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Self-organizing feature maps for solving location–allocation problems with rectilinear distances
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Computers & Operations Research 31 (2004) 1017 – 1031
www.elsevier.com/locate/dsw
Self-organizing feature maps for solving location–allocation problems with rectilinear distances Kuang-Han Hsieha;∗ , Fang-Chih Tienb a
b
Department of Management Science, Chinese Military Academy, FengShan, Kaohsiung 803, Taiwan, ROC Department of Industrial Engineering and Management, National Taipei University of Technology, 1 Section 3, Chung-Hsiao E. Rd., Taipei, Taiwan, ROC
Abstract This study deals with solving uncapacitated location–allocation (LA) problems with rectilinear distances by using a method based on Kohonen self-organizing feature maps (SOFMs). By treating LA problems as clustering problems, this method has the advantage of extracting the structure of the input data by a self-organizing process based on adaptation rules. In this paper, a heuristic method is constructed by using SOFMs with a guided re3ning procedure, and its performance is compared with simulated annealing. The experimental results using the proposed guided re3ning procedure to reinforce the SOFM method show that the proposed method is excellent in terms of quality of solution and speed of computation. In addition, the experimental results suggest that SOFMs may provide an excellent approach when generating initial solutions for other heuristic or exact algorithms. This conjecture is made because most of the solutions yielded by SOFM are close to the optimal solution in all experiments. Scope and purpose Given the location of a set of customers with di7erent demands, the LA problem is to select the locations of a number of supply centers to serve the customers and to decide the corresponding allocation of the customers to supply centers under a given optimization criterion. The LA problem may arise in 3re stations, hospitals, police stations, telecommunication networks, and warehouses relative to production facilities and customers. In addition, the recent rapid growth of demand for supply chain management, which can decrease the total cost, while improving the quality of goods and services of various organizations, has drawn signi3cant research attention. As an important part of supply chain management, the LA problem is related to the facility management service, logistics and distribution, order entry and customer service operations. Therefore, it should be indicated much more than before for organizations to improve their competitive advantage. In solving the LA problem, the costs of transportation between customers and supply centers are proportional to ∗
Corresponding author. Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan, ROC. Tel.: +886-223627688; fax: +886-223627688. E-mail addresses: [email protected], [email protected] (K.-H. Hsieh), [email protected] (F.-C. Tien). 0305-0548/$ - see front matter ? 2003 Published by Elsevier Ltd. doi:10.1016/S0305-0548(03)00049-2
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an appropriately determined distance, e.g. the Euclidean distance or rectilinear distance. The location of supply centers and the allocation of customers to supply centers must be considered simultaneously. The capacities of supply centers in LA problems can either be 3xed or treated without any limitation if the capacity of each supply center can be adjusted according to the amount of allocated service demands of customers, which are referred to as capacitated and uncapacitated problems, respectively. In addition, the assumption of no interaction between supply centers makes LA problems di7erent from p-hub problems. Currently, existing algorithms for solving the LA problems cannot meet the demands of rapidly changing industrial or market environments, since they are either time consuming or provide poor-quality solutions. Therefore, in this article, we propose a heuristic method using Kohonen’s feature maps, which are arti3cial neural networks capable of mapping the distribution of input data to speci3ed neurons, with a guided re3ning procedure to solve uncapacitated LA problems. In addition, the investigation of appropriateness of using Kohonen’s feature maps (called SOFMs) as an initial solution-generating method for another heuristic or exact method is also provided. ? 2003 Published by Elsevier Ltd. Keywords: Location–allocation problem; Combinatorial optimization problem; Neural networks; Self-organizing map
1. Introduction Given an optimization criterion and a set of customers (demand centers) with known demands, the location–allocation (LA) problem is to select the locations of a number of supply centers to serve customers, and to decide the corresponding allocation of the customers to supply centers. To focus on the LA problem discussed in this research, some characteristics are provided 3rst: (1) every customer shall be allocated to the closest supply center; (2) customers are serviced independently; (3) there is no interaction between supply centers; (4) installation costs are not considered; and (5) demand is concentrated in a set of discrete locations of customers. In addition, in order to tackle the problem arising in the context of a large number of urban location problems and facilities location problems within a factory [1,2], the rectilinear distance is adopted for distance measurement and the criterion is to minimize the total weighted distances between supply centers and customers. Moreover, the number of independent, uncapacitated supply centers is assumed to be known in advance. This kind of problem is usually called the minisum (LA) problem. With minor notational changes, the formulation reported by Liu [3] is used to illustrate this problem: Minimize
=
N M
Zij Di d(wj ; Xi )
j=1 i=1
Subject to
M
Zij = 1
(1) for i = 1; : : : ; N;
j=1
Zij ∈ {0; 1};
(2)
where is the total cost per unit time, Xi = (xi ; yi ); i = 1; : : : ; N , are N known points, representing customer locations, on a planar coordinate system, wj = (aj ; bj ); j = 1; : : : ; M , are M unknown points, representing locations of supply centers, and can be located anywhere on the planar coordinate
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system, Di ; i = 1; : : : ; N , are the frequency of service demands per unit time for N customers, 1 if customer i is assigned to supply center j Zij = for i = 1; : : : ; N; j = 1; : : : ; M; 0 otherwise and d(wj ; Xi ) = |xi − aj | + |yi − bj | is the rectilinear distance between location of supply center, wj , and customer location, Xi . According to Cooper [4], when M and N are large, LA problems are diLcult combinatorial optimization problems. He developed an exact solution procedure [5] and two heuristic methods [6] to solve these problems. The previously proposed methods [4,7,8] start with an initial set of suppliers, allocate each customer to the closest supply center, and then recalculate the locations of suppliers. By using new locations of suppliers, customers are re-allocated. This cycle is repeated until the allocation or cost in two successive iterations remains unchanged. This kind of approach is called alternate location–allocation (ALA). Recently, many researchers [9,10] have adopted the simple but powerful ALA computing procedure as a part of their algorithms. The methods developed so far for LA can be classi3ed into three categories: branch and bound algorithms (BBA) [11,12]; combinatorial optimization techniques [3,13]; and specially designed algorithms [2,9,14]. Kuenne and Soland [11] utilize BBA for the case of two supply centers. Marucheck and Aly [12] assume bivariate uniform density functions for demands and then use BBA to minimize weighted rectilinear distance. Regarding combinatorial optimization techniques, Liu et al. [3] develop a simulated annealing (SA) method to solve the LA problem with rectilinear distance measurement. Gamal and Salhi [13] use Tabu Search to solve LA problems. Gong et al. [15] and Houck et al. [16] adopt genetic algorithms for solving LA problems. Brimberg et al. [17] enhance some heuristics [16,18–20] to solve LA problems. In the 3elds of specially designed algorithms, Sherali and Shetty [2] develop a convergent cutting plane algorithm for solving both small- and large-sized problems. Love and Juel [14] propose 3ve methods based on the special properties of LA problems, two of which provide the optimal solutions for all 102 test problems. Adopting Euclidean and square Euclidean distance measurements, Lozano et al. [9] apply self-organizing feature maps (SOFMs) with an ALA re3ning procedure for obtaining the local optimal solution under the continuous demand condition that the demand is spread over a given continuous area according to a given probability. This work of Lozano et al. [9] is the only research applying SOFM to solve the LA problem which focuses on the validation of using SOFM for LA problem. Most of these approaches mentioned, exact solution and heuristic methods, for solving large LA problems (large M; N ) using rectilinear distances, are either time consuming or provide only local optimal solutions. The work of Lozano et al. [9] shows that the SOFM may be a potential heuristic method to meet the requirements of both quality of solution and speed of computation. However, adopting ALA as the re3ning procedure to subsequently solve LA problems in their work yields worse results compared with those obtained by other powerful heuristic methods. The disadvantage of adopting ALA in the work of Lozano et al. [9] will be explained below in the section of proposed methods. Therefore, a good re3ning procedure should be proposed to replace ALA procedure to improve the resulting SOFM results so that they can compete with other powerful methods. In this research, an SOFM arti3cial Neural Network with a new guided re3ning procedure (SOFMGR) is proposed for solving large LA problems. The SOFM method adopted in this research is the same as that reported by Lozano et al. [9], except that the distance measurement is changed to rectilinear and the continuous demand condition is replaced by discrete demand, which is concentrated on the point of customer location. The rest of this paper is organized as follows. The structure of the
