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Modified Rank Order Clustering Algorithm Approach by Including Manufacturing Data
4th IFAC International Conference on Intelligent Control andConference Automationon Sciences 4th IFAC International 4th 4th IFAC IFAC International International Conference Conference on onAvailable online at www.sciencedirect.com June 1-3, 2016. Reims, France Intelligent Control and Automation Intelligent Control Control and and Automation Automation Sciences Sciences Intelligent Sciences June 1-3, 2016. Reims, France June 1-3, 2016. Reims, June 1-3, 2016. Reims, France France
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Modified Rank Order Clustering Algorithm Approach by Including Manufacturing Modified Modified Rank Rank Order Order Clustering Clustering Algorithm Algorithm Approach by by Including Including Manufacturing Manufacturing Data Approach Data Data Nagdev Amruthnath Nagdev Nagdev Amruthnath Nagdev Amruthnath Amruthnath
Tarun Gupta Tarun Tarun Gupta Tarun Gupta Gupta
IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI USA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 49009 USAUSA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: [email protected]) [email protected]) Abstract: A modified rank order clustering (MROC) method based on weight and data reorganization Abstract: A rank (MROC) based weight data has been developed to facilitate theclustering needs of real worldmethod manufacturing environment. is designed Abstract: A modified rank order clustering (MROC) method based on weight and data reorganization Abstract: A modified modified rank order order clustering (MROC) method based on on weight and and MROC data reorganization reorganization has been developed to facilitate the needs of real world manufacturing environment. MROC is to optimize the manufacturing process based on important independent variables with weights and has been developed to facilitate the needs of real world manufacturing environment. MROC is designed has been developed to facilitate the needs of real world manufacturing environment. MROC is designed designed to optimize manufacturing based on important independent with weights reorganize data that helps cells where each cellvariables would have to optimize the manufacturing process based on important independent variables with weights and to optimizethethe themachine-component manufacturing process process based onform important independent variables withapproximately weights and and reorganize the that cells each would approximately the same work load. The developed data algorithm usingform a heuristics minimizes number bottlenecks for the reorganize the machine-component data that helps form cells where each cell would have approximately reorganize the machine-component machine-component data that helps helps form cells where where each cell cell wouldofhave have approximately the same work load. The developed algorithm using a heuristics minimizes number of bottlenecks for cellular solution without human input (necessary in King {1980)), while ensuring comparable machine the the the same same work work load. load. The The developed developed algorithm algorithm using using aa heuristics heuristics minimizes minimizes number number of of bottlenecks bottlenecks for for the the cellular solution without human input (necessary in King {1980)), while ensuring comparable machine utilizations in each cell. This paper describes our proposed algorithm and a solution to the machine cell cellular cellular solution solution without without human human input input (necessary (necessary in in King King {1980)), {1980)), while while ensuring ensuring comparable comparable machine machine utilizations in each cell. This paper describes our proposed algorithm and solution to the machine cell design process for the realThis world manufacturing environment. utilizations in cell. paper describes proposed utilizations in each each cell. This paper describes our our proposed algorithm algorithm and and aaa solution solution to to the the machine machine cell cell design process for the real world manufacturing environment. design process for the real world manufacturing environment. design process for the real world manufacturing environment. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Rank order clustering, Production Flow Analysis, Data Reorganization Keywords: Rank order clustering, Production Flow Analysis, Data Reorganization Keywords: Rank order clustering, Production Flow Analysis, Data Reorganization Keywords: Rank order clustering, Production Flow Analysis, Data Reorganization 1. INTRODUCTION 1. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION Over fifty years rank-order clustering (ROC) algorithm has Over fifty years rank-order clustering (ROC) algorithm has matured with its implementation in many Over fifty years rank-order (ROC) algorithm has Over fifty years rank-order clustering clustering (ROC)domains. algorithmKing has matured with its implementation in many domains. King (1980) first introduced the notion of ROC in the domain of matured with its implementation in many domains. King matured with its implementation in many domains. King (1980) first introduced the notion of ROC in the domain manufacturing for clustering machines into machine cells (1980) first introduced the notion of ROC in the domain of (1980) first introduced the notion of ROC in the domain of of manufacturing for clustering machines into machine cells simultaneous formation of part families that need to be manufacturing for clustering machines into machine cells manufacturing for clustering machines into machine cells simultaneous of families that need to be assigned to oneformation of the machine clusters. Ideally simultaneous formation of part families that need to be simultaneous formation of part part families that each needpart to and be assigned to one of the machine clusters. Ideally each part and its part family has a unique assignment when it comes to assigned assigned to to one one of of the the machine machine clusters. clusters. Ideally Ideally each each part part and and its part family has a unique assignment when it comes assigning to one and only one cell while getting completely its to its part part family family has has aa unique unique assignment assignment when when it it comes comes to to assigning to one and only one cell while getting completely processed inside the assigned machine cell. A machine cell is assigning to one and only one cell while getting completely assigning to one and only one cell while getting completely processed inside the assigned machine cell. A machine cell is constituted of all the those machines that are part of the machine processed assigned machine cell. A cell processed inside inside the assigned machine cell. A machine machine cell is is constituted of all those machines that are part of the machine cell. However, this historic ROC algorithmic method has constituted of all those machines that are part of the machine constituted of all those machines that are part of the machine cell. this ROC algorithmic has since been applied in various domains–image processing cell. However, this historic ROC algorithmic method has cell. However, However, this historic historic ROC algorithmic method method has since been applied in various domains–image processing (Patel & Stonhem 1992), Bromley (1966) Jiang et al. (2004) since been applied in various domains–image processing since been applied in various domains–image processing (Patel & 1992), Jiang et to name few. Chandrasekheran Rajgopalan found (Patel & Stonhem 1992), Bromley (1966) Jiang et al. (2004) (Patel & aStonhem Stonhem 1992), Bromley Bromley& (1966) (1966) Jiang (1986) et al. al. (2004) (2004) to name a few. Chandrasekheran & Rajgopalan (1986) limitations with ROC presented originally in King (1980) and to found to name name aa few. few. Chandrasekheran Chandrasekheran & & Rajgopalan Rajgopalan (1986) (1986) found found limitations with ROC presented originally in King (1980) proposed a MODROC. Regardless whether ROC or limitations and limitations with with ROC ROC presented presented originally originally in in King King (1980) (1980) and and proposed a MODROC. Regardless whether ROC MODROC is used to design cellular manufacturing systems proposed or proposed aa MODROC. MODROC. Regardless Regardless whether whether ROC ROC or or MODROC is used to design cellular manufacturing systems there are several decisions that have needed human input MODROC is used to design cellular manufacturing MODROC is used to design cellular manufacturing systems systems there are several decisions that have needed human input requiring expert’sdecisions knowledge and needed thus elements of there that human there are are several several decisions that have have needed human input input requiring expert’s knowledge and thus elements subjectivity remains in the process of arriving at a feasible requiring expert’s knowledge and thus elements of requiring expert’s knowledge and thus elements of of subjectivity remains in the process of arriving at a feasible cellular solution. Some of these decisions are, desired subjectivity remains in the process of arriving at a feasible subjectivity remains in the process of arriving at a feasible cellular of decisions are, maximum cell size,Some maximum number of parts a part cellular solution. Some of these decisions are, desired cellular solution. solution. Some of these these decisions are,in desired desired maximum cell size, maximum number of parts aaa part family; and also selecting one of several solutions the maximum in part maximum cell cell size, size, maximum maximum number number of of parts parts in infrom part family; and also selecting one of several solutions from final matrix of the iterative process. As mentioned earlier, family; the family; and and also also selecting selecting one one of of several several solutions solutions from from the the final matrix of the iterative process. As mentioned earlier, ROC is one several approaches to machine cell part final earlier, final matrix matrix of of the the iterative iterative process. process. As As mentioned mentionedand earlier, ROC is one of several approaches to machine cell and part family for cellular manufacturing. Burbidge (1963, ROC one approaches to cell part ROC is isformation one of of several several approaches to machine machine cell and and part family formation for cellular manufacturing. Burbidge (1963, 1975) developed machine cell formation methodology based family formation for cellular manufacturing. Burbidge (1963, family formation for cellular manufacturing. Burbidge (1963, 1975) developed machine cell methodology based on production analysis (PFA). As interest among 1975) developed machine cell formation methodology based 1975) developedflow machine cell formation formation methodology based on production flow analysis (PFA). As interest among manufacturing systems research community grew due to on production flow analysis (PFA). As interest among on production flow analysis (PFA). As interest among manufacturing systems research community grew rapidly changing industrial environment primarily manufacturing systems research community grew due to manufacturing systems research community grew due due to to rapidly industrial environment primarily to transition from mass production to small volume large due variety rapidly changing industrial environment primarily due to rapidly changing changing industrial environment primarily due to transition mass small large variety production, more cellular to manufacturing transition from mass production to small volume large variety transition from from massofproduction production to small volume volume was largegaining variety production, more of cellular manufacturing was gaining ground. One of the simpler method for forming cells was the production, production, more more of of cellular cellular manufacturing manufacturing was was gaining gaining ground. One of the simpler method for forming cells was the result of McCauley’s Similarity Coefficient Method (SCM), ground. One of the simpler method for forming cells was ground. One of the simpler method for forming cells was the the result of McCauley’s Similarity Coefficient Method (SCM), result of McCauley’s Similarity Coefficient Method (SCM), result of McCauley’s Similarity Coefficient Method (SCM),
