Tokyo Tech

Optimization of inventory transportation
strategy, considering the procurement ofgoods by the affected population
21st Summer Course
Behavior Modeling in Transportation Networks
Tokyo Institute of Technology

Riki Kawase1

Background
Distribution
Center (DC)Disaster Base(DB)Tokyo Techaffected area
Evacuation
Center (EC)

Relief Goods Transportation Network Humanitarian Organizations (HOs; e.g.,governments) need to supply relief goodsto ECs.

Bad Reports3) (Examples)(a) Queue by rationing 1)

causes Surge Demand ➡ Queue Congestion
causes Supply Constraints ➡ Inventory Congestion

Should rethinkrelief goods flow(b) Inventory congestion 2)2016 Kumamoto Earthquakes2Humanitarian Logistics (HL)

Tokyo Tech

Definition“ the process of planning, implementing and controlling the efficient, cost-effective flow andstorage of goods and materials, as well as related information, from the point of origin to
the point of consumption for the purpose of alleviating the suffering of vulnerable people. ”

HL with restoration process of Information Infrastructure
Information
HL Strategy

Focus

Disaster

Delay

Restoration

Unreliable
Push*

Pull

Kawase (2021)(almost)complete*Supply corresponding
to predicted demand**Supply corresponding
to demand information

Pull**
this research

Time

What preparations (e.g., staffing) should be done for the pull strategy?
What is the efficient, cost-effective flow and storage in the pull strategy?

3

Challenges of Pull Strategy

Tokyo Tech(Almost) Perfect Information but Supply Constrained In the Kumamoto Earthquakes, direct
supply was the solution alternative.
 However, a large-scale disaster wouldalso disrupt direct supply.

Observation (in the Kumamoto Earthquakes)

Supply Constrained Some victims made trips toacquire food and essential goods.

Suggest the potential for
improvement through
Demand-Supply Approach

Direct SupplyMethodology
 Formulate an optimization problem
for the demand-supply approach Numerical evaluate theperformance of the approach

Demand-Supply Approach4

Previous Research

Tokyo TechPrevious models of network flow optimization for humans and goods
 discard Queue Congestion and Inventory Congestion.
 are computationally inefficient.
Typical Assumptions in Logistics Optimization
 Humans and Goods can move in free-flow travel time (FFTT).
 Humans/goods are serviced/consumed immediately on arrival.
Previous Models Considering Only Flow Congestion, not Queue Congestion Muggy and Stamm (2020) formulate MPEC* s.t. congestion game.
 Gutjahr and Dzubur (2016) formulate BLPP s.t. optimal destination*Mathematical Programming withchoice.
Previous Models Considering Queue Congestion

Equilibrium Constraints**Bilevel Programming Problems

 Espejo-Díaz and Guerrero (2021), Fikar et al.(2018) present anagent-based simulation optimization framework.
 Limitation is computational efficiency.

5

Previous Research

Tokyo TechPrevious models of network flow optimization for humans and goods

Research Gap

 discardQueueCongestion
and InventoryCongestion.
 Optimal
networkflow for humans(traffic) and goods
could
HL problems, supplyconstraints.
 areattack
computationallyinefficient.

 There is no optimization model to meet both computationally
Typicalefficient
Assumptionsin Logistics
Optimizationand queue
congestion
considerations.
 Humans and Goods can move in free-flow travel time.

Research
Objective
Humans/goods
are serviced/consumed immediately on arrival.
 Formulate an optimization problem that is (potentially)
Previous
Models Considering
Only Flow
not Queue
Congestion
computationallyefficientand Congestion,
describes queue
congestion.
* Muggyand Stamm Propose
BLPP (2020) formulate MPEC s.t. congestion game.

