ࡱ> oqn]fG ^bjbjَ Xy]86 wf (vvvvvvv$zyn{v-vW WWW@vvWWU] iA v~ d%Ou A NONLINEAR MATHEMATICAL PROGRAMMING MODEL TO CELL FORMATION IN GROUP TECHNOLOGY Bo~enko Bili Ivica Ve~a University of Split, Faculty of electrical engineering, mechanical engineering and naval architecture, R. Boakovia bb, 21000 Split Keywords: group technology, cellular manufacturing, nonlinear mathematical programming ABSTRACT This paper deals with a model based on nonlinear mathematical programming to solve the machine component grouping problem for the design cells in a manufacturing system. An algorithm has been proposed to solve this mathematical model. The results obtained with this algorithm have been tested and compared with the some well-known algorithms. 1. INTRODUCTION New technologies are rapidly developing in todays competitive manufacturing environment and, consequently, customers preferences constantly fluctuate. Hence, managers seek new production approaches having more flexibility and productivity. One such approach is the group technology (GT). GT is a manufacturing philosophy that identifies and exploits the similarity of parts and processes in design and manufacturing. It is a philosophy with broad applicability, potentially affecting all areas of a manufacturing organization. One specific application of GT is cellular manufacturing. The first step in the preliminary stage of cellular manufacturing system design is cell formation, generally known as a machine-part cell formation or a machine-component grouping problem. Cellular manufacturing is a system in which a large number of common parts are grouped together and produced in a cell consisting of all the machines that are needed to produce that group. The cell is a group of processes designed to make a family of parts in a flexible way. One-piece movement of parts occurs within cells. Small-lot movement of parts occurs between cells. The workers in the cells are multiprocessor: they can run more than one process, and they can run different kinds of processes. The group technology approach is a philosophy that exploits the proximity among the attributes of given objects. Cellular manufacturing is an application of group technology in manufacturing. Cellular manufacturing involves processing a collection of similar parts (part families) on dedicated cluster of machines or manufacturing processes (cells). 2. DESIGN OF CELLULAR MANUFACTURING SYSTEMS 2.1 Problem formulation The cell formation problem in cellular manufacturing systems is the decomposition of the manufacturing systems into cells. Part families are identified such that they are fully processed within a machine group. The processing requirements of the parts are commonly represented by a binary (zero-one) matrix known as a part-machine matrix  EMBED Equation.3 : EMBED Equation.3 where p - parts, i.e. rows ( EMBED Equation.3 ), m - machines, i.e. columns ( EMBED Equation.3 ). The cells are formed to capture the inherent advantages of GT such as reduced setup times, reduced in-process inventories, improved product quality, shorter lead time, reduced tool requirements, improved productivity, better overall control of operations, etc. The common disadvantages are lower machine and labor utilization and higher investment due to duplication of machine and tools (Singh, 1993). The problem of cell design is a very complex exercise with wide ranging implications. Normally, cell design is understood as the problem of identifying a set of part types that are suitable for manufacture on a group of machines. However, there are a number of other strategic level issues such as level of machine flexibility, cell layout, type of material handling equipment, types and number of tools and fixtures, etc. that should be considered as part of the cell design problem. Further, any meaningful cell design must be compatible with the tactical/operational goals such as high production rate, low WIP, low queue length at each work