社会技術研究論文集 Vol.14, 156-166, June 2017

A MULTI-HIERARCHY FACILITY LOCATION PROBLEM WITH ROUTING UNDER
DEMAND UNCERTAINTY: AN APPLICATION TO RELIEF DISTRIBUTIONS
Rattanaporn KASEMSRI1・Kazushi SANO2・Amila JAYASINGHE3
1Doctoral Student, Graduate School of Civil and Environmental Engineering, Nagaoka University of Technology(E-mail:rattanaporn7777@gmail.com)
2Professor, Civil and Environmental Engineering, Nagaoka University of Technology (E-mail:sano@nagaokaut.ac.jp)
3Doctoral Student, Graduate School of Civil and Environmental Engineering, Nagaoka University of Technology(E-mail:amilabjayasinghe@gmail.com)

Demand fluctuation, known as demand uncertainty, usually occurs under several circumstances not only inordinary logistics firm but also humanitarian logistics. An objective of this study is to tackle the facility locations andallocations under demand uncertainty in a case study for a relief distribution. The demand uncertainty is handled byrobust counterpart in Robust Optimization (RO) called an ellipsoidal uncertainty set, which is a novel approach thathas never been fully applied so far solving on a multi-facility location network. Herein, we analyze and compare
three distinct network designs to find the most cost efficient and robust network. The best model is selected throughcontinuous improvement by routing optimization, leading to delivery cost reduction.

Key Words: multi-hierarchy, demand uncertainty, facility locations, Robust Optimization

1.

INTRODUCTION

to solve facility location problems by varying the size both box
and ellipsoid. They found that the box uncertainty model
performs poorly with respect to balancing robustness with themodel objective function. However, their model only
considered a single layer of facility location. On the contrary,
this study focuses on the fully multi-hierarchy facility location
network, which is more complicated but more realistic. Min etal.5) found that there is a difference between location withrouting and classical location: while the first one serves
customers through tours, the second one serves the customers
through radial trips.
In such a background, this study attempts to contribute to theprevious research by considering the worst case scenarios for
multi-facility location problems under demand uncertainty.
Accordingly, the main objective of the study is to tackle thefacility location and allocation problem with demand
uncertainty by using an ellipsoidal uncertainty set approach,which has never been fully applied so far. This allows to solve
the multi-hierarchy facility location network problem, andsubsequently building vehicle routing from the allocations
derived from location problem with demand uncertainty. The
objectives of the study can be summarized as follows:
1. To apply ellipsoidal uncertainty set, a novel approach tosolve fully multi-hierarchy facility location network
problems.
2. To manage the facility location problem in the context ofboth deterministic and uncertain demand, andsubsequently compare solutions’ robustness of five

Humanitarian logistics have been indicated as significant
issues in several terms of natural disaster operations andmanagement. Recently, researchers have extensively workedon the most appropriate location of medical centers and shelters
to provide the evacuees quick access. Lin et al.2) found thatprioritizing items for delivery and an extensive time period areimportant factors in humanitarian logistics. Accordingly, toimprove logistics efficiency, they proposed the location of
temporary depots to be around the disaster-affected area, alongwith the required vehicles and resources. In contrast, our study
intends to design the depot locations by considering the cost
efficiency and whether demand is met. Real situations usually
involve uncertainty, that is, parameter fluctuation.
Holguín-Veras3) has inferred from the Tohoku experience that,in order to improve future response efforts, disaster planners
should design worst case scenarios from small disasters tomassive destruction. Therefore, this study also stresses theimportance of such improvements in case the demand becomes
uncertain. Furthermore, we aim to improve the network byconsidering the practical level of vehicle routing problems after
obtaining the locations at a strategic level are obtained underdemand uncertainty, setting up a two-stage facility location
network with a route planning problem. Baron and Milner4)
have carried on the similar works on facility location by using
the Robust Optimization approach. The potential of box
uncertainty set and ellipsoidal uncertainty set was investigated156

社会技術研究論文集 Vol.14, 156-166, June 2017

Robust Optimization. Snyder7) surveyed the facility location
under uncertainty. The author classified several papers by theirapproach to uncertainty. Those are stochastic location problems
and robust location problems. However, there is no
multi-facility location problem solved through robust
counterpart. Accordingly, we attempt to address the problem by
employing a robust counterpart of the ellipsoidal uncertainty set,
which is a novel approach that has never been fully applied so
far to solve on multi-facility location problems.
In addition, according to the definition of solitary facility
location problem, the classical location serves customers
through radial trips. They do not serve reasonably through tours
as in more realistic operational circumstances. Therefore, theroute building is conducted to improve the distribution network
onward.

