Using the Ant Algorithm to Derive Pareto Fronts ... - Columbia University

Using the Ant Algorithm to Derive Pareto Fronts for Multiobjective Siting of Emergency Service Facilities Nan Liu, Bo Huang, and Xiaohong Pan commensurable and cannot be integrated into a single objective. With that observation, the notion of Pareto optimality has been introduced instead of the optimality concept in single-objective optimization (3). However, Pareto optimal solutions cannot be uniquely determined, that is, usually there exists a set of solutions that all satisfy Pareto optimality, which form the Pareto front in the solution space. Hence, decision makers usually cannot find a best solution that dominates all the others. On the contrary, they may prefer to seek a Pareto front, which is formulated by the Pareto optimal solutions, so that they can evaluate the trade-offs among these solutions and make decisions accordingly (4–7 ). The ant algorithm is an adaptive construction heuristic that combines with a local search measure to direct its search in the solution space. It is a family of metaheuristics, which can be implemented to solve different types of nonpolynomial deterministic problems, for example, the traveling salesman problem and the vehicle routing problem (VRP) (8). However, so far it has rarely been used to solve MO optimization problems. This research explores its application to the MO siting of ESFs by deriving the Pareto front for the problem. This paper is organized as follows. The next section, which reviews the related literature, is followed by a detailed discussion of the proposed MO-ant algorithm. Then a case study on MO siting of fire stations is presented. Finally, the proposed method is summarized and its future development suggested.

Efficient and timely response during accidents has received increased attention from practitioners and researchers. The siting of emergency service facilities (ESFs) plays a crucial role in determining the efficiency of safety protection and emergency response. This paper explores a novel multiobjective ant algorithm for the siting of ESFs. With the aid of the geographic information system, the algorithm finds a population of solutions, uses Pareto ranking to sort these solutions, and derives the Pareto front. It is demonstrated that the algorithm successfully captures a pool of nondominated solutions and thereby provides decision makers with a set of alternative solutions. The case study also demonstrates how decision makers may choose one “best” solution from the pool according to their preference or determinant criteria.

Efficient and timely response during accidents has always been a heated area for practitioners and researchers. In the wake of the September 11, 2001, terrorist attacks, security and emergency response issues have received increased attention. Emergency service facilities (ESFs), for example, hospitals, fire stations, and police stations, are equipped with necessary personnel and paraphernalia for providing prerequisite support and saving life and property in the event of an accident. The siting of emergency facilities plays a crucial role in determining the efficiency of safety protection and emergency response. ESFs should be sited in such a strategic way that they can serve as many areas as possible in a reasonable time during daily operations, and there should be an efficient cooperation among them in time of necessity. ESF siting problems are dedicated to providing the optimal locations of single or multiple ESFs in a civilization system based on subjectively defined criteria, especially when the analysis is extended to practical scenarios, for example, irregular geographical constraints that are hard to be configured exist; or the problem is “wicked,” illdefined, or semistructured (1, 2); or the size of the problem is out of computational limits. In many cases the problems are made even more complicated because multiple, often conflicting, objectives are involved. It is realized that among such multiobjective (MO) optimization problems, multiple objectives under consideration are often non-

LITERATURE REVIEW ESF siting problems have been well studied by a number of researchers during the past 30 years. Marianov and ReVelle provide a general review of the related models and methods (9). They also point out certain important issues on siting ESFs (servers), for example, the number of servers to be sited, the longest time for which customers involved in an emergency event can afford to wait, the definition of coverage, the actions to be taken when servers are not available, the balanced allocation of workload to each server, and the data availability. They argue that once all the issues mentioned above are addressed, a solution method may be chosen to “solve” the emergency system design problem as it is finally characterized.

N. Liu, Grado Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. B. Huang, Department of Geomatics Engineering, University of Calgary, Calgary, Alberta, T2N 1N4, Canada. X. Pan, Department of Civil Engineering, National University of Singapore, 1 Engineering Drive 2, E1A #08-25, Singapore 117576.

Emergency Facility Siting Models In general, emergency facility siting models fall into two categories: deterministic models and probabilistic models. Deterministic models do not consider the probabilities of servers being busy and are usually formulated in integer linear programming problems with objectives

Transportation Research Record: Journal of the Transportation Research Board, No. 1935, Transportation Research Board of the National Academies, Washington, D.C., 2005, pp. 120–129.

