0
■ Regression speed < 0
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18
- Optimal Pull-mode Support -
Numerical Analysis
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Settings19
Parameter setting
▪Prediction demand 𝐷 𝑡 = −0.15 𝑡 − 𝑇 .
▪The number of simulations is 5,000.
▪case-a「𝑅 𝑧1 𝑡 , 𝑧2 𝑡 = 1」,case-b「𝑅 𝑧1 𝑡 , 𝑧2 𝑡 ≠ 1 」
RS and DC share prediction errors not share
▪Comparing the objective functions of the push 𝑉𝑝𝑢𝑠ℎ and
pull 𝑉𝑝𝑢𝑙𝑙 𝑘1, 𝑘2 with the Monte Carlo simulation.
Analysis method
Parameter setting
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Results : Comparing Push and Pull20
The sufficient conditions for control strategy change (push-mode to pull-mode) is
𝑘1 = 𝑘2, that is the centralized information system is restored.
Under the decentralized information system, pull-mode may not be effective.
𝑅 𝑧1 𝑡 , 𝑧2 𝑡 = 1(case−a)
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21
Conclusion
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Conclusion22
■ Dynamic optimization approach analyzed the mathematical
properties of the optimal control strategy.
⁃ Push-mode: direct supply from the RS to the shelter is optimal
transportation strategy. DC should not be used.
⁃ Pull-mode: the sufficient condition for pull-mode to be optimal
is to restore the centralized information system. Otherwise,
pull-mode may make a not always good result.
Summary
Future work
■ General network analysis (e.g., many-to-many network) can
consider significant elements such as the single point of failure.
■ Considering optimal day-to-day recovery dynamics.
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