Reengineering of the Emergency Service System under Generalized

Disutility

Marek Kvet and Jaroslav Janacek

Faculty of Management Science and Informatics, University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovakia

Keywords: Emergency Medical Service, System Reengineering, Generalized Disutility, Radial Formulation.

Abstract: Emergency medical service system structure is defined by deployment of service providing centers, number

of which is usually limited. The objective of the designer is to minimize the total discomfort of all system

users. Thus, the problem often takes the form of a weighted p-median problem. Since population and

demands for service change in time and space, current service center deployment may not meet the

requirements of the users and service providers neither. In this paper, we introduce a mathematical model

for system reengineering under the generalized disutility, which follows from the idea that the individual

user’s disutility comes from more than one located service center. At the moment of current demand

occurrence, the nearest service center may be unavailable due to satisfying another arisen demand.

Presented approach constitutes an extension of previously developed methods, where only the nearest center

was taken as a source of individual user’s demand satisfaction.

1 INTRODUCTION

The emergency service systems as the medical

emergency system, system of fire brigades and

police stations are designed for given geographical

area to satisfy the demand of population living in the

area for more secure life. The associate service is

provided from a given number of service centers and

the overwhelming objective used for the design

evaluation is the average time necessary to deliver

service from a service center to the user location, at

which a demand for service has occurred.

Host of models consider that serviced population

is concentrated to a finite number of dwelling places

of the considered area. Frequency of the demand

occurrence is proportional to the number of

inhabitants of the given town or village. A finite set

of possible service center locations is assumed and

also, the assumption is made that a user demand is

serviced from the nearest located service center.

This way, the weighted p-median problem

formulation is used to the emergency service system

design and solving the underlying problem to

optimality (Current et al., 2002, Doerner et al., 2005,

Ingolfsson et al., 2008, Jánošíková, 2007). The

original model was based on the location-allocation

decision variables and constraints (Current et al.,

2002), where an occurring demand is assigned to

exactly one possible center location. As concerns

usage of a general IP-solver, 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 of

computer 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. Then another approach using so-called radial

formulation was developed to avoid the particular

assignment of user’s locations to the located service

centers. The radial approach successfulness 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). To accelerate the

p-median problem solving process, an approximate

approach has been developed (Janáček and Kvet,

2013). This approach uses an approximation of a

common time distance between a service center

location and a user by some pre-determined time

Kvet, M. and Janacek, J.

Reengineering of the Emergency Service System under Generalized Disutility.

DOI: 10.5220/0006621400850093

In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems (ICORES 2018), pages 85-93

ISBN: 978-989-758-285-1

distances and gives near to optimal results in the

case of integer time distances.

A bit different situation occurres, when

reengineering of a current emergency service system

is performed. The necessity of system updating

ussualy follows from the fact that distribution of

demands for service has been developping in time

and space and thus, the originally determined center

locations do not suit both serviced population and

providers operating the service centers. Contrary to

the original system design, the current service

providers suggest changes in the center deployment

and their suggestion may be in a conflict with public

interests. That is why the system administrator

permits system reengineering only subject to some

formal rules, which are intented to prevent

worsening the service accessibility. The considered

formal rules are quantified by a maximal number of

provider’s centers, which are allowed to change their

locations and by the maximal distance between a

current center location and a possible new location.

Generally, addition of constraints may considerably

spoil the computational time necessary to obtain the

optimal solution of the problem. The study (Kvet

and Janáček, 2016) showed, that they do not impact

the computational time, when a user demand is

serviced from the nearest located center.

In this paper, we deal with more general model

of the emergency medical system design under

reengineering. We assume that service of a user

demand is provided from the nearest center only if

the center is not occupied by servicing a former

demand. Otherwise, the user’s demand is serviced

from the nearest unoccupied center. Initial

emergency system design considering the failing

centers was studied by (Snyder and Daskin, 2005)

and the associated radial formulation was presented

in (Kvet, 2014). Nevertheless, the reengineering of

service system with failing centers has not been

studied yet. Therefore, we focus on the influence of

the formal rule constraints on best possible service

availability in the service system and on the

associated computational process convergency.