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of minimizing cost and maximizing covering or other measures of merits. However, probabilistic models take explicit account of the probabilities of servers being busy to compute the amount of redundancy actually needed (9). In other words, they use explicit probabilistic constraints inside the mathematical programming models, most of which are nonlinear. Because the case study to be introduced later in this paper follows the principle of deterministic models, the following review will focus on those types of models. The first model on emergency service covering is probably the location set covering problem (LSCP) (10, 11). The LSCP seeks to site the minimum number of servers in such a way that all demand nodes are covered by at least one server within a standard time or distance. However, it may make use of excessive resources to cover all points of demand, no matter how small or remote. Church and ReVelle proposed the maximal covering location problem (MCLP), in which the economic budget is reflected as a constraint on the number of servers to be positioned (12). The MCLP seeks the placement of a fixed number of servers (probably insufficient to cover all demand nodes) to maximize the coverage of the demand nodes. The importance of each demand node is represented by a weight value, for example, population or calls for emergency service. The most general formulation of the model types mentioned above is known as the facility location, equipment emplacement technique (FLEET) model (13), which determines the locations of a limited number of engine companies (i.e., pumper brigades) and truck companies (i.e., ladder brigades) as well as the fire stations that house them. The objective of the model is to maximize the population covered by an engine company within the engine company distance standard and a truck company within the truck company distance standard. In this model the coverage is gained by simultaneously siting two types of service in relation to their respective distance standards. The FLEET model formulations assume that all servers are available at all times; however, this may not always be true because congestion may occur in real operations. Deterministic models can be developed to address the congestion issue, and so they are also called redundant coverage optimization models. Redundant coverage models seek to locate servers in such a way that a demand node can be served by more than one server within the distance limit. Daskin and Stern formulated a model to maximize the redundant coverage given a fixed number of servers, in which the redundant coverage is measured as the difference between the number of servers stationed within the distance standard and the minimum number required for coverage (14).

Optimal Siting of Fire Stations The optimal location of fire stations has been extensively studied, and a range of models has been developed. Doeksen and Oehrtman used a general transportation model based on alternative objective functions to obtain optimal fire station locations for a rural fire system (15). The different objectives used to obtain the optimal sites were minimizing response time to the fire spot, minimizing total mileage for fighting rural or county fires, and maximizing protection of per dollar’s worth of burnable property. Plane and Hendrick used the max covering distance concept to develop a hierarchical objective function for the set-covering formulation of the fire station location problem (16 ). The objective function permitted the simultaneous minimization of the number of fire stations and the maximization of the existing fire stations within the minimum total number of stations.

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Hogg used a set-covering technique, which minimizes the total number of fire appliance journey times to fires for any given number of fire stations and applied that to the city of Bristol (17 ). Badri et al. underline the need for a multiobjective model in determining fire station locations (18). The authors used a multiple criteria modeling approach via integer goal programming to evaluate potential sites in 31 subareas in the state of Dubai. Their model determines the location of fire stations and the areas they are supposed to serve. It considers 11 strategic objectives that incorporate travel times and travel distances from stations to demand sites and also other costrelated objectives and criteria that are technical and political in nature. Tzeng and Chen used a fuzzy multiobjective approach to determine the optimal number and sites of fire stations in Taipei’s international airport (19). A genetic algorithm (GA) was used to solve the problem and was compared with the enumeration method. The results bear evidence to the fact that a GA is suitable for solving such location problems. Nevertheless, its efficiency still remains to be verified by means of large-scale problems. Most of the aforementioned researchers employed the discrete location model or its variations to site fire stations. The modeling techniques and solution algorithms of that category of problems have been methodically reviewed in Mirchandani and Francis (20) and in Daskin (21). Although multiple objectives have been considered in many research projects, most researchers just used a scalarization method to derive a subjectively defined best solution (3). The method of generating multiple nondominant Pareto optimal alternatives has rarely been addressed in facility-siting problems. As mentioned before, practitioners may, however, prefer the latter method because they could have a further evaluation of the multiple alternatives before they make their final decision. Xiao et al. developed a GA to generate multiple alternatives for MO site-search problems, but they dealt only with the siting of a single facility rather than multiple facilities simultaneously and did not consider any coverage objective (7 ). In that regard, this research addresses the problem of MO siting of multiple ESFs and proposes a quality ant algorithm to generate multiple alternatives for decision makers.

