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Two sequence-pattern, matching-based, flow analysismethods for multi-flowlines layout designYING-CHIN HO a , CHING-EN C. LEE b & COLIN L. MOODIE aa School of Industrial Engineering, Grissom Hall, Purdue University , West Lafayette, IN,47907, USAb Department of Industrial Engineering, National Chiao-Tung University , Hsin-Chu, Taiwan,Republic of China.Published online: 07 May 2007.

To cite this article: YING-CHIN HO , CHING-EN C. LEE & COLIN L. MOODIE (1993) Two sequence-pattern, matching-based,flow analysis methods for multi-flowlines layout design, International Journal of Production Research, 31:7, 1557-1578, DOI:10.1080/00207549308956809

To link to this article: http://dx.doi.org/10.1080/00207549308956809

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INT. 1. PROD. RES., 1993, VOL. 31, No.7, 1557-1578

Two sequence-pattern, matching-based, flow analysis methods formulti-flowlines layout design

YING-CHIN HOt, CHING-EN C. LEE"and COLIN L. MOODIEt§

In optimizing the layout design of a multi-product assembly environment, theanalysis of tbe material flow is a vital ingredient. In a multi-product productionenvironment, products are usually grouped together into families. Products of thesame family group usually involve similar operations; however, it is common tosee some difference in the operations and the operational sequences among theseproducts. The design technique for the layout of multi-product flowlines shouldbe able to resolve the differenceamong the operation sequence of these products.The ideal material flow in a good layout design should be mostly in-sequenceflow. In-sequence flow usually has the benefits of smaller flow distance, easiercontrol of the production process and easier material handling. This paperproposes two flow analysis methods that are designed to provide a better flowanalysis for the design of multi-products flowline. The first method adopts thetraditional line structure for analysis. The second method proposes a networkstructure for the analysis of the material flow within the system.

1. IntroductionWith the rapid development of technology, the market becomes more versatile.

Today, the life cycle of a new product model is much shorter, and product variety isbroader. As a result, a production environment is no longer for high-volumeproduction, instead the production environment is multi-produce and small- tomedium-volume orientated.

In a multi-product flowline, there are four types of flow movement that can beobserved (Aneke and Carrie 1986). They are repeat operation, in-sequence, by-passmovement, and backtrack movement (Fig. I).

Figure I. Four different flow movements within a multi-product flowline: repeat (R),in-sequence (I), by-passing (BP), and backtracking (BT) (Aneke and Carrie 1986).

Revision received July 1992.·Department of Industrial Engineering, National Chiao-Tung University, Hsin-Chu,

Taiwan, Republic of China.tSchool of Industrial Engineering, Grissom Hall, Purdue University, West Lafayette,

IN 47907, USA.§To whom correspondence should be addressed.

0020-7543/93 SI()-()() © 1993 Taylor & Francis Ltd.

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1558 Yinq-Chin Ho et al.

Of these four flow movements, the in-sequence flow is the most desirable becauseof its unidirectional movement. However, because of the variations in the number ofoperations in the operation sequences and in the operations themselves, it is unlikelythat a routeing (sequence of workstations/machines) will accommodate all thesequences without incurring backtracking movements. Backtracking flow is the leastdesirable movement since it complicates the flow and violates the flow direction.Repeat operations can be treated as an aggregate operation, and thus can be ignoredin flowline analysis. By-pass flow doesn't violate the direction of flow although itcreates sequence jumping that may complicate the flow.

A layout that has more in-sequence flow movement usually has better perfor­mance in the following areas: less flow distance; easier material handling; and moreefficient production. On the other hand, a layout with a lot of backtrackingmovements usually has greater flow distance, and a more difficult and complexmaterial handling problem. The main purpose of flow analysis is to achieve a logicallayout configuration where the flow movements in the layout will be mostly in­sequence and undirectional.

2. Literature reviewThe early studies of material flow analysis are for single machine/workstation

cases which only allow one machine of each type to be included in a flowline.Some well-known models are summarized here. Noy (1957) proposed a

sequence-demand method which is probably the earliest multi-product flowlineanalysis technique. Singleton (1962) presented a simplified approach by convertingall the routeings to a common length scale and analysing them based on their meanpositions. Moore (1962) described a similar approach called tile sequence-demandtechnique by calculating the mean demand position for each operation on the line.Hollier (1963) discussed several methods that arrange workstations in line so that itcan achieve one of the following two objectives: (I) maximizing in-sequencemovements; or (2) minimizing the amount of backtracking. The limitation that onlyone machine of each type is allowed in a flowline prohibits a general solution. In anactual production design problem, duplicate machines are very common, and also,in a modern production environment, allowing duplicate machines in a productionline increases the flexibility of the production system.

Carrie (1975) discussed a more general type of problem which allows more thanone machine of each type in a flowline. This problem is more complicated because ofthe consideration of the workload of each operation and the capacity limit of themachines in the evaluation when additional machines are included. Carrie (1975)proposed a three-stage procedure for constructing a multi-product flowline. Thisprocedure first constructs a single line which consists of sufficient workstations thatno backtracking movements can occur. It then eliminates uneconomical work­stations and re-routes the operations to other workstations in the line. The re­routeing procedure incurs backtracking movements. Usually, several alternativesolutions can be obtained after the elimination process. Finally, computer simula­tions are employed to compare the alternative solutions and the better one is thenselected. The elimination of unjustified workstations is highly dependent on thedesigner's experience and personal preference. Thus, this subjective eliminationapproach becomes inefficient once there are a large number of choices.

