An Improved Harmony Search Algorithm for Emergency Inspection Scheduling Nikos Ath. Kallioras+, Nikos D. Lagaros1+ and Matthew G. Karlaftis* +
Institute of Structural Analysis & Seismic Research, National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece, e-mail: {
[email protected],
[email protected]}
*
Department of Transportation Planning and Engineering, National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece, e-mail:
[email protected]
Abstract: The ability of nature inspired search algorithms to efficiently handle combinatorial problems and their successful implementation in many fields of engineering and applied sciences has led to the development of new, improved algorithms. In this paper, an improved harmony search algorithm (IHS) is presented, while a holistic approach in solving the problem of post-disaster infrastructure management is also proposed. The efficiency of IHS is compared with the algorithms of particle swarm optimization, differential evolution, basic harmony search and the pure random search procedure, when considered solving the districting problem that is the first part of post-disaster infrastructure management. While the ant colony optimization algorithm is employed for solving the associated routing problem. The comparison is based on the quality of the results, the computational demands, and the sensitivity on the algorithmic parameters. Keywords: Nature inspired search algorithms; emergency infrastructure inspections; districting & scheduling problems; harmony search; particle swarm optimisation; differential evolution; ant colony optimization.
1.
INTRODUCTION
Catastrophic events may severely affect the structural integrity and operation of civil infrastructures, leading to both immediate and long-term economic losses for the surrounding communities and even the national or global economy. Impacts and associated risks originated from natural disasters can be mitigated through careful planning. Disaster management is a multi-stage process which begins with the predisaster planning and system improvement, and extends to post-disaster system response, recovery and reconstruction. Pre-disaster planning stage involves strategic
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Corresponding author
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decision-making for risk assessment and management, infrastructure improvements to reduce vulnerability, enhanced human and physical system resilience, and emergency plans. Post-disaster stage involves tactical and operational decision-making for providing critical emergency, recovery and re-construction services, to support society. Following such disasters, local communities and search-rescue crews are faced with rapidly degrading infrastructure networks that may result in much slower response times, delays in population evacuation, and significant complications in infrastructure repair. Recent advances on computational engineering and optimization have enabled the transition from traditional trial-and-error procedures to fully automated ones, where search algorithms are used. This is mostly attributed to the rapid development of metaheuristic search algorithms, which imitate biological evolution and combine the concept of survival of the fittest with random operators. Minimising inspection times in post-disaster management requires optimal scheduling of the inspection crews which is a complex combinatorial problem. In this work an improved harmony search algorithm (IHS) is proposed for solving the districting problem where an urban area is decomposed into optimal areas equal to the number of available inspection crews. In addition, a new, modified districting problem formulation is presented that reduces the size of the problem, making the implementation of the algorithms computationally efficient and more robust. The improved harmony search algorithm is applied in two cities: (i) the city of Patras and (ii) the city of Thessaloniki, both located in Greece. The proposed improved algorithm is compared with three well established nature inspired algorithms and the random search (RS) procedure when implemented for solving the districting problem. In particular, the basic formulation of the harmony search (HS), the particle swarm optimization (PSO) and the differential evolution (DE) algorithms are employed. The ant colony optimization (ACO) algorithm is also used for defining the optimal route of every crew inside its area of responsibility. The combination of the
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algorithms employed for solving the districting problem with ACO adopted for solving the routing problem, aims to minimize the necessary time for inspecting all structures in the urban area.
2.
POST-DISASTER INFRASTRUCTURE MANAGEMENT
2.1 An Overview Following a catastrophic event (earthquake, hurricane, etc) it is critical to assess the integrity of structure and infrastructure systems in order to minimize impacts. Therefore, structures and infrastructures have to be inspected for detecting possible damages and thus to organize the support, recovery and rehabilitation process. Before beginning this process, inspection has to be completed; to this end, inspection time is crucial and therefore an operational plan for minimizing it is crucial. The formulation of this plan is divided in two main steps: (i) dividing the area affected by the disaster into smaller districts (districting problem) and (ii) defining the optimal routes in each district (routing problem). A successful operational plan requires almost the same time for inspecting each district. Due to the importance of minimizing the adverse impacts from natural disasters, the literature has systematically dealt with what are considered the four main steps in disaster response [1]: First, mitigation; this includes assessing seismic hazards [2], probabilistic damage projection [3,4], and decision support systems for integrating the emergency process [5,6]. Second, preparedness, which has mainly focused on preparing infrastructure networks for dealing with potential disasters and for accommodating evacuation needs [7-12]. Third, response, which has evolved around two main research paths: (i) planning the response-relief logistics operations [13-16], and, (ii) assessing the performance of the infrastructure system following the natural hazard [17-20]. Fourth, recovery operations which have attracted limited attention despite their importance in practice; for example, work has concentrated on infrastructure element
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protection [21], general assessment of relief performance [22], and fund allocation for infrastructure repairs following disasters [23,24]. As it is obvious from the previous brief review of the literature, there has been significant interest in many areas of emergency response, including relief logistics and preparedness. However, the crucial aspect of minimizing infrastructure inspection time following a disaster has hardly – if at all – been addressed. This step, with its obvious importance in practice, has not been addressed possibly because of the difficulty in collecting appropriate data and the difficulty in formulating and solving the associated mathematical problem. This is the main goal of this paper; to formulate and solve the problem of organizing post disaster infrastructure inspection.
