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

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

3Chairs lined up to read "Paper, bread, water, SOS“

Transshipment processed inefficiently by human power

Many damaged roads

Kumamoto Earthquake (2016)

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 ?

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

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.

7

Modeling

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

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

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

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

12

- Optimal Push-mode Support -

Mathematical Analysis

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 → ∞.

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

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)

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

[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

18

- Optimal Pull-mode Support -

Numerical Analysis

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

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)

21

Conclusion

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.

Reference23

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Thank you for your attention.

Riki Kawase: r-kawase@stu.kobe-u.ac.jp