demand scenario assumptions.
3. To evaluate the total delivery cost efficiency on different
networks. Accordingly, we analyze the model on three
networks differing in structures and truck sizes.
4. To select the best network and improve it by making
practical uses of routing.
The expected results from the facility location model withrouting are defined as follows:
1. To search the appropriate locations of depots for relief
items’ distribution in Miyagi prefectures.
2. To allocate the transportation link flow at each network
configuration.
3. To minimize the total delivery cost which includes
transportation cost, facility opening cost, andtransshipment cost.
4. To seek a practical routing to reduce the total delivery
cost.
In upcoming sections, we state the problem, and present
model structure and assumptions, as well as results andconclusions.

2.

3.

MODEL STRUCTURE
ASSUMPTIONS

AND

DATA

3.1. Model structure
The problem is designed for three different network
structures. We categorized the three networks based on network
configurations and dispatched truck sizes. The first network
element consists of the locations with serviceable supports(suppliers). The second network element is the central relief
depot in case of double hierarchies. The third element isconstituted by the relief depots for double hierarchy and therelief depot in case of single hierarchy. The latter two network
locations are unknown and need to be defined in the mostefficient way. Finally, areas subject to a natural disaster are
called shelters, which can be defined known locations as
demands. The transportation truck sizes are 10-ton trucks and4-ton trucks. Networks specification is described below.

PROBLEM STATEMENT

The 2011 Tohoku’ Earthquake in Japan, represents our casestudy. We focus on Miyagi prefecture, the most affected areawith a huge number of evacuees. As it is widely known, the2011 Tohoku earthquake and tsunami had enormous impacts
both as a humanitarian crisis, while also leaving economic
aftermath. Moreover, the situation generated the several costs
during and after disaster like reconstruction cost, rescue cost,
logistics cost, etc. The logistics cost is quantified by Nagurney
et al.6) as approximately 80 percent of overall operation
responding costs. Therefore, the cost efficiency should be oneof the many aspects to be considered. Because of this, thisstudy would like to examine the logistics cost efficiency. An
improved supply distribution cost can reduce the expenditure
for the whole operation cost during the amelioration period. A
bottle of water is considered to be a requisite item forpreliminary succor. Although total delivery cost minimization
is not the only aspect to be considered in humanitarian logistics,
it is a good criterion to compare the results for distinct network
systems.
In fact, post disaster circumstances usually involve
fluctuations in the number of evacuees and imprecise
predictions. This study aims to handle the facility location
problems or mixed integer problems under demand uncertainty.
The methodology to deal with this demand fluctuation is（1）Network 1
This network is determined by three network elements:
suppliers, relief depots and shelter demands. The relief depot
candidate sites are located inside the affected areas. The relief
items are dispatched from suppliers to relief depots by using

Fig.1 The single-hierarchy network framework and 10-4-ton truck
delivery157

社会技術研究論文集 Vol.14, 156-166, June 2017

terms of distance. Further, there were relatively less evacueeson the top and bottom sites located near Sendai city. The lowest
number of evacuees (5 evacuees) was recorded in the
Marumori town where connects under Sendai. Exempt thesementions before a number of evacuees are quite small whencomparing with Sendai and Ishinomaki. These amounts aresome hundreds of people for each shelter.
The study not only examines a single demand model but
also multi-demand scenarios are examined. These demand
scenarios are used in order to analyze the sensitivity of threedifferent network structures. The demand is determined for five
scenarios which deviate from the historical case in both sides
optimistic and pessimistic. These five demand scenarios are
separated as less than historical demand by 20 percent (S1) and
10 percent (S2), historical demand (S3), and more than actual
demand by 10 percent (S3) and 20 percent (S4) respectively.

10-ton trucks. Then, 4-ton trucks are used for portage of therelief items from relief depots to shelter demands.
（2）Network 2This network is determined by four network elements:
suppliers, central relief depots, relief depots and shelter
demands. The central relief depot candidates are supposed to belocated inside the affected areas. The 10-ton trucks are meant totransport relief items from suppliers to central relief depots andfrom central relief depots to relief depots. Then, relief items arecarried from relief depots to shelter demands by using 4-tontrucks.

