1. the 7 th Online World Conference on Soft Computing in Industrial Applications (WSC7) PROCESS-ORIENTED PLANT LAYOUT DESIGN USING A FUZZY SET DECOMPOSITION ALGORITHM E. Avineri a, P. Vaish b a Technion - Israel Institute of Technology, Faculty of Civil Engineering, Haifa 32000, Israel, E-mail : avineri@internet-zahav.netavineri@internet-zahav.net b Oracle Software India ltd., 5 th Floor, Cyber Gateway, Hi-Tec City, Madha Pur, Hyderabad - 500081, India, E-Mail: vaishprakhar@yahoo.comvaishprakhar@yahoo.com

2. 1 INTRODUCTION Efficient plant layout is critical to achieve and maintain manufacturing competitiveness. 20% to 50% of total operating cost is spent on Material Handling (MH) and an appropriate facilities design can reduce this cost by at least 10% to 30% Process-oriented layout: physical arrangement of given number of machines (or facilities) within a given configuration.

3. 1 INTRODUCTION Facility Layout Problem (FLP) considers the amount of load or number of trips multiplied with distance-related costs, and looks for a solution which would minimize the value of function F: where (traffic) ij is the traffic between machines i and j; (distance) f(i)f(j) is the distance between machine locations and M is the number of machines.

4. 1 INTRODUCTION Rectilinear Distance Model Distance between i and j is: d ij = | x i - x j | + | y i - y j | NP-completeness of the FLP: impractical to search for optimal solutions. Fuzzy sets theory was found to be a powerful method of approximate reasoning. The number of different layout patterns, in which M machines are placed (in N locations or cells (N  M)

5. 2. MATHEMATICAL FORMULATION The distance between machines i and j, d ij, is inversely proportional to m ij because more is the mass nearer the machines should be. Similarly d jk is inversely proportional to m jk. d ik  d ij +d jk. 1/m ik  1/m ij + 1/m jk m ik  {m ij *m jk }/{m ij +m jk } If m ij >0,m jk >0, and m ik =0 then no mass moves from i to k. But indirectly mass moves from machine i to machine k via machine j. Connection matrix [M ij ], m ij  0

6. 2. MATHEMATICAL FORMULATION Thus: m ik /P max  (m ij /P max * m jk /P max }/{m ij /P max + m jk /P max )  i,j | m ij +m jk > 0 (*) This forms the principle of fuzzy set algorithm that mass transfer between no machines is zero as assumed in conventional layout algorithms.  j m ij all mass transported from machine i to other machines  i m ij all mass transported to machine j from all other machines P max = Max{  j m ij,  i m ij }

7. 2. MATHEMATICAL FORMULATION A fuzzy relation matrix M’ can be defined. Each of its elements represents the membership value of a pair of machines to the fuzzy set “machines that should be placed close to each other”, as calculated by (*) and its extensions. In other words, The higher the membership value is, the nearer the machines should be placed. The total mass transfer between machines i and j is m ’ ij +m ’ ji. By adding the element min{m ’ ij, m ’ ji } to the element max{m ’ ij, m ’ ji } we will get the matrix M ’’ ij. The larger the value of m ’ ij +m ’ ji, the closer the machines i and j should be placed.

8. 2. MATHEMATICAL FORMULATION Applying the fuzzy decomposition algorithm to M ’’ ij, the decomposition of fuzzy relation may be summarized as: R=    *R  0  2  R  1  R  2 where R  denotes a relation between all machine pairs with m ’’ ij > , and  *R  means all the elements of relation R  multiplied with . The basic structure of the algorithm is as follows:

9. 2. MATHEMATICAL FORMULATION Step 1: Construct the modified fuzzy relation matrix M ’’ ij. Step 2: Set  ;  Step 3: Let R 1 be the set of all pairs of machines; Let R 2 =  Step 4: Set R 3 = R 1 -R 2 (the set of pairs that haven’t been considered yet) Step 5: Set R 2 be a subset of R 3, including all the pairs of machines to be considered for placing in the current iteration (m ’’ ij   ) Step 6: Place the machine pairs that are included in set R 2 ; Step 7: Set R 1 =R 1  R 2 ; Set R 3 =R 3 -R 2 Step 8 If R 3 is an empty set (no more machines to be placed) then stop, otherwise –Set  =  -  ; Step 9: Go to Step 4.

10. 3 EXAMPLE A to F machines. Factory floor: 2 * 3. The flow mass (in Kg./Hour) between machines: Thus: P max = Max{  j m ij,  i m ij } = Max{60,70) = 70

11. 3 EXAMPLE the matrix M ij /P max : From i A B C D E F To j A ---0.143 0 0 0 0 B 0 ---0.214 0 0 0 C 0.286 0 ---0.286 0 0 D 0.571 0 0 ---0.3570.071 E 0 00.457 0 --- 0 F 0 0 0 00.214 ---

12. 3 EXAMPLE From i A B C D E F To j A ---0.1430.086 0.0140.003 0.001 B 0.122 ---0.2140.1220.0260.008 C 0.2860.095 ---0.2860.1590.057 D 0.5710.1140.201 ---0.3570.071 E 0.1760.0210.4570.176 ---0.011 F 0.0290.0040.1460.0290.214 --- the maximum value of the path lengths between all pairs of machines is calculated. These values are summarized in the following matrix:

13. 3 EXAMPLE By adding the element min(m ’ ij,m ’ ji ) to the element max(m ’ ij,m ’ ji ) we will get the matrix M ’’ ij as following:

14. 3 EXAMPLE When the total mass flow from/to a specific machine x,  i m ’’ ix +  j m ’’ xj, is large, we should consider to place it in the most approachable position

15. 3 EXAMPLE

18. 4 COMPARISON WITH THE OPTIMAL SOLUTION The total cost (number of movements) of the final arrangement solution recommended by the fuzzy set decomposition method is 247 for the first alternative and 217 for the second one. The second alternative is also the optimal solution of this problem.

19. 5 CONCLUSION Fuzzy set based algorithms can form a firm basis for the design of plant layout. They provide a basic reasoning for the machine layout in the plant which otherwise are subject to a complex process.

20. QUESTIONS ?