** s.t. optimal
 Gutjahr
Dzubur(2016)
formulate BLPP
• andUpper:
Optimize
preparations(e.g., staffing,stockpiles)*Mathematical Programming withdestination choice.
• Lower: Optimize dynamic network flowusing
Equilibrium Constraints**Bilevel
OptimalControl
Problem
Programming
Problems
Previous Models Considering Queue Congestion Numerical evaluate the performance of the demand-supply Espejo-Díazapproach. and Guerrero (2021), Fikar et al.(2018) present anagent-based simulation optimization framework.
 Show preliminary results on a simple network.
 Limitation is computational efficiency.

6Modeling7

Model Framework
Relief Goodsinventory

𝐼𝐼1 (𝑡𝑡)

𝑡𝑡: FFTT

𝐼𝐼2 (𝑡𝑡)

Human (Traffic, Victim)
queue𝑞𝑞(𝑡𝑡)𝑂𝑂𝜅𝜅1Service rate 𝜅𝜅2A Parallel Link Network

 To minimize queue congestion andinventory congestion,
 (Upper) Determine preparation
strategies {𝐼𝐼1 0 , 𝐼𝐼2 0 , 𝜅𝜅1 , 𝜅𝜅2 } and (Lower) Determine demand andsupply dynamics.
One victim needs only one item.

Relief Goods Transportation Network𝑡𝑡

Tokyo Tech𝐷𝐷 (Supply) HOs supply DB, or supply
EC via DB.
 Between DB and EC, relief
goods can move in FFTT.
 (Demand) Victims receive relief
goods at DB or EC.
 Between DB and EC, victims
can move in FFTT.
 Total relief demand for victims
is uncertain. Denote thestochastic process as 𝑄𝑄(𝑡𝑡) andthe sample path as 𝑞𝑞(𝑡𝑡).

8Variables*an inventory if positive;
a shortage otherwise

Stock (State Variables)

Relief Goods (c.f. Kawase et al., 2019)
𝐼𝐼1 (𝑡𝑡)
𝑆𝑆1 𝑡𝑡∈ [0, 𝑆𝑆1 ]𝐼𝐼𝐼𝐼(𝑡𝑡)

𝑡𝑡: FFTT𝐼𝐼2 (𝑡𝑡)

𝑆𝑆2 𝑡𝑡 𝑆𝑆2 (𝑡𝑡 − 𝑡𝑡)∈ [0, 𝑆𝑆2 ]𝜅𝜅1 𝑡𝑡 − 𝑡𝑡𝜅𝜅2 𝑡𝑡Relief Goods Transportation Network𝑞𝑞(𝑡𝑡)𝑂𝑂𝑡𝑡

𝑥𝑥2 (𝑡𝑡)

𝑞𝑞 𝑡𝑡 − 𝜈𝜈(𝑡𝑡)

𝑥𝑥1 (𝑡𝑡)𝜅𝜅1

𝜅𝜅2A Parallel Link Network𝐷𝐷

𝐼𝐼1 (𝑡𝑡), 𝐼𝐼2 (𝑡𝑡) : Net inventory* at nodes
𝐼𝐼𝐼𝐼(𝑡𝑡) : In-transit inventory

Flow (Control Variables)
𝑆𝑆1 𝑡𝑡 : Inflow per unit-time at DB
𝑆𝑆2 𝑡𝑡 : Outflow per unit-time at DB
𝑆𝑆2 (𝑡𝑡 − 𝑡𝑡) : Inflow per unit-time at EC𝜅𝜅1 𝑡𝑡 , 𝜅𝜅2 (𝑡𝑡) : Demand per unit-time

Human (c.f. Akamatsu and Nagae, 2007)𝜈𝜈 𝑡𝑡 ∈ [0, 𝑞𝑞(𝑡𝑡)]Tokyo Tech

One victim needs only one item.

