A hybrid AI-based particle bee algorithm for facility layout optimization ...

Engineering with Computers (2012) 28:57–69 DOI 10.1007/s00366-011-0216-z

ORIGINAL ARTICLE

A hybrid AI-based particle bee algorithm for facility layout optimization Min-Yuan Cheng • Li-Chuan Lien

Received: 11 November 2010 / Accepted: 21 March 2011 / Published online: 13 April 2011  Springer-Verlag London Limited 2011

Abstract Facility layout (FL) design presents a particularly interesting area of study because of its relatively high level of attention to aesthetics and usability qualities, in addition to common engineering objectives such as cost and performance. However, this generally presents a difficult combinatorial optimization problem for engineers. Swarm intelligence, an approach to decision making that integrates collective social behavior models such as the bee algorithm (BA) and particle swarm optimization (PSO), is being increasingly used to resolve various complex optimization problems. In order to integrate BA global search ability with the local search advantages of PSO, this study proposes a new optimization hybrid swarm algorithm—the particle bee algorithm (PBA) which imitates the intelligent swarming behavior of honeybees and birds. This study also proposes a neighborhood-windows technique for improving searching efficiency as well as a self-parameterupdating technique for preventing trapping into a local optimum in high-dimensional problems. This study compares PBA performance against BA and PSO performance in practical FL problem. Results show PBA performance is comparable to those of BA and PSO and can be efficiently employed to solve practical FL problem with high dimensionality. Keywords Facility layout  Swarm intelligence  Bee algorithm  Particle swarm optimization  Particle bee algorithm M.-Y. Cheng  L.-C. Lien (&) Department of Construction Engineering, National Taiwan University of Science and Technology, 106 Taipei, Taiwan e-mail: [email protected] M.-Y. Cheng e-mail: [email protected]

1 Introduction Facility layout (FL) problems are particularly interesting, because in addition to common engineering objectives such as cost and performance, facility design is especially concerned with aesthetics and usability qualities of a layout [1]. The FL problem identifies a feasible location for a set of interrelated objects that meet all design requirements and maximizes design quality in terms of design preferences while minimizing total cost associated with interactions between these facilities. Pairwise costs usually reflect transportation costs and/or inter-facility adjacency preferences [1, 2]. FL problems arise in the design of hospitals, service centers and other facilities [3]. However, all such problems are known as ‘‘NP-hard’’ and because of the combinatorial complexity, it cannot be solved exhaustively for reasonably sized layout problems [3]. For n facilities, the number of possible alternatives is n!, which gives huge numbers even for small n values. When 10 facilities are involved, possible alternatives number well over 3,628,800, and 15 facilities have possible alternatives numbering in the billions. In practical application, however, a project with n = 15 is still considered small [3]. Due to FL problem complexity, many algorithms have been developed to generate practical solutions. Algorithms can be classified into layout improvement, entire layout and partial layout categories [4]. Layout improvement algorithms require an existing layout, with new (improved) layouts generated by relocating the facilities in order to improve the layout [5]. Entire layout algorithms select and place one facility each time according to a predetermined selection and allocation order [6]. Partial improvement algorithms place a particular facility at every possible location in order to generate all possible partial layout alternatives, from which the best layout may be selected.

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The procedure is repeated for all subsequent facilities until all facilities are located [4]. This study also contributes to the area of partial improvement algorithms. In the past, knowledge and artificial intelligence-based methods have been applied to solving FL problems, e.g., knowledge-based systems have been developed to provide users with problem-specific heuristic knowledge to facilitate appropriate facility allocations [7, 8]. For the AI-based algorithms, Elbeitagi and Hegazy [9] used a hybrid neural network to identify optimal site layout. Yeh [3] applied annealed neural networks to solve construction site-level FL problems. Other well-known algorithms, e.g., tabu search (TS), simulated annealing (SA) and genetic algorithms (GA) are used widely to solve site layout problems. TS is a local search method, which is used for the laying out of multi-floor facilities [10]. SA is a method for solving combination problems generally applied to the layout design of multi-objective facilities [11]. Gero and Kazakov [12] incorporated the concept of genetic engineering into the GA system for solving building space layout problems. Li and Love [13] and Osman et al. [14] used GA to solve site layout problems in unequally sized facilities. The objective functions of the above-mentioned algorithms were to optimize the interaction between facilities, such as total inter-facility transportation costs and frequency of inter-facility trips. Hegazy and Elbeltagl [15] developed a comprehensive system for site layout planning based on GA. Elbeitagi et al. [16] presented a practical model for schedule-dependent site layout planning in construction by combining a knowledge-based system, fuzzy logic and GA. Although previous research focused on solving different optimization problems by applying algorithms under different constraints, the quality of solutions were limited by the capability of the algorithms. Swarm intelligence (SI) has been of increasing interest to research scientists in recent years. SI was defined by Bonabeau et al. [17] as any attempt to design algorithms or distributed problem-solving devices based on the collective behavior of social insect colonies or other animals. Bonabeau et al. focused primarily on the social behavior of ants [18], fish [19], birds [20] and bees [21], etc. However, the term ‘‘swarm’’ can be applied more generally to refer to any restrained collection of interacting agents or individuals. Although bees swarming around a hive is the classical example of ‘‘swarm’’, swarms can easily be extended to other systems with similar architectures. A few models have been developed to model the intelligent behaviors of honeybee swarms and applied to solve combinatorial type problems. Yang [22] presented a virtual bee algorithm (VBA) that is effective when applied to function optimization problems. However, while the proposed algorithm was similar to GA, it was much more efficient due to the parallelism of multiple independent

