Use of memetic algorithms to solve a stochastic location model: health resources for diabetics in some provinces of
Castilla-León
USE OF MEMETIC ALGORITHMS TO SOLVE A
STOCHASTIC LOCATION MODEL: HEALTH
RESOURCES FOR DIABETICS IN SOME PROVINCES OF
CASTILLA-LEÓN
Jesús F. Alegre 1, Ada Álvarez 2, Silvia Casado 1 and Joaquín A. Pacheco 1
1 Department of Applied Economics, University of Burgos,
2 Facultad de Ingeniería Mecánica y Eléctrica,
Universidad Autónoma de Nuevo León, México;
RESUMEN
This paper applies a memetic algorithm, MA, to solve a stochastic facility
location problem. This MA is constructed by combining a genetic algorithm with a local
search operator. The problem consists on determining the best locations in which to
place health resources where patients who have suffered a diabetic coma can be
attended. We compare our memetic algorithm with a genetic algorithm. Experimental
results show that our memetic algorithm obtains the best results.
Keywords: Genetic algorithm, local search, location, memetic algorithm,
stochastic model.
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1. INTRODUCTION
In this work, a hybrid metaheuristic strategy to solve a special facility location
problem is proposed.
The objective of this work consists on determining the sites to install resources
where patients who have suffered a diabetic coma can be attended, as well as the
assignment of potential customers to those resources.
It is widely known that if a patient does not recover from a hypoglycemia, he/she
loses consciousness, he/she goes into a diabetic coma and he/she has to be attended in a
short period of time which is named critical time (for example less than 20 minutes),
otherwise damage may be devastating (neuropathy, blindness, amputations, etc.).
Therefore, if the patient is treated rapid enough, no serious damage will appear, on the
contrary, while it more delays the medical aid over the critical time more severe will be
the damages, including death (see www.fundaciondiabetes.org).
In a deterministic situation the model would be described by the classical
Maximum Set Covering Problem (MSCP) (Love et al 1988). This model deals with the
problem of locating facilities under deterministic conditions. In order to address the
situation previously described we have introduced in this model the probability of a
patient suffer a diabetic coma and the probability of a patient that have gone into a
diabetic coma gets permanent damage.
For solving this special facility location problem we propose a hybrid
metaheuristic based on a memetic algorithm, a hybrid of genetic algorithm and local
search. We compare this hybrid algorithm with a genetic algorithm.
Computational tests have been done in order to test the efficiency of our hybrid
metaheuristic using the instances of the well-known library OR-lib (Beasley, 1990) and
real data refer to some provinces of Castilla y Leon, Spain. Because of these provinces
have a limited budget less than 10 facilities are considered.
The paper is organized as follows: in the next section we describe the model
formulation and
section 3 describes the hybrid algorithm proposed as well as its
elements. In Section 4 we show the computational results with hypothetical instances
and with real data. Finally, Section 5 is devoted to the conclusions.
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Use of memetic algorithms to solve a stochastic location model: health resources for diabetics in some provinces of
Castilla-León
2. MODEL FORMULATION
In a deterministic situation the model would be described by the classical
Maximum Set Covering Problem (MSCP). The traditional maximum covering location
model, which minimizes the number of patients who are attended in a period of time
longer than the critical time, may be formulated as follows (Love et al 1988):
Minimize
∑ q ·r
i
(1)
i
i
subject to:
∑z
j
=l
(2)
j =1..n
∑
z j + ri ≥ 1
i = 1…m;
(3)
zj ∈ {0,1}
j = 1...n;
(4)
ri ∈ {0,1}
i = 1…m;
(5)
j / tiji ≤ t _ critical
where: m = number of demand nodes; n = number of potential facility sites; l =
number of facilities to be located; qi = demand in node i, i=1...m.;
t ij
i
= time to arrive
from node i to its assigned facility at node ji; t_critical = maximum time to attend
patients; zj =1 if any facility is located at node j; zj =0 if a facility is not located at
node j; ri=1 if the facility assigned to node i is more than t_critical; ri=0 otherwise
The former model deals with the problem of locating facilities under
deterministic conditions. In order to address more realistic situations we have
introduced randomness in the following way: Let Yi be the random variable that
represents the number of patients who suffer a diabetic coma in node i (i=1...m). We
assume that this variable follows a binomial distribution with parameters qi and π, the
probability of diabetic coma for a patient, Yi ∼ Bin(qi , π).
The probability φ of permanent damage for a patient who suffers a diabetic coma
depends on the time t elapsed between he/she suffers a diabetic coma and he/she is
attended in the closest health centre; specifically let be φ (t ) =
(t ) 2
, if t <
(t _ critical ) 2
t_critical; and 1 otherwise.
