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Architectural layout optimization using annealed neural network
Automation in Construction 15 (2006) 531 – 539 www.elsevier.com/locate/autcon
Architectural layout optimization using annealed neural network I-Cheng Yeh Department of Civil Engineering, Chung-Hua University, Hsin Chu, Taiwan 30067, ROC Accepted 31 July 2005
Abstract The facility layout problem is concerned with finding feasible locations for a set of interrelated objects that meet all design requirements and maximize design quality in terms of design preferences. The contribution of this paper is a new framework, named annealed neural network, for efficiently finding competitive solutions for the facility layout problem. This framework arises from the combination of Hopfield neural networks and simulated annealing. The first is a representation model of the layout problem and the second is a search algorithm for finding the optimum or near optimum solutions. The annealed neural network combines characteristics of the simulated annealing algorithm and the Hopfield neural network. Annealed neural network exhibits the rapid convergence of the neural network, while preserving the solution quality afforded by simulated annealing. Strategies for setting reasonable penalty factor in objective function and temperature in simulated annealing procedure were proposed. A case study of a hospital building with 28 facilities was employed to demonstrate that this model is rather efficient to solve the architectural layout problem, and it is amenable to fast computation for large layout problems. D 2005 Elsevier B.V. All rights reserved. Keywords: Architecture; Layout; Annealed neural network; Simulated annealing; Combinatorial optimization
1. Introduction The facility layout problem is concerned with finding feasible locations for a set of interrelated objects that meet all design requirements and maximize design quality in terms of design preferences such as minimizing the total cost associated with the interactions between these facilities. The given pairwise costs usually reflect transportation cost and/or adjacency preferences between facilities [1,2]. Architectural layout problem is particularly interesting because in addition to common engineering objectives such as cost and performance, architectural design is especially concerned with aesthetic and usability qualities of a layout, which are generally more difficult to describe formally [2]. In this paper we discuss facility layout design formulated as a Quadratic Assignment Problem (QAP). This is a difficult combinatorial optimization problem of great importance, reaching beyond formal architectural design [3]. Versions of this problem occur in many environments, such as hospital layout and service center layout, as well as in other engineering
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applications, such as VLSI placement and design. All of these problems are known to be NP-hard. For this reason, most of the approaches in the literature are based on heuristics [1]. Because of the combinatorial complexity, it cannot be solved exhaustively for reasonably sized layout problems. Several heuristic strategies have been developed to find solutions without searching the design space exhaustively. Liggett and Mitchell [4] use a constructive placement strategy followed by an iterative improvement strategy. In this method, space is allocated for rooms one at a time based on the best probable design move at each step. Other researchers have used stochastic algorithms for search [3,5]. An overview and history of the automated layout problem can be found in Liggett [6]. It has been argued that with the possibility of finding good solutions quickly using heuristic methods such as genetic algorithm and simulated annealing [7,8], the search for exact solutions may not be worthwhile. This argument holds merit since we are dealing with models that are abstractions and not themselves reality. Therefore, it may be not necessary to have provably optimum solutions except in rather rare circumstances, when the model precisely matches the situation and large expenditures of time and money are involved [9]. Recently simulated annealing (SA) has been proposed as a
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general technique for attempting to solve combinational optimization problems. Neural networks is a powerful method of optimization which relies on developing systems that demonstrate selforganization and adaptation in a similar, though simplified, manner to the way in which biological systems work. Hopfield neural network [10], a kind of artificial neural network, has been proposed as a model of computation to solve a wide variety of discrete combinatorial optimization problems. The contribution of this paper is a new framework, named annealed neural network, for efficiently finding competitive solutions for the facility layout problem. This framework arises from the combination of Hopfield neural networks and simulated annealing. The first is a representation model of the layout problem and the second is a search algorithm for finding the optimum or near optimum solutions. The annealed neural network combines characteristics of the simulated annealing algorithm and the Hopfield neural network. Annealed neural network exhibits the rapid convergence of the neural network, while preserving the solution quality afforded by simulated annealing. This paper describes an effort that applies annealed neural network to generate architectural layout alternative. In the following sections, Section 2 introduces architectural layout problem. Section 3 describes the annealed neural network model. Section 4 describes the model using annealed neural network to solve the problem. In Section 5, a case study of a hospital building with 28 facilities is employed to illustrate the practical applications and to demonstrate that this model is rather efficient to solve the architectural layout problem. In addition, the effects of three control parameters on the model are experimented. Finally, Section 6 gives summary and conclusions. 2. Architectural layout problem The QAP can be defined in the following way: n distinct objects N = {1, 2,. . ., n} are to be placed uniquely in m distinct sites M = {1, 2,. . ., m}, where m 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 [3]. For an architectural layout problem, facilities can be viewed as architectural functions. Sites can be defined as spaces allocated for specific architectural functions. It can be argued that the two cases ‘‘m = n’’ and ‘‘m > n’’ essentially are equivalent since 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 the building is designed to exactly house the n facilities in its free cells. Of course, there may also be other cells in the building, but they are in the context regarded as inadmissible, perhaps because they are reserved for other purposes [9]. In the present work, the formulation of the architectural layout problem is 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 objective considered in this paper including adjacency of objects, distance between objects, availability of space for object location, positions of objects in relation to others. The layout constraints measures the feasibility of the layout, 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. Then, the problem is formulated as follows XX Maximize F ¼ Bxi Cxi x
þ
I
i
XXXX x
i
y
I I I
Bxi Byj Aij Dxy
j
ð1Þ
Subjected to Byi ¼ 0 if Bxi ¼ 1 and ymx Bxj ¼ 0 if Bxi ¼ 1 and jmi where F = objective function; d xi = permutation matrix variable (is 1 if facility X is assigned on site i); C xi = layout preference of assigning facility X on site i; A ij = site-neighboring index, if site i is a neighbor to site j, A ij = 1, if site i is not a neighbor to site j, A ij = 0, and A ii = 0; D xy = interactive preference of assigning facility X on the neighboring site of facility Y, and D xx = 0. In the objective function, the first term accounts for the total layout preferences of assigning facilities on sites, and the second term represent the total interactive preferences between facilities. If a pre-determined site i is very unsuitable to a predetermined facility X, based on size or shape, this problem can be solved by assigning a very low layout preference C xi . 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 D xy. Therefore, Eq. (1) is rather practical for architectural layout problem. Each layout alternative can also be represented by a n n permutation matrix whose rows and columns are labeled by facilities and by sites, respectively. The permutation matrix has only one 1 in each row and each column, with the remaining elements being 0. Fig. 1 shows a permutation matrix, where facility A is assigned on the site 3, facility B is assigned on the site 1, facility C is assigned on the site 4, facility D is assigned on the site 5, and facility E is assigned on the site 2. 1
2
3
4
5
A
0
0
1
0
0
B
1
0
0
0
0
C
0
0
0
1
0
D
0
0
0
0
1
E
0
1
0
0
0
Fig. 1. Permutation matrix with five facilities.
