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Ш.УПРАВЛ1ННЯ
YAK 681.32
A FUZZY-TABU SEARCH APPROACH FOR THE PLANT LAYOUT PROBLEM
Turkbey Orhan, Alabas Cigdem
Расходы на систему погрузочно-разгрузочных работ составляют почти 20-50% общей себестоимости производственной системы. Вследствие этого, проблема размещения предприятий - одна из важнейших задач. Но, так как проблема размещения предприятий - это NP-полная проблема, имеющая комбинаторную структуру, размещение предприятий является плохо исследованной областью. В настоящей работе для проблемы размещения предприятий, которая формализуется как квадратичная проблема назначения, используется запрещающий алгоритм поиска, который является одним из интеллектуальных эвристических методов. Отличный от других известных подходов запрещающего поиска, разработанный алгоритм использует теорию нечетких множеств. Разработанный алгоритм запрещающего поиска сравнивается с классическим алгоритмом запрещающего поиска на тестовых задачах различной размерности и методом случайного поиска согласно определенным критериям эффективности функционирования типа "качество решения и номер найденной точки в пространстве решений". Согласно полученным результатам показано, что в отношении задачи размещения предприятий нечеткий запрещающий алгоритм поиска превосходит другие алгоритмы.
Materials handling system cost expenditures make up nearly 20-50% of the total production cost of a manufacturing system. Owing to this, the plant layout problem is one of the subjects which are discussed mostly. But, because the plant layout problem is a NP-complete problem and has a combinatorial structure, the plant layout is still an investigation area. In the content of this study, for the plant layout problem, which is formalized as Quadratic Assignment Problem, Tabu Search Algorithm, which is one of the Smart Heuristic Techniques, is used. Different from other tabu search/search approaches in the literature, the developed algorithm uses Fuzzy Set Theory. The developed Tabu Search Algorithm in this study is compared with Classical Tabu Search Algorithm on test problems in various sizes and Random Search Method according to some performance criteria such as "solution quality and the searched point number in the solution space". According to received results, it's seen that in the plant layout problem Fuzzy-Tabu Search Algorithm is superior to other algorithms.
1. INTRODUCTION
Plant Layout Problem (PLP) has a strategical location in the success of manufacturing systems. The main reason for this fact is that, 20-50% of the total manufacturing cost originates from material management expenses. A good solution for the PLP will assist the general productivity of the system. On the contrary, a bad layout will lead to accumulation of mid stores, collidation of material transport systems, increment of preparation times and finally will lead to forma-
tion of long tails in the system (Jajodia et al. 1992). PLP may require high cost investments in long term. Moreover, relocation or improvement of a present plant/facility is not only expensive and time consuming but also this prevents activities of the workers and material flow (Sule, 1988). But the strategical importance of plant layout design should not be neglected.
The most popular formulation used in solution of PLPs is Quadratic Assignment Problem (QAP). The other formulations used for this purpose are quadratic set covering, linear integer programming, mixed integer programming, graph theoretic models (Alvarenga et al. 2000).
Koopmans and Beckman (1957) formulized PLP as QAP for the first time. The reason for naming this problem as "quadratic assignment" is that, the variables of the object function are second degree polinomial functions and the limits also resemble linear assignment problem's limits. The aims of QAP is to assign n facilities (machines, departments or work stations) to n places optimally by minimizing the material transport cost, which is defined as multiplication of work flow and the distance travelled, as much as possible.
QAP formulation is given in equation 1.
Minimize
III I f^xx.
tp jq ч pq
i = 1 p = 1 j = 1 q = 1
Subject to: I Xj = 1 Vi and j
j = 1 n
r I Xj = 1 Vj and j
(1)
xtj = 0 or 1 .
ij
In the model given above x,, decision variable will be 1 if
lJ
facility i is assigned to area j , otherwise it will be 0. The
work flow between facilities i and p is expressed as fip , the
ip
distance between areas J and q is expressed as d,q . The
object function in this problem is quadratic and it has a non-convex structure. There are more than one local optimum solutions for QAP and there are n! solution points in the solution space.
n
n
n
n
n
126
ISSN 1607-3274 "Радюелектрошка. 1нформатика. Управл1ння" № 2, 2001
2. LITERATUE SURVEY FOR QAP
QAP is a NP-complete problem. Gilmore (1963), Lawler (1963), Burkard and Stratmann (1978), Bazaraa and Elshafei (1979), Bazaraa and Sherali (1980) developed procedures that give the optimum results for this problem. But the calculation ability of the optimum procedures provides the solution of the layout problems which have at most 20 facilities (Resende et al. 1995).
Because of the combinatorial structure of the PLP and the inadequency of the optimal procedures in solution of the large sized problems, researchers directed themselves to development of the heuristic algorithms on this subject. These heuristics are classified in two, as construction and improvement methods. Construction methods start the operation with an empty area and locate the facilities by choosing them consequently till the layout is completed. On the other hand in the improvement methods, the operation starts on a present layout. This initial layout can be a result of structuring algorithm or be selected randomly. Improvement methods try to reach better layout designs by changing the places of the facility units reciprocally on the basis of the present layout. Foulds (1983), Kusiak (1990) and Burkard (1990) presented detailed surveys on heuristic algorithms.
Researchers are using smart heuristic approaches for QAP recently because smart heuristic techniques such as Genetic Algorithms (GAs), Simulated Annealing (SA), Tabu Search (TS) and Artificial Neural Networks have been applied to many problems till that time since they were developed and they gave good solutions for these problems.
SA, which is presented as independent from each other by Kirkpatrick et al.(1983) and Cerny (1985), is a probabilistic method imitating the physical annealing procedure and many researchers used this method in order to solve various PLPs. Kim and Kim (1998), Chwif et al. (1998), Wang et al. (1998), Bazarganlari and Kaebernick (1997), Meller and Bozer (1996), Yip and Pao (1994) are the researchers who have applied SA to PLPs most recently.