Aneke and Carrie (1986) presented a travel-chart method. This method, similarto Hollier's link-analysis, constructs the flowline from both ends simultaneously.

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Sequence-pattern matching-based flow analysis 1559

This construction method is usually good in situations where a family of productshave similar initial and final workstation requirements, and processes in which thedifferences occur mainly in the intermediate processes. Like the early method ofCarrie (1975), this method does not provide a clear-cut method for the eliminationof unjustified workstations.

3. Proposed algorithmsThere are two solution procedures proposed in this paper. The first one is the

heuristic pattern matching algorithm. This method uses the traditional line structureas a basis for flowline construction. The construction process of the heuristic patternmatching method emphasizes sequence similarity among products and uses sequencesimilarity information to construct the flowline.

The second method is a network-based pattern matching algorithm. This methodproposes an alternative network structure for the construction of the multi-flowlines.Like the heuristic pattern matching method, it also considers sequence similarityinformation in the network construction process. Both of these models are generalworkstation cases where more than one machine of each type is allowed within themulti-product flowlines.

4. Heuristic pattern matchingThere are three main stages in the heuristic pattern matching method. The first

stage is the selection stage where a candidate product is selected. The sequence of thecandidate product is accommodated into the constructed flowline, which is a flowpath that consists of different operations (machines/workstations). The selection ofthe candidate product is mainly based on the sequence similarity between theproduct sequence and the sequence of the constructed flowline. The product that hasthe highest sequence similarity with the constructed flowline is selected. A sequencesimilarity measurement called 'compliant index' is introduced in this method. Thesecond stage is the construction stage where a new flowline is constructed bymodifying its sequence so that the sequence of the candidate product can be includedinto the flowline. The third stage is the elimination stage where the elimination of theinfeasible or uneconomic workstations/machines from the constructed flowline isperformed.

4.1. Compliant index and sequence similarity coefficientIn the proposed algorithm, a processing sequence similarity measurement called

the sequence similarity coefficient is used. In order to calculate the sequencesimilarity coefficient (CO), a compliant index is defined first. The compliant index ofthe sequence of a product compared with a flow path is determined by the number ofoperations in the sequence of the product that have either 'in-sequence' or 'by­passing' relationship with the sequence of the flow path.

There are two kinds of compliant indexes: forward compliant index andbackward index. These two compliant indexes can be calculated by comparing theoperation sequence of the product with the sequence of the flow path forwards andbackwards. The process of calculating the forward compliant and backwardcompliant indexes are presented in the flow charts of Figs. 2a and b. Once thecompliant indexes of both directions have been calculated, the sequence similaritycoefficient of this product can be calculated by dividing the sums of both compliantindexes by twice the number of operations in this product.

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Search for thefirst occurrence of anOJforwards from thefirstoperation(beginning) of the path.

no. .-----~===:...----,

Search for thefirst occurrence of anOJforwards from themarked operationof the path.

• Clear the old mark in the path.• Put a markon theOJfound.·SetCF=CF+ 1.

·SetCF=O&j= 1.• Let OJ stand for the J th operationof product i.

(a)

Search forthefirst occurrence of anOJbackwards from the lastoperation (end) of the path.

no. r-----..:===~--___,

Search forthefirst occurrence of anOJbackwards fromthemarked operationof the path.

• Set CB = 0 & j = no. of operationsof product i.

• Let OJstand for the J th operationof product i.

• Clear the old mark in the path.• Puta markon the OJ found.• Set CB = CB + 1.

(b)

Figure 2. (a) Procedure for calculating the forward compliant index between the sequence ofa product and a flow path, (b) Procedure for calculating the backward compliant indexbetween the sequence of a product and a flow path.

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where CO;CF i

CB;Ni

Sequence-pattern matching-based flow analysis

CO;=(CF;+CBi)/(2 x N;)

sequence similarity coefficient of product i= compliant index in forward direction= compliant index in backward direction= number of operations in product i

1561

(I)

An example is given here to illustrate the calculation of the sequence similaritycoefficient (CO) value. Suppose we have a flow path which is 1-2-3-2-4-5-6. Thenumbers in the flow path can be operation numbers or workstation numbers.Suppose we are given a product A whose operation sequence is 1-3-4-5-2. Beforecalculating the CO value of product A with the flow path, the compliant indexesneed to be calculated.

The forward compliant index is obtained by comparing the operation sequenceof product A with the sequence of the flow path sequentially in a forward direction.The comparison process is illustrated in Fig. 3. First of all, the first operation ofproduct A, OPI, can be found in the flow path. As we continue searching from OPIin the flow path, we can find the second operation of product A, OP3, in the flowpath. OP4 can also be found after OP3 in the flow path, and OP5 after OP4.However, no OP2 can be found after OP5 in the flow path. The forward compliantindex is 4, since OPI, 3, 4, and 5 can be found.

The backward compliant index can be obtained in a similar manner to theforward one, except that the comparing and searching direction is backward. Thecomparison process for obtaining the backward compliant index is shown in Fig. 4.Searching backwards, we can find the last operation of product A, OP2, in the flowpath. Continuing the search backwards from OP2, we can find neither OP5 nor OP4in the flow path. However, we can find OP3 before OP2, and OPI before OP3 in theflow path. The backward compliant index is 3. Since the total number of operationsof product A is 5, applying formula (I), the CO value of product A with the flowpath is calculated as O·7.