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
3. Annealed neural network Hopfield neural networks [10] contain many simple computing elements, which cooperatively traverse the energy surface defined by the energy function to find a local or global minimum. In spite of its novel perspective, however, the Hopfield approach to the solution of these optimization problems is limited by its inability to ferret out global minimum with a single invocation of the algorithm. It is effectively a steepest-descent procedure that settles at the nearest local minimum. A global minimum is insured only by initializing the algorithm at a sufficient number of different starting points. In addition, Hopfield neural networks have several parameters that need to be selected, and often, carefully tuned for a network to produce a sensible computation [10]. Simulated annealing is a probabilistic hill-climbing search algorithm, which finds a global minimum of energy function by combining gradient descent with a random process. This combination allows, under certain conditions, changes to state that actually increases energy function, thus providing SA with a mechanism to escape from local minimum. Although changes to state that decrease energy function are always accepted, a move that causes an increases will be taken with a Boltzmann probability. Though, SA has been effective in many practical problems such as traveling salesman problem (TSP), in terms of the quality of solutions, it requires unacceptably large computing times [11,12]. A new neural network model, named annealed neural network, which merges many features of simulated annealing and Hopfield neural networks, was proposed [11,12]. Annealed neural network exhibits the rapid convergence of the neural network, while preserving the solution quality afforded by simulated annealing. The annealed neural networks have been proposed as a model of combinational optimization problem such as the TSP problem and graph-partitioning problem [11,12]. The simulated neural network algorithm can be written as follows Set initial state variables, S, on network; Set a starting temperature T; Calculate initial energy function E; Repeat For i = 1 to n do Calculate energy gap DE i ; Calculate new state variable S i = exp( DE i / T); Continue Calculate new energy function E; Lower temperature T; Until energy function is stationary.
533
those solutions with high design preference; and (2) it must be high for only those solutions that satisfied the layout constraints. Therefore, the total objective function can be given as follows: Maximize F ¼ F1 þ F2 þ F3 XX XXXX ¼ Vxi Cxi þ Vxi Vyj Aij Dxy x
k
I
i
XXX x
i
x
I
y
i
ð2Þ
Vxi Vyj
ymx
where F 1 = evaluation function of layout preference; F 2 = evaluation function of interactive preference; F 3 = evaluation function of constraint violation; V xi = state variable, a probability of assigning facility X on site i. Each variable V xi is looked upon as the probability of finding facility X currently assigned on site i of the alternative; k = penalty factor. 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. Eq. (2) can be rewritten as energy function: Minimize E ¼ F:
ð3Þ
The energy gap can be calculated as follows: DExi ¼ E ðVxi ¼ 1Þ E ðVxi ¼ 0Þ XX ¼ Cxi Vyj Aij Dxy þ k y
I I
j
The objective function of the architectural layout problem must satisfy two requirements: (1) It must be high for only
I
X
Vyi
ð4Þ
ymx
The probability of assigning facility X on site i, V xi , can be calculated by Boltzmann distribution and the following normalization operation [11,12]: expð DExi =T Þ Vxi ¼ X : exp DExj =T
ð5Þ
j
The normalization operation in (5) guarantees that each facility will be assigned to one site only, so the n facilities will be assigned to the n sites. 4.2. Penalty factor The setting for the penalty factor determines the importance of the penalty term of the energy function. Setting it to a small value emphasizes the term concerned with preference and leads to high preference, which is, unfortunately, invalid. AlternaSmall
Small
Small
1
2
3
Middle
Middle
Middle
4
5
6
Large 7
4. Architectural layout using annealed neural network 4.1. Energy function
I I I
j
Fig. 2. Plane of the standard floor of the hospital.
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I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
1
2
3
4
5
6
gap = Max C xi Min C xi ; DD max = maximum interactive preference gap =Max D xy Min D xy. By increasing the value of the penalty coefficient, i.e., is by making the last term in (2) relatively more important, we can increase the rate of success, but then the quality of the solutions will not be as good.
7
1
0.0 1.0 0.8 1.0 0.8 0.8 0.8
2
1.0 0.0 1.0 0.8 1.0 0.8 0.8
3
0.8 1.0 0.0 0.8 0.8 1.0 1.0
4
1.0 0.8 0.8 0.0 1.0 0.8 0.8
5
0.8 1.0 0.8 1.0 0.0 1.0 0.8
6
0.8 0.8 1.0 0.8 1.0 0.0 1.0
7
0.8 0.8 1.0 0.8 0.8 1.0 0.0
4.3. Simulated annealing temperature
Fig. 3. Site-neighboring index matrix for the same floor.
tively, setting a large value makes the penalty so stiff that the neural network will converge to any feasible solution regardless of its design preference. If two facilities try to occupy the same site, the penalty term in energy function should be incurred to prevent this type of a facility layout alternative. This is ensured by setting the penalty factor slightly larger than the maximum layout preference gap plus the maximum interactive preference gap.
I
k ¼ n ðDCmax þ DDmax Þ
ð6Þ
where n = penalty coefficient, n > 0. Generally, n should be approximately equal to 1; DC max = maximum layout preference
At a very high temperature, each facility will be assigned equally across each site. As the temperature is reduced, the facilities will begin to coagulate in particular sites, which will hopefully minimize the energy function. In general, the cooling procedure is proposed as follows
I
T ð t þ 1Þ ¼ a T ð t Þ
ð7Þ
where a = temperature cool coefficient, a 1. Generally, a = 0.9– 1.0. The initial temperature is very important for convergence speed. At a too high initial temperature, convergence will be very slow, while at a too low initial temperature, convergence will be very quick, but the quality of the solutions will not be as good. In this paper, a formula to determine a
Table 1 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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Endocrine metabolism Digestion Cardiology Chest department Kidney department Blood department Immunity rheumatism Pestilence Medical department Rehabilitation Pediatrics Orthopedics Eye ophthalmology Neuropsychiatry Obstetrics gynecology Dermatology Pediatrics surgery Head psychiatry Thoracic cardiac Anaplastic department Surgery department
Small Large Large Middle Small Small Small Small Small Small Large Large Middle Middle Middle Middle Small Small Small Middle Small
– – – – – – – 1–3 2–4 2–4 3–4 – – – 2–4 – – 1–3 – 1–2 –
14 6,9 19 19 6,23 2,5,7,19 6,8 7 2,27 20 15,17 20 – 1,18,26 11,17 20 11,15 14,26 3,4,6 10,12,16 –
– – – – – – – 11,17,21 – 11,17 8,10,18 – – – 28 – 8,10,20,28 11,20 – 11,17,18,27 8
22 23 24 25 26 27 28
Rectum surgery Urological surgery Otolaryngology Dentistry Neurosurgery Family medicine Radiosurgery tumor
Small Small Middle Small Small Small Small
– – – 3–4 – 2–4 1–3
– 5 – – 14,18 9 –
2,5,15,16 1,4,5,11,16,27 4,5,6,10 2,3,9 1,2,3,8,9,15,16,19 3,8 10,21,28 5,6,10,16,23,24 4,5,23,24 3,7,8,12,18,21,26,27 2,24,25,27 10,21,25,26 20,26 – 1,5,23,27 1,2,5,8,23,24,26 21,27 10,27 5,21 13,21,25 7,10,12,17,19,20,22, 23,24,26,27 21,23 8,9,15,16,21,22,27 8,9,11,16,21,27 11,12,20,26 10,12,13,16,21,25 2,10,11,15,17,18,21, 23,24 7
– – – – – 20 11,15,17
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
Iteration 0
2
4
6
8
10
12
14
16
18
20
1000
0 -500 -1000
Site
Evaluation Function
500
Layout Preference Interaction Preference Constraint Violation Total Objective
-1500 -2000 -2500
Fig. 4. Iteration of the evaluation function.