2.2 Addressing the Districting Problem Urban areas are composed of structural (or building) blocks (SBk) that are defined by their nodal coordinates (Xk, Yk) while there are specific terms related to the construction for each SBk that have to be respected. Such terms are: the use of land (uL), the building factor (fB), the maximum built area (AB,max) and the coverage factor (fC). The use of land determines the usage of the constructions allowed at a specific structural block (houses, factories, etc.). The building factor is the ratio between the maximum permitted area of constructions for the specific SBk, divided by the area of the structural block, while the maximum permitted area of constructions for SBk defines AB,max. For example, if the structural block area is equal to 1000 m2 and the corresponding SBk is equal to 1.6, then AB,max is equal to 1000×1.6 = 1600 m2. Figure 1 depicts an example of neighbouring structural blocks with various areas, building factors and thus different maximum built areas AB,max. The districting problem aims to partition the urban area into groups of structural blocks, called districts, having almost the same sum of the corresponding maximum built areas AB,max. As it will be presented in the next section, the inspection demand Di is used in the formulation that is defined according to expression:
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(i) nSB
Di D(k ), i 1, k 1
N IC and D(k ) A(k ) f B (k )
(1)
(i ) where nSB is the number of structural blocks assigned to the ith district while the
structural block total area and the corresponding building factor for the kth structural block is equal to A(k) and fB(k). Furthermore, it is assumed that the allowed building factor is covered.
3.
PROBLEM FORMULATION
The main objective of the districting problem in post-disaster management is to define areas of responsibility for available inspection crews; the problem is formulated as a nonlinear programming optimization problem as follows [25]:
D(k ) d ( SBk , Ci ) min U tr i 1 k 1 U in (i ) N IC nSB
(2)
(i ) where NIC is the number of the available inspection crews, nSB is the number of
structural (or building) blocks assigned to the ith district (frequently referred in the infrastructure literature as inspection crew), d(SBk,Ci) is the distance between the SBk building block and Ci is the starting block of the crew responsible for the ith group of structural blocks, Uin is the inspection speed of the inspection crews, and Utr is the travelling speed of the inspection crews. D(k) is inspection “demand” for the kth building block defined as the product of the building block total area A(k) times the building factor fB(k) (i.e. the structured percentage of the area). Thus, the districting problem is formulated as a discrete unconstrained nonlinear optimization problem where the objective is to define which inspection crew is responsible for which block. Therefore, the design variables s [1, nSB ] are integers denoting the inspection crew to which each built-up block has been assigned to. In this study, a modified formulation of the one presented in Eq. (2) that aims to reduce the variance on the inspection time between the available inspection crews, is examined. In the formulation of Eq. (2), working hours for the crews might not be the 5
same. This can be corrected by introducing the variance of the working hours between the crews into the objective function. The modified districting problem is defined as follows:
D(k ) d (SBk , Ci ) min IC U tr i 1 k 1 U in (i ) N IC nSB
(3)
where σIC is the value of the standard deviation of the working hours of all inspection crews. The second step in post-disaster management is solving the prioritization problem defined for each crew; for this purpose the starting point of the inspection route for each crew should be defined first. In both formulations presented above (Eqs. (2) and (3)) the design variables are equal to the total number of the structural blocks and the kth design variable denotes which inspection crew serves the kth structural block. Thus, the starting point for the ith inspection crew is defined as the geometric centre for the ith district which is defined from the nodal coordinates of the structural blocks belonging to it. On the other hand, the centre of weight (CW) for every building block is calculated from its nodal coordinates (Xk, Yk), while all Euclidean distances are calculated with respect to this centre CW. According to a second modification of the districting problem, the design variables are equal to the number of the inspection crews and the ith design variable denotes which structural block is the starting point for the ith inspection crew. This modification leads to a simplification of the districting problem since the number of the design variables are reduced from nSB to NIC. In this new modified formulation the definition of the area of responsibility (district) is performed through a three step procedure (see Figure 2). In Step 1, the starting points of the inspection crews are defined (this is provided by the values of the design variables according to the second modification, see Figure 2b for the case of three inspection crews). In Step 2, the definition of their areas of responsibility is performed based on the distances between the starting points and all the structural blocks (Figure 2c).Therefore, each building
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block is assigned to the crew whose starting position is closer to it. In Step 3, the starting points are corrected by redefining their positions as the centre of weight of each district (Figure 2d), and then Step 2 is performed again in order to correct the area of responsibility for each district. In this stage, every block is reassigned to the nearest starting point, from the updated starting block, inspection crew (Figure 2e). The optimal routing (also called scheduling) problem of the inspection crews is formulated as a travelling salesman problem (TSP). TSP is represented by a weighted graph G=(N,A), where N is the set of nodes and A is the set of connections/paths that connect all N nodes. A cost function is assigned to paths between two nodes (i and j), represented by the distance between the two nodes di,j (ij). A solution of the TSP is a permutation p= [p(1), …, p(N)] T of the node indices [1, …, N], as every node must not appear more than once in a solution. The solution that minimizes the total length L(p) given by: N 1
L( p) d p ( i ), p ( i 1) d p ( N ), p (1)
(4)
i 1
is noted as the optimal one. The corresponding scheduling problem is defined as follows:
nSB 1 min d ( SBk , SBk 1 ) d ( SBn( i ) , SB1 ) , i 1,..., N IG SB k 1 (i )
(5)
where d(SBk, SBk+1) is the distance between the kth and kth+1 building blocks. The desired result is to locate the shortest route between all structural blocks assigned to each inspection crew.
4.
METAHEURISTIC SEARCH ALGORITHMS
For the needs of this study, four metaheuristic search algorithms have been considered for solving the districting problem and one for the routing. In this section, the basic harmony search, particle swarm optimization, differential evolution and ant colony
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optimization algorithms are briefly discussed while the improved harmony search algorithm is described in the next section.
