Abstract Long-term health care facilities have gained an important role in today’s healthcare environments, due to the global trend of aging of human population. This paper con-siders the problem of network design in health-care systems, named the Long-Term CareFacility Location Problem (LTCFLP), which deals with determining locations for long-termcare facilities among given potential sites. The objective is to minimize the maximal numberof patients assigned to established facilities. We have developed an efficient hybrid method,based on combining the Evolutionary Approach (EA) with modified Variable NeighborhoodSearch method (VNS). The EA method is used in order to obtain a better initial solutionthat will enable the VNS to solve the LTCFLP more efficiently. The proposed hybrid al-gorithm is additionally enhanced by an exchange local search procedure. The algorithm isbenchmarked on a data set from the literature with up to 80 potential candidate sites andon large-scale instances with up to 400 nodes. Presented computational results show thatthe proposed hybrid method quickly reaches all optimal solutions from the literature and inmost cases outperforms existing heuristic methods for solving this problem.
Keywords Hybrid algorithm · Evolutionary method · Variable neighborhood search ·Health care systems · Location problems · Network design
This research was supported by Serbian Ministry of Education, Science and TechnologicalDevelopment under the grants no. 174010 and 47017.
M. Maric · Z. Stanimirovic (�)Faculty of Mathematics, University of Belgrade, Studentski trg 16/IV, 11000 Belgrade, Serbiae-mail: [email protected]
The application of operational research in health-care has developed significantly during thepast decade, regarding the number of fields of application as well as the number of paperspublished in this area. The operational research applications include wide range of topicsfrom optimizing the use of health-care resources to finding a balance between health-careservices provided for patients and efficiency for its providers.
There are many research articles on combinatorial optimization problems which arisefrom designing and optimizing health-care systems and health-care service delivery. In pa-pers by Boffey et al. (2003) and Galvao et al. (2002), authors consider the problem of locat-ing perinatal facilities in the municipality of Rio de Janeiro. Their research lead to the devel-opment of an uncapacitated, three-level hierarchical model, denoted as the “basic model”.Boffey et al. (2003) point out that this model omits some significant practical aspects such ascapacity limits, aggregation of adjacent neighborhoods, political boundaries and social fac-tors. The authors propose an improvement of the idealized “basic model” by incorporatingsome capacity constraints, which showed to be important in practice.
Huang et al. (2010) consider a facility location problem for large-scale emergencies. Theauthors propose a variation of the p-center problem with the additional assumption that afacility at a node fails to respond to demands from the node. A dynamic programming ap-proach is used for the location on a path network and an efficient algorithm is proposed foroptimal locations on a general network. Bish et al. (2011) propose a decision support toolfor hospital evacuation and emergency response. The authors develop the Hospital Evacu-ation Transportation Model (HETM) that can assist in the planning and operational phasesof an evacuation. The proposed integer programming model HETM considers the allocationof patients, categorized by criticality and care requirements, to a limited fleet of vehicles ofvarious capacities and medical capabilities, to be transported to appropriate receiving hos-pitals considering the current available space in each hospital for each category of patient.The results of computational experiments in Bish et al. (2011) showed that the HETM canbe solved in a reasonable amount of time, which makes it useful for both planning andoperational services of emergency management.
Bilsel and Davutyan (2011) analyze the operational performance of Turkish rural gen-eral hospitals and adopt a directional distance approach to help improving performance onboth input and output space. They provide some estimates of the trade-off involved whenhealth care quality inadequacy is reduced. Galvao et al. (2006a, 2006b) deal with both unca-pacitated and capacitated problems of balancing loads of maternal and perinatal health-carefacilities in Rio de Janeiro and propose 3-level hierarchical models with a bi-criterion objec-tive function. The authors apply Lagrangian heuristics for solving these models. The resultsobtained with the uncapacitated model produced a good spatial distribution of the perina-tal health care facilities at all 3 levels of the hierarchy. The capacitated model was furtherused in a case study that allowed capacity planning issues to be analyzed. There are variousemergency-service models that capture stochastic aspects related to demand or travel time,see Boffey et al. (2003), Geroliminis et al. (2011), Snyder (2006). A detailed review of fa-cility location modeling applications in health care systems and other emergency servicesmay be found in papers (Brotcone et al. 2003; Marianov and ReVelle 1995; ReVelle 1989;Goldberg 2004).