reasonable initial temperature for facility X, T x is proposed as follows: 0 1 B expð DExi =Tx Þ C C MaxB @ X exp DE =T A ¼ Pmax xj
x
where P max = maximum initial probability in a normalization set; and Pmax ¼ g Pavg ð9Þ
I
where g = maximum initial probability coefficient, 1 < g < n; P avg = average probability, P avg = 1 / n; n = the number of neurons in a normalization set, i.e., the number of facilities (sites). The reason is that the reasonable initial temperature should be able to raise the maximum initial probability in a normalization set to a reasonable value, for example, from double to triple average probability, i.e., g = 2– 3. 4.4. Facility layout with simulated annealing procedure The architectural layout procedure using annealed neural network is summarized as follows:
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
Fig. 6. Permutation matrix of the hospital layout design (iteration = 3).
Calculate initial energy function E using (3); Repeat For X = 1 to n do For i = 1 to n do Calculate DE xi using (4); Continue For i = 1 to n do Calculate V xi using (5); Continue Continue Calculate new energy function E using (3); Calculate new temperature T x for each facility X using (7); Until energy function is stationary. 4.5. Neural network model The above procedure can be modeled as a neural network as follows. Each facility is represented as a row of n neurons. One
Site
Set initial state variables, V xi , on network; Set a starting temperature T x for each facility X using (8);
Site
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Facility
ð8Þ
j
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
535
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
Facility Fig. 5. Permutation matrix of the hospital layout design (iteration = 0).
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 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
Facility Fig. 7. Permutation matrix of the hospital layout design (iteration = 6).
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Site
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
Site
536
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
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
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
Facility
Facility
Fig. 8. Permutation matrix of the hospital layout design (iteration = 9).
Fig. 10. Permutation matrix of the hospital layout design (iteration = 12, binary solution).
and only one such neuron in a row may be set to one (all others must be set to zero). This neuron set to one indicates the site that a specific facility is assigned. Hence we make a network composed of n identical neurons and denote them by a pair of indices, x referring to the facility and i referring to the site in the alternative. For example, neuron U xi indicates the state that the facility X assigned on site i. Let each connection be identified by four subscripts, i.e., W xiyj indicates the link between neuron U xi and neuron U yj , and each threshold be identified by double subscripts, i.e.,h xi indicates the threshold for neuron U xi . Let 8 < Aij Dxy ; if xmy and imj ð10Þ Wxiyj ¼ k; if xmy and i ¼ j : 0; otherwise
Then the energy function on Eq. (3) can be rewritten as follows XXXX XX Vxi hxi þ Vxi Vyj Wxiyj : ð12Þ E¼ x
i
Site
ð11Þ
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
x
i
y
j
I I
immunity rheumatism
pestilence
head psychiatry
anaplastic department
eye ophthalmo-l ogy
radiosurgery tumor department
blood department
thoracic cardiac
neurosurgery
dermatology
endocrine metabolism
neuro-psych iatry
kidney department
surgery
urological surgery
4th floor
I
hxi ¼ Cxi
I
rehabilitation
3rd floor
orthopedics
2nd floor
cardiology chest department
dentistry
pediatrics surgery
family medicine
rectum surgery
digestion
medical department
obstetrics and gynecology
otolaryngology
1st floor
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
Facility Fig. 9. Permutation matrix of the hospital layout design (iteration = 12).
pediatrics
Fig. 11. A layout design alternative from ARCHPLAN (alternative preference = 856.0).
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
II
It is easy to find that Eq. (12) is similar to Hopfield energy function. Furthermore, a positive value for W xiyj lowers the energy function if both neurons U xi and U yj are on and thus makes it more likely that the final state will include those two neurons in the on state. Conversely, a negative value lowers the likelihood of the two neurons being in the on state, in effect penalizing such a configuration. In a similar fashion, a positive value for h xi lowers the energy function if the neuron U xi is off and thus makes it more likely that the final state will include the neuron in the off state. Conversely, a negative value lowers the likelihood of the neuron being in the on state, in effect penalizing such a configuration. 5. Case study: hospital layout design In this section, a C program called ArchPlan is built on personal computer to implement the annealed neural network algorithm, and a case study of a 4-story hospital building with 28 facilities is employed to demonstrate ArchPlan in practical applications. Each floor contains 7 sites. The plane of the immunity rheumatism
pestilence
radiosurgery tumor
rehabilitation
anaplastic department
thoracic cardiac
blood department
neurosurgery
head psychiatry
neuro-psych iatry
chest department
endocrine metabolism
dentistry
surgery
4th floor
cardiology eye ophthalmology
3rd floor
orthopedics
1.00 0.90
Feasible Solution Ratio
Comparing Eq. (12) with Hopfield energy function [10] X 1 XX E¼ Si hi Si Sj Wij ð13Þ 2 i j i
0.60 0.50 0.40 0.30 0.20
0.00
1
1.5
2
2.5
3
3.5
Penalty Coefficient Fig. 13. Feasible solution ratio under various penalty coefficients.
standard floor of the hospital is shown in Fig. 2. The siteneighboring index for the same floor is shown in Fig. 3. If the two facilities are not at the same floor, the site-neighboring index is 0.6 when their floor difference is 1 floor, is 0.2 when their floor difference is 2 floors, and is 0.0 when their floor difference is 3 floors. The facilities and their layout and interactive preference are shown in Table 1. These design preferences are as follows: & If the space requirement of the facility is ‘‘small’’, and it is assigned at a ‘‘small’’, ‘‘middle’’, ‘‘large’’ space, then its layout preference is 10, 0, 10, respectively. & If the space requirement of the facility is ‘‘middle’’, and it is assigned at a ‘‘small’’, ‘‘middle’’, ‘‘large’’ space, then its layout preference is 0, 10, 0, respectively. & If the space requirement of the facility is ‘‘large’’, and it is assigned at a ‘‘small’’, ‘‘middle’’, ‘‘large’’ space, then its layout preference is 10, 0, 10, respectively. & If the facility is assigned at 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.
family medicine
urological surgery
rectum surgery pediatrics
1 floor otolaryngo-lo gy
obstetrics and gynecology
Fig. 12. The best layout design alternative from ARCHPLAN (alternative preference = 870.0).