MULTIOBJECTIVE ANT ALGORITHM FOR EMERGENCY FACILITY SITING Ant algorithms, inspired by nature, simulate the capability of an ant colony to locate the shortest path between its nest and the food source while searching for food. Ant algorithms use a special feature, the pheromone matrix, to record the historical information collected by ants and use this information in further exploration in the solution space. They have been applied to solve a variety of problems, for example, the quadratic assignment problem (22), VRP (23), scheduling problems (24), and telecommunications network problems (25). However, the role of ant algorithms in MO siting analysis as generators of alternatives has scarcely been investigated. In ant algorithms, a feasible solution is formulated by the movements of ants. The information contained in the pheromone matrix, in which each entry corresponds to a position ants can move to, controls the movements of ants. The larger the entry value in the pheromone matrix, the higher the probability that ants may move to that corresponding position. A general schematic structure of the ant algorithms is shown in Figure 1. It can be extended to suit the need of MO siting analysis of ESFs. But before that is discussed, the concepts of Pareto optimum and Pareto ranking are introduced; these concepts provide a basis for evaluating different MO solutions.

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ers (28–30). The strategy, the main component of which is Pareto ranking, is illustrated by the following pseudo code:

Initialization Phase Initialize the pheromone matrix Randomly generate a certain number of solutions Iteration Phase (Until Stop Criteria are Reached) Construct new solutions Perform local search Update the best found solution Update the pheromone matrix (Other operations may be included according to different versions of ant algorithms) End Output the result

Pareto-Ranking ()

The notion of Pareto optimum is from Pareto’s original work (26 ). Consider a k-objective min optimization problem:

for i = 1 to sol_num /* initially, all unranked */ rank[i] = −1 remain = sol_num curr_rank = 1 while (remain > 0) do /* search all unranked individuals */ { for i = 1 to popsize /* non-dominated with curr rank */ if (rank[i] == −1) if (not IsDominated(i, curr rank)) { rank[i] = curr rank remain = remain-1 } curr rank = curr rank + 1 }

min f ( x ) = [ f1 ( x1 ), f2 ( x2 ), f3 ( x3 ), . . . , fk ( xk )]T

IsDominated(ix, curr_rank)

FIGURE 1

Schematic structure of ant algorithms.

Pareto Optimum

(1)

{

subject to the following: g(x ) ≥ 0

(2)

h (x) = 0

(3)

bool flag = FALSE for i = 1 to sol_num do if (i != ix and (rank[i] = −1 or rank[i] = curr_rank)) { k1 = 0 /* count # dominating obj */ for j = 1 to num_obj if (obj[ix] [j] < obj[i] [j]) /* a dominating obj */ k1 = k1 + 1 if (k1 == 0) /* ix is dominated */ { flag = TRUE i = sol_num + 1 } } return flag

where –x = the decision vector, g– = the inequality constraint, and – h = the equality constraint. A solution vector –x * is said to dominate another solution vector –x if and only if ∀i, fi ( x * ) ≤ fi ( x ) ∧ ∃i, fi ( x * )i < fi ( x ), i ∈ {1, 2, 3, . . . , k} A solution is called Pareto optimal if no other solutions dominate it. In practical MO optimization problems, it is usually impossible to find a unique solution that dominates all other solutions. Instead it is expected that a number of Pareto optimal (or called nondominated) solutions can be found. A Pareto front is formed by those Pareto optimal solutions, in which an increase in one objective will surely cause a decrease in one or more other objectives. The goal of MO optimization is to generate feasible solutions that are on, or close to, the Pareto front (7 ). From those alternatives situated on or around the Pareto front, decision makers can make their final decision (4, 5). Pareto Ranking Traditional ant algorithms are used to solve single-objective optimization problems. A common measure to convert an MO optimization problem into a single objective one is to use scalarization methods, for example, the weighting method or minimax method (3). However, those methods cannot efficiently characterize the Pareto front for an MO optimization problem. To overcome them, Goldberg first proposed a selection strategy based on Pareto dominance (27); the strategy has since been modified and improved by other research-