In this paper, we provide a reader with a radial

model of emergency service system reengineering

with failing centers under rules imposed by the

system administrator. We perform a computational

study to find whether real-sized instances of the

problem are solvable using a common IP-solver.

The remainder of the paper is organized as

follows. The next section is devoted to the radial

model formulation, in which temporarily failing

centers are considered. In Section 3, the

administrator auxiliary rules are introduced. Section

4 contains a description of experiments. The

conclusion summarizes obtained findings and

contains possible directions of a further research.

2 REENGINEERING OF A

SERVICE SYSTEM WITH

FAILING CENTERS

To describe the problem of the users’ disutility

minimization by changing the deployment of centers

belonging to one considered provider, we introduce

J as a finite set of all users (dwelling places), where

denotes a volume of expected demand of user jJ.

Let I be a finite set of possible center locations.

Symbol d

denotes the integer network time distance

between locations i and j, where i, j I



J. The

maximal relevant distance is denoted by m. The

current emergency service center deployment is

described by two disjoint sets of located centers I



and I



I, where I

contains p centers of the

considered provider, which performs updating of his

part of the system and I

is the set of centers

belonging to the other providers. Locations from I

stay unchanged. The center locations from I

can be

relocated within the set I

= I- I

In this paper, the generalized disutility perceived

by a user is modelled by a sum of weighted time

distances from the r nearest located centers. The

probabilities q

for k=1..r are positive real values,

which meet the following inequalities q

≥ q

≥ …

≥ q

and depend only on the order of distances from

the user to the r nearest centers. The k-th value can

be proportional to the probability of the case that the

k-1 nearest centers are occupied and the k-th nearest

center is available (Jankovič, 2016, Snyder and

Daskin, 2005).

We introduce coefficients a

for each pair i, j

iI and jJ, where a

= 1 if and only if d



s and

= 0 otherwise for s= 0, 1, …, m-1.

To describe decisions on new center deployment,

we introduce series of decision variables, where

binary variable y

defined for each iI

takes the

value of one, if a service center is to be located at i

and it takes the value of zero otherwise. To express

the total distance necessary for user demand

satisfaction, we introduce auxiliary zero-one

variables x

jsk

for jJ, s0, ..., m-1, k1, ..., r to

model the disutility contribution value of the k-th

nearest service center to the user j. The variable x

jsk

takes the value of 1 if the k-th smallest disutility

contribution for the customer jJ is greater than s

and it takes the value of 0 otherwise. Then the

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

expression x

j0k

+ x

j1k

+ … + x

jm-1k

constitutes the k-th

smallest distance from the user j to a located center.

If this k-th smallest distance is denoted by d

ik(j)

, then

the expression of d

ik(j)

by the auxiliary 0-1 variables

jsk

is clearly reported on the following figure.

Figure 1: Expression of the k-th smallest distance from the

user j to a located center by the auxiliary 0-1 variables x

jsk

for s0, ..., m-1.

Using the above introduced structures and

decision variables, we suggest the following model.

 





 Jj

jskkj

xqbMinimize

0 1

(1)

pytoSubject







(2)

1...,,0,









msJjfor

rayax

ijjsk

(3)

Iifory  }1,0{

(4)

rkmsJjfor

jsk

...,,1,1...,,0,

}1,0{





(5)