Average of Objective Function
kidney department
st
pediatrics surgery
0.70
0.10
digestion
medical department
0.80
880
2nd floor dermatology
537
870 860 850 840 830 820 810 800 790 780
1
1.5
2
2.5
3
3.5
Penalty Coefficient Fig. 14. Average of objective function under various penalty coefficients.
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539 880
880
870
870
Average of Objective Function
Maximum of Objective Function
538
860 850 840 830 820 810 800 790 780 1
1.5
2
2.5
3
860 850 840 830 820 810 800 790 780 1.75
3.5
Penalty Coefficient
2
2.25
2.5
2.75
3
3.25
Maximum Initial Probability Coefficient
Fig. 15. Maximum of objective function under various penalty coefficients.
The direct evaluation with 1,000,000 feasible alternatives yields an average alternative preference (T standard deviation) of 354.6 T 53.3 with the maximum being 652.0. The evaluation with a greedy algorithmic method in [13] yielded an alternative preference of 646.0. The problem is resolved by annealed neural network with the following parameters: & penalty coefficient = 2.0 & maximum initial probability coefficient = 3.0 & temperature cool coefficient = 0.9. The solutions, based on 30 experiments, have an average alternative preference (Tstandard deviation) of 826.2 T 22.3 with the maximum being 870.0, and a convergence success rate of 93%. Each experiment takes about 6 s on a Pentium 3 notebook computer. Iteration of the evaluation function of one of the experiments (alternative preference = 856.0) is shown in Fig. 4. The evaluation functions of layout preference, interactive preference, constraint violation, and total objective in Eq. (2) increase monotonously. Its permutation matrix at the 0th, 3rd, 6th, 9th, and 12th iteration are shown in Figs. 5 – 9. The variations of the permutation matrixes showed that the neurons in the annealed neural network
Fig. 17. Average of objective function under various maximum initial probability coefficients.
competed with one another, and finally the winners composed of the near optimum solution. The binary permutation matrix at the 12th iteration is shown in Fig. 10. Its layout alternative based on the binary permutation matrix is shown in Fig. 11. It is easy to find that the alternative satisfies most design preferences. The best layout alternative (alternative preference = 870.0) in the 30 experiments is shown in Fig. 12. The alternative shows a better design quality according to on the design preferences shown in Table 1. Based on alternative preference, it can be concluded that the annealed neural network is superior to the direct evaluation with 1,000,000 feasible alternatives and a greedy algorithmic method. For the reason of comparison, the above mentioned implementation is reimplemented by the cases applying various parameters including penalty coefficient and maximum initial probability coefficient. Their results based on 30 experiments are shown in Figs. 13– 18. From the results, some observations can be given as follows: & Penalty coefficient (refer to Figs. 13– 15) The results show that convergence success rate increases monotonically but the average alternative preference decreases monotonically with the increase of the penalty coefficient. Therefore, penalty coefficient from 1.5 to 3.0 is suggested to
1.00 880
Maximum of Objective Function
Feasible Solution Ratio
0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 1.75
2
2.25
2.5
2.75
3
3.25
Maximum Initial Probability Coefficient Fig. 16. Feasible solution ratio under various maximum initial probability coefficients.
870 860 850 840 830 820 810 800 790 780 1.75
2
2.25
2.5
2.75
3
3.25
Maximum Initial Probability Coefficient Fig. 18. Maximum of objective function under various maximum initial probability coefficients.
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
computation time (sec)
1000.0
B ij
real predicted
C xi D xy
100.0
10.0
1.0
0
0.1
10
20
30
40
50
60
number of facilities
Fig. 19. Computation time under various numbers of facilities.
get high alternative preference and high convergence success rate. Besides, the maximum alternative preference is largely independent of the penalty coefficient. & Maximum initial probability coefficient (refer to Figs. 16 –18) The results show that the convergence success rate increases monotonically but the average alternative preference decreases monotonically with the increase of the maximum initial probability coefficient. Therefore, maximum initial probability coefficient from 2.5 to 3.0 is suggested to get high alternative preference and high convergence success rate. Besides, the maximum alternative preference is largely independent of the maximum initial probability coefficient. To evaluate the combinatorial complexity of the algorithm, a set of experiments with various numbers of facilities was implemented. The results in Fig. 19 show that the combinatorial complexity of the algorithm is about n 4. 6. Conclusions This study led to the following conclusions: 1. The QAP is an example of problems, which belong to the class of NP-hard problems in terms of their complexity. The advantage of our approach is that it is largely independent of the size of the search space unlike many other approaches. Therefore it is amenable to fast computation for large layout problems. 2. This new algorithm can be used to find competitive layout with relatively little computational effort. This is advantageous for a user who wishes to consider several competitive layouts rather than simply using the mathematically optimal layout. 3. Our computational experience shows that the choice of penalty coefficients has a significant impact on the layout obtained using our algorithm. Therefore the role of penalty coefficients should be the subject of future research. Notation A ij Site-neighboring index, if site i is a neighbor to site j, then A ij =1, else A ij =0
E F n P avg P max Si T U xi V xi Wij a DE DE xi n g k hi
539
Permutation matrix variable (is 1 if element i is assigned on position j) Layout preference of assigning facility X on site i Interactive preference of assigning facility X on the site-neighboring facility Y Energy function Objective function Number of facilities (sites) Average probability Maximum initial probability State variable (is 1 if neuron i is on and 0 otherwise) Simulated annealing temperature Neuron denoted that assigning facility X on site i State variable, a probability of assigning facility X on site i Weight on the connection between neuron i and neuron j Temperature cool coefficient Energy gap Energy gap of neuron U xi Penalty coefficient Maximum initial probability coefficient Penalty factor Threshold for neuron i
References [1] M.F. Anjos, A. Vannelli, A new mathematical programming framework for facility layout design, (2002) http://www.optimization-online.org/DB HTML/2002/03/454.html. [2] J.J. Michalek, R. Choudhary, P.Y. Papalambros, Architectural layout design optimization, Engineering Optimization 34 (5) (2002), 461 – 484. [3] R. Jagielski, J.S. Gero, A genetic programming approach to the space layout problem, in: R. Junge (Ed.), CAAD Futures, Kluwer, Dordrecht, 1997, pp. 875 – 884. [4] R.S. Liggett, W.J. Mitchell, Optimal space planning in practice, Computer-Aided Design 13 (5) (1981) 277 – 288. [5] J.H. Jo, J.S. Gero, Space layout planning using an evolutionary approach, Artificial Intelligence in Engineering 12 (1998) 149 – 162. [6] R.S. Liggett, Automated facilities layout: past, present and future, Automation in Construction 9 (2000) 197 – 215. [7] I.D. Tommelein, R.E. Levitt, T. Confrey, SightPlan expertments: alternate strategies for site layout design, Journal of Computing in Civil Engineering, ASCE 5 (1) (1991) 42 – 63. [8] S. Kirkpatric, C. Gelatt, M. Vecchi, Optimization by simulated annealing, Science 220 (4598) (1983) 671 – 680. [9] P.M. Hahn, J. Jrarup, A hospital facility layout problem finally solved, Journal of Intelligent Manufacturing 15 (5/6) (2000) 1 – 11. [10] J. Hopfield, D. Tank, Neural computations of decisions in optimization problems, Biological Cybernetics 51 (1985) 141 – 152. [11] D.E. Van den Bout, T.K. Miller, A traveling salesman objective function that works, Proceedings of International Joint Conference on Neural Networks 1989, Washington, DC, 1989, pp. 299 – 303. [12] D.E. Van den Bout, T.K. Miller, Graph partitioning using annealed neural networks, IEEE Transaction on Neural Networks 1 (2) (1990) 192 – 203. [13] I-Cheng Yeh, Construction site layout using annealed neural network, Journal of Computing in Civil Engineering, ASCE 9 (3) (1995) 201 – 208.