} The function of Pareto-ranking() examines each solution to determine whether it will be dominated by others. According to the dominance relationships among solutions, this function classifies all solutions into a number of classes; the nondominated solutions are in the uppermost class. The solutions in a lower class, for example, the class with curr_rank as 3, are dominated by at least one solution in this class’s upper classes, that is, the class with curr_rank as 1 or 2. Solutions in the same class cannot dominate each other. The function of IsDominated(ix, curr_rank) is a subfunction that determines whether a specified solution is dominated by other unclassified solutions [these solutions will have a higher curr_rank (i.e., in a lower class) as compared with the classified solutions]. The function compares each objective of a solution, denoted as –x , to those of others. If it finds a solution in which all objectives are less than those of the solution –x , then the solution –x is dominated and the function returns a boolean value TRUE. The pseudocode of the function illustrated here is for a min optimization problem, and the contrary is for a max case. The Pareto optimum of the algorithm above is theoretically different from its original form (7 ). The Pareto optimum here refers to

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the current optimal solutions, that is, those solutions in the uppermost class, and the nondominance applies only to the current solution set. The solution population is expanded by including the Pareto optimal solutions in the solution set found in each iteration. And the Pareto front is determined by the Pareto optimal solutions in the final solution population. Geographical Representation of Emergency Facility Siting Problems In this paper, to define the ESF siting problem, it is assumed that the study area is represented by a grid map, in which the map features, for example, land areas, roads, and buildings, are characterized by certain tactic cells. All the cells are of the same size and each cell stores the information about its location, which is uniquely determined by the row and column number of this cell. The grid map can be attained easily by the rasterization function in a geographical information system (GIS) (31). Through the use of a grid map, an ESF siting problem can be defined easily. For example, the restriction of irregular geographical boundaries can be done by recording only those cells in the feasible region; a coverage objective can be evaluated by counting the number of cells in the coverage constraint.

Multiobjective Ant Algorithm The MO ant algorithm uses a number of ants to represent the locations of ESFs to be sited one by one. It controls the movement of these ants on the grid map according to certain rules so that they can help find the optimal locations of emergency facilities the user wants to site. The flowchart of the algorithm is shown in Figure 2. As do other ant algorithms, MO-ant has the following four key components: pheromone matrix and its updating policies, solution construction rules, local search measures, and evaporation mechanism (8).

matrix is a two-dimensional matrix corresponding to the grid system, for example, an m × n matrix is a grid cell matrix with m rows and n columns. Each cell of the matrix is filled with a positive float value called pheromone value representing the “desirability” of choosing the corresponding cell (i, j) on the grid system (i is the row number, j is the column number) as a location for one of the ESFs to be sited. Those cells representing the infeasible areas, for example, a body of water, are assigned the value of zero, thus avoiding their selection as the candidate sites for the ESFs. MO-ant determines the locations of the ESFs by controlling the artificial ants to detect the desirability and directing them to move to those desirable cells. The probability of an ant choosing one cell is the function of the desirability of that cell. The larger the desirability of a cell, the higher the ant’s probability of moving to that cell. At the beginning of the algorithm, the pheromone matrix is initialized. Because there is no initial information contained in the matrix at that stage, each entry of the matrix is assigned with the same initial value. The pheromone matrix is updated by means of the iterations in the MO-ant’s running process. The update policy of MO-ant adopts a merit-based procedure, which fortifies the pheromone value of a cell according to the solutions it builds up. The better solution a cell builds, the greater the pheromone value of it will be enhanced. The matrix update policy is based on the rationale that the cells forming good solutions have larger probabilities of constituting the components of the optimal solution. The mathematical formulation of the update policy is as follows: τ ij (t + 1) = τ ij (t ) + Δτ  xij (t ) rank(i, j ) where τij(t) = pheromone value of cell (i, j) at iteration t; xij(t) = binary variable, which equals 1 if the cell (i, j) is included in a solution found at iteration t, zero otherwise; Δτ = basis enhancement level; and rank(i, j) = rank of the solution the cell (i, j) builds.