The objective function (1) expresses the expected

volume of transportation performance denoted by

generalized disutility in this paper. Constraint (2)

preserves constant number of centers belonging to

the considered part of the emergency service system

under reengineering. For given pair of user j and a

distance value s, constraints (3) assure relation

between the set of location variables y

, i I

and the

sum of auxiliary variables x

jsk

over range 1, …, r of

subscript k. If no center is located in the radius s,

then the sum of auxiliary variables x

jsk

equals to r. If

exactly k ≤ r centers is located in the radius s, then

the sum of variables equals to r-k due to

minimization process, which presses down the

values of the variables x

jsk

. If the sum of variables

jsk

equals to k<r, then the variables x

js1

, …, x

jsr-k

equal to zero and remaining variables equal to one

due to the used optimization process and decreasing

values of the coefficients q

, …, q

The objective function value of the optimal

solution of the problem (1)-(5) gives expected total

length or time of trips from the service centers to the

demand locations necessary for satisfaction of all

demands for service. This objective function value

explanation holds subject to assumption that the

coefficients q

, …, q

correspond to the probability

values expressing that the k-th nearest center is the

first available (unoccupied) service center. The next

assumption is that demand volume b

is proportional

to the number of trips necessary for the demand

satisfaction. The model (1)-(5) is much more

realistic than the original approach based on the

simple weighted p-median problem, which

corresponds to the case of r=1. The bigger accuracy

of the model (1)-(5) is paid for by higher complexity

of the solved problem, which may issue to enormous

increase of computational time. A question emerges

here, which limit of accuracy presented by the value

of r pays off regarding the increase of computational

time. As a solution of the problem (1)-(5) is discrete

and the values of probabilities q

sharply decrease,

influence of increasing value of r may appear

negligible behind some limiting value r*.

3 REENGINEERING UNDER

AUXILIARY CONSTRAINTS

As mentioned in Section 1, the administrator of the

system sets up parameters of rules to prevent a

designer of new center deployment from increasing

provider’s benefit at the expense of the system users.

The rules are easy to evaluate and check. That is

why the studied rules have a simple form. The first

rule limits the total number w of the provider’s

centers, which can be moved. The second rule limits

the distance between original and new location of a

service center.

To be able to formulate the rules in a concise

way, we derive several auxiliary structures using

Figure 2. We assume that all points 1-11 represent

system users and the black points 2, 3, 9 and 11

represent current service center locations.

Figure 2: Simple example of reengineering restrictions.

Let N

={iI

: d



D} denote the set of all

possible center locations, to which the center tI

can be moved. If we consider the example depicted

on Figure 2, we can observe that the center located

Reengineering of the Emergency Service System under Generalized Disutility

at the point 9 can be moved to 6, 8, 10 and 13 or stay

unchanged. Thus, the set N

= {6, 8, 9, 10, 13}.

Similarly, N

= {3, 4, 6}. Additionally, let S

={tI

iN

} denote a set of all centers of the considered

provider, which can be moved to iI

. Here S

= {3,

9}. Realize that tN

and iS

for tI

and iI

and

thus I



Now, we introduce series of decision reallocation

variables. The variable u

{0, 1} for tI

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. Using the above introduced structures and

variables we suggest the following model extension.

wpy







(6)

Itforu







(7)

Iiforyu







(8)

tLti

NiItforu  ,}1,0{

(9)

Constraint (6) limits the number of changed

center locations by the constant w. Constraints (7)

allow moving the center from the current location t

to at most one other possible location in the radius

D. Constraints (8) 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.

Based on our experience, we have to raise the

question of technical solvability of the formulated

problem (1)-(9). We ask whether a commercial

solver based on the branch-and-bound technique is

able to find the exact solution of a real-sized

problem in acceptable time. Consequence of

structural constraint addition to some model is

always matter of question from the point of

computational process convergence. Furthermore,

we have to realize that even if the administrator’s

rules are established to defend users’ interests, they

represent further restriction of the set of feasible

solutions. This phenomenon may lead to a less

possible benefit (higher disutility) for the average

user. That is why, the dependence of the optimal

objective function value on setting of parameters w

and D is worth to study.

4 NUMERICAL EXPERIMENTS

This section is devoted to the results of numerical

experiments performed in the optimization software

FICO Xpress 8.0, 64-bit. The experiments were run

on a PC equipped with the Intel® Core™ i7 5500U

2.4 GHz processor and 16 GB RAM.

Used benchmarks were derived from real

emergency health care system, which was originally

implemented in eight regions of Slovak Republic.

