Robust Emergency System Design using Reengineering Approach

Marek Kvet and Jaroslav Janáček

Faculty of Management Science and Informatics, University of Žilina,

Univerzitná 8215/1, 010 26 Žilina, Slovakia

Keywords: Robust Emergency System Design, Detrimental Scenarios, Reengineering Approach.

Abstract: A robust emergency service system is usually designed so that the deployment of given number of service

centers minimizes the maximal value of objective functions corresponding with the specified detrimental

scenarios. If the problem is solved by any solving technique based on the branch-and-bound method, the min-

max link-up constraints cause bad convergence of the associated computational process. Within this paper,

we try to overcome the drawback following from the link-up constraints by usage of an iterative process

applied to a series of surrogate problems. The surrogate problems represent a simple emergency system

reengineering under a given scenario and chosen values of reengineering parameters. The results of the

surrogate problems are used for considerable reduction of the initial set of possible service center locations.

The robust emergency service system is obtained as the optimal solution of the reduced problem. We provide

the reader with a comparison of the original min-max problem solution to the suggested approach.

1 INTRODUCTION

The emergency system design problem is a

challenging task for system designer focusing on

satisfaction of future demands of the system users in

case of emergency. Emergency service system

performance is considerably influenced by

deployment of the service centers, which send

emergency vehicles to satisfy demands on service at

the system users’ locations. The number of service

providing centers is usually limited. As the quality

characteristic of the design corresponds to an average

response time of the system on a demand raised by

a user, then the emergency service system design can

be tackled as the weighted p-median problem, which

was studied in (Current et al., 2002, Ingolfsson et al.,

2008, Jánošíková, 2007, Snyder and Daskin, 2005).

As far as the usage of a general IP-solver is

concerned, the size of the solved integer

programming problem must be taken into account. In

the real problems, the number of serviced users takes

the value of several thousands, and the number of

possible service center locations can take this value

as well (Avella et al., 2007). The number of possible

service center locations seriously impacts the

computational time and the memory demands due to

used branch-and-bound method, which stores the

unfathomed nodes of the inspected searching tree for

the further processing. That is why the direct attempt

at solving the problem described by a location-

allocation model often fails, when larger instances are

solved by a commercial IP-solver. Mentioned

weakness has led to the development of so-called

radial approach, successfulness of which is based on

the fact that there is only finite set of radii, which must

be taken into account (Elloumi et al., 2004, García et

al., 2011, Janáček, 2008). Simultaneously, several

heuristic and approximate approaches have been

developed to get a good solution of the problem in a

short time (Doerner, K.F., et al. 2005, Gendreau, M.

and Potvin, J., 2010).

When the emergency service system is designed,

the designer must take into account that the response

time might be impacted by various random events

caused by partial disruptions of the road network.

That is why; the system resistance to such critical

events is demanded. Host of approaches to increasing

the emergency system resistance (Correia and

Saldanha da Gama, 2015, Kvet and Janáček, 2017b,

Pan et al., 2014, Scaparra and Church, 2015) are

based on incorporating possible failure scenarios into

model of the robust service system design problem.

Focusing on the objective function value of the robust

system design, the most frequently used objective

function consists in minimizing the maximal

objective function of the individual instances

172

Kvet, M. and Janá

cek, J.

Robust Emergency System Design using Reengineering Approach.

DOI: 10.5220/0008946401720178

In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 172-178

ISBN: 978-989-758-396-4; ISSN: 2184-4372

corresponding with particular scenarios. It follows

that the min-sum objective function used in the

classical weighted p-median problem is replaced by

the min-max criterion. The associated min-max

model uses link-up constraints to limit from above the

individual scenario min-sum objectives by their upper

bound corresponding to the objective function of the

min-max model. In addition, incorporating

the scenarios into the mathematical programming

model causes that the model magnifies its size

proportionally to the cardinality of the scenario set.

Both the model structure and its magnified size

represent an undesirable burden of the computational

process of most available IP-solvers. Thus,

complementary approximate approaches to the

robustness constitute a big challenge to operational

researchers and professionals in applied informatics

(Janáček and Kvet, 2017, Kvet and Janáček, 2017a,

Kvet and Janáček, 2017b). In this paper, we present

an attempt to the robust emergency system design.