GAs, which is another smart heuristic algorithm, imitates the "survival of the f?ttest" principle of the nature. Holland (1975), DeJong (1975) and Goldberg (1989) are avant-gardes of GAs. GAs have been used in order to solve various PLPs till today. Zhang et al. (2000), Al-Hakim (2000), Gau and Meller (1999), Kochhar and Heragu (1999), Hamamoto et al. (1999), Tam and Chan (1998), Mak et al. (1998), Suresh et al. (1995) applied GAs on PLPs successfully recently.
TS, which is designed modernly by Glover (1989,1990), is a smart heuristic technique assisting the local search procedures in order to be saved from optimal traps. Abninnour-Helm and Hadley (2000), Alvarenga et al. (2000), Chittra-tanawat and Noble (1999), Chiang and Chiang (1998), Chiang and Kouvelis (1996) are the most recent studies on TS for PLPs.
Artificial Neural Networks, which take origin from studies of McCulloch and Pitts (1943) and Rosenblatt (1962), imi-
tates the working principle of the human brain. Rossin et al. (1999), Chung (1999), Kazurhiro et al. (1996), Liao (1994) applied Artificial Neural Networks on PLP. On the other hand, Badiru and Arif (1996) and Raoot and Rakshit (1994) applied Fuzzy Set (FS) (Zadeh, 1965) based approaches on PLP.
3. TABU SEARCH AND FUZZY SET THEORY
The philosophy of TS method is claimed by Glover and Greenberg (1989) and Hansen (1986) for the first time independently from each other. But, this name is given to the method by Glover (1989,1990) who brought the method to its present modern appearance. TS shows the route to neighborhood search methods in order to be saved from local optimal traps.
TS technique begins the search operation by any beginning solution. Adjacent solutions received from the small changes done on this solution makes up the localization of the present solution and conversion of a solution into another solution is named as moving. Classical neighborhood search methods moves to another solution with better aim values from the present neighborhood solution, they choose this solution as the new present solution. This operation is repeated till all of the adjacent solutions in the neighborhood have a worser aim value and subsequently the solution found in the end becomes a local optimal point.
TS technique uses various methods in order to save the local search procedures from local traps. Primarily, the best solution in the present neighborhood of the solution is chosen as the new solution though it's worser than the present solution. But the neighborhood solutions received by the prohibited movements are not included in this selection operation. TS uses tabu list(s) in order to make the opposite of a movement done before. For doing this, while moving to a solution from the present solution another movement changing this movement into its previous solution is listed in the tabu list and prohibited, in a way a tabu is made. The movements listed in the tabu list stay in this list during a certain iteration time and later on they are let to be done by ending their tabu situation. This mechanism forming the short term memory of TS prevents formation of cycles during the search and provides saving from local optimal traps by changing the route of the search. Search operation continues with the selection of the adjacent solution which has the best aim value and is not a tabu as the new present solution from the neighborhood of the present solution. These operations are repeated repeatedly till a stopping condition such as maximum iteration number is provided. Moreover, under some special circumstances TS lets some movements to be made although they are tabu. Among these special circumstances, tabu demolishment principle, which is used most widely, is let to be done when tabu has a better aim value than the one of the best solution that is received by a movement throughout the search.
On the other hand some properties of the movements
made are kept in the long term memory of the tabu. In the long term memory usually the permanent and temporariness frequencies of the movements-meaning the iteration number selected for forming the present solution and iteration number of taking out of the present solution are kept. This frequency information is used for strengthening the search by focusing it on certain areas and/or for varifying the search by directing it to different areas in the solution space. In an efficient TS algorithm strengthening and varifying strategies carry a big importance. TS have been applied on many problems since it's claimed till now and quite a lot good results are obtained. Glover and Laguna (1993) give a related literature survey on the application areas of this method.
Fuzzy Set Theory (FST) was spelled in mid-60s for the first time. Zadeh (1965) set up bases of this theory. FST is developed in order to manage with the uncertainities which are not statistical in structure. The difference between fuzzy and classic sets is their membership degrees. A member in the classic set is a member of this set or not, thus membership function is a (0,1) binary function. On the contrary, in the fuzzy set membership,function can take any value between 0 and 1. In this case, a fractional membership degree defines the partial membership of this member for that set. This property is very useful when there are not clear borders defining the main aspects of the set. One of the main advantages of the fuzzy set function is that, it provides degreed passage from the situation of being a member of the set to the situation of not being a member (Hassanein and Cherlopalle, 1999).
In this study the reason for the necessity of assuming TS and FST together is the systematic path showing in TS. The strengthening and diversity strategies used in TS depend on intensifying on good solution areas and reaching different solution areas. But the balance between these two strategies is important for the performance of the algorithm. Thus, there're no definite answers for these questions such as "how much does the research intensify on the good solution areas ? and when one should direct towards the different solution areas ?". Essentially, these questions are internally related and their answers vary depending on the course of the search. That's why in this study, a new fuzzy-tabu search algorithm is developed by using a pre-defined fuzzy membership func-
tion in order to find out the answer to these questions during the search.
4. FUZZY-TABU SEARCH ALGORITHM FOR QAP
In this study for the PLP which is formulized as QAP, an heuristic algorithm depending on tabu search is developed. This algorithm is completely different from the other TS applications in the literature because it contains FS properties. The area structure used by the developed FTS, management of tabu_list and the determination of fuzzy tabu tenures are presented in the following parts.
4.1 Region structure
In the FTS algorithm, any solution vector in the solution space shows which machine (or facility, unit) is appointed to which area. The sequence of machine numbers and the areas where these machines are appointed to are explained according to the coding structure used in FTS algorithm. As an example, for a 5 size PLP, x = {2, 1, 3, 5, 4} is the solution vector, the placement of machine 1 to area 2, machine 2 to area 1, machine 3 to area 3, machine 4 to area 5 and machine 5 to area 4 are shown. Here it's assumed that all of the sizes of all machines are suitable for placing to every area.
According to the region structure the adjacents of a present solution are received by the inter-change movements which are possible to do on this solution, that means the places of the chosen machines are changed reciprocally. Adjacent solutions received by the inter-change movements of x-vector are given in figure 1 as a sample.