3.3. Supplier locations and facility locations
This section specifies the network element locations. These
locations are identified by latitude and longitude coordinates.
Our study assumes that locations are easily accessible byexpressway or main road. Absent significant distance, locations
might also be a group of sites. First, the locations of suppliers
are determined. The Miyagi Prefectural Government Report8)
shows that in Japan there are 29 sites available for serving theaffected areas with goods after the disaster. However, wegrouped these 29 sites into nine sites of suppliers by using theadjacent zone ideal. The locations of supplier locations areillustrated in Figure 5(a).
Then, the locations of candidate central depots and candidate
depot locations are defined. The assumption is that the central
depots are inside the affected areas or in Miyagi prefecture.
These are assigned to cover the areas of the next layer or depots.
The central depots should respond to the next layer necessities
by considering the zoning characteristics. The locations ofcandidate central depots are illustrated in Figure 5(b). Then, thecandidate depot locations are indicated in 11 places. These
locations are specified by clustering zoning areas and demand
size in Figure 5(c). As mentioned in the previous section, thedifferent colors of depot candidates are supposed to correspondto different demand zones.

Fig.2 The two-hierarchy network framework and 10-10-4-ton truck
delivery（3）Network 3This network is a duplicate structure of network 2 in terms ofthe number of network configurations and their locations.
However, there is a difference in terms of truck size, as here thetransportation from central relief depots to relief depots is
assigned to 4-ton trucks instead of 10-ton trucks.

Fig.3 The two hierarchies network framework and 10-4-4-ton truck
delivery

3.2. Shelter locations and number of evacuees
This study used data published in the Miyagi Prefectural
Government Report8). The number of evacuees (Figure 4) is
considered to estimate the amount of relief items demanded
and their locations. We classified locations into zones
distinguished by different colors in Figure 4. The data indicatedthat the number of evacuees was very high in flat areas and thecapital city. The highest demand, approximately 70,000, wasrecorded in Sendai. Therefore, we divided Sendai into five
demand sites. Although the number of victims in Ishinomaki
was also high, the demand site was not subdivided because theflat area is quite small. Hence there is no significant impact in

3.4. The total delivery cost
We assume that total delivery costs can be separated into
three parts: travel cost, facility opening cost and transshipmentcost. Firstly, the travel cost is measured by the fuelconsumption rate based on 4-ton trucks and 10-ton trucks.
Moreover, the travel cost also includes driver’s salary under the158

社会技術研究論文集 Vol.14, 156-166, June 2017

3.5. The travel costThis cost changes according to the distance and energy
consumption rate. At the beginning, the energy consumption
rate for 4-ton trucks and 10-ton trucks was set to 7.69 and 11.54
yen per kilometer respectively. The travel cost is computed
below and considers travel cost from length, driver salary,purchase truck cost and insurance cost. The driver cost is
defined per hour of driving with an upper limit of eight hours

working time limitation, purchase truck cost per day andvehicle insurance per day. Secondly, the facility opening cost isdetermined.
This cost is measured as the average price of building rental
for firm, and it differs depending on the zone. Finally, thetransshipment cost is set about three times the value of an
ordinary business firm (15,000 yen per ton per day) for loading
and unloading goods.

Fig.4 The amount of demand in each shelter and zone clustering (in the map)(a) Supplier locations(b) Central depot locations
Fig.5 The supplier and facility locations

159(c) Depot locations

社会技術研究論文集 Vol.14, 156-166, June 2017,

per day. Purchase truck cost and vehicle insurance cost are
divided into one day units.
a. Large truck size (10-ton)*driver salary= 1,250 yen per hour
*purchase truck cost = 2,063 yen per day*vehicle insurance cost = 1,032 yen per day
b. Small truck size (4-ton)*driver salary= 1,000 yen per hour
*purchase truck cost = 1,375 yen per day*vehicle insurance cost = 688 yen per day

and: The capacity at the candidate central
depots (j=1,2,3… ): The capacity at the candidate depots( =1,2,3… )Sets of parameters: The amount of items at the supply: The demand at the affected area,,

: The travel cost between
, and

,
,
,

3.6. Facility opening cost
The facility opening cost or fixed cost includes construction
cost, electricity cost and water supply cost as well as the rentalbuilding cost based on websites in @Nifty10). This cost ismeasured as the average price of a storage building rental ineach area of central depot candidates. This implies that the
average price of building rental is a function of the storage size.
However, several severe destructions including road disruptionsin the affected areas after the disaster need to be taken into
account. Therefore, it becomes very difficult to determine thesuitable locations for candidate central depots and depots.
Moreover, after the disaster there is also a lack of necessary
resources like human resources, electricity supplies, watersupplies, fuel energy and particularly land areas. Therefore, theopening cost for the facility location is set 10% higher than theoriginal.