Pheromone Matrix and Updating Policies The pheromone matrix is a mechanism used in the ant algorithm to store the historical “good” information. In MO-ant the pheromone

The update policy is applied to all solutions found in a local search. The rank of a solution is attained by the aforementioned Pareto-ranking () function.

Initialization of the pheromone matrix

Stop criteria reached?

Yes

No Construct new solutions Local search Pareto ranking of newly found solutions Update of the solution population Update of the pheromone matrix Evaporation

FIGURE 2

( 4)

Flowchart of MO-ant algorithm.

Pareto ranking of the solution population

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Solution Construction

Evaporation

The solution is constructed on the basis of the pheromone matrix. The construction is implemented as a linear search through a roulette wheel with slots weighted in proportion to cell values in the pheromone matrix. Simply stated, the probability of choosing the cell (i, j) as a location for one of the emergency facilities to be sited is calculated as follows:

Evaporation is a commonly used measure in some other ant algorithms, for example, the ant colony system, to force ants to forget the “bad” information collected before and prevent the algorithm from falling into a local optimum (32). Toward the end of each iteration, the evaporation mechanism is activated in MO-ant and controlled by a parameter called evaporation ratio. This results in a reduction of the cell values of the pheromone matrix. For example, if the evaporation ratio equals 10%, then the value of each cell in the pheromone matrix will be reduced to 90% of its original value.

Pij (t ) =

τ ij (t ) ∑ ∑ τij (t ) i

(5)

j

where Pij(t) is the probability of choosing the cell (i, j) at iteration t and τij(t) is the pheromone value of the cell (i, j) at iteration t. Because the algorithm starts with a uniform pheromone matrix, all feasible cells have the same probability of being chosen. It means that all the ants randomly choose a feasible cell and then form the initial solution.

Two-Phase Local Search The local search is performed immediately after the newly constructed solution is obtained. A novel two-phase local search algorithm has been developed in MO-ant. The first phase of the local search is called the neighborhood random search (NRS), which is conducted for a specific number of iterations. Within a single iteration of NRS, the ants randomly move from their current cells to other cells within a limited distance, for example, 3 km. The objectives are then revaluated and recorded. Subsequent to the first-phase local search, a Pareto ranking process is conducted and then the second-phase local search is applied to those solutions whose rank is one. The second-phase local search is called the adaptive enumeration neighborhood search (AENS). In AENS, each of the ants moves to every cell within a certain distance from its current cell while keeping the other ants fixed in their original cells. If the total sum of the objectives has been improved, the ant enters the cell that improves the sum of the objectives and restarts a new AENS. Note that if the total sum of the objectives is improved, the new solution cannot be dominated by the old one and it possibly dominates the old one, and thus it is expected to be “better” than the old one. The AENS is a thorough and rigorous local search method because it continues until no movements of the ants can improve the sum of the objectives. But there is a “myopic” characteristic associated with the AENS in that it considers just the effect of moving only one ant, while not taking into account the interactive effect of moving multiple ants. Thus it might lose a better solution that could be obtained only by moving multiple ants simultaneously. However, the choice of AENS is attributed to computational complexity. For example, supposing that an ant has n alternative cells to move to, the computational complexity of using the AENS will be on the order of 6n; however, if moving multiple ants is considered, the computational complexity will be on the order of nm (where m is the number of ESFs to be sited), which can be insupportable if n is large. After the local search, the newly found solutions are ranked using Pareto-ranking(). Those solutions with Pareto rank as one are appended into the solution population. Note that the stop criterion of this algorithm is the size of the solution population. When the size of the solution population exceeds a certain number, the algorithm stops.