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proposed problem-solving procedure is described 3rst. Then, the details of SOFMGR procedure for LA problems are introduced. Finally, the performance of SOFMGR is compared with other methods. 2. Proposed methods The purpose of this paper is to study the ability of SOFM for solving large LA problems. Therefore, we not only present a procedure, which can solve large LA problems quickly and provide good solutions, but also demonstrate the potential capability of SOFM for providing good initial solutions for other heuristic or exact algorithms. It is well known that LA problems can be treated as clustering problems. More precisely, the problem-solving process can be viewed as properly clustering N customers into M groups supplied by the associated M supply centers to achieve the minisum criterion. The SOFM methods are adopted in this research for the following advantages. First, SOFM methods can mimic distributed inputs by using an ordered array, and the array is native for clustering usage. As reported by Mangiameli et al. [21], SOFM methods are superior to hierarchical clustering methods. The other advantage of SOFM methods is that they are less time consuming (computing time proportional to M × N ). However, as mentioned by Lozano et al. [9], a 3ne-tuning procedure is necessary to re3ne the obtained local optimal solution. In their research, an ALA process is applied. However, even with the clustering advantage of SOFM, the ALA process is still incapable of yielding results meeting the solutions obtained by other powerful heuristic methods. Sometimes, although an SOFM method yields a good (initial) solution that is topologically very close to the optimal solution, the ALA procedure is still incapable of 3nding the optimal solution because there may be some local optimums on the ALA searching path due to the complexity of the searching space. On the contrary, the most powerful heuristic methods for solving LA problems or other complex combinatorial optimization problems have one common good property/function which can decrease or prevent the possibility of optimum search stocking in the local optimums. Therefore, we develop a new re3nement procedure, rather than using the scheme proposed by Lozano et al. to 3nd the local optimal solution. In the proposed re3ning process, the idea of decreasing the chance of stocking in local optimums is comprehended and a special property proposed by Wendell and Hurter [22] is applied in the guided search for an optimal solution. The proposed system constructed by SOFM process and the re3ning process can be described as input SOFM training process output input Re3ning process output W→ →W → →W ; (3) (Xi & Di mapping) (guided search) where W is the initial set of locations of supply centers, W = {wj ; ∀j = 1; : : : ; M }, W is the set of locations of supply centers after the SOFM process, and W is the set of locations of supply centers after the re3ning process. The following sections describe the proposed SOFM process and the re3ning process in detail. 2.1. Self-organizing feature maps SOFMs can map distributed input patterns into an ordered array of competing output units to extract the global structure of the input data. By this property, the locations of the N customer, Xi
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for i = 1; : : : ; N , are used as input patterns to form a training set, and M output units of SOFM will be the locations of supply centers, wj for j = 1; : : : ; M . In addition, the demands of N customers can be represented by the repeating frequency in a training set. During the training process of an SOFM, the location of supply centers, wj , is updated to reinforce the proximity between the distribution of the input locations/demands of the N customers and the con3guration of the weights. It is worth noting that the computing time of SOFM corresponding to di7erent sizes of M and N is proportional to M × N . The SOFM adopted in this research is the basic SOFM known as a Kohonen map [10]. For more general ideas of the construction and learning process of SOFM for solving LA problems (with continuous demand), please refer the work of Lozano et al. [9]. In the proposed SOFM method, some parts which di7er from the work of Lozano et al. [9] are brieNy mentioned below. To handle the demands of customers in SOFM methods, the following technique is proposed. A training set, in which each Xi ; i = 1; : : : ; N , shows exactly Di times in the training set to 3t the N demands criterion, so that the set including i=1 Di inputs is formed by randomly arranging the sequence of all Xi . In this way, the training set contains the probability distribution of customer demands and it will not cause any extra computing in the training stage. The other way mentioned by Lozano et al. [9] to handle the demands of customers is to present the inputs, i , to the network X N according to the discrete probability distribution constructed by p(Di ) = Di = k=1 Dk . However, a random generator and a large lookup table are needed for the implementation of probability distribution inferring process in the second method. Moreover, the necessary amount of inputs may be huge, leading to a time-consuming inferring process. It is clear that the proposed method is faster than the method of Lozano et al. [9]. Regarding the inNuence of using rectilinear distance measurement in SOFM methods, the best matching, wj , in competitive learning process should be selected based on the rectilinear distance measurement. No modi3cation of any other functions is necessary in the SOFM methods. 2.2. Re5ning process In the proposed re3nement process, a special property proposed by Wendell and Hurter [22] for LA problems with rectilinear distance measurement is used in the guided search. This property shows that, in location problems incorporating rectilinear distance, it is suLcient to consider the “intersection points’ in the convex hull as candidates for optimal locations. The “intersection points” are the intersections of all vertical and horizontal lines through the locations of customers. By this property, the problem can be simpli3ed so that there are N 2 candidate locations for M optimal locations of supply centers. The other important property that should be included is the decrease in the chance of optimum search stocking in the local optimal location. Including these two properties, the re3ning process is constructed as follows. Since the LA problem can be treated as a clustering problem, there are two ways to improve the yielded locations of supply centers for minisum criterion. One way is to allocate a customer into a di7erent cluster (group), and the other way is to move the location of a supply center to a new possible location. Both these ways may lead to a new allocation. Based on these two ideas, together with the properties mentioned above, a re3ning process is proposed as follows: 1. Use W SOFM obtained by SOFM to allocate the customers to M clusters.
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2. For each cluster of customers, compute the new supply center, W , and total cost (the computing sub-procedure of supply centers is shown in Appendix). 3. Assign to ∗ (∗ = ); W to W ∗ (W ∗ = W ) and initialize count (count = 0): 4. If the computing time does not exceed the pre-speci3ed time limit, do 4.1. Generate a random sequence of cluster indexes, S = {s(1); s(2); : : : ; s(M )}, where s(j) ∈ {1; 2; : : : ; M }, j ∈ {1; 2; : : : ; M } and s(j) = s(k) if j = k. 4.2. For each cluster, s(j); j = 1; : : : ; M , do 4.2.1. Regarding the location of supply center of current cluster, ws( j) , 3nd the nearest customer outside the current cluster. 4.2.2. Re-allocate the customer found to the current cluster. 4.2.3. Compute the location of supply centers of current cluster, ws( j) , and the location of supply center of the cluster to which the re-allocated customer originally belonged, wk ; k ∈ {S − {s(j)}}. 4.2.4. If count ¿ M , for two supply centers obtained in step 4.2.3, generate two random numbers, ri ; i = 1; 2; 0 6 ri 6 1 and do 4.2.4.1. If r1 ¿ 0:5, move ws( j) along the X -axis to the next right candidate location in cluster s(j). If r1 ¡ 0:5, move ws( j) to the next left candidate location. 4.2.4.2. If r2 ¿ 0:5, move wk along the Y -axis to the next upper candidate location in cluster k. If r2 ¡ 0:5, move wk to the next lower candidate location. (It should be noted that if there is no candidate location on the attempted moving direction in steps 4.2.4.1 and 4.2.4.2, the candidate location on the opposite direction is utilized.) 4.2.5. Re-allocate customers according to the new obtained locations of supply centers, W , and compute new total cost, . 4.2.6. If ¡ , do 4.2.6.1. Update W by W and by . 4.2.6.2. If ¡ ∗ , update W ∗ by W and ∗ by . 4.2.6.3. Let count = 0 and go to step 4. 4.2.7. If ¿ ; count = count + 1. 4.2.8. If count ¿ M 2 , update W by W and go to step 4. 5. Return W ∗ and ∗ . The process of re-allocating customers shown in steps 1 and 4.2.5 is conducted by assigning each customer to the supply center which has the minimal rectilinear distance to the customer. In the above procedure, steps 4.2.1– 4.2.3 conduct the re-allocation process of a customer. This process is repeated until a local optimum is found. However, this process is not capable of making a search jump out of the local optimum. Therefore, whenever a local optimum is identi3ed which is achieved by trying out all clusters using steps 4.2.1– 4.2.3 without any improvement in (count ¿ M ), the location of the supply center moving process is triggered (shown in step 4.2.4). By moving to an adjacent candidate location, the new locations of supply centers may lead the search out of a local optimum when the re-allocation of customers is conducted. If the moving processes of the supply center location still does not work (the ¡ condition cannot be satis3ed within M 2 attempts), then W is updated by steps 4.2.1– 4.2.3 and 4.2.4 without ¡ constraint.
