A Placement Algorithm for Efficient Generation of Superior ... - CiteSeerX

Fifth International Conference on Operations and Quantitative Management

October 25-27, 2004

A Placement Algorithm for Efficient Generation of Superior Decision Alternatives in Layout Design Abdul-Rahim Ahmad, Otman Basir, Systems Design Eng g., University of Waterloo, Canada Muhammad Hasan Imam, Civil Engineering, Ummul Qura University, Saudi Arabia Khaled Hassanein, DeGroote School of Business, McMaster University, Canada The Layout Design is a complex problem that requires advanced decision analysis and support techniques. In this regard, swift and easy generation as well as analysis of better -quality decision alternatives is a favored approach. Various metaheuristics can be employed for efficient generation of superior layout decision alternatives. However, the efficiency and efficacy of such approaches mainly depend on the efficiency of the placement heuristics. More importantly, the quality of decision alternatives is mainly influenced by the effectiveness of module placement heuristic used. Here, one such efficient and robust placement heuristic is proposed and compared with some popular existing heuristics in terms of speed, space utilization, and quality of outcome. Quantitative as well as qualitative fitness evaluations are used to provide a more rational evaluation and comparison scheme. The proposed algorithm consistently and speedily furnishes layout decision alternatives carrying higher aesthetic value. Such a development is likely to facilitate productive utilization of resources. Keywords: Facilities/VLSI Layout, Cutting/Packing, Metaheuristics, Placement Heuristics, BL-Algorithm

1. Introduction Layout Design (LD) is the assembly of modules in a specified space for given preferences and constraints. It is a common problem in such faculties as Facilities, VLSI, and Newspaper layout designs as well as in various Cutting and Packing industries. Recently, the pervasiveness of the Web and Mobile services has brought some interesting and challenging layout design applications such as E-store and E-Learning sites [1], [2]. Naturally, such applications differ from general LD problems in terms of subjective nature of preferences because of a diverse target population. Nevertheless, such subjective and uncertain decision preferences and constraints also exist in other applications that are conventionally treated as hard optimization problems. Such complexity and subjectivity augment the enormity of task in automating the LD process. As such, computer-based algorithms cannot replace human intuition [2], [13]. In addition, various existing models employed in automated layout design are known to be NP -Hard in strong sense [2], [7]. Consequently, the goal of producing an optimal or superior solution in reasonable time quickly becomes elusive with the increase in the size of problem. Nevertheless, automated layout design systems can be helpful in generation, elaboration, articulation, enumeration, and ranking of a large number of competing Layout Alternatives [1], [3], [5], [13]. For instance, an intelligent and interactive system for decision support in layout design, that offers immense promise in this important area, has recently been presented [1], [2]. Indeed, such an automatic and knowledge-based generation and analysis of decision alternatives is critical to any layout decision process [11], [13]. Intuitively, various heuristic and metaheuristic techniques have considerable role to play in such an automated system. Indeed, past studies have demonstrated the effectiveness of such approaches. Such methods involve dealing with LD as a packing problem by generating an Ordering of Modules and a Placement Heuristic for placing modules in the specified order [2], [3], [6 ]. Consequently, an efficient and effective placement algorithm is critical to such methodology. Nevertheless, the existing placement heuristics lack the requisite efficiency and efficacy required in most scenarios. Accordingly, continual research has been made towards developing superior heuristics. In this regard, we propose a new placement algorithm for efficiently obtaining decision alternatives that are superior in both quantitative and qualitative terms. The comparison with other existing popular placement algorithms, such as the Bottom-Left or BL strategy [9], Improved-BL or IBL [10], MERA [3], etc. demonstrates the superiority of the new algorithm. The rest of the paper is organized as follows. Section 2 provides a brief overview of solution methodologies used to tackle the layout design problem. Section 3 deals with some existing as well as the proposed placement heuristics for layout design. Section 4 provides comparison between proposed and existing placement algorithms using both quantitative and qualitative fitness appraisal scheme. Section 5 concludes the paper with some future research directions.