Architectural layout optimization using annealed neural network I-Cheng Yeh Department of Civil Engineering, Chung-Hua University, Hsin Chu, Taiwan 30067, ROC Accepted 31 July 2005
Abstract The facility layout problem is concerned with finding feasible locations for a set of interrelated objects that meet all design requirements and maximize design quality in terms of design preferences. The contribution of this paper is a new framework, named annealed neural network, for efficiently finding competitive solutions for the facility layout problem. This framework arises from the combination of Hopfield neural networks and simulated annealing. The first is a representation model of the layout problem and the second is a search algorithm for finding the optimum or near optimum solutions. The annealed neural network combines characteristics of the simulated annealing algorithm and the Hopfield neural network. Annealed neural network exhibits the rapid convergence of the neural network, while preserving the solution quality afforded by simulated annealing. Strategies for setting reasonable penalty factor in objective function and temperature in simulated annealing procedure were proposed. A case study of a hospital building with 28 facilities was employed to demonstrate that this model is rather efficient to solve the architectural layout problem, and it is amenable to fast computation for large layout problems. D 2005 Elsevier B.V. All rights reserved. Keywords: Architecture; Layout; Annealed neural network; Simulated annealing; Combinatorial optimization
1. Introduction The facility layout problem is concerned with finding feasible locations for a set of interrelated objects that meet all design requirements and maximize design quality in terms of design preferences such as minimizing the total cost associated with the interactions between these facilities. The given pairwise costs usually reflect transportation cost and/or adjacency preferences between facilities [1,2]. Architectural layout problem is particularly interesting because in addition to common engineering objectives such as cost and performance, architectural design is especially concerned with aesthetic and usability qualities of a layout, which are generally more difficult to describe formally [2]. In this paper we discuss facility layout design formulated as a Quadratic Assignment Problem (QAP). This is a difficult combinatorial optimization problem of great importance, reaching beyond formal architectural design [3]. Versions of this problem occur in many environments, such as hospital layout and service center layout, as well as in other engineering
E-mail address: [email protected]. 0926-5805/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.autcon.2005.07.002
applications, such as VLSI placement and design. All of these problems are known to be NP-hard. For this reason, most of the approaches in the literature are based on heuristics [1]. Because of the combinatorial complexity, it cannot be solved exhaustively for reasonably sized layout problems. Several heuristic strategies have been developed to find solutions without searching the design space exhaustively. Liggett and Mitchell [4] use a constructive placement strategy followed by an iterative improvement strategy. In this method, space is allocated for rooms one at a time based on the best probable design move at each step. Other researchers have used stochastic algorithms for search [3,5]. An overview and history of the automated layout problem can be found in Liggett [6]. It has been argued that with the possibility of finding good solutions quickly using heuristic methods such as genetic algorithm and simulated annealing [7,8], the search for exact solutions may not be worthwhile. This argument holds merit since we are dealing with models that are abstractions and not themselves reality. Therefore, it may be not necessary to have provably optimum solutions except in rather rare circumstances, when the model precisely matches the situation and large expenditures of time and money are involved [9]. Recently simulated annealing (SA) has been proposed as a
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I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
general technique for attempting to solve combinational optimization problems. Neural networks is a powerful method of optimization which relies on developing systems that demonstrate selforganization and adaptation in a similar, though simplified, manner to the way in which biological systems work. Hopfield neural network [10], a kind of artificial neural network, has been proposed as a model of computation to solve a wide variety of discrete combinatorial optimization problems. The contribution of this paper is a new framework, named annealed neural network, for efficiently finding competitive solutions for the facility layout problem. This framework arises from the combination of Hopfield neural networks and simulated annealing. The first is a representation model of the layout problem and the second is a search algorithm for finding the optimum or near optimum solutions. The annealed neural network combines characteristics of the simulated annealing algorithm and the Hopfield neural network. Annealed neural network exhibits the rapid convergence of the neural network, while preserving the solution quality afforded by simulated annealing. This paper describes an effort that applies annealed neural network to generate architectural layout alternative. In the following sections, Section 2 introduces architectural layout problem. Section 3 describes the annealed neural network model. Section 4 describes the model using annealed neural network to solve the problem. In Section 5, a case study of a hospital building with 28 facilities is employed to illustrate the practical applications and to demonstrate that this model is rather efficient to solve the architectural layout problem. In addition, the effects of three control parameters on the model are experimented. Finally, Section 6 gives summary and conclusions. 2. Architectural layout problem The QAP can be defined in the following way: n distinct objects N = {1, 2,. . ., n} are to be placed uniquely in m distinct sites M = {1, 2,. . ., m}, where m 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 [3]. For an architectural layout problem, facilities can be viewed as architectural functions. Sites can be defined as spaces allocated for specific architectural functions. It can be argued that the two cases ‘‘m = n’’ and ‘‘m > n’’ essentially are equivalent since 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 the building is designed to exactly house the n facilities in its free cells. Of course, there may also be other cells in the building, but they are in the context regarded as inadmissible, perhaps because they are reserved for other purposes [9]. In the present work, the formulation of the architectural layout problem is 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 objective considered in this paper including adjacency of objects, distance between objects, availability of space for object location, positions of objects in relation to others. The layout constraints measures the feasibility of the layout, 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. Then, the problem is formulated as follows XX Maximize F ¼ Bxi Cxi x
þ
I
i
XXXX x
i
y
I I I
Bxi Byj Aij Dxy
j
ð1Þ
Subjected to Byi ¼ 0 if Bxi ¼ 1 and ymx Bxj ¼ 0 if Bxi ¼ 1 and jmi where F = objective function; d xi = permutation matrix variable (is 1 if facility X is assigned on site i); C xi = layout preference of assigning facility X on site i; A ij = site-neighboring index, if site i is a neighbor to site j, A ij = 1, if site i is not a neighbor to site j, A ij = 0, and A ii = 0; D xy = interactive preference of assigning facility X on the neighboring site of facility Y, and D xx = 0. In the objective function, the first term accounts for the total layout preferences of assigning facilities on sites, and the second term represent the total interactive preferences between facilities. If a pre-determined site i is very unsuitable to a predetermined facility X, based on size or shape, this problem can be solved by assigning a very low layout preference C xi . 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 D xy. Therefore, Eq. (1) is rather practical for architectural layout problem. Each layout alternative can also be represented by a n n permutation matrix whose rows and columns are labeled by facilities and by sites, respectively. The permutation matrix has only one 1 in each row and each column, with the remaining elements being 0. Fig. 1 shows a permutation matrix, where facility A is assigned on the site 3, facility B is assigned on the site 1, facility C is assigned on the site 4, facility D is assigned on the site 5, and facility E is assigned on the site 2. 1
2
3
4
5
A
0
0
1
0
0
B
1
0
0
0
0
C
0
0
0
1
0
D
0
0
0
0
1
E
0
1
0
0
0
Fig. 1. Permutation matrix with five facilities.