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3. Performance of the proposed method The proposed method is implemented using MatLab version 5.3 with six existing test problems. In the six test problems, problem 1 (N = 16) and problem 2 (N = 30) were generated by Love and Morris [23]. Problem 3 (N = 20), problem 4 (N = 35), and problem 5 (N = 100) were generated by Love and Juel [14]. Problem 6 (N = 150) was generated by Liu and et al. [3]. All experiments were run on a Pentium-III 600-MHz-based IBM PC with 640 MB RAM. Since the objective of this research is to study the ability of SOFM for solving LA problems, the details of parameter studies are excluded from this paper. However, some general setting is provided, as follows. For the SOFM training stage, the feature map is set to a linear array, [1; : : : ; M ], and the parameters use the most frequently used setting in MatLab. More speci3cally, the default settings for the two-phase training process in MatLab are: ordering phase learning rate, 0.9; ordering phase steps (epochs), 1000; tuning phase learning rate, 0.02; topology function adopts ‘hextop’ and distance function uses ‘mandist’. For more detailed information, please refer to the MatLab Help documentation. The training results of SOFM for various sizes of M and epoch settings are shown in Tables 1–3. It should be noted that the computing time and quality of solutions could be improved by adjusting the combinations of various parameters. In Tables 2 and 3, the solutions of total cost obtained from SOFM show that the computing time is robust to the size of M . This excellent property makes the SOFM-based method especially Table 1 Average solutions and run time of SOFM averaged over 20 runs (1000 epochs)
Average solutions () Average run time (s)
Problem 1
Problem 2
Problem 3
Problem 4
2:1659 × 108 12.67
5:4875 × 108 18.57
1:5854 × 108 11.89
6:1823 × 108 20.63
Table 2 Average solutions and run time of SOFM for problem 5, averaged over 25 runs 1000 Epochs
2000 Epochs
3000 Epochs
Average solution ()
Average run time (s)
Average solution ()
Average run time (s)
Average solution ()
Average run time (s)
M =2 N = 100
1:6331 × 109
47.86
1:6171 × 109
51.54
1:6099 × 109
55.08
M =3 N = 100
1:2665 × 109
47.63
1:2615 × 109
51.27
1:2614 × 109
55.26
M =4 N = 100
1:1244 × 109
47.83
1:1052 × 109
51.49
1:1130 × 109
55.06
M =5 N = 100
1:0028 × 109
48.11
9:8070 × 108
51.98
9:6490 × 108
55.77
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Table 3 Average solutions and run time of SOFM for problem 6, averaged over 25 runs 1000 Epochs
2000 Epochs
3000 Epochs
Average solution ()
Average run time (s)
Average solution ()
Average run time (s)
Average solution ()
Average run time (s)
M =2 N = 150
2:6103 × 109
71.32
2:5718 × 109
74.87
2:5683 × 109
78.51
M =3 N = 150
2:1050 × 109
71.33
2:0848 × 109
75.01
2:0875 × 109
78.57
M =4 N = 150
1:8141 × 109
71.32
1:7947 × 109
75.08
1:7930 × 109
78.87
M =5 N = 150
1:6108 × 109
71.25
1:5588 × 109
75.08
1:5597 × 109
78.68
suitable for problems with large M . Another advantage is that the computing time is robust to the size of epoch speci3ed. These advantages make the solutions obtained from SOFM very suitable for use as initial solutions for other heuristics or exact algorithms. Some results shown in Table 3 are worth noting. When M = 3 and N = 150, or when M = 5 and N = 150, average solutions of the total cost of 3000 epochs are worse than 2000 epochs. This phenomenon is due to the movement of location of supply centers to the corresponding nearest customer locations in order to computing the cost. However, since the optimal (sub-optimal) solution locates on the nearby “intersections” but not on the nearest “intersection” of the locations of supply centers obtained by SOFM process, moving the location of supply centers to the corresponding nearest customer locations may not show exactly the best results obtained at di7erent training stages (epochs). In fact, the optimal/sub-optimal solution located on the nearby “intersections” of solutions yielded by SOFM is a key point of this research. If this point is true, it is reasonable that SOFM may be an excellent approach to generate initial solutions for supply center locations, not only for the proposed re3ning procedure, but also for the other heuristic or exact algorithms to solve LA problems. To validate this inference, the proposed guided re3ning procedure, which uses the results of SOFM as initial solutions, should not only be able to 3nd the optimal solutions with great probability, but also 3nd optimal solutions very fast for most testing problems if the initial solutions of locations of supply centers are close to the optimal solutions of locations of supply centers. In the following paragraphs, we will show that these requirements for validation are ful3lled. To evaluate the performance of the proposed SOFM method with guided re3ning procedure, the results obtained by the proposed method are compared to those yielded by the SA algorithm proposed by Liu et al. [3], which is known both as a fast heuristic for large problems and also as having a high probability of 3nding the optimal solution. The algorithm and parameter setting for the SA method used in our experiments are all the same as those reported by Liu et al. [3], except that the stop criterion is changed to a time limit rule (the annealing schedule remains unchanged).
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Table 4 Solutions and run times using SOFMGR and SA over 20 runs No. of Problem Stop Optimal test size times solution problem (s) ()
SOFMGR (1000 epochs in SOFM training)
1
M =3 N = 16
20
191 354 085
a
a
13.80
a
a
2.13
2
M =2 N = 30
40
516 254 969
a
a
19.61
a
a
9.04
3
M =3 N = 20
50
140 236 644
a
a
12.75
a
1:4249 × 108
6.05
4
M =2 N = 35
50
598 084 656
a
a
23.25
a
5:9846 × 108 13.79
a
SA
Best Average Average run Best Average solution () solution () time solution solution for getting () () best solution (s)
Average run time for getting best solution (s)
Same value as the optimal solution.
Table 5 Percentages of obtaining the best known solution and maximal run time (in s) to obtain the best known solution using SOFMGR and SA over 20 runs No. of Stop SOFMGR test time (1000 epochs problem (s) Percentage of obtaining optimal solution
Percentage of worst solution away from optimal solution
Maximal run time Percentage of for getting best obtaining solution optimal solution (all within)
Percentage of worst solution away from optimal solution
Maximal run time for getting best solution (all within)
1 2 3 4
0 0 0 0
16.75 22.07 13.94 28.36
0 0 5.63 0.37
4.02 12.77 8.44 17.31
20 40 50 50
100 100 100 100
in SOFM training)
SA
100 100 70 75
Of the six test problems, the 3rst four are small problems and their optimal solutions are known, whereas the remaining two problems are large (N ¿ 100 customers) and their optimal solutions are unknown. The data in Tables 4 and 5 compare the results from the proposed SOFMGR method and from the SA method for problems 1– 4. As shown in Tables 4 and 5, the best solutions for total cost of both methods in 20 runs, under randomly generated initial setting, are equal to the optimal solutions for every problem within the
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Table 6 Solutions and run times (in s) using SOFMGR and SA for problem 5 over 25 runs No. of Problem Stop SOFMGR SA test size time (2000 epochs in SOFM training) problem Best Average Average run time Best Average solution () solution for getting best solution () solution () solution
Average run time for getting best solution
5
M =2 500 N = 100
1 583 327 772
a
53.54
a
a
5
M =3 500 N = 100
1 234 118 203
a
64.67
a
1:2361 × 109 388.74
5
M =4 500 N = 100
1 060 697 798
a
87.94
a
1:0729 × 109 465.12
5
M =5 500 N = 100
919 515 658
9:1952 × 109 140.13
a
9:2698 × 109 477.12
a
263.90
Same value as the best solution found in both SOFMGR and SA methods.
speci3ed time limits. However, the SA method fails to achieve the optimal solutions in all runs for problems 3 and 4, though it uses less computing time than the SOFMGR method for these small problems. The average run times to obtain the best solutions in the proposed guided re3ning procedure only are 1.1, 1.0, 0.9 and 2:6 s. These times are obtained by subtracting the average run time in Table 1 from the average run time to obtain the best solution in Table 4, for problems 1– 4. These results shows that the initial solutions of the locations of supply centers yielded by SOFM are close to the optimal solutions so that the proposed re3ning procedure can 3nd the optimal solutions very quickly for all test runs for problems 1– 4. Since the optimal solutions for various M sizes are unknown for problem 5, the best solutions of total cost, de3ned as the best solutions obtained by both SOFMGR and SA methods, are used to compare SOFMGR and SA. As shown in Table 6, the best solutions achieved by both methods in 25 runs are equal for various sizes of M within the speci3ed time limits. However, the percentage of the best solutions achieved by the SA method decreases when M increases (refer Table 7). In addition, the SOFMGR method has shorter run times than the SA method. Moreover, the high percentage of achieving optimal solutions and the short average run time to obtain the best solution in guided re3ning procedure shows that the quality of initial solutions of locations obtained by SOFM is good. The average run times for getting the best solution using the guided re3ning procedure for problem 5 (2 s for M = 2 case, 13:4 s for M = 3 case, 16:5 s for M = 4 case, 88:2 s for M = 5 case) are obtained by subtracting the average run time in Table 2 from the average run time for getting the best solution in Table 8. For problem 6, the best solutions achieved by SOFMGR method are better than the best results yielded by the SA method for all four di7erent sizes of M (as shown in Tables 8 and 9). In addition,
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Table 7 Percentages of obtaining the best known solution and maximal run time (in s) to obtain the best known solution using SOFMGR and SA for problem 5 over 25 runs No. of test problem
Problem size
5
M =2 N = 100
5
Stop time
SOFMGR (2000 epochs in SOFM training)
SA
Percentage of obtaining best known solution
Percentage of worst solution away from best known solution
Percentage obtaining best known solution
Percentage of worst solution away from best known solution
Maximal run time for getting best solution (all within)
500
100
0
57.33
100
0
309.76
M =3 N = 100
500
100
0
95.65
12
0.20
429.16
5
M =4 N = 100
500
100
0
220.83
20
2.46
499.86
5
M =5 N = 100
500
96
0.02
389.29
4
4.22
499.60
Maximal run time for getting best solution (all within)
Note: Percentage of worst solution away from optimal solution is de3ned as (worst solution/optimal solution)−1.