2. Layout Design Problem This research draws from an extensive range of work domains including layout design, cutting/packing, optimization, metaheuristics, decision-making, etc. Accordingly, an extensive examination of literature is outside

122

Fifth International Conference on Operations and Quantitative Management

October 25-27, 2004

the scope of this paper. Nevertheless, we present a cursory overview from the automated layout decision pers pective for reference purposes. Furthermore, we use the rectangular packing problem for our illustrations. The layout design problem has been with us since long and received considerable attention in various LD problem domains. It has been variously referred to as facilities layout, topology optimization, plant layout, machine layout, block placement, macro cell placement, layout optimization, etc. Various mathematical formulations for tackling the LD problem have been proposed in the literature. The most popular of such formulations include the quadratic assignment problem or QAP, the two-dimensional bin -packing problem or 2D-BPP, and the quadratic set-covering problem or QSC. However, the QAP does not allow control over the shape of the modules in the resulting layout and QSC requires a large number of user inputs for every module under consideration. Furthermore, such approaches result in prohibitive computational cost due to requisite discretization of modules and packing space. These and other laggings make such formulations somewhat incompatible for the application to general LD problems where a large number modules with fixed shape and size needs to be considered. Consequently, here we use an approach similar to oriented two-dimensional finite bin packing for elaboration and comparison of existing and proposed placement heuristics. It should be noted that even a small change in the position of a module in a feasible direction creates a new packing pattern without any genuine alteration in the topological arrangement [2], [11]. As such, the search space is infinite even for very small size problems. Consequently, more emphasis is given to the soft optimization methods. As already pointed out, the existing mathematical models offer little practical benefit in dealing with problems of some genuine significance due to the prohibitive size of associated mathematical program. Such core issues as subjectivity and vagueness of the layout design fitness evaluation objectives, preferences, and constraints further exacerbate the situation. Accordingly, development of some fast and effective heuristics and metaheuristics that reliably provide superior decision alternatives is the major focus in this area. Recent metaheuristics that have shown promising results include Genetic Algorithms, Simulated Annealing, Naïve Evolution, and Random Search [2] , [3], [8], [9], [10]. Other solution techniques include Binary mixed Integer-Programming, Tree Search, the Network Decomposition, etc. Furthermore, some analytical techniques that deal with continuous design space with minimal compu tational requirements are available [11]. However, such analytical techniques do not produce results comparable with advanced heuristic techniques.

3. Placement Algorithms It is important to effectively limit the infinite decision space in layout design to some realistically tractable subspace of solution topologies [6]. Such tractability can be obtained by using some deterministic or semi deterministic module placement algorithms that take one module at a time from a given sequence of modules and determine its location in the packing. Indeed, the computational cost of such a meta-search based layout optimization process depends on the cost of the module placement heuristic [6], [9]. Consequently, an efficient module placement decision strategy that generates superior layout decision alternatives is critical for the effectiveness of such efforts [2]. 3.1 BL-Algorithm Recently, the BL placement algorithm has drawn considerable attention from researchers [6], [9], [10] . It requires placing a module at the bottom-most and left-most feasible location. The process is demonstrated in Figure 1. Following are the main steps involved in the basic BL algorithm: Let Blocks = No. of Modules at hand for Placement. 1) Place module M 1 at the bottom-left corner of the bin 2) For K = 2 to Blocks Shift module MK alternately, beginning from upper-right corner of packing space, as far as possible to the bottom and then as far as possible to the left. Next K 3) Stop if no room for more modules.

The popularity of the BL is derived from the simplicity of the underlying notion, ease of implementation, and speed of execution [2], [3], [6]. However, BL has inherent shortcomings including poor space utilization and inability to obtain some simple optimal solutions [9], [10]. For instance, it possible to have an optimal packing pattern of n modules that even fulfills the BL-condition but we cannot write out a permutation for the BLalgorithm corresponding to it. Simply stated, the optimal packing pattern may be obtained by the BL-algorithm

123

Fifth International Conference on Operations and Quantitative Management

October 25-27, 2004

even if all permutations are enumerated as the case in example shown in Figure 2. In addition, BL is not very effective in incorporating such qualitative design consideration as symmetry, balance, equilibrium, coherence, homogeneity, etc. [2]. y