We based our approximate approach on replacing the

computational process of the huge original problem

solution with a series of solving processes of the

much simpler problems. Each of the simpler

problems represents the problem of reengineering

(Brotcorne, L. et al., 2003, Guerriero, F. et al., 2016,

Schneeberger, K. et al. 2016) of some original service

center deployment under a given scenario (Kvet and

Janáček, 2018). This approach enables us to identify

the most important changes in the service center

deployment to react on the individual scenarios.

Having inspected all considered scenarios, we can

reduce the set of possible center locations and then

solve much smaller min-max problem with the

original set of scenarios.

The remainder of this paper is organized as

follows: Section 2 is devoted to the description of

original min-max robust design of emergency system,

in which all scenarios and possible center locations

are taken into account either as fixed for center

location or either free or forbidden for locating

a service center. The approach based on system

reengineering process used for identification of

the important locations is explained in Section 3. The

next section contains a description of the complete

iterative approach. The fifth Section contains the

overview of performed numerical experiments and

yields brief comparative analysis of designed service

center deployments. The results and findings are

summarized in Section 6.

2 ROBUST EMERGENCY

SYSTEM DESIGN PROBLEM

The robust emergency system design problem can be

modelled using the following data structures and

decision variables. Symbols J and I will denote the set

of users’ locations and the set of possible service

center locations respectively. The set I will be

partitioned into three subsets F1, F0 and V, where set

F1 contains the locations, in which a service center

must be located. Set F0 consists of center locations,

where no center can be temporarily located, and V is

the set of possible locations, from which p service

centers must be chosen. Symbol b

denotes the

number of users sharing the location j. Symbol U

denotes the set of considered failure scenarios.

The response time following from the distance

between locations i and j under a specific scenario

uU is denoted as d

iju

. In this paper, we consider that

each value of d

iju

is integer and less than or equal to

the maximal value D

max

. As we want to make the

system resistant to the individual detrimental

scenarios, the objective function of the robust system

design turns into minimizing the maximal objective

function of the individual scenarios.

Complexity of location problems with limited

number of facilities to be deployed and the necessity

to solve large instances of the problem led the radial

formulation of the problem, which could considerably

accelerate the associated solving process (Kvet and

Janáček, 2017b). As this concept proved to be a

suitable tool, we decided to use the radial formulation

also for the robust emergency system design.

To complete the associated mathematical model,

we introduce the following decision variables. The

variable y

{0,1} models the decision on service

center location at the location iV. The variable takes

the value of 1 if a service center is located at i and it

takes the value of 0 otherwise. In the robust problem

formulation, the variable h denotes the upper bound

of the objective functions over the set U of scenarios.

Let us define v= D

max

-1. Next, auxiliary zero-one

variables x

jsu

for s = 0 … v and uU are introduced

to complete the radial model. The variable x

jsu

takes

the value of 1, if the response time of the nearest

service center to the user at j  J under the scenario

uU is greater than s and it takes the value of 0

otherwise. Then the expression x

j0u

+ x

j1u

+ … + x

jvu

constitutes the value of response time d

ju*

under the

scenario uU. We introduce a zero-one constant a

iju

under the scenario uU for each triple [i, j, s], where

iV



F1, jJ, s[0..v]. The constant a

iju

is equal to

1, if the response time d

iju

of a center located at i on a

Robust Emergency System Design using Reengineering Approach

173

user located at j is less than or equal to s, otherwise

iju

is equal to 0. Then the model, in which the

maximum of the objective function values over

the set U is minimized, follows.

inimize h

(1)

,0,1 ,

jsu iju i iju

iV iF

Subject to x a y a

for j J s , ,v u U







 





(2)







(3)

jjsu

jJ s

bx hforuU









(4)

{0, 1}

for i V



(5)

0,0,1,

jsu

for j J s , ,v u U 

(6)

0h 

(7)

The objective function (1) represents the upper

bound of all objective function values over the

individual scenarios. The constraints (2) ensure that

the variables x

jsu

are allowed to take the value of 0, if

at least one center is located in radius s from the user

location j. The constraint (3) limits the number of

service centers located in V by p. The link-up

constraints (4) ensure that each perceived disutility is

less than or equal to the upper bound h. As the

obligatory constraints (6) are concerned, only values

zero and one are expected in any feasible solution.