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Jesús F. Alegre, Ada Álvarez, Silvia Casado and Joaquín A. Pacheco
φ(t)
1
Diabetic coma
t_ critical
t
Figure 1.- Probability of permanent damage
Let’s denote φi = φ (tiji ) , that is, φi is the probability of permanent damage in a
patient with diabetic coma in location i. Let Xi be the random variable that represents
the number of patients that after a diabetic coma suffer permanent damage in location i.
This variable depends on Yi in the following way:
h
p (Xi = a / Yi = h) =  φ ia (1 − φ i ) h − a
a
In the problem of locating facilities with the goal of minimizing the expected
value of the number of patients who suffer permanent damage as a consequence of a
diabetic coma, the objective function can be stated as follows:
m
Minimize
∑ E( X )
i
(6)
i =1
qi
Note that E(Xi) =
∑ E( X
i
/ Yi = h)· p(Yi = h) . Therefore it is easy to find that
h =1
E(Xi) = φi qi π and the objective (6) can be written as
Minimize
m
m
m
i =1
i =1
ri =1
i =1
ri = 0
∑ φ i qi π = ∑ qi π + ∑ φ i q i π
(7)
Because of the presence of φi in the objective function (7), the constraints (2)-(5)
are not enough for the complete formal description of the problem and new constraints
and variables should be added to ensure the assignment of each location i to its closest
facility (ji). Nevertheless, we will not describe the more complex mathematical
formulation, because metaheuristic strategies will be used to solve it, and these
strategies are not based on the mathematical formulation. Note that each solution is
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Use of memetic algorithms to solve a stochastic location model: health resources for diabetics in some provinces of
Castilla-León
perfectly defined by the set of l locations where facilities are installed, and the objective
function can be easily calculated from this set by (a) finding, for every location i, its
closest facility ji and b) calculating the values φi that appear in the objective function .
If we compare the objective function of the deterministic model (1) and the
objective function of this last problem (7) we can conclude that in (1) only the
minimization of the “not-covered patients” (who are further than t_critical from their
closest facility)is considered. But, even though, the risk of permanent damage for the
“cover patients” is less, it is important to take it into account. For this reason we argue
that (7) is more complete and real than (1).
3. MEMETIC ALGORITHM
Memetic algorithms ,MAs, (Moscato, 1989) is a population –based approach for
optimization that tries to synergetically use all possible information about the problem
being addressed. This knowledge is generally, but not restricted to, some sort of
previously-known heuristics or local search algorithms.
MAs are metaheuristics that take advantage of evolutionary operators in
determining interesting regions of the search space, as well as the local search operator
rapidly finding good solutions in small regions of the search space (Merz and
Freisleben, 1999). This method is proved to be of practical success in a variety of
problem domains and in particular for the approximate solution of NP optimization
problems.
In the last few years , many researches have become more attracted to this
methodology as a way out of some limitations present in other approaches. This is
particularly the case with genetic algorithms (GAs) since MAs are orders of magnitude
faster than GAs in some problem domains. For these reason , MAs are sometimes called
hybrid genetic algorithms, Lamarkian evolutionary algorithms , or knowledgeaugmented. Other denominations include genetic local search, parallel genetic
algorithms, or genetic search agents. These denominations are misleading but the MAs
are easily recognizable from the so-called “standars GAs” since in MAs the new
solutions (or “children”) are created form highly evolved “parents” of previous
generations.
In this paper we propose the following memetic algorithm for the special
facility location problem described previously:
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Jesús F. Alegre, Ada Álvarez, Silvia Casado and Joaquín A. Pacheco
Memetic Algorithm
Generate individuals for the initial population
Apply local search to improve each individual
Repeat
Select pairs of parents for crossover
Crossover: Produce children from each pair of parents
Perform mutation on each child.
Apply local search to each child
Population replacement: Select children for the population in the next
generation
Until predefined number of generations are completed.
The implementation of MA is described below:
Generate individuals for the initial population : The initial population (n_pob
solutions) is generated randomly .
Apply local search to improve each individual: Our improvement method is
based on two procedures designed by Mladenovic et al. (2001), named Alternate and
Interchange procedures. They were adapted by Mladenovic et al. from the well-known
heuristics of Mulvey and Beck (1984) for the p-median problem to the p-center
problem.