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
3. Annealed neural network Hopfield neural networks [10] contain many simple computing elements, which cooperatively traverse the energy surface defined by the energy function to find a local or global minimum. In spite of its novel perspective, however, the Hopfield approach to the solution of these optimization problems is limited by its inability to ferret out global minimum with a single invocation of the algorithm. It is effectively a steepest-descent procedure that settles at the nearest local minimum. A global minimum is insured only by initializing the algorithm at a sufficient number of different starting points. In addition, Hopfield neural networks have several parameters that need to be selected, and often, carefully tuned for a network to produce a sensible computation [10]. Simulated annealing is a probabilistic hill-climbing search algorithm, which finds a global minimum of energy function by combining gradient descent with a random process. This combination allows, under certain conditions, changes to state that actually increases energy function, thus providing SA with a mechanism to escape from local minimum. Although changes to state that decrease energy function are always accepted, a move that causes an increases will be taken with a Boltzmann probability. Though, SA has been effective in many practical problems such as traveling salesman problem (TSP), in terms of the quality of solutions, it requires unacceptably large computing times [11,12]. A new neural network model, named annealed neural network, which merges many features of simulated annealing and Hopfield neural networks, was proposed [11,12]. Annealed neural network exhibits the rapid convergence of the neural network, while preserving the solution quality afforded by simulated annealing. The annealed neural networks have been proposed as a model of combinational optimization problem such as the TSP problem and graph-partitioning problem [11,12]. The simulated neural network algorithm can be written as follows Set initial state variables, S, on network; Set a starting temperature T; Calculate initial energy function E; Repeat For i = 1 to n do Calculate energy gap DE i ; Calculate new state variable S i = exp( DE i / T); Continue Calculate new energy function E; Lower temperature T; Until energy function is stationary.
533
those solutions with high design preference; and (2) it must be high for only those solutions that satisfied the layout constraints. Therefore, the total objective function can be given as follows: Maximize F ¼ F1 þ F2 þ F3 XX XXXX ¼ Vxi Cxi þ Vxi Vyj Aij Dxy x
k
I
i
XXX x
i
x
I
y
i
ð2Þ
Vxi Vyj
ymx
where F 1 = evaluation function of layout preference; F 2 = evaluation function of interactive preference; F 3 = evaluation function of constraint violation; V xi = state variable, a probability of assigning facility X on site i. Each variable V xi is looked upon as the probability of finding facility X currently assigned on site i of the alternative; k = penalty factor. 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. Eq. (2) can be rewritten as energy function: Minimize E ¼ F:
ð3Þ
The energy gap can be calculated as follows: DExi ¼ E ðVxi ¼ 1Þ E ðVxi ¼ 0Þ XX ¼ Cxi Vyj Aij Dxy þ k y
I I
j
The objective function of the architectural layout problem must satisfy two requirements: (1) It must be high for only
I
X
Vyi
ð4Þ
ymx
The probability of assigning facility X on site i, V xi , can be calculated by Boltzmann distribution and the following normalization operation [11,12]: expð DExi =T Þ Vxi ¼ X : exp DExj =T
ð5Þ
j
The normalization operation in (5) guarantees that each facility will be assigned to one site only, so the n facilities will be assigned to the n sites. 4.2. Penalty factor The setting for the penalty factor determines the importance of the penalty term of the energy function. Setting it to a small value emphasizes the term concerned with preference and leads to high preference, which is, unfortunately, invalid. AlternaSmall
Small
Small
1
2
3
Middle
Middle
Middle
4
5
6
Large 7
4. Architectural layout using annealed neural network 4.1. Energy function
I I I
j
Fig. 2. Plane of the standard floor of the hospital.
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I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
1
2
3
4
5
6
gap = Max C xi Min C xi ; DD max = maximum interactive preference gap =Max D xy Min D xy. By increasing the value of the penalty coefficient, i.e., is by making the last term in (2) relatively more important, we can increase the rate of success, but then the quality of the solutions will not be as good.
7
1
0.0 1.0 0.8 1.0 0.8 0.8 0.8
2
1.0 0.0 1.0 0.8 1.0 0.8 0.8
3
0.8 1.0 0.0 0.8 0.8 1.0 1.0
4
1.0 0.8 0.8 0.0 1.0 0.8 0.8
5
0.8 1.0 0.8 1.0 0.0 1.0 0.8
6
0.8 0.8 1.0 0.8 1.0 0.0 1.0
7
0.8 0.8 1.0 0.8 0.8 1.0 0.0
4.3. Simulated annealing temperature
Fig. 3. Site-neighboring index matrix for the same floor.
tively, setting a large value makes the penalty so stiff that the neural network will converge to any feasible solution regardless of its design preference. If two facilities try to occupy the same site, the penalty term in energy function should be incurred to prevent this type of a facility layout alternative. This is ensured by setting the penalty factor slightly larger than the maximum layout preference gap plus the maximum interactive preference gap.