McCauley (1972) followed by McCormick et. al. (1976). McCauley by et. The SCM is(1972) basedfollowed on establishing similarity coefficient for McCauley (1972) followed by McCormick et. al. (1976). McCauley (1972) followed by McCormick McCormick et. al. al. (1976). (1976). The SCM is based on establishing similarity coefficient for each pair of machines derived from the same m x n (mThe SCM is based on establishing similarity coefficient The SCM is based on establishing similarity coefficient for for each pair of machines derived from the same m x n (mmachines & n-parts) machine-component matrix also used in each pair of machines derived from the same m x n (meach pair of machines derived from the same m x n (mmachines & n-parts) machine-component matrix also used ROC and MODROC methods. These similarity coefficients machines & n-parts) machine-component matrix also used in machines & n-parts) machine-component matrix also used in in ROC and MODROC methods. These similarity coefficients are arranged in m x n similarity matrix with one half of the ROC and MODROC methods. These similarity coefficients ROC and MODROC methods. These similarity coefficients are m similarity matrix half matrix be thein of the otherwith half.one Guided by the an are arranged in m x n similarity matrix with one half of the are arranged arranged inmirror mx xn nimage similarity matrix with one half of of the matrix be the mirror image of the other half. Guided an overall measure of similarity among machines in each cell matrix by an matrix be be the the mirror mirror image image of of the the other other half. half. Guided Guided by by the an overall measure of similarity among machines in each cell the clustering process continues until all machines cells converge overall overall measure measure of of similarity similarity among among machines machines in in each each cell cell the the clustering process continues all cells converge into one large cluster whichuntil almost always we not clustering process continues until all machines cells converge clustering process continues until all machines machines cellswould converge into one large cluster which almost always we would not want for obvious reasons. The solution to the clustering into one large cluster which almost always we would into one large cluster which almost always we would not not want for obvious reasons. The solution to the clustering process is identified at a carefully selected threshold value want for obvious reasons. The solution to the clustering want for obvious reasons. The solution to the clustering process is aaa carefully threshold value that varies between 0at 1 (same selected as similarity coefficient process is identified at carefully selected threshold value process is identified identified atand carefully selected threshold value that varies between 0 and 1 (same as similarity coefficient value for a pair of machines). Often the selection of suitable that varies between 0 and 1 (same as similarity coefficient that varies between 0 and 1 (same as similarity coefficient value pair of Often selection of threshold until the similarity is established value for pair of machines). Often the selection of suitable value for for aaavalue pair waits of machines). machines). Often the the matrix selection of suitable suitable threshold value waits until the similarity matrix is established or machine cluster formation process has begun. Needless to threshold threshold value value waits waits until until the the similarity similarity matrix matrix is is established established or machine cluster formation process has begun. Needless say decisions regarding cell size and number of cells is or to or machine machine cluster cluster formation formation process process has has begun. begun. Needless Needless to to say decisions regarding cell size and number of cells required to be made with experts’ input and that too without say is say decisions decisions regarding regarding cell cell size size and and number number of of cells cells is is required to be made with experts’ input and that too without any notion of cell performance. Often the only performance required to be made with experts’ input and that too required to be made with experts’ input and that too without without any notion of cell performance. Often the only performance criteria thatof usedperformance. is number ofOften intercellular that a any the performance any notion notion ofiscell cell performance. Often the only only moves performance criteria that is used is number of intercellular moves particular solution will result into; this holds good as long criteria that is used is number of intercellular moves that criteria that is used is number of intercellular moves that thatasaaa particular solution will result into; this holds good as long as product as well process information particular solution will into; this good long as particularmix solution willasresult result into;routing this holds holds good as as did longnot as product mix as well as process routing information did not change. Moreover, binary matrix (or machineproduct as as process information did product mix mix as well wellthe as standard process routing routing information did not not change. the standard binary matrix (or machinecomponent matrix) did account for neither change. Moreover, the standard binary matrix (or machinechange. Moreover, Moreover, the not standard binary matrix the (or frequency machinecomponent matrix) did not account for neither the frequency of visits to a specific machine nor the order. Gupta & component component matrix) matrix) did did not not account account for for neither neither the the frequency frequency of visits to a specific machine nor the order. Gupta Seifoddini (1990) developed a more comprehensive SCM to of & of visits visits to to aa specific specific machine machine nor nor the the order. order. Gupta Gupta & & Seifoddini (1990) developed a more comprehensive SCM to incorporate part routing sheet information and Seifoddini (1990) developed a more comprehensive SCM to Seifoddini (1990) developed a more comprehensive also SCMthe to incorporate part routing sheet information and also the projected volume incorporate part sheet information also the incorporateproduction part routing routing sheet information information toand andhelp alsoassist the projected production volume information to help assist estimating true work load on each of the machine cell to projected production volume information to help assist projected production volume information to help assist estimating true work load on each of the machine cell provide a measure of cell performance for manufacturing estimating true work load on each of the machine cell to estimating true work load on each of the machine cell to to provide aaa measure of system needs. Yet,performance the decisionfor for manufacturing forming cells provide measure of cell performance for manufacturing provide reporting measure of cell cell performance for manufacturing system Yet, the forming cells was stillreporting based onneeds. subjective about thefor threshold system reporting needs. Yet, the decision for forming cells system reporting needs. Yet,input the decision decision for formingvalue, cells was still based on subjective input about the threshold value, cell size and number of cells in the desired solution. This was was still still based based on on subjective subjective input input about about the the threshold threshold value, value, cell size and number of cells in the desired solution. leads us to infer that as far as machine cell formation is cell This cell size size and and number number of of cells cells in in the the desired desired solution. solution. This This leads us to infer that as far as machine cell formation is concerned many decisions in the process still remain for the leads us to infer that as far as machine cell formation leads us to infer that as far as machine cell formation is is concerned many decisions in the process still remain for the experts to determine after the adopted machine cell formation concerned many decisions in the process still remain for the concerned many decisions in the process still remain for the experts experts to determine after the adopted machine cell formation experts to to determine determine after after the the adopted adopted machine machine cell cell formation formation
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machines behave exactly same. In this context, the value of the product refers to cycle time of the product or monthly volume of the product likewise the value of the machine refers to cycle time of the machine, reliability of the machine and setup time of the machine. These values play an important role while creating cells. In our research we found that, in ROC method there is no consideration of real time data of either machines or part numbers such as cycle time, volume, set up time considered in creating the cells.
procedure will present a solution, typically more than one; and again the decision maker is to deploy his/her expertise to choose one that best meets the objectives. The objective behind weight-based rank order clustering (ROC) algorithm is to create workload balanced machine cells and associated part numbers. Traditional ROC algorithm forms machine cells purely based on machine-component matrix solely. Our algorithm generates machine cell solution with balanced workload identified with similar weight ratio. This task is accomplished by assigning weight to part numbers and machines and then rearranging the data based on these weights by concepts of Bond Energy algorithm. The concept of weights will be discussed in detail in later sections. ROC algorithm is performed on this reorganized data. The cells are formed when no more iterations can be conducted satisfying the stopping rule.