For each self-governing region, i.e. Bratislava (BA),

Banská Bystrica (BB), Košice (KE), Nitra (NR),

Prešov (PO), Trenčín (TN), Trnava (TT) and Žilina

(ZA), all cities and villages with corresponding

number of inhabitants b

were taken into account.

The coefficients b

were rounded to hundreds. 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 vary from 87 to

664 locations. In all solved instances, the network

distance from a user to the located center was taken

as the user´s disutility.

An individual experiment was organized so that

the optimal solution of the reengineering problem

(1)-(5) was obtained first. The value of r was set to 7

and the associated coefficients q

for k=1, …, r were

set in percentage in the following way: q

= 77.063,

= 16.476, q

= 4.254, q

= 1.593, q

= 0.47, q

0.126, and q

= 0.018. These values follow from a

simulation model of existing emergency medical

service system in Slovakia (Jankovič, 2016). To

enrich the pool of benchmarks, for each self-

governing region ten instances were created in such

a way that they differ in the list of located service

centers operated by the considered provider. The

average results are summarized in Table 1. The left

part of this table contains the basic benchmark

characteristics. The total number of possible service

center locations regardless the service providers is

reported in the column denoted by |I|. The value of

TNC represents the total number of located centers.

The average percentage rate of the provider’s centers

is reported in the column denoted by “Prov. [%]”.

The right part of the table contains the results of the

model (1)-(5). The average computational time in

seconds of ten instances solved for each region is

reported in the column denoted by “CT [s]”. The last

column “ObjF” contains the average values of the

objective function (1).

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

Table 1: Average results of numerical experiments for

each self-governing region. The value of r was set to 7.

Reg.

|I|

TNC

Prov. [%]

CT [s]

ObjF

55.1

0.5

28087.8

515

44.9

43.6

47706.5

460

46.0

30.4

48490.9

350

50.7

10.8

52024.6

664

44.3

50.5

61070.2

276

52.9

5.1

36800.9

249

49.6

6.1

43986.1

315

46.8

6.1

45341.2

The results indicate that the reengineering of the

emergency service system under generalized

disutility for r=7 from the point of service provider

does not represent a hard solvable problem. It can be

observed that the radial formulation enables to get

the optimal solution within 1 minute. In spite of this

useful feature, the second portion of experiments

was performed to find out, whether a lower value of

r will have significant influence on the resulting

solution from the point of the objective function

value. As we have mentioned in Section 2, we

assume that the influence of increasing value of r

may appear negligible behind some limiting value

r*. To verify this hypothesis and to find a suitable

value of r*, we have solved all instances with

different values of r. If r<7, then the coefficient q

was computed according to (10) as a complement of

the coefficients q

for k=1, …, r-1 to the value of

100, i.e. the sum of q

for k=1, …, r must equal 100.

100









(10)

The dependency of average computational time

on the value of r was studied first. We assume that

the computational time grows with increasing value

of r, because it affects the number of variables and

the model size as well. Our expectation has been

confirmed by the results summarized in Table 2.

Each row represents the average results of ten

instances for each region and the columns are used

for different setting of parameter r. The last row

contains the average values of all instances. The

dependency of average computational time on the

value of r is shown also on Figure 3.

Table 2: Average computational time in seconds of the

solving process depending on r for each region.

Reg.

r = 1

r = 2

r = 3

r = 4

r = 5

r = 6

0.1

0.2

0.3

6.5

8.3

11.5

19.3

27.1

33.7

6.0

7.0

8.9

11.6

15.7

18.3

2.1

2.6

3.6

6.7

6.8

8.4

20.6

22.8

26.1

31.8

38.8

47.0

1.4

1.8

3.2

2.8

3.4

4.3

1.2

1.7

2.3

3.1

4.2

5.2

1.7

2.0

2.6

3.3

4.4

5.2

AVG

4.96

5.79

7.29

9.87

12.58

15.29

Figure 3: Dependency of average computational time in

seconds on the number r.