station, high machine utilization, etc. Figure 1a presents part-machine matrix for a system that consists of four parts and four machines. Row and column permutation of matrix yields a block diagonal matrix A that consists of two diagonal blocks, i.e. cells. This example illustrates a perfect decomposition. Figure 1b presents an imperfect decomposition, because parts 2 and 3 require processing by both the machine groups. Also, not all the machines are required for processing by every part assigned to cells 1 and 2. The objective of decomposition of the manufacturing systems into cells is: 1. to have a minimum voids (i.e. minimum number of zeros inside the cells) 2. to have a minimum exceptional elements (i.e. minimum number of ones outside the cells).  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 incidence matrixdecompositioned matrixincidence matrixdecompositioned matrixa) Perfect decompositionb) Imperfect decompositionFig. 1. Example of perfect and imperfect decomposition Numerous research papers have appeared in literature for cell formation. These methods are based on the following approaches (Singh, 1993): classifications and coding systems, machine-component group analysis, similarity coefficients, mathematical programming, heuristics, and artificial intelligence methods (knowledge-based systems, neural networks, genetic algorithms, fuzzy clustering) 2.2 Goodness of solution Solution of practical problems without adequate generalization and theoretical formulations is useful only in specific situations. Such results cannot usually be applied to another problem. Group technology has suffered from this malady for a long time. When researchers and practitioners claim success with a certain method it is necessary to evaluate the result on an absolute quantitative scale. The practical use of such a measure is that subjective individual claims can be compared objectively. In the case of block diagonalization of zero-one matrices such a quantitative measure can be used as an objective function to be maximized. Thus a need for a function or a quantitative criterion for comparing different block diagonal forms cannot be emphasized. The criteria available in the published literature are grouping efficiency ( (Chandrasekharan and Rajagopalan, 1986), grouping efficacy (ef (Kumar and Chandrasekharan, 1990.), and grouping measure (g (Adil et al.1997.). Grouping efficiency:  EMBED Equation.3  (1)  EMBED Equation.3   EMBED Equation.3  In the (1) (1 is the ratio of number of 1s to the total number of elements is the diagonal blocks, (2 the ratio of zeros to the total number of elements in the off-diagonal blocks. While (1 represents within cell utilization, (2 represents intercellular movements. Grouping efficacy  EMBED Equation.3  can be calculated as follows:  EMBED Equation.3  (2)  EMBED Equation.3   EMBED Equation.3  Grouping measure  EMBED Equation.3  can be calculated as follows:  EMBED Equation.3  (3) Grouping measure has relative great value if solution has fewer voids and fewer exceptional elements. c - cells, i.e. diagonal blocks ( EMBED Equation.3 ), e - number of ones in part-machine matrix, ed - number of ones in the cells formed (diagonal blocks), eo - number of exceptional elements, i.e. number of ones in the off-diagonal blocks, ev - number of voids (zeros) in the diagonal blocks, Mc - number of machines assigned to cell c, Pc - number of parts allocated to cell c, w - weighting factor ( EMBED Equation.3 ) that fixes the relative importance between voids and intercellular movements. 