Stock (State Variables)𝑥𝑥1 (𝑡𝑡), 𝑥𝑥2 (𝑡𝑡) : Queue

Flow (Control Variables)𝜈𝜈 𝑡𝑡 : Inflow per unit-time to DB𝜅𝜅1 𝑡𝑡 , 𝜅𝜅2 (𝑡𝑡) : Outflow per unit-time

𝑂𝑂: Need relief goods 𝐷𝐷: Get relief goods

9Dynamics#𝐼𝐼1 ≥ 0𝑡𝑡
Cumulative curve at DB

#𝑥𝑥2
𝐼𝐼𝐼𝐼

𝐼𝐼2 ≥ 0𝐼𝐼2 < 0𝑑𝑑𝐼𝐼1 /𝑑𝑑𝑑𝑑 = 𝑆𝑆1 (𝑡𝑡) −𝑆𝑆2 (𝑡𝑡) −𝜅𝜅1 𝑡𝑡 − 𝑡𝑡 , 𝐼𝐼1 0 = 𝐼𝐼01 (1)𝑑𝑑𝑥𝑥1 /𝑑𝑑𝑑𝑑 = 𝜈𝜈(𝑡𝑡)𝑡𝑡𝑡𝑡−𝜅𝜅1 (𝑡𝑡) ,𝑥𝑥1 0 = 0 (2)𝜅𝜅1 if 𝑥𝑥1 𝑡𝑡 > 0𝜅𝜅1 (𝑡𝑡) = � 𝜅𝜅1 if 𝑥𝑥1 𝑡𝑡 = 0 and 𝜈𝜈 𝑡𝑡 ≥ 𝜅𝜅1
𝜈𝜈(𝑡𝑡) if 𝑥𝑥1 𝑡𝑡 = 0 and 𝜈𝜈 𝑡𝑡 < 𝜅𝜅1∫ 𝑆𝑆2 𝑑𝑑𝑑𝑑

∫ 𝑆𝑆2 𝑡𝑡 − 𝑡𝑡 𝑑𝑑𝑑𝑑

𝑑𝑑𝐼𝐼2 /𝑑𝑑𝑑𝑑 = 𝑆𝑆2 (𝑡𝑡 − 𝑡𝑡)(3)∫ (𝑞𝑞 − 𝜈𝜈)𝑑𝑑𝑑𝑑∫ 𝜅𝜅2 𝑑𝑑𝑑𝑑

−𝜅𝜅2 𝑡𝑡 ,𝑑𝑑𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 = 𝑆𝑆2 (𝑡𝑡) −𝑆𝑆2 𝑡𝑡 − 𝑡𝑡 ,𝑑𝑑𝑥𝑥2 /𝑑𝑑𝑑𝑑 = 𝑞𝑞 𝑡𝑡 − 𝜈𝜈(𝑡𝑡)𝑡𝑡
Cumulative curve at ECTokyo Tech∫ 𝜅𝜅1 𝑡𝑡 − 𝑡𝑡 𝑑𝑑𝑑𝑑

∫ 𝜈𝜈𝜈𝜈𝜈𝜈

𝐼𝐼1 < 0𝑥𝑥1

∫ 𝜅𝜅1 𝑑𝑑𝑑𝑑

∫ 𝑆𝑆1 − 𝑆𝑆2 𝑑𝑑𝑑𝑑

−𝜅𝜅2 (𝑡𝑡),𝐼𝐼2 0 = 𝐼𝐼02
𝐼𝐼𝐼𝐼 0 = 0𝑥𝑥2 0 = 0(4)

(5)
(6)

if 𝑥𝑥2 𝑡𝑡 > 0𝜅𝜅2if 𝑥𝑥2 𝑡𝑡 = 0 and 𝑞𝑞 𝑡𝑡 − 𝜈𝜈(𝑡𝑡) ≥ 𝜅𝜅2 (7)𝜅𝜅2𝜅𝜅2 (𝑡𝑡) = �𝑞𝑞 𝑡𝑡 − 𝜈𝜈(𝑡𝑡) if 𝑥𝑥2 𝑡𝑡 = 0 and 𝑞𝑞 𝑡𝑡 − 𝜈𝜈(𝑡𝑡) < 𝜅𝜅2*(t) is omitted. 10