When studying the impact of r on the resulting

system design, we have evaluated Hamming

distance of the vectors of resulting location variables

obtained for various values of parameter r.

Generally, Hamming distance of two 0-1 vectors y

and z is defined by the expression (11). The average

results are reported in Table 3.

 

( , )

HD y z







(11)

Table 3: Average Hamming distance from the optimal

solution obtained for r=7 computed for each region.

Reg.

r = 1

r = 2

r = 3

r = 4

r = 5

r = 6

5.2

2.0

0.6

0.4

0.0

12.6

11.0

3.6

0.6

0.4

0.0

11.8

6.0

3.4

1.4

0.6

10.2

6.6

2.0

0.4

0.0

11.8

7.4

4.0

0.0

0.6

0.0

8.0

2.6

0.6

0.2

0.0

8.4

3.8

0.4

1.4

0.0

12.4

4.0

3.2

1.0

0.2

0.0

AVG

10.05

5.43

2.23

0.68

0.23

0.08

The dependency of average Hamming distance

from the optimal solution obtained for r=7 on the

number r of service providing centers for each

system user is shown also on Figure 4.

Reengineering of the Emergency Service System under Generalized Disutility

Figure 4: Dependency of Hamming distance from the

optimal solution obtained for r=7 on the number r of

service providing centers for each system user.

The reported results show that the suitable value

of r* is 3. Thus, 3 nearest located service centers are

enough to be taken into account when emergency

system reengineering under generalized disutility.

As shown, the service center deployment for r=3

differs from the service center deployment obtained

for r=7 only in one center on the average.

The last characteristics studied in this portion of

experiments consists in the objective function value.

For each system design obtained for particular value

of r=1, 2, …, 6, the objective function (1) with r=7

and the full set of coefficients q

was computed. This

value was compared to the objective function value

obtained for r=7 and the gap was evaluated. Here,

the gap is defined as a percentage difference of two

objective function values, where the objective

function value for r=7 was taken as the base. The

average values of gaps of ten instances computed for

each self-governing region are reported in Table 4,

which follows the structure of previous tables. To

find a suitable value of r*, the gaps lower than 0.1

percent are marked.

Table 4: Average gap from the optimal solution obtained

for r=7.

Reg.

r = 1

r = 2

r = 3

r = 4

r = 5

r = 6

2.52

0.23

0.02

0.01

0.00

6.19

0.43

0.07

0.00

2.88

0.21

0.11

0.01

0.00

2.31

0.55

0.06

0.00

5.19

0.62

0.04

0.00

2.81

0.24

0.04

0.01

0.00

2.60

0.32

0.01

0.02

0.00

4.33

0.69

0.05

0.00

AVG

3.73

0.43

0.05

0.01

0.00

The dependency of average gap from the optimal

solution obtained for r=7 on the number r of service

providing centers for each system user is shown also

on Figure 5.

Figure 5: Dependency of average gap from the optimal

solution obtained for r=7 on the number r of service

providing centers for each system user.

The detailed analysis of presented results shows

that if we use three nearest service providing centers

for each system user instead of seven, we can obtain

very similar results and save more than one half of

computational time. That is why, the next portion of

experiments was performed with r=3. This new

portion of experiments was aimed at the additional

constraints (6)-(9) and their impact on the solving

process characteristics, mainly computational time.

This new set of experiments was organized so

that the maximal radius D, in which a center can be

moved, was fixed at the value of 15 and the maximal

number w of centers allowed to change their

locations was set to p/4, p/2, 3p/4, and p

respectively. It must be realized, that the parameter p

represents the total number of centers operated by

considered provider, who performs reengineering.

Dependency of average computational time in

seconds computed for 10 instances of each region is

reported in Table 5. Each row of the table represents

one region and the columns are devoted to different

settings of w.

Table 5: Average computational time in seconds for each

region and different values of w. Parameter D was 15.