3. A NONLINEAR MATHEMATICAL PROGRAMMING MODEL TO CELL FORMATION This section presents nonlinear mathematical programming model for cell formation developed by Adil et. al. (1997) The model will select the appropriate number of cells in production system. The proposed model is based on the objective of minimizing the weighted sum of voids and exceptional elements in the matrix  EMBED Equation.3 :  EMBED Equation.3  (4) Constraints:  EMBED Equation.3  (5)  EMBED Equation.3  (6)  EMBED Equation.3 . (7) Parameters: w - weighting factor of exceptional elements  EMBED Equation.3 , (EMBED Equation.3) = weighting factor of voids. Variables: EMBED Equation.3 EMBED Equation.3 The first term of the objective function (4) captures the contribution of the exceptional elements and the second term represents the contribution of voids. Constraints (5) ensure that each part is allocated to only one cell. Constraints (6) guarantee that each machine is assigned to a cell. Constraints (7) guarantee that variables x and y are binary variables. The nonlinearity of the model arises from the product terms of the variables xpc i ymc. Changing weights for voids end exceptional elements gives the designer the flexibility of forming large loose cells (more voids but fewer exceptional elements) or small tight cells (fewer voids and more exceptional elements). The problem of cell formation, by the application of the described model, can be solved by an algorithm, instead by a group of equations. The model becomes linear if one of the variable sets has known values. First, one set of variables (for example ymc) is fixed and values of other set of variables (xpc) are obtained. Then using the values of xpc so obtained, ymc are obtained, and so on until a convergence of objective function is reached. So, solution is to be found by iterative scheme. Determination of values xpc is referred to assignment of machines to cells, and the determination of values ymc is referred to allocation of parts to the cells. For example, set of the ymc variables are fixed at value Ymc  EMBED Equation.3 . The result is a allocation submodel: EMBED Equation.3 (8) where  EMBED Equation.3  (9) Variable xpc can be solved for each part p (EMBED Equation.3) independently. Then, set EMBED Equation.3 such that:  EMBED Equation.3  (10) The remaining variables EMBED Equation.3 ( EMBED Equation.3 ). Explanation: Assume that machine assignment to the cell is given. In that case variable ymc is known. Let the number of machines assigned in the cell c be denoted NMc. For every part p (row of matrix) one should calculate the objective function contribution for that part. If part p is allocated to cell c the contribution is Bpc. Let NMp denotes the number of machines required for processing part p (i.e. total number of ones in the row p). The number of ones in the intersection of row p with all machine columns assigned to cell c is NPpc. This allocation will give the number of voids and the number of exceptional elements in the cell c for the part p:  EMBED Equation.3  (11)  EMBED Equation.3  (12) The objective function value contribution for part p allocated to cell c:  EMBED Equation.3  (13) Procedure described for the part p should be repeat for every station c. The optimal allocation of the part p is to a cell c* for which the contribution Bpc is minimum. Then, set of the xpc variables are fixed at value Xpc  EMBED Equation.3 . The result is a assignment submodel:  EMBED Equation.3  (15) where  EMBED Equation.3  (16) Variable ymc can be solved for each machine m (EMBED Equation.3) independently. Then, set EMBED Equation.3 such that:  EMBED Equation.3  (17) The remaining variables EMBED Equation.3 ( EMBED Equation.3 ). Further procedure should be carried out in a similar way as described for the allocation procedure. Assume that part allocation to the cell is given. In that case variable xpc is known. Let the number of parts allocated in the cell c be denoted NPc. For every machine m (column of matrix) one should calculate the objective function contribution for that machine. If machine m is assigned to cell c the contribution is Dmc. Let NPm denotes the number of parts, which are processing on the machine m (i.e. total number of ones in column m). The number of ones in the intersection of column m (i.e. machines m which is assigned to cell c) with all parts p allocated to