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bees. VBA was tested on two functions with two parameters, single-peaked and multi-peaked, respectively. Results show that the VBA is significantly more efficient than GA. Karaboga et al. [23] presented an artificial bee colony (ABC) algorithm and expanded its experimental results [24]. It has been pointed out that the ABC algorithm outperforms GA for functions exhibiting multi-modality or uni-modality. Pham et al. [21] presented an original bee algorithm (BA) and applied to two standard functional optimization problems with two and six dimensions. Results demonstrated BA able to find solutions very close to the optimum, showing that BA generally outperformed GA. Ozbakir et al. [25] developed a modified BA [21] to solve generalized assignment problems (GAP) that presented an ejection chain neighborhood mechanism. Results found that the proposed BA offers the potential to solve GAP. However, while BA [21] offers the potential to conduct global searches and uses a simpler mechanism in comparison with GA, it is weak in local searching and does not record past searching experiences during the optimization search process. For instance, a flock of birds may be thought of as a swarm whose individual agents are birds. Particle swarm optimization (PSO), which has become quite popular for many researchers recently [26, 27], models the social behavior of birds [21]. PSO is a population-based stochastic optimization technique that is well adapted to the optimization of nonlinear functions in multi-dimensional space. PSO consists of a swarm of particles moving in a search space of potential problem solutions. Every particle has a position vector representing a candidate solution to the problem and a velocity vector. Moreover, each particle contains a small memory that stores its own best position so far and a global best position obtained through communication with neighbor particles. PSO potentially used in local searching, and records past searching experiences during optimization search process. However, it converges early in highly discrete problems [28]. Hence, in order to improve BA and PSO, this study proposed an improved optimization hybrid swarm algorithm called the particle bee algorithm (PBA) that imitates a particular intelligent behavior of bird and honeybee swarms and integrates their advantages. In addition, this study also proposed a neighborhood-windows (NW) technique for improving PBA search efficiency and proposed a self-parameter-updating (SPU) technique for preventing trapping into a local optimum in high-dimensional problems. This study compares PBA algorithm performance with that of BA [21] and PSO for practical FL problem in construction engineering. The FL problem is described in Sect. 2, and the BA, PSO and PBA are introduced in Sect. 3. Section 4 presents a case study of a hospital building with 28 facilities, which illustrates practical application of

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the developed model and demonstrates that PBA is relatively effective at solving the FL problem. Results obtained are presented and discussed in Sect. 5. Finally, Sect. 6 provides a summary and conclusions.

2 FL problem The FL design formulated as a Quadratic Assignment Problem (QAP) represents a difficult combinatorial optimization problem of great importance, the scope of which goes well beyond normal facility design [3]. The QAP can be defined in the following way: n distinct objects N = {1, 2, …, n} are placed uniquely in m distinct sites M = {1, 2, …, m}, where m C n. In other words, we want to find a one-to-one mapping of the set of facilities N into a set of locations M. The QAP can be stated as the task of finding the minimum cost F(A) of allocation A, usually with some constraints to be satisfied as well [29]. For FL problems, facilities may be viewed in terms of facility functions. Sites can be defined as spaces allocated for specific facility functions. It can be argued that the two cases ‘‘m = n’’ and ‘‘m [ n’’ are essentially equivalent, as m - n ‘‘dummy’’ facilities can be added if m is strictly greater than n. With the straightforward interpretation of QAP in mind, m = n means that a building is designed to house n facilities exactly within its free cells. Of course, there may also be other cells in the building, but they are, in context, regarded as inadmissible, perhaps because they are reserved for other purposes [30]. In the present work, the formulation of the FL problem holds that a set of facilities needs to be located on the site while optimizing layout objectives and satisfying a set of layout constraints. The layout objective represents the goodness of the functional interactions of the facility with other facilities. Some of the layout objectives considered in this study include adjacency of objects, distance between objects, availability of space for object location and positions of objects in relation to others. Layout constraints measure layout feasibility, i.e., each site should be assigned with one and only one facility, and each facility should be assigned on one and only one site. The problem may then be formulated as follows (1): Minimize n X n n X n X n X n X 1 X ¼ Bxi  Cxi þ Bxi  Byj  Aij  Dxy F x¼1 i¼1 x¼1 i¼1 y¼1 j¼1 ð1Þ Subjected to Byi ¼ 0