4.2. Heuristic pattern matching flow analysis algorithmThe detailed steps of heuristic pattern matching are discussed below. An example

will be presented in the next section to demonstrate how the algorithm works.

(I) Select the product with the largest number of operations. Use the sequenceof this product as the initial constructed flowline (CFL).

(2) For every remaining product, calculate its sequence similarity coefficientwith CFL.

(3) The selection of the next product is based on its CO value. A product withhigher CO value means its sequence is more similar to the sequence of CFL.

Flow Path: rPl . 0/077pS ·OP6

Product A: OPl - OP3 - OP4 . OPS - OP2•Comparing Direction

Figure 3. Comparison process for obtaining forward compliant index.

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1562 Yinq-Chin H0 et al.

Thus, the product with the largest CO value will be selected as the candidateproduct.

(4) The sequence of the candidate product is then accommodated into thesequence of the current CFL. Two tentative CFLs can be constructed bymodifying the current CFL in forward and backward directions. Themodifying procedure for each direction is presented in flow chart style inFigs 5 (a) and (b).

(5) Check the feasibility of the machines/workstations in both tentative CFLs,and eliminate the unjustified machines/workstations. Select a tentative CFLusing the following rules.

(5.1) If only one tentative CFL satisfies the workstation availability con­straint, the qualified one is selected.

Flow Path: OPl • OP2 - OP3 • OP2 • OP4 - OPS • OP6

Product A: lPl .!.OP4 • O~P2..Comparing Direction

Figure 4. Comparison process for obtaining backward compliant index.

·Sctj~1.

• Let OJ stands for the J th operationof product i.

no. Search Cor the first occurrence of anOJ forwards fromme first operation(beginning) of the path.

(a)

Search for the first occurrence of anOJ forwards from Ihc markedoperation of the path,

• If there is a mark in the path, clearthisold mark.

• Puta markon theOJ found.

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(b)

Sequence-pattern matching-based flow analysis

• Setj = no. of operations of product i.• Let OJ 'lands for the J th operation

of product i.

no. Search forthe first occurrence of anOJ backwards from the lastoperation (end) of the path.

Search for thefirstoccurrence of anOJ backwards from the markedoperation of the path.

• If there is a mark in the path, clearthis old mark.

• Puta mark on theOJfound.

• lnsen OJ before the markedoperation.

• Clear theold mark.

1563

Figure 5. (a) Forward modification procedure of a flow path for the heuristic patternalgorithm, (b) Backward modification procedure of a flow path for the heuristicpattern algorithm.

(5.2) If both tentative CFLs satisfy the availability constraints, the one thatneeds fewer newly added workstations is selected.

(5.3) If both tentative CFLs fail to satisfy the constraints, the one thatcreates smaller backtracking flow distance, after the elimination of theunjustified workstations/machines, is selected.

(5.4) If there is a tie, arbitrarily select one.

(6) The selected CFL is the new CFL. Repeat (2)-(5) until all the sequences ofthe products have been accommodated into CFL.

5. Example 1An example is given here to demonstrate the construction procedure of the

flowline in the heuristic pattern matching algorithm. The workstation data and theproduct data of this example are given in Fig. 6. The decision process is illustrated inFig. 7.

Initially, product 4 is selected since it has the largest number of operations. Thesequence of product 4 is the initial constructed flowline (CFL) which is 2-3-4-5-6-7.In the second iteration, the CO values are computed from sequences of the productsand the CFL constructed in the first iteration. As can be seen, product I is selected

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1564 Yinq-Chin H0 et al.

because it has the highest sequence similarity coefficient (CO) value. The modifiedCFL remains the same since the sequence of product I is subsumed (CO I = I) by thesequence ofCFL. This means all of the operations in product I can be found in CFLand they are either in 'in-sequence' or 'by-passing' relationship with CFL. The valueof flow distance is updated from 500 to 750. In the third iteration, product 2 isselected as the next candidate. Two tentative CFLs are generated. Since bothtentative CFLs are identical, one of them can be ignored. The new tentative CFLsatisfies the workstation availability constraint, therefore it can be used as the newCFL. Finally, in the fourth iteration, product 3 is selected. At this iteration, twotentative CFLs are created. Both tentative CFLs fail to meet the workstationavailability constraint. The elimination of unjustified workstation(s) is thus per­formed. For the first tentative CFL, one of the two workstations 3 has to beremoved in order to satisfy the availability constraint. Two possible sequences, 1-2­3-4-5-6-7 and 1-3-2-4-5-6-7, can be obtained after one of the two workstations 3is eliminated from the first tentative CFL. Since sequence 1-3-2-4-5-6-7 creates lessbacktracking flow distance (100) than the sequence 1-2-3-4-5-6-7 (150), andrequires less in-sequence (including by-pass) flow distance (2100 versus 2300),sequence 1-3-2-4-5-6-7 is a better choice in the first tentative CFL. Similarly, forthe second tentative CFL, one of the two workstations 2 needs to be removed. Aftersimilar analysis, the sequence (1-3-2-4-5-6-7) is the better choice obtained from thesecond tentative CFL. Therefore, the new CFL for this iteration is 1-3-2-4-5-6-7.