4.1 Harmony Search The harmony search algorithm was inspired by Zong Woo Geem in 2001 by the improvisation procedure of musicians in Jazz bands [26]. Musicians correspond to decision variables while pitch range of every instrument corresponds to the value range of decision variables. Musical harmony produced by the band corresponds to a specific solution vector whereas acceptance and likeness by the audience is equivalent to the objective function value. The HS algorithm consists of five steps: Parameter initialization: where the optimization problem is determined along with the HS algorithm parameters (HMS is the harmony memory size that corresponds to the number of solution vectors that are stored in the memory, HMCR defines the harmony memory considering rate and PAR is the pitch adjusting rate). Harmony memory initialization: In this step, the harmony memory (HM) is initialized with HMS randomly generated solution vectors. New harmony improvisation: In this step, a new harmony vector is generated by means of the following three procedures: random selection, memory consideration and pitch adjustment. Harmony memory update: If the new generated harmony vector is better than the worst harmony vector of the HM, with reference to the objective function value, the worst harmony is replaced by the new harmony vector. After all iterations are completed, the best solution found, according to the objective function value, is chosen.
4.2 Particle Swarm Optimization In particle swarm optimization proposed by Kennedy and Eberhart [27], multiple candidate solutions coexist and collaborate simultaneously. Each solution is called “a particle” having a position and a velocity in the multidimensional design space while a population of particles is called a swarm. A particle adjusts its velocity and position
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according to its own “experience” as well as the experience of neighbouring particles. A particle's experience is built by tracking and memorizing the best position encountered. PSO combines local search (through self-experience) with global search (through neighbouring experience), attempting to balance exploration and exploitation. Each particle maintains its two basic characteristics, velocity and position, in the multidimensional search space that are updated as follows:
v j (t 1) wv j (t ) c1r1
s Pb,j s j (t ) c2r2 s Gb s j (t )
s j (t 1) s j (t ) v j (t 1)
(6) (7)
where vj(t) denotes the velocity vector of particle j at time t, sj(t) represents the position vector of particle j at time t, vector sPb,j is the personal best ever position of the jth particle, and vector sGb is the global best location found by the entire swarm. The acceleration coefficients c1 and c2 indicate the degree of confidence in the best solution found by the individual particle (c1 - cognitive parameter) and by the whole swarm (c2 - social parameter), respectively, while r1 and r2 are two random vectors uniformly distributed in the interval [0,1].
4.3 Differential Evolution In 1995, Storn and Price [28] proposed a new floating point evolutionary algorithm for global optimization and named it differential evolution. DE utilizes a population of NP parameter vectors si,g (i=1,..,NP) for each generation g. According to the variant implemented, a donor vector vi,g+1 is generated first according to:
vi , g 1 sr1 , g F (sr2 , g - sr3 , g )
(8)
before defining the ith parameter vector si,g+1. Integers r1, r2 and r3 are chosen randomly from the interval [1,NP] while i r1, r2 and r3. F is a real constant value, called mutation factor, which controls the amplification of the differential variation (s r2 , g - s r3 , g ) and is defined in the range [0, 2]. In the next step the crossover operator is applied by generating the trial vector ui,g+1 = [u1,i,g+1,u2,i,g+1,…,uD,i,g+1] T which is defined from the 9
elements of the vector si,g and the elements of the donor vector vi,g+1 whose elements enter the trial vector with probability CR as follows:
v j ,i ,g 1 if rand j ,i CR or j I rand u j ,i ,g 1 s j ,i ,g if rand j ,i CR or j I rand i 1,2,..., NP and j 1,2,..., n
(9)
where rand j ,i U [0,1], I rand is a random integer from [1,2,...,n] that ensures that
vi ,g 1 si ,g . The last step of the generation procedure is the implementation of the selection operator where the vector si,g, is compared to the trial vector ui,g+1:
ui , g 1 if F (ui , g 1 ) F (si , g ) si , g 1 si , g otherwise i 1, 2,..., NP
(10)
where F (s ) is the objective function to be optimized, while without loss of generality the implementation described in Eq. (10) corresponds to minimization.
4.4 Ant Colony Optimization The ant colony optimization algorithm [29] is a search algorithm inspired by studying the behaviour of ant colonies in their natural environment. In ACO, a colony of artificial ants searches for the best path inside a weighted graph. Consider a population of m ants; initially, the ants are randomly positioned on different nodes of the graph. At each step, ant k follows a random action choice rule, called random proportional rule, in order to decide which of the nodes will be visited next. While defining the route, ant k positioned at node i, maintains a memory Mk containing all nodes previously visited. This memory is used for defining the feasible neighbourhood Nki that contains the nodes that have not been visited by ant k yet. The probability with which ant k, positioned at node i, chooses to move to node j is defined as follows:
p k i, j
( i , j ) (i , j )
(
N ik
i,
) (i , )
, if j N ik
(11)
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where τi,j is the amount of pheromone between nodes i and j, α is a parameter controlling the influence of pheromone τi,j, ηi,j is a heuristic information that is available a priori, denoting the desirability of the path between nodes i and j and it is given by the following expression:
i , j
1 di , j
(12)
while in Eq. (11) β is a parameter controlling the influence of the path’s desirability ηi,j. According to Eq. (12), the heuristic desirability of moving from node i to node j is inversely proportional to the distance between nodes i and j. The probability of choosing a particular connection i,j increases with the value of the pheromone trail τi,j and the heuristic information value ηi,j. When all ants have completed their routes, the pheromone concentration for each connection between i and j nodes, is updated for the next iteration t+1 as follows: m
i , j (t 1) 1 i , j (t ) ik, j (t ), (i, j ) A
(13)
k 1
where ρ is the rate of pheromone evaporation, A is the set of paths (edges or connections) that fully connects the set of nodes and Δτki,j(t) is the amount of pheromone ant k has deposited on connections it has visited during its tour Tk and it is given by:
ik, j
Q if connection (i,j ) belongs to T k k L( T ) 0 otherwise
(14)
The coefficient ρ must be less than 1.0 in order to avoid unlimited accumulation of trail [29], while Q is constant. In general, connections used by many ants and are parts of short tours, receive more pheromone due to more deposition and less evaporation and are therefore more likely to be chosen by ants in future iterations of the algorithm.