,,,: The maximum working time of drivers
between and , and , and,: The distance between
and

,

4.1.

and

: The opening depot cost at and: The transshipment cost at and: The capacity of truck between and ,and , and

,

,

and ,

and ,

and ,

: The travel time between and , and, and: The energy consumption rate of truck sizes: The driver salary of truck size s: The truck cost of truck size s
Objective function

Classical location problem+

3.7. Transshipment cost
Transshipment cost is assumed to be approximately 15,000
yen per ton per day. This cost is three times the typical businessfirm cost for loading or unloading of goods at facility sites. This
cost will generate when the set of facility sites is selected tooperate. This cost includes salary, bonus, social insurance andpension saving of human resources. This value is obtained fromthe firm Logistic Behavior Survey (PWRI)9).

+

+

+

+

(1)

+

When

4.

=
=

+
+

+
+

+
=
Decision variables+

MATHEMATICAL FORMULATION

Indices: Set of the supplier nodes ( ) ( =1,2,3… ): Set of the candidate central depots ( ) ( =1,2,3… ): Set of the candidate depots ( ) ( =1,2,3… ): Set of the demand nodes or shelters ( ) ( =1,2,3… )TS: Set of the truck size (s)Notations, ,: The flow of items from and ,

=

(5)

1,
=

160

0,

ℎ

0,

ℎ

1,

Subject to
and

(2)
(3)
(4)

(6)

社会技術研究論文集 Vol.14, 156-166, June 2017

≤

(7)

≥

(8)

≤

(9)

≤

(10)

≤

(11)

≥

(12)

≤

from suppliers to, respectively, depots and to demand ,are the same. Constraint (7) guarantees that total flows fromsuppliers to central depots are not higher than the amountof serving goods at suppliers . Constraint (8) ensures that thesum of link flows from to does not exceed the total
availability of goods at opening central depots . Constraint (9)limits the total amount of link flows from toto thecapacity of opening central depots . Constraint (10) limits thesum of link flows from to to the availability of goods at
depots . Constraint (11) ensures that the total amount of link
flows from depots to demand must not be higher thanthe capacity of next network configuration or depots .
Constraint (12) confirms that the total amount of goods fromdepots satisfies demand . Constraints (13), (14), and (15)prohibit that the amount of a commodity exceeds the
maximum truck volume. Constraints (16), (17), and (18) limit
the total amount of driving hours to the maximum working
time. Constraint (19) confirms that each link flow from site
to , to , and to needs to define some amount ofgoods. Finally, constraint (20) specifies that both decisionvariablesandare binary variables, taking value 1 if the(13)

facility is located at sites
≤

(14)

≤

(15)

Decision variables
1,

ℎ

=

,
0,

≤

and , and 0 otherwise.

(21)

ℎ

Location problem with routing

(16)

+
≤

(17)
+

≤

+

+

(22)

(18)

, ,
≥0
Y , Z ∈ 0,1 for all j and k

+

(19)
(20)

+

Additionally subject to
The objective function (1) minimizes the total delivery cost
as the sum of transportation cost, facility cost andtransshipment cost. Formulation (2), (3) and (4) represent travel
cost parameter functions including travel cost and operation
cost.
and
in (5) and (6) are decision variables to locate≥1

the central depots and depots . Note that (1) is completely
performed for network 2 and network 3 through both decisionvariablesandin (5) and (6), respectively. However,there are no central depots−

in network 1, therefore the links

161

(23)

≤

(24)

=0

(25)

社会技術研究論文集 Vol.14, 156-166, June 2017

̅This part refers to the second stage of this study that is
improvement through routing. The variables in the location
problem, namely locations and allocations, were determined in
the primary stage. In other words, the decision ,andvariables, , in (22) are derived from the optimization

≥

(28)

This study focuses on the demand uncertainty parameter,
which deviates from the historical value of the uncertain
parameters. Demand uncertainty is expanded in the region of
the ellipsoidal uncertainty set. The demand parameter in (26) isin (1) under demand uncertainty (26, 27, and 28). Then, thedecision ,and variables, ,in (27) turn into