if Bxi ¼ 1

and

y 6¼ z

Bxi ¼ 0

if Bxi ¼ 1

and

j 6¼ i

where F = objective function; Bxi = permutation matrix variable (is 1 if facility X is assigned on site i); Cxi = layout preference of assigning facility X on site i; Aij = site-neighboring index, if site i is a neighbor to site j, Aij = 1, if site i is not a neighbor to site j, Aij = 0, and Aii = 0; Dxy = interactive preference of assigning facility X on the neighboring site of facility Y, and Dxx = 0. In the objective function, the first term accounts for total layout preferences for assigning site facilities, and the second represents total interactive preferences between facilities. If a predetermined site i is very unsuitable for a predetermined facility X, based on size or shape, this problem can be solved by assigning a very low layout preference Cxi. If a facility X is very unsuitable to be close to a facility Y, based on functional interference, this problem can be solved by assigning a very low interactive preference Dxy. Therefore, (1) is rather practical for FL problems. Each layout alternative can also be represented by a n 9 n permutation matrix, whose rows and columns represent facilities and sites, respectively. The permutation matrix allows a single value of 1 in each row and each column, with all remaining elements valued as 0. Figure 1 shows a permutation matrix, where facility A is assigned on site 3, facility B is assigned on site 1, facility C is assigned on site 4, facility D is assigned on site 5, and facility E is assigned on site 2.

3 Hybrid swarm algorithm: the PBA 3.1 BA BA is an optimization algorithm inspired by the natural foraging behavior of honeybees, which is tailored to find an optimal solution [31]. BA flowchart is shown in Fig. 2. BA [21] requires the setting of a number of parameters, including number of scout bees (n), number of elite sites selected from n visited sites (e), number of best sites out of n visited sites (b), number of bees recruited for elite e sites (n1), number of bees recruited for best b sites (n2), number of bees recruited for other visited sites (r), and neighborhood (ngh) of bees dance search and stopping criterion.

Layout

Facility 1 A B C D E

0 1 0 0 0

2 0 0 0 0 1

3 1 0 0 0 0

4 0 0 1 0 0

5 0 0 0 1 0

Fig. 1 Permutation matrix with five facilities

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Step (5)

Step (6)

Step (7) Fig. 2 Bee algorithm flowchart

Step (1)

Step (2) Step (3)

Step (4)

Initialize scout bees BA starts with n scout bees placed randomly in the search space. Evaluate fitness Start the loop and evaluate scout bee fitness. Select elite sites (e) from scout bees Bees that have the highest fitness are chosen as elite bees, and sites they visit are chosen for neighborhood search. Recruit bees (n1) start neighborhood dance search The algorithm conducts searches in the neighborhood of selected sites, assigning more recruit bees to dance near to elite sites. Recruit bees can be chosen directly according to the fitness associated with their specific dance sites (2): xid ðt þ 1Þ ¼ xid ðtÞ  ðRand  0:5Þ  2 þ xid ðtÞ ð2Þ where xi is ith x and i = 1 to n; d is dimension in xi and d = 1 to D, t is iteration; xid(t ? 1) is dth dimension in ith x and in t ? 1 iteration; xid(t) is dth dimension in ith x and in t iteration; Rand is a uniformly distributed real random number within the range 0–1; n is number of scout bees.

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Step (8)

Select best sites (b) from scout bees Otherwise, elite bees with the highest fitness are chosen as best bees, and sites they visit are chosen for neighborhood search. Recruit bees (n2) start neighborhood dancing search The algorithm conducts searches in the neighborhood of the selected sites, assigning more recruit bees to dance near the best sites. Recruit bees can be chosen directly according to the fitness associated with dancing sites (2). Elite bees differ from best bees as the former focus on local search in order to search the local optimum solution, and the latter focus on global search in order to avoid missing other potential global optimum solutions. Alternatively, fitness values are used to determine the elite/best bees selected. Dancing searches in the neighborhood of elite and best sites that represent more promising solutions are made more detailed by recruiting more bees to follow them than others. Recruit random bees (r) for other visited sites The remaining bees in the population are assigned randomly around the search space scouting for new potential solutions. Convergence? Throughout step (3) to step (7), such differential recruitment is a key BA operation. However, in step (8), only bees with the highest fitness for the current iteration will be selected for the following iteration. While there is no such restriction in nature, it is introduced here to reduce the number of points to be explored. These steps are repeated until a stopping criterion is met that determines whether bees are to be abandoned or memorized.