3. LITERATURE REVIEW There are several other approaches to clustering and machine cell formation. One commonly used approach is Similarity Coefficient Method (SCM). SCM is one of the methods used to form the machine cells in group technology applications. Compared to the other methods, SCM incorporates more flexibility into the machine-component grouping process and more easily lends itself to the computer application, Rajagopalan & Batra (1975). The new model improves the existing models based on SCM by dealing with the duplication of bottleneck machines and by employing special data storage and analysis techniques which greatly simplify the machine-component grouping process, Krishnanada & Chincholkar (2004). The duplication process in the new model is based on the number of inter-cellular moves. Duplication starts with the machine generating the largest number of inter-cellular moves and continues until no machine generates more inter-cellular moves than specified by a threshold value. By changing the threshold value, alternative solutions can be examined. The new model employs the bit-level data storage technique to reduce the storage and computational requirements of the machinecomponent grouping process.
In the next section of this paper we define rank order clustering algorithm with its pros and cons. The literature review in support of this research is presented in section III. In section IV and V we present our weight based rank order clustering algorithm and the results obtained. In section IV we present our weight and data reorganization approach with modified rank ordering clustering algorithm. An analysis with important results is included in section V. We have also performed sensitivity analysis to verify the results and analyse the robustness of our new approach that is included in section VI.
2. RANK ORDER CLUSTERING ALGORITHM Rank order clustering algorithm is also called as production flow algorithm is used to create cells to accommodate part numbers to specific machines. Although in manufacturing, machines are capable of running different part numbers, it is important to route them to create a specific flow of part numbers through assigned machines. This also improves productivity and eliminates cross line flow. Rank order clustering algorithm functions as follows
Manufacturing has always been an area where having a competitive edge in the market has a strong foot in the market. To have this competitive edge it’s important to have low product cost, on time delivery and quality of the product. To achieve this competitive edge it is important to design a reliable, lower lead time and cost effective manufacturing process. One such process is directing right products through right machines.
2.1 Algorithm Step 1: Create an n*m matrix bij (binary number for part and machine). Where, n is parts and m is machines m-j Step 2: For each row of i compute, ij *2 Step 3: Rearrange the rows in descending order based on the computed numbers n-i Step 4: For each row of j compute, ij *2 Step 5: Rearrange the columns in descending order based on the computed numbers Step 6: Repeat step 1 until there is no change is observed in step 3 and 5 Step 7: Stop
There has been various research based on rank order clustering in incorporating an algorithm within an algorithm. Some of these approaches are distance measure based approach, graphical approach, direct clustering, hierarchical clustering, data reorganization approach. All the research from the past is more concentrated towards optimizing the rank order clustering process and unassigned part or machines, reducing the complexity of the iteration process by grouping the parts and machines as the hierarchy increases, reorganizing the rows and columns after the iteration to cater the needs of respective environment and towards specific environment. Specific environment based research such as distance measure based approach which highlights more about conveyance of product between processes. Another such research is Problem decomposition and data reorganization where the data is organized by clustering the
This algorithm works well in an ideal manufacturing environment where all the products have same value and all machines run exactly the same. In real world it highly unlikely where the entire product have same weight or all 139
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data based on similar values which is again concise to transportation or conveyance environment.
4.2 Weight Assignment The weights are assigned to each part based on its total cycle time and the production volume. Since, these weights have different units, they are normalized by converting the weights into percentage. The formula for this normalization is as below.
4. MODIFIED RANK ORDER CLUSTERING ALGORITHM In our approach, we take rank order clustering (ROC) algorithm to next level by incorporating manufacturing data in the process for creation of cells. In a generic understanding, this approach can be used to create a balanced flow structure involving man, machine, method and money. In our weight based approach we provide the best optimal solution in cell creation irrespective on environment, and any number of dependent variables (also called as dependent factors) affecting the solution of the problem.
Cycle Time (sec) Volume (pcs)
10 11527
15 13720
Raw Part Weight Data 16 17 10 413 276 4152
Sum 13 1938
129 43979
Normalizing Data into Percentage Cycle Time (%) 0.093023 0.077519 0.116279 0.124031 0.131783 0.077519 0.085271 0.116279 0.077519 0.100775 Volume (%) 0.14755 0.262093 0.311956 0.009401 0.006278 0.094405 0.044523 0.047957 0.031773 0.044065
1 1
Normalized Part Weight 0.120286 0.169806 0.214117 0.066716 0.069031 0.085962 0.064897 0.082118 0.054646 0.07242
1
Part weight
Weight and data reorganization based rank clustering algorithm is an approach where the weights are assigned to either part numbers or machines or both and the data is reorganized in descending order of their weights. This data is iterated using rank order clustering algorithm and then the cells are formed. The weights assigned in our approach based on the most common factors of a manufacturing process. These factors are cycle time, monthly volume, design, availability, ranking for a part structure; reliability, setup time, utilization, manpower, inter-process distance for a machine and other generic factors. These weights are later normalized. This process is discussed in detail later section. Based on the results in section V our approach is more in line with real world scenario and can be modified based on operational environment. The algorithm for Weight and data reorganization based rank clustering is as follows.
12 6489
11 1958
15 2109
10 1397
Table 1. Normalized cycle time and normalized volume data 4.3 Equations The above data is normalized using the formula nwi` = [CTi / ∑CTi] + [Volumei / ∑Volumei] Where, i = part numbers CT is cycle time Volume is the part volume processes through that machine nwi is normalized part weight nwi = [CTi / ∑CTi] + [Volumei / ∑Volumei] nwi`
4.1 Algorithm Step 1: Develop weight factors for part and machines i w and jw Step 2: If there are more than weight factors then, convert each weight factor into percentage and then sum it. Else assign the weight to the part numbers Step 3: Create an n*m matrix bij (binary number for part and machine). Where, n is parts and m is machines Step 4: Rearrange the parts and machines in descending order based on weights m-j Step 5: For each row of i compute, ij *2 Step 6: Rearrange the rows in descending order based on the computed numbers n-i Step 7: For each row of j compute, ij *2 Step 8: Rearrange the columns in descending order based on the computed numbers Step 9: Repeat step 1 until there is no change is observed in step 3 and 5 Step 10: Stop
5. RESULTS AND ANAYSIS 5.1 Matrix solutions For the data analysis, Rank Order clustering method as well as Modified Rank Order Clustering was solved for 3 different levels. The levels are as follows M=N M>N M < N where, M is number of machines N is number of parts This was performed to demonstrate how these two procedures perform differently in the same environment; the difference is attributed to the additional data used and the modified ROC algorithm. In real world, there is high possibility that reflect the scenario and catering these needs is one of the most important focus of our research. After the final iteration, the weights are used in creation of balanced cells with equal or similar weights. Table 2 below represents results of weight distribution of two different methods applied to thirty (30) randomly generated problem data sets.
Since the data is reorganized based on the operating environment before applying rank order clustering algorithm, the cells formed from matrix obtained in the final iteration is of same or similar weight. Because of this, the bottlenecks on machines are minimized as the product flows to machines are equally divided among cells.