In this paper we consider a discrete optimization problem, named the Long-Term CareFacility Location Problem (LTCFLP), which arises from designing the network of long-term health care facilities. We assume that the set J of potential facilities is given and that nofacility is previously established. Each facility involves a certain number of patients to whom
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a long-term health care service has to be provided. The set of potential facility sites coincideswith the set of patient groups, which means that a facility may be located at one of thelocations of the patient groups. The maximum number K of facilities to be established is pre-determined, and inequality K ≤ |J | holds. No capacity restrictions (for example, the numberof sickbeds or limited medical stuff) on established facilities are assumed. The objective is tolocate certain number of long-term facility sites, such that the maximum load of establishedfacilities is minimized. The problem involves a single allocation scheme, which means thateach patient group is assigned to exactly one, previously established facility. In addition, itis assumed that each patient group is assigned to its closest established long-therm healthcare facility. Fixed costs for locating facility sites are not considered in this model.
The main difference between the LTCFLP and the classical uncapacitated facility loca-tion problem UFLP (Kuehn and Hamburger 1963) is in the objective function. The choiceof objective function of the LTCFLP model is appropriate for this specific application, sincethe patients would naturally choose the nearest health care facility, and therefore, the riskis to have too many patients assigned to a facility. Some applications of this model mightalso include the design of network for mobile or satellite receivers or antennas, location ofschools, shopping centers, waste collection areas, etc. In these applications, it is natural thatcustomers will be allocated to the nearest established center for technical reasons.
The Long-Term Care Facility Location Problem was first considered by Kim and Kim(2010), inspired by the problem of establishing a system of long-term medical care facil-ities in Korea. The authors first developed the Modified Add-Drop-Interchange heuristicalgorithm (MADI) by adopting and modifying the add, drop and interchange methods fromFeldman et al. (1966), Kuehn and Hamburger (1963), Teitz and Bart (1968) and applyingthem sequentially. The solution generated by MADI heuristic was used as an upper boundin the branch and bound method (BnB). Results of the experiments on the real-life instancewith |J | = 33 nodes and 15 instances with up to |J | = 70 nodes showed that the BnB al-gorithm produced optimal solutions on problem instances with up to |J | = 40 nodes, whilethe MADI heuristics obtained solutions with certain gaps from the optimal ones. Among15 problems with 50–70 patient groups, the BnB method solved 10 problem to optimality,while the MADI heuristic produced solutions with significant gaps from the optimal/bestknown solutions. In other cases, neither BnB nor CPLEX 10.0 reached optimal solutionafter 24 hours of computational run.
An evolutionary-based algorithm (EA) for solving the LTCFLP is proposed in Stan-imirovic et al. (2012). The EA from Stanimirovic et al. (2012) uses binary representationof solutions and appropriate evolutionary operators. It also employs several strategies forcorrecting and keeping the individuals feasible, preserving the diversibility of individualsand preventing EA to finish in local optimum. The results of the EA implementation are pre-sented on a real-life instance with 33 nodes presented in Kim and Kim (2010) and instanceswith up to 80 nodes introduced in Stanimirovic et al. (2012). The algorithm is additionallybenchmarked on the newly generated set of AP-based hub instances with up to 400 poten-tial facility locations with two types of demands and different levels of K . The performanceof the proposed EA is compared with the exact BnB method and the MADI heuristic fromKim and Kim (2010) on the available common set of test instances and optimal solutionspresented in Stanimirovic et al. (2012).
In this paper we propose a hybrid algorithm, named VNS-EA, which involves evolution-ary approach (EA) nested in the variant of Variable Neighborhood Search (VNS) frame forsolving the LTCFLP. As the evolutionary part of the algorithm we use evolutionary-basedapproach from Stanimirovic et al. (2012), which was successfully applied to this problem.The best solution obtained by the EA is used as initial solution for the VNS algorithm. Con-ducted computational experiments clearly indicate that the VNS-EA hybridization showed
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to be successful for solving the LTCFLP in the sense of both solution quality and the CPUtime. The obtained results also indicate that the proposed VNS-EA method outperforms theEA approach from Stanimirovic et al. (2012), especially for larger and large-scale problemdimensions.
2 Mathematical formulation
In this paper, we use the mixed integer formulation presented in Kim and Kim (2010). Theformulation uses the following notation:
J = the set of locations of patient groups and potential long-term care facility locations,ai = the number of patients in patient group i ∈ J ,dij = the distance between the location of a patient group i and a candidate facility loca-tion j ,K > 0 the maximum number of facilities to be located,M > 0 large constant.
Decision binary variable xij ∈ {0,1}, i, j ∈ J takes the value of 1 if patient group i isallocated to a facility location j and 0 otherwise. Variable yj ∈ {0,1}, j ∈ J is equal to 1 ifa facility is located at node j , and 0 otherwise. Non-negative variable Lmax represents themaximum load of an established long-term care facility site.