Table 8 Solutions and run times (in s) using SOFMGR and SA for problem 6 over 25 runs No. of Problem Stop SOFMGR SA test size time (2000 epochs in SOFM training) problem Best Average Average run time Best solution () solution () for getting best solution () solution
Average solution ()
Average run time for getting best solution
6
M =2 500 N = 150
2 515 820 112
a
103.20
2:5200 × 109 2:5394 × 109 487.06
6
M =3 500 N = 150
1 998 514 195
a
78.25
2:1299 × 109 2:2261 × 109 415.64
6
M =4 500 N = 150
1 715 275 017 1:7159 × 109 127.38
1:9454 × 109 2:1029 × 109 429.01
6
M =5 500 N = 150
1 487 070 093 1:4871 × 109 161.87
1:9014 × 109 2:0018 × 109 420.68
a
Same value as the best solution found in both SOFMGR and SA methods.
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Table 9 Percentages of obtaining the best known solution and maximal run time (in s) to obtain the best known solution using SOFMGR and SA for problem 6 over 25 runs No. of Problem Stop SOFMGR test size time (s) (2000 epochs problem Percentage of obtaining best known solution
in SOFM training)
SA
Percentage of worst solution away from best known solution
Maximal run time for getting best solution (all within)
Percentage of obtaining best known solution
Percentage of worst solution away from best known solution
Maximal run time for getting best solution (all within)
5
M =2 500 N = 150
100
0
184.70
0
2.24
499.54
5
M =3 500 N = 150
100
0
85.97
0
25.87
499.26
5
M =4 500 N = 150
96
0.96
272.52
0
33.06
499.90
5
M =5 500 N = 150
96
0.002
289.30
0
46.23
499.89
Note: Percentage of worst solution away from optimal solution is de3ned as (worst solution/optimal solution)−1.
the SOFMGR method has a shorter run time than the SA method. Since the stop time setting (500 s) does not seem long enough for SA to achieve the stopping criterion speci3ed in Liu et al. [3], 3ve extra experimental runs (M = 5 and stop time setting of 1 h) were conducted for SA method for more investigation. However, the results yielded by these extra runs are still worse than the solutions obtained by the SOFMGR method at a seconds setting of 500. Two other aspects that should be carefully observed are the very high percentage of achieving optimal solutions and the short average run time to obtain the best solution in the guided re3ning procedure. When these two things occur simultaneously, it is reasonable to conclude that the initial location solutions for supply centers obtained by SOFM are close to the best solutions. The average run times for obtaining the best solution by the guided re3ning procedure for problem 6 are 28:3 s for M = 2 case, 3:2 s for M = 3 case, 52:3 s for M = 4 case, and 86:8 s for M = 5 case. These results are obtained by subtracting the average run time in Table 3 from the average run time for obtaining the best solutions in Table 8. From the above experimental results (Tables 5, 7 and 9), we observe that the percentage achieving optimal or the best known solutions is never less than 96% for the proposed method. In addition, the average run times for obtaining the best solutions for total cost in guided re3ning procedure are short. These results show that the SOFM method provides excellent initial solutions for supply center locations for subsequent guided re3ning process. In Fig. 1, the typical example of problem 1 demonstrates that the solution of SOFM can indeed provide a very good initial solution for supply center locations for the guided re3ning process. Since the guided re3ning process, which only applies simple guided random searches on the neighborhoods of solutions yielded by SOFM, not only has a
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Fig. 1. Typical results of SOFM and SOFMGR for problem 1.
very high probability of 3nding the optimal or best known solution but also 3nds the solution very quickly, we may infer that most of the solutions for supply center locations obtained by SOFM are close to the optimal solution. Therefore, other sophisticated heuristic or exact algorithms may also adopt SOFM as the initial solution-generating method to produce good starting locations of supply centers and improve the computing speed.
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4. Conclusions From the above experimental results, we have shown that the proposed SOFM with a guided re3ning (SOFMGR) procedure performs better in solving the minisum location–allocation problems with rectilinear distance. For large location–allocation problems, the proposed method outperforms the SA method proposed by Liu et al. [3], which is known not only as a fast heuristic for large problems but also as one with a high probability of 3nding optimal solutions, yielding high-quality solutions and with eLcient computation. In addition, some results shown in this research indicate that the SOFM may be used as an excellent procedure to 3nd initial solutions for supply center locations for other heuristic or exact algorithms. Regarding the experimental results, although the SOFM computation time is robust regarding the size of M, the time consumed by the re3ning process does increase as M increases. Therefore, further research may concentrate on developing a more sophisticated algorithm for re3ning the results yielded by the SOFM method. Appendix A. 1. Within each cluster, s(j); j = 1; : : : ; M , there are Ls( j) customers. For each cluster, s(j), do 1.1 Sort Xi in cluster, s(j), by X coordinates in ascending order. Then the sorted Xi with associated Di can be expressed as Xm and Dm ; m = 1; : : : ; L(j) . Find a Dc that satis3es c −1
Dm ¡
m=1
Ls( j) c 1 Di 6 Dm : 2 i=1 m=1
(A.1)
Then, the X coordinate of supply center (ws( j) ) is the X coordinate of Xc . 1.2 Sort Xi in cluster, s(j), by Y coordinates in ascending order. Then the sorted Xi with associated Di can be expressed as Xn and Dn ; n = 1; : : : ; Ls( j) . Find a Dc that satis3es c −1 n=1
Dn ¡
Ls( j) c 1 Di 6 Dn : 2 i=1 n=1
(A.2)
Then, the Y coordinate of supply center (ws( j) ) is the Y coordinate of Xc . 2. Return ws( j) Ls( j) If 12 i=1 Di = cm=1 Dm occurs in Eq. (A.1), the X coordinates of supply centers can be arbitrarily located between the X coordinates of Xc and Xc+1 . In this research, (Xc + Xc+1 )=2 is used. The same technique can be applied for similar situations which occur in computing the Y coordinates of supply centers. References [1] Francis RL, White JA. Facility layout and location—an analytical approach. Englewood Cli7s, NJ: Prentice-Hall, 1974. p. 212.
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[2] Sherali AD, Shetty CM. The rectilinear distance location–allocation problem. AIIE Transactions 1977;9(2):136–43. [3] Liu C-M, Kao R-L, Wang A-H. Solving location–allocation problems with rectilinear distances by simulated annealing. Journal of the Operational Research Society 1994;45:1304–15. [4] Cooper L. Location–allocation problems. Operations Research 1963;11:331–43. [5] Cooper L. The transportation–location problems. Operations Research 1972;20:94–108. [6] Cooper L. Heuristic methods for location–allocation problem. SIAM Review 1964;6:37–53. [7] Vergin RC, Rogers JD. An algorithm and computational procedure for locating economic facilities. Management Science 1967;13:240–54. [8] Eilon S, Watson-Gandy CDT, Christo3des N. Distribution management: mathematical modeling and practical analysis. New York: Hafner, 1971. [9] Lozano S, Guerrero F, Onieva L, Larraneta J. Kononen maps for solving a class of location–allocation problems. European Journal of Operational Research 1998;108:106–17. [10] Kohonen T. Self-organized formation of topologically correct feature maps. Biological Cybernetics 1982;43:59–69. [11] Kuenne RE, Soland RM. Exact and approximate solutions to the multisource weber problem. Mathematical Programming 1972;3:193–209. [12] Marucheck AS, Aly AA. An eLcient algorithm for the location–allocation problem with rectangular regions. Naval Research Logistics Quarterly 1981;28:309–23. [13] Gamal MDH, Salhi S. Constructive heuristics for the uncapacitated continuous location–allocation problem. Journal of the Operational Research Society 2001;52:821–9. [14] Love RF, Juel H. Properties and solution methods for large location–allocation problems. Journal of the Operational Research Society 1982;33:443–52. [15] Gong D, Gen M, Xu W, Yamazaki G. Hybrid evolutionary method for obstacle location–allocation. Computers Industrial Engineering 1995;29(1– 4):525–30. [16] Houck CR, Joines JA, Kay MG. Comparison of genetic algorithms, random restart and two-opt switching for solving large location–allocation problems. Computers Operations Research 1996;23:587–96. [17] Brimberg J, Hansen P, MladenoviXc N, Taillard ED. Improvements and comparison of heuristics for solving the uncapacitated multisource weber problem. Operations Research 2000;48:444–60. [18] Bongartz I, Calamai PH, Conn AR. A projection method for lp norm location–allocation problems. Mathematical Programming 1994;66:283–312. [19] Brimberg J, MladenoviXc N. Solving the continuous location–allocation problems with tabu search. Studies in Locational Analysis 1996;8:23–32. [20] Brimberg J, MladenoviXc N. Variable neighbourhood algorithm for solving the continuous location–allocation problem. Studies in Locational Analysis 1996;10:1–10. [21] Mangiameli P, Chen S-K, West D. A comparison of SOM neural network and hierarchical clustering methods. European Journal of Operational Research 1996;93:402–17. [22] Wendell RE, Hurter Jr. AP. Location theory, dominance, and convexity. Operations Research 1973;21:314–20. [23] Love RF, Morris JG. A computation procedure for the exact solution of location–allocation problems with rectangular distances. Naval Research Logistics Quarterly 1975;22:441–53. Kuang-Han Hsieh is an Assistant Professor in the Department of Management Science at Chinese Military Academy, Taiwan. He received the B.S. degree in Industrial Engineering from Chung Yuan Christian University, Taiwan, and the M.S. and Ph.D. degrees in industrial engineering from the University of Missouri-Columbia, USA. His research interests are in the areas of combinatorial optimization, arti3cial neural networks, genetic algorithms, and computer vision systems. Fang-Chih Tien is an Associated Professor in the Department of Industrial Engineering and Management at National Taipei University of Technology, Taiwan. He received the B.S. degree in Industrial Engineering from Chung Yuan Christian University, Taiwan, and the M.S. and Ph.D. degrees in industrial engineering from the University of Missouri-Columbia, USA. His research interests are in the areas of computer vision systems, combinatorial optimization, correspondence problems, genetic algorithms, and arti3cial neural networks.
