BinN International Research Seminar, Sep. 23, 2019

Optimal Control Strategy for Relief SupplyBehavior after a Major Disaster
Riki Kawase
Doctoral Student, Department of Civil Engineering, Kobe University

Joint work with Dr. Urata and Prof. Iryo

Background: Humanitarian Logistics

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Humanitarian Logistics is important for minimizing thedamage between the rescue and restoration period after a disaster.
Difficulties■ 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)
Relief suppliers(RS)⋮Distribution centers(DCs)

Shelters⋮⋮

Material flow
Information flow

SupplyBuffer

DemandKumamoto Earthquake (2016)

Many damaged roads

Transshipment processed inefficiently by human power
Chairs lined up to read "Paper, bread, water, SOS“34Background: Control Strategy
Control Strategy in Japan[Push-mode support (sequence control)][Pull-mode support (feedback control)]

Demand forecasting

Demand feedback

Past disasters

Pre-stock■ Long push-mode support caused
the gap between supply (meals)
and demand (evacuees).

Push

PulltimeMeals

Research Question
When should the control strategy
change from push to pull ?

Evacuees

PushPullNumber of evacuees and meals supplied after the Kumamoto Earthquake

Background: Humanitarian Logistics
Upper logistics
Relief suppliers(RS)

Lower logistics
Distribution centers(DCs)

Shelters⋮⋮⋮

Material flow
Information flow

SupplyBuffer

Demand■ DCs can adjust gaps by holding inventories on the implicit
assumption of supply and information availability.
⁃ Push-mode⁃ not available⁃ Decentralized Pull-mode⁃ limited available⁃ Centralized Pull-mode⁃ fully available

PushborderPull5Purpose and Methodology

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Research Question: When should the push- be changed to pull-mode ?
■ We drive the sufficient condition to change control strategy frompush-mode to pull-mode.
Purpose: Mathematical properties of the optimal control strategy■ Focus on information availability■ Mathematically analyze the optimal push-mode (no information)and the optimal pull-mode (decentralized/centralized information).
Methodology■ Dynamic optimization approach using the stochastic optimal
control theory considering Demand uncertainty.
■ The Bayesian updating process can model two control strategies.
∞
not availableupdating interval = 0~∞ (limited available)0
fully available

Modeling7

8Definition
𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝑆𝑖𝑗 (𝑡): Supply rate from node 𝑖 to node 𝑗 at time 𝑡𝑆𝑡𝑎𝑡𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝐼𝑁 𝑡 : Net inventory level in the shelter at time 𝑡
𝐼𝑖 𝑡 : Inventory level in node 𝑖 at time 𝑡𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟
𝐷 𝑡 : Predicted demand rate at time 𝑡 (𝑑𝐷/𝑑𝑡 ≤ 0)𝑧(𝑡): Standard wiener process
𝐼2
2
𝑇:
The end of timeDC𝑏:
Unsatisﬁed cost coefﬁcientℎ𝑖′ : Inventory cost coefﬁcient(0 < ℎ1′ < ℎ2′ < ℎ3′ < 𝑏)𝑐:𝑟𝑖𝑗 :𝐼1𝐼𝑁Handling cost coefﬁcient
Lead time from node 𝑖 node 𝑗

1
RS(𝑟13 < 𝑟12 + 𝑟23 = 𝑟123 )Supply Chain Network (SCN)

𝑟13𝑆133
ShelterInformation Updating Algorithm

9■ Predicted Demand 𝐷 𝑡 follows the normal distribution.
𝐷(𝑡) 𝑑𝑡 = 𝐷 𝑡 𝑑𝑡 + 𝐷 𝑆𝐷 𝑡 𝑑𝑧(𝑡)[Dynamics]

𝐷 𝑡 𝑑𝑡~𝑁 𝐷 𝑡 𝑑𝑡,[Distribution]

𝐷 𝑆𝐷 𝑡2𝑑𝑡■ 𝐷 𝑡 is updated to subjective demand 𝐷𝑙 (𝑡) based on information
𝐷 𝑡 , applying the Bayesian Estimation at an interval 𝑘𝑙 (𝑘1 ≥ 𝑘2 ).
■ 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 )𝑡 = 𝑘1

𝐷2DC2DC𝐷𝐷1RS

𝐷2

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ShelterPush-mode support

𝐷𝐷1RS

3ShelterPull-mode support

Stochastic Optimal Control Problem
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 Problem
Object Function[Net inventory holding cost][Inventory holding cost][Inventory handling cost][Outflows at destination 𝑗][Costfunctions]▪ 𝑇𝐶𝑆𝑙 𝑡 : Changes in the inventory level in node 𝑙⁃ min 𝑇𝐶𝑆𝑙 𝑡 = 0 (∴ no transshipment)

State Equation (Inventory Dynamics)