while ( ) is the demand deviating from historical values.
parameters to plug in (22). Subsequently, the variablein
Given the uncertain demand, we consider the sets(21) is added to generate the tours at the third hierarchy. The
around the historical values, where is a scaling
objective function (22) minimizes the total delivery cost ofparameter for the ellipsoid radius andis the constraint. The
location with routing problem. Constraint (23) ensures that thesize of the ellipsoid can increase by the ellipsoid radius ordelivery route is linked to the opening depots. Constraint (24)the constraint . Increasing the size of the ellipsoid will makefixes the capacity of trucks at depots. Constraint (25) requires
the model more robust against (more) uncertainty, however atthat the trucks ought to leave every point that they visited.
the cost of a worse solution. Moreover, if the size of theellipsoid becomes too large then the model might become
4.2. Mathematical with Robust Formulation by using infeasible. In accordance with the solution feasibility, we set
Robust Counterpart
equal to 1/150 – 1 and use
equal to 1 to restrict the region
This part of the study focuses on the multi-source andaround historical demand. We determine that the interval range
multi-layer aspects of the facility location problem withof demand ( − ) is equivalent to the maximum truck
uncertainty demand by considering the ellipsoidal uncertainty
capacity which is still in the feasible solution area. Then, theset within the robust optimization approach. Ben-Tal andvariables from the deterministic demand model in (1) are used
15)
Nemirovski considered the ellipsoidal uncertainty set within (27) to represent the uncertain demand model. Thesevariables give the solutions in the worst case under the
linear programming. Kouvelis and Yu16) discussed the robust
constraint (28) determines as ellipsoidal uncertainty set.
discrete optimization and its applications. They proposed an
The robust counterpart in Robust Optimization (RO) is
approach to find a solution that minimizes the worst caseprovided by AIMMS software, which has been more recently
performance under a set of scenarios for the data. Bertsimas
applied to handle parameter uncertainty. RO is designed toand Brown17) proposed a methodology to construct uncertainty
meet
some
major
challenges
associatedwithsets for robust linear optimization based on decision maker risk
18)
uncertainty-affected optimization problems; to operate under
preferences. Josef gave an overview of the state-of-the-art
lack of full information on the nature of uncertainty, to modeland recent advances in mixed integer optimization to solve
the problem in a form that can be solved efficiently, and toplanning and design problems in industry processes. Stochastic
provide guarantees about the performance of the solutions. RO
programming for continuous LP problems is now included in
is an uncertainty modeling approach suitable for a situation
most optimization packages, and there is encouraging progress
where the uncertainty ranges are known while the distribution
in the ﬁelds of stochastic MILP and robust MILP. Ben-Tal,
19)
not necessarily is. Typically, some inputs take an uncertain
Bertsimas and Brown proposed a soft robust model forvalue anywhere between fixed minimum and maximum. This
optimization under ambiguity.
demand uncertainty can present how the worst case is when we(26)
Ellipsoidal:=≤
×
consider the fluctuation of the demand. RO is very suitable formany problems as only simple inputs about data uncertainty
̅are required as there are no scenarios or distribution functions
̅+̅
need to be defined. The advantage of RO models is that theygrow only slightly when uncertainty is added. As a result, the̅model can be solved efficiently.
(27)
+
̅ ̅ +
+
The objective function containing the integer variables, iscalled Mixed Integer Program (MIP). In addition, one̅constraint is quadratic, thus the model becomes a Mixed++
Integer Program Quadratically-Constrained Program (MIQCP).

162

社会技術研究論文集 Vol.14, 156-166, June 201735,695,605 yen, 28,639,608 yen, and 29,552,784 yen in case ofdeterministic demand and 39,265,851 yen, 32,226,419 yen,and 33,054,993 yen in the uncertain demand case. Network 2
and network 3 lessened by 21.94 percent and 18.83 percent
respectively in the deterministic case. Similarly, they also
lessened by 19.69 percent and 17.18 percent in the uncertain
case. The total delivery cost is mostly generated by the travel
cost (approximately more than 90 percent), and its rapid
increase is due to the amount of transportation.
When comparing network 2 and network 3, all demand
scenarios in network 2 can be reduced by 1.19 percent, 2.79
percent, 6.06 percent, 2.49 percent, and 1.71 percent,respectively. These results indicate that not only network
configurations, but also truck size operations have significant
influence on the total delivery cost function. Using 10-ton
trucks to deliver from suppliers to central depots and from
central depots to depots has a significant positive effect on cost
reduction. Therefore, we can infer that network 2 meets withthe criteria of cost performance.
Figure 7 illustrates the sensitivity analysis on the objectivefunction for each network. In the range of sensitivity values
comparison, network 1 presents a wider range and higher
sensitivity than the others, meaning that network 1 is less robust
than the other networks. The range scale of network 1 wasapproximately 0.1 to 9 million yen, while there isapproximately 3 to 5 million yen of sensitivity for network 2
and network 3. In particular, when comparing the averagesensitivity, network 1 is at 4,708,289 and 4,943,401 in term of
deterministic and uncertain demand, while the corresponding
values for network 2 are only 3,001,504 and 2,495,848.
Moreover, the average sensitivity of network 2 is also lower
than network 3, which is 3,325,625 and 2,552,370 for
deterministic and uncertain demand, respectively. These results