Using the notation mentioned above, the problem can be formulated as:
minLmax (1)
subject to:∑
j∈J
xij = 1 for every i ∈ J (2)
xij ≤ yj for every i, j ∈ J (3)∑
l∈J
dilxil ≤ dij + M(1 − yj ) for every i, j ∈ J (4)
∑
j∈J
yj ≤ K (5)
∑
i∈J
aixij ≤ Lmax for every j ∈ J (6)
xij ∈ {0,1} for every i, j ∈ J (7)
yj ∈ {0,1} for every j ∈ J (8)
The objective function (1) minimizes the maximum load of established facilities. Con-straints (2)–(3) ensure that each patient group is allocated to exactly one, previously estab-lished facility location. Each patient group is assigned to its nearest located facility, whichis provided by constraint (4). By constraint (5), we put an upper limit K on the number offacilities to be established. Constraint (6) represents the lower bound for the objective vari-able Lmax, which is the maximum load of established facilities for load balancing. Finally,constraints (7) and (8) indicate binary nature of variables xij and yj respectively.
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VNS-EA hybrid methodInitialization:
Define the representation of solutions;Define distance � and neighborhood structures N_k, i = 1, . . . ,N;Choose the stopping criteria: max_iter_EA, max_iter_VNS;
Evolutionary part EA for finding a good initial solution I for the VNS:Generate the initial population P;iter=1;while iter ≤ max_iter_EA
For each solution S ∈ P do Objective_Function(S);Selection;Crossover;Mutation;iter=iter+1;
end whileVNS part of the hybrid algorithm:
S = Exchange_Local_Search(I );p = the number of established facilities in S;N = min(p, n − p), N ≤ K, p ≤ K;iter=1;while (iter ≤ max_iter_V NS)
neighborhood N1 (k = 1), otherwise set k = k + 1;iter=iter+1;
end whileend while
Fig. 1 The VNS-EA scheme
3 Proposed VNS-EA implementation
For many combinatorial optimization tasks, it has been shown that it is essential to incor-porate some improvement method into VNS concept to yield effective optimization tools,see Brimberg et al. (2000), Festa et al. (2001), Hansen et al. (2001), Loudin and Boizumault(2001), Ochi et al. (2001), Mladenovic and Hansen (1997). Hybrid VNS-based algorithmsare often based on the incorporation of an improvement or repair heuristic into traditionalVNS concept, such as: tabu search, greedy heuristic, local search, enumeration method, etc.For a successful application of a VNS-based method, it is important to obtain a good qual-ity initial solution quickly. This becomes more obvious when solving large-scale instancesof an NP-hard combinatorial optimization problem. In this study, we use an evolutionaryalgorithm in order to obtain a better initial solution for a variant of the VNS approach forsolving the LTCFLP efficiently. The choice of evolutionary heuristic was motivated by obvi-ous potential of evolutionary-based methods that were previously applied for solving variouslocation problems, see Cruz et al. (2010), Kratica et al. (2011), Maric (2010), Stanimirovic(2008), Stanimirovic et al. (2012), etc.
In the initialization part of the VNS-EA hybrid approach we define the encoding of so-lutions, distance measure and neighborhood structures (see Fig. 1). The proposed VNS-EAalgorithm uses the binary representation of solutions, which means that each solution is rep-resented by a binary string of length n = |J |. Each bit in the code corresponds to one node inthe network: bit 1 in the ’s code denotes that facility is located at that particular node, while0 indicates it is not. From the solution’s code we obtain the indices of established facili-ties, which further give us the values of variables xkk , k = 1,2, . . . , n. Since a patient groupcan be assigned only to one established facility, the values of xik , i, k = 1, . . . , n, i �= k are
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Exchange_Local_Search(S)Let L ⊆ J be the set of established facilities in solution S;for each j_out ∈ L
for each j_in ∈ J\ L
ind (j_in)=1;ind (j_out)=0;temp=Objective_Function();ind (j_in)=0;ind (j_out)=1;value=Objective_Function();if temp < value
easily determined by comparing corresponding distances dik . The objective value is simplyevaluated by comparing loads of established facilities and determining the maximal one.
We say that the distance between two solutions S and S ′ in the search space X is equal tok if and only if they differ in k locations. The metric function can be defined as �(S,S ′) =|S\S ′| = |S ′\S|,∀S,S ′ ∈ X. By neighborhood Nk(S) we denote the set of solutions suchthat exactly k locations of facilities from the current solution S are replaced by k others.More precisely, S ′ ∈ Nk(S) ⇔ �(S,S ′) = k. The parameters of stopping criteria for the EAand VNS are also set in the initialization stage.