Cost* This study assumes thatTotal Cost for a certain time horizon [0, 𝑇𝑇] *𝑇𝑇(8)𝐶𝐶 = � 𝑐𝑐 𝑡𝑡 𝑑𝑑𝑑𝑑 + 𝜙𝜙(𝑇𝑇)0

the terminal cost 𝜙𝜙(𝑇𝑇) is 0.

max{0, 𝐼𝐼1 (𝑡𝑡)}Increase in Total Cost per unit-time, 𝑐𝑐 𝑡𝑡

𝑐𝑐 𝑡𝑡 = 𝑐𝑐𝐼𝐼1 𝑡𝑡 + 𝑐𝑐𝐼𝐼2 𝑡𝑡 + 𝑐𝑐𝐼𝐼𝐼𝐼 𝑡𝑡 + 𝑐𝑐𝑥𝑥1 𝑡𝑡 + 𝑐𝑐𝑥𝑥2 (𝑡𝑡)(9)
Net Inventory Cost, 𝑐𝑐𝐼𝐼1 𝑡𝑡 + 𝑐𝑐𝐼𝐼2 𝑡𝑡
𝑐𝑐 𝑡𝑡 = max 0, 𝐼𝐼 𝑡𝑡 − min{0, 𝐼𝐼 (𝑡𝑡)} (10) − min{0, 𝐼𝐼2 (𝑡𝑡)}𝐼𝐼1

1

𝑐𝑐𝐼𝐼2 𝑡𝑡 = max 0, 𝐼𝐼2 𝑡𝑡

In-transit Inventory Cost, 𝑐𝑐𝐼𝐼𝐼𝐼 𝑡𝑡
𝑐𝑐𝐼𝐼𝐼𝐼 𝑡𝑡 = 𝑆𝑆2 𝑡𝑡 ⋅ 𝑡𝑡

− m𝑖𝑖𝑖𝑖{0, 𝐼𝐼1 (𝑡𝑡)}𝑐𝑐𝐼𝐼1 𝑡𝑡𝑡𝑡

1− min{0, 𝐼𝐼2 (𝑡𝑡)} (11)(12)

Total Travel Time, 𝑐𝑐𝑥𝑥1 𝑡𝑡 + 𝑐𝑐𝑥𝑥2 𝑡𝑡(13)
𝑐𝑐𝑥𝑥1 𝑡𝑡 = 𝑥𝑥1 𝑡𝑡 /𝜅𝜅1 + 𝑡𝑡 𝜈𝜈(𝑡𝑡)𝑐𝑐𝑥𝑥2 𝑡𝑡 = 𝑥𝑥2 𝑡𝑡 /𝜅𝜅2 (𝑞𝑞 𝑡𝑡 − 𝜈𝜈 𝑡𝑡 ) (14)𝑥𝑥1 /𝜅𝜅1 𝑡𝑡
𝑥𝑥1 𝜅𝜅
1