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5.
IMPROVED HARMONY SEARCH ALGORITHM
In the past several modifications of the basic harmony search algorithm have been presented in the literature.
5.1 Literature Survey In particular, Mahdavi et al. [30] presented a method for generating new solution vectors that enhances accuracy and convergence rate of the algorithm and has been applied in benchmark structural engineering problems. Ling et al. [31] proposed a dynamic procedure for determining the algorithm parameters and has been applied in slope stability analysis. Jaberipour and Khorram [32] presented two new versions of HS algorithm for solving continuous optimization problem where procedures for adjusting bandwidth were implemented; while they have been applied to mathematical function minimization and structural engineering optimization problems. Geem and Sim [33] presented a parameter-setting-free technique which was applied in three examples in order to assess its robustness. Jaberipour and Khorram [34] presented a mixed-discrete harmony search approach for solving nonlinear optimization problems with non-convex and non-differentiable objective functions and integer, discrete and continuous design variables. Omran et al. [35] proposed three variants of the HS algorithm investigating the effect of using opposition-based learning and quadratic interpolation; and have been applied to benchmark functions. Sarvari and Zamanifar [36] presented a new step procedure to create new harmony vectors and introduced the bandwidth parameter; the new algorithm was implemented to constrained and unconstrained mathematical and engineering problems. Yadav, et al. [37] proposed an improved harmony search algorithm for optimal scheduling of diesel generators in oil rig platforms, where it was found that the proposed algorithm has better accuracy and convergence rate by improving the local search ability. This improvement was achieved by fine-tuning the algorithm’s parameters during new generations. Afshari, et al. [38] presented an improved harmony search algorithm in well placement optimization using 12
streamline simulation. Degertekin [39] proposed two improved harmony search algorithms and implemented them to sizing optimization of truss structures. The first one presented better accuracy and convergence rate than HS by updating parameter values during iterations. The second one presented better performance than HS by removing one of the parameters and updating the value of another parameter through iterations. In this work, an improved harmony search algorithm is proposed aiming to solve the districting problem where an urban area is decomposed into optimal areas equal to the number of available inspection crews. The main objective of the studies mentioned above, was mainly to define the best values of two algorithmic parameters (PAR and BW that is an arbitrary distance bandwidth for continuous design variables). Such a procedure requires a large number of independent runs to be performed which leads to higher computing demands. The algorithm proposed in this study does not use the parameters PAR and BW, thus no trial and error procedure is required in order to define their values.
5.2 The Proposed Algorithm The HS algorithm consists of five steps: parameter initialization; harmony memory initialization; new harmony improvisation; harmony memory update; and termination criterion check. In an attempt to improve the quality of results, reduce computational load and diminish the sensitivity of the algorithm with reference to the HS parameters, we propose certain improvements, particularly regarding the general structure of the algorithm and new harmony improvisation step. In the basic form of HS, a new harmony vector is generated by following three rules: random selection, memory consideration and pitch adjustment. In the IHS algorithm a new harmony vector is improvised following two procedures: random selection and memory consideration; the pitch adjustment procedure is not used. The second difference is the implementation of the two procedures. All the elements of the new harmony vector are improvised with the same procedure, while in the basic HS a different procedure is implemented for 13
each element of the vector. Thus, the HM memory fills with “good” solution vectors faster than in the basic HS algorithm. Parameter initialization: In the first step, the optimization problem is determined where n is the number of decision variables (equivalent to the number of available inspection crews), while siL si siU , i 1,2,..., n determines the range of the ith decision variable’s value. The HS algorithm parameters are also defined in this step: HMS is the harmony memory size that corresponds to the number of solution vectors that are stored in the memory and HMCR defines the harmony memory considering rate. Harmony memory initialization: In the second step, the harmony memory (HM) is initialized with HMS randomly generated solution vectors defining which building block is assigned to each crew:
s11 2 s HM 1 ... HMS s1
s21 s22 ... s2HMS
s31... s1n s32 ... sn2 ... ... HMS HMS s3 ... sn
(15)
Inspection crews are randomly placed in the city and each building block is assigned to the nearest crew. New harmony improvisation: In the third step, a new harmony vector is generated by means of the following two procedures: random selection and memory consideration. According to the random selection procedure, the ith design variable (starting point of the ith inspection crew) is selected randomly with probability 1-HMCR (0≤ HMCR≤1); according to the memory consideration it is randomly chosen from HM memory with probability HMCR. The two procedures used for generating a new harmony vector can be implemented both to continuous and discrete optimization problems. Thus, for discrete optimization problems the new harmony vector generation is implemented as follows: si [ siL , siU ] with probability 1 HMCR siNew 1 2 HMS si HM {si , si ,..., si } with probability HMCR
(16)
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where each new solution vector sNew is defined according to either the first or the second functions of Eq. (16). Harmony memory update: If the new generated harmony vector is better than the worst harmony vector of the HM, with reference to the objective function value, the worst harmony is replaced by the new harmony vector. After all iterations are completed, the best solution found according to the objective function value is chosen and the routing problem of Eqs. (4) and (5) is formulated and solved for each district. The basic steps of IHS algorithm are presented in Figure 3.