Whenever the uncertainty set of a mixed-integer robust
problem is an ellipsoidal, the robust counterpart can bereformulated as a mixed-integer second-order cone program(MISOCP) which is a special case of MIQCP. This study uses
a well-known powerful solver namely Gurobi (GUROBI
website)20), available for MISOCP. The algorithm used by
GUROBI solvers is parallel SOCP barrier algorithm.

5.

COMPUTATINAL RESULTS

Results illustrate outcomes for both deterministic anduncertain model. The anticipation results for both
circumstances are the location of the facility and the total
delivery cost of the three different network structures. As
mentioned before, each network structure includes five demand
scenarios, corresponding to five expectation results for each.
We compare the total delivery cost of the three networks andindicate the best network structure by cost performance andnetwork robustness. Then, we present the sensitivity analysis
and compare the robustness of the three networks. Furthermore,
we choose the best among the three networks to establish thevehicle routing designs.
The locations of opened facility for each network of both
deterministic demand and uncertain demand are indicated inTable 1. From the results, we found that the network
configurations and their systems have an impact on the total
delivery cost on both deterministic demand and uncertain
demand (see Figure 6). It can be seen that network 2 andnetwork 3, which are defined for two layers of facility, areobviously preferable in term of cost performance whencompared to network 1, which is single layer. The average total
delivery costs of the three networks are, respectively

Table 1 The opened facility locations for each network of both deterministic demand (left) and uncertainty demand (right)

Central depot
Kesennuma
Tome
Sendai(Kumagane)
Yamamoto

Network
Network 1
Network 2
Network 3
S1 S2 S3 S4 S5 S1 S2 S3 S4 S5 S1 S2 S3 S4 S5
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1

Network 1
Network 2
Network 3
Depot
S1 S2 S3 S4 S5 S1 S2 S3 S4 S5 S1 S2 S3 S4 S5Tome and Kurihara
1 1 1
1 1 1
1 1 1
Kesennuma
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
North Ishinomaki
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1South Ishinomaki
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1South East Ishinomaki 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Taiwa Town
North West Sendai
North East Sendai
Central Sendai
South East Sendai
South Sendai

Network 1
Central depot
Kesennuma
Tome
Sendai(Kumagane)
Yamamoto

Network
Network 2

Network 3

S1 S2 S3 S4 S5 S1 S2 S3 S4 S5 S1 S2 S3 S4 S5
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1

Network 1
Network 2
Network 3
Depot
S1 S2 S3 S4 S5 S1 S2 S3 S4 S5 S1 S2 S3 S4 S5Tome and Kurihara
1 1 1 11 1 1 11 1 1
Kesennuma
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
North Ishinomaki
1 1 1 1 1
1 1 1 11 1 1 1
South Ishinomaki
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1South East Ishinomaki 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Taiwa Town
North West Sendai
North East Sendai
Central Sendai
South East Sendai
South Sendai

163

社会技術研究論文集 Vol.14, 156-166, June 2017(a) Deterministic demand(b)

Uncertainty demandFig.6 The total delivery cost

Objective value (yen)Millions

Deterministic

in the deterministic case, showing the advantage of using RO.
In the comparison of deterministic demand and uncertain
demand, network 2 and network 3 were similar that by using
robust optimization to handle the uncertainty demandillustrated more robustness than ordinary deterministic demand.
In addition, the fluctuation in sensitivity between deterministic
demand and uncertain demand of network 2 was less than in
network 3, meaning that network 2 is more robust than theother networks.
Moreover, we not only locate the optimal facility sites, butalso allocate the optimal link flows. Results indicate that we
can improve the best network among the three networks bymaking practical use of routing. According to the above
comparisons, by considering cost efficiency and model
robustness, network 2 offers the best outcome when the
demand becomes uncertain. Consequently, we choose scenario
3 of this network as a representative to improve such an
outcome through practical uses of vehicle routing problems.
We focus on the last hierarchy of the network elements, fromdepots to shelters, and apply it to routing problems. The
optimized routing network of each opened facility is illustrated
in Figure 8. There are five depots selected at the optimal
solutions, and each depot responds to its link flow allocation.
The routing, which visits more than one shelter in one trip,
consists instead of a single visit trip when link flows are lessthan truck capacity. We found that the total delivery cost is
reduced for every opened facility site, by approximately 30
percent (Table 2).