In table 2 above, M represents number of machines and N is number of parts for each of the thirty data sets. Cell weight ratio represents how well the cells are balanced. If the cell 140
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balance ratio is 1, then the cells are balanced equally. For example, if the cell balance ratio is 0.70, then the cell is balanced 70% and there is an imbalance of 30%. Iterations represent the number of iterations required to arrive at final solution. Cell 1, Cell 2 and Cell 3 represents their cell weights after creation of their cells. Mach Parts # M 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
N 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 8 8 8 8 8 8 8 8 8 8
10 10 10 10 10 10 10 10 10 10 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10
ROC
141
Sensitivity analysis is performed to understand the robustness of the above particular model as well as to understand the relationship between input and output of our mathematical model. Since, our data is variable sensitivity analysis would be used at 3 different levels by altering the part weight to understand the behavior of our model. The 3 different levels are as follows
MROC
1. Use standard defined weight defined above 2. Changing the volume of the parts 3. Changing the cycle time of the part The analysis set up is as follows
Cell Weight %Itirations Cell 1 Cell 2 Cell 3 Cell Weight %Itirations Cell 1 Cell 2 Cell 3 0.83 3 0.24 0.4 0.36 0.90 1 0.37 0.29 0.34 0.64 2 0.26 0.52 0.22 0.83 1 0.4 0.33 0.26 0.81 2 0.28 0.41 0.31 0.89 1 0.37 0.38 0.26 0.83 2 0.4 0.37 0.23 0.83 1 0.4 0.38 0.22 0.66 1 0.5 0.29 0.206 0.66 1 0.5 0.28 0.214 0.88 2 0.37 0.26 0.35 0.93 1 0.36 0.34 0.3 0.77 2 0.37 0.43 0.19 0.83 1 0.4 0.29 0.3 0.72 1 0.46 0.34 0.194 0.79 1 0.4 0.34 0.206 0.66 1 0.504 0.292 0.203 0.66 1 0.504 0.292 0.203 0.81 2 0.37 0.41 0.214 0.80 1 0.369 0.417 0.214 0.76 1 0.65 0.34 0.76 1 0.65 0.34 0.75 2 0.67 0.33 0.76 1 0.65 0.34 0.81 1 0.61 0.38 0.89 1 0.44 0.56 0.88 1 0.56 0.43 0.99 1 0.5 0.49 0.88 2 0.56 0.43 0.90 1 0.552 0.44 0.85 1 0.58 0.41 0.89 1 0.56 0.44 0.87 2 0.57 0.42 0.93 1 0.54 0.46 0.84 1 0.59 0.4 0.81 1 0.61 0.38 0.87 1 0.57 0.42 0.83 1 0.66 0.44 0.83 2 0.6 0.39 0.98 1 0.49 0.51 0.77 1 0.64 0.35 0.77 1 0.64 0.35 0.98 2 0.51 0.49 0.76 1 0.65 0.34 0.96 1 0.52 0.48 0.96 1 0.52 0.48 0.84 1 0.59 0.4 0.91 1 0.55 0.45 0.99 1 0.49 0.5 0.98 1 0.49 0.51 0.97 1 0.51 0.48 0.93 1 0.54 0.46 0.97 1 0.48 0.51 0.99 1 0.49 0.5 0.92 1 0.54 0.45 0.93 1 0.46 0.54 0.93 1 0.54 0.46 0.98 1 0.51 0.49 0.70 1 0.71 0.29 0.90 1 0.55 0.44
Level 1 Level 2 Level 3 CT (Sec) Volume(Pcs) Part weight CT (Sec) Volume(Pcs) Part weight CT (Sec) Volume(Pcs) Part weight 10 12 6489 0.12028547 12 600 0.05333307 6 6489 0.09702966 9 10 11527 0.16981087 10 500 0.04444422 12 11527 0.17756281 8 15 13720 0.21412307 15 12000 0.19456828 5 13720 0.17536338 7 16 413 0.06671093 16 413 0.06671093 12 413 0.05120705 6 17 276 0.06902933 17 5000 0.12273679 12 276 0.04964949 5 10 4152 0.08596404 10 3000 0.07286688 19 4152 0.12084776 4 11 1958 0.06489628 11 1700 0.06196307 8 1958 0.05326838 3 15 2109 0.08211689 15 900 0.06837169 7 2109 0.05110914 2 10 1397 0.05464227 10 100 0.0398966 15 1397 0.07402212 1 13 1938 0.07242084 13 12000 0.18681635 13 1938 0.07242084 #
Table 4. Normailized raw data that is varied for three levels
Table 5 is the initial matrix setup or data setup for Sensitivity analysis. This is used at 3 different levels in performing the analysis. In table 6 we can compare the behavior of MROC approach and the data obtained at setup 3 levels. The data obtained are positive at level 1 and level 3 but, there is a negative difference at level 2 although, the difference is only -4%. Parts
Table 2. Cell balance data collection for the 30 problems obtained from both ROC and MROC process.
5.2 Weight distribution Performing a weight distribution analysis for our data at their 3 different levels is very important as to understand its behavior at these levels. This also provides a clear view of their performance which is one of the main scope of our research. The results at 3 different levels are as follows Level
ROC
MROC
CBD
M=N
76%
81%
5%
M>N
83%
87%
4%
M
90%
91%
1%
Machine s
512 256 128 64 32 16 8 4 2 1
10 9 8 7 6 5 4 3 2 1
512 256 128 64 32 16 8 4 2 1 10 9 8 7 6 5 4 3 2 1 1 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 0 1 0 1 0 1 1 0 1 0 1 519 392 831 892 263 707 178 515 566 241
726 480 281 81 203 203 448 738 702 693
Table 3. Weight distribution of final matrix obtained using three different levels
Table 5. Initial matrix used for performing sensitivity analysis
From the above table 3, we can analyze that the load balance at their 3 difference levels. CBD is the Cell Balance difference between ROC approach and MROC approach. It is important to notice that, our new approach presented has proved to provide higher cell balance at all three levels with inclusion of real world manufacturing data in creation of cells.
From sensitivity analysis using 3 different levels, we can conclude that the MROC approach is more efficient in creation machine cells with balanced loads. Even when the weights are altered between the ranges of 5% to 25% the overall cell weight ratio for MROC approach is higher than ROC approach. Sensitivity analysis also provides enough evidence to conclude, MROC approach is more efficient and robust in changing environments.
5.3 Sensitivity analysis
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6. CONCLUSIONS
Chan, H.M. and Milner, D. A.,“direct clustering algorithm for group formation in Cellular manufacturing”, Journal of Manufacturing Systems, Vol. 1, pp.65-75, 1982.
In our research, we were able to include real time manufacturing data in creation of product flow for cellular manufacturing based on weight assignment for parts and data reorganization approach has proved not only robust and efficient but also a methodology for creation of load balanced cells while minimizing bottlenecks. In conclusion based on the results presented above MROC provided more efficient cell formation by overcoming the deficiencies of ROC approach. Level
Cells
ROC Cell Weight
CWR
Cells
10
9 7
78%
2 3
1 4
4
5
10
8 2 0.48
7
9
3 0.42
10
5
4
4
5
10
9
7
King, J.R., “Machine-component grouping in production flow analysis: an approach using a rank order clustering algorithm”, IJPR Vol. 18 Issue 2, 1980, 213-232.
0.47
McCauley, J., “Machine Grouping for efficient production”, Production Engineering, 1972, 51, 53-60.
2 0.55
8
9
McCormick, W. T., Schweitzer, P.J., and White, T. E., “Problem decomposition and data reorganization by clustering technique”, Operation Research, 20, 9931008.
0.52
3
8
7
84%
2 3 1
91%
9
3
3
Chandrasekharan, B.S., “MODROC – Modified ROC for Group Technology” IJPR, Vol 24, Issue 5, 1986, 12211233.
0.43
1
95%
2
6
0.41
6
8
1
Jiang, D., Tang, C. & Zhang, A., “Cluster Analysis for Gene Expression Data: A Survey”, IEEE Transactions on Knowledge and Data Engineering, Vol. 16, No 11, November 2004.
10 0.36
7
2
85%
6
5
6
Bromley, D. B., “Rank Order Cluster Analysis”, The British Journal of Mathematical and Statistical Psychology, 1966, Vol. 19, Part 1 105-123.
0.58
3
8
1
Patel & T. J. Stonhem, Texture Image Classification and Segmentation using RANK-order Clustering, IEEE, 1992 0-8186-2920-7/92
CWR
2 0.639
9 1
P. Krishnananda Rao, and A. M. Chincholkar, A distance measure based approach for solving group technology cell, Formation problem”, (2004)
8
7 6
MROC Cell Weight
R, Rajagopalan, J. L. Batra, “Design of cellular production system – A graph theoretic approach”, International Journal of Production Research, Vol. 13, No.6, pp.567579, 1975
87%
6 10
0.37
1
5
5
4
4
Gupta, T. and Seifoddini, “Production data based Similarity Coefficient Method for Machine Cells and Part Family Formation Heuristics”, IJPR 1990, No 7, 1247-1269
0.393
Table 6. Cell weight distribution for ROC and MROC process for three levels set for performing sensitivity analysis
REFERENCES Burbidge, J.L. (1975). “Production Flow Production Engineering, 1963, 42, 742
Analysis”,
W. T. McCornick, J. P. J. Schweitzer, and T. W. White, “Problem decomposition and data reorganization by a clustering technique”, Operations Research, Vol.20, No. 5, pp. 993-1009, 1972 C. Dimopoulos, and N. Mort, “A hierarchical clustering methodology based on genetic programming for the solution of simple cell formation problems”, International Journal of Production research, Vol. 39, No. 1, pp.1-19, 2001.