6.
NUMERICAL TESTS
In order to assess the performance of the HS, IHS, PSO, DE algorithms and the RS procedure along with the ACO algorithm, they are implemented into post-disaster management problems. In particular, two test cases are considered based on an imaginary seismic event for the cities of Patras and Thessaloniki in Greece. Following this imaginary seismic event, all structures need to be inspected for restoration of the urban activities. Therefore, the first step of the post-disaster management procedure is to optimally assign the structural blocks into the available inspection crews (districting problem), while the second step is the solution of the route-scheduling problem (inspection prioritization) for all districts defined in the first problem. Patras is a medium-sized city while Thessaloniki is the second largest city of Greece. The city of Patras is composed by 112 grouped structural blocks while the total built area is equal to 18.559.676 m2. On the other hand, the city of Thessaloniki is composed of 471 grouped structural blocks and its built area is equal to 154.205.128 m2. The difference in size of the two cities will offer information on the performance of the proposed IHS algorithm in medium and larger scale problems. For the city of Patras, two problems are examined varying on the number of available inspection crews. Increasing the number of available inspection crews, the complexity of the problem is also increased offering us useful information regarding the performance of the 15
algorithms. In such a problem the computational cost is proportional to the number of the building blocks; therefore, grouping the building blocks into larger ones, significant reduction on the computational demand is achieved. The grouping is applied into neighbouring building blocks with similar build up percentages. For an unbiased comparison of the algorithms the termination criterion for all implementations examined in this study is the same (200,000 function evaluations).
6.1 Post-disaster Management Applied to the City of Patras The city of Patras is composed by NSB=112 grouped structural blocks with various areas and built-up percentages (see Figure 4). Two different problems (A and B) are examined with the hypothesis that ten and thirty crews are available for performing the inspection. By separating them into two 8-hour shift work-groups, NIC = 5, inspection crews are available for 16 inspection hours per day (denoted as problem A) and NIC = 15, inspection crews are available for 16 inspection hours per day (denoted as problem B). 6.1.1 Sensitivity analysis of the problem formulation In the first part of the study performed for the city of Patras two different problem formulations are examined: in the first one (denoted as IMPL1) the implementation of the IHS to the districting problem of Eq. (2) is considered, while the second one (denoted as IMPL2) corresponds to the implementation of the same algorithm (i.e. IHS) to the districting problem of Eq. (3). Sensitivity analysis is performed on both problem formulations in order to examine the influence of parameter values on the robustness of the IHS algorithm in solving the districting problem. The performance of the metaheuristics depends on the selection of the algorithmic parameters. For comparative reasons, a sampling method named Latin hypercube sampling (LHS) is used [40] to generate combinations of the parameters of the algorithms. LHS ensures that all regions of the sample space of the parameters will be sampled. For each implementation, 32
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independent runs are carried out, corresponding to the different sets of the parameters. The parameters sampled with LHS for HS are HMS, HMCR and PAR while for IHS are HMS and HMCR, Table 1 depicts the combinations of the parameters. HMS defines the size of the memory used by HS for storing solution vectors defined in the range of [5, 20]. HMCR (0 ≤ HMCR ≤ 1) is the probability for random selection and PAR (0 ≤ PAR ≤ 1) is the probability for performing changes in solutions obtained from HM [41]. The parameters of the ACO algorithm are not sampled and the best values identified in [25] are used, in particular the size of the population of the ant colony m = 65, α = 0.91 that is the parameter that controls the influence of the pheromone amount, β = 0.97 that is the parameter that controls the influence of the desirability of a connection, ρ = 0.21 that is the rate of pheromone evaporation and Q = 0.306 that corresponds to the rate of pheromone deposition by ants [42]. Since the inspection time varies between the crews, the maximum time required between the inspection crews is considered as the time required for the inspection procedure of the entire city. Figure 5 depicts the time required for the inspection of the building blocks of the city of Patras for each implementation (IMPL1 and IMPL2). These results are used for evaluating the performance of the two formulations. An indicator of the quality of the results is the difference between the working hours of the different inspection crews. As can be seen in Figure 5, IMPL2 (i.e. implementation of the objective function, Eq. (3)), provides better results than IMPL1. It can also be seen that IHS (both in IMPL1 and IMPL2) is not sensitive to the parameters of the algorithm since IHS converges to the same optimal solution in almost all the combinations of parameters regardless the objective function considered. On the other hand, Figure 6 presents the variance of the inspection time between the crews for the two implementations. IHS algorithm presents small variance values regardless the objective function used (IMPL1 and IMPL2, respectively). In both implementations, IHS converges to optimized solutions with almost no variance