Computers & Operations Research 31 (2004) 1017 – 1031
www.elsevier.com/locate/dsw
Self-organizing feature maps for solving location–allocation problems with rectilinear distances Kuang-Han Hsieha;∗ , Fang-Chih Tienb a
b
Department of Management Science, Chinese Military Academy, FengShan, Kaohsiung 803, Taiwan, ROC Department of Industrial Engineering and Management, National Taipei University of Technology, 1 Section 3, Chung-Hsiao E. Rd., Taipei, Taiwan, ROC
Abstract This study deals with solving uncapacitated location–allocation (LA) problems with rectilinear distances by using a method based on Kohonen self-organizing feature maps (SOFMs). By treating LA problems as clustering problems, this method has the advantage of extracting the structure of the input data by a self-organizing process based on adaptation rules. In this paper, a heuristic method is constructed by using SOFMs with a guided re3ning procedure, and its performance is compared with simulated annealing. The experimental results using the proposed guided re3ning procedure to reinforce the SOFM method show that the proposed method is excellent in terms of quality of solution and speed of computation. In addition, the experimental results suggest that SOFMs may provide an excellent approach when generating initial solutions for other heuristic or exact algorithms. This conjecture is made because most of the solutions yielded by SOFM are close to the optimal solution in all experiments. Scope and purpose Given the location of a set of customers with di7erent demands, the LA problem is to select the locations of a number of supply centers to serve the customers and to decide the corresponding allocation of the customers to supply centers under a given optimization criterion. The LA problem may arise in 3re stations, hospitals, police stations, telecommunication networks, and warehouses relative to production facilities and customers. In addition, the recent rapid growth of demand for supply chain management, which can decrease the total cost, while improving the quality of goods and services of various organizations, has drawn signi3cant research attention. As an important part of supply chain management, the LA problem is related to the facility management service, logistics and distribution, order entry and customer service operations. Therefore, it should be indicated much more than before for organizations to improve their competitive advantage. In solving the LA problem, the costs of transportation between customers and supply centers are proportional to ∗
Corresponding author. Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan, ROC. Tel.: +886-223627688; fax: +886-223627688. E-mail addresses: [email protected], [email protected] (K.-H. Hsieh), [email protected] (F.-C. Tien). 0305-0548/$ - see front matter ? 2003 Published by Elsevier Ltd. doi:10.1016/S0305-0548(03)00049-2
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an appropriately determined distance, e.g. the Euclidean distance or rectilinear distance. The location of supply centers and the allocation of customers to supply centers must be considered simultaneously. The capacities of supply centers in LA problems can either be 3xed or treated without any limitation if the capacity of each supply center can be adjusted according to the amount of allocated service demands of customers, which are referred to as capacitated and uncapacitated problems, respectively. In addition, the assumption of no interaction between supply centers makes LA problems di7erent from p-hub problems. Currently, existing algorithms for solving the LA problems cannot meet the demands of rapidly changing industrial or market environments, since they are either time consuming or provide poor-quality solutions. Therefore, in this article, we propose a heuristic method using Kohonen’s feature maps, which are arti3cial neural networks capable of mapping the distribution of input data to speci3ed neurons, with a guided re3ning procedure to solve uncapacitated LA problems. In addition, the investigation of appropriateness of using Kohonen’s feature maps (called SOFMs) as an initial solution-generating method for another heuristic or exact method is also provided. ? 2003 Published by Elsevier Ltd. Keywords: Location–allocation problem; Combinatorial optimization problem; Neural networks; Self-organizing map
1. Introduction Given an optimization criterion and a set of customers (demand centers) with known demands, the location–allocation (LA) problem is to select the locations of a number of supply centers to serve customers, and to decide the corresponding allocation of the customers to supply centers. To focus on the LA problem discussed in this research, some characteristics are provided 3rst: (1) every customer shall be allocated to the closest supply center; (2) customers are serviced independently; (3) there is no interaction between supply centers; (4) installation costs are not considered; and (5) demand is concentrated in a set of discrete locations of customers. In addition, in order to tackle the problem arising in the context of a large number of urban location problems and facilities location problems within a factory [1,2], the rectilinear distance is adopted for distance measurement and the criterion is to minimize the total weighted distances between supply centers and customers. Moreover, the number of independent, uncapacitated supply centers is assumed to be known in advance. This kind of problem is usually called the minisum (LA) problem. With minor notational changes, the formulation reported by Liu [3] is used to illustrate this problem: Minimize
=
N M
Zij Di d(wj ; Xi )
j=1 i=1
Subject to
M
Zij = 1
(1) for i = 1; : : : ; N;
j=1
Zij ∈ {0; 1};
(2)
where is the total cost per unit time, Xi = (xi ; yi ); i = 1; : : : ; N , are N known points, representing customer locations, on a planar coordinate system, wj = (aj ; bj ); j = 1; : : : ; M , are M unknown points, representing locations of supply centers, and can be located anywhere on the planar coordinate
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system, Di ; i = 1; : : : ; N , are the frequency of service demands per unit time for N customers, 1 if customer i is assigned to supply center j Zij = for i = 1; : : : ; N; j = 1; : : : ; M; 0 otherwise and d(wj ; Xi ) = |xi − aj | + |yi − bj | is the rectilinear distance between location of supply center, wj , and customer location, Xi . According to Cooper [4], when M and N are large, LA problems are diLcult combinatorial optimization problems. He developed an exact solution procedure [5] and two heuristic methods [6] to solve these problems. The previously proposed methods [4,7,8] start with an initial set of suppliers, allocate each customer to the closest supply center, and then recalculate the locations of suppliers. By using new locations of suppliers, customers are re-allocated. This cycle is repeated until the allocation or cost in two successive iterations remains unchanged. This kind of approach is called alternate location–allocation (ALA). Recently, many researchers [9,10] have adopted the simple but powerful ALA computing procedure as a part of their algorithms. The methods developed so far for LA can be classi3ed into three categories: branch and bound algorithms (BBA) [11,12]; combinatorial optimization techniques [3,13]; and specially designed algorithms [2,9,14]. Kuenne and Soland [11] utilize BBA for the case of two supply centers. Marucheck and Aly [12] assume bivariate uniform density functions for demands and then use BBA to minimize weighted rectilinear distance. Regarding combinatorial optimization techniques, Liu et al. [3] develop a simulated annealing (SA) method to solve the LA problem with rectilinear distance measurement. Gamal and Salhi [13] use Tabu Search to solve LA problems. Gong et al. [15] and Houck et al. [16] adopt genetic algorithms for solving LA problems. Brimberg et al. [17] enhance some heuristics [16,18–20] to solve LA problems. In the 3elds of specially designed algorithms, Sherali and Shetty [2] develop a convergent cutting plane algorithm for solving both small- and large-sized problems. Love and Juel [14] propose 3ve methods based on the special properties of LA problems, two of which provide the optimal solutions for all 102 test problems. Adopting Euclidean and square Euclidean distance measurements, Lozano et al. [9] apply self-organizing feature maps (SOFMs) with an ALA re3ning procedure for obtaining the local optimal solution under the continuous demand condition that the demand is spread over a given continuous area according to a given probability. This work of Lozano et al. [9] is the only research applying SOFM to solve the LA problem which focuses on the validation of using SOFM for LA problem. Most of these approaches mentioned, exact solution and heuristic methods, for solving large LA problems (large M; N ) using rectilinear distances, are either time consuming or provide only local optimal solutions. The work of Lozano et al. [9] shows that the SOFM may be a potential heuristic method to meet the requirements of both quality of solution and speed of computation. However, adopting ALA as the re3ning procedure to subsequently solve LA problems in their work yields worse results compared with those obtained by other powerful heuristic methods. The disadvantage of adopting ALA in the work of Lozano et al. [9] will be explained below in the section of proposed methods. Therefore, a good re3ning procedure should be proposed to replace ALA procedure to improve the resulting SOFM results so that they can compete with other powerful methods. In this research, an SOFM arti3cial Neural Network with a new guided re3ning procedure (SOFMGR) is proposed for solving large LA problems. The SOFM method adopted in this research is the same as that reported by Lozano et al. [9], except that the distance measurement is changed to rectilinear and the continuous demand condition is replaced by discrete demand, which is concentrated on the point of customer location. The rest of this paper is organized as follows. The structure of the