Once the workstations and the sequence in the constructed flow path have beendetermined, the layout can be designed. Figure 8 suggests three possible logicallayout configurations. It seems that choice (c), a branching flowline, is a betterchoice (if a bi-directional shuttle conveyor is used between workstations 2 and 3)

2 3 4 5 6 7

Product Demand #cIOp. Sequence (in WS #)

1 50 3 2-4-72 100 4 1-2-5-63 150 5 1-3-2-4-74 100 6 2-3-4-5-6-7

Figure 6. Operation sequences and demands of products in example I.

Decision Process

hera- Select In-Seq Back- tentativetion C01 CO2 C03 C04 Product Flow Track CFL CFL

1 .- -- - -- 4 500 0 ----- 2-3-4-5-6-7

2 1 314 3/5 - 1 750 0 ----- 2-3-4-5-6-7

3 C-2-3-4-5-6-7 1-2-3-4-5-6-7-- 314 3/5 -- 2 1250 0 1-2-3-4-5-6-7

4 -- -- 4/5 -- 3 2100 100 C-3-2-3-4-5-6-7 1-3-2-4-5-6-71-2-3-2-4-5-6-7

Figure 7. Decision process of example I.

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Sequence-pattern matching-based flow analysis

(a)

1565

Prod

(b) (c)

Figure 8. Three possible workstation/machine layout configurations for example I:(a) straight flowline with backtracking; (b) U-shaped flowline with backtracking;(c) branching flowline.

Product Demand Operation Sequence (in w.s. #)

A 20 2-3-4-6-8-9-7

B 10 14-2-3-4-5-10-11-12

C 15 2-4-6-8-9-13

D 10 1-2-3-5-11-12

Figure 9. Product demands and operation sequence data for the illustrative example.

than the other two choices because of their backtracking flow. In a more detailedlayout design, other factors such as physical cell space, transferring and loading/unloading devices, dispatching sequences, and possible parallel operations also needto be considered.

6. Network-based flow analysisUnlike the traditional line construction approach, which constructs a flowline by

insertion and expansion of the operation sequence of the flowline, the network-basedapproach constructs a flow network by branching and expansion of the flow. Themain idea behind the network method is that branching of flow results in less flowdistance and more flexibility in the routeing selection for each product. This routeingflexibility is especially important in a flexible production environment. Theseadvantages can be demonstrated by the following example. In the example, there arefour product types with their demand and operation sequences information sum­marized in Fig. 9.

A possible flow analysis result, using the heuristic pattern matching method, is1-14-2-3-4-6-8-9-13-7-5-10-11-12. A linear machine configuration (Fig. 10)incurs extra flow distance when compared with a network configuration (see Fig.11). In this example, the flow distance for the network configuration is 350, while the

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1566 Yinq-Chin Ho et al.

flow distance for the linear layout configuration is 480. As a result of branching ofthe flow, some flow distance can be saved in the network structure because of lessby-passing flow movements.

Another advantage of the network machine/workstation layout configuration isthe independence of each flow branch. If there is a bottleneck machine or abreakdown in any branch, then it won't have a domino effect on the other branches.This independence is very important in achieving flexibility of a layout design.Although one can deduce a network layout configuration from the result of the lineconfiguration analysis and the sequence information of the products, it is verytedious when the sequence information becomes complex.

A typical flow network is presented in Fig. 12. A node in a flow networkrepresents a machine or a workstation, the directed arc presents the material flowdirection. A head node is defined as a node without any 'ancestor' nodes. A tail nodeis defined as a node without any 'descendant' nodes. For example, in the network ofFig. 12, node I is the head node, and nodes 6, 4 and 8 are tail nodes. A path in thenetwork is defined as a sequence of nodes which starts at a head node and ends at atail node. For example, path J-2-3-4 is a path in the network of Fig. 12.

There are three main stages in network-based flow analysis. The first stage is theselection stage where the next product is selected. The second stage is the selection ofthe best path for modification. The third stage is the modification stage where theselected path is modified so that the sequence of the selected product is included anda new network is then constructed.

6.1. Basic algorithm of network-based flow analysisThe following details illustrate the procedures of the network-based flow analysis

algorithm. The basic concept of the network-based algorithm is similar to that of theheuristic pattern matching method, i.e. both of them consider sequence similarity inthe construction of the flow path. Three assistant approaches which can make theoverall computation more efficient and economic when coupled with the basicnetwork-based algorithm are presented in the following section. An example is givenin the next section to illustrate how the basic algorithm of the network-based flow

Figure 10. A linear layout configuration for the illustrative example.

Figure II. A possible network logical layout configuration of the illustrative example.

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Sequence-pattern matching-based flow analysis

Figure 12. A typical network.

1567

analysis works. In this algorithm, it is assumed that the operation sequences of theproducts are presented in workstation type numbers, and the 'F' and 'B' in thenotations such as CF, CB, MWF, MWB, AWF, AWB, UWF, and UWB refer to thedirections being compared, where 'F' indicates a forward direction and 'B' abackward direction.

(I) Select the product with the highest demand. Use the operation sequence ofthe product as the initial network.