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between the working hours of the inspection crews. While Figure 7 depicts the total number of iterations required for converging to the optimal solution. As can be seen in Figure 7, IMPL2 requires more iterations to achieve the optimal solution compared to IMPL1. The inspection time for IMPL1 is 1537.26 hours for all combinations of the parameters of the algorithm; the mean time between the 5 inspection crews varies from 1238.18 to 1238.19 hours while the inspection time for all crews varies from 986.07 to 1537.26 hours. On the other hand, for the case of IMPL2 the inspection time varies from 1288.55 to 1310.44 hours; the mean time varies from 1238.17 to 1238.22 hours while the inspection time for all crews varies from 1145.72 to 1310.44 hours. In particular, Figure 8 depicts the optimal districting and routing solutions obtained for the case of IMPL1 while Figure 9 presents the optimal districting and routing solutions that are obtained with the implementation of IHS and ACO algorithms to the formulation of Eq. (3). The above mentioned results indicate that IHS is capable of solving the districting problems of Eq. (2) and (3). In addition, although implementing the formulation IMPL1 requires less iterations compared to IMPL2 to converge to the optimal solution, a significant variance in the working hours of the inspection crews is noticed. This variance is reduced with the use of the objective function presented in Eq. (3). Despite the increased complexity of the formulation IMPL2, IHS manages to solve the problem and to provide a better-quality solution as seen in Figure 9. 6.1.2 Performance of four metaheuristics in solving the districting problem In the second part of the study performed for the city of Patras, four different nature inspired algorithms and the pure random search are examined with reference to their performance when implemented to solve the districting problem of the city of Patras. In particular IHS, HS, PSO and DE algorithms are applied and compared in two problems (A and B) with 5 and 15 inspection crews. A random search procedure is also 18
implemented and compared with the above mentioned algorithms. The objective function used for all algorithms is the one used in the formulation of Eq. (3). The performance of the metaheuristics is influenced by the selection of their parameters. In order to study the influence of the parameters for each metaheuristic algorithm, 32 combinations of the parameters are generated by means of LHS. The resulting optimization runs for dealing with the districting problem considered for defining the parameters of the four metaheuristics and the random search procedure are equal to 5 algorithms × 32 combinations = 160 optimization runs. The termination criterion used is the maximum number of function evaluations that was set equal to 200,000 for all algorithms. The parameters that are identified for each algorithm are: (i) For PSO, the number of particles NP defined in the range of [50, 200], the inertia weight w defined in the range of [0.01, 0.7], while the cognitive parameter c1 and social parameter c2 both defined in the range of [0, 1], the combinations considered are provided in Table 2. (ii) For DE, the population size NP defined in the range of [50, 200], the probability CR, the constant F and the control variable λ all defined in the range of [0, 1], the combinations considered are provided in Table 3. In order to ensure the properness of the parameter values used for all the algorithms mentioned before, the combinations of parameters used are generated with the use of LHS technique. The use of LHS ensures that every area of the sample space of each parameters will definitely be sampled. LHS generates 32 different sets of the parameters of each algorithm, one for every test run. Similar to the first part of the study performed for the city of Patras, the maximum time required between the inspection crews is considered as the necessary time for completing the inspection of the entire city. In the first problem examined (Problem A), 5 crews are considered for the inspection of the city of Patras. Figure 10 depicts the time required for the inspection of the building blocks of the city for each algorithm. The difference between the working hours required for performing the inspection in the
19
32 different solutions for each algorithm is used in evaluating the quality of the results. In Table 4 the maximum, minimum, average, standard deviation and the coefficient of variation (COV) of the required time is depicted. Although DE is slightly better with reference to the average inspection time, as it can be seen IHS is not sensitive to the parameters of the algorithm since it always converges to almost the same optimum solution, contrary to the other algorithms where significant differences between the optimum solutions are observed. The inspection time for IHS varies from 1288.55 to 1310.44 hours with an average value of 1295.01 hours, the standard deviation is equal to 5.45 and COV is equal to 0.42%. For the case of HS, the inspection time varies from 1249.42 to 1298.97 with an average value of 1282.12 hours, the standard deviation is equal to 14.08 and COV is equal to 1.10%.For the case of PSO, the inspection time varies from 1250.17 to 2071.43 with an average value of 1428.25 hours, the standard deviation is equal to 168.60 and COV is equal to 11.81%. For the case of DE, the inspection time varies from 1250.17 to 1294.94 with an average of 1257.66 hours, the standard deviation is equal to 12.57 and COV is equal to 1.00%. In the second problem examined (Problem B), 15 crews are considered for the inspection of the city of Patras. Figure 11 depicts the time required for the inspection of the building blocks of the city for each algorithm. As it can be seen in Table 5, although DE is also slightly better compared to the other metaheuristics with reference to the average inspection time, similar to problem A, IHS is not sensitive to the parameters of the algorithm as it concludes to almost the same optimum solution. The other algorithms present significant differences between the optimum solutions achieved for the 32 independent runs. The inspection time for IHS varies from 574.17 to 728.29 hours with an average value of 608.77 hours, the standard deviation is equal to 37.49 and COV is equal to 6.16%. For HS, the inspection time varies from 591.66 to 769.62 with an average value of 647.89, the standard variation is equal to 42.37 and COV is equal to 6.54%. For the case of PSO, the inspection time varies from 497.77 to
20
1130.73 with an average value of 784.60, the standard variation is equal to 143.13 and COV is equal to 18.24%. For the case of DE, the inspection time varies from 455.010 to 643.51 with an average value of 523.23, the standard variation is equal to 54.87 and COV is equal to 10.49%. Worth mentioning also that although the basic HS is slightly better compared to IHS with respect to the average inspection time for the case of five inspection crews, when the complexity of the problem increases (i.e. for the case of the 15 inspection crews) the superiority of the proposed IHS algorithm is clear. For the purposes of this work a personal computer that consists of the Intel Core 2 Quad Q6600 2.4 GHz with 4 physical cores was used. In the last part of this numerical investigation the computational cost of the metaheuristics is examined for 200,000 function evaluations. As it can be seen in Table 6 PSO requires more computing time compared to all other metaheuristics for both Problems A and B while HS and HIS require almost the same time for both problems, while the computing time required for solving Problem B is increased by 10% compared to Problem A.