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proposed problem-solving procedure is described 3rst. Then, the details of SOFMGR procedure for LA problems are introduced. Finally, the performance of SOFMGR is compared with other methods. 2. Proposed methods The purpose of this paper is to study the ability of SOFM for solving large LA problems. Therefore, we not only present a procedure, which can solve large LA problems quickly and provide good solutions, but also demonstrate the potential capability of SOFM for providing good initial solutions for other heuristic or exact algorithms. It is well known that LA problems can be treated as clustering problems. More precisely, the problem-solving process can be viewed as properly clustering N customers into M groups supplied by the associated M supply centers to achieve the minisum criterion. The SOFM methods are adopted in this research for the following advantages. First, SOFM methods can mimic distributed inputs by using an ordered array, and the array is native for clustering usage. As reported by Mangiameli et al. [21], SOFM methods are superior to hierarchical clustering methods. The other advantage of SOFM methods is that they are less time consuming (computing time proportional to M × N ). However, as mentioned by Lozano et al. [9], a 3ne-tuning procedure is necessary to re3ne the obtained local optimal solution. In their research, an ALA process is applied. However, even with the clustering advantage of SOFM, the ALA process is still incapable of yielding results meeting the solutions obtained by other powerful heuristic methods. Sometimes, although an SOFM method yields a good (initial) solution that is topologically very close to the optimal solution, the ALA procedure is still incapable of 3nding the optimal solution because there may be some local optimums on the ALA searching path due to the complexity of the searching space. On the contrary, the most powerful heuristic methods for solving LA problems or other complex combinatorial optimization problems have one common good property/function which can decrease or prevent the possibility of optimum search stocking in the local optimums. Therefore, we develop a new re3nement procedure, rather than using the scheme proposed by Lozano et al. to 3nd the local optimal solution. In the proposed re3ning process, the idea of decreasing the chance of stocking in local optimums is comprehended and a special property proposed by Wendell and Hurter [22] is applied in the guided search for an optimal solution. The proposed system constructed by SOFM process and the re3ning process can be described as input SOFM training process output input Re3ning process output W→ →W → →W ; (3) (Xi & Di mapping) (guided search) where W is the initial set of locations of supply centers, W = {wj ; ∀j = 1; : : : ; M }, W is the set of locations of supply centers after the SOFM process, and W is the set of locations of supply centers after the re3ning process. The following sections describe the proposed SOFM process and the re3ning process in detail. 2.1. Self-organizing feature maps SOFMs can map distributed input patterns into an ordered array of competing output units to extract the global structure of the input data. By this property, the locations of the N customer, Xi
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for i = 1; : : : ; N , are used as input patterns to form a training set, and M output units of SOFM will be the locations of supply centers, wj for j = 1; : : : ; M . In addition, the demands of N customers can be represented by the repeating frequency in a training set. During the training process of an SOFM, the location of supply centers, wj , is updated to reinforce the proximity between the distribution of the input locations/demands of the N customers and the con3guration of the weights. It is worth noting that the computing time of SOFM corresponding to di7erent sizes of M and N is proportional to M × N . The SOFM adopted in this research is the basic SOFM known as a Kohonen map [10]. For more general ideas of the construction and learning process of SOFM for solving LA problems (with continuous demand), please refer the work of Lozano et al. [9]. In the proposed SOFM method, some parts which di7er from the work of Lozano et al. [9] are brieNy mentioned below. To handle the demands of customers in SOFM methods, the following technique is proposed. A training set, in which each Xi ; i = 1; : : : ; N , shows exactly Di times in the training set to 3t the N demands criterion, so that the set including i=1 Di inputs is formed by randomly arranging the sequence of all Xi . In this way, the training set contains the probability distribution of customer demands and it will not cause any extra computing in the training stage. The other way mentioned by Lozano et al. [9] to handle the demands of customers is to present the inputs, i , to the network X N according to the discrete probability distribution constructed by p(Di ) = Di = k=1 Dk . However, a random generator and a large lookup table are needed for the implementation of probability distribution inferring process in the second method. Moreover, the necessary amount of inputs may be huge, leading to a time-consuming inferring process. It is clear that the proposed method is faster than the method of Lozano et al. [9]. Regarding the inNuence of using rectilinear distance measurement in SOFM methods, the best matching, wj , in competitive learning process should be selected based on the rectilinear distance measurement. No modi3cation of any other functions is necessary in the SOFM methods. 2.2. Re5ning process In the proposed re3nement process, a special property proposed by Wendell and Hurter [22] for LA problems with rectilinear distance measurement is used in the guided search. This property shows that, in location problems incorporating rectilinear distance, it is suLcient to consider the “intersection points’ in the convex hull as candidates for optimal locations. The “intersection points” are the intersections of all vertical and horizontal lines through the locations of customers. By this property, the problem can be simpli3ed so that there are N 2 candidate locations for M optimal locations of supply centers. The other important property that should be included is the decrease in the chance of optimum search stocking in the local optimal location. Including these two properties, the re3ning process is constructed as follows. Since the LA problem can be treated as a clustering problem, there are two ways to improve the yielded locations of supply centers for minisum criterion. One way is to allocate a customer into a di7erent cluster (group), and the other way is to move the location of a supply center to a new possible location. Both these ways may lead to a new allocation. Based on these two ideas, together with the properties mentioned above, a re3ning process is proposed as follows: 1. Use W SOFM obtained by SOFM to allocate the customers to M clusters.
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2. For each cluster of customers, compute the new supply center, W , and total cost (the computing sub-procedure of supply centers is shown in Appendix). 3. Assign to ∗ (∗ = ); W to W ∗ (W ∗ = W ) and initialize count (count = 0): 4. If the computing time does not exceed the pre-speci3ed time limit, do 4.1. Generate a random sequence of cluster indexes, S = {s(1); s(2); : : : ; s(M )}, where s(j) ∈ {1; 2; : : : ; M }, j ∈ {1; 2; : : : ; M } and s(j) = s(k) if j = k. 4.2. For each cluster, s(j); j = 1; : : : ; M , do 4.2.1. Regarding the location of supply center of current cluster, ws( j) , 3nd the nearest customer outside the current cluster. 4.2.2. Re-allocate the customer found to the current cluster. 4.2.3. Compute the location of supply centers of current cluster, ws( j) , and the location of supply center of the cluster to which the re-allocated customer originally belonged, wk ; k ∈ {S − {s(j)}}. 4.2.4. If count ¿ M , for two supply centers obtained in step 4.2.3, generate two random numbers, ri ; i = 1; 2; 0 6 ri 6 1 and do 4.2.4.1. If r1 ¿ 0:5, move ws( j) along the X -axis to the next right candidate location in cluster s(j). If r1 ¡ 0:5, move ws( j) to the next left candidate location. 4.2.4.2. If r2 ¿ 0:5, move wk along the Y -axis to the next upper candidate location in cluster k. If r2 ¡ 0:5, move wk to the next lower candidate location. (It should be noted that if there is no candidate location on the attempted moving direction in steps 4.2.4.1 and 4.2.4.2, the candidate location on the opposite direction is utilized.) 4.2.5. Re-allocate customers according to the new obtained locations of supply centers, W , and compute new total cost, . 4.2.6. If ¡ , do 4.2.6.1. Update W by W and by . 4.2.6.2. If ¡ ∗ , update W ∗ by W and ∗ by . 4.2.6.3. Let count = 0 and go to step 4. 4.2.7. If ¿ ; count = count + 1. 4.2.8. If count ¿ M 2 , update W by W and go to step 4. 5. Return W ∗ and ∗ . The process of re-allocating customers shown in steps 1 and 4.2.5 is conducted by assigning each customer to the supply center which has the minimal rectilinear distance to the customer. In the above procedure, steps 4.2.1– 4.2.3 conduct the re-allocation process of a customer. This process is repeated until a local optimum is found. However, this process is not capable of making a search jump out of the local optimum. Therefore, whenever a local optimum is identi3ed which is achieved by trying out all clusters using steps 4.2.1– 4.2.3 without any improvement in (count ¿ M ), the location of the supply center moving process is triggered (shown in step 4.2.4). By moving to an adjacent candidate location, the new locations of supply centers may lead the search out of a local optimum when the re-allocation of customers is conducted. If the moving processes of the supply center location still does not work (the ¡ condition cannot be satis3ed within M 2 attempts), then W is updated by steps 4.2.1– 4.2.3 and 4.2.4 without ¡ constraint.