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Modified Rank Order Clustering Algorithm Approach by Including Manufacturing Modified Modified Rank Rank Order Order Clustering Clustering Algorithm Algorithm Approach by by Including Including Manufacturing Manufacturing Data Approach Data Data Nagdev Amruthnath Nagdev Nagdev Amruthnath Nagdev Amruthnath Amruthnath
Tarun Gupta Tarun Tarun Gupta Tarun Gupta Gupta
IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI USA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 49009 USAUSA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: [email protected]) IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: IEEEM Department, Western Michigan University, MI 49009 USA (e-mail: [email protected]) [email protected]) Abstract: A modified rank order clustering (MROC) method based on weight and data reorganization Abstract: A rank (MROC) based weight data has been developed to facilitate theclustering needs of real worldmethod manufacturing environment. is designed Abstract: A modified rank order clustering (MROC) method based on weight and data reorganization Abstract: A modified modified rank order order clustering (MROC) method based on on weight and and MROC data reorganization reorganization has been developed to facilitate the needs of real world manufacturing environment. MROC is to optimize the manufacturing process based on important independent variables with weights and has been developed to facilitate the needs of real world manufacturing environment. MROC is designed has been developed to facilitate the needs of real world manufacturing environment. MROC is designed designed to optimize manufacturing based on important independent with weights reorganize data that helps cells where each cellvariables would have to optimize the manufacturing process based on important independent variables with weights and to optimizethethe themachine-component manufacturing process process based onform important independent variables withapproximately weights and and reorganize the that cells each would approximately the same work load. The developed data algorithm usingform a heuristics minimizes number bottlenecks for the reorganize the machine-component data that helps form cells where each cell would have approximately reorganize the machine-component machine-component data that helps helps form cells where where each cell cell wouldofhave have approximately the same work load. The developed algorithm using a heuristics minimizes number of bottlenecks for cellular solution without human input (necessary in King {1980)), while ensuring comparable machine the the the same same work work load. load. The The developed developed algorithm algorithm using using aa heuristics heuristics minimizes minimizes number number of of bottlenecks bottlenecks for for the the cellular solution without human input (necessary in King {1980)), while ensuring comparable machine utilizations in each cell. This paper describes our proposed algorithm and a solution to the machine cell cellular cellular solution solution without without human human input input (necessary (necessary in in King King {1980)), {1980)), while while ensuring ensuring comparable comparable machine machine utilizations in each cell. This paper describes our proposed algorithm and solution to the machine cell design process for the realThis world manufacturing environment. utilizations in cell. paper describes proposed utilizations in each each cell. This paper describes our our proposed algorithm algorithm and and aaa solution solution to to the the machine machine cell cell design process for the real world manufacturing environment. design process for the real world manufacturing environment. design process for the real world manufacturing environment. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Rank order clustering, Production Flow Analysis, Data Reorganization Keywords: Rank order clustering, Production Flow Analysis, Data Reorganization Keywords: Rank order clustering, Production Flow Analysis, Data Reorganization Keywords: Rank order clustering, Production Flow Analysis, Data Reorganization 1. INTRODUCTION 1. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION Over fifty years rank-order clustering (ROC) algorithm has Over fifty years rank-order clustering (ROC) algorithm has matured with its implementation in many Over fifty years rank-order (ROC) algorithm has Over fifty years rank-order clustering clustering (ROC)domains. algorithmKing has matured with its implementation in many domains. King (1980) first introduced the notion of ROC in the domain of matured with its implementation in many domains. King matured with its implementation in many domains. King (1980) first introduced the notion of ROC in the domain manufacturing for clustering machines into machine cells (1980) first introduced the notion of ROC in the domain of (1980) first introduced the notion of ROC in the domain of of manufacturing for clustering machines into machine cells simultaneous formation of part families that need to be manufacturing for clustering machines into machine cells manufacturing for clustering machines into machine cells simultaneous of families that need to be assigned to oneformation of the machine clusters. Ideally simultaneous formation of part families that need to be simultaneous formation of part part families that each needpart to and be assigned to one of the machine clusters. Ideally each part and its part family has a unique assignment when it comes to assigned assigned to to one one of of the the machine machine clusters. clusters. Ideally Ideally each each part part and and its part family has a unique assignment when it comes assigning to one and only one cell while getting completely its to its part part family family has has aa unique unique assignment assignment when when it it comes comes to to assigning to one and only one cell while getting completely processed inside the assigned machine cell. A machine cell is assigning to one and only one cell while getting completely assigning to one and only one cell while getting completely processed inside the assigned machine cell. A machine cell is constituted of all the those machines that are part of the machine processed assigned machine cell. A cell processed inside inside the assigned machine cell. A machine machine cell is is constituted of all those machines that are part of the machine cell. However, this historic ROC algorithmic method has constituted of all those machines that are part of the machine constituted of all those machines that are part of the machine cell. this ROC algorithmic has since been applied in various domains–image processing cell. However, this historic ROC algorithmic method has cell. However, However, this historic historic ROC algorithmic method method has since been applied in various domains–image processing (Patel & Stonhem 1992), Bromley (1966) Jiang et al. (2004) since been applied in various domains–image processing since been applied in various domains–image processing (Patel & 1992), Jiang et to name few. Chandrasekheran Rajgopalan found (Patel & Stonhem 1992), Bromley (1966) Jiang et al. (2004) (Patel & aStonhem Stonhem 1992), Bromley Bromley& (1966) (1966) Jiang (1986) et al. al. (2004) (2004) to name a few. Chandrasekheran & Rajgopalan (1986) limitations with ROC presented originally in King (1980) and to found to name name aa few. few. Chandrasekheran Chandrasekheran & & Rajgopalan Rajgopalan (1986) (1986) found found limitations with ROC presented originally in King (1980) proposed a MODROC. Regardless whether ROC or limitations and limitations with with ROC ROC presented presented originally originally in in King King (1980) (1980) and and proposed a MODROC. Regardless whether ROC MODROC is used to design cellular manufacturing systems proposed or proposed aa MODROC. MODROC. Regardless Regardless whether whether ROC ROC or or MODROC is used to design cellular manufacturing systems there are several decisions that have needed human input MODROC is used to design cellular manufacturing MODROC is used to design cellular manufacturing systems systems there are several decisions that have needed human input requiring expert’sdecisions knowledge and needed thus elements of there that human there are are several several decisions that have have needed human input input requiring expert’s knowledge and thus elements subjectivity remains in the process of arriving at a feasible requiring expert’s knowledge and thus elements of requiring expert’s knowledge and thus elements of of subjectivity remains in the process of arriving at a feasible cellular solution. Some of these decisions are, desired subjectivity remains in the process of arriving at a feasible subjectivity remains in the process of arriving at a feasible cellular of decisions are, maximum cell size,Some maximum number of parts a part cellular solution. Some of these decisions are, desired cellular solution. solution. Some of these these decisions are,in desired desired maximum cell size, maximum number of parts aaa part family; and also selecting one of several solutions the maximum in part maximum cell cell size, size, maximum maximum number number of of parts parts in infrom part family; and also selecting one of several solutions from final matrix of the iterative process. As mentioned earlier, family; the family; and and also also selecting selecting one one of of several several solutions solutions from from the the final matrix of the iterative process. As mentioned earlier, ROC is one several approaches to machine cell part final earlier, final matrix matrix of of the the iterative iterative process. process. As As mentioned mentionedand earlier, ROC is one of several approaches to machine cell and part family for cellular manufacturing. Burbidge (1963, ROC one approaches to cell part ROC is isformation one of of several several approaches to machine machine cell and and part family formation for cellular manufacturing. Burbidge (1963, 1975) developed machine cell formation methodology based family formation for cellular manufacturing. Burbidge (1963, family formation for cellular manufacturing. Burbidge (1963, 1975) developed machine cell methodology based on production analysis (PFA). As interest among 1975) developed machine cell formation methodology based 1975) developedflow machine cell formation formation methodology based on production flow analysis (PFA). As interest among manufacturing systems research community grew due to on production flow analysis (PFA). As interest among on production flow analysis (PFA). As interest among manufacturing systems research community grew rapidly changing industrial environment primarily manufacturing systems research community grew due to manufacturing systems research community grew due due to to rapidly industrial environment primarily to transition from mass production to small volume large due variety rapidly changing industrial environment primarily due to rapidly changing changing industrial environment primarily due to transition mass small large variety production, more cellular to manufacturing transition from mass production to small volume large variety transition from from massofproduction production to small volume volume was largegaining variety production, more of cellular manufacturing was gaining ground. One of the simpler method for forming cells was the production, production, more more of of cellular cellular manufacturing manufacturing was was gaining gaining ground. One of the simpler method for forming cells was the result of McCauley’s Similarity Coefficient Method (SCM), ground. One of the simpler method for forming cells was ground. One of the simpler method for forming cells was the the result of McCauley’s Similarity Coefficient Method (SCM), result of McCauley’s Similarity Coefficient Method (SCM), result of McCauley’s Similarity Coefficient Method (SCM),