𝑐𝑐𝑥𝑥1 𝑡𝑡

𝑐𝑐𝐼𝐼2 𝑡𝑡

Tokyo Tech

𝑐𝑐𝐼𝐼𝐼𝐼 𝑡𝑡

𝑥𝑥2 /𝜅𝜅2𝑥𝑥2 𝜅𝜅

2𝑐𝑐𝑥𝑥2 𝑡𝑡

11

Modified Cost

Tokyo Tech

Problems involving delays are not easy to solve.
 Define modified cost 𝐶𝐶̂ so that the problem does not include delays.
̂ When 𝐼𝐼1 𝑡𝑡 ≥ 0 and 𝐼𝐼2 𝑡𝑡 ≥ 0 ∀𝑡𝑡 ∈ [0, 𝑇𝑇], 𝐶𝐶 = 𝐶𝐶.
Modified Net Inventory Cost, 𝑐𝑐𝐼𝐼̂ 1 𝑡𝑡 + 𝑐𝑐𝐼𝐼̂ 2 𝑡𝑡𝑐𝑐𝐼𝐼̂ 1 𝑡𝑡 = max 𝐼𝐼𝑁𝑁1 𝑡𝑡 , 0 − min{0, 𝐼𝐼𝑁𝑁1 𝑡𝑡 } + 𝑐𝑐𝑥𝑥1 𝑡𝑡
𝑐𝑐𝐼𝐼̂ 2 𝑡𝑡 = 𝐼𝐼𝑁𝑁2 𝑡𝑡 − 𝑐𝑐𝐼𝐼𝐼𝐼 𝑡𝑡(15)

𝑑𝑑𝐼𝐼𝐼𝐼1 /𝑑𝑑𝑑𝑑 = 𝑆𝑆1 (𝑡𝑡) −𝑆𝑆2 (𝑡𝑡) −𝜈𝜈 𝑡𝑡 , 𝐼𝐼𝐼𝐼1 0 = 𝐼𝐼01
𝑑𝑑𝐼𝐼𝐼𝐼2 /𝑑𝑑𝑑𝑑 = 𝑆𝑆2 (𝑡𝑡)−𝜅𝜅2 𝑡𝑡 , 𝐼𝐼𝐼𝐼2 0 = 𝐼𝐼02

(17)

(16)

𝐼𝐼𝐼𝐼1 (𝑡𝑡), 𝐼𝐼𝑁𝑁2 (𝑡𝑡) : Potential Net inventory, including unarrived supply and demand=𝑐𝑐𝐼𝐼1 𝑡𝑡(18)+𝑐𝑐𝑥𝑥1 𝑡𝑡−max 𝐼𝐼𝑁𝑁1 𝑡𝑡 , 0min{0, 𝐼𝐼𝑁𝑁1 𝑡𝑡 }12

Bilevel Programming Problem

Tokyo TechLower-level: Linear Optimal Control Problem
̂ 01 , 𝐼𝐼02 , 𝜅𝜅1 , 𝜅𝜅2 , {𝑞𝑞(𝑡𝑡)}]
min[𝐶𝐶|𝐼𝐼𝑿𝑿,𝑼𝑼s. t.

Eq. 1 − 7 , 12 − (18)𝑇𝑇𝐶𝐶̂ = � 𝑐𝑐̂ 𝑡𝑡 𝑑𝑑𝑑𝑑
0

𝑿𝑿: = {𝐼𝐼1 (𝑡𝑡), 𝐼𝐼2 𝑡𝑡 , 𝐼𝐼𝐼𝐼1 (𝑡𝑡), 𝐼𝐼𝐼𝐼2 𝑡𝑡 ,𝐼𝐼𝐼𝐼 𝑡𝑡 , 𝑥𝑥1 𝑡𝑡 , 𝑥𝑥2 (𝑡𝑡)|∀𝑡𝑡 ∈ 0, 𝑇𝑇 }𝑼𝑼 ≔ {𝑆𝑆1 𝑡𝑡 , 𝑆𝑆2 𝑡𝑡 , 𝜈𝜈 𝑡𝑡 ,𝜅𝜅1 𝑡𝑡 , 𝜅𝜅2 (𝑡𝑡)|∀𝑡𝑡 ∈ 0, 𝑇𝑇 }𝑐𝑐̂ 𝑡𝑡 = 𝑐𝑐𝐼𝐼̂ 1 𝑡𝑡 + 𝑐𝑐𝐼𝐼̂ 2 𝑡𝑡 + 𝑐𝑐𝐼𝐼𝐼𝐼 𝑡𝑡 + 𝑐𝑐𝑥𝑥1 𝑡𝑡 + 𝑐𝑐𝑥𝑥2 (𝑡𝑡)

 Given preparation strategies {𝐼𝐼01 , 𝐼𝐼02 , 𝜅𝜅1 , 𝜅𝜅2 } and a demand sample path{𝑞𝑞(𝑡𝑡)}, the optimal control is obtained numerically.