3.1 Evolutionary part of the hybrid algorithm
As the evolutionary part of the VNS-EA method, we use the evolutionary approach EA pro-posed in Stanimirovic et al. (2012) that showed to be successful for solving the LTCFLP.The EA concept from Stanimirovic et al. (2012) involves appropriate solutions’ encoding,evolutionary operators and uses parameter values adapted to the problem under considera-tion. In this subsection we provide a short description of the EA part of the hybrid algorithm,while more details can be found in paper Stanimirovic et al. (2012).
In order to provide better quality of the genetic material in the initial population, the prob-ability of generating bits with the value of 1 in the initial genetic codes is set to prob = K
n.
The following operators are used in the evolutionary part:
– fine grained tournament selection (Filipovic 2003), with real parameter avgtour = 6.4;– standard one-point crossover, performed with probability crossrate = 0.85;– mutation with frozen bits (Stanimirovic 2008), with basic mutation rate of 0.4/n for non-
frozen bits and 1.6/n. for frozen bits;
Several strategies applied in order to improve the algorithm’s efficiency:
– steady-state generation replacement scheme with elitist strategy;– incorrect individuals (with p > K ones in the genetic code) are corrected by changing
p − K ones to zeros from the end of genetic code;– duplicate individuals are discarded from the population;– the appearance of individuals with the same objective function but different genetic codes
is limited;
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Shaking_Phase(S, k)X_in = ∅;X_out = ∅;for (int i = 0; i < k; i++)
Let L ⊆ J be the set of established facilities in solution S;j_in = Random (J \ (L ∪ X_in))
value=Infinity;for each j_out ∈ (L \ X_out)
ind (j_in)=1;ind (j_out)=0;temp=Objective_Function();ind (j_in)=0;ind (j_out)=1;if temp < value
The EA operators (selection, crossover and mutation) are iteratively applied until themaximal number Gmax = 100 of EA generations is reached. The obtained best EA solutionI represents good starting point for the VNS part of the proposed hybrid method.
3.2 VNS frame of the hybrid method
After the evolutionary phase, we introduce a modification of the VNS method, which hasbeen adapted to the problem under consideration. Before the main VNS cycle, we applythe Exchange Local Search procedure—ELS on the best solution I obtained in the EAphase. This local search procedure is also incorporated into the main VNS cycle as well.By using the strategy of multiple ELS applications, we achieve additional improvements ofthe current best solution and direct the algorithm to promising neighborhoods. The basicconcept of hybrid evolutionary algorithm VNS-EA for solving the LTCFLP is shown in theFig. 1.
In the implemented Exchange Local Search procedure, we first consider the set of es-tablished facilities L in a solution S. Let p = |L| be the number of established facilitiesin a solution S. We try to exchange an established facility l ∈ L with a non-establishedone j ∈ J \ L and look for the improvement of the objective function. Note that we searchthrough all possible exchanges, while the number of located facilities p remains the same.The obtained (improved) solution S is the subject to the main VNS cycle of the hybridapproach. The scheme of the ELS procedure is presented in Fig. 2.
Let us denote by p the number of established facilities in the solution S and let N =min(p,n − p), N ≤ K , p ≤ K . As the Fig. 1 shows, in each iteration of the VNS we repeatthe following procedure. We set k = 1 and explore neighborhood Nk(S) of the solution S.In the Shaking Phase, we try to exchange exactly k facility locations in the solution S. Inthis procedure, new k locations to be included are chosen randomly, while k establishedlocations are excluded in such a way that objective function has the best possible value. Thedetailed pseudocode of Shaking Phase is given in Fig. 3.
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Let S ′ ∈ Nk(S) be the new solution obtained in the Shaking Phase. The next step isto perform the Exchange Local Search with S ′ as a starting point in order to obtain thebest possible improvement S ′′. If the obtained local optimum S ′′ is better than the currentoptimum, we move to the solution S ′′ and set k = 1 again. If the new solution is not betterthan the incumbent one, we set k = k + 1, and repeat the whole procedure looking forimprovement by exchanging k + 1 locations. The steps described above are repeated untilk > N . The VNS part is iteratively applied until the maximal number of VNSmax = 1000iterations is reached.
4 Computational results
In this section, computational results of the VNS-EA method and comparisons with existingalgorithms are presented. All experiments were carried out on an Intel Core i7-860 2.8 GHzwith 8 GB RAM memory under Windows 7 Professional operating system. The VNS-EAimplementation was coded in C� programming language. In order to provide more reliabletest results, the VNS-EA was run 20 times for each considered problem instance.