Reg.

w = p/4

w = p/2

w = 3p/4

w = p

0.11

0.12

4.36

6.40

5.36

5.32

4.44

5.95

5.77

5.09

2.02

2.20

2.75

2.78

9.78

9.76

9.79

9.83

1.55

1.64

1.68

1.73

1.30

2.30

1.47

1.52

1.74

1.65

1.63

1.66

AVG

3.16

3.75

3.57

3.50

The reported results show that different settings

of w do not significantly affect the computational

process, because the value of w limits only the

number of possible service center location changes

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

and thus, the number of variables and constraints is

independent on w.

The objective function values can be compared

in Table 6. Even if parameter r was set to 3 in all

solved models, the objective function values were

computed for r=7 based on the resulting service

center deployment.

Table 6: Average objective function values for each region

and different values of w. Parameter D was 15. The

objective function value was recomputed for r=7 and the

whole set of probability coefficients q

Reg.

w = p/4

w = p/2

w = 3p/4

w = p

28607.0

28334.8

50676.2

50433.7

50430.4

51141.3

50916.9

50913.4

53995.8

53482.7

53471.5

63791.3

63532.1

63526.0

37286.6

37225.5

45670.3

44915.7

44733.6

47278.1

46673.0

46634.3

The last portion of experiments was aimed at

exploration of the impact of parameter D on the

solving process complexity. Here, the parameter w

was set to its maximal value p, i.e. all centers

operated by the provider could change their current

locations. The average computational times in

seconds computed for each self-governing region

and given values of D are reported in Table 7, which

has the same structure as previous tables.

Table 7: Average computational time in seconds for each

region and different values of D. Parameter w was set to

its maximal value, i.e. w=p.

Reg.

D = 5

D = 10

D = 15

D = 20

D = 25

0.04

0.08

0.12

0.16

0.17

0.90

3.13

5.32

10.54

15.38

1.02

2.61

5.09

7.41

8.81

0.42

1.11

2.78

6.91

5.31

1.91

4.93

9.83

15.85

19.06

0.40

1.05

1.73

2.16

2.96

0.28

0.79

1.52

1.99

2.44

0.45

0.97

1.66

2.14

2.81

AVG

0.68

1.83

3.50

5.89

7.12

The results reported in Table 7 have confirmed

our expectation that the parameter D has a direct

impact on the computational process. As it can be

observed, the average computational time grows

with increasing value of D, i.e. with increasing

radius, in which current center can be removed. This

phenomenon has a simple explanation. The bigger is

the radius for center location change, the higher is

the number of its possible new locations. As we can

see in constraints (6)-(9), this parameter defines the

number of decision variables and it directly affects

the model size. Therefore, the solving process for

higher distance D takes longer time. Finally, the

dependency of objective function value on the

parameter D is shown in Table 8.

Table 8: Average objective function values for each region

and different values of D. Parameter w=p. The objective

function value was recomputed for r=7 and the whole set

of probability coefficients q

Reg.

D = 5

D = 10

D = 15

D = 20

D = 25

29563.0

28798.3

28334.8

28255.0

28136.1

52115.0

50635.4

50430.4

49429.6

49130.0

52111.9

51398.4

50913.4

50412.6

49959.5

56153.7

54360.2

53471.5

52674.5

52422.9

66115.8

65081.5

63526.0

63070.1

62444.1

37714.0

37320.5

37225.5

37148.4

37009.5

46162.7

45395.9

44733.6

44114.7

44078.5

48712.7

47763.3

46634.3

46230.4

46115.0

As far as the objective function value expressed

by generalized disutility is concerned, the achieved

results indicate that the higher is the value of D, the

better solution can be obtained. The radius D defines

the set of all new possible locations of a center and

thus, it affects the possibility for obtaining better

results. More elements in the set N

for each tI

mean more candidates for new center locations and

bigger possible change of current center deployment,

which can bring better service accessibility for

system users.