From (2), BA dependence on random search makes it relatively weak in local search activities. Also, BA does not have past searching records of PSO capabilities. 3.2 PSO PSO is an optimization algorithm inspired by the natural foraging behavior of birds to find an optimal solution [20]. In PSO, a population of particles starts to move in a search space by following current optimum particles and changing their positions in order to find the optimum. The position of a particle refers to a possible solution of the function to be optimized. Evaluating the function by the particle’s position provides the fitness of that solution. In every iteration, each particle is updated by following the best current

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particle solution achieved so far (local best) and the best of the population (global best). When a particle takes part of the population as its topological neighbors, the best value becomes a local best. Particles tend to move toward good areas in the search space in response to information spreading through the swarm. A particle moves to a new position calculated by the velocity updated at each time step t by (3). Equation (4) is then used to calculate the new velocity, as the sum of the previous position and the velocity: xid ðt þ 1Þ ¼ xid ðtÞ þ vid ðt þ 1Þ

ð3Þ

Start (1) Initial particles with position and velocity (n) (2) Evaluate fitness

(3) Update Gbest particle No

(5) Update particles

where xi is ith x and i = 1 to n; vi is ith v; d is dimension in xi or v and d = 1 to D; t is iteration; xid(t) is dth dimension in ith x and in t iteration; vid(t ? 1) is dth dimension in ith v and in t ? 1 iteration; xid(t ? 1) is dth dimension in ith x and in t ? 1 iteration; n is number of particles:

(6) Convergence ? Yes

vid ðt þ 1Þ ¼ w  vid ðtÞ þ c1  Rand  ½Pid ðtÞ  xid ðtÞ þ c2  Rand  ½Gd ðtÞ  xid ðtÞ

End

ð4Þ where vid(t) is dth dimension in ith v and in t iteration; w is inertia weight and controls the magnitude of the old velocity vid(t) in the calculation of the new velocity; Pid(t) is dth dimension in ith local best particle and in t iteration; Gd(t) is dth dimension global best particle in t iteration; c1 and c2 determine the significance of Pid(t) and Gd(t); Rand is a uniformly distributed real random number within the range 0–1. Furthermore, vid at any time step of the algorithm is constrained by parameters vmax and vmin. The swarm in PSO is initialized by assigning each particle to a uniformly and randomly chosen position in the search space. Velocities are initialized randomly in the range vmax to vmin. Particle velocities on each dimension are clamped to a maximum velocity vmax. If the velocity of that dimension exceeds vmax or vmin (user-specified parameters), dimension velocity is limited to vmax or vmin. PSO flowchart is shown in Fig. 3. Step (1)

Step (2) Step (3)

Step (4)

Step (5)

(4)Update Pbest particles

Initialize particles PSO starts with n particles being randomly introduced with respective positions and velocities into the search space. Evaluate fitness Start the loop and evaluate particle fitness. Update Gbest particle The algorithm updates global best particle through problem iterations. Update Pbest particles The algorithm updates local best particles through the current problem iteration. Update particles using steps (3) and (4)

Fig. 3 Particle swarm optimization flowchart

Step (6)

The algorithm updates particles using (3) and (4). Convergence? The above steps are repeated until the stop criterion is met.

However, while PSO may be employed in local search and has a track record of experience being used in optimization search processes, it tends to achieve early convergence in highly discrete problems [28]. 3.3 Proposed PBA In order to integrate BA global search ability with the local search advantages of PSO, this study proposed an optimization hybrid swarm algorithm, the PBA, based on the intelligent behaviors of bird and honeybee swarms. For improved BA local search ability, PSO global search ability and to seek records past experience during optimization search process, this study reconfigures the neighborhood dance search [21] as a PSO search [20]. Based on cooperation between bees (BA) and birds (PSO), the proposed algorithm improves BA neighborhood search using PSO search. Therefore, PBA employs no recruit bee searching around ‘‘elite’’ or ‘‘best’’ positions (as BA does). Instead, a PSO search is used for all elite and best bees. In other words, after PSO search, the number of ‘‘elite’’, ‘‘best’’ and ‘‘random’’ bees equals the number of scout bees. In PBA, the particle bee colony contains four groups, namely (1) number of scout bees (n), (2) number of elite sites selected out of n visited sites (e), (3) number of best

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Start

Itr 1

Itr 2

Itr 3

Itr 4

Itr 5

Itr 6

Itr 7

20

(1) Initial scout bees (n)

15

(2) Evaluate fitness (3) Select Elite bees from scout bees (e)

(5) Select Best bees from scout bees (b)

(4) PSO procedure (Pelite) by neighborhood windows

(6) PSO proceduce (Pbest) by neighborhood windows

X2 value

10 5 0 -15

-5

-5

5

15

25

-10 -15 -20

(7) Random assign scout bees (r)

X1 value Fig. 5 The behavior of NW in PSO

(8) Self-parameter update No

as shown in (5). Thus, after xid(t ? 1) is substituted into (3) and (4), the NW ensures PSO searching within the designated xidmin and xidmax. In other word, if the sum of xid(t ? 1) exceeds xidmin or xidmax then xid(t ? 1) is limited to xidmin or xidmax. Figure 5 shows NW behavior in PSO (two dimensions):