According to inter-change movement mechanism, if n shows the size of the problem (area/number of machines), n(n - 1)/2 adjacent solutions of any solutions may be received. FTS algorithm, which is a repeatedly method, investigates the region of the present solution that's made up of n(n - 1)/2 adjacent solution in each iteration. Also, for big sized problems (n > 20), it's possible to limit adjacent solutions of the region width in one of its subsets.
Table 1 - Adjacent solutions received from inter-change movement of x = {2, 1, 3, 5, 4}
Inter-Changes 1 2 3 4
1 Adjacents received from inter-change of machine 2 x1' ={1,2,3,5,4} x2' ={3,1,2,5,4} x3' ={5,1,3,2,4} x4' ={4,1,3,5,2}
2 Adjacents received from inter-change of machine 1 x5' ={2,3,1,5,4} x6' ={2,5,3,1,4} x7' ={2,4,3,5,1} Same as (1,1)
3 Adjacents received from inter-change of machine 3 x8' ={2,1,5,3,4} x9' ={2,1,4,5,3} Same as (1,2) Same as (2,1)
4 Adjacents received from inter-change of machine 5 x10' ={2,1,3,4,5} Same as (1,4) Same as (2,3) Same as (3,1)
4.2 Tabu list management
In the FTS the movements which ruin a preformed interchange movement that converts the present solution into its previous sitution, are prohibited along a certain repetition number which is shown by tabu-tenure (theyre taken into tabu class), thus this provides reaching different solution regions by changing the direction of the search in the solution space. This principle lets the FTS algorithms to be saved from the local best traps. For this purpose, a tabu list named as tabu_list[i, j] (i = 1, 2, ..., n , j = 1, 2,..., n ) is kept.
Tabu_list[i,j] method can be explained shortly as; the
present solution x[k] (k = 1, 2,..., n ) shows any sequence of machines (style of location to the areas) and the adjacent solution obtained by the inter-change between machine a in area p and machine b in area q is accepted as the new present solution. In this situation, tabu_list[i,j] is updated as follows in order to show the iter repetition number for the purpose of prohibiting the departures of machines a and b from the newly assaigned q and p areas sequentially,
(tabu_list[a, q] = iter) and (tabu_list[b,p] = iter).
Tabu_list [ i, j] takes the value of the first iteration since an inter-change movement has become a tabu and during the following repetition number as tabu_tenure the related movement becomes the tabu. Thus, in the following reviews of the algorithm, the inter-change of machine c in area r and machine d in area s becomes tabu if one of the unequal-nesses is provided.
(iter < tabu_list[c, r] + tabu_tenure) or (iter < tabu_list[d, s] + tabu_tenure).
Lets explain the tabu_list [ i, j] list management with an example; the present solution shows sequence x[k] , x[ 1 ] = 2, x[ 2 ] = 1, x[ 3 ] = 3, x [ 4] = 5, x [ 5] = 4 (that means x = {2, 1, 3, 5, 4}) and the adjacent solution received by inter-change of machines 2nd and 5th in the first iteration is accepted as the new present solution. In this situation these assignings below occur:
Tabu_list[2, 5] = 1, Tabu_list[ 4, 1 ] = 1 , x[1] = 4 , x[5] = 2 .
In the second repetition, if the adjacent solution obtained from the inter-change of machines 2 and 5 in the present solution provides one of the two inequations "(2 < tabu_list[2, 5] + tabu_tenure ) or
(2 < tabu_list[5, 4] + tabu_tenure)", it's a tabu. Here, because tabu_list[2, 5] = 1 this adjacent solution will be a tabu when the tabu_tenure is 1 or a value more than 1. In the FTS algorithm which value will be taken by the tabu_tenure is determined according to a fuzzy membership function. This subject will be emphasized in the next part.
4.3 Fuzzy-Tabu tenure
The studies made on TSA up to today show that the tabu_tenure has a rather efficacy on the performance of the algorithm. Choosing the tabu_tenure short leads to focussing of the rearch to certain areas in the solution space (that means strengthening it locally) while choosing it longer leads to direction of the search towards different areas in the solution space (that means diversity in general). But, if the tabu_tenure is chosen shorter than it has to be, the same solutions will be reached at the end of a certain number of repetitions, thus cycles will be formed, on the contrary if it's chosen longer than normal, the quality of the solution will decrease because we will not be able to quit from bad areas in the solution space. In both conditions the solutions will be far away from the global optimum. Due to this fact, in the literature, we encounter the studies in which dynamic tabu tenures are used instead of constant tabu tenures. Beside this, strengthening and diversity strategies are considered together in the tabu search algorithm. Using both of these two strategies and the balance between them is really important in terms of algorithm's success.
In this study, a different approach is developed in order to determine the tabu_tenure. The basic logic of this approach is both to diverse and strengthen the direction of the search by choosing the tabu_tenure shorter for the opposite of the movements which are repeated seldomly during the search and by choosing the tabu_tenure longer for the opposite of the movements which are repeated frequently. But at this point the relative answers to these question "How often? or How much frequent?" show the necessity of using the fuzzy set theory. At the end of the trials, it's understood that the fuzzy membership function which forms a same-directioned linear relation between the assignment frequency of the machines to areas and tabu_tenure, is suitable for the related plant layout problem. Tabu_tenure_lower shows the lower limit of tabu_tenure, tabu_tenure_upper shows the upper limit of tabu_tenure and ^^ shows the frequency of the
assignment of machine i and J which is normalized between 0 and 1, the choosen fuzzy membership function is shown in equation 2. Graph of the fuzzy membership function is given in figure 1.
tabu_tenurei, = tabu_tenure_lower + (2)
iJ
+ (ui; - 1)(tabu_tenure_lower - tabu_tenure_upper).
iJ
The tabu_tenure of an inter-change movement which attains machine i to area J is calculated by equation 2. In order to determine Ui; membership value, a frequency [ i, J ]
iJ
(i = 1, ..., n , J = 1, ..., n ) list of the accomplished attainments that are done by the inter-change movement of each iteration is formed. This list, which works as a counter, is updated as below after a movement, making the new present solution in which inter-change between machine a on area p and machine b on area q occurs.
ß
Tabu tenure
Tabu tenure lower Tabutenureupper
Figure 1 - Fuzzy membership function for determination of tabu tenures
frequency[a, q] = frequency[a, q] + 1 , frequency[ b, p] = frequency[ b, p] + 1 .