I
k ¼ n ðDCmax þ DDmax Þ
ð6Þ
where n = penalty coefficient, n > 0. Generally, n should be approximately equal to 1; DC max = maximum layout preference
At a very high temperature, each facility will be assigned equally across each site. As the temperature is reduced, the facilities will begin to coagulate in particular sites, which will hopefully minimize the energy function. In general, the cooling procedure is proposed as follows
I
T ð t þ 1Þ ¼ a T ð t Þ
ð7Þ
where a = temperature cool coefficient, a 1. Generally, a = 0.9– 1.0. The initial temperature is very important for convergence speed. At a too high initial temperature, convergence will be very slow, while at a too low initial temperature, convergence will be very quick, but the quality of the solutions will not be as good. In this paper, a formula to determine a
Table 1 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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Endocrine metabolism Digestion Cardiology Chest department Kidney department Blood department Immunity rheumatism Pestilence Medical department Rehabilitation Pediatrics Orthopedics Eye ophthalmology Neuropsychiatry Obstetrics gynecology Dermatology Pediatrics surgery Head psychiatry Thoracic cardiac Anaplastic department Surgery department
Small Large Large Middle Small Small Small Small Small Small Large Large Middle Middle Middle Middle Small Small Small Middle Small
– – – – – – – 1–3 2–4 2–4 3–4 – – – 2–4 – – 1–3 – 1–2 –
14 6,9 19 19 6,23 2,5,7,19 6,8 7 2,27 20 15,17 20 – 1,18,26 11,17 20 11,15 14,26 3,4,6 10,12,16 –
– – – – – – – 11,17,21 – 11,17 8,10,18 – – – 28 – 8,10,20,28 11,20 – 11,17,18,27 8
22 23 24 25 26 27 28
Rectum surgery Urological surgery Otolaryngology Dentistry Neurosurgery Family medicine Radiosurgery tumor
Small Small Middle Small Small Small Small
– – – 3–4 – 2–4 1–3
– 5 – – 14,18 9 –
2,5,15,16 1,4,5,11,16,27 4,5,6,10 2,3,9 1,2,3,8,9,15,16,19 3,8 10,21,28 5,6,10,16,23,24 4,5,23,24 3,7,8,12,18,21,26,27 2,24,25,27 10,21,25,26 20,26 – 1,5,23,27 1,2,5,8,23,24,26 21,27 10,27 5,21 13,21,25 7,10,12,17,19,20,22, 23,24,26,27 21,23 8,9,15,16,21,22,27 8,9,11,16,21,27 11,12,20,26 10,12,13,16,21,25 2,10,11,15,17,18,21, 23,24 7
– – – – – 20 11,15,17
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
Iteration 0
2
4
6
8
10
12
14
16
18
20
1000
0 -500 -1000
Site
Evaluation Function
500
Layout Preference Interaction Preference Constraint Violation Total Objective
-1500 -2000 -2500
Fig. 4. Iteration of the evaluation function.
reasonable initial temperature for facility X, T x is proposed as follows: 0 1 B expð DExi =Tx Þ C C MaxB @ X exp DE =T A ¼ Pmax xj
x
where P max = maximum initial probability in a normalization set; and Pmax ¼ g Pavg ð9Þ
I
where g = maximum initial probability coefficient, 1 < g < n; P avg = average probability, P avg = 1 / n; n = the number of neurons in a normalization set, i.e., the number of facilities (sites). The reason is that the reasonable initial temperature should be able to raise the maximum initial probability in a normalization set to a reasonable value, for example, from double to triple average probability, i.e., g = 2– 3. 4.4. Facility layout with simulated annealing procedure The architectural layout procedure using annealed neural network is summarized as follows:
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
Fig. 6. Permutation matrix of the hospital layout design (iteration = 3).
Calculate initial energy function E using (3); Repeat For X = 1 to n do For i = 1 to n do Calculate DE xi using (4); Continue For i = 1 to n do Calculate V xi using (5); Continue Continue Calculate new energy function E using (3); Calculate new temperature T x for each facility X using (7); Until energy function is stationary. 4.5. Neural network model The above procedure can be modeled as a neural network as follows. Each facility is represented as a row of n neurons. One
Site
Set initial state variables, V xi , on network; Set a starting temperature T x for each facility X using (8);
Site
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Facility
ð8Þ
j
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
535
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
Facility Fig. 5. Permutation matrix of the hospital layout design (iteration = 0).
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 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
Facility Fig. 7. Permutation matrix of the hospital layout design (iteration = 6).
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Site
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
Site
536
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
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
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
Facility
Facility
Fig. 8. Permutation matrix of the hospital layout design (iteration = 9).
Fig. 10. Permutation matrix of the hospital layout design (iteration = 12, binary solution).
and only one such neuron in a row may be set to one (all others must be set to zero). This neuron set to one indicates the site that a specific facility is assigned. Hence we make a network composed of n identical neurons and denote them by a pair of indices, x referring to the facility and i referring to the site in the alternative. For example, neuron U xi indicates the state that the facility X assigned on site i. Let each connection be identified by four subscripts, i.e., W xiyj indicates the link between neuron U xi and neuron U yj , and each threshold be identified by double subscripts, i.e.,h xi indicates the threshold for neuron U xi . Let 8 < Aij Dxy ; if xmy and imj ð10Þ Wxiyj ¼ k; if xmy and i ¼ j : 0; otherwise
Then the energy function on Eq. (3) can be rewritten as follows XXXX XX Vxi hxi þ Vxi Vyj Wxiyj : ð12Þ E¼ x
i
Site
ð11Þ
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
x
i
y
j
I I
immunity rheumatism
pestilence
head psychiatry
anaplastic department
eye ophthalmo-l ogy
radiosurgery tumor department
blood department
thoracic cardiac
neurosurgery
dermatology
endocrine metabolism
neuro-psych iatry
kidney department
surgery
urological surgery
4th floor
I
hxi ¼ Cxi
I
rehabilitation
3rd floor
orthopedics
2nd floor
cardiology chest department
dentistry
pediatrics surgery
family medicine
rectum surgery
digestion
medical department
obstetrics and gynecology
otolaryngology
1st floor
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
Facility Fig. 9. Permutation matrix of the hospital layout design (iteration = 12).
pediatrics
Fig. 11. A layout design alternative from ARCHPLAN (alternative preference = 856.0).
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
II
It is easy to find that Eq. (12) is similar to Hopfield energy function. Furthermore, a positive value for W xiyj lowers the energy function if both neurons U xi and U yj are on and thus makes it more likely that the final state will include those two neurons in the on state. Conversely, a negative value lowers the likelihood of the two neurons being in the on state, in effect penalizing such a configuration. In a similar fashion, a positive value for h xi lowers the energy function if the neuron U xi is off and thus makes it more likely that the final state will include the neuron in the off state. Conversely, a negative value lowers the likelihood of the neuron being in the on state, in effect penalizing such a configuration. 5. Case study: hospital layout design In this section, a C program called ArchPlan is built on personal computer to implement the annealed neural network algorithm, and a case study of a 4-story hospital building with 28 facilities is employed to demonstrate ArchPlan in practical applications. Each floor contains 7 sites. The plane of the immunity rheumatism
pestilence
radiosurgery tumor
rehabilitation
anaplastic department
thoracic cardiac
blood department
neurosurgery
head psychiatry
neuro-psych iatry
chest department
endocrine metabolism
dentistry
surgery
4th floor
cardiology eye ophthalmology
3rd floor
orthopedics
1.00 0.90
Feasible Solution Ratio
Comparing Eq. (12) with Hopfield energy function [10] X 1 XX E¼ Si hi Si Sj Wij ð13Þ 2 i j i
0.60 0.50 0.40 0.30 0.20
0.00
1
1.5
2
2.5
3
3.5
Penalty Coefficient Fig. 13. Feasible solution ratio under various penalty coefficients.