(9) Convergence ? Yes End

xidmin  xid ðt þ 1Þ  xidmax

Fig. 4 Particle bee algorithm flowchart

sites out of n visited sites (b), and (4) number of bees recruited for the other visited sites (r). The first half of the bee colony consists of elite bees, and the second half includes the best and random bees. The particle bee colony contains two parameters, i.e., number of iteration for elite bees by PSO (Pelite) and number of iteration for best bees by PSO (Pbest). PBA flowchart is shown in Fig. 4. Step (1)

Step (2) Step (3)

Step (4)

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Initialize scout bees PBA starts with n scout bees being randomly placed with respective positions and velocities in the search space. Evaluate fitness Start the loop and evaluate scout bee fitness. Select elite sites (e) from scout bees Elite sites are selected for each elite bee, whose total number is equal to half the number of scout bees. Elite bees initiate the PSO procedure by Pelite iteration for NW In this step, new particle bees from elite and best bees are produced using (3). Elite and best bee velocity updates are performed as indicated in (4). This study further proposed a NW technique to improve PSO searching efficiency

Step (5)

Step (6)

Step (7)

Step (8)

ð5Þ

where xi is ith x and i = 1 to n; d is dimension in xi and d = 1 to D; t is iteration; xid(t ? 1) is dth dimension in ith x and in t ? 1 iteration; n is number of particles. Select best sites (b) from scout bees Best sites are selected for each best bee, the total number of which equals one-quarter of the number of scout bees. Best bees start the PSO procedure using the NW Pbest iteration In this step, new particle bees from elite and best bees are produced using (3). Elite and best bee velocity updates are acquired using (4). The NW technique improves PSO search efficiency, as shown in (5) and Fig. 5. Recruit random bees (r) for other visited sites The random bees in the population are assigned randomly around the search space scouting for new potential solutions. The total number of random bees is one-quarter of the number of scout bees. SPU for elite, best and random bees Furthermore, in order to prevent being trapped into a local optimum in high-dimensional problems, this study proposed a solution, i.e., the SPU technique, the idea for which came

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if finesses of elite, best or random bees are both improved using (6) and improved over previous finesses, the new finesses are memorized. In step (3) through step (8), this differential recruitment is a key operation of the PBA. The Rastrigin function is provided as a sample for observing the proposed optimization procedure behavior in this study. Rastrigin function formula and figure are shown, respectively, in (9) and Fig. 6. In this study, while the NW carries out the PSO search of the local optimization at the start of each iteration search space, BA random bees and the SPU technique control the global optimization at the end of each iteration search space. Therefore, in the PBA procedure, the proposed optimization search techniques may search across each iteration search space based on that particular search space’s potential. The procedure of this study combines NW, BA’s random bees and the SPU technique, as shown in Fig. 7:

from Karaboga [23]. Equation (6) shows the SPU equation: xidðnewÞ ¼ xidðcurÞ þ 2  ðRand  0:5Þ  ðxidðcurÞ  xjk Þ

ð6Þ

j ¼ int(Rand  nÞ þ 1

ð7Þ

k ¼ intðRand  dÞ þ 1

ð8Þ

where xi is ith x and i = 1 to n; d is dimension in xi and d = 1 to D; xid(cur) is dth dimension in ith x and in current solution; xid(new) is dth dimension in ith x and in new solution; Rand is a uniformly distributed real random number within the range 0–1; j is the index of the solution chosen randomly from the colony as shown in (7), k is the index of the dimension chosen randomly from the dimension as shown in (8); n is number of scout bees. In step (8), after elite, best and random bees have been distributed based on finesse, finesses are checked to determine whether they are to be abandoned or memorized using (6). Therefore,

~Þ ¼ f ðx

2 X

ðx2id  10 cosð2pxid Þ þ 10Þ

ð9Þ

d¼1

where -5.12 B xid B 5.12; f(0) = 0. Besides, this study uses common five benchmark functions [23] such as Rastrigin, Sphere, Griewank, Rosenbrock and Ackley [(9)–(13)] to verify the performance of NW and SPU as shown in Table 1. From Table 1, only Rosenbrock function’s result shows that ‘‘PBA without NW and SPU’’ is better than ‘‘PBA with NW’’. The other function’s result shows that ‘‘PBA with NW’’ is better than ‘‘PBA without NW and SPU’’. Besides, the other function’s result also shows that ‘‘PBA with NW and SPU’’ is better than ‘‘PBA with NW’’. The results show that ‘‘PBA with NW and SPU’’ and ‘‘PBA with NW’’

Fig. 6 The Rastrigin function

Fig. 7 Optimization procedure behavior in each iteration

Itr 2 Itr 7

X2 value

Itr 1 Itr 6

Itr 3

Itr 4

Itr 5

X1 Maximum value X2 Maximum value

25

25

20

20

15

15

10

10 X

5 0 -15

-5

-5

X1 Minimum value X2 Minimum value

5 0

5

15

25

-5

Itr 1

Itr 2

Itr 3

Itr 4

Itr 5

Itr 6

Itr 7

-10

-10

-15

-15

X1 value

Iteration

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Table 1 Validation of the performance of NW and SPU in PBA