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3. Performance of the proposed method The proposed method is implemented using MatLab version 5.3 with six existing test problems. In the six test problems, problem 1 (N = 16) and problem 2 (N = 30) were generated by Love and Morris [23]. Problem 3 (N = 20), problem 4 (N = 35), and problem 5 (N = 100) were generated by Love and Juel [14]. Problem 6 (N = 150) was generated by Liu and et al. [3]. All experiments were run on a Pentium-III 600-MHz-based IBM PC with 640 MB RAM. Since the objective of this research is to study the ability of SOFM for solving LA problems, the details of parameter studies are excluded from this paper. However, some general setting is provided, as follows. For the SOFM training stage, the feature map is set to a linear array, [1; : : : ; M ], and the parameters use the most frequently used setting in MatLab. More speci3cally, the default settings for the two-phase training process in MatLab are: ordering phase learning rate, 0.9; ordering phase steps (epochs), 1000; tuning phase learning rate, 0.02; topology function adopts ‘hextop’ and distance function uses ‘mandist’. For more detailed information, please refer to the MatLab Help documentation. The training results of SOFM for various sizes of M and epoch settings are shown in Tables 1–3. It should be noted that the computing time and quality of solutions could be improved by adjusting the combinations of various parameters. In Tables 2 and 3, the solutions of total cost obtained from SOFM show that the computing time is robust to the size of M . This excellent property makes the SOFM-based method especially Table 1 Average solutions and run time of SOFM averaged over 20 runs (1000 epochs)
Average solutions () Average run time (s)
Problem 1
Problem 2
Problem 3
Problem 4
2:1659 × 108 12.67
5:4875 × 108 18.57
1:5854 × 108 11.89
6:1823 × 108 20.63
Table 2 Average solutions and run time of SOFM for problem 5, averaged over 25 runs 1000 Epochs
2000 Epochs
3000 Epochs
Average solution ()
Average run time (s)
Average solution ()
Average run time (s)
Average solution ()
Average run time (s)
M =2 N = 100
1:6331 × 109
47.86
1:6171 × 109
51.54
1:6099 × 109
55.08
M =3 N = 100
1:2665 × 109
47.63
1:2615 × 109
51.27
1:2614 × 109
55.26
M =4 N = 100
1:1244 × 109
47.83
1:1052 × 109
51.49
1:1130 × 109
55.06
M =5 N = 100
1:0028 × 109
48.11
9:8070 × 108
51.98
9:6490 × 108
55.77
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Table 3 Average solutions and run time of SOFM for problem 6, averaged over 25 runs 1000 Epochs
2000 Epochs
3000 Epochs
Average solution ()
Average run time (s)
Average solution ()
Average run time (s)
Average solution ()
Average run time (s)
M =2 N = 150
2:6103 × 109
71.32
2:5718 × 109
74.87
2:5683 × 109
78.51
M =3 N = 150
2:1050 × 109
71.33
2:0848 × 109
75.01
2:0875 × 109
78.57
M =4 N = 150
1:8141 × 109
71.32
1:7947 × 109
75.08
1:7930 × 109
78.87
M =5 N = 150
1:6108 × 109
71.25
1:5588 × 109
75.08
1:5597 × 109
78.68
suitable for problems with large M . Another advantage is that the computing time is robust to the size of epoch speci3ed. These advantages make the solutions obtained from SOFM very suitable for use as initial solutions for other heuristics or exact algorithms. Some results shown in Table 3 are worth noting. When M = 3 and N = 150, or when M = 5 and N = 150, average solutions of the total cost of 3000 epochs are worse than 2000 epochs. This phenomenon is due to the movement of location of supply centers to the corresponding nearest customer locations in order to computing the cost. However, since the optimal (sub-optimal) solution locates on the nearby “intersections” but not on the nearest “intersection” of the locations of supply centers obtained by SOFM process, moving the location of supply centers to the corresponding nearest customer locations may not show exactly the best results obtained at di7erent training stages (epochs). In fact, the optimal/sub-optimal solution located on the nearby “intersections” of solutions yielded by SOFM is a key point of this research. If this point is true, it is reasonable that SOFM may be an excellent approach to generate initial solutions for supply center locations, not only for the proposed re3ning procedure, but also for the other heuristic or exact algorithms to solve LA problems. To validate this inference, the proposed guided re3ning procedure, which uses the results of SOFM as initial solutions, should not only be able to 3nd the optimal solutions with great probability, but also 3nd optimal solutions very fast for most testing problems if the initial solutions of locations of supply centers are close to the optimal solutions of locations of supply centers. In the following paragraphs, we will show that these requirements for validation are ful3lled. To evaluate the performance of the proposed SOFM method with guided re3ning procedure, the results obtained by the proposed method are compared to those yielded by the SA algorithm proposed by Liu et al. [3], which is known both as a fast heuristic for large problems and also as having a high probability of 3nding the optimal solution. The algorithm and parameter setting for the SA method used in our experiments are all the same as those reported by Liu et al. [3], except that the stop criterion is changed to a time limit rule (the annealing schedule remains unchanged).
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Table 4 Solutions and run times using SOFMGR and SA over 20 runs No. of Problem Stop Optimal test size times solution problem (s) ()
SOFMGR (1000 epochs in SOFM training)
1
M =3 N = 16
20
191 354 085
a
a
13.80
a
a
2.13
2
M =2 N = 30
40
516 254 969
a
a
19.61
a
a
9.04
3
M =3 N = 20
50
140 236 644
a
a
12.75
a
1:4249 × 108
6.05
4
M =2 N = 35
50
598 084 656
a
a
23.25
a
5:9846 × 108 13.79
a
SA
Best Average Average run Best Average solution () solution () time solution solution for getting () () best solution (s)
Average run time for getting best solution (s)
Same value as the optimal solution.
Table 5 Percentages of obtaining the best known solution and maximal run time (in s) to obtain the best known solution using SOFMGR and SA over 20 runs No. of Stop SOFMGR test time (1000 epochs problem (s) Percentage of obtaining optimal solution
Percentage of worst solution away from optimal solution
Maximal run time Percentage of for getting best obtaining solution optimal solution (all within)
Percentage of worst solution away from optimal solution
Maximal run time for getting best solution (all within)
1 2 3 4
0 0 0 0
16.75 22.07 13.94 28.36
0 0 5.63 0.37
4.02 12.77 8.44 17.31
20 40 50 50
100 100 100 100
in SOFM training)
SA
100 100 70 75
Of the six test problems, the 3rst four are small problems and their optimal solutions are known, whereas the remaining two problems are large (N ¿ 100 customers) and their optimal solutions are unknown. The data in Tables 4 and 5 compare the results from the proposed SOFMGR method and from the SA method for problems 1– 4. As shown in Tables 4 and 5, the best solutions for total cost of both methods in 20 runs, under randomly generated initial setting, are equal to the optimal solutions for every problem within the
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Table 6 Solutions and run times (in s) using SOFMGR and SA for problem 5 over 25 runs No. of Problem Stop SOFMGR SA test size time (2000 epochs in SOFM training) problem Best Average Average run time Best Average solution () solution for getting best solution () solution () solution
Average run time for getting best solution
5
M =2 500 N = 100
1 583 327 772
a
53.54
a
a
5
M =3 500 N = 100
1 234 118 203
a
64.67
a
1:2361 × 109 388.74
5
M =4 500 N = 100
1 060 697 798
a
87.94
a
1:0729 × 109 465.12
5
M =5 500 N = 100
919 515 658
9:1952 × 109 140.13
a
9:2698 × 109 477.12
a
263.90
Same value as the best solution found in both SOFMGR and SA methods.