Robustness

10
9
8
7
6
5
4
3
2
1
0
1

2

3

4

5

Demand scenarios

Millions

Objective value (yen)Network 1
Deterministic

Robustness

1

3

6
5
4
3
2
1
0
2

4

5

Demand scenarios

Network 2

Millions

Objective value (yen)Deterministic

Robustness

6
5
4
3
2
1
0
1

2

3

4

5

Demand scenarios

Network 3
Fig.7 The model sensitivity analysis of deterministic demand anduncertainty demandindicate that network 2 is the most robust network among thethree networks. In addition, the average sensitivity of uncertain
demand for both network 2 and network 3 is more robust than164

社会技術研究論文集 Vol.14, 156-166, June 2017

Fig.8 The routing operation of network 2, scenario

uncertainty set is a novel approach that has never been fully
applied to solve facility location problems. Accordingly, themain objective of the study is to tackle the facility locations andallocations with an uncertain demand function, as well as toimprove the solution by integrating with vehicle routing
problems. We analyze the three network structures including
one single-layer facility network and two two-layer facility
networks with distinct truck sizes (large trucks and small
trucks). The uncertainty model helps the planner identify
trade-offs between the inability to recover full costs for excess
link flow and satisfying the demand.
First, the results from our calculation demonstrate that the
network configurations influence total delivery costs. It can beseen clearly that the total delivery cost of network 2 andnetwork 3 can reduce because the travel cost reduces, eventhough more facility cost and transshipment cost is required.
Furthermore, results indicate that the travel cost has moresignificant influence than the facility opening cost. Moreover,truck size operation is important when the demand is highenough. This study found that large trucks are appropriate todeliver both inbound and outbound from the central depots. To
apply the model, we suggest to establish the central depots andTable 2 The total delivery cost reduction by using LRP
Selected opening
Location problem of
depotsechelon 3
Tome and Kurihara
7,027,454
Kesennuma
6,640,056
North Ishinomaki
3,903,580
South Ishinomaki
2,731,894
South East Ishinomaki
3,124,383
Sum of echelon 3
23,427,368
Travel cost of echelon
1 and 2
6,662,229
Opening cost
677,732
Transhipment cost
1,263,564
Total delivery cost
32,030,892

6.

Improved byrouting
4,163,944
3,704,803
1,766,211
1,739,353
2,587,476
13,961,787

Percentage of
cost reduction
40.75
44.21
54.75
36.33
17.18

6,662,229
677,732
1,263,564
22,565,312

29.55

CONCLUSION AND FUTURE WORKS

This study contributes to the previous research byconsidering the worst case scenarios of multi-facility location
problems under uncertain demand. We justify the uncertainty
on demand by the fact that it is quite difficult to predict postdisaster demand. Therefore, our study determines the region of
uncertain demand as an ellipsoid uncertainty set that is suitablefor our situation where only the uncertainty ranges are known
and not necessitate the distribution. Moreover, the ellipsoid165

社会技術研究論文集 Vol.14, 156-166, June 2017

REFERENCES

to using large trucks to deliver both inbound and outbound.
Furthermore, we show that networks are robust when the
demand becomes uncertain or unknown. Here, we tested fivedifferent demand scenarios in each network based on the actual
number of evacuees during post disaster. After solving for theuncertain demand by using robust optimization, the results
prove that the structural networks have an effect on the model
robustness. The two hierarchies for facilities provide morerobustness than the single hierarchy case. Moreover, theuncertain demand model is more robust than the deterministic
demand model. Accordingly, we conclude that the Robust
Optimization has an advantage in solving uncertain models.
In addition, we improve the distribution network by routing
design. The improved facility location with routing has anadvantage in saving approximately 30 percent of the total
delivery cost. Therefore, this study can help the decision makerplan for the efficient post disaster distribution network when
demand turn out to be uncertainty.
The historical data was considered an example because of
the simplicity entailed by the real geography. The model in thisstudy represents the majority of demand. Therefore, thelimitations of this study involve the data analysis and network
assumptions. The results and findings are rather effects withthis data. The two layers of facility network and the large truck
size operation assumptions are probably an advantage when
they match with the right demand. However, we tested themodel with five demand scenarios, when they become less andhigh, and results from these assumptions and operations ensurethat the model responds properly. The model provides the same
trend of objective values. In addition, according to the model is
MISOCP thus it is not able to guarantee global optimization.
However, we use Gurobi solver, which has better
performance20).
Finally, we discuss the interrelated aspects to improve thiswork in the future as follows: (1) we are developing the
integrated Location-Routing problem (LRP) explained by
Hamidi et al.21), in which the LRP integrates location, allocation,
and also routing problems to design the network efficiency; (2)we have not considered other parameters that could fluctuateduring humanitarian logistics, for example supply amount, unittransportation cost, facility opening cost, etc. Therefore, notonly the uncertain demand but also such kind of parameters
should be considered simultaneously; (3) future work could
also consider the multi-objective facility location routing
problem. The model should be more reasonable byinvestigating both cost and time indicators simultaneously.
Once the uncertainty of demand is incorporated in the model,
related robustness is evaluated.