Function

PBA without NW and SPU

PBA with NW

PBA with NW and SPU

Rastrigin (10D) Avg fitness

7.71

5.77

5.34E-4

Std fitness

2.49

2.77

0.001

Avg fitness

2.86E-19

1.67E-19

3.53E-20

Std fitness

5.41E-19

3E-19

7.37E-20

Avg fitness

0.136

0.099

0.059

Std fitness

0.067

0.030

0.013

2.32E-3 0.005

7.36E-3 0.013

2.62E-5 7.75E-5

Sphere (10D)

Griewank (10D)

Rosenbrock (10D) Avg fitness Std fitness Ackley (10D) Avg fitness

0.116

0.116

9.49E-10

Std fitness

0.347

0.347

1.89E-9

are significantly improving ‘‘PBA without NW and SPU’’: f ðxÞ ¼

D X

x2i

ð10Þ

i¼1

where -100 B xid B 100; f(0) = 0 ! D X 1 2 f ðxÞ ¼ ðxi  100Þ 4,000 i¼1  ! D Y xi  100 pffi  cos þ1 i i¼1

ð11Þ

where -600 B xid B 600; f(0) = 0 f5 ðxÞ ¼

D1 X

100ðxiþ1  x2i Þ2 þ ðxi  1Þ2

ð12Þ

i¼1

where -30 B xid B 30; f(0) = 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 u D u1 X @ f ðxÞ ¼ 20 exp 0:2t x2 A n i¼1 i ! D 1X  exp cosð2pxi Þ þ 20 þ e n i¼1

4 Case study and experiment

ð13Þ Step (9)

123

In PBA, scout bees are used to classify both elite and best bees. Classification is controlled by scout bee fitness and optimized by control parameters called ‘‘Pelite’’ and ‘‘Pbest’’, which are important PBA control parameters. In PBA, the idea of Pelite for elite bees gives a higher potential to search optimization solutions. The idea of Pbest for best bees gives a second opportunity to search optimization solutions because luck continues to play a role in resource identification. Therefore, in this study, Pelite is always larger than Pbest. In a robust search process, exploration and exploitation processes must be carried out together. In PBA, while elite bees (Pelite) implement the exploitation process in the search space, best bees (Pbest) and random bees control this process. A sensitivity analysis for Pelite and Pbest was proposed by Cheng and Lien [32] in 2010.

where -32 B xid B 32; f(0) = 0. Convergence? In this step, only the bee with the highest fitness will be selected to form the next bee population. These steps are repeated until the stop criterion is met and bees are selected to be abandoned or memorized.

A case study of a four-storey hospital building with 28 facilities was employed to demonstrate a FL plan in practical applications [3]. Each floor contained seven sites. The plane of the standard floor of the hospital is shown in Fig. 8. The site-neighboring index for the same floor is

Fig. 8 Plane of the standard floor of hospital

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interactive preference are shown in Table 3. Design preferences are as follows [3]:

Table 2 Site-neighboring index matrix for facilities on a floor Facility

Facility A

B

C

D

E

F

G

A

0.0

1.0

0.8

1.0

0.8

0.8

0.8

B

1.0

0.0

1.0

0.8

0.8

0.8

0.8

C

0.8

1.0

0.0

0.8

1.0

1.0

1.0

D

1.0

0.8

0.8

0.0

0.8

0.8

0.8

E

0.8

1.0

0.8

1.0

0.0

1.0

0.8

F

0.8

0.8

1.0

0.8

1.0

0.0

1.0

G

0.8

0.8

1.0

0.8

1.0

1.0

0.0

•

•

•

• shown in Table 2. If two facilities were not on the same floor, the site-neighboring index was designated by the value 0.6 when the floor difference is one floor, 0.2 when the floor difference is two floors, and 0.0 when the floor difference is three floors. The facilities and their layout and

•

If the space requirement of the facility is ‘‘small’’, and it is assigned in a ‘‘small’’, ‘‘middle’’, or ‘‘large’’ space, then its layout preference is 10, 0, or 10, respectively. If the space requirement of the facility is ‘‘middle’’, and it is assigned in a ‘‘small’’, ‘‘middle’’, or ‘‘large’’ space, then its layout preference is 0, 10, or 0, respectively. If the space requirement of the facility is ‘‘large’’, and it is assigned in a ‘‘small’’, ‘‘middle’’, or ‘‘large’’ space, then its layout preference is 10, 0, or -10, respectively. If the facility is assigned on a floor where it should not be assigned, then -10 is added to its layout preference. The interactive preference is 10 for the two facilities that must be close to each other, 5 for the two facilities that should be close to each other, and -10 for the two facilities that must not be close to each other.

Table 3 Design preferences of the hospital layout problem No.