 Apply existing computationally efficient solution method (c.f.
Chavanasporn and Ewald, 2012) for linear optimal control problem.

Upper-levelmin

𝐼𝐼01 ,𝐼𝐼02 ,𝜅𝜅1 ,𝜅𝜅2

̂ 𝑼𝑼]
E[𝐶𝐶|𝑿𝑿,s. t.

Lower-level Problem and𝐼𝐼1 𝑡𝑡 ≥ 0, 𝐼𝐼2 𝑡𝑡 ≥ 0, 𝜅𝜅1 + 𝜅𝜅2 = 𝜅𝜅, {𝑄𝑄(𝑡𝑡)} Generate a finite number of sample paths 𝑞𝑞(𝑡𝑡) of stochastic process 𝑄𝑄(𝑡𝑡) Evaluate the expected total cost approximately

 Find the optimal 𝐼𝐼01 , 𝐼𝐼02 , 𝜅𝜅1 , 𝜅𝜅2 in a brute-force fashion

13

Numerical Experiments

14

Numerical Experiments: Setting

Tokyo Tech The number of sample paths 𝑞𝑞(𝑡𝑡) is 500.

 𝑑𝑑𝑞𝑞 𝑡𝑡 = 𝜎𝜎𝑞𝑞 𝑑𝑑𝑑𝑑(𝑡𝑡), 𝑧𝑧 𝑡𝑡 ~𝑁𝑁(0, 𝑡𝑡), 𝑞𝑞(0) = {3,4,5,6,7}, 𝜎𝜎𝑞𝑞 = {0, 0.05, 0.1}
 𝑇𝑇 = 100

 FFTT, 𝑡𝑡 = {5,10,20}
 𝑆𝑆1 = 10, 𝑆𝑆2 = 3,5,7

 𝜅𝜅 = 5, 𝜅𝜅1 = {0,1,2,3,4,5}

500 Sample Paths (𝑞𝑞(0) = 5, 𝜎𝜎𝑞𝑞 = 0.05, then 𝑞𝑞 𝑡𝑡 ~𝑁𝑁(𝑞𝑞 0 , 𝜎𝜎𝑞𝑞2 𝑡𝑡))15

E[𝑞𝑞 𝑡𝑡 ]E[𝑞𝑞 𝑡𝑡 ]E 𝑞𝑞 𝑡𝑡𝜎𝜎𝑞𝑞𝜎𝜎𝑞𝑞

Tokyo TechOptimal 𝜅𝜅1= 𝑞𝑞 0

Numerical Experiments: Result𝜎𝜎𝑞𝑞

Optimal 𝜅𝜅1 when FFTT 𝑡𝑡 = 5 (left: 𝑆𝑆2 = 3, center: 𝑆𝑆2 = 5, right: 𝑆𝑆2 = 7)

 The larger mean E 𝑞𝑞 𝑡𝑡

and variance 𝜎𝜎𝑞𝑞2 𝑡𝑡 of demand increase optimal 𝜅𝜅1 .

 When supply constraint is relaxed (𝑆𝑆2 = 7), optimal 𝜅𝜅1 is likely to be 0.

Suggest that in a typical disaster situation(𝐄𝐄 𝒒𝒒 𝒕𝒕 , 𝝈𝝈𝟐𝟐𝒒𝒒 𝒕𝒕 are large, supply constraint is tight),
optimal 𝜿𝜿𝟏𝟏 > 𝟎𝟎 (i.e., demand-supply approach is effective). 16

E[𝑞𝑞 𝑡𝑡 ]E[𝑞𝑞 𝑡𝑡 ]𝜎𝜎𝑞𝑞𝜎𝜎𝑞𝑞

Tokyo TechOptimal 𝜅𝜅1

E[𝑞𝑞 𝑡𝑡 ]Numerical Experiments: Result𝜎𝜎𝑞𝑞Optimal 𝜅𝜅1 when 𝑆𝑆2 = 7(left: FFTT 𝑡𝑡 = 5, center:𝑡𝑡 = 10, right:𝑡𝑡 = 20)

 FFTT 𝑡𝑡 between DB and EC increases, then optimal 𝜅𝜅1 tends to increase, and the ratio of users’ cost (total travel time for victims) increase.