MULTIOBJECTIVE SITING OF FIRE STATIONS IN SINGAPORE Background Information Singapore has 18 fire stations positioned around the island, each having the basic equipment of at least one fire engine, one red rhino (light fire attack vehicle), and one ambulance. The effectiveness of the fire stations in covering the transportation routes of hazardous materials (HAZMATs) through Singapore is of primary concern in this case study. It is imperative for authorities to be forearmed to tackle a crisis situation arising out of HAZMAT transportation, for instance, explosion and crashing. That requires a proper assessment of the existing fire stations in regard to their location and their ability to promptly reach the accident sites along the transportation routes. The Singapore Civil Defense Force (SCDF) has approved specific routes for transporting HAZMATs and other petroleum products in Singapore (Figure 3). These routes (termed SCDF routes) keep away from densely populated areas and water catchment areas, and HAZMAT transportation is allowed only between 7 a.m. and 7 p.m., when sufficient daylight exists for remedying an accident. The vehicles are not allowed to travel along expressway tunnels, which may otherwise lead to major pile-ups during accidents. According to SCDF, the targeted response time is 8 min, from the moment of receipt of an emergency call to that of the arrival of a fire engine at the accident site. However, with the recent heightened consideration of security, this project aims to reduce the response time from 8 to 5 min with the addition of six fire stations. Other objectives considered in this project include determining a suitable distance between the fire stations and maximizing the areas that can be served by fire stations within 6 min. To be more specific, the three objectives are stated as follows: • Maximize the coverage of the routes not covered by existing fire stations. Some sections of the SCDF routes cannot be served within 5 min by the existing fire stations. The objective is to site six new fire stations throughout Singapore to maximize the coverage of these parts of SCDF routes within 5 min. • Achieve a reasonable distance between fire stations. The intention here is to obtain optimal coverage and efficient cooperation among fire stations. Investigations by local authorities revealed that the distance between one fire station and the fire station nearest to it must be from 1 to 9 km. That is a reasonable distance; it is not so long that it prevents efficient cooperation among stations nor is it so short that it causes overlapping and redundancies of services.

Liu, Huang, and Pan

FIGURE 3

125

Existing fire stations and SCDF routes in Singapore.

• Maximize the area that can be served by fire stations within 6 min. The third goal is to maximize the coverage of the uncovered land by means of the additional fire stations. Above and beyond combating the HAZMAT accidents on the SCDF routes, fire stations will have to render a great many additional services to places located elsewhere. This project therefore takes into account those urban and suburban areas nonreachable in 6 min by the existing fire stations.

Methodology

Construction of Two-Level Grids The whole Singapore map is converted into a raster form. The cell size is of foremost consideration when converting a continuous map into a raster map. Two scalars of the cell size with respect to two raster systems (macro and micro) are used to keep the computational burden within a tolerable range, simultaneously ensuring data accuracy. The area for siting the fire stations is represented by a macro raster map (125 rows × 215 columns) with a larger cell size of 200 m. This larger cell size was reached by considering the present customary size of Singaporean fire stations and their surroundings, 200 m × 200 m. The macromap is used to locate additional fire stations to reduce the computational burden, hence reducing processing time. The micro raster map employs a smaller cell size, no greater than the width of SCDF routes, 25 m. The micromap snaps its extent to the macromap, thus ensuring that the microgrid coincides with the macrogrid. The micromap is used to determine the coverage of fire stations on SCDF route cells and land cells in Singapore. In Figure 4, the macrogrids of 200 m × 200 m are represented by the larger squares. The microgrids of 25 m × 25 m are the smaller squares located within the larger macrosquares. Those microroute and land cells inside the macrocells falling under the coverage (buffer) of a fire station are shown in black. The uncovered microroutes and land cells are shown in grey.

Calibration of Response Time Function According to the SCDF, the fire engines should reach any section of SCDF routes within 8 min. The response time function of a fire station can be estimated as follows (33): T = t0 + K  r

(6)

where T = response time of fire station; r = distance in kilometers; t0 (min) = operational readiness time (time taken for the fire engine to leave the fire station on receiving the call), which is 1.0 min given by the SCDF; and K = traffic impedance factor. An experiment was conducted in a GIS environment to estimate the value of K using data obtained from local fire stations and transport authorities. The smallest radius of the fire station buffer covering all SCDF routes was estimated to be 5.30 km. Then K was calculated by substituting 5.3 km for r and 8.0 min for T in the response time function. A conservative approach is adopted to estimate K, thus making the approximation more reliable. The final estimated response time function is as follows: T = 1.0 + 1.32  r

( 7)

MO Optimization Model The MO optimization model of the problem is formulated as follows: max[μ1 ( L ), μ 2 ( L ), μ 3 ( L )]T L

(8)

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Macro Grid (200m) Route Cells

Fire Station j

Fire Station i

Coverage

ce tan Dis

Land Cells

Micro Grid (25m) FIGURE 4

Macro- and microgrids.

