Fuzzy multi-criteria emergency service centers selection problem in extreme environment

Fuzzy multi-criteria emergency service centers selection
problem in extreme environment
Gia Sirbiladze, Bidzina Midodashvili and Bidzina Matsaberidze
Department of Computer Science, I. Javakhishvili Tbilisi State University, Tbilisi, Georgia
March 21-23, 2019

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In this work we consider one fuzzy multi-objective optimization problem for
selection of emergency facility locations.
Optimal location planning for service centers and timely servicing from emergency
service centers to the demand points are key tasks of the emergency
management system.
We develop a fuzzy multi-attribute decision making approach for the service
centers location selection problem under extreme environment for which a fuzzy
probability aggregation operators’ approach is used.

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The formation of expert’s input data for construction of attributes is an important
task of the centers’ selection problem. To decide on the location of service
centers, it is assumed that a set of candidate sites already exists. This set is
denoted by = {1
, 2
, … ,
}, where we can locate service centers and
let = 1
, 2
, … ,
be the set of all attributes which define CCs selection.
For example:
S1 -- ”post disaster access by public and special transport modes
to the candidate site”;
S2 -- ” post disaster security of the candidate site from accidents, theft and
vandalism”;
S3 -- ” post disaster connectivity of the location with other modes of
transport (highways, railways, seaport, airport etc.)”;
S4 -- ”costs in vehicle resources, required products and etc. for the
location of CCs in candidate site”;

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Let us assume that = 1
, 2
, … ,
is the set of all demand points. Let
�
= {�
1
, �
2
, … , �

} be the fuzzy weights of attributes. For each expert
from
invited group of experts (emergency service dispatchers) = {1
, 2
, … ,
} let
�

be the fuzzy positive rating (see Table 1) in triangular fuzzy numbers (TFN)
of his/her evaluation for each candidate site
, = 1, … , m, with respect to each
attribute 𝑗𝑗
, j = 1, … , n.
Table 1. Fuzzy terms and their fuzzy ratings.

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In fuzzy set theory conversion scales are applied to transform fuzzy terms into
TFNs. We use a rating scale of 1 - 9 as a semantic form of fuzzy terms of linguistic
variables such as attribute's valuation, importance value of an attribute and
interaction index of attributes.
Our task is to build aggregation operators’ approach, which for each candidate site

, = 1, … , m, aggregates presented objective and subjective data into scalar
values – site’s selection ranking index. This aggregation we define by the TFCA
(Triangular Fuzzy Choquett Averaging) operator:
̃

≡ ̃

= 𝑇𝑇(�
𝑖
, �
𝑖
, … , �
𝑖𝑖
),
where (�
𝑖
, �
𝑖
, … , �
𝑖𝑖
) are aggregated arguments of experts’ evaluations –
(�
𝑖
, �
𝑖
, … , �
𝑖𝑖
).

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The proposed framework of location planning for candidate sites consists of the
following steps (the complete presentation of the scheme is limited here):
Step 1: Selection of location attributes.
Step 2: Selection of candidate location sites
Step 3: Locations evaluation using fuzzy aggregation approach.
Step 3.1: Assignment of ratings to the attributes with respect to the candidate
sites.
Step 3.2: Computation of aggregated fuzzy ratings for the attributes and the
candidate sites. Aggregation of fuzzy weights of attributes.
Step 3.3: Computation and Normalization of the aggregated fuzzy decision matrix.
Step 4: Identification of constructive TFVFM which takes into account attributes
aggregated fuzzy importance values and attributes fuzzy interactions.
Step 5: Computation of selection ranking index of candidate sites by the TFCA
operator.

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Let us consider the fuzzy multi-objective optimization model for the location set
covering problem.
Let = 1
, 2
, … ,
be a Boolean decision vector, which defines some selection
from candidate sites = {1
, 2
, … ,
} for facility location. We can build sites’
selection ranking index as a linear sum of triangular fuzzy values - ̃
𝑗𝑗
𝑗𝑗
. As a
result, we obtain fuzzy objective function for selection ranking index of candidate
sites ∑𝑗𝑗=1
̃
𝑗𝑗
𝑗𝑗
. Its maximization selects a group of sites with the maximal
selection ranking index from admissible covering selections.
A classical facility location set covering problem “tries” to minimize the total cost
needed for opening of service centers - ∑𝑗𝑗=1
𝑗𝑗
𝑗𝑗
, where 𝑗𝑗
is a cost necessary for
opening of the candidate site - 𝑗𝑗
.

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Assume that we know in advance the approximate number of people needed to
make a candidate site operate as a service center. This number is denoted by

, = 1, 2, … , . Our goal is to locate service facilities centers with minimal
number of agents needed to operate the opened service centers - ∑𝑗𝑗=1
𝑗𝑗
𝑗𝑗
.
The problem aims at locating service facilities in minimal travel time from
candidate sites. In extreme environment for emergency planning a radius of
service center is defined based not on distance but on maximum allowed time for
movement, since the rapid help and servicing is crucial for demand points in such
situations.
Suppose that experts evaluated movement fuzzy times between demand points
and candidate sites ̃
𝑗𝑗
, cc-costs 𝑗𝑗
and number of cc-agents 𝑗𝑗
.

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Correspondingly, a set of candidate sites
, covering customer

∈ = 1
, 2
, … ,
, is defined as

= {𝑗𝑗
∈ : 𝑃𝑃𝑃𝑃𝑃𝑃( ̃
𝑗𝑗
where is a minimal possibilistic level (selected by administrators) which
covering condition ̃
𝑗𝑗
Thus we can state fuzzy multi-objective facility location set covering problem:
max

1
= ∑𝑗𝑗=1
̃
𝑗𝑗
𝑗𝑗
,
min 2
= ∑𝑗𝑗=1
𝑗𝑗
𝑗𝑗
, (1)
min 3
= ∑𝑗𝑗=1
𝑗𝑗
𝑗𝑗
,
∑∈
𝑗𝑗
≥ 1, (2)
= 1, 2, … , ; = 1, 2, … , ; 𝑗𝑗
∈ 0, 1 .