Computational experiments were first performed on a small-size LTCFLP instance with|J | = 33 nodes, which was derived from real situation in Korea and introduced in Kim andKim (2010). Potential facility locations are given by their (x, y) coordinates in the plane,together with forecasted number of patients assigned to each location. The maximal numberof facilities to be located—K takes seven values: (4,8,12,16,20,24,28). The results of theproposed VNS-EA implementation, BnB method and heuristic approach from Kim and Kim(2010), and pure EA method from Stanimirovic et al. (2012) are presented in Table 1. Notethat experiments in Kim and Kim (2010) are carried out on a Pentium 3.2 GHz processor,while the tests of the EA from Stanimirovic et al. (2012) are carried out on the same machineas the VNS-EA.
Column headings used to present computational results in tables throughout this sectionhave the following meaning:
– Instance’s parameters: number of nodes n = |J |, demand type (for modified AP instancesonly) and the maximal number of facilities to be established K ;
– Optimal solution on the current instance Opt.Sol. obtained by CPLEX 12.1 solver andtaken from Stanimirovic et al. (2012);
– Total CPLEX 12.1 running time CPUt (in seconds);– The best value of the VNS-EA method VNS-EAbest , with mark opt in cases when it coin-
cides with the optimal solution;– Running time CPUstart (in seconds) in which the VNS-EA reaches the optimal/best known
solution for the first time;– Total running time CPUend (in seconds) of the VNS-EA;– Average gap agap (in percents), which is calculated as average percentage deviation of
the VNS-EAbest solution from the optimal/best solution through all VNS-EA runs;– Standard deviation σ (in percents) of the VNS-EAbest solution calculated in respect to the
optimal/best solution through all VNS-EA runs;– The best value EAbest of the EA method from Stanimirovic et al. (2012), with mark opt in
cases when the EA reached optimal solution;– Total running time CPUt (in seconds) of the EA;– The total running time BnBt (in seconds) of the BnB method proposed in Kim and Kim
(2010);– The best solution of heuristic method Heurbest from Kim and Kim (2010);
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Table 1 Results and comparisons on small-size instance |J | = 33
K Opt.Sol. CPUt (s) VNS-EAbest CPUstart(s) CPUend(s) agap(%) σ (%)
4 3610 19.85 opt 0.56 1.61 0.0 0.0
8 1993 15.14 opt 0.53 3.28 0.0 0.0
12 1440 7.55 opt 0.58 5.11 0.0 0.0
16 1079 4.92 opt 0.5 7.53 0.0 0.0
20 1079 3.68 opt 0.43 4.17 0.0 0.0
24 1079 3.03 opt 0.42 1.91 0.0 0.0
28 1079 2.04 opt 0.4 1.14 0.0 0.0
avg – 8.030 opt 0.489 3.536 0.0 0.0
K Opt.Sol. EAbest(Stanimirovicet al. 2012)
CPUt (s) BnBt (s)
(Kim andKim 2010)
Heurbest(Kim andKim 2010)
Heurgap(%) Heurt (s)
4 3610 opt 0.216 1.14 3646 1.0 0.05
8 1993 opt 0.217 101.69 2001 0.4 0.09
12 1440 opt 0.227 171.92 1440 0.0 0.17
16 1079 opt 0.289 68.45 1127 4.4 0.27
20 1079 opt 0.212 179.14 1079 0.0 0.25
24 1079 opt 0.22 76.69 1079 0.0 0.30
28 1079 opt 0.339 6.45 1079 0.0 0.25
avg – opt 0.246 86.497 – 0.829 0.197
– The percentage gap Heurgap (in percents) of the Heurbest from the optimal one;– The total running time Heurt (in seconds) of heuristic method Heur.
As it can be seen from Table 1, the proposed VNS-EA quickly reaches all optimal solu-tions in maximal 7.53 seconds of total CPU time and detects optimal solution for the firsttime in up to 0.4 seconds. The total running times are significantly shorter compared to thetime of CPLEX 12.1 solver, which was run the same machine as the VNS-EA. Comparedto the pure EA from Stanimirovic et al. (2012), the total VNS-EA running time is longer,because these small size instances are very easy to solve and EA finds optimal solution infew iterations. In the VNS-EA method, the EA phase runs 1000 generations, before the al-gorithm enters in the VNS cycle with initial solution that is already optimal. These smallsize instances do not represent a good benchmark example to compare the VNS-EA and thepure EA. The differences in the performance of these two methods become more obvious asthe dimension of test problem grows.