All the experiments presented above were aimed

primarily at studying the model solvability and the

sensitivity of the associated computational process

on different model parameters. Besides some

interesting findings and suitable settings of

parameters, we also present the emergency system

characteristics in the next paragraphs. The following

table contains the comparison of current service

center deployment to the results of suggested

reengineering model, which was configured as

follows. Based on the above presented results, the

parameter r was set to 1 (simple disutility) and 3

(generalized disutility). In the experiments with the

generalized disutility, the associated probability

coefficients q

= 77.063, q

= 16.476 and q

= 6.461

were used. It must be noted that the objective

function reported in the table was recomputed for

r=7 and the whole set of probability values reported

at the beginning of this section. The maximal

number w of centers operated by the considered

service provider, which are allowed to change their

current location, was set to the cardinality of the set

, i.e. all considered provider’s centers could be

Reengineering of the Emergency Service System under Generalized Disutility

moved. The value 15 limited the radius D, in which

a center could be relocated. This initial value of D

corresponds to the rule applied in the emergency

health care system of the Slovak Republic (Kvet and

Janáček, 2016). Table 9 contains the average results

of 10 instances solved for each self-governing

region. The objective function value corresponding

to the current service center deployment is reported

in the column denoted by “Current ObjF”. The right

part of the table is dedicated to the results of

suggested reengineering problems. The abbreviation

“ObjF” denotes the objective function value of the

emergency system design obtained by solving the

reengineering model. Finally, the value of Imp was

computed to show possible improvement of the

objective function value expressed by the

generalized disutility, which can be achieved by

relocating of some service centers. Its value was

computed as a percentage difference between

objective function values of the current service

center deployment and the new system design

resulting from the model. The objective function

value of current deployment was taken as the base.

Table 9: Comparison of current service center deployment

to the results of reengineering model for r=1 (simple

disutility) and r=3 (generalized disutility). The

reengineering parameters were set at w = p and D = 15.

Reg.

Current

ObjF

r=1

r=3

ObjF

Imp. [%]

ObjF

Imp. [%]

29792

28810

3.30

28335

4.89

52510

51094

2.70

50430

3.96

52786

51894

1.69

50913

3.55

56759

54440

4.09

53472

5.79

67037

65807

1.83

63526

5.24

38625

38091

1.38

37226

3.62

472163

45569

3.48

44734

5.25

49324

47566

3.56

46634

5.45

The reported results show that the emergency

system reengineering may bring a considerable

improvement of service accessibility for system

users expressed by general disutility. The average

values of Imp indicate that the objective function

value corresponding to the system design can be

reduced up to 6 percent. The achieved results also

confirm the usefulness of suggested reengineering

model, because it enables us to obtain better system

design from the point of service accessibility. It is

obvious from the comparison of the case r=1 to r=3

that the usage of generalized disutility leads to such

solutions, which are approximately by 2 percent

better than those, which can be obtained by usage of

simple disutility model.

5 CONCLUSIONS

This paper was focused on emergency medical

system reengineering under generalized disutility,

which follows the idea that the associated service

can be provided from more than one nearest located

centers. Presented generalized disutility makes the

model more realistic by taking into account possible

temporarily unavailability of service centers. In our

computational study we have found, that three

nearest located centers are enough to be considered

in the objective function value, because the accuracy

of the result is satisfactory. The second part of

experiments was aimed at additional constraints,

which define some new restrictions to service center

location changes. Based on reported results we can

conclude that we have constructed a very useful tool

for emergency medical system reengineering under

generalized disutility from the point of service

provider. Presented model is easy to be implemented

and solved in common optimization environment

equipped with the branch-and-bound method or

other technique to integer programming problems.

Future research in this field may be aimed at

using of the suggested modelling technique in such

situations, where the time distances are influenced

by randomly occurring failures in the underlying

transportation network.

ACKNOWLEDGEMENTS

This work was supported by the research grants

VEGA 1/0518/15 “Resilient rescue systems with

uncertain accessibility of service”, VEGA 1/0463/16

“Economically efficient charging infrastructure

deployment for electric vehicles in smart cities and

communities”, and APVV-15-0179 “Reliability of

emergency systems on infrastructure with uncertain

functionality of critical elements”.

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