4 8

5

6

3

4

7

4 1

Dead Area

3

2 1

2 x

Figure 1: Poor Space Utilization with BL. Figure 2: Optimal Packing Not Possible with BL. Furthermore, BL is more appropriate for minimizing the height of the packing pattern. However, if two packing patterns have the same height or consume equal amount of space then their fitness values are same. Nevertheless, one of the packing patterns might be deemed ‘superior’ based on other objectives. Moreover, most cutting and packing applications require optimizing the contiguous remainder, discussed later on, rather than the height of packing. The BL-algorithm is not very useful in optimizing the contiguous remainder as can be seen from Section 4 as well as shown in past studies [2], [3]. In addition, the BL algorithm converges modules at the bottom-left corner of the packing space that might not be a useful strategy in many cases. 3.2 Improved BL-Algorithm Various improvement plans have been proposed for the BL such as Improved-BL or IBL [10] . Such improved strategies consist of refinement in placement decisions in the Step 2 of BL by giving priority to a shift towards bottom and some allowance for module rotation. Such improvement schemes are quite popular in practice. Consequently, we have also included IBL in our comparison analyses. However, our implementation of IBL does not involve rotation of modules in accordance with our intended work domain that involves only oriented modules. It should be noted that even these improved algorithms encounter such problems as lack of aesthetic value and dead -area surrounded by placed modules that could not be utilized. Furthermore, much like BL, IBL is also not suited for optimizing the contiguous remainder or providing decision alternatives with higher aesthetic value. 3.3 Proposed Algorithm The proposed module placement algorithm is motivated by the fact that for any given packing space the number of modules available for placement is a small integer. In addition, if we restrict our packing decisions only to the corners of the ‘in -place’ modules then for a given module there are only O( n) possible locations. Consequently, the combinatorial intractability should not deter the use of some fast pseudo -exhaustive search, aiming at improving the space utilization as well as the layout quality, in a hierarchical manner [2], [3]. Here we outline the proposed module placement algorithm called Min imization of Enclosure under Gravitational Attraction (MEGA) algorithm. The name implies the underlying notion where a reduction of the minimization of sum (or weighted sum) of inter-module distances in the packing pattern is sought during all placement decisions. It is a thoughtful variation of a reportedly very efficient and effective algorithm called MERA (or Minimization of Enclosing Rectangle Area) in order to incorporate inter-module interactions in the optimization process [3]. The refinement part in the placement algorithm is a simple pseudo-exhaustive exploration of solution space in which only corners of in-place modules are investigated as potential locations for an in -coming module. This pseudo-exhaustive placement technique is summarized in the pseudo-code given below. In this pseudo-code, index K denotes four corners of an in-place module (MJ) and index L denotes four corners of an in-coming module (MI). The Step 2 proceeds by investigating the placeme nt opportunities for each of the four corners of an in -coming module at each of the four corners of all in -place modules and seeking a minimum of the sum of all inter-module distances between centroids (d i,j). Such hierarchical approach should result in an increased complexity of the algorithm. The computational complexity of BL-algorithm is O(n 2) as each in-coming module can be shifted a maximum of i times and each shift is constrained by one of the i–1 already placed modules or by the boundaries of the bin. However, in case of MEGA, each in-coming module can be placed at a maximum of 16i possible corner points (a very loose bound)

124

Fifth International Conference on Operations and Quantitative Management

October 25-27, 2004

where i–1 modules are already in place. As such, theoretically, MEGA also has the same O(n2) computational cost as the case for BL and IBL [9], [10]. It should be noted that such a small increase in computational complexity is deemed quite reasonable and tolerable when weighted against the potential benefits, as demonstrated by the results in Section 4. Let: n = No. of Modules at hand for Placement NPlaced = No. of Modules Already Placed newOBJ = Sum of Inter-module Distances 1) Place module 1 at the bottom-left corner of the page 2) Set OBJ to a big value 3) FOR I = 2 to n FOR J = 1 to NPlaced FOR K = 1 to 4 FOR L = 1 to 4 Place corner B of MK on corner A of MJ Check Overlap conditions Check Boundary conditions IF both conditions satisfied THEN

Calculate the newOBJ =

i= I

NPlaced

i =1

j =1

∑ ∑d

i, j

IF newOBJ is less than OBJ THEN OBJ = newOBJ Save placement of module MI ENDIF ENDIF END L END K END J END I 4) Stop if no room for more modules.

4. Results and Discussions A computer program is written in Visual BASIC to implement the BL, IBL, and MEGA algorithms. The resultant computer program is used on Intel Xeon 3.06 GHz processor and 256MB of RAM under Windows XP. The comparative analyses are shown in Tables 1 through 5. The most challenging and application specific task in layout evaluation is the definition of the fitness function [2]. Any heuristic or metaheuristic approach uses fitness function to differentiate between ‘superior’ and ‘inferior’ alternatives. The ideal, though somewhat impractical, course would be to let layout designers establish the fitness through visual evaluation. As such, researchers have used metrics like minimization of Packing Height, maximization of Module Tightness (or the percent of enclosing rectangle area wasted) and the maximization of Contiguous Remainder or CR (the largest contiguous unused portion of the bin still available for further packing). Such quantitative metrics afford some degree of space utilization [2] , [3], [8], [9], [10] . However, the industrial practice is to use contiguous remainder as the fitness metric in most cutting and packing applications [2], [9], [10]. Consequently, to make our comparisons more realistic, we opted to use CR for our studies. Nevertheless, the bulk of the existing literature uses minimization of height as an alternate for contiguous remainder as height is relatively easy to quantify or calculate in an automated manner.