standard floor of the hospital is shown in Fig. 2. The siteneighboring index for the same floor is shown in Fig. 3. If the two facilities are not at the same floor, the site-neighboring index is 0.6 when their floor difference is 1 floor, is 0.2 when their floor difference is 2 floors, and is 0.0 when their floor difference is 3 floors. The facilities and their layout and interactive preference are shown in Table 1. These design preferences are as follows: & If the space requirement of the facility is ‘‘small’’, and it is assigned at a ‘‘small’’, ‘‘middle’’, ‘‘large’’ space, then its layout preference is 10, 0, 10, respectively. & If the space requirement of the facility is ‘‘middle’’, and it is assigned at a ‘‘small’’, ‘‘middle’’, ‘‘large’’ space, then its layout preference is 0, 10, 0, respectively. & If the space requirement of the facility is ‘‘large’’, and it is assigned at a ‘‘small’’, ‘‘middle’’, ‘‘large’’ space, then its layout preference is 10, 0, 10, respectively. & If the facility is assigned at 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.
family medicine
urological surgery
rectum surgery pediatrics
1 floor otolaryngo-lo gy
obstetrics and gynecology
Fig. 12. The best layout design alternative from ARCHPLAN (alternative preference = 870.0).
Average of Objective Function
kidney department
st
pediatrics surgery
0.70
0.10
digestion
medical department
0.80
880
2nd floor dermatology
537
870 860 850 840 830 820 810 800 790 780
1
1.5
2
2.5
3
3.5
Penalty Coefficient Fig. 14. Average of objective function under various penalty coefficients.
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539 880
880
870
870
Average of Objective Function
Maximum of Objective Function
538
860 850 840 830 820 810 800 790 780 1
1.5
2
2.5
3
860 850 840 830 820 810 800 790 780 1.75
3.5
Penalty Coefficient
2
2.25
2.5
2.75
3
3.25
Maximum Initial Probability Coefficient
Fig. 15. Maximum of objective function under various penalty coefficients.
The direct evaluation with 1,000,000 feasible alternatives yields an average alternative preference (T standard deviation) of 354.6 T 53.3 with the maximum being 652.0. The evaluation with a greedy algorithmic method in [13] yielded an alternative preference of 646.0. The problem is resolved by annealed neural network with the following parameters: & penalty coefficient = 2.0 & maximum initial probability coefficient = 3.0 & temperature cool coefficient = 0.9. The solutions, based on 30 experiments, have an average alternative preference (Tstandard deviation) of 826.2 T 22.3 with the maximum being 870.0, and a convergence success rate of 93%. Each experiment takes about 6 s on a Pentium 3 notebook computer. Iteration of the evaluation function of one of the experiments (alternative preference = 856.0) is shown in Fig. 4. The evaluation functions of layout preference, interactive preference, constraint violation, and total objective in Eq. (2) increase monotonously. Its permutation matrix at the 0th, 3rd, 6th, 9th, and 12th iteration are shown in Figs. 5 – 9. The variations of the permutation matrixes showed that the neurons in the annealed neural network
Fig. 17. Average of objective function under various maximum initial probability coefficients.
competed with one another, and finally the winners composed of the near optimum solution. The binary permutation matrix at the 12th iteration is shown in Fig. 10. Its layout alternative based on the binary permutation matrix is shown in Fig. 11. It is easy to find that the alternative satisfies most design preferences. The best layout alternative (alternative preference = 870.0) in the 30 experiments is shown in Fig. 12. The alternative shows a better design quality according to on the design preferences shown in Table 1. Based on alternative preference, it can be concluded that the annealed neural network is superior to the direct evaluation with 1,000,000 feasible alternatives and a greedy algorithmic method. For the reason of comparison, the above mentioned implementation is reimplemented by the cases applying various parameters including penalty coefficient and maximum initial probability coefficient. Their results based on 30 experiments are shown in Figs. 13– 18. From the results, some observations can be given as follows: & Penalty coefficient (refer to Figs. 13– 15) The results show that convergence success rate increases monotonically but the average alternative preference decreases monotonically with the increase of the penalty coefficient. Therefore, penalty coefficient from 1.5 to 3.0 is suggested to
1.00 880
Maximum of Objective Function
Feasible Solution Ratio
0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 1.75
2
2.25
2.5
2.75
3
3.25
Maximum Initial Probability Coefficient Fig. 16. Feasible solution ratio under various maximum initial probability coefficients.
870 860 850 840 830 820 810 800 790 780 1.75
2
2.25
2.5
2.75
3
3.25
Maximum Initial Probability Coefficient Fig. 18. Maximum of objective function under various maximum initial probability coefficients.
I.-C. Yeh / Automation in Construction 15 (2006) 531 – 539
computation time (sec)
1000.0
B ij
real predicted
C xi D xy
100.0
10.0
1.0
0
0.1
10
20
30
40
50
60
number of facilities
Fig. 19. Computation time under various numbers of facilities.
get high alternative preference and high convergence success rate. Besides, the maximum alternative preference is largely independent of the penalty coefficient. & Maximum initial probability coefficient (refer to Figs. 16 –18) The results show that the convergence success rate increases monotonically but the average alternative preference decreases monotonically with the increase of the maximum initial probability coefficient. Therefore, maximum initial probability coefficient from 2.5 to 3.0 is suggested to get high alternative preference and high convergence success rate. Besides, the maximum alternative preference is largely independent of the maximum initial probability coefficient. To evaluate the combinatorial complexity of the algorithm, a set of experiments with various numbers of facilities was implemented. The results in Fig. 19 show that the combinatorial complexity of the algorithm is about n 4. 6. Conclusions This study led to the following conclusions: 1. The QAP is an example of problems, which belong to the class of NP-hard problems in terms of their complexity. The advantage of our approach is that it is largely independent of the size of the search space unlike many other approaches. Therefore it is amenable to fast computation for large layout problems. 2. This new algorithm can be used to find competitive layout with relatively little computational effort. This is advantageous for a user who wishes to consider several competitive layouts rather than simply using the mathematically optimal layout. 3. Our computational experience shows that the choice of penalty coefficients has a significant impact on the layout obtained using our algorithm. Therefore the role of penalty coefficients should be the subject of future research. Notation A ij Site-neighboring index, if site i is a neighbor to site j, then A ij =1, else A ij =0
E F n P avg P max Si T U xi V xi Wij a DE DE xi n g k hi
539
Permutation matrix variable (is 1 if element i is assigned on position j) Layout preference of assigning facility X on site i Interactive preference of assigning facility X on the site-neighboring facility Y Energy function Objective function Number of facilities (sites) Average probability Maximum initial probability State variable (is 1 if neuron i is on and 0 otherwise) Simulated annealing temperature Neuron denoted that assigning facility X on site i State variable, a probability of assigning facility X on site i Weight on the connection between neuron i and neuron j Temperature cool coefficient Energy gap Energy gap of neuron U xi Penalty coefficient Maximum initial probability coefficient Penalty factor Threshold for neuron i
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