Frequency[i,j] list is converted into u... membership
v
function values after normalizing it between 0 and 1. The formula given in equation 3 is used for this operation.
= frequency[i, j] - min (frequency[i, j]) (3) max (frequency[i,j]) - minfreq uency[i, j]).
4.4 Steps of FTS algorithm
FTS algorithm, which is an iterational algorithm, starts (runs) the search by considering the randomly chosen beginning solution as the present solution and determines its adjacent solutions in the region of the present solution according to the inter-change movement mechanism explained in part 4.1. Among these adjacent the one which has the best aim function value, that means the lowest material handling cost and which is not a tabu regarding the fuzzy tabu_tenure in part 4.3 according to the given tabu list management in part 4.2 is chosen as the new present solution. But, when tabu destroying criterion is taken into account, if the value of the aim function of the adjacent solution, which is obtained by an in-change movement classified as tabu, is better than the value of the best aim function found during the search, this adjacent solution is accepted as the new present solution even if it is a tabu. The tabu_list[i,j] and frequency[i,j] (i = 1, ..., n , j = 1 ,..., n ) lists are updated after choosing the new present solutions. All of these procedures are repeated till the maximum iteration number, which is determined before, is reached. The steps of FTS algorithm showing the f aim function is given below.
Step 1. Choose an x initial solution randomly and assign it as the present solution:
xpres = x .
The best solution xt
= x„
the best "pres ' Give the initial value to the iteration counter: iter = 1.
Start the search with empty tabu and frequency lists.
Step 2. Find the adjacent of the Xpres solution according to inter-change movement mechanism.
xkkm , k = 1, ..., n(n - 1)/2.
Step 3. For k = 1, ..., n(n - 1)/2 , calculate f(x) values. Assume the adjacent solution which has the best f( xajj) value and is not a tabu or provides f(xVdjj) <f(xthe best) criteria is x<as}j .
Step 4. Update the present solution: xpres = x^) .
If f(xpres) < f(xthe besti) , xthe best xpres .
Step 5. Update the tabu and frequency lists.
Convert the frequency list into membership function by normalizing it.
Step 6. iter = iter + 1 .
If iter > max_iter , stop, if not go back to step 2.
5. TRIAL STUDY
In order to show the efficacy of the developed FTS algorithm, this algorithm is compared with Classical Tabu Search (CTS) and Random Search (RS) algorithms, which use constant tabu_tenure. The only difference between CTS and FTS algorithms is that CTS algorithm considers a constant tabu_tenure after pre-trials. CTS algorithm considers the same principles completely with algorithm. The only difference between them is that tabu tenures are determined according to fuzzy membership function in FTS algorithm while a constant tabu_tenure is used is CTS algorithm. The basic aim in comparison of FTS and CTS algorithms is to determine whether using the fuzzy tabu_tenure increases the efficacy of fuzzy search method on QAP or not. On the other hand in RS algorithm, which is the simplest heuristic search method, a certain sampling is done from the solution space randomly and the solution with the lowest aim function value among these solutions is accepted as the result of the related trial. The sampling size, which is the only parameter of RS algorithm, is accepted equal to the number of solutions searched by FTS and CTS algorithms in solution space in order to provide the comparison on the same base.
In order to compare the performances of the algorithm, their small and large sized QAPs we produced randomly. For small problems with 5, 6, 7, 8 and 9 sizes,the optimum solutions are found by the enumeration method. Inter-plant distance and flow matrices for some of these small-scaled problems are given in appendix 1. Optimum solutions for 10, 20 and 30 sized problems are not known. Also, in each size there are 5 different QAP (different distance and work flow matrices). According to this, FTS, CTS, RS algorithms are compared on 8 X 5 = 40 QAPs. But, before the comparisons for the purpose of determining the parameter values of CTS
1
0
and FTS algorithms, the situation, in which the algorithms have the highest success for the chosen different parameter values, is searched. The pre-trials made show that,CTS and FTS algorithms have a higher success when their parameters depend on problem size (n ). In CTS algorithm tabu_tenure is used as the only parameter. In FTS algorithm the parameters are the upper and lower limits of the tabu_tenure function which is given in equation (2) (tabu_tenure_lower, tabu_tenure_upper). The average successes of CTS and FTS algorithms in different parameter levels determined according to the problem size are given in table 2 and table 3 sequentially. For the mentioned success the obtained average aim value is considered as criterion.
Table 2 - Average success of CTS algorithm in different parameter levels
N Tabu_ tenure
n/3 n/2 n 2n
5 15602,6 15602,6 15557,0 15557,0
6 16149,8 16149,8 15978,0 15978,0
7 25070,8 25021,0 24671,2 24613,0
8 29222,0 29158,4 28518,0 28518,0
9 43107,8 42969,0 42833,8 42695,0
10 43719,2 43719,2 43719,2 43470,0
20 702533,0 698096,8 692972,0 694188,2
30 480507,8 480785,2 479428,6 480155,6
In table 2, it is seen that the best result is obtained for 510 sized problems when the tabu_tenure used in CTS algorithm is taken as two times of problem size. The success of CTS algorithm is higher when tabu_tenure is considered as equal to problem size for 20 and 30 sized problems.
Table 3 - Average success of FTS algorithm in different parameter levels
It's understood that in 5-10 sized problems tabu_tenure_lower and tabu_tenure_upper parameters can be chosen as n/2 and 2n or n and 2n from table 3which is formed in order to determine the parameters of FTS algorithm. In our study n/2 and 2n values are chosen. On the other hand, for 20 sized problems the best success is obtained when tabu_tenure_lower and tabu_tenure_upper parameters are chosen as n/2 and 2n , for 30 sized problems the best success is obtained when they're chosen as n/2 and n .
After determining the parameter values, algorithms are tried on 40 different QAP for 5 times for each of them. Though FTS and CTS algorithms have deterministic structures, trials are made by choosing the starting solutions randomly. For problems with 5, 6, 7, 8 and 9 facilities the stopping condition of FTS and CTS algorithms are considered as finding the best solution or the highest repetition number that is 1.000. The highest repetition number is chosen 1.500 for size 10, 2.500 for 20 and 3.500 for 30. The stopping condition for RS algorithm is considered as the condition when the sample size is equal to the number of solutions searched by FTS and CTS in the related problems.