(2) Select from the remaining products the product with the biggest demand.(3) List all the possible alternative paths in the current network.(4) For every path generated in step 3, calculate the compliant indexes, CF and

CB, using the methods illustrated in Figs 2 (a) and (b).(5) Calculate the number of missing workstations of each path, MWF and

MWB in both directions. The number of missing workstations in anydirection is defined as the difference between the number of operations inthe candidate product Pi and the compliant index of that direction.

(6) Of the missing workstations in every path and direction, i.e. MWF andMWB, find the number of available workstations AWF and AWB and theunavailable workstations UWF and UWB, where MWF = AWF + UWFand MWB = AWB + UWB.

(7) Select the best choice (path and modification direction) to modify using thefollowing selection rules:

(7.1) Select the choice with the smallest Uw, which can be UWF or UWB.(If UWF is smaller than UWB, then the modification direction isforward, otherwise it is backward).

(7.2) If there are ties in the values of Uw, select the one with the smallestMW(MWF or MWB) from the ties.

(7.3) If there are ties in the values of MW, select one with the biggestcompliant index from the ties.

(7.4) If there are ties in the values of compliant index, then arbitrarily selectone from the tied choices.

(8) Modify the path using the modification procedure. There are two modifica­tion procedures. One is for the forward modification, the other is for thebackward modification. They are presented in the flow diagrams in Figs 13aand b. The S, in the flow diagrams is the 'adjusted' sequence of thecandidate produce i. It is the sequence after the elimination of those'missing workstations' that are not available.

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1568 Ying-Chin Ho et al.

For example, suppose we have a product with a sequence 1-2-3-4-5-6.The selected path is 2-4-6-7-9, and modification direction is forward. Aftercomparing the sequence and the forward path, we find that there are three'missing workstations' (i.e. MWF = 3). These missing workstations are WS"

WS 3, and wSs. If only ws 3 and wSs are available, then the 'adjusted' sequencewill be 2-3-4-5-6 after the elimination of ws, from the sequence. We usethis 'adjusted' sequence to modify the selected path.

(9) Repeat (2)-(9) until all of the sequences of the products have been includedin the network.

7. Example 2In this section, an example will be given to illustrate the basic algorithm of the

network-based flow analysis. In the example, there are eight different types ofworkstations and four different products. The workstation and product informationis listed in Fig. 14.

<Sct j e t ,

• LetSi be the adjusted sequenceofproduct i.

• Let OJ stands for the J th operation of Si .• Let's call the path P.

yes.

Search for the firstoccurrence of anOJ forwards from thefirstoperation(beginning) of Palh P.

Arc thenode withthenewtagand the node with the old tag

both in path P1• Create anarc lO connect menode

no. with the old tag and the nodeL- ~ with the new tag.

• Clear the old tag.

Search for me firstoccurrence of anOJ forwards from the markedoperationof the path P.

• I f thereis a mark in the path.clear thisold mark.

• Put a mark on me OJfound.• Puta lag on OJ.

(a)

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Sequence-pattern matching-based flow analysis 1569

• If there is a mark in me path,clear thisold mark.

• Puta markon theOJ found.• Put a lag on OJ.

• LeI Si be the adjusted sequence ofproduct i.

• Set j =no. of operations in Si.• Let OJ stands for the J th operation of Si.• Let's call the path P.

yes.

Searchfor the first occurrence of anOJ backwards from the last operation(end) of path P.

no.

Search for the firstoccurrence of anOJ backwards from the markedoperation of path P.

Select operation OJ from Si.

(b)

yes. r--+ Clear the nld lag. I-----I~

• Create an arc to connectthenodeno. withtheold tagandthe node

1--------1~ with me new lag.• Clear me old lag.

Figure 13. (a) Flow chart for the forward modification process of a path in a network,(b) Flow chart for the backward modification process of a path in a network.

Initially (Fig. 15), product 4 is selected because it has the highest demand. Theinitial configuration is created based on the sequence of product 4. At the seconditeration (Fig. 16), product I is selected because it has the highest demand among theremaining products. All the paths from the resulting network of the first iteration arecollected. Since the initial structure is a line, there is only one path that can be foundin the initial result. The compliant indexes, the number of missing workstations, thenumber of the available workstations and the number of the unavailable work­stations are then calculated. The number of available workstations is calculated bychecking how many workstations among the missing workstations have positiveavailability. There are two choices IF and IB that can be generated from the onlypath found at iteration 2. These two choices have the same operation content, buttheir comparing and future modification directions are different (one is forward, theother is backward). Both of the modification alternatives (\ F and IB) at iteration 2

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1570 ring-Chin Ho et al.

are equally good and will produce the same results, therefore one is arbitrarilyselected. Product 2 is selected at the third iteration. There are four different pathsthat can be found in the current network, and since each one of them has twomodification directions, there will be eight modification choices totally. Choice 4F isselected for modification because it has the smallest number of unavailable work­stations and the smallest number of missing workstations according to step 7 of thealgorithm. The detailed process and the result of the third iteration can be seen inFig. 17. Similar operations are performed at the last iteration where the last product(product 3) is selected (Fig. 18).

8. Making network-based Row. analysis algorithm more efficientAs one can see in example 2, the number of paths may become very large as the

network becomes more complicated. The total number of paths of a flow networkdepends not only on factors such as the number of branches, the number of headnodes, and the number of tail nodes, but also on the way the branches interact withone another (for example the way the branches overlap each other). As shown inFig. 19, although both networks A and B have the same number of branches, thenumber of paths in these two networks are different (four paths for A, three pathsfor B).