6.2 Post-disaster Management Applielems d to the City of Thessaloniki Thessaloniki is composed by NSB = 471 grouped structural blocks with various areas and built-up percentages (see Figure 12). For this test case two different implementations are examined. In the first one the combination of IHS (districting problem of Eq. (3)) with ACO (routing problem of Eq. (5)) is implemented. In the second implementation the combination of HS (districting problem of Eq. (3)) with ACO (routing problem of Eq. (5)) is implemented. Due to the variance between the inspection time required by the available crews, the maximum time required by the inspection crews is also considered as the necessary time for the inspection of the entire city to finish. In the test case examined, 20 crews are considered for inspecting the city of Thessaloniki. Assuming that the crews work into two 8-hour shift work-groups, a total of 10 crews are available for 16 hours inspection per day. Figure 13 depicts the time required for the inspection of the building 21
blocks of the city for each algorithm. These results are used for evaluating the performance of the algorithms. The quality of the results is indicated by the difference between the working hours required for completing the inspection for the 32 independent runs performed for each algorithm. As can be seen in Table 7, IHS is not sensitive to the parameters of the algorithm since it converges to almost the same optimum solution. On the other hand, HS presents significant differences between the optimum solutions achieved. The inspection time for IHS varies from 5478.61 to 6034.62 hours with an average value of 5678.24 hours, the standard deviation is equal to 154.29 and COV is equal to 2.72%. For the case of HS, the inspection time varies from 5751.64 to 6643.47 with an average value of 6097.35, the standard variation is equal to 260.34 and COV is equal to 4.27%. This larger-scale test case and the results presented in Figure 14 show that IHS can achieve significantly better results compare to HS when dealing with the districting problem. It can also be seen that IHS produces results of lesser variation than HS as it appears to be less sensitive to its parameters. This confirms the finding of the Patras test example that when the complexity of the problem increases the superiority of the proposed IHS algorithm becomes obvious.
7.
CONCLUSIONS
In this study, the efficiency of nature inspired algorithms in dealing with difficult combinatorial optimization problems and in particular the districting and routing problems involved into a post-disaster management procedure is examined. Furthermore, a holistic approach on post-disaster infrastructure management is proposed and assessed with reference to the original one found in the literature. A number of significant conclusions can be drawn concerning the effectiveness of four nature inspired search algorithms when dealing with the post-disaster management problem. In particular, the performance of the improved harmony search algorithm is compared to the performance of the differential evolution algorithm, the particle swarm 22
optimization algorithm and the pure random search procedure; while the new formulation of the districting problem is also assessed. For an unbiased comparison of the four algorithms, a number of independent runs are performed corresponding to different combinations of the algorithmic parameters generated by the Latin hypercube sampling technique. A general conclusion that can be drawn from this study is that nature inspired algorithms can become very robust for solving difficult combinatorial optimization problems while improvements can further advance the performance of the algorithms. In particular, the improved version of the harmony search algorithm, presented in this study, manages to locate a much better solution than the basic harmony search algorithm by using a different structure while the modified formulation of the districting problem further improves the solution obtained. It is also noted that the improved harmony search algorithm is not sensitive with respect to parameter values, as it always yields almost the same solutions. The improved harmony search algorithm appears to present less variation in its results compared to the harmony search, the differential evolution and the particle swarm optimization. In the modified formulation, the districting problem is improved by defining the starting points of the inspection crews first and then the areas of responsibility rather than defining the areas of responsibility first and then the starting points. The modified formulation is simpler, thus improving the computational efficiency of the algorithms. An additional feature of the new objective function is that it incorporates the need for similar working hours between the inspection crews. Worth mentioning also that in the test cases examined, it was observed that the difference on the performance increases in proportion to the complexity of the problem. In particular, for the city of Patras corresponding to a total built area of 18.559.676 m2, 1310.44 hours of inspection are required with the use of 10 crews working in two 8 hour shifts (5 crews per shift) or 728.29 hours of inspection with the use of 30 crews in two 8 hour shifts (15 crews per shift). For the city of
23
Thessaloniki corresponding to a total built area of 154.205.128 m2, the corresponding inspection time is equal to 6034.62 hours with the use of 20 crews in two 8 hour shifts (10 crews per shift). Future research can be focused on real-time solution where the data that affect the solution are received dynamically.