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4)
5)

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11)
12)
13)
14)

15)
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17)
18)

19)
20)
21)

166

Van Wassenhove, L.N. Blackett Memorial Lecture: Humanitarian
aid logistics: supply chain management in high gear, Journal of the
Operational Research Society, 57, 475-489, 2006.
Lin, Y.-H., Batta, R., Rogerson, P., Blatt, A., Flanigan, M. Locationof temporary depots to facilitate relief operations after anearthquake, Socio-Economic Planning Sciences, 46, 112-123,
2012
Holguín-Veras, J., Taniguchi, E., Jaller, M., Aros-Vera, F., Ferreira,F., and Thompson, R. The Tohoku disasters: Chief lessons
concerning the post disaster humanitarian logistics response andpolicy implications. Transportation Research Part A, 69, 2014, pp.
86–104.
Baron, O., Milner, J., & Naserldin, H. Facility Location: A Robust
Optimization Approach. Production and operations Management,
20(5), 2011.
Min, H., Jayaraman, V., & Srivastava, R. Combined
location-routing problems: A synthesis and future research
directions. European Journal of Operational Research, 108(1),1-15, 1998.
Nagurney, A. SCH-MGMT 597LG Humanitarian Logistics and
Healthcare Spring 2012, Presentation, 2012
Synder, L.V. Facility location under uncertainty: A review. IIE
Transactions, Vol. 38(7), pp. 537-554, 2006.
Miyagi Prefectural Government. Earthquake Damage Information,
Report (in Japanese), 2012.
Hosoya, R., Sano, K., Ieda, H., Kato, H., Fukuda, A. Evaluation of
Logistic Policies in the Tokyo Metropolitan Area Using A
Micro-Simulation Model for Urban Goods Movement. Journal of
the Eastern Asia Society for Transportation Studies, Vol.5, 2003.
http://myhome.nifty.com/rent/
Bramel, J., Simchi-Levi, D. The Logic of Logistics: Theory,
Algorithms, and Applications for Logitics Managemen, New York,USA, 1999
Indra-payoong, N. Discrete Optimization in Transport andLogistics. Bangkok, Thailand (in Thai), 2005
Caunhye, A.M., Nie, X., Pokharel, S. Optimization models inemergency logistics: A literature review, Socio-Economic
Planning Sciences, 46, 4-13, 2012.
Mete, HO., Zabinky, ZB. Stochactic optimization of medical
supply location and distribution in disaster management,International Journal of Production Economics, 126(1), 76-84,
2010.
Ben-Tal, A., Nemirovski, A. (2000) Robust solution of Linear
Programming problems contaminated with uncertain data. Math.
Program. 88, 411-424
Kouvelis, P., & Yu, G. (1997) Robust discrete optimization and its
applications. Kluwer Academic Publishers, Norwell, MA.
Bertsimas, D., Brown, D.B. (2009) Constructing Uncertainty Sets
for Robust Linear Optimization. Operations Research, 57(6), p.
1483-1495.
Josef, K. (2004) Solving Planning and Design Problems in theProcess Industry Using Mixed Integer and Global Optimization.
Special Edition of Annals of Operations Research, State-of-the-Art
IP and MIP, p. 31-61
Ben-Tal, A., Bertsimas, D., Brown, D.B. (2010) A Soft Robust
Model for Optimization under Ambiguity. Operations Research,
58(4), 2(2), 1220-1234.
http://www.gurobi.com/products/features-benefits
Hamidi, M., Farahmand, K., & Sajjadi, S. R. Modeling afour-layer location-routing problem. International Journal ofIndustrial Engineering Computations, 3(1), 43-52, 2012a.

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