McCauley (1972) followed by McCormick et. al. (1976). McCauley by et. The SCM is(1972) basedfollowed on establishing similarity coefficient for McCauley (1972) followed by McCormick et. al. (1976). McCauley (1972) followed by McCormick McCormick et. al. al. (1976). (1976). The SCM is based on establishing similarity coefficient for each pair of machines derived from the same m x n (mThe SCM is based on establishing similarity coefficient The SCM is based on establishing similarity coefficient for for each pair of machines derived from the same m x n (mmachines & n-parts) machine-component matrix also used in each pair of machines derived from the same m x n (meach pair of machines derived from the same m x n (mmachines & n-parts) machine-component matrix also used ROC and MODROC methods. These similarity coefficients machines & n-parts) machine-component matrix also used in machines & n-parts) machine-component matrix also used in in ROC and MODROC methods. These similarity coefficients are arranged in m x n similarity matrix with one half of the ROC and MODROC methods. These similarity coefficients ROC and MODROC methods. These similarity coefficients are m similarity matrix half matrix be thein of the otherwith half.one Guided by the an are arranged in m x n similarity matrix with one half of the are arranged arranged inmirror mx xn nimage similarity matrix with one half of of the matrix be the mirror image of the other half. Guided an overall measure of similarity among machines in each cell matrix by an matrix be be the the mirror mirror image image of of the the other other half. half. Guided Guided by by the an overall measure of similarity among machines in each cell the clustering process continues until all machines cells converge overall overall measure measure of of similarity similarity among among machines machines in in each each cell cell the the clustering process continues all cells converge into one large cluster whichuntil almost always we not clustering process continues until all machines cells converge clustering process continues until all machines machines cellswould converge into one large cluster which almost always we would not want for obvious reasons. The solution to the clustering into one large cluster which almost always we would into one large cluster which almost always we would not not want for obvious reasons. The solution to the clustering process is identified at a carefully selected threshold value want for obvious reasons. The solution to the clustering want for obvious reasons. The solution to the clustering process is aaa carefully threshold value that varies between 0at 1 (same selected as similarity coefficient process is identified at carefully selected threshold value process is identified identified atand carefully selected threshold value that varies between 0 and 1 (same as similarity coefficient value for a pair of machines). Often the selection of suitable that varies between 0 and 1 (same as similarity coefficient that varies between 0 and 1 (same as similarity coefficient value pair of Often selection of threshold until the similarity is established value for pair of machines). Often the selection of suitable value for for aaavalue pair waits of machines). machines). Often the the matrix selection of suitable suitable threshold value waits until the similarity matrix is established or machine cluster formation process has begun. Needless to threshold threshold value value waits waits until until the the similarity similarity matrix matrix is is established established or machine cluster formation process has begun. Needless say decisions regarding cell size and number of cells is or to or machine machine cluster cluster formation formation process process has has begun. begun. Needless Needless to to say decisions regarding cell size and number of cells required to be made with experts’ input and that too without say is say decisions decisions regarding regarding cell cell size size and and number number of of cells cells is is required to be made with experts’ input and that too without any notion of cell performance. Often the only performance required to be made with experts’ input and that too required to be made with experts’ input and that too without without any notion of cell performance. Often the only performance criteria thatof usedperformance. is number ofOften intercellular that a any the performance any notion notion ofiscell cell performance. Often the only only moves performance criteria that is used is number of intercellular moves particular solution will result into; this holds good as long criteria that is used is number of intercellular moves that criteria that is used is number of intercellular moves that thatasaaa particular solution will result into; this holds good as long as product as well process information particular solution will into; this good long as particularmix solution willasresult result into;routing this holds holds good as as did longnot as product mix as well as process routing information did not change. Moreover, binary matrix (or machineproduct as as process information did product mix mix as well wellthe as standard process routing routing information did not not change. the standard binary matrix (or machinecomponent matrix) did account for neither change. Moreover, the standard binary matrix (or machinechange. Moreover, Moreover, the not standard binary matrix the (or frequency machinecomponent matrix) did not account for neither the frequency of visits to a specific machine nor the order. Gupta & component component matrix) matrix) did did not not account account for for neither neither the the frequency frequency of visits to a specific machine nor the order. Gupta Seifoddini (1990) developed a more comprehensive SCM to of & of visits visits to to aa specific specific machine machine nor nor the the order. order. Gupta Gupta & & Seifoddini (1990) developed a more comprehensive SCM to incorporate part routing sheet information and Seifoddini (1990) developed a more comprehensive SCM to Seifoddini (1990) developed a more comprehensive also SCMthe to incorporate part routing sheet information and also the projected volume incorporate part sheet information also the incorporateproduction part routing routing sheet information information toand andhelp alsoassist the projected production volume information to help assist estimating true work load on each of the machine cell to projected production volume information to help assist projected production volume information to help assist estimating true work load on each of the machine cell provide a measure of cell performance for manufacturing estimating true work load on each of the machine cell to estimating true work load on each of the machine cell to to provide aaa measure of system needs. Yet,performance the decisionfor for manufacturing forming cells provide measure of cell performance for manufacturing provide reporting measure of cell cell performance for manufacturing system Yet, the forming cells was stillreporting based onneeds. subjective about thefor threshold system reporting needs. Yet, the decision for forming cells system reporting needs. Yet,input the decision decision for formingvalue, cells was still based on subjective input about the threshold value, cell size and number of cells in the desired solution. This was was still still based based on on subjective subjective input input about about the the threshold threshold value, value, cell size and number of cells in the desired solution. leads us to infer that as far as machine cell formation is cell This cell size size and and number number of of cells cells in in the the desired desired solution. solution. This This leads us to infer that as far as machine cell formation is concerned many decisions in the process still remain for the leads us to infer that as far as machine cell formation leads us to infer that as far as machine cell formation is is concerned many decisions in the process still remain for the experts to determine after the adopted machine cell formation concerned many decisions in the process still remain for the concerned many decisions in the process still remain for the experts experts to determine after the adopted machine cell formation experts to to determine determine after after the the adopted adopted machine machine cell cell formation formation
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machines behave exactly same. In this context, the value of the product refers to cycle time of the product or monthly volume of the product likewise the value of the machine refers to cycle time of the machine, reliability of the machine and setup time of the machine. These values play an important role while creating cells. In our research we found that, in ROC method there is no consideration of real time data of either machines or part numbers such as cycle time, volume, set up time considered in creating the cells.
procedure will present a solution, typically more than one; and again the decision maker is to deploy his/her expertise to choose one that best meets the objectives. The objective behind weight-based rank order clustering (ROC) algorithm is to create workload balanced machine cells and associated part numbers. Traditional ROC algorithm forms machine cells purely based on machine-component matrix solely. Our algorithm generates machine cell solution with balanced workload identified with similar weight ratio. This task is accomplished by assigning weight to part numbers and machines and then rearranging the data based on these weights by concepts of Bond Energy algorithm. The concept of weights will be discussed in detail in later sections. ROC algorithm is performed on this reorganized data. The cells are formed when no more iterations can be conducted satisfying the stopping rule.