 Non-negative constraints on 𝐼𝐼1 𝑡𝑡 and 𝐼𝐼2 𝑡𝑡 could be causing supplyside costs to be greater than optimum.

17

Conclusions

Tokyo Tech

 Formulation
 Formulate BLPP to evaluate the performance of demandsupply approach• Upper: Optimize preparations (e.g., staffing, stockpiles)• Lower: Optimize dynamic network flow for humans andgoods
 Reformulate lower-level one as a (non-delayed) optimal
control problem by modifying net inventory cost
 Numerical Experiments Demand-supply approach would be effective in a
typical disaster situation.
 However, the proposed model is likely to overestimate
supply-side costs. The optimality of the reformulated
problem, optimal control problem, should be evaluated.

18References

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Case Studies
1.

Asahi Shimbun Digital, April 19, 2016, https://www.asahi.com/special/0417earthquake/gallery/102.html,
accessed 15 Sep. 2022. (in Japanese)2.

Sompo Japan Insurance Inc.,災害時に救援物資が被災者に届かない理由とは, https://www.sompo-japan.co.jp//media/SJNK/files/hinsurance/logistics/news/2016/b_news123.pdf?la=ja-JP, accessed 15 Sep. 2022. (inJapanese)

3.

Kawase, R., Urata, J., Iryo, T. The POST Disaster Humanitarian Logistics: Case Studies of the Tohoku
Disasters and Kumamoto Earthquake. Journal of Japan Society of Civil Engineers, Ser. D3 (InfrastructurePlanning and Management), vol.76, No.5, pp.I_987-I_999, 2021. (in Japanese)

Network flow optimization for humans and goods
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Muggy, L., and Heier Stamm, J. L. Decentralized beneficiary behavior in humanitarian supply chains:
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Gutjahr, W. J., and Dzubur, N. Bi-objective bilevel optimization of distribution center locations considering
user equilibria. Transportation Research Part E: Logistics and Transportation Review, vol.85, pp.1-22, 2016.

6.

Espejo-Díaz, J. A., and Guerrero, W. J. A multiagent approach to solving the dynamic postdisaster relief
distribution problem. Operations Management Research, Vol.14, No.1, pp.177-193, 2021.

7.

Fikar, C., Hirsch, P., and Nolz, P. C. Agent-based simulation optimization for dynamic disaster relief
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8.

Akamatsu, T., and Nagae, T. Dynamics Ramp Control Strategies for Risk Averse System Optimal Assignment. Allsop, Belland Heydecker (eds.): Transportation and Traffic Theory, pp.87-111, 2007.

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Kawase, R., Urata, J., and Iryo, T. Optimal Control Strategy for Relief Supply Considering Information andDemand Uncertainty after a Major Disaster. IFAC-PapersOnLine, Vol.52, No.6, pp.31-38, 2019.

10. Kawase, R. Mathematics for Optimization of Post-Disaster Humanitarian Logistics (Doctoral dissertation),
2021. (in Japanese)19

ReferencesTokyo TechOptimal Control Problem
11. Chavanasporn, W., and Ewald, C. O. A numerical method for solving stochastic optimal control problems
with linear control. Computational Economics, vol.39, No. 4, pp.429-446, 2012.
12. Huschto, T. Numerical Methods for Random Parameter Optimal Control and the Optimal Control of
Stochastic Differential Equations (Doctoral dissertation), 2014.

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