cell c is NMmc. This assignment will give the number of voids and the number of exceptional elements in the cell c for the machine m:  EMBED Equation.3  (18)  EMBED Equation.3  (19) The objective function value contribution for machine m assigned to cell c:  EMBED Equation.3  (20) Procedure described for the machine m should be repeat for every station p. The optimal assignment of the machine m is to cell c* for which the contribution Dmc is minimum. Described procedure (the assignment and allocation submodel) is solved iteratively until convergence is achieved. Figures 2-5 present detailed algorithm steps.  EMBED CorelDRAW.Graphic.9  Fig. 2. Basic structure of the algorithm EMBED CorelDRAW.Graphic.9  Fig. 3. Initial assignment of machines in the cells  EMBED CorelDRAW.Graphic.9  EMBED CorelDRAW.Graphic.9 Fig. 4. Part allocationFig. 5. Machine assignment4. APPLICATION Example 1 (S. K. Mukhopadhyay and A. Gopalakrishnan, 1995): Part-machine matrix: 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 Number of parts, P: 10 Number of machines, M: 10 Number of '1's in matrix, e: 24 Upper limit on the number of cells to be formed, C: 11 wCedeoev(gr .0 0 0 24 0 .000 .1 6 17 7 0 .708 .2 6 17 7 0 .708 .3 6 17 7 0 .708 .4 6 17 7 0 .708 .5 5 21 3 4 .715 .6 4 22 2 6 .702 .7 4 23 1 8 .700 .8 4 23 1 8 .700 .9 4 23 1 8 .700 1.0 3 24 0 10 .706 5 2 3 4 8 6 9 7 10 1 ------------------------------------ 3 | 1 | 0 0 0 0 1 0 0 0 0 | |---+------------ | 2 | 0 | 0 1 1 1 | 0 0 0 0 0 | 7 | 0 | 0 1 0 1 | 0 0 0 0 0 | 9 | 0 | 1 1 1 1 | 0 0 0 0 0 | 10 | 0 | 1 1 1 0 | 0 0 0 0 0 | | -------------+------- | 8 | 0 0 0 0 0 | 1 1 | 0 0 0 | | -------+------- | 5 | 0 0 0 0 0 0 0 | 1 1 | 0 | 6 | 0 0 0 0 0 0 0 | 1 1 | 1 | | -------+---| 1 | 0 0 0 0 0 0 0 0 1 | 1 | 4 | 0 0 0 0 0 0 0 0 0 | 1 | ------------------------------------ 2 7 9 10 3 8 1 4 5 6 ---------------------------------- 2 | 0 0 1 1 | 0 0 0 0 0 0 | 3 | 1 1 1 1 | 0 0 0 0 0 0 | 4 | 1 0 1 1 | 0 0 0 0 0 0 | 8 | 1 1 1 0 | 0 0 0 0 0 0 | |------------|------- | 5 | 0 0 0 0 | 1 0 | 0 0 0 0 | 6 | 0 0 0 0 | 1 1 | 0 0 0 0 | 9 | 0 0 0 0 | 0 1 | 0 0 0 0 | | ------|------------| 1 | 0 0 0 0 0 0 | 1 1 0 1 | 7 | 0 0 0 0 0 0 | 0 0 1 1 | 10 | 0 0 0 0 0 0 | 1 0 1 1 | ----------------------------------The best solution in respect to grouping measure (w = .5) ((gr = 0.715, (ef = 0.75, ( = 0.90)The best solution which was given by Mukhopadhyay and Gopalakrishnan and which is provided by the shown algorithm when w = 1.0 ((gr = 0.706, (ef = 0.706)Example 2 (M. P. Chandrasekharan and R. Rajagopalan, 1986 - II): Part-machine matrix: 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 Number of parts, P: 20 Number of machines, M: 35 Number of '1's in matrix, e: 135 Upper limit on the number of cells to be formed, C: 36 The best solution in respect to grouping measure (w = 0.7): 8 14 16 19 22 26 34 1 3 5 15 17 20 23 25 29 2 7 10 12 13 18 24 27 31 4 6 9 11 21 28 30 32 33 35 -------------------------------------------------------------------------------------------------------------- 5 | 1 1 1 0 0 0 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 6 | 1 1 1 1 1 1 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 9 | 1 1 0 1 1 1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 10 | 1 1 1 1 1 1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 20 | 1 1 0 1 0 1 0 | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | |---------------------+--------------------------- | 1 | 0 0 0 0 0 0 0 | 1 1 0 0 0 1 1 1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 3 | 0 0 0 0 0 0 0 | 1 1 1 1 1 0 0 0 1 | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 | 7 | 0 0 0 0 0 0 0 | 1 1 1 1 1 1 1 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 8 | 0 0 0 0 0 0 0 | 1 1 1 1 1 1 1 1 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 17 | 0 0 0 0 0 0 0 | 1 1 1 1 1 0 1 1 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | ----------------------------+---------------------------- | 2 