3. LITERATURE REVIEW There are several other approaches to clustering and machine cell formation. One commonly used approach is Similarity Coefficient Method (SCM). SCM is one of the methods used to form the machine cells in group technology applications. Compared to the other methods, SCM incorporates more flexibility into the machine-component grouping process and more easily lends itself to the computer application, Rajagopalan & Batra (1975). The new model improves the existing models based on SCM by dealing with the duplication of bottleneck machines and by employing special data storage and analysis techniques which greatly simplify the machine-component grouping process, Krishnanada & Chincholkar (2004). The duplication process in the new model is based on the number of inter-cellular moves. Duplication starts with the machine generating the largest number of inter-cellular moves and continues until no machine generates more inter-cellular moves than specified by a threshold value. By changing the threshold value, alternative solutions can be examined. The new model employs the bit-level data storage technique to reduce the storage and computational requirements of the machinecomponent grouping process.
In the next section of this paper we define rank order clustering algorithm with its pros and cons. The literature review in support of this research is presented in section III. In section IV and V we present our weight based rank order clustering algorithm and the results obtained. In section IV we present our weight and data reorganization approach with modified rank ordering clustering algorithm. An analysis with important results is included in section V. We have also performed sensitivity analysis to verify the results and analyse the robustness of our new approach that is included in section VI.
2. RANK ORDER CLUSTERING ALGORITHM Rank order clustering algorithm is also called as production flow algorithm is used to create cells to accommodate part numbers to specific machines. Although in manufacturing, machines are capable of running different part numbers, it is important to route them to create a specific flow of part numbers through assigned machines. This also improves productivity and eliminates cross line flow. Rank order clustering algorithm functions as follows
Manufacturing has always been an area where having a competitive edge in the market has a strong foot in the market. To have this competitive edge it’s important to have low product cost, on time delivery and quality of the product. To achieve this competitive edge it is important to design a reliable, lower lead time and cost effective manufacturing process. One such process is directing right products through right machines.
2.1 Algorithm Step 1: Create an n*m matrix bij (binary number for part and machine). Where, n is parts and m is machines m-j Step 2: For each row of i compute, ij *2 Step 3: Rearrange the rows in descending order based on the computed numbers n-i Step 4: For each row of j compute, ij *2 Step 5: Rearrange the columns in descending order based on the computed numbers Step 6: Repeat step 1 until there is no change is observed in step 3 and 5 Step 7: Stop
There has been various research based on rank order clustering in incorporating an algorithm within an algorithm. Some of these approaches are distance measure based approach, graphical approach, direct clustering, hierarchical clustering, data reorganization approach. All the research from the past is more concentrated towards optimizing the rank order clustering process and unassigned part or machines, reducing the complexity of the iteration process by grouping the parts and machines as the hierarchy increases, reorganizing the rows and columns after the iteration to cater the needs of respective environment and towards specific environment. Specific environment based research such as distance measure based approach which highlights more about conveyance of product between processes. Another such research is Problem decomposition and data reorganization where the data is organized by clustering the
This algorithm works well in an ideal manufacturing environment where all the products have same value and all machines run exactly the same. In real world it highly unlikely where the entire product have same weight or all 139
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data based on similar values which is again concise to transportation or conveyance environment.
4.2 Weight Assignment The weights are assigned to each part based on its total cycle time and the production volume. Since, these weights have different units, they are normalized by converting the weights into percentage. The formula for this normalization is as below.
4. MODIFIED RANK ORDER CLUSTERING ALGORITHM In our approach, we take rank order clustering (ROC) algorithm to next level by incorporating manufacturing data in the process for creation of cells. In a generic understanding, this approach can be used to create a balanced flow structure involving man, machine, method and money. In our weight based approach we provide the best optimal solution in cell creation irrespective on environment, and any number of dependent variables (also called as dependent factors) affecting the solution of the problem.
Cycle Time (sec) Volume (pcs)
10 11527
15 13720
Raw Part Weight Data 16 17 10 413 276 4152
Sum 13 1938
129 43979
Normalizing Data into Percentage Cycle Time (%) 0.093023 0.077519 0.116279 0.124031 0.131783 0.077519 0.085271 0.116279 0.077519 0.100775 Volume (%) 0.14755 0.262093 0.311956 0.009401 0.006278 0.094405 0.044523 0.047957 0.031773 0.044065
1 1
Normalized Part Weight 0.120286 0.169806 0.214117 0.066716 0.069031 0.085962 0.064897 0.082118 0.054646 0.07242
1
Part weight
Weight and data reorganization based rank clustering algorithm is an approach where the weights are assigned to either part numbers or machines or both and the data is reorganized in descending order of their weights. This data is iterated using rank order clustering algorithm and then the cells are formed. The weights assigned in our approach based on the most common factors of a manufacturing process. These factors are cycle time, monthly volume, design, availability, ranking for a part structure; reliability, setup time, utilization, manpower, inter-process distance for a machine and other generic factors. These weights are later normalized. This process is discussed in detail later section. Based on the results in section V our approach is more in line with real world scenario and can be modified based on operational environment. The algorithm for Weight and data reorganization based rank clustering is as follows.
12 6489
11 1958
15 2109
10 1397
Table 1. Normalized cycle time and normalized volume data 4.3 Equations The above data is normalized using the formula nwi` = [CTi / ∑CTi] + [Volumei / ∑Volumei] Where, i = part numbers CT is cycle time Volume is the part volume processes through that machine nwi is normalized part weight nwi = [CTi / ∑CTi] + [Volumei / ∑Volumei] nwi`
4.1 Algorithm Step 1: Develop weight factors for part and machines i w and jw Step 2: If there are more than weight factors then, convert each weight factor into percentage and then sum it. Else assign the weight to the part numbers Step 3: Create an n*m matrix bij (binary number for part and machine). Where, n is parts and m is machines Step 4: Rearrange the parts and machines in descending order based on weights m-j Step 5: For each row of i compute, ij *2 Step 6: Rearrange the rows in descending order based on the computed numbers n-i Step 7: For each row of j compute, ij *2 Step 8: Rearrange the columns in descending order based on the computed numbers Step 9: Repeat step 1 until there is no change is observed in step 3 and 5 Step 10: Stop
5. RESULTS AND ANAYSIS 5.1 Matrix solutions For the data analysis, Rank Order clustering method as well as Modified Rank Order Clustering was solved for 3 different levels. The levels are as follows M=N M>N M < N where, M is number of machines N is number of parts This was performed to demonstrate how these two procedures perform differently in the same environment; the difference is attributed to the additional data used and the modified ROC algorithm. In real world, there is high possibility that reflect the scenario and catering these needs is one of the most important focus of our research. After the final iteration, the weights are used in creation of balanced cells with equal or similar weights. Table 2 below represents results of weight distribution of two different methods applied to thirty (30) randomly generated problem data sets.
Since the data is reorganized based on the operating environment before applying rank order clustering algorithm, the cells formed from matrix obtained in the final iteration is of same or similar weight. Because of this, the bottlenecks on machines are minimized as the product flows to machines are equally divided among cells.