speci3ed time limits. However, the SA method fails to achieve the optimal solutions in all runs for problems 3 and 4, though it uses less computing time than the SOFMGR method for these small problems. The average run times to obtain the best solutions in the proposed guided re3ning procedure only are 1.1, 1.0, 0.9 and 2:6 s. These times are obtained by subtracting the average run time in Table 1 from the average run time to obtain the best solution in Table 4, for problems 1– 4. These results shows that the initial solutions of the locations of supply centers yielded by SOFM are close to the optimal solutions so that the proposed re3ning procedure can 3nd the optimal solutions very quickly for all test runs for problems 1– 4. Since the optimal solutions for various M sizes are unknown for problem 5, the best solutions of total cost, de3ned as the best solutions obtained by both SOFMGR and SA methods, are used to compare SOFMGR and SA. As shown in Table 6, the best solutions achieved by both methods in 25 runs are equal for various sizes of M within the speci3ed time limits. However, the percentage of the best solutions achieved by the SA method decreases when M increases (refer Table 7). In addition, the SOFMGR method has shorter run times than the SA method. Moreover, the high percentage of achieving optimal solutions and the short average run time to obtain the best solution in guided re3ning procedure shows that the quality of initial solutions of locations obtained by SOFM is good. The average run times for getting the best solution using the guided re3ning procedure for problem 5 (2 s for M = 2 case, 13:4 s for M = 3 case, 16:5 s for M = 4 case, 88:2 s for M = 5 case) are obtained by subtracting the average run time in Table 2 from the average run time for getting the best solution in Table 8. For problem 6, the best solutions achieved by SOFMGR method are better than the best results yielded by the SA method for all four di7erent sizes of M (as shown in Tables 8 and 9). In addition,
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Table 7 Percentages of obtaining the best known solution and maximal run time (in s) to obtain the best known solution using SOFMGR and SA for problem 5 over 25 runs No. of test problem
Problem size
5
M =2 N = 100
5
Stop time
SOFMGR (2000 epochs in SOFM training)
SA
Percentage of obtaining best known solution
Percentage of worst solution away from best known solution
Percentage obtaining best known solution
Percentage of worst solution away from best known solution
Maximal run time for getting best solution (all within)
500
100
0
57.33
100
0
309.76
M =3 N = 100
500
100
0
95.65
12
0.20
429.16
5
M =4 N = 100
500
100
0
220.83
20
2.46
499.86
5
M =5 N = 100
500
96
0.02
389.29
4
4.22
499.60
Maximal run time for getting best solution (all within)
Note: Percentage of worst solution away from optimal solution is de3ned as (worst solution/optimal solution)−1.
Table 8 Solutions and run times (in s) using SOFMGR and SA for problem 6 over 25 runs No. of Problem Stop SOFMGR SA test size time (2000 epochs in SOFM training) problem Best Average Average run time Best solution () solution () for getting best solution () solution
Average solution ()
Average run time for getting best solution
6
M =2 500 N = 150
2 515 820 112
a
103.20
2:5200 × 109 2:5394 × 109 487.06
6
M =3 500 N = 150
1 998 514 195
a
78.25
2:1299 × 109 2:2261 × 109 415.64
6
M =4 500 N = 150
1 715 275 017 1:7159 × 109 127.38
1:9454 × 109 2:1029 × 109 429.01
6
M =5 500 N = 150
1 487 070 093 1:4871 × 109 161.87
1:9014 × 109 2:0018 × 109 420.68
a
Same value as the best solution found in both SOFMGR and SA methods.
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Table 9 Percentages of obtaining the best known solution and maximal run time (in s) to obtain the best known solution using SOFMGR and SA for problem 6 over 25 runs No. of Problem Stop SOFMGR test size time (s) (2000 epochs problem Percentage of obtaining best known solution
in SOFM training)
SA
Percentage of worst solution away from best known solution
Maximal run time for getting best solution (all within)
Percentage of obtaining best known solution
Percentage of worst solution away from best known solution
Maximal run time for getting best solution (all within)
5
M =2 500 N = 150
100
0
184.70
0
2.24
499.54
5
M =3 500 N = 150
100
0
85.97
0
25.87
499.26
5
M =4 500 N = 150
96
0.96
272.52
0
33.06
499.90
5
M =5 500 N = 150
96
0.002
289.30
0
46.23
499.89
Note: Percentage of worst solution away from optimal solution is de3ned as (worst solution/optimal solution)−1.
the SOFMGR method has a shorter run time than the SA method. Since the stop time setting (500 s) does not seem long enough for SA to achieve the stopping criterion speci3ed in Liu et al. [3], 3ve extra experimental runs (M = 5 and stop time setting of 1 h) were conducted for SA method for more investigation. However, the results yielded by these extra runs are still worse than the solutions obtained by the SOFMGR method at a seconds setting of 500. Two other aspects that should be carefully observed are the very high percentage of achieving optimal solutions and the short average run time to obtain the best solution in the guided re3ning procedure. When these two things occur simultaneously, it is reasonable to conclude that the initial location solutions for supply centers obtained by SOFM are close to the best solutions. The average run times for obtaining the best solution by the guided re3ning procedure for problem 6 are 28:3 s for M = 2 case, 3:2 s for M = 3 case, 52:3 s for M = 4 case, and 86:8 s for M = 5 case. These results are obtained by subtracting the average run time in Table 3 from the average run time for obtaining the best solutions in Table 8. From the above experimental results (Tables 5, 7 and 9), we observe that the percentage achieving optimal or the best known solutions is never less than 96% for the proposed method. In addition, the average run times for obtaining the best solutions for total cost in guided re3ning procedure are short. These results show that the SOFM method provides excellent initial solutions for supply center locations for subsequent guided re3ning process. In Fig. 1, the typical example of problem 1 demonstrates that the solution of SOFM can indeed provide a very good initial solution for supply center locations for the guided re3ning process. Since the guided re3ning process, which only applies simple guided random searches on the neighborhoods of solutions yielded by SOFM, not only has a
K.-H. Hsieh, F.-C. Tien / Computers & Operations Research 31 (2004) 1017 – 1031
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Fig. 1. Typical results of SOFM and SOFMGR for problem 1.
very high probability of 3nding the optimal or best known solution but also 3nds the solution very quickly, we may infer that most of the solutions for supply center locations obtained by SOFM are close to the optimal solution. Therefore, other sophisticated heuristic or exact algorithms may also adopt SOFM as the initial solution-generating method to produce good starting locations of supply centers and improve the computing speed.
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4. Conclusions From the above experimental results, we have shown that the proposed SOFM with a guided re3ning (SOFMGR) procedure performs better in solving the minisum location–allocation problems with rectilinear distance. For large location–allocation problems, the proposed method outperforms the SA method proposed by Liu et al. [3], which is known not only as a fast heuristic for large problems but also as one with a high probability of 3nding optimal solutions, yielding high-quality solutions and with eLcient computation. In addition, some results shown in this research indicate that the SOFM may be used as an excellent procedure to 3nd initial solutions for supply center locations for other heuristic or exact algorithms. Regarding the experimental results, although the SOFM computation time is robust regarding the size of M, the time consumed by the re3ning process does increase as M increases. Therefore, further research may concentrate on developing a more sophisticated algorithm for re3ning the results yielded by the SOFM method. Appendix A. 1. Within each cluster, s(j); j = 1; : : : ; M , there are Ls( j) customers. For each cluster, s(j), do 1.1 Sort Xi in cluster, s(j), by X coordinates in ascending order. Then the sorted Xi with associated Di can be expressed as Xm and Dm ; m = 1; : : : ; L(j) . Find a Dc that satis3es c −1
Dm ¡
m=1
Ls( j) c 1 Di 6 Dm : 2 i=1 m=1
(A.1)
Then, the X coordinate of supply center (ws( j) ) is the X coordinate of Xc . 1.2 Sort Xi in cluster, s(j), by Y coordinates in ascending order. Then the sorted Xi with associated Di can be expressed as Xn and Dn ; n = 1; : : : ; Ls( j) . Find a Dc that satis3es c −1 n=1
Dn ¡
Ls( j) c 1 Di 6 Dn : 2 i=1 n=1
(A.2)
Then, the Y coordinate of supply center (ws( j) ) is the Y coordinate of Xc . 2. Return ws( j) Ls( j) If 12 i=1 Di = cm=1 Dm occurs in Eq. (A.1), the X coordinates of supply centers can be arbitrarily located between the X coordinates of Xc and Xc+1 . In this research, (Xc + Xc+1 )=2 is used. The same technique can be applied for similar situations which occur in computing the Y coordinates of supply centers. References [1] Francis RL, White JA. Facility layout and location—an analytical approach. Englewood Cli7s, NJ: Prentice-Hall, 1974. p. 212.
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