Average results of the problems with the best results obtained by the algorithms are given in table 4 while the data in table 4 are more generalized according to problem sizes by taking their average in table 5. Again in table 6 the best layout plans for small size problems are given. The average results of large size problems with an unknown best solution are seen in table 7. Here are the abbreviations used in these tables: SSS is the size of solution space, OMHC is the optimum material handling cost, OPDC is the optimum percentage of deviation from cost, SPSS is the searched percentage of solution space and FMHC is the found material handling cost.
It's seen in table 4 that FTS algorithm can find the best solution in all trials done for small sized problems by searching only a small part of the solution space. CTS algorithm again reaches the best solution in all trials done except the problems 7/5 and 8/4. But the solution space search percentage of CTS algorithm reaches to higher values than that of FTS algorithm as the size of the problem gets larger. On the other hand RS algorithm shows a worser performance than FTS and CTS algorithms do in terms of not only the deviation from the best solution but also the solution space search percentage. Although RS algorithm found the best solution in all trials for problems with 5 size, it searched nearly the whole solution space. Table 5, which is formed according to the averages of the data on basic of size, shows the higher success of FTS algorithms than that of CTS and RS algorithms in all small problem sizes.
N tabu_tenure_lower - tabu_tenure_upper
n / 3 — n / 2 n/3 - n n/2 - n n/2 - n 2 n - n 2
5 15602,6 15557,0 15557,0 15557,0 15557,0
6 16107,2 15978,0 15978,0 15978,0 15978,0
7 24826,4 24807,6 24807,6 24613,0 24613,0
8 29158,4 28654,8 28881,2 28518,0 28518,0
9 42969,0 42833,8 42833,8 42695,0 42695,0
10 43719,2 43533,8 43702,4 43470,0 43470,0
20 697941,2 693958,2 694002,8 692881,6 694495,8
30 479572,2 479690,6 479557,2 480604,0 481149,6
Table 4 - Average success of the algorithms in all small sized test problems
QAP FTS CTS RS
Size/No SSS OMHC OPDC% SPSS% OPDC% SPSS% OPDC% SPSS%
5/1 120 317 0 11,83 0 9,5 0 100
5/2 120 5708 0 2,33 0 2,33 0 95,50
5/3 120 15557 0 3,83 0 3,83 0 100
5/4 120 9828 0 5,67 0 5,67 0 100
5/5 120 12198 0 4,67 0 4,67 0 89,01
6/1 720 18444 0 2,06 0 1,42 0,62 100
6/2 720 15978 0 1,61 0 1,50 0,27 88,64
6/3 720 19292 0 1,56 0 1,56 0,03 70,40
6/4 720 7953 0 3,64 0 6,19 1,36 94,58
6/5 720 17930 0 2,36 0 3,14 0,16 100
7/1 5040 14597 0 0,33 0 0,87 1,03 15,17
7/2 5040 26875 0 0,42 0 0,79 0,34 18,12
7/3 5040 24613 0 0,52 0 0,44 1,37 16,62
7/4 5040 24896 0 0,47 0 0,54 1,20 15,30
7/5 5040 21652 0 0,54 0,31 4,59 2,25 18,21
8/1 40320 34684 0 0,09 0 0,19 2,39 2,48
8/2 40320 32000 0 0,19 0 0,42 3,59 2,48
8/3 40320 28518 0 0,40 0 0,14 4,93 2,48
8/4 40320 31870 0 0,09 0,09 0,59 3,44 2,48
8/5 40320 25698 0 0,07 0 0,16 8,06 2,48
9/1 362880 30076 0 0,04 0 0,05 6,08 0,28
9/2 362880 42787 0 0,02 0 0,02 4,22 0,28
9/3 362880 34875 0 0,005 0 0,009 6,19 0,28
9/4 362880 36620 0 0,007 0 0,010 4,27 0,28
9/5 362880 42695 0 0,008 0 0,010 3,65 0,28
Table 5 - Average success of the algorithms according to small problem sizes
Size FTS CTS RS
Ave.OPDC Ave.SPSS Ave.OPDC Ave.SPSS Ave.OPDC Ave.SPSS
5 0 5,667 0 5,200 0 96,902
6 0 2,246 0 2,762 0,488 90,724
7 0 0,456 0,062 1,446 1,238 16,684
8 0 0,168 0,018 0,300 4,482 2,480
9 0 0,016 0 0,020 4,882 0,280
Average 0 1,711 0,016 1,946 2,218 41,414
Table 6 - The best layout plans for small sized problems
Таблица 7 - Average material handling costs found by the algorithms in big sized test problems
Size/No The best layout plan Size/No The best layout plan
5/1 5 1 3 2 4 7/4 2 1 5 7 6 3 4
5/2 1 5 2 4 3 7/5 7 6 2 4 1 3 5
5/3 4 3 2 5 1 8/1 2 5 7 4 3 1 8 6
5/4 3 5 1 4 2 8/2 7 8 3 6 5 1 4 2
5/5 3 1 4 5 2 8/3 5 1 8 6 4 3 7 2
6/1 3 4 1 2 5 6 8/4 7 6 5 1 2 4 3 8
6/2 2 1 3 4 6 5 8/5 6 5 2 1 4 7 3 8
6/3 4 1 3 2 5 6 9/1 1 3 6 9 4 8 5 7 2
6/4 3 4 2 1 5 6 9/2 5 3 9 8 7 1 4 6 2
6/5 6 1 2 3 4 5 9/3 9 1 4 7 5 3 6 2 8
7/1 7 6 5 4 2 1 3 9/4 8 2 1 6 4 7 3 9 5
7/2 1 7 4 5 3 6 2 9/5 7 4 2 5 3 8 6 1 9
7/3 5 3 1 4 2 6 7
In table 7 It's seen that RS algorithm has a rather worser success rate than FTS and CTS algorithms in all problem sizes in the comparisons made on big sized problems. Success rates of FTS and CTS algorithms are same in problems with size 10 except the problem 5. FTS algorithm is more efficient than CTS algorithm for problem 5. In the table 7, it's again seen that the usage of the fuzzy tabu tenures gives more successful results for problems with size 20. Similarly, FTS algorithm finds better results than CTS algorithm does in problems with size 30 except problem 5. The average cost of the solutions found by CTS algorithm in problem 5 with size 30 is much more lower. For big sized problems, RS algorithm
Size/No FTS CTS RS
10/1 45157,0 45157,0 47621,8
10/2 43470,0 43470,0 46094,4
10/3 44282,0 44282,0 47735,8
10/4 42252,0 42252,0 45416,0
10/5 54287,0 54347,8 57124,0
20/1 692881,6 692972,0 1050269,6
20/2 210781,0 211294,8 226100,4
20/3 211765,6 212208,4 228372,0
20/4 202466,2 202912,0 219157,8
20/5 204522,0 205455,0 221068,2
30/1 470554,4 471890,0 507328,4
30/2 482664,6 482770,6 519662,4
30/3 442195,6 443678,8 481238,0
30/4 356066,2 356505,6 390911,2
30/5 479557,2 479428,6 519897,0
has rather worser success but FTS algorithm can reach much more better results than CTS algorithm can do in general.