The total number of paths in the network can be calculated by first dividing thenetwork into several independent sections so that each section is singly-linked to its

Workstation 1 2 3 4 5 6 7 8

# Available 2 2 1 2 3 2 1 1

Product Demand #of Op. Sequence (in WS #)

1 150 5 3-2-4-5-6

2 100 5 3-2-5-6-4

3 50 6 3-2-1-4-6-8

4 200 5 1-2-4-6-7

Figure 14. Workstation and product data for example 2.

iteration l'

Decision: Selectproduct 4 andbuild the initial line.

l"lial Networ1< (a line here):

Figure 15. Iteration I of example 2.

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Sequence-pattern matching-based flow analysis

iteration 2-

Available Ws. after Iteration 1:

Workstation 1 2· 3 4 5 6 7 8

# Available 1 1 1 1 3 1 0 1

Candidate Product: 1.Product Sequence: 3-2-4-5-6 # 01Op. = 5.

Compliant # of Missing #of Avail- # of Unavall-Dire- Index Ws.[(2).# ableWs. ableWs.Path Conlent ction (1) ofop - (1) ) (3) (4)_(2)- (3)

F 3 5 -3c 2 2 2-2·01 1-2-4-6-7

B 3 5-3.2 2 2-2=0

Decision: Since choices 1F and 1B are equally good (according to the selectionrules of step 7). arbitrarily select 1F or 1B. Modify path 1 from enherdirection (F or B) so fhat the sequence ot product 1 can be included.

Updated Network:

Figure 16. Iteration 2 of example 2.

1571

adjacent section(s). After that, for each section, the number of ways (sub-paths)crossing it is found. The total number of alternative paths is the overall product ofthe number of ways of crossing in each section. For example, in Fig. 20, the networkcan be divided into four sections. There is only one way to go from section I tosection II that is through node 3. So are section II to III and section III to IV. Thereare two sub-paths (1-2 and 10-2) crossing section I. There are three sub-paths forsection II, four for section III, and two for section IV. The total number of paths inthe network will be 48 (2 x 3 x 4 x 2). The more branches, head nodes, and tail nodesthat a network has, the more paths can be generated from the network.

The required computational time at each iteration increases as the number ofalternative paths increases. There are several ways that can help cut down thecomputational time: cutting down the problem size, decreasing the chance ofbranches occurring in the network, and considering only the significant paths. Threeassistant approaches, which can be used along with the basic network-basedalgorithm, are devised to achieve these goals.

8.1. Grouping technologyApplying grouping technology before the network flow analysis can bring two

advantages. First, we can divide a large problem into several smaller ones (groups orcells) and solve the smaller ones individually. Second, it is expected that theoperation sequences of products in the same group will be more similar, and as aresult the chances of branching can be decreased. This is even more true if the

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1572 Ying-Chin Ho et al.

applied grouping technique considers the sequence similarities of the products.Coupling the GT technique with the network-based flow analysis algorithm can thusreduce the problem size and make the problem easier to handle, and as a resultalleviates the potential computational explosion problem.

8.2. Elimination by subsumptionSome calculation can be saved by subsumption among the paths. We define that

a path, A, subsumes another path, B, if the CO value of B with A is I. In otherwords, all the elements in path B can be found in path A and they are in the desiredorder in both directions. This saving is due to the fact that a path is at least as goodas the one it subsumes. In other words, if a path can be selected as the best choice formodification, so can those paths that subsume it and the same result can be obtainedfrom them. Thus, only those paths that are not subsumed by any other paths need tobe generated and considered. For example, in iteration 3 of example 2 (Fig. 17), path

iteration 3-

Available Ws. afterIteration 2:

Workstation 1 2 3 4 5 6 7 8

# Available 1 1 0 1 2 1 0 1

Candidate Product: 2.Product Sequence: 3·2·5-6-4 # of Op. _ 5.

Content Dire- Ocmptant "of Missing II of Avail- # of Unavail-Path ction Index Ws.[(2).# ableWs, ableWs.

(1) atop' (1) I (3) (4).(2)·(3)

F 2 5·2 =3 2 3·2_1, '·2·4·6·7B 2 5·2 =3 2 3·2-1

F 3 5·3 = 2 1 2·1.12 ' ·2·4·5·6·7

B 2 5·2.3 2 3·2=1

F 3 5·3_2 2 2 ·2_03 3·2·4·6·7

B 3 5·3=2 2 2 ·2=0

F 4 5·4 = 1 1 1 ·1 ~ 04 3·2·4·5·6·7

B 3 5 ·3=2 2 2 ·2=0

Decision: Since 4F is the best choice [according to selection rule 7.3), select 4F.Modify path 4 in forward direction so that the sequence of product 2can be included intothe network.

Updated Network:

Figure 17. Iteration 3 of example 2.

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Sequence-pattern matching-based flow analysis

lIer8t1on 4'

Available Ws. alter "eralion 3:

Workstation 1 2 3 4 5 6 7 8

# Available 1 1 0 0 2 1 0 1

Candldale Product: 3.Product Sequence: 3·2·1·4·6-8 #of Op. a 6.

o;,e- CompUant "of Missing # ofAvail- 'at Unavail-Path Content ction Index Ws.[(2).' ableWs. ableWs.