Facility (medical function)

Space requirement

Floor where it should not be assigned

The facility must be close to

The facility should be close to

The facility must not be close to

1

Endocrine metabolism

Small

2

Digestion

Large

–

14

2, 5, 15, 16

–

–

6, 9

1, 4, 5, 11, 16, 27

3

Cardiology

Large

–

–

19

4, 5, 6, 10

–

4

Chest department

Middle

5

Kidney department

Small

–

19

2, 3, 9

–

–

6, 23

1, 2, 3, 8, 9, 15, 16, 19

6

Blood department

Small

–

–

2, 5, 7, 19

3, 8

–

7 8

Immunity rheumatism Pestilence

Small Small

– 1–3

6, 8 7

10, 21, 28 5, 6, 10, 16, 23, 24

– 11, 17, 21

9 10

Medical department

Small

2–4

2, 27

4, 5, 23, 24

–

Rehabilitation

Small

2–4

20

3, 7, 8, 12, 18, 21, 26, 27

11

Pediatrics

11, 17

Large

3–4

15, 17

2, 24, 25, 27

8, 10, 18

12

Orthopedics

Large

–

20

10, 21, 25, 26

–

13

Eye ophthalmology

Middle

–

–

20, 26

–

14

Neuropsychiatry

Middle

–

1, 18, 26

–

–

15

Obstetrics gynecology

Middle

2–4

11, 17

1, 5, 23, 27

28

16

Dermatology

Middle

–

20

1, 2, 5, 8, 23, 24, 26

–

17

Pediatrics surgery

Small

–

11, 15

21, 27

8, 10, 20, 28

18

Head psychiatry

Small

1–3

14, 26

10, 27

11, 20

19

Thoracic cardiac

Small

–

3, 4, 6

5, 21

–

20

Anaplastic department

Middle

1–2

10, 12, 16

13, 21, 25

11, 17, 18, 27

21

Surgery department

Small

–

–

7, 10, 12, 17, 19, 20, 22, 23, 24, 26, 27

8

22 23

Rectum surgery Urological surgery

Small Small

– –

– 5

21, 23 8, 9, 15, 16, 21, 22, 27

– –

24

Otolaryngology

Middle

–

–

8, 9, 11, 16, 21, 27

–

25

Dentistry

Small

3–4

–

11, 12, 20, 26

–

26

Neuron surgery

Small

–

14, 18

10, 12, 13, 16, 21, 25

–

27

Family medicine

Small

2–4

9

2, 10, 11, 15, 17, 18, 21, 23, 24

20

28

Radiosurgery tumor

Small

1–3

–

7

11, 15, 17

123

66

Engineering with Computers (2012) 28:57–69

The objective function of the FL problem must satisfy two requirements, namely: (1) It must be high for only those solutions with high design preference and (2) it must be high only for those solutions that satisfy layout constraints. Therefore, this study gives the total objective function modified form Yeh [3] as follows (14):

Table 4 Parameter values used in the experiments BA [21]

PSO

PBA [32]

n

100

n

100

n

100

e

n/2

w

0.9–0.4

e

n/2

b

n/4

v

xmin/10–xmax/10

b

n/4

r

n/4

r

n/4

n1

2

w

0.9–0.4

n2

1

v

xmin/10–xmax/10

Pelite

15

Pbest

9

n population size (colony size), e elite bee number, b best bee number, r random bee number, n1 elite bee neighborhood number, n2 best bee neighborhood number, w inertia weight, v limit of velocity for PSO, Pelite PSO iteration of elite bees, Pbest PSO iteration of best bees

Table 5 The result of three algorithms Std

Best

Worst

BA [21]

4.71E-11

2.08E-11

1.77E-11

9.30E-11

PSO

1.01E-10

5.55E-11

3.18E-11

2.52E-10

PBA [32]

3.97E-11

1.92E-11

1.41E-11

7.24E-11

PBA

0

200

BA

400

PSO

600

800

1000

1.E-11

1.E-10

1.E-09

Mean

Mean best value

Algorithms

where F1 = evaluation function of layout preference; F2 = evaluation function of interactive preference; F3 = evaluation function of constraint violation. The first term accounts for the layout preference, the second term accounts for the interactive preference, and the third term maintains feasibility by acting as a repulsive force that discourages two facilities from occupying the same site on an alternative; Bij = permutation matrix variable; Cij = layout preference of assigning facility i on site j; Dik = interactive preference of assigning facility i on the neighboring site of facility k. In terms of PBA performance analysis, this study compared favorably with the BA conducted by Pham [21] and the PSO that generates control parameter values as shown in Table 4.

5 Results and discussion

Iteration

Fig. 9 Evolution of mean best values for architecture layout problem

Fig. 10 PBA best layout design

This study used BA, PSO and PBA to conduct 30 experiment runs with Table 4 values. Table 5 and Fig. 9 show the evolved results of the FL problem. In Table 5, PBA attained the best evolution of mean fitness, standard, best and worst result that were better than either BA or PSO.