In figure 2 the convergence velocities of FTS, CTS, RS algorithms on a layout problem with 20 facilities are given. As it is seen RS algorithm converges to a much more worser solution than FTS and CTS algorithms do. FTS algorithm reaches much more better solutions than CTS algorithm does even if it finds more costing solution than CTS does at the beginning of the search.
Figure 2 - Convergences of FTS, CTS and RS algorithms for a problem with 20 facilities
Cost 53000 52000 51000 50000 49000 48000 47000 46000 45000 44000
]
11 1 t
t ♦ t A
J IUI*a* A t (It < > t«t . fl "
h II Hi MirilintttUITITni'illllllllljfilir.'ll M "'mini IIHi ÄKW iiti.iii[iiiiiii'v;jin iiitrviiii;
4-"-prp-VT»*—
i i i i i i
FTS CTS
0 2000 4000 6000 8000 10000 12000 14000 Iteration
Figure 3 - Search behaviours of FTS and CTS algorithms
In figure 3 search behaviors of FTS and CTS algorithms on a layout problem with 10 facilities are seen. As understood from the figure the search in CTS algorithm intensifies at certain regions while it is directed towards different regions in FTS algorithm and eventually it reaches to better solutions than CTS algorithm does. Thus the efficacy of the fuzzy tabu tenures are shown as the only difference between these two algorithms is the usage of the fuzzy tabu_tenure.
6. SUMMARY
Because facility layout problem has a combinatorial structure and it is NP-complete, researchers are directed towards using heuristic approaches for facility layout. The first approaches applied to facility layout problem, which is modelled as QAP generally, have former or improver structures. These heuristics or their combinations can find a best but local solution for the problem. Due to this fact, intelligent heuristic techniques such as genetic algorithms, simulated annealing, tabu search and artificial neural networks are applied to QAPs and really good results are obtained.
In this study a tabu search algorithm which uses fuzzy tabu tenures is developed and it's tried on various sized QAPs. The developed algorithm is different from the tabu search approaches in the literature in terms of the usage of fuzzy tabu tenures. The developed tabu search algorithm is compared with classical tabu search algorithm, which uses static tabu_tenure and random search algorithm. Thus it is aimed to show whether using the fuzzy tabu_tenure will give better results or not. The obtained results show that fuzzy tabu search algorithms is more dominant than other algorithms in terms of the quality of the solutions found and the number of the points searched in solution space.
In the continuation of this study we aim to investigate whether tabu search algorithm using fuzzy tabu tenures is also effective on some other NP-hard problems or not. Also,
whether different membership functions for tabu tenures will increase the success of the algorithm or not will be investigated.
APPENDIX 1
Material handling cost matrices for some small sized problems.
Problem 5/1: Distance and flow matrix.
1 2 3 4 5 1 2 3 4 5
1 - 2 6 10 7 1 - 0 3 5 10
2 - 8 10 9 2 - 11 0 6
3 - 3 4 3 - 8 9
4 - 12 4 - 7
5 - 5 -
Problem 6/2: Distance and flow matrix.
1 2 3 4 5 6 1 2 3 4 5 6
1 - 97 22 92 11 57 1 - 16 43 42 9 37
2 - 56 12 70 60 2 - 33 11 25 45
3 - 52 13 15 3 - 16 37 35
4 - 23 24 4 - 35 36
5 - 76 5 - 16
6 - 6 -
Problem 7/3: Distance and flow matrix.
12 3 4 5 6 7 1 2 3 4 5 6 7
1 - 70 85 74 22 15 18 1 - 45 18 16 16 19 2
2 - 51 55 76 86 36 2 - 32 24 36 2 33
3 - 70 39 40 84 3 - 32 12 16 36
4 - 45 77 11 4 - 17 22 44
5 - 84 54 5 - 31 32
6 - 57 6 - 45
7 - 7 -
Problem 8/4: Distance and flow matrix.
[9]
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
1 - 83 13 66 71 48 28 46 1 - 9 6 44 34 21 18 40
2 - 91 72 94 62 58 81 2 - 12 30 23 25 2 10
3 - 31 44 53 58 64 3 - 10 17 16 32 34
4 - 93 17 64 44 4 - 28 13 18 43
5 - 28 67 97 5 - 12 40 28
6 - 92 53 6 - 17 29
7 - 26 7 - 23
8 - 8 -
Problem 9/5: Distance and flow matrix.