(1) ofop '(1) ) (3) (4).(2)· (3)

F 3 6·3a3 2 3·2-11 1·2-4·6·7

8 3 6·3=3 2 3·2.1

F 3 6·3=3 2 3·2.12 1·2·4·6·4

8 3 6·3=3 2 3· 2 a 1

3 F 3 6·3 =3 2 3·2=11·2-4·5·6·7

8 3 6·3=3 2 3·2.1

F 3 6-3.3 2 3-2 = 14 1-2·4-5-6-4

8 3 6·3-3 2 3-2.1

F 4 6·4.2 2 2-2=05 3·2-4·6-7

8 4 6·4=2 2 2-2=0

F 4 6·4=2 2 2·2=06 3·2-4·6-4

8 4 6·4a2 2 2-2.0

F 4 6·4=2 2 2·2·07 3·2·4·5·6·7

8 4 6·4 = 2 2 2-2.0

F 4 6-4=2 2 2·2=08 3-2-4·5·6·4

8 4 6·4.2 2 2-2.0

Decision: Since choices 5F,58,6F,68,7F,78,8F, and 88 are equally good(according to Ihe selection rules in step 7), arbilrarily select one fromIhem. Modify Ihe palh 01the selected choice so that the sequence ofproduct 3 can be included In Ihe network.

Updaled Network:

Figure 18. Iteration 4 of example 2.

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1574 Ying-Chin H0 et al.

I is subsumed by path 2, and path 3 by path 4, thus only paths 2 and 4 need to begenerated and considered, paths I and 3 can be eliminated from consideration.

To demonstrate the saving that can be earned from path subsumption, considerthe example in Fig. 21. There are 192 (2 x 2 x 4 x 3 x 2 x 2) different paths in thenetwork. The number of those paths which are not subsumed by any other paths canbe calculated in a similar manner to the calculation of the number of total paths inthe network. The difference is that for each section only those sub-paths that are notsubsumed by any other sub-paths in the same section are counted. For example, insection I, since sub-path 2-11-3 subsumes sub-path 2-3, only sub-path 2-11-3 needsto be counted. Similarly, in section III, only two sub-paths (4-15-16-8 and 4-6-11­7-8) need to be considered. With the subsumption consideration, only 16(2 x I x 2 x 2 x I x 2) paths need to be generated and considered. The rest of thepaths are subsumed by these 16 paths and can be ignored. A lot of computationalsaving can be realized in this example. The amount of saving depends on the numberof branches and the ways they interact with one another. Usually, a lot ofcomputational time can be saved as the network becomes bigger.

8.3. Interactive deepening methodThe name and idea are taken from an AI searching algorithm where instead of

listing and testing all the possible paths before selecting the best one, we onlygenerate some of the alternatives and test them. If a desirable path is found fromthose generated, it is selected and the modification made. If none of the pathsgenerated so far is satisfactory, some more of the alternatives are generated. Thisgenerating and testing process continues until a desirable choice is found. Thenumber of alternative paths generated each time should be between one and the totalnumber of all the alternative paths in the network (if it is equal to the total numberof all the alternative paths in the network, then it deteriorates to the original basicalgorithm). One disadvantage with this approach is that the selected choice may notbe the best one, but one that is acceptable to the user.

8.4. DiscussionThese three assistant approaches can be used with one another. However, some

evaluation needs to be done before the interactive deepening approach can be used.

Figure 19. An illustrative example for the effects of the interaction among branches on thenumber of paths.

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Sequence-pattern matching-based flow analysis 1575

I II III IV

Figure 20. An illustrative example for the calculation of the total number of paths in anetwork.

Figure 21. An illustrative example of the computational saving with the subsumptionelimination approach.

The evaluation is whether the saving from the interactive deepening method is worththe risk of not achieving optimality, even with the assistance of group technologyand the subsumption consideration. Most of the time, a problem will become veryeasy once GT has been applied and the subsumptions among paths are utilized. It issuggested by the authors that the first two assistant approaches be applied beforedeciding to use the third one.

9. ComparisonIn this section, the proposed algorithm is compared with other methods. Since

the proposed methods are developed for handling general machine problems, we willcompare the proposed methods with other general algorithms. The most recentpublished general machine algorithm is by Aneke and Carrie (\ 986), and is used forcomparison.

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1576 Ying-Chin Ho et al.

Two test cases are examined for comparison. The first test case is taken fromAneke and Carrie (1986). The second test example is example 2 in the previoussection. Because the method of Aneke and Carrie (1986) doesn't consider theworkstation availability, while the proposed algorithms consider workstation avail­ability and use this information in the selection rules and the elimination process ofinfeasible workstations, in order to have a fair comparison, we will assume that theworkstation resource is unlimited for all test cases. The performance measure usedhere is total flow distance. In order to calculate flow distance, it is assumed that thelength of the links connecting two workstations is one.

The data and the comparison of the first test case is in Fig. 22. The comparisonof the second test case is in Fig. 23, and the product data of the second test case canbe found in Fig. 14.