From Table 1 we notice that the VNS-EA reaches optimal solutions in shorter total CPUtime compared to BnB method from Kim and Kim (2010), since Pentium 3.2 GHz con-figuration (used for BnB and Heur testings in Kim and Kim 2010) has around 1.5 slowerperformance compared to the one used in this paper (see SPEC fp2006 and fp2000 bench-marks at www.spec.org). The proposed VNS-EA has obviously better performance com-pared to the Modified Add-Drop-Interchange heuristic algorithm Heur regarding both solu-tion’s quality and computational time. The heuristic Heur doesn’t achieve optimal solutionsfor K = 4,8,16 and produces average gap of 0.829 %. We could carry out even detailedperformance comparison of all three methods if we had more test instances available. In the
100T-50 2034.180 2.44 721.58 0.401 0.697 2034.180 329.65 same
100T-60 1762.489 1.7 720.11 0.064 0.152 1769.981 339.272 same
100T-70 1529.787 3.31 380.83 0.072 0.313 1529.787 356.265 same
100T-80 1490.351 2.33 123.11 0.000 0.000 1490.351 325.574 same
100T-90 1490.351 0.33 30.06 0.000 0.000 1490.351 345.664 same
avg – 1.96 378.37 0.826 0.965 – 308.930 –
paper (Kim and Kim 2010), the authors also performed experiments on problem instanceswith up to |J | = 70 nodes, but these instances are not publicly available.
In our experimental study, we further used modified medium and larger AP instanceswith 50 ≤ n ≤ 200 nodes that are described and used in Stanimirovic et al. (2012). In thisdata set, each node represents potential long-term health care facility location or a patientgroup. Two types of demands are considered: loose—L and tight—T, which gives two mod-ified AP instances for the same number of nodes. Both tight and loose demands for eachparticular node are generated regarding the sum of incoming and outcoming flow to/from anode and the total flow in the network. Tight instances involve lower demands assigned to
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Table 4 Results and comparisons on AP instances (n = 110,120,130)
130T-100 2995.209 23.700 810.320 1.891 0.438 2995.209 286.271 same
130T-110 2995.209 1.450 266.890 0.000 0.000 2995.209 276.658 same
130T-120 2995.209 0.200 45.040 0.000 0.000 2995.209 292.346 same
avg – 426.790 1040.539 1.0610 0.803 – 253.040 –
nodes, which generally results in lower objective function value compared to loose variants(with few exceptions). In the considered modified AP instances, the values of K were set tointegers from the interval [20, |J | − 10].
In Table 2, we present computational results of the proposed VNS-EA approach on mod-ified AP instances 50 ≤ n ≤ 80 and comparisons with results of the pure EA from Sta-nimirovic et al. (2012). Optimal solutions presented in Table 2 have been obtained fromStanimirovic et al. (2012). Mark “–” in column Opt.Sol. means that no optimal solutionwas obtained due to memory or time limits. Note that experiments in Stanimirovic et al.(2012) have been performed on the same machine as the VNS-EA tests in this research. Onall tested instances with 50 ≤ n ≤ 80 nodes, the VNS-EA running times were significantlyshorter compared to CPLEX 12.1 solver, in cases when CPLEX gave solutions.