subject to the following: μ( L ) =

xi ( L ) x +i

∀i = 1, 3

μ 2 ( L ) = min{x2 (l )

( 9)

∀l ∈ L}

(10)

In Equation 10, ⎧1 ⎪ U U ⎪⎪( D − dl ) ( D − D) x 2 (l ) = ⎨ L L ⎪( dl − D ) ( D − D ) ⎪ ⎪⎩0

cells representing the routes uncovered by the buffer of existing fire stations within 5 min. Also, the optimistic value x3+ in Objective 3 is the sum total of all land cells not covered by the buffer of current fire stations within 6 min. Equation 10 furnishes the normalization function of Objective 2. The function calculates the minimal achievement level of the proposed fire stations as the overall achievement level of Objective 2. The achievement level of an individual fire station is a segmented linear function of its distance from its nearest counterpart as shown in Equation 11.

if dl = D if DU ≥ dl > D if D > dl ≥ D L

Computation Results and Analysis (11)

otherwise

In Equations 8, 9, 10, and 11, L = set (solution) that represents the locations of new fire stations; µi(L) = normalization function of objective i, which turns the objective value to its achievement level (a real number between 0 and 1); xi (L) = value of objective i given a solution L; xi+ = optimistic value of objective i; x2(l) = achievement level of Objective 2 with respect to the fire station located at l; dl = distance between the fire station located at l and its nearest counterpart; D = desired distance between two fire stations; DU = upper bound of the distance between two fire stations; and D L = lower bound of the distance between two fire stations. Equation 9 is the normalization function of Objectives 1 and 3. The optimistic value x+1 in Objective 1 is the total number of all grid

The algorithm was coded in C++ on a desktop PC with Intel PIV processor (2.80 GHz) and 216 MB of RAM running WinXP. The parameters of MO-ant are set as follows: The initial pheromone value is set as 1.0. The basis enhancement level is set as 6.0. The NRS is limited to 100 iterations, and the search radius is 3 km. The scope of the AENS is within 0.5 km of the original ant location. The evaporation ratio is 10%. The solution population size is limited to 500. Eight independent runs using different rand seed generators were conducted; the statistics of the computational results are presented in Table 1. Table 1 shows that the computational results of eight independent runs are comparably good in regard to the criteria used, for example, mean achievement level value, the max–min achievement level value of all nondominated solutions, and the sum of the mean achievement level values. Particularly, for the sum of the mean achievement level values, the maximum deviation to its mean of eight runs is within 3%. This indicates that the MO-ant algorithm works stably and is quite robust in all eight independent runs. The average computational time of eight runs is about 4,061 s, which is reasonable because of the large size and complex nature of the problem. For the sake of comparison, 10,000 random solutions are generated and Pareto ranked. Those random solutions with Pareto rank as

Liu, Huang, and Pan

TABLE 1

–µ 1 –µ 2 –µ 3

Max–min Sum Num Time (s)

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Statistics of Computational Results 1

2

3

4

5

6

7

8

Avg

0.6297 0.6534 0.5606 0.6207 1.8437 29 3624

0.6076 0.6736 0.5686 0.6131 1.8498 44 3821

0.6152 0.7082 0.5162 0.6484 1.8396 58 4380

0.5785 0.6752 0.6205 0.6402 1.8742 50 3842

0.5689 0.6513 0.6083 0.5938 1.8832 42 4628

0.5367 0.7480 0.5985 0.6606 1.8285 45 3467

0.6240 0.6812 0.5864 0.6232 1.8916 58 4538

0.6276 0.6903 0.6008 0.6396 1.9187 43 4187

0.5985 0.6852 0.5825 0.6300 1.8662 46.125 4061

NOTE: –µ : mean value of the ith achievement levels of all nondominated solutions after a run i Max–min: maximum of the minimum achievement level among three objectives of all nondominated solutions after a run Sum: sum of the mean value of three objectives Num: total number of nondominated solutions after a run Time: computation time of a run Avg: average value of the eight independent runs

one are selected to draw a Pareto front. To make the picture clearer, those nondominated solutions form a triangular mesh (the left void mesh in Figure 5). Without loss of generality, the computational results from Experiment 6 (Table 1) was taken to make comparisons. There are a total of 45 nondominated solutions found by MO-ant algorithm in the final solution population (500 solutions). See Table 2 for more details. These solutions may constitute a candidate pool for decision makers. The Pareto front of these solutions is also drawn in Figure 5 and represented by a triangular surface (the right solid surface in Figure 5). The comparison between the random solutions and MO-ant solutions suggests the clear effectiveness of the latter approach. Figure 5 clearly shows that the MO-ant Pareto front outperforms the random one, which indicates that MO-ant provides a much better Pareto front. In this study MO-ant is able to generate the Pareto front for MO ESF siting problems. That suggests a promising future for incorporating