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 1 1 1 1 | 0 0 0 0 0 0 0 0 0 0 | 4 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 0 1 1 0 1 1 0 | 0 0 0 0 0 0 0 0 0 0 | 13 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 0 1 1 0 1 0 0 | 0 0 0 0 0 0 0 0 0 0 | 14 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 1 1 1 1 | 0 0 0 0 0 0 0 0 0 0 | 18 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 1 1 1 1 1 0 1 | 0 0 0 0 0 0 0 0 0 0 | | ----------------------------+------------------------------| 11 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 1 1 1 0 1 | 12 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 0 0 0 1 0 | 15 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 1 1 0 0 0 | 16 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 1 1 1 0 0 | 19 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | 1 0 1 1 1 1 1 1 0 0 | -------------------------------------------------------------------------------------------------------------- (eo = 2, ev = 42, ed = 133, (gr = 0.745, (ef = 0.751) 5. CONCLUSION This paper deals with a nonlinear mathematical programming model, which is intended to solve the machine component grouping problem for the design cells in a manufacturing system. For the solution of the mathematical problem there has been proposed the algorithm for which a programme in the DELPHI programme language was designed.The algorithm identifies natural groupings present in the part-machine matrix. The results obtained with this algorithm have been tested and compared with the some well-known algorithms. The algorithm was tested on some examples available in literature. In most cases, the obtained results were the same or better in comparison with the results obtained by other methods or models. Only in few cases the results obtained by this algorithm were worse. Like with other methods and models the quality of solution becomes poorer as the problem being solved becomes bigger. Weight w has the biggest influence upon the quality of the solution obtained by this algorithm while initial solution, i.e. initial machine assignment has less influence. Changing weights w ( EMBED Equation.3 ) for voids end exceptional elements, gives the designer the flexibility of forming large loose cells (more voids but fewer exceptional elements) or small tight cells (fewer voids and more exceptional elements). The best results were obtained when weight factor was in the range 0.5-0.7. Counting time is insignificant. The proposed model also identifies part/machines that, if not assigned to a cell, can enhance the block diagonalization. These elements are known as external part/machine elements. 6. LITERATURE 1. N. Singh: Design of cellular manufacturing systems: An invited review, European Journal of Operational Research, 1993, 69, pp. 284-291. 2. M. P. Chandrasekharan and R. Rajagopalan: An ideal seed non-hierarchical clustering algorithm for cellular manufacturing, International Journal of Production Research, 1986, 24 (2), pp. 451-464. 3. G. K. Adil, D. Rajamani and D. Strong: Assignment allocation and simulated annealing algorithms for cell formation, IIE Transactions, 1997, 29, pp. 53-67. 4. C. Suresh Kumar and M. P. Chandrasekharan: Grouping efficacy: a quantitative criterion of goodness of block diagonal forms of binary matrices in group technology, International Journal of Production Research, 1990, 28 (2), pp. 233-243. 5. S. K. Mukhopadhyay and A. Gopalakrishnan: A vector analitic (VECAN) method for solving the machine-part grouping problem in GT, International Journal of Production Research, 1995, 33 (3), pp. 795-818. 6. M. P. Chandrasekharan and R. Rajagopalan: MODROC: an extension of rank order clustering for group technology, International Journal of Production Research, 1986, 24 (5), pp. 1221-1233.  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QALISTNobj flgs bboxeqobbx eqeq1usdnLISTlgoblodazz0HLPTXvz d..x%GX2DV LIST\trfltrfdPPP. &??uALISTNobj flgs bbox;(obbx ;(;0(usdnLISTlgoblodazz0HLPTXvz d..x%0FDV LIST\trfltrfdPPP. &?D?r|!ALISTNobj flgs bboxDL  obbx DL  DL 0 usdnLISTlgoblodazz0HLPTXvz d..x%H2bDV LIST\trfltrfdPPP. &?D?