Both VNS-EA and pure EA reach optimal solutions, which were previously obtainedby CPLEX 12.1 solver in short CPU time. On instances for which CPLEX 12.1 produceda solution but didn’t confirm its optimality, both VNS-EA and EA approaches producedthe same solution. In cases when no optimal solution was obtained by CPLEX 12.1, theVNS-EA and EA reach the same best solutions. In average, the total VNS-EA runningtime is CPUend = 103.953 seconds, while the pure EA had approximately two times slowerperformance CPUt = 216.056 seconds. However, the average time in which the VNS-EA
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Table 5 Results and comparisons on AP instances (n = 200)
200T-150 1445.452 285.620 3607.550 0.000 0.000 1445.451 550.732 same
200T-160 1445.452 110.090 2724.070 0.000 0.000 1445.451 493.84 same
200T-170 1445.452 61.500 1524.550 0.000 0.000 1445.451 529.524 same
200T-180 1445.452 4.900 427.000 0.000 0.000 1445.451 522.44 same
200T-190 1445.452 0.290 109.950 0.000 0.000 1445.451 550.144 same
avg – 1058.961 3170.851 1.108 0.879 – 466.555 –
obtained the best solution for the first time is very short CPUstart = 1.96 seconds. The av-erage gap and standard deviation of the VNS-EA solution over all instances presented inTable 2 are 0.963 % and 1.020 % respectively
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Table 6 Results and comparisons on large-scale instances (n = 300,400)
400T-280 3359.675 1763.770 3668.950 2.123 1.225 3359.675 1042.131 same
400T-300 3359.675 1244.200 3666.320 6.685 5.250 3359.675 1093.365 same
400T-340 3359.675 253.390 3649.910 0.795 1.589 3359.675 1027.304 same
400T-380 3359.675 9.090 1693.050 0.000 0.000 3359.675 1121.522 same
avg – 1693.050 1880.206 2.779 2.396 – 758.686
Tables 3 and 4 contain the results of the VNS-EA approach and comparisons with theEA on instances with n = 90,100 and n = 110,120,130 nodes respectively. From the pre-sented results, it can be seen that for instances with n = 90,100, the hybrid VNS-EA methodoutperforms the EA in the sense of solution quality: for 6 instances the VNS-EA improvedsolutions obtained by the EA, while on 24 instances they produced solutions of the samequality (see Table 3). As the problem dimension grows, the advantage of the proposed hy-brid method, in the sense of solutions’ quality, becomes more obvious. On tested modifiedAP instances with n = 110,120,130 nodes (60 in total), the VNS-EA outperformed the EAin 21 cases (35 %), the EA produced better solutions in only 7 cases (12 %), and on 32instances (53 %) both methods obtained the same solutions. Results presented in Table 5 forproblem instances with n = 200 nodes, show that the hybrid method VNS-EA was betteron 18 instances, that was 50 % of the number of tested instances, while the EA gave bettersolutions in 4 cases (11 %) only.
In Table 6 we present results of the VNS-EA and comparisons with the pure EA methodon large scale instances with n = 300 and n = 400 nodes, considered in Stanimirovic et al.(2012). These instances were derived from the largest AP instance with n = 200 nodes,and involve two types of patient demands (T and L) and different integer values of K . On48 tested instances of size n = 300 and n = 400, the VNS-EA gave the best solution on 18(38 %) instances, the EA on 14 (29 %) instances, while in 16 cases (43 %) the solutionswere the same. The total running time of the hybrid VNS-EA method is up to 3765 seconds.
Regarding the running times, from Tables 5 and 6, it can be noticed that for larger prob-lem dimensions, the VNS-EA becomes slower compared to the pure EA. This is due to thefact that the VNS-EA method first runs through certain number of EA generations before itenters the VNS part. Another reason for prolonging the CPU runs is Exchange Local Search,which is applied before and within the main VNS cycle, which needs additional CPU timeto improve the best solution. However, the total VNS-EA running times are reasonably shortwhile the percentage of the obtained improvements is significant.
In order to decrease the VNS-EA running times, we have experimented with limitednumber of exchanges in the Exchange Local Search phase. We have involved a parameterr ≥ 1, which denotes how often we perform exchanges within the ELS procedure. Moreprecisely, from the list of all possible exchanges of an established facility from L with anon-established one from J \L, we perform every r-th exchange, starting from an exchange
Ann Oper Res
Table 7 Experimental results with different local search parameter r
Inst r VNS-EAbest CPUstart(t) CPUend(t) agap(%) σ (%)
200T-60 50 3327.186 131 195.76 2.079 1.551
200T-60 25 3262.348 203.65 336.78 2.676 1.788
200T-60 20 3276.921 154.15 261.52 3.758 2.904
200T-60 15 3227.814 317.2 502.55 2.718 1.679
200T-60 10 3284.526 303.14 454.57 2.684 3.786
200T-60 8 3139.699 505.22 979.26 3.876 1.884
200T-60 4 3082.148 1082.48 1932.85 2.710 2.342
200T-60 2 3070.740 1960 3605.66 1.341 1.118
200T-60 1 3056.090 1802.770 3624.180 2.001 1.621