MO-ant into a decision support system. In addition, the performance of MO-ant can be improved in many ways, including fine-tuning the parameters and modifying certain components of the algorithm. In the current case, changing the solution construction function may be helpful to the improvement of the algorithm. Other solution construction functions can be found in Stützle and Dorigo (8). Finally, how to choose a “best” solution from the pool of these nondominant solutions is an important concern to decision makers. However, there appears to be no best among these solutions because they cannot dominate each other. To sort out this problem, decision makers may have their own subjective preference or determinant criteria. For instance, they may try to find a solution with the maximum average achievement level or with a maximum of the minimum achievement level. The solution with a maximum of the minimum achievement level is in bold font in Table 2. This provides an example on the way decision makers can make use of these nondominated solutions.

Miu3

Miu1

FIGURE 5

3-D view of Pareto fronts.

Miu2

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TABLE 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Nondominated Solutions Found by MO-Ant

CONCLUSION

µ1

µ2

µ3

0.501779 0.506863 0.559227 0.443315 0.415353 0.398576 0.429080 0.701576 0.695984 0.692425 0.324352 0.324860 0.419420 0.595831 0.519065 0.587189 0.754448 0.737672 0.761057 0.745297 0.472801 0.691408 0.677682 0.691408 0.575496 0.474835 0.499746 0.432639 0.443823 0.422979 0.434164 0.405694 0.405694 0.663955 0.669548 0.677173 0.494154 0.583630 0.494154 0.494154 0.494154 0.494154 0.494154 0.422979 0.426029

0.762423 0.818878 0.801190 0.671954 0.758357 0.834020 0.667878 0.600000 0.671954 0.754988 0.846586 0.846586 0.804751 0.750000 0.762423 0.754988 0.506637 0.532624 0.600000 0.600000 0.846586 0.600000 0.651388 0.610233 0.769804 0.779563 0.762423 0.851136 0.878051 0.851136 0.878051 0.878051 0.942686 0.761187 0.662414 0.667878 0.720824 0.710469 0.762423 0.775914 0.762423 0.762423 0.775914 0.804751 0.846586

0.666737 0.644074 0.610519 0.784339 0.784056 0.789256 0.788676 0.353784 0.348840 0.337016 0.770890 0.744247 0.671991 0.705514 0.684651 0.723274 0.494465 0.507914 0.476130 0.486183 0.556174 0.397283 0.464840 0.377519 0.401264 0.675393 0.680058 0.395123 0.381674 0.393215 0.379766 0.460654 0.438941 0.660615 0.647124 0.645718 0.724543 0.767813 0.761745 0.761901 0.761745 0.764083 0.761901 0.726109 0.575403

This paper proposes a novel MO-ant algorithm and illustrates its application to the MO ESF siting problem on a raster data structure with the aid of GIS. The concepts of Pareto optimum and Pareto ranking are also introduced. The MO-ant algorithm uses Pareto ranking to sort the solutions searched out and is able to generate the Pareto front for the problem. A case study of siting an additional six fire stations in Singapore with three predefined objectives is presented. It is found that the MO-ant algorithm successfully captures a pool of nondominated solutions that build up the Pareto front. Computational experiments reveal that the algorithm is able to produce a set of quality alternative solutions for the MO ESF siting problems. It should be noted that the proposed MO-ant algorithm is a generic algorithm, which can be adjusted easily to solve other MO facility siting problems.

Alternatively, they may find another solution using different criteria and compare the trade-offs between the two. The pool of these nondominated solutions furnishes a global view of the possible “good” solutions, but how to choose the “best” from them is highly dependent on the decision makers’ point of view based on their practical situations.

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