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We illustrate the effectiveness of the constructed fuzzy optimization model by the
numerical example.
Let us consider an emergency management administration of Tbilisi city (Georgia)
that wishes to locate some fire stations with respect to timely servicing of critical
infrastructure objects. Assume that there are 30 demand points (critical
infrastructure objects) and 8 candidate facility sites (fire stations) in the urban
area.
Let us have 4 experts from Emergency Management Agency (EMA) of the city of
Tbilisi for evaluation of the travel times and ratings values of candidate sites,
interaction indexes and importance values of attributes.

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The travel times between demand points and candidate sites are evaluated in
triangular fuzzy numbers (evaluated in minutes, omitted because the matrix of
fuzzy travel times has large dimensions). According to the standards of the EMA,
let the principle of location fire stations be that the fire station can reach the area
edge within 5 minutes after receiving the dispatched instruction. Therefore, we set
covering radius = 5 minutes. Minimal possibilistic level is equal to 0.9.
Based on the semantic form (Table 1) experts’
, (k = 1, 2, 3, 4), ratings �

for
each candidate site
, (i = 1,…,8), with respect to each attribute 𝑗𝑗
, (j = 1,…,10),
interaction indexes between attributes and weights for each attribute are
aggregated by the steps 1- 5 (all data are omitted).

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A software is developed, in which the steps 1-5, mathematical programing
problem (1)-(2) and -constraint approach are realized.
Movement fuzzy times between demand points and candidate sites ̃
𝑗𝑗
, i=1, 2, …,
30; j = 1, 2, …, 8, evaluated by experts, define the subsets of service demand
points 𝑗𝑗
, j = 1, 2, …, 8.
At the end, the Fuzzy Multi-objective Emergency Service Facility Location Set
Covering Problem (1) – (2) is constructed, where the matrix of covering
constraints B is a concatenation of vectors 𝑗𝑗
in which the covering of demand
point is presented by “1” and opposite by “0” (omitted here because of large
dimensions).

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The problem takes the following form
1
= ̃
1
1
+ ̃
2
2
+ ̃
3
3
+ ̃
4
4
+ ̃
5
5
+ ̃
6
6
+ ̃
7
7
+ ̃
8
8
⇒ max ,
2
= 351
+ 472
+ 553
+ 394
+ 705
+ 626
+ 467
+ 578
⇒ min ,
3
= 271
+ 192
+ 313
+ 184
+ 235
+ 296
+ 257
+ 208
⇒ min ,
≥ 1,1,1,1,1,1,1,1 ,
≡ 1
, … , 8
, 𝑗𝑗
∈ 0, 1 , = 1, … , 8,
where fuzzy selection ranking indexes ̃

of candidate sites are given by Table 2.

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Table 2: Fuzzy selection ranking indexes of candidate sites
1
δ 2
δ 3
δ 4
δ 5
δ 6
δ 7
δ 8
δ
0,0845 0,0822 0,1181 0,0868 0,0951 0,0653 0,1127 0,1169
0,4926 0,4903 0,4991 0,5513 0,4772 0,4497 0,5386 0,4777
1,4506 1,4386 1,4275 1,4752 1,4045 1,3892 1,5820 1,3238

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After checking all 28 = 256 possible variants of constraints of the problem we
receive the following twelve coverings:
: {1
, 5
, 8
}, : {2
, 3
, 7
}, : {3
, 4
, 7
},
𝐶𝐶: 3
, 5
, 8
, 𝐶𝐶: 3
, 6
, 7
, : 3
, 6
, 8
,
: 3
, 7
, 8
, : 4
, 6
, 7
, 𝐶𝐶: 1
, 2
, 5
, 7
,
𝐶𝐶: 1
, 5
, 6
, 7
, 𝐶𝐶: 2
, 4
, 5
, 8
,
𝐶𝐶: 4
, 5
, 6
, 8
.

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In the TFCA aggregation example there exist 6 Pareto solutions - {Cov1, Cov2,
Cov3, Cov6, Cov7, Cov8}.
It is clear that increasing the service centers selection ranking index of covering in
Pareto solutions gives more worse level of the second objective function - the total
cost needed for opening of service centers or of the third objective function -
number of agents needed for operating the opened service centers.
The decision on the choice of fire stations as service centers depends on the
decision making person’s preferences with respect to risks of administrative
actions. Of course, fire elimination is an extreme process and the reliability of
choice of service centers is envisaged in Pareto solutions.

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Notwithstanding, if we had to consider the problem without TFCA aggregations
(interacting attributes), i.e. we would have two-criteria (2
, 3
) classical FLSC
problem, then we would have only one Pareto solution Cov3, having minimal cost
and number of agents. This covering also enters into Pareto solution of fuzzy-
FLSC problem, but its selection ranking index is rather low.
As we see the candidate sites’ selection ranking index aggregates not only direct
factors with their interactions, but also experts’ evaluations of such factors as
closeness to demand points and central bases, quality of infrastructure, security
issues and so on. These factors are decisive in planning distribution networks in
extreme environments.

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Thank you for your attention!

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Fuzzy multi-criteria emergency service centers selection problem in extreme environment

Fuzzy multi-criteria emergency service centers selection problem in extreme environment

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Fuzzy multi-criteria emergency service centers selection problem in extreme environment

Fuzzy multi-criteria emergency service centers selection problem in extreme environment

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