Figure 3: Three Packing Patterns with same Height

125

Fifth International Conference on Operations and Quantitative Management

October 25-27, 2004

It should be noted that such rigid quantitative metrics fall short of capturing such qualitative aspects as layout symmetry [2], [4], [12]. Such limitations of these rigid and myopic metrics can be elaborated using Figure 3. It is evident that the three given packing patterns have same height; nevertheless, patterns B and C are visibly more desirable than pattern A in a bin-packing context. In the same league, patterns B and C have same CR. However, pattern C might be more desirable than the pattern B keeping in view that a large portion of contiguous remainder in pattern B is not quite contiguous in practical sense. Consequently, we also used subjective evaluation of layout quality by layout design experts for obtaining a more realistic comparison scheme. In short, our quantitative analysis uses Contiguous Remainder and qualitative analysis uses human intuition and aesthetic perception in judging the relative fitness of layouts. As such, we requested a couple of experienced researchers in the layout design area to evaluate and rank top ten layouts on a scale of 1 to 10 with regard to the layout symmetry. We used 25- and 50-module problems (P25 and P50) reported by Jakobs (1996) and Liu (1999), respectively, as well as a 100-module problem (P100) reported by Ahmad et al. (2004a) for comparison and demonstration purposes. Problem

Objective

Tech. BL

Wins

Best

Worst

Mean

Std.Dev.

0

573.0

464.0

529.78

23.13

(Ideal = 600) Higher the Better MERA

IBL

5 54

584.0 590.0

470.0 526.0

547.13 569.84

21.14 11.86

MEGA

41

598.0

530.0

567.03

12.97

BL IBL (Ideal = 10) MERA Higher the Better MEGA

0

4.5

1.0

2.525

1.127

0

4.0

1.75

2.725

0.749

5

7.0

3.5

4.9

1.149

5

7.5

3.0

5.05

1.347

CR P25 Bin =40x30 Optimal=40x15

Quality

Table 1 : Comparison of BL, IBL, MERA, and MEGA for 100 random sequences in a 25-module problem. Problem

Objective

P50 Bin =40x40 Optimal=40x15

Tech. BL CR IBL (Ideal = 1000) Higher the Better MERA MEGA

Wins

Best

Worst

Mean

Std.Dev.

0 4

981.0 980.0

875.0 913.0

933.6 953.6

16.55 12.49

73

987.0

961.0

976.8

7.24

23

993.0

949.0

969.4

8.76

0

6.0

2.75

4.43

0.965

0

7.0

2.75

4.53

1.309

5 5

8.5 8.0

4.5 4.5

6.5 6.48

1.111 1.102

BL IBL

Quality

(Ideal = 10) Higher the Better MERA

MEGA

Table 2: Comparison of BL, IBL, MERA, and MEGA for 100 random sequences in a 50-module problem. Problem

Objective CR (Ideal = 5000) Higher the Better

P100 Quality (Ideal = 10) Higher the Better

Tech.

Wins

Best

Worst

Mean

Std.Dev.

BL IBL MERA MEGA

0 0

3117.8 3147.3

879.3 1591.7

1995.7 2491.9

442.4 361.6

66

4604.8

3694.2

4100.6

163.6

34

4364.3

3368.0

3996.4

150.2

BL

0

4

1.5

2.65

0.75

IBL MERA MEGA

0 8

3.5 6

2.0 2.5

2.75 4.05

0.77 1.17

2

5.5

2.5

3.80

1.23

Table 3: Comparison of BL, IBL, MERA, and MEGA for 100 random sequences in a 100-module problem.