123456789
1 - 87 22 24 89 25 61 17 81
2 - 21 91 21 62 75 69 71
3 - 73 73 59 83 61 52
4 - 69 53 22 21 69
5 - 69 87 68 98
6 - 59 21 29
7 - 25 23
8 - 90
9 -
1 2 3 4 5 6 7 8 9
1 - 16 25 17 32 31 37 29 2
2 -3 43 33 23 36 20 31
3 - 44 23 30 12 29 11
4 -67 9 22 18
5 - 20 40 30 24
6 - 59 25 17
7 - 45 31
8 - 44
9 -
[1]
[2]
[3]
[4]
[5]
[6]
[7]
REFERENCES
Abninnour-Helm, S. & Hadley, S.W. (2000). Tabu search based heuristics for multi-floor facility layout. International Journal of Production Research, 38(2), 365-383. Al-Hakim, L. (2000). On solving facility layout problems using genetic algorithms. International Journal of Production Research, 38(11), 2573-2582.
Alvarenga, A.G., Negreiros-Gomes, F. & Mestria, M. (2000). Metaheuristic methods for a class of the facility layout problems. Journal of Intelligent Manufacturing, 11, 421-430. Badiru, A.B. & Arif, A. (1996). FLEXPERT: Facility layout expert system using fuzzy linguistic relationship codes. 11E Transactions, 28(4), 295-308.
Bazaraa, M.S. & Elshafei, A.N. (1979). An exact branch and bound procedure for quadratic assignment problems. Naval Research Logistic Quarterly, 26, 109-121. Bazaraa, M.S. & Sherali, A.N. (1980). Bender's partitioning scheme applied to a new formulation of the quadratic assignment problem. Naval Research Logistic Quarterly, 27(1), 2941.
BazarganLari, M. & Kaebernick, H. (1997). An approach to the machine layout problem in a cellular manufacturing environment. Production Planning & Control, 8(1), 41-55. Burkard, R.E. & Stratmann, K.H. (1978). Numerical investigation on quadratic assignment problems. Naval Research Logistic Quarterly, 25, 129-148.
[10 [11 [12 [13
[14
[15
[16
[17
[18
[19
[20 [21 [22
[23 [24 [25 [26
[27
[28
[29 [30
[31
[32
[33 [34
[35
[36 [37 [38
[39
[40
Burkard, R.E. (1990). Locations with spatial interactions: the quadratic assignment problem. Discrete Location Theory, (Editors: Mirchandani, P.B., Francis, R.L.), Wiley, Berlin. Cerny, V., 1985, Thermodynamical approach to traveling salesman problem: an efficient simulation algorithm, Journal of Optimization Theory and Applications, 45(1), 41-51. Chiang, W.C. & Chiang, C. (1998). Intelligent local search strategies for solving facility layout problems. European Journal of Operational Research, 106(2-3), 457-488. Chiang, W.C. & Kouvelis, P. (1996). An improved tabu search heuristic for solving facility layout design problems. International Journal of Production Research, 34(9), 2565-2585. Chittratanawat, S., & Noble, J.S. (1999). An integrated approach for facility layout, P/D location and material handling system design. International Journal of Production Research, 37(3), 683-706.
Chung, Y.K. (1999). A neuro-based expert system for facility layout construction. Journal of Intelligent Manufacturing, 10(5), 359-385.
Chwif, L., Barretto, M.R.P. & Moscato, L.A. (1998). A solution to the facility layout problem using simulated annealing. Computers in Industry, 36(1-2), 125-132.
De Jong, K.A. (1975). An analysis of the behaviour of a class of genetic adaptive systems. PhD Thesis, University of Michigan.
Foulds, L.R. (1983). Techniques for facilities layout: deciding which pairs of activities should be adjacent. Management Science, 29(12), 1414-1426.
Gau, K.Y. & Meller, R.D: (1999). An iterative facility layout algorithm. International Journal of Production Research, 37(16), 3739-3758.
Gilmore, P.C. (1963). Optimal and sub-optimal algorithms for the quadratic assignment problem. SIAM Journal, 10(2), 205313.
Glover, F., 1989, Tabu search - part I, ORSA Journal on Computing, 1(3), 190-206.
Glover, F., 1990, Tabu search - part II, ORSA Journal on Computing, 2(1), 4-32.
Glover, F., Greenberg, H.J., 1989, New approaches for heuristic search: a bilateral linage with artificial intelligence, European Journal of Operational Research, 39, 119-130. Glover, F., Laguna, M., 1993, Modern Heuristic Techniques For Combinatorial Problems, Blackwell Syntefic Publications. Glover, F., Laguna, M., 1997, Tabu Search, Kluwer Academic Publishers, Boston.
Goldberg, D.E. (1989). Genetic algorithms in search, optimization, and machine learning. Addison-Wesley. Hamamoto, S., Yih, Y. & Salvendy, G. (1999). Development and validation of genetic algorithm-based facility layout-a case study in the pharmaceutical industry. International Journal of Production Research, 37(4), 749-768.
Hansen, P., 1986, The steepest ascent mildest descent heuristic for combinatorial programming, Congress on Numerical Methods in Combinatorial Optimization, Capri, Italy. Hassanein, A.A. & Cherlopalle, V. (1999). Fuzzy sets theory and range estimating. 1999 AACE International Transactions, 4, 1-9.
Holland, J.H. (1975). Adaptation in natural and artificial systems. University of Michigan Press.
Jajodia, S., Minis, I., Harhalakis, G. & Proth, J.M. (1992). CLASS: Computerized layout solutions using simulated annealing. International Journal of Production Research, 30(1), 95-108.
Kazurhiro, T., Sunil, B. & Yoshiyasu, T. (1996). A neural network approach to facility layout problems. European Journal of Operational Research, 83(3), 556-570.
Kim, J.G. & Kim Y.D. (1998). A space partitioning method for facility layout problems with shape constraints. IIE Transactions, 30(10), 947-957.
Kirkpatrick, S., Gelatt Jr., C.D., Vecchi, M.P., 1983, Optimization by simulated annealing, Science, 220(4598), 671-680. Kochhar, J.S. & Heragu, S.S. (1999). Facility layout design in a changing environment. International Journal of Production Research, 37(11), 2429-2446.
Koopmans, T.C. & Beckman, M.J. (1957). Assignment problems and the location of economic activities. Econometrica, 25(1), 53-76.
Kusiak, A. (1990). Intelligent manufacturing systems. Prentice-Hall, Englewood Cliffs, NJ.
Lawler, E.L. (1963). The quadratic assignment problem. Management Science, 9(4), 586-599.