Nevertheless, it can be seen that the model has

integrality property concerning the variables x

jsu

. It

follows that the relevant values of x

jsu

in the optimal

solution will be equal to one or zero without imposing

binary constraints upon these variables.

For purpose of conciseness, we introduce the

denotation of the set of resulting (optimal) service

center locations as IR(U,F1,F0). The set includes

service centers from both F1 and V.

3 EMERGENCY SERVICE

SYSTEM REENGINEERING

PROBLEM

The emergency system reengineering was originally

studied in (Kvet and Janáček, 2018), where the radial

model of the problem was also employed. The basic

idea follows from the analysis of current service

center deployment, which may not be optimal due to

changing demands and development of the

underlying transportation network.

To describe the problem of the system average

response time minimization by changing the

deployment of centers belonging to a given sub-set of

the located centers. Let I be a finite set of all possible

center locations. As above, the response time

following from the distance between locations i and j

under a specific scenario uU is denoted as d

iju

. The

current emergency service center deployment is

described by union of two disjoint sets of located

centers L and F1, where I

contains p centers under

reconstruction and F1 is the set of fixed centers.

The center locations from I

can be relocated within

the set I- F1-F0, where F0 is the set of temporarily

forbidden locations.

When reengineering of an emergency service

system is performed, the administrator of the system

sets up parameters of rules to prevent a designer of

new center deployment from changes, which can be

perceived by system users as obnoxious. We consider

two formal rules within this study. The first rule limits

the total number w of the centers, which locations can

be changed. The second rule limits the time distance

between current and newly suggested location of

a service center by the given value D. To be able to

formulate the rules in a concise way, we derive

several auxiliary structures.

Let N

={i I- F1-F0: d



D} denote the set of all

possible center locations, to which the center tL can

be moved subject to limited length of the move.

Additionally, symbol S

={tL: iN

} denotes a set of

all centers of L, which can be moved to i I- F1-F0

subject to the mentioned limitation. Now, we

introduce series of decision reallocation variables,

which model the decisions on moving centers from

their original positions to new ones. The variable

{0, 1} for tL and iN

takes the value of one, if

the service center at t is to be moved to i and it takes

the value of zero otherwise.

For the given scenario uU the problem can be

formulated as follows.

jJ s

inimize b x







(8)

10 1

,0,1

js iju i iju

iIF F iF

Subject to x a y a

for j J s , ,v

  











(9)

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

174

iIF F

 





(10)

ypw







(11)

ufortL







(12)

ti i

uy foriIFF







(13)

{0, 1} ,

ti t

ufortLiN

(14)

{0, 1} 1 0

yforiIFF

(15)

0,0,1

for j J s , ,v

(16)

As the constraints (9), (10) and formulae together

with used decision variables were explained in the

previous section, we restrict explanation only on

remainder of the above model. Constraint (11) limits

the number of changed center locations by

the constant w. Constraints (12) allow moving the

center from the current location t to at most one other

possible location in the radius D. Constraints (13)

enable to bring at most one center to a location i

subject to condition that the original location of the

brought center lies in the radius D. These constraints

also assure consistency among the decisions on move

and decisions on center location.

To be concise in the next explanation, we

introduce the denotation of the set of resulting

(optimal) service center locations for given scenario

u as I(u,F1,F0,L,w,D). The set also includes the

service centers from F1.

4 APPROXIMATE APPROACH

TO THE ROBUST SERVICE

SYSTEM DESIGN

To formulate the suggested approximate algorithm

for solving of the robust service system design, we

employ the procedures IR(U,F1,F0) and

I(u,F1,F0,L,w,D) introduced in Section 2 and Section

3 respectively. We assume that the data structures J,

}, I, U, {d

iju

} introduced in Section 2 are given and

the number p of centers to be located is also known.

As the set U of scenarios contains one special

scenario b corresponding to standard conditions, we

start the process of designing the robust system with

solving the weighted p-median problem for time

distances {d

ijb

}. This problem can be described by the

model (8)-(10), (15), (16).The resulting set of p center

locations will be denoted as L.