126

Fifth International Conference on Operations and Quantitative Management

October 25-27, 2004

We employed a Random Search approach for our comparative studies by generating 100 random sequences of modules. As already mentioned, Random Search and Naïve Evolution are among the most effective search strategies for layout design problems [3], [8], [9], [10]. The relative performance of the BL, IBL, MERA, and MEGA placement strategies for 100 random sequences of each problem instance is depicted in Table 1, 2, and 3. It can be seen that MERA and MEGA consistently outperform BL and IBL by large margins in terms of both space utilization and quality of outcome. The performance gain is more pronounced for larger problems. Although MERA furnis hed superior solutions more consistently, the best solution obtained through MEGA is marginally better in terms of CR for moderate size problems. 10

15

Problem Size 25 50

100

BL 0.00019 0.00279 0.00705 0.01716 0.4152 Tech. MERA 0.00841 0.01271 0.05207 0.2863 13.361 MEGA 0.00916 0.0188 0.09322 1.0249 61.011 Table 4: Average Time Elapsed (in seconds) per 100 iterations for BL, MERA, and MEGA. The mean CPU time taken by BL, MERA, and MEGA for 100 random sequences of the 100-module problem is shown in Table 4. Apparently, the mean time taken by MERA or MEGA increases exponentially with the increase in size of the problem. However, our experience has demonstrated that just one random module sequence almost always, if not always, furnishes a layout with MEGA -alone that is superior to the best obtained after hundreds of evaluations of BL or IBL. It happened in more than 70% random sequences of 50-module problem and in more than 80% random sequences of the 100 -module problem during repeated experiments. Moreover, a module sequence sorted in descending order of length of the longer side provides a quite superior decision alternative in both quantitative and qualitative terms, as evident from Table 5. It implies that only one evaluation of MEGA is enough to beat such existing algorithms as BL and IBL by a wide margin. Outcomes of such an ordered sequence for our 100-module problem are shown in appendix, with trimmed bin tops, for visual comparison purposes. Problem

Objective CR

Tech. BL+DL IBL+DL

(Ideal = 5000) Higher the Better MERA+DL

Fitness 3479.3 (30.5%) 3787.5 (24.3%) 4636.1 (7.3%)

MEGA+DL 4422.3 (11.5%)

P100

BL+DL IBL+DL (Ideal = 10) MERA+DL Higher the Better MEGA+DL Quality

2.0 2.25 4.75 4.25

Table 5: Comparison using a Decreasing Length sequence for a 100-module problem (difference from optimal CR is shown in parentheses). In addition, the resulting superior quality and diversity of layout decision alternatives obtained through MERA and MEGA sanction a relatively higher computational cost a worthwhile trade-off. Moreover, the performance of BL and IBL is known to deteriorate dramatically with the increase in the problem size as can be seen from results in Tables 1, 2 and 3, and as demonstrated by a series of earlier studies [3], [8], [9], [10] . In contrast, MERA and MEGA result in substantially higher performance gains for larger problems; thus, furnishing another cogent incentive for resorting to a seemingly costly approach. In short, MEGA not only results in better quality alternatives but also provides those faster than BL or IBL. It demonstrates that the computational cost of MEGA is not prohibitive in very efficiently achieving results better than BL or IBL. The comparison of MERA and MEGA is a bit tricky enterprise. It can be seen that MERA provides higher CR more frequently than MEGA. The same is true in terms of layout quality. However, the best outcome of MEGA for moderate size problems is marginally better than that of MERA. This marginal superiority could translate into substantial overall benefit when used in conjunction with such metaheuristic search approaches as Genetic Algorithms, which depend on the quality of intermediate solutions to define and refine future search operations. Furthermore, keeping in view that MERA is known to be very effective in generating layouts with

127

Fifth International Conference on Operations and Quantitative Management

October 25-27, 2004

very high CR, such seemingly marginal performance improvement is actually a significant development. Furthermore, MEGA seems to be more appropriate for applications where inter-module interaction calls for compact packing while minimizing the total inter-module distance, such as the wiring length in some VLSI layout design. However, la yout design using MERA in such a scenario would require an auxiliary fitness evaluation and optimization mechanism that could result in reduced efficiency of the overall process. For instance, one effective methodology would be to employ metaheuristics in conjunction with MERA; however, such metaheuristics require a large number of evaluations even at the onset of the process. In contrast, only few random sequences of MEGA could provide a very good, if not the best, solution. In addition, the less frequent but marginal improvement in CR provided by MEGA in larger problems could also translate into ample benefits in terms of speed and performance with metaheuristic-based search techniques. Such comparisons of MERA and MEGA furnish a very interesting future research direction. However, we want to emphasize that our preliminary studies show that MERA is more consistent in furnishing layouts with higher space utilization and layout quality. Briefly, MEGA results in an efficient, effective, and superior layout optimization than such existing algorithms as BL and IBL. Furthermore, MEGA holds promise to beat MERA in terms of speed and space utilization when used with such effective metaheuristic search approaches as Genetic Algorithms and Simulated Annealing. Moreover, MEGA is quite simple to understand and implement; as such, the adoption of MEGA by the scientific community in future studies would be easy.