200T-70 50 2866.605 138.82 262.43 4.569 4.369
200T-70 25 2831.822 220.34 399.65 2.950 1.610
200T-70 20 2830.874 205.27 356.75 4.089 3.677
200T-70 15 2812.532 366.73 598.29 3.880 2.839
200T-70 10 2863.965 286.66 476.98 3.139 3.143
200T-70 8 2727.677 1102.21 1868.91 3.384 1.699
200T-70 4 2709.895 2227.81 3581.38 2.199 1.401
200T-70 2 2717.532 1532.13 3608.13 2.840 1.281
200T-70 1 2701.661 1349.790 3625.180 1.610 1.468
200T-130 50 1652.203 227.47 494.54 7.537 7.096
200T-130 25 1650.062 286.66 557.89 6.656 8.086
200T-130 20 1652.203 262.85 532.49 12.714 12.853
200T-130 15 1650.062 352.25 657.96 9.558 14.454
200T-130 10 1650.062 354.59 885.95 8.982 11.416
200T-130 8 1590.279 381.77 1079.84 10.554 14.425
200T-130 4 1590.279 789.45 2217.98 4.808 3.277
200T-130 2 1593.908 896.61 2511.6 5.150 4.193
200T-130 1 1590.279 746.530 3612.940 1.793 1.566
200L-50 50 8894.673 72.95 122.65 3.521 2.205
200L-50 25 8767.258 106.47 161.54 3.615 2.278
200L-50 20 8610.270 146.2 235.35 3.713 2.069
200L-50 15 8648.892 154.34 222.85 3.082 2.127
200L-50 10 8624.609 169.53 406.28 3.353 3.145
200L-50 8 8362.252 636.31 1347.1 3.306 1.619
200L-50 4 8361.697 1328.6 3046.25 2.318 1.338
200L-50 2 8352.375 1515.37 3605.65 1.466 1.504
200L-50 1 8319.796 1674.530 3622.820 1.241 0.997
200L-100 50 4899.604 252.900 441.610 6.391 5.162
200L-100 25 4894.320 285.340 538.250 7.976 9.553
200L-100 20 4812.799 304.570 703.100 5.646 4.509
200L-100 15 4754.321 287.510 795.180 5.288 4.461
200L-100 10 4752.583 424.610 846.230 6.807 3.813
Ann Oper Res
Table 7 (Continued)
Inst r VNS-EAbest CPUstart(t) CPUend(t) agap(%) σ (%)
on randomly chosen position {0,1, . . . , r − 1}. The parameter r may be increased if onewants to look for a better solution or decreased if there is a time limit.
In computational experiments presented in Tables 1–6, we considered all possibilitiesto exchange an established facility with a non-established one (see pseudocode of the ELSprocedure on Fig. 2). This corresponds to the case of r = 1. In order to investigate the effectof increasing the value of parameter r , we have chosen several larger problem instanceswith 200, 300 and 400 nodes with different values of K . We have varied the values ofparameter r from 50 to 2 and run the VNS-EA with these values. The results of computationalexperiments are shown in Table 7. The best solutions obtained on the considered instancesare bolded.
As it can be seen from Table 7, the VNS-EA produced best solutions for r = 1, i.e. whenall possible exchanges in the ELS procedure are considered. As it was expected, for all con-sidered instances, the solutions obtained for larger values of r are worse compared to the bestones, with few exceptions when the best values were reached for r > 1 (see 200T − 130,r = 4,8; 300L − 200, r = 25 and 300T − 160, r = 2). The average gap and standard devia-tion values generally increase as the value of parameter r increases. From column CPUend ,we can see that total running times are generally longer for smaller value of r . However, thevalues of CPUstart do not decrease linearly as the values of r increase—in some cases theyare similar or even longer compared to the case of r = 1. This means that in some casesthe VNS-EA needs more time to escape from a neighborhood of a local minimum and findbetter solution. The obtained computational results show that the strategy of involving pa-rameter r , which controls how deep we go with exchanges in the ELS procedure, is usefulin cases when one needs to obtain a good quality solution in a limited amount of time.
5 Conclusions
This paper deals with the Long-Term Care Facility Location Problem (LTCFLP) that arisesfrom designing a health care system. We propose a hybrid metaheuristic method for solvingthe LTCFLP, which combines an Evolutionary Algorithm and a modification of the VariableNeighborhood Search, additionally enhanced by an Exchange Local Search procedure. Thebest solution of the evolutionary part of the algorithm, obtained after certain number of EAgenerations, is used as initial solution of the VNS method. The Exchange Local Search isused both before and within the VNS cycle and its role is to direct the algorithm to promisingregions of the search space. The hybrid VNS-EA shows to be very efficient in obtainingpreviously known optimal solutions on smaller and medium size problem instances up to80 potential locations. For larger and large-scale problem dimensions (90 ≤ n ≤ 400), theapplied VNS-EA approach produces high-quality solutions in reasonable amount of CPUtime and in most cases, outperforms the pure EA method in the sense of the solution quality.The presented computational results indicate that the combination of the two robust heuristicapproaches, such as Variable Neighborhood Search and Evolutionary Algorithm, results inpowerful hybrid metaheuristic method that could be applied for solving similar discretelocation problems as well.
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