Ț&ALISTNobj flgs bbox@p obbx @p 0@p usdnLISTlgoblodazz0HLPTXvz d..x%(B,aDV LIST\trfltrfdPPP. &?D?(,ALISTNobj flgs bboxCްobbx CްC0ްusdnLISTlgoblodazz0HLPTXvz d..x%xCmDV LIST\trfltrfdPPP &?D?C1ALISTNobj flgs bboxʞ obbx ʞ ʞ0 usdnLISTlgoblodazz0HLPTXvz d..x%B;DV LIST\trfltrfdPPP. &?D?2ALISTNobj flgs bboxcEobbx cEcE0usdnLISTlgoblodazz0HLPTXvz d..x%=7?DV LIST\trfltrfdPPP. &?D?,s4ALIST`obj flgs bbox1huobbx 1hu1hcuusdnLISTlgobloda0HLPTX d..SV h6 S(7O6DD DDD1V LIST\trfltrfdPPPh &?8A? LISTjobj flgs bboxQpobbx QpMlusdnLISTlgobloda0HLPTX d..SV V b8b877DDD DDDD V LIST\trfltrfdPPP. &?D? LIST&obj spndflgs bbox*ˡobbx *ˡ*ˡusdnLIST$lgobloda||8X\`dhltx| d..>=xdftil0??LIST\trfltrfdPPP ??txsm??STOPLIST6obj spndflgs bbox{Ȼobbx {Ȼ{ȻusdnLIST$lgobloda||8X\`dhltx| d..>x<aftil0??LIST\trfltrfdPPPq ??txsm??xResultsLISTLobj flgs bbox5[P`@+obbx 5[P`@+MC8gX$usdnLISTlgoblodaxx0HLPTX`x(d..@SV <DDLIST\trfltrfdPPPq Fmܙ?Fmܙ㿼̰Fmܙ?Fmܙ?0ORLISTobj spndflgs bbox& nobbx & n& nusdnLIST$lgobloda||8X\`dhltx| d..>X<H_ftil0??LIST\trfltrfdPPPN ?P?`@txsm?P?`@X `FP(i-1)=FP(i) FM(i-1)=FM(i)LISTobj spndflgs bbox&obbx &&usdnLIST$lgobloda||8X\`dhltx| d..>7\ftil0??LIST\trfltrfdPPPN ?~?2o&JH?Д+ wALISTBobj flgsbbox>.Dnobbx >.DnͥiusdnLISTlgoblodann,@DHLPnd..SV 3٩ukDLIST\trfltrfdPPP & lP?S%. VB#JH?IALISTBobj flgsbbox>obbx >usdnLISTlgoblodann,@DHLPnd..SV <Q _DLIST\trfltrfdPPPq & lP?c!ʹVB#JH?(ANvALIST,grp flgsbboxeVbobbx eVb9W ^usdnLISTlgobloda00 (,0d.@?LIST\trfltrfdPPPX?#9 A?@!@LISTBobj flgsbbox~YVbobbx ~YVbW ^usdnLISTlgoblodann,@DHLPnd..SV p<p+`Zftil0??LIST\trfltrfdPPPq ? ?txsm\? ?/.STEP 6: Evaluation and comparation of resultsLISTobj spndflgs bbox;;ۨ>obbx ;;ۨ>;;ۨ>usdnLIST$lgobloda||8X\`dhltx| d..> ;Xftil0??LIST\trfltrfdPPPq ?(?HxAtxsm?(?HxA STEP 5: Machine assignmentLISTobj spnd@flgs bboxMG @ obbx MG @ MG @ usdnLIST$lgobloda||8X\`dhltx| d..>0:ftil0??LIST\trfltrfdPPPq ??xo$Atxsm@??xo$ASTEP 4: Part allocation i=i+1LISTobj spnd flgs bboxz y;5 obbx z y;5 z y;5 usdnLIST$lgobloda||8X\`dhltx| d..>( ftil0??LIST\trfltrfdPPPq ?[?)Atxsm2 ?[?)A"$STEP 3: Initial assignment C=M+1, LIST<obj flgs bbox>,obbx >,\ 16usdnLISTlgoblodahh,@DHLPhd..SV P8LIST\trfltrfdPPPq j1R?{ A 6?@LIST<obj flgs bbox V-$obbx V-$8)usdnLISTlgoblodahh,@DHLPhd..SV H7LIST\trfltrfdPPPq j1R?{ L~? WALIST<obj flgs bbox 1"$obbx  1"$ 'BusdnLISTlgoblodahh,@DHLPhd..SV @6LIST\trfltrfdPPPq j1R?{ < Kk??h0`ftil0??LIST\trfltrfdPPPh ??;0Atxsm0 ??;0Ah. :`````````@````@`` `STEP 2: Initialization i=0 IN=999999 FP(i)=- IN FM(i)=- INLIST<obj flgs bboxH-8obbx H-8 *1VusdnLISTlgoblodahh,@DHLPhd..SV 3JdLIST\trfltrfdPPPh /?|܍1ʺftil0??LIST\trfltrfdPPPh ?\f?1Atxsm ?\f?1A܍  w = 0,...,1LIST<obj flgs bboxw oe'Gobbx w oe'G YQjEBusdnLISTlgoblodahh,@DHLPhd..SV h! SLIST\trfltrfdPPPN XU?!L"]?Tw 2ALISTfobj spnd flgs bbox5k!Ϯobbx 5k!Ϯ5k!Ϯusdn^LIST$lgobloda||8X\`dhltx| d..>܍x/ɺftil0??LIST\trfltrfdPPP ?m?z@3Atxsm ?m?z@3A܍ STEP 1: a(p,m)LISTfobj flgs bbox/L{|obbx /L{|'Q\usdnLISTlgobloda,@DHLPd..SV - ) DDDH DDDHDLIST\trfltrfdPPP& 1?p`S@` LISTfobj flgs bbox1;Cobbx 1;C) ausdnZLISTlgobloda,@DHLPd..SV  ) DDDH DDDHDLIST\trfltrfdPPP& 1?p`S9h)?@m3ALISTBobj flgs bboxkv2SovQobbx kv2SovQt2SwQusdnLISTlgoblodann,@DHLPnd..p$gDLIST\trfltrfdPPP &?L&?ԸLISTBobj flgs bboxv)v'obbx v)v't)w'usdnXLISTlgoblodann,@DHLPnd..+gDLIST\trfltrfdPPP &?'?'4ALISTHobj flgs bboxO1_tobbx O1_tmdqousdnLISTlgoblodatt0HLPTXpt d..*OXy LIST\trfltrfdPPPN ?8NĢ@h?0<LISTHobj flgs bboxs.'Dobbx s.'DU3 1?usdnWLISTlgoblodatt0HLPTXpt d..8)OXy LIST\trfltrfdPPPN ?8Nd@h?4r.5ALISTobj flgs bbox3q)Ifobbx 3q)If3q)IfusdntLISTJlgobloda4PTX\` d.'--0D.0D.- DDDLwy ftil0??LIST\trfltrfdPPP ??0Q`ALISTobj flgs bbox.obbx ..usdnLISTJlgobloda4PTX\` d.(g"|g-&R-&R"|g"| DDDLwy ftil0??LIST\trfltrfdPPP ?=?0Q`%ALIST,obj spndflgs bbox xmobbx xm xmusdn,LIST$lgobloda||8X\`dhltx| d..>P#Xɺy ftil0??LIST\trfltrfdPPP ?G? 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