Liao, T.W. (1994). Design of line-type cellular manufacturing systems for minimum operating and material-handling costs. International Journal of Production Research, 32(2), 387-397. Mak, K.L., Wong, Y.S. & Chan, F.T.S. (1998). A genetic algorithm for facility layout problems. Computer Integrated Manufacturing Systems, 11(1-2), 113-127.
McCulloch, W.S., & Pitts, W. (1943). A logical calculus of the ideas immanent in nervous activity.
[41] Meller, R.D. & Bozer, Y.A. (1996). A new simulated annealing algorithm for the facility layout problem. International Journal of Production Research, 34(6), 1675-1692.
[42] Raoot, A.D. & Rakshit, A. (1994). A fuzzy heuristic for the quadratic assignment formulation to facility layout problem. International Journal of Production Research, 32(3), 563-581.
[43] Resende, M.G.C., Ramakrishnan, K.G. & Drezner, Z. (1995). Computing lower bound for the quadratic assignment problem with an interior point algorithm for linear programming. Operations Research, 43(5), 781-791.
[44] Rosenblatt, F. (1962). Principles of neurodynamics. Washington, DC: Spartan Books.
[45] Rossin, D.F., Springer, M.C. & Klein, B.D. (1999). New complexity measures for the facility layout problem: an emprical study using traditional and neural network analysis. Computers & Industrial Engineering, 36(3), 585-602.
[46] Sule, D.R. (1988). Manufacturing facilities. PWS-KENT, MA.
[47] Suresh, G., Vinod, V.V. & Sahu, S. (1995). A genetic algorithm for facility layout. International Journal of Production Research, 33(12), 3411-3423.
[48] Tam, K.Y. & Chan, S.K. (1998). Solving facility layout problems with geometric constraints using parellel genetic algorithms: experimentation and findings. International Journal of Production Research, 36(12), 3253-3272.
[49] Wang, T.Y., Lin, H.C. & Wu, K.B. (1998). An improved simulated annealing for facility layout problems in cellular manufacturing systems. Computers & Industrial Engineering, 34(2), 309-319.
[50] Yip, P.P.C & Pao, Y.H. (1994). A guided evolutionary simulated annealing approach to the quadratic assignment problem. IEEE Transactions on System Man and Cybernetics, 24(9), 1383-1387.
[51] Zadeh, L.A. (1965). Fuzzy sets. Information and Control, 8(3), 338-353.
[52] Zhang, G.Q., Xue, J. & Lai, K.K. (2000). A genetic algorithm based heuristic for adjacent paper-real layout problem. International Journal of Production Research, 38(14), 3343-3356.
УДК 681. 513: 519. 7
КОДИРОВАНИЕ СИГНАЛОВ В ЗАДАЧЕ НЕЙРОЗМУЛЯЦИИ
Е.В.Бодянский, Н.Е.Кулишова
В статье рассматриваются вопросы создания адаптивных искусственных нейронных сетей, работающих в реальном времени. Для обработки параметров, значения которых лежат в различных диапазонах, предложены кодирующие и декодирующие нейроны. Приведена схема нейроэмуляции объекта с применением сети, включающей нейроны-кодеры и декодеры.
В статтг розглядаються питання створення адаптивних нейронних мереж, якг працюють в реальному часг. Для обробки параметргв, значення яких лежать в ргзних диапазонах, пропонуються кодуючг та декодуючг нейрони. Наведено схему нейроемуляцп об'екту гз вживанням мережг, що мгстить нейрони-кодери та декодери.
Questions of real-time adaptive neuron networks creation are researching in this paper. Parameters of real objects lies in different intervals. To process such values koding and decoding neurons are proposed. Sceme of object's neuroemulation using network with koding and decoding neurons is given.
ВВЕДЕНИЕ
Задача управления динамическими стохастическими объектами в условиях неопределенности сформировала одно из ключевых направлений в современной теории автоматического управления (ТАУ) - аппарат адаптивных систем, среди которых широкое распространение получили непрямые системы, или адаптивные системы управления с идентификатором в цепи обратной связи (АСИ) [1]. В этих системах параллельно объекту управления подключается настраиваемая модель, параметры которой уточняются в реальном времени с помощью адаптивного идентификатора, реализующего ту или иную рекуррентную процедуру оценивания. Полученные оценки параметров объекта используются далее для расчета управляющих воздействий. В качестве основного недостатка таких систем можно отметить то, что процедура их синтеза в зна-
чительной мере опирается на линеаризацию объекта управления, а в качестве настраиваемых моделей используются линейные по параметрам структуры.
Необходимость управления существенно нелинейными объектами в условиях не только параметрической, но и структурной неопределенности привела к созданию принципиально новых алгоритмов и методов управления, связанных с нейросетевыми технологиями, и сформировала новое направление в ТАУ, известное как нейроуправление [2-9]. В рамках этого подхода широкое распространение получила схема управления с нейроконтроллером и нейро-эмулятором, который фактически выполняет функцию адаптивного идентификатора [7, 10-12]. В процессе обучения нейроэмулятора, в нем формируется информация, используемая далее нейроконтроллером для формирования управляющих сигналов. Естественно, что такие системы, в значительной мере обладающие интеллектуальными свойствами, характеризуются большими функциональными возможностями, чем ставшие уже традиционными АСИ.
1. ЦЕНТРИРОВАНИЕ И НОРМИРОВАНИЕ
ПЕРЕМЕННЫХ В РЕАЛЬНОМ ВРЕМЕНИ
Некоторые проблемы, связанные с синтезом нейроэму-ляторов, определяются необходимостью предварительной обработки поступающих на них сигналов, несущих информацию о входных и выходных переменных объекта. Такая обработка предполагается и в некоторых схемах АСИ с целью повышения скорости сходимости процесса идентификации [1] и предусматривает центрирование и нормирование наблюдаемых переменных объекта.
При этом для объекта управления с т входами (к) , I = 1, 2,..., т и с р выходами у1 (к) , I = 1, 2,...,р
136
ISSN 1607-3274 "Радюелектрошка. 1нформатика. Управл1ння" № 2, 2001