Then we set parameters w and D of the algorithm

at chosen values from the ranges [1 .. p] and [1 .. D

max

]

respectively. We set F1=



and F0=



. The

suggested algorithm consists of two following steps.

1. For each u



U-{b} compute the set

One(u)= I(u,F1,F0,L,w,D).

2. Set F1=

⋂

()

∈{}

and

F0=−

⋃

()

()

and compute

Output= IR(U,F1,F0).

The objective function value of the resulting

center deployment Output can be enumerated

according to (17).



()maxmin: :

j iju

Output b d i Output u U

















(17)

5 NUMERICAL EXPERIMENTS

The presented numerical experiments were focused

on comparison of the presented approximate

approach to the exact method of the robust emergency

system design from the points of computational time

and the solution accuracy. We performed the

numerical experiments using the optimization

software FICO Xpress 8.3 (64-bit, release 2017). The

experiments were run on a PC equipped with the

Intel® Core™ i7 5500U processor with the

parameters: 2.4 GHz and 16 GB RAM.

The used benchmarks were derived from the real

emergency health care system, which was originally

implemented in seven regions of Slovak Republic.

For each self-governing region from the following list

of region names followed by their abbreviations, all

cities and villages with corresponding population b

were taken into account. The mentioned list contains

Bratislava (BA), Banská Bystrica (BB), Košice (KE),

Nitra (NR), Prešov (PO), Trenčín (TN), Trnava (TT)

and Žilina (ZA). The coefficients b

were rounded to

hundreds. In the benchmarks, the set of communities

represents both the set J of users’ locations and the set

I of possible center locations as well. The cardinalities

of these sets are reported in Table 1 together with the

number p of located centers. The network time -

Robust Emergency System Design using Reengineering Approach

175

distances from a user to the nearest located center

were derived from the real transportation network.

Due to the lack of scenario benchmarks for the

experiments, the problem instances used in our

computational study were created in the way used in

(Janáček and Kvet, 2016). There were selected one

quarter of matrix rows so that these rows

corresponded to the biggest cities concerning the

number of system users. Then same of them were

chosen randomly and the associated time distance

values were multiplied by the randomly chosen

constant from the numbers 2, 3 and 4. The rows,

which were not chosen by this random process, stay

unchanged. This way, 10 different scenarios were

generated for each self-governing region.

The first series of experiments was performed so

that the exact model (1) - (7) was used to obtain the

optimal solution of the robust emergency system

design problem. The achieved results of the first

series are reported in Table 1. The computational

times in seconds are given in the column denoted by

CT and the optimal objective function values are

reported in the column denoted by ObjF

robust

Table 1: Results of the exact approach for robust service

system designing applied on the self-governing regions of

Slovakia.

Region |I| p CT ObjF

robust

BA 87 9 52.8 25417

BB 515 52 1605.0 18549

KE 460 46 1235.5 21286

NR 350 35 11055.1 24193

PO 664 67 3078.2 21298

TN 276 28 616.6 17535

TT 249 25 563.8 20558

ZA 315 32 1304.7 23004

The next series of experiments was performed

with the goal to find a suitable setting of the

parameters w and D used in the approximate

approach, see model (8) – (16) of the reengineering

process. For this study, the benchmark Žilina (ZA)

was used. In this portion of experiments, the

parameter p was set at the value 32 reported in Table

1. The maximal radius D was fixed at one of the

values 5, 10, 15, 20 and 25 and the maximal number

w of centers allowed to change their locations was set

to p/4, p/2, 3p/4, and p respectively. The results of this

series of experiments are summarized in Table 2.