In table 2 above, M represents number of machines and N is number of parts for each of the thirty data sets. Cell weight ratio represents how well the cells are balanced. If the cell 140
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balance ratio is 1, then the cells are balanced equally. For example, if the cell balance ratio is 0.70, then the cell is balanced 70% and there is an imbalance of 30%. Iterations represent the number of iterations required to arrive at final solution. Cell 1, Cell 2 and Cell 3 represents their cell weights after creation of their cells. Mach Parts # M 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
N 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 8 8 8 8 8 8 8 8 8 8
10 10 10 10 10 10 10 10 10 10 8 8 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10
ROC
141
Sensitivity analysis is performed to understand the robustness of the above particular model as well as to understand the relationship between input and output of our mathematical model. Since, our data is variable sensitivity analysis would be used at 3 different levels by altering the part weight to understand the behavior of our model. The 3 different levels are as follows
MROC
1. Use standard defined weight defined above 2. Changing the volume of the parts 3. Changing the cycle time of the part The analysis set up is as follows
Cell Weight %Itirations Cell 1 Cell 2 Cell 3 Cell Weight %Itirations Cell 1 Cell 2 Cell 3 0.83 3 0.24 0.4 0.36 0.90 1 0.37 0.29 0.34 0.64 2 0.26 0.52 0.22 0.83 1 0.4 0.33 0.26 0.81 2 0.28 0.41 0.31 0.89 1 0.37 0.38 0.26 0.83 2 0.4 0.37 0.23 0.83 1 0.4 0.38 0.22 0.66 1 0.5 0.29 0.206 0.66 1 0.5 0.28 0.214 0.88 2 0.37 0.26 0.35 0.93 1 0.36 0.34 0.3 0.77 2 0.37 0.43 0.19 0.83 1 0.4 0.29 0.3 0.72 1 0.46 0.34 0.194 0.79 1 0.4 0.34 0.206 0.66 1 0.504 0.292 0.203 0.66 1 0.504 0.292 0.203 0.81 2 0.37 0.41 0.214 0.80 1 0.369 0.417 0.214 0.76 1 0.65 0.34 0.76 1 0.65 0.34 0.75 2 0.67 0.33 0.76 1 0.65 0.34 0.81 1 0.61 0.38 0.89 1 0.44 0.56 0.88 1 0.56 0.43 0.99 1 0.5 0.49 0.88 2 0.56 0.43 0.90 1 0.552 0.44 0.85 1 0.58 0.41 0.89 1 0.56 0.44 0.87 2 0.57 0.42 0.93 1 0.54 0.46 0.84 1 0.59 0.4 0.81 1 0.61 0.38 0.87 1 0.57 0.42 0.83 1 0.66 0.44 0.83 2 0.6 0.39 0.98 1 0.49 0.51 0.77 1 0.64 0.35 0.77 1 0.64 0.35 0.98 2 0.51 0.49 0.76 1 0.65 0.34 0.96 1 0.52 0.48 0.96 1 0.52 0.48 0.84 1 0.59 0.4 0.91 1 0.55 0.45 0.99 1 0.49 0.5 0.98 1 0.49 0.51 0.97 1 0.51 0.48 0.93 1 0.54 0.46 0.97 1 0.48 0.51 0.99 1 0.49 0.5 0.92 1 0.54 0.45 0.93 1 0.46 0.54 0.93 1 0.54 0.46 0.98 1 0.51 0.49 0.70 1 0.71 0.29 0.90 1 0.55 0.44
Level 1 Level 2 Level 3 CT (Sec) Volume(Pcs) Part weight CT (Sec) Volume(Pcs) Part weight CT (Sec) Volume(Pcs) Part weight 10 12 6489 0.12028547 12 600 0.05333307 6 6489 0.09702966 9 10 11527 0.16981087 10 500 0.04444422 12 11527 0.17756281 8 15 13720 0.21412307 15 12000 0.19456828 5 13720 0.17536338 7 16 413 0.06671093 16 413 0.06671093 12 413 0.05120705 6 17 276 0.06902933 17 5000 0.12273679 12 276 0.04964949 5 10 4152 0.08596404 10 3000 0.07286688 19 4152 0.12084776 4 11 1958 0.06489628 11 1700 0.06196307 8 1958 0.05326838 3 15 2109 0.08211689 15 900 0.06837169 7 2109 0.05110914 2 10 1397 0.05464227 10 100 0.0398966 15 1397 0.07402212 1 13 1938 0.07242084 13 12000 0.18681635 13 1938 0.07242084 #
Table 4. Normailized raw data that is varied for three levels
Table 5 is the initial matrix setup or data setup for Sensitivity analysis. This is used at 3 different levels in performing the analysis. In table 6 we can compare the behavior of MROC approach and the data obtained at setup 3 levels. The data obtained are positive at level 1 and level 3 but, there is a negative difference at level 2 although, the difference is only -4%. Parts
Table 2. Cell balance data collection for the 30 problems obtained from both ROC and MROC process.
5.2 Weight distribution Performing a weight distribution analysis for our data at their 3 different levels is very important as to understand its behavior at these levels. This also provides a clear view of their performance which is one of the main scope of our research. The results at 3 different levels are as follows Level
ROC
MROC
CBD
M=N
76%
81%
5%
M>N
83%
87%
4%
M
90%
91%
1%
Machine s
512 256 128 64 32 16 8 4 2 1
10 9 8 7 6 5 4 3 2 1
512 256 128 64 32 16 8 4 2 1 10 9 8 7 6 5 4 3 2 1 1 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 0 1 0 1 0 1 1 0 1 0 1 519 392 831 892 263 707 178 515 566 241
726 480 281 81 203 203 448 738 702 693
Table 3. Weight distribution of final matrix obtained using three different levels
Table 5. Initial matrix used for performing sensitivity analysis
From the above table 3, we can analyze that the load balance at their 3 difference levels. CBD is the Cell Balance difference between ROC approach and MROC approach. It is important to notice that, our new approach presented has proved to provide higher cell balance at all three levels with inclusion of real world manufacturing data in creation of cells.
From sensitivity analysis using 3 different levels, we can conclude that the MROC approach is more efficient in creation machine cells with balanced loads. Even when the weights are altered between the ranges of 5% to 25% the overall cell weight ratio for MROC approach is higher than ROC approach. Sensitivity analysis also provides enough evidence to conclude, MROC approach is more efficient and robust in changing environments.
5.3 Sensitivity analysis
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6. CONCLUSIONS
Chan, H.M. and Milner, D. A.,“direct clustering algorithm for group formation in Cellular manufacturing”, Journal of Manufacturing Systems, Vol. 1, pp.65-75, 1982.
In our research, we were able to include real time manufacturing data in creation of product flow for cellular manufacturing based on weight assignment for parts and data reorganization approach has proved not only robust and efficient but also a methodology for creation of load balanced cells while minimizing bottlenecks. In conclusion based on the results presented above MROC provided more efficient cell formation by overcoming the deficiencies of ROC approach. Level
Cells
ROC Cell Weight
CWR
Cells
10
9 7
78%
2 3
1 4
4
5
10
8 2 0.48
7
9
3 0.42
10
5
4
4
5
10
9
7
King, J.R., “Machine-component grouping in production flow analysis: an approach using a rank order clustering algorithm”, IJPR Vol. 18 Issue 2, 1980, 213-232.
0.47
McCauley, J., “Machine Grouping for efficient production”, Production Engineering, 1972, 51, 53-60.
2 0.55
8
9
McCormick, W. T., Schweitzer, P.J., and White, T. E., “Problem decomposition and data reorganization by clustering technique”, Operation Research, 20, 9931008.
0.52
3
8
7
84%
2 3 1
91%
9
3
3
Chandrasekharan, B.S., “MODROC – Modified ROC for Group Technology” IJPR, Vol 24, Issue 5, 1986, 12211233.
0.43
1
95%
2
6
0.41
6
8
1
Jiang, D., Tang, C. & Zhang, A., “Cluster Analysis for Gene Expression Data: A Survey”, IEEE Transactions on Knowledge and Data Engineering, Vol. 16, No 11, November 2004.
10 0.36
7
2
85%
6
5
6
Bromley, D. B., “Rank Order Cluster Analysis”, The British Journal of Mathematical and Statistical Psychology, 1966, Vol. 19, Part 1 105-123.
0.58
3
8
1
Patel & T. J. Stonhem, Texture Image Classification and Segmentation using RANK-order Clustering, IEEE, 1992 0-8186-2920-7/92
CWR
2 0.639
9 1
P. Krishnananda Rao, and A. M. Chincholkar, A distance measure based approach for solving group technology cell, Formation problem”, (2004)
8
7 6
MROC Cell Weight
R, Rajagopalan, J. L. Batra, “Design of cellular production system – A graph theoretic approach”, International Journal of Production Research, Vol. 13, No.6, pp.567579, 1975
87%
6 10
0.37
1
5
5
4
4
Gupta, T. and Seifoddini, “Production data based Similarity Coefficient Method for Machine Cells and Part Family Formation Heuristics”, IJPR 1990, No 7, 1247-1269
0.393
Table 6. Cell weight distribution for ROC and MROC process for three levels set for performing sensitivity analysis
REFERENCES Burbidge, J.L. (1975). “Production Flow Production Engineering, 1963, 42, 742
Analysis”,
W. T. McCornick, J. P. J. Schweitzer, and T. W. White, “Problem decomposition and data reorganization by a clustering technique”, Operations Research, Vol.20, No. 5, pp. 993-1009, 1972 C. Dimopoulos, and N. Mort, “A hierarchical clustering methodology based on genetic programming for the solution of simple cell formation problems”, International Journal of Production research, Vol. 39, No. 1, pp.1-19, 2001.
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