Initial Condition⁃ 𝑇𝐶𝑆𝑙 𝑡 ≠ 0 means that the inventory level should change (∴ handling cost).
𝑆𝑖𝑗 𝑡 − 𝑟𝑖𝑗𝑗

⁃ min 𝑇𝐶𝑆𝑙 𝑡 means supply constraints for node 𝑙𝑇𝑆𝑙𝑗 𝑡11

Mathematical Analysis- Optimal Push-mode Support -12

Assumption
Cost functions

𝑓𝐼 𝑥 = 𝑓𝐵 𝑥 = 𝑓𝑆 𝑥 = 𝑥 𝛼 , 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 𝑆12𝑡 < 𝐸 𝑆13𝑡 .
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
Optimal Inventory𝑉 ∗ = 𝑉(𝐼∗ , 𝐼𝑁 ∗ , 𝑆 ∗ )■ Positive inventory level■ Decreasing function■ The limit value is 0>𝟎*

■ 𝐼 𝑡 asymptotically approaches 0 from initial value 𝐼 (0) > 0.
→ Solve 𝐼 ∗ (0) = argmin𝐼 0 𝑉 ∗Pre-stock> 0, therefore 𝐼 ∗ 0 = 0Lemma 1
There is no need to pre-store at DC and to add stock, 𝐼 ∗ 𝑡 = 0.

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Optimal Control : 𝑺𝟏𝒋 𝒕
Optimal supply rate from RS

∗S12𝑡

∗
S13𝑡

*𝑡 ∈ 𝑟123 − 𝑟13 , 𝑇 − 𝑟123

*Differentiate 𝐸 𝑆 𝑡

𝑟13 < 𝑟123

as follows:<0We obtain,∗
When 𝑑𝐸 𝑆 𝑡 /𝑑𝑡 < 0, we obtain 𝐸 𝑆12𝑡∗
< 𝐸 𝑆13𝑡

Lemma 2∗
"Direct Supply" is eﬀective, 𝐸 𝑆12𝑡(∵ 𝑟13 < 𝑟123 )∗< 𝐸 𝑆13𝑡 .

16Optimal Control : 𝑺𝟏𝒋 𝒕
Optimal supply rate from RS

∗
S12𝑡

∗
S13𝑡

*
*

Lemma 1 𝑡 ∈ 𝑟123 − 𝑟13, 𝑇 − 𝑟123
There is no need to pre-store at DC and to add stock, 𝐼 ∗ 𝑡

Differentiate 𝐸 𝑆 𝑡

𝑟13 < 𝑟123= 0.

as follows:

Lemma 2∗
"DirectSupply"
iseﬀective,𝐸𝑆𝑡
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We obtain,Theorem 1∗
When 𝑑𝐸 𝑆 𝑡 /𝑑𝑡 < 0, we obtain 𝐸 𝑆12𝑡<0∗< 𝐸 𝑆13 𝑡 .

∗< 𝐸 𝑆13𝑡(∵ 𝑟13 < 𝑟123 )Lemma DC
2 is unnecessary for sequence control strategy.
∗∗"Direct Supply" is eﬀective, 𝐸 𝑆12𝑡 < 𝐸 𝑆13𝑡 .

17Optimal Control : 𝑰𝑵 𝒕
Optimal net inventory[Dynamics][Optimal]■ Long-term expected value is 0
■ Regression speed < 0𝐼𝑁 𝑡■ Dynamics of 𝐼𝑁 𝑡 is
■ Assume 𝑏 → ∞Theorem 2
Maintaining inventory at the shelter isthe optimal when the penalty cost is
suﬃciently high, 𝐼𝑁 ∗ 𝑡 > 0 𝑖𝑓 𝑏 → ∞.

0𝑏→∞𝑡𝐼𝑁 𝑡

0𝑡 + Δ𝑡Numerical Analysis- Optimal Pull-mode Support -18

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Settings
Analysis method▪ Comparing the objective functions of the push 𝑉 𝑝𝑢𝑠ℎ andpull 𝑉 𝑝𝑢𝑙𝑙 𝑘1 , 𝑘2 with the Monte Carlo simulation.
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 」
not share
RS and DC share prediction errors
Parameter setting

Results : Comparing Push and Pull
𝑅 𝑧1 𝑡 , 𝑧2 𝑡

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= 1(case−a) 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.

Conclusion

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Conclusion

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Summary■ Dynamic optimization approach analyzed the mathematicalproperties of the optimal control strategy.
⁃ Push-mode: direct supply from the RS to the shelter is optimaltransportation 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.
Future work

■ General network analysis (e.g., many-to-many network) canconsider significant elements such as the single point of failure.
■ Considering optimal day-to-day recovery dynamics.

Reference

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Thank you for your attention.
Riki Kawase: r-kawase@stu.kobe-u.ac.jp

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