Thoracic cardiac

Radio surgery tumor

Blood department

Pestilence

Chest department

Immunity rheumatism

Urological surgery

Rectum surgery

Dentistry

Rehabilitation

Surgery department

Neuron surgery

Head psychiatry

th

4 floor

Cardiology

3rd floor

Orthopedics Ana plastic department Endocrine metabolism

2nd floor

Otolaryngology Neuron psychiatry Medical department

Eye ophthalmology Kidney department

Dermatology Pediatrics surgery

st

1 floor

Digestion Pediatrics

123

Obstetrics and gynecology

Family medicine

Engineering with Computers (2012) 28:57–69 Fig. 11 Yeh [3] best layout design

67 Immunity rheumatism

Pestilence

Radio surgery tumor

Eye ophthalmology

Rehabilitation

Ana plastic department

Thoracic cardiac

Blood department

Neuron surgery

4th floor

Cardiology

3rd floor

Orthopedics Head psychiatry

Neuron psychiatry

Chest department

Endocrine metabolism

Dentistry

Surgery department

Dermatology

Kidney department

Urological surgery

Medical department

Family medicine

Rectum surgery

Otolaryngology

Obstetrics and gynecology

2nd floor

Digestion

1st floor

Pediatrics Pediatrics surgery

Table 6 Results of layout facility score

Sort by Floor Floor

1st

Total score

2nd

Total score

3rd

Total score

4th

Total score Summary

PBA Yeh [3] Facility No. Score Facility No. Score 9 35 9 25 5 20 27 40 17 35 22 10 17 25 11 30 15 40 25 24 15 35 27 30 2 35 11 40 225 200 1 25 1 20 26 40 25 10 18 20 21 15 14 40 16 25 13 15 25 5 16 25 23 10 2 20 24 5 170 125 23 20 19 30 22 20 6 20 25 10 26 35 20 40 18 0 10 10 14 30 21 25 4 20 12 35 12 15 160 150 19 40 7 30 28 15 25 8 6 40 28 15 13 15 8 15 4 25 10 5 20 15 25 7 3 30 3 15 190 120 745 595

Sort by Facility No. Facility No. PBA Yeh [3] Variation 1 25 20 5 2 35 20 15 3 30 15 15 4 25 20 5 5 20 25 -5 6 40 20 20 7 25 30 -5 8 15 25 -10 9 35 25 10 10 10 5 5 11 30 40 -10 12 35 15 20 13 15 15 0 14 40 30 10 15 40 35 5 16 25 25 0 17 35 25 10 18 20 0 20 19 40 30 10 20 40 15 25 21 25 15 10 22 20 10 10 23 20 10 10 24 5 25 -20 25 10 10 0 26 40 35 5 27 30 40 -10 28 15 15 0 Summary 745 595 150

The best layout alternatives proposed by PBA and Yeh [3] are shown in Figs. 10 and 11. The best layout design alternatives proposed by PBA and Yeh [3] were 1.41E-11 and 3.48E-11, calculated using (14). On the first floor,

there are two same facilities for having same space size on this floor that compared to reference. On the second floor, there are two same facilities for having same space size on this floor that compared to reference. On the third floor,

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68

there is one same facility for having same space size on this floor that compared to reference. On the fourth floor, there are two same facilities for having same space size on this floor that compared to reference. On the first floor, there are three same facilities for having not same space size on this floor that compared to reference. On the second floor, there is 0 same facilities for having not same space size on this floor that compared to reference. On the third floor, there is 0 same facilities for having not same space size on this floor that compared to reference. On the fourth floor, there are two same facilities for having not same space size on this floor that compared to reference. Besides, there are ten same facilities for having same space size but having not on the same floor that compared to reference. Therefore, there are only six facilities not having any relationship. Thus, in this study, there are totally 22 same facilities having relationship from 28 facilities. So that this study uses Fig. 10 is rather similar to Fig. 11. In order to analyze the difference in results obtained, respectfully, by PBA and Yeh [3], this study calculated the layout of each independent facility function score using Table 3, as shown in Table 6. From Table 6, the total independent score of PBA for each floor is better than Yeh [3], and only certain layout facility function scores for the PBA (e.g., ‘‘Kidney Department’’, ‘‘Immunity Rheumatism’’, ‘‘Pestilence’’, ‘‘Pediatrics’’, ‘‘Otolaryngology’’ and ‘‘Family Medicine’’) fall below Yeh [3]. Therefore, the PBA proposed in this study can be efficiently employed to solve practical construction engineering problems for FL with high dimensionality.

6 Conclusion In the previous section, the performance of the PBA was compared with BA and PSO for practical FL construction engineering problems. In the evolution of mean and best fitness, PBA achieved 3.97E-11 and 1.41E-11, scores significantly better than BA’s 4.71E-11 and 1.77E-11 and PSO’s 1.01E-10 and 3.18E-11, demonstrating better PBA performance than attained by either BA or PSO for this FL problem. In facility function score analysis, results demonstrate PBA able also to solve practical construction engineering problem for FL with high dimensionality efficiently.

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