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TABLES Table 1. Parameters combinations for sensitivity analysis of the HS and IHS algorithms HMS 11 20 11 12 15 17 8 6 20 18 9 10 7 16 7 8 15 5 5 16 10 13 12 17 14 9 19 13 6 18 19 14
HMCR 0.090 0.139 0.002 0.098 0.269 0.240 0.049 0.231 0.072 0.186 0.146 0.173 0.250 0.016 0.260 0.208 0.108 0.059 0.044 0.198 0.075 0.165 0.151 0.189 0.289 0.122 0.221 0.128 0.293 0.027 0.272 0.036
PAR 0.171 0.853 0.913 0.698 0.585 0.251 0.739 0.463 0.638 0.985 0.055 0.658 0.235 0.815 0.121 0.397 0.299 0.081 0.314 0.525 0.414 0.002 0.201 0.904 0.373 0.793 0.962 0.601 0.490 0.142 0.548 0.771
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Table 2. Parameters combinations for sensitivity analysis of the PSO algorithm NP 158 109 142 61 119 102 121 150 128 174 179 188 155 96 85 64 70 59 164 133 171 193 197 148 113 82 186 135 99 77 90 52
c1 0.648 0.108 0.774 0.408 0.364 0.132 0.970 0.400 0.796 0.314 0.049 0.731 0.281 0.011 0.817 0.562 0.492 0.092 0.459 0.163 0.617 0.907 0.584 0.954 0.247 0.204 0.525 0.299 0.668 0.707 0.881 0.863
c2 0.387 0.112 0.184 0.431 0.524 0.716 0.829 0.147 0.976 0.883 0.259 0.075 0.028 0.306 0.595 0.729 0.226 0.451 0.053 0.645 0.313 0.193 0.358 0.562 0.852 0.751 0.475 0.578 0.668 0.967 0.918 0.798
w 0.474 0.362 0.105 0.157 0.541 0.137 0.391 0.491 0.054 0.278 0.664 0.239 0.198 0.209 0.033 0.638 0.420 0.333 0.628 0.698 0.576 0.568 0.446 0.256 0.014 0.414 0.079 0.167 0.339 0.300 0.611 0.514
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Table 3. Parameters combinations for sensitivity analysis of the DE algorithm NP 106 113 139 151 185 200 156 172 56 96 85 176 73 125 148 189 192 64 161 77 118 79 137 178 132 166 68 98 104 123 91 50
λ 0.648 0.108 0.774 0.408 0.364 0.132 0.970 0.400 0.796 0.314 0.049 0.731 0.281 0.011 0.817 0.562 0.492 0.092 0.459 0.163 0.617 0.907 0.584 0.954 0.247 0.204 0.525 0.299 0.668 0.707 0.881 0.863
CR 0.387 0.112 0.184 0.431 0.524 0.716 0.829 0.147 0.976 0.883 0.259 0.075 0.028 0.306 0.595 0.729 0.226 0.451 0.053 0.645 0.313 0.193 0.358 0.562 0.852 0.751 0.475 0.578 0.668 0.967 0.918 0.798
F 0.942 0.779 0.187 0.194 0.896 0.653 0.079 0.524 0.710 0.921 0.022 0.118 0.232 0.283 0.404 0.971 0.330 0.555 0.144 0.730 0.565 0.470 0.858 0.371 0.274 0.809 0.824 0.618 0.667 0.428 0.459 0.057
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Table 4. Patras test case - Statistical analysis of IHS, HS, DE, PSO algorithms (Problem A – 5 Crews) Minimum Time (h) Maximum Time (h) Average Time (h) Standard Deviation (h) COV (%)
IHS 1288.55 1310.44 1295.01 5.45 0.45
HS 1249.42 1298.97 1282.12 14.08 1.10
DE 1250.17 1294.94 1257.66 12.57 1.00
PSO 1250.17 2071.43 1428.25 168.60 11.80
Table 5. Patras test case - Statistical analysis of IHS, HS, DE, PSO algorithms (Problem B – 15 Crews) Minimum Time (h) Maximum Time (h) Average Time (h) Standard Deviation (h) COV (%)
IHS 574.17 728.29 608.78 37.49 6.16
HS 591.66 769.62 647.89 42.38 6.54
DE 455.01 643.51 523.23 54.87 10.49
PSO 497.77 1130.73 784.60 143.13 18.24
Table 6. Computational time for solving the two problems (in seconds) Method PSO DE HS IHS
Problem A 354.99 190.32 123.36 122.91
Problem B 383.85 210.36 135.05 135.70
Table 7. Thessaloniki test case - Statistical analysis of IHS and HS algorithms (10 Crews) Minimum Time (h) Maximum Time (h) Average Time (h) Standard Deviation (h) COV (%)
HIS 5478.61 6034.62 5678.24 154.29 2.72
HS 5751.64 6643.47 6097.35 260.34 4.27
30
FIGURES
Figure 1. Building blocks, Area, fB, AB,max
(a)
(b)
(c)
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(d)
(e)
Figure 2. Four step procedure for defining the area of responsibility
Step 1: Initialize Parameters
Step 2: Initialize HM
F(s): objective function si: design variable n: number of design variables HMS: number of solution vectors in HM HMCR: HM considering rate TC: termination criterion
For i=1 to HMS Random generation of the solution vector si Calculate F(si)
Step 3: Improvise new harmony Step 4: Harmony memory update
Step 5: Check of convergence
For i=1 to n Random selection or memory consideration according to Eq. (16) Calculate F(si)
Yes
Is better
No Yes Satisfied
Include new harmony and Exclude worst harmony
Repeat Step 3
Terminate Computations
No
Repeat Steps 3,4
Figure 3. Flowchart of the improved harmony search algorithm
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Figure 4. City of Patras - Subdivision into structural blocks
IMPL1
IMPL2
1550
Inspection Time (hours)
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Combination of Parameters
Figure 5. City of Patras -Minimum inspection time achieved for each combination of the parameters
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IMPL1
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Combination of Parameters
Figure 6. City of Patras -Standard deviation between the inspection crews
IMPL1
IMPL2
2.00E+05 1.80E+05
Number of Iterations
1.60E+05 1.40E+05 1.20E+05 1.00E+05 8.00E+04
6.00E+04 4.00E+04 2.00E+04
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Combination of Parameters
Figure 7. City of Patras -Number of iterations for achieving the best solution for each combination of the parameters
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Figure 8. City of Patras -Implementation 1- (a) Subdivision into inspection areas and (b) Best route for each inspection crew
Figure 9. City of Patras -Implementation 2- (a) Subdivision into inspection areas and (b) Best route for each inspection crew
35
IHS
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2100 2000
Inspection Time (hours)
1900 1800 1700 1600 1500 1400 1300
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Combination of Parameters
Figure 10. City of Patras -Problem A: Minimum inspection time required IHS
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Combination of Parameters
Figure 11. City of Patras -Problem B: Minimum inspection time
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Figure 12. City of Thessaloniki- Subdivision into structural blocks
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IHS
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6800 6600
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6400 6200 6000 5800 5600 5400 5200 0
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Combination of Parameters
Figure 13. City of Thessaloniki-Minimum inspection time achieved for each independent run
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Figure 14. City of Thessaloniki - Subdivision into inspection areas
39