Alternate procedure
Repeat
For each facility j of S, determine the subset of points U j ⊂ U that have j
as the closest facility
Solve the 1-center problem for each subset U j
Let S’ be the set of solutions of these l problems, and f’ its value
If f’ < f let S = S’ and f = f’
until no more changes take place in S
Interchange procedure
Repeat
For each j ∈ V \ S and k ∈ S calculate the value of the objective function
vjk if the facility moved from k to j
Calculate vj*k* = min { vjk / j ∈ V-S and k ∈ S }
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Use of memetic algorithms to solve a stochastic location model: health resources for diabetics in some provinces of
Castilla-León
If vj*k* < f then, let S = S-{k*}, S = S ∪ {j*} and f = vj*k*
until no further improvement
(where vjk is the value of the objective function if the facility moved from k to j).
Parent selection: a number of n_sel parents are selected with a probability
proportional to the fitness. The fitness of an individual is the difference between the
maximum value of the objective function in the population and the value of this
function for this individual.
Crossover: It’s an uniform crossover. Let. S and S’ the two parent solutions
selected: S = (V, F, V,..., F ) y S’ = (F, F, V, ..., V ). The children are composed with the
common genes or elements between the two parent solutions and the rest of the
elements are obtained randomly with one restriction: the total number of facilities must
be l .
Mutation: The genes of each new child can change or mutate with a low
probability, p_mut. A random value between (0,1) is generated. If this value is smaller
than p_mut the change is done. It consists on locating a facility in this place if there isn,t
or remove it otherwise. Because of the total number of resources is l, a change V→F
leads to another F→V and vice versa.
Improvement of the “children solutions”: After generating children solutions
they are improved applying the procedure of local search , that is the addition of
alternate procedure plus interchange procedure.
Population replacement: “Children solutions” are inserted in the population
replacing the worst “parent solutions”.
4. COMPUTATIONAL RESULTS
In order to assess the efficiency of the strategy suggested for this problem,
computational tests were carried out. We have compared our memetic algorithm (it is a
hybrid algorithm between a genetic algorithm and a local search procedure) and a
genetic algorithm. These tests were conducted on a Pentium IV a 2Ghz computer.
In order to measure the quality of the solutions obtained by our algorithm we
have implemented a Genetic algorithm (GA). The implementation of these algorithms
(GA and our MA) was done in Pascal and compiled using the Delphi 5.0 compiler.
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Jesús F. Alegre, Ada Álvarez, Silvia Casado and Joaquín A. Pacheco
Genetic algorithm generates randomly a initial population of solutions and then
uses the same crossover and mutation operators than memetic algorithm previously
described.
4.1. Hypothetical instances
Firstly, we have used instances from the OR-Lib for p-median problem
corresponding to values l≤ 10. Instances with l ≤10 have been chosen because this work
is aimed at solving problems with real data and the health authorities cannot consider
more than 10 new units due to budget restrictions.
For these computational experiences we have considered t_critical=15, 20 and
25 and π=1/10; qi is generated randomly for values ranging form 1 to 100 (these values
are available for interested readers).
In these examples U = V, which means that the facility locations match users.
Table 1 shows the following results: Columns (GA) and (MA) show the expected value
of the number of patients who suffer permanent damage, as a consequence of a diabetic
coma, obtained by each metaheuristic strategy. Columns (T.B.GA) and (TBMA) show
computational time (in seconds) till best solution is found by GA and MA respectively.
The total computational time for these strategies was limited to 10 seconds (without any
improvement).
Table 1. Results for the instances of OR-Lib with t_critical=15, 20 and 25 and
l≤10 (U=V)
Or
1
1
1
2
2
2
3
3
3
6
6
6
7
7
7
11
11
11
8
t_critical
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
GA
378
355
331
284
251
209
296
244
203
683
559
400
487
335
195
872
554
285
TBGA
30
1.19
1.25
2.55
2.44
2.48
2.72
2.48
2.42
4.91
5.25
5.09
9.34
9.80
9.27
4.37
3.69
6.30
MA
378
355
328
284
251
209
295
244
203
683
556
397
480
326
195
726
412
203
TBMA
0.05
0.44
0.02
0.14
0.39
0.14
0.03
8.28
0.05
2.16
0.14
7.50
7.11
10.23
0.87
0.26
10.55
1.78
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Use of memetic algorithms to solve a stochastic location model: health resources for diabetics in some provinces of
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12
12
16
16
16
17
17
17
21
21
21
22
22
22
26
26
26
27
27
27
31
31
31
32
32
32
35
35
35
36
36
36
38
38
38
39
39
39
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
792
479
232
918
416
160
725
289
99
909
307
87
872
320
85
762
231
64
562
147
32
546
90
13
459
86
13
291
45
8
375
62
12
288
45
11
196
29
10
10.14
4.12
1.36
1.70
1.64
6.01
10.33
9.97
8.30
7.12
6.70
5.94
8.70
10.20
10.28
8.02
7.27
0.61
1.55
10.67
10.41
3.33
10.37
2.28
8.61
6.98
7.94
8.13
5.91
0.30
5.39
4.00
0.64
10.23
0.59
4.36
9.42
2.14
10.47
490
220
94
610
253
82
400
132
31
616
170
37
458
115
21
452
112
22
281
55
6
243
27
1
196
22
1
161
18
1
145
14
0
124
13
0
51
2
0
3.53
5.84
0.48
2.56
1.55
6.78
8.30
2.55
1.53
1.50
0.80
10.69
3.27
5.06
6.28
4.56
0.80
4.89
12.97
4.59
9.88
1.30
1.16
5.86
11.41
12.92
11.75
1.84
3.64
8.75
6.70
11.19
8.80
7.47
3.80
5.84
11.66
9.80
1.31
In table 1 we can observe that MA obtains the best results in all the cases.