5. Conclusion The layout design is a difficult and frequently encountered decision problem in a range of work domains. In this paper, a new efficient, effective, and robust module placement strategy has been proposed for obtaining superior layout decision alternatives. Studies demonstrate the power of the new placement strategy to outperform some popular existing strategies in speed as well as quality of outcome. It is particularly suited for applications where aesthetic values, space utilization, as well as inter-module interactions bear importance to decision makers. Such strengths make it quite pertinent for the use in some automated and knowledge-based decision support tool for layout design. We believe that this research would result in increased efficiency and productivity of layout designers as well as facilitation of future research in related areas. Acknowledgements: Authors are grateful to an anonymous technical support specialist at VIP -PlanOptTM (www.PlanOpt.com) for valuable electronic discourse resulting in an improved understanding of the underlying dynamics in layout design.

6. References [1]

[2] [3]

[4]

[5]

[6] [7] [8] [9]

Ahmad, A.R., Basir, O., Hassanein, K., (2003), “Fuzzy Inferencing in the Web Page Layout Design’” In J. Bézivin, J. Hu, Z. Tari (Eds.), Proc. of the 1st Workshop on Web Services: Modeling, Architecture and Infrastructure (WSMAI’03), Angers, France, pp. 33-41. Ahmad, A.R., Basir, O., Hassanein, K., (2004a), “Intelligent Decision Support System for Layout Design”, Proc. of the 9 th Asia-Pacific Decision Sciences Institute Conference (APDSI’04), July 2004, Seoul, Korea. Ahmad, A.R., Basir, O., Hassanein, K., (2004b), “Improved Placement Algorithm for Layout Optimization”, Proc. of the 2nd Int’l Industrial Engineering Conf. (IIEC’04), Dec. 2004, Riyadh, Saudi Arabia, to appear. Ahmad, A.R., Basir, O., Hassanein, K., (2004c), “Decision Preferences, Constraints, and Evaluation Objectives in Layout Design: A Review of Modeling Techniques”, ICOQM-V, Oct. 2004, Seoul, Korea, this volume . Ahmad, A.R., O. Basir, K. Hassanein, “Adaptive User Interfaces in Intelligent Multimedia Educational Systems: Issues and Trends”, Proc. of Int’l Conf. on Electronic Business (ICEB’04), Dec. 2004, Beijing, China, to appear. Dowsland, K.A., Vaid, S., Dowsland, W.B., (2002), “An algorithm for polygon placement using a bottomleft strategy”, Euro. J. of Op. Res., 141, pp. 371-381. Garey, M.R., Johnson, D.S., (1979), Computers and Intractability, W.H. Freeman Press, NY. Hopper, E., Turton, B.C.H., (2001), “An Empirical Investigation of meta-heuristic and heuristic algorithms for a 2D packing problem”, Euro. J. of Op. Res., 128, pp. 34-57. Jakobs, S., (1996), “On genetic algorithms for packing of polygons”, Euro. J. of Op. Res., 88, pp. 165-181.

128

Fifth International Conference on Operations and Quantitative Management

October 25-27, 2004

[10]

Liu, D., Teng, H., (1999), “An improved BL-algorithm for genetic algorithm of the orthogonal packing of recta ngles”, Euro. J. of Op. Res., 112, pp. 413-420.

[11]

Mir, M., Imam, M.H., (2001), “A hybrid optimization approach for layout design of unequal-area facilities”, Computers and Industrial Engineering, 39. Ngo, D.C.L., (2001), “Measuring the aesthetic elements of screen designs”, Displays, 22, pp. 73-78. Tompkins, J.A., White, J.A., Bozer, Y.A., Tanchoco, J.M.A., (2002), Facilities Planning, 3rd Ed. (John Wiley Inc.).

[12] [13]

APPENDIX – A:

Figure A.1: BL + DL.

Figure A.2: IBL + DL.

Figure A.3 MERA + DL.

Figure A.4: MEGA + DL

129