Table 2: Detailed results of numerical experiments for the

self-governing region of Žilina: computational study of the

impact of individual parameters on the results accuracy.

w D CT SCT [%] ObjF

approx

gap [%] HD

8 5 231.6 82.25 23411 1.77 14

8 10 430.9 66.98 23377 1.62 14

8 15 502.8 61.46 23236 1.01 6

8 20 516.8 60.39 23359 1.54 10

8 25 529.5 59.42 23359 1.54 10

16 5 234.8 82.00 23411 1.77 14

16 10 455.1 65.12 23377 1.62 14

16 15 516.0 60.45 23236 1.01 6

16 20 529.9 59.38 23359 1.54 10

16 25 542.6 58.41 23359 1.54 10

24 5 232.5 82.18 23411 1.77 14

24 10 434.6 66.69 23377 1.62 14

24 15 508.5 61.02 23236 1.01 6

24 20 531.3 59.28 23359 1.54 10

24 25 534.6 59.02 23359 1.54 10

32 5 231.6 82.25 23411 1.77 14

32 10 427.8 67.21 23377 1.62 14

32 15 509.2 60.97 23236 1.01 6

32 20 531.4 59.27 23359 1.54 10

32 25 534.8 59.01 23359 1.54 10

Each row of the table corresponds to one setting

of the parameters w and D. In this portion of

experiments, several characteristics were studied.

The computational time in seconds is reported in the

column denoted by CT. Since the approximate

approach proved to be much faster than the exact one,

the percentage save of computational time SCT was

computed. Here, the computational time of the exact

approach was taken as the base. Furthermore, the

objective function associated with the obtained

service center deployment is reported in the column

denoted by ObjF

approx

. To evaluate the accuracy of

suggested approximate method, the value of gap was

also computed. It expresses the difference between

the objective function values of the exact and

approximate models. The objective value of the exact

approach was taken as the base. The value of gap is

reported also in percentage. Finally, the resulting

service center deployments were compared in the

terms of Hamming distance of the vectors of location

variables y. This value is denoted by HD.

It can be seen that the lowest computational time

of the approximate method was reached for the

settings w = p/4 and D = 5. As the associated gap was

acceptable, we used this setting in the third series of

experiments, which was performed for each self-

governing region. The obtained results are reported in

Table 3, where the same denotations as in Tables 1

and 2 were used. The table is divided into two

sections denoted by EXACT and APPROXIMATE.

The first section contains the results from Table 1 for

bigger comfort of the readers. The second section

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

176

contains the results obtained by the approximate

approach for each of considered benchmarks.

Table 3: Comparison of the approximate approach to the

exact approach applied on the self-governing regions of

Slovakia. Parameters of the approximate approach were set

in this way: w = p/4, D = 5.

EXACT APPROXIMATE

CT ObjF

robust

CT ObjF

approx

gap HD

BA 52.8 25417 11.8 26197 3.07 4

BB 1605.0 18549 679.8 18861 1.68 12

KE 1235.5 21286 633.4 21935 3.05 16

NR 11055.1 24193 274.2 24732 2.23 14

PO 3078.2 21298 1601.9 21843 2.56 20

TN 616.6 17535 223.1 17851 1.80 10

TT 563.8 20558 152.4 20980 2.05 10

ZA 1304.7 23004 231.6 23411 1.77 14

6 CONCLUSIONS

This paper was focused on mastering dimensionality

of the robust emergency system design problem using

commercial IP-solver. The robustness follows the

idea of making the system resistant to various

randomly occurring detrimental events. The original

approach with the min-max objective function value

proved to be extremely time consuming due to the

fact, that the min-max link-up constraints cause bad

convergence of the branch-and-bound method. This

obstacle can be overcome by presented approximate

solving method, which is based on reengineering

approach applied on individual scenarios. The

approximate approach enables to obtain the resulting

robust service center deployment in the

computational time, which is much less than half of

the computational time demanded by the exact

approach. As concerns the accuracy of the resulting

solution, it can be observed that the approximate

method is very satisfactory. Thus, we can conclude

that we have presented a very useful tool for robust

service system designing.

The future research in this field could be aimed at

other approximate techniques, which will enable to

reach shorter computational time under the

acceptable solution accuracy. Another future research

goal could be focused on mastering the presented

problem with larger set of detrimental scenarios.

ACKNOWLEDGEMENT

This work was supported by the research grants

VEGA 1/0342/18 "Optimal dimensioning of service

systems", VEGA1/0089/19 “Data analysis methods

and decisions support tools for service systems

supporting electric vehicles” and APVV-15-0179

"Reliability of emergency systems on infrastructure

with uncertain functionality of critical elements".

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