4.2. Real Data
In this sub-section we show the results obtained from the tests performed with
actual problems. The data refer to different provinces of Castilla Leon, northern Spain.
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Jesús F. Alegre, Ada Álvarez, Silvia Casado and Joaquín A. Pacheco
The data were supplied by the Local Public Health Authorities (www.jcyl.es). The
objective is to analyze the best locations for a series of diabetes units that are chosen
from a set of potentially suitable towns.
A time matrix (in minutes) was created with the source locations, and the ones
that could potentially host the diabetes services. To calculate traveling times, we used
road data provided by the Spanish Center of Geographical Information (CNIG), and we
estimated different speeds depending on the kind of road (national, regional, local roads,
etc.). The road network data were used to calculate the time matrix using Dijkstra’s
algorithm. The data employed (source and destination towns, time matrix, diabetes
cases per location, etc.) are available to readers upon request.
In these real cases U≠V. The values of m and n for each one and the results are
shown in Table 2. The value of critical time that has been considered is t_critical=20.
The results and the computational time to obtain the best solution by each
algorithm (in these case tabu search and memetic algorithms )are shown in Table 2.
Table 2. Results for real instances and t_critical=20
M
n
l
MA
TBMA
Ávila
248
156
5
44
0.44
Ávila
248
156
7
6
3.00
Ávila
248
156
10
0
5.19
Burgos
452
152
5
59
0.78
Burgos
452
152
7
29
4.72
Burgos
452
152
10
12
4.08
León
211
184
5
300
1.02
León
211
184
7
129
1.59
León
211
184
10
39
11.27
Salamanca
362
150
5
154
0.33
Salamanca
362
150
7
61
5.28
Salamanca
362
150
10
18
5.84
In table 2 we can observe that MA obtains the best solutions in all the cases.
5. CONCLUSIONS
In this paper a hybrid metaheuristic to solve a special facility location problem is
designed. This hybrid metaheuristic is a memetic algorithm (MA) which is constructed
by combining a genetic algorithm with a local search operator. We compare this
memetic algorithm with a genetic algorithm. The computational experiencies that have
been made shown that MA obtains the best results.
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Use of memetic algorithms to solve a stochastic location model: health resources for diabetics in some provinces of
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6. REFERENCES
• Beasley, J. E. (1990) “OR-Library: Distributing Test Problems by Electronic Mail”.
Journal of the Operational Research Society, vol. 41, pp. 1069-1072.
• Diabetes foundation: web page www.fundaciondiabetes.org
• Love, R.F., Morris, J.G. and Wesolowsky, G.O. (1988) “Facilities Location: Models
and Methods”. North Holland.
• Merz P. and Freisleben B.(1999). “A comparison of memetic algorithms, tabu
search, and ant colonies for the quadratic assignment problem”. In proceedings of the
1999 Congress on Evolutionary Computation, pages 2063-2070.
• Mladenović, N., Labbe, M. and Hansen, P. (2001) “Solving the p-center Problem
with Tabu Search and Variable Neighborhood Search”. Les Cahiers du GERAD, G2000-35.
• Moscato , P. (1989).” On Evolution, Search, Optimization, Genetic Algorithms and
Martial Arts: Toward Memetic Algorithms”. Caltech Concurrent Computation
Program, C3P Report 826.
• Mulvey, J.M. and Beck, M.P. (1984) "Solving Capacitated Clustering Problems".
European Journal Operations Research, 18 (3), pp. 339- 348.
• Pitsoulis, L.S. and Resende, M.G.C., (2002). “Greedy Randomized Adaptive Search
Procedures”. In Handbook of Applied Optimization, P. M. Pardalos and M. G. C.
Resende (Eds.), Oxford University Press, pp. 168-182.
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