From the test experiments, it is observed that the method of Aneke and Carrie(1986) is sensitive to the initial and the final operations of the products. This isbecause their method selects the operations from either end of the products, andadds the selected operations to the corresponding end of the flowline. The disadvan­tage of this is that the products may travel through many unnecessary intermediateworkstations, resulting in larger flow distance. On the other hand, the heuristicpattern matching and the network-based algorithms emphasize the overall sequencesimilarity in their building processes, and they are less sensitive to the initial andfinal operations of the products, which explain their better performances. From the

Produet Demand Nocf op. Operation Sequence(inws.#)

1 250 7 3 - 5 - 8 - 9 - 11 - 12 - 13

2 5 11 1 - 2 - 4 - 5 - 3 - 9 - 4 -7 - 12 - 11 - 13

3 40 , 9 1 - 2 - 3 - 5 -4 -7 - 12 -11- 13

4 30 5 8-3-5-11-13

5 4 6 4 - 5 - 7 - 6 - 11 - 13

6 8 5 3-4-10-11-13

7 200 13 I - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 -10 - 11 -12 - 13

(a)

Melhod FlowSequence Result TotalFlow-Distance

Aneke & Carrie 1·2·3·4·5·6-4 -7.8-3·9·4.7 -937909861 6 - 5· 12- 10·11 - 12· 13

Heuristic Pauern 1·2·8-3·4-5·3 ·9-4-6·7·6·12-8792

Matching 8 - 9 - 10- 11 ·12·13

7 10

.-1'. ANetwork-Based 4~8A-IIN3Flow Analysis 4661

1-2-3-5-34-\)1/V

8 4 6

(h)

Figure 22 (a) Product data of the example from Aneke and Carrie (1986), (h) Result of thecomparison.

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Sequence-pattern matching-based flow analysis

Method FlowSequence Result TotalFlow-Distance

Aneke& Carrie3-1-2-1-5-6-4-5-6-8-7 4050

[1986)

Heuristic Pattern3-2-1·2·4-5-6 -7-4-8Matching 3150

1 8

Network-Based A /Flow Analysis 1-2-4-6-7 2150

/ V'3 5 4

Figure 23. Result of comparison of example 2.

1577

results of the network-based algorithm, it can be seen that flow distance can be savedfrom the branching flow.

10. ConclusionTwo flow analysis approaches were proposed in this paper. Unlike previous

methods, our proposed methods provide consideration of workstation availabilityand sequence similarity in the construction process, and an elimination process forinfeasible workstations.

The network-based approach provides more layout information than the lineapproach. It also has the advantages of less flow distance and more flow indepen­dence. However, the selection process may become complicated once' the networkbecomes large. The complexity of a network can be measured by the number ofalternative paths that can be generated from the network. The total number ofalternative paths, as explained earlier, can be calculated by

TI n ii= 1 ... S

(where S is the total number of singly linked sections, and n, is the number ofsub-paths in section I), which can be affected by the number of branches, thenumbers of head nodes and tail nodes, and by the way that the branches interactwith one another.

Ideally, given the number of available workstations, a network will have thelargest number of paths if all of the available workstations are built into the networkand all of them are branching nodes, i.e. nodes that create branches when built intothe network. An upper bound of a network problem, given that the total number ofavailable workstations is known (M), can thus be derived (2M

) . However, this worstcase can never happen, because in a typical network the number of branching nodeswill be much smaller than M. Also, the effect a branching node has on the totalnumber of paths is not always a factor of two. This can be realized from thenetworks A and B of Fig. 19. Assume that node 8 in both networks are added afternode I to node 7. Both networks have two alternative paths before node 8 is added.After node 8, a branching node, is added in both networks. The number of paths innetwork A increase from 2 to 4, a two-fold increase, but not in network B where it

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1578 Sequence-pattern matching-based flow analysis

increases from 2 to 3. Therefore, the complexity of a network should be much lessthan that of the worst case. Also, the complexity of a problem can be furtherreduced by the three suggested assistant methods which can save a lot of computa­tional time. For example, the subsumption elimination process can reduce thenumber of sub-paths that need to be considered at each section, and as a resultreduces the number of paths that need to be generated and considered.

The decision of which method to use depends on the problem and the expectedgoal. If the sequences of the products are dissimilar and a network approach isadopted, then, as suggested in the first assistant approach, a more sequence­conscious grouping preparation should be conducted before the network-based flowanalysis. GT techniques break down a large problem into smaller ones and makethem easier to handle. On the other hand, if there is not much difference between thesequences of products, the network approach can be used right away. The proposedline approach can be used in any situation, however, it provides little layoutinformation.

ReferencesANEKE, N. A. G., and CARRIE, A. S., 1986, A design technique for the layout of multi-product

flowlines. International Journal of Production Research, 24, (3), 471-481.CARRIE, A. S., 1975, Layout of multi-product lines. International Journal of Production

Research, 13, (6) 451-557.Ho, Y. c., 1991, A heuristic sequence pattern and network based flow analysis for flexible

assembly systems. Master's Thesis, Purdue University, West Lafayette, IN, USA.HOLLIER, R. H., 1963, The layout of multi-product lines. International Journal of Production

Research, 2, (2), 47-57.LEE, C. E. c., 1991, An integrated methodology for the analysis and design of cellular flexible

assembly systems. PhD Dissertation, Purdue University, West Lafayette, IN USA.MOORE, J. M., 1962, Plant Layout and Design (New York: Macmillan).Nov, P. c., 1957, Make the right plant layout-mathematically. American Machinist, March,

121-125.SINGLETON, W. T., 1962, Optimum sequencing of operations for batch production. Work

Study and Industrial Engineering, March, 100-110.

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