Research Article A Mathematical Programming Model for Cell

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 285759 9 pageshttpdxdoiorg1011552013285759

Research ArticleA Mathematical Programming Model for Cell FormationProblem with Machine Replication

Reza Raminfar12 Norzima Zulkifli1 and Mohammadreza Vasili2

1 Department of Mechanical and Manufacturing Engineering Universiti Putra Malaysia Selangor Malaysia2 Department of Industrial Engineering Lenjan Branch Islamic Azad University Esfahan Iran

Correspondence should be addressed to Reza Raminfar raminfariaulnacir

Received 12 December 2012 Accepted 8 February 2013

Academic Editor Ricardo Perera

Copyright copy 2013 Reza Raminfar et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Cell formation (CF) is a crucial aspect in the design of cellular manufacturing (CM) systemsThis paper develops a comprehensivemathematical programmingmodel for the cell formation problem where product demands cell size limits sequence of operationsmultiple units of identical machines machine capacity or machine cost are all considered In this model the intercell moves arerestricted to be unidirectional from one cell to the downstream cells without backtracking The proposed model is investigatedthrough several numerical examples To evaluate the solution quality of the proposed model it is compared with some well-knowncell formation methods from the literature by using group capability index (GCI) as a performance measure The results andcomparisons indicate that the proposed model produces solution with a higher performance

1 Introduction

Group technology (GT) can be defined as a manufacturingphilosophy aims at identifying similar parts and groupingthem together to take advantage of their similarities in themanufacturing and design [1] Cellular manufacturing (CM)is an application of GT which has emerged as a promisingalternative manufacturing system [2] It provides an envi-ronment to meet todayrsquos production requirements wheremanufacturing systems are often required to be reconfiguredto respond to changes in product design and demand [3]CM offers the advantages of simplified material flows fasterthroughput reduced setup times reduced inventory bettercontrol over the shop floor and lower scrap rates [4]

Thedesign and implementation of an effectiveCMsysteminvolves many issues such as machine part cell formation(CF) production planning layout design and schedulingThe CF problem can be defined as ldquoif the number types andcapacities of production machines the number and types ofparts to be manufactured and the routing plans andmachinestandards for each part are known which machines and theirassociated parts should be grouped together to form cellrdquo

[5] Numerous algorithms heuristic and nonheuristicmodelshave been developed in the literature for solving the CFproblem

The CF problems can be classified into binary andcomprehensive problems depending on whether or not pro-cessing times and the machine capacities are consideredThe binary problem arises if the part demands are unknownwhen the CM system is being developed Some examplesof the binary problems can be found in Askin et al [6]Chen and Cheng [7] and Chan and Milner [8] If the partdemand can be accurately predicted processing time andmachine capacities have to be included in the analysis Thisgives rise to comprehensive problems [9] Some examplesof comprehensive models can be found in Logendran [10]Zolfaghari and Liang [11] Raminfar et al [12] and Defershaand Chen [13]

The primary objective of this paper is to present acomprehensive model to solve the CF problem The restof this paper is organized as follows Literature review andprimary definitions of CF problem are addressed in Section 2Detailed description of the problem and the proposed modelare given in Section 3 Numerical examples are presented in

2 Journal of Applied Mathematics

Table 1 Machine-part matrix

Machine types Part type1 2 3 4 5

1 1 1 12 1 1 13 1 14 1 1

Section 4 to illustrate the proposed model Discussions toverify the model and conclusions are presented in Sections5 and 6 respectively

2 An Overview of CellFormation (CF) Problem

A variety of methods have been proposed by researchers forsolving the CF problem General overviews on the methodsof solving the CF problem can be found in Singh [14] Selim etal [1] and Papaioannou and Wilson [2] These methods canbe classified based on the proceduresformulations employedto form manufacturing cells and also the part families Selimet al [1] identified three solution strategies The first classwhich is referred to as part families identification (PFI)begins the cell formation process by identifying the familiesof parts first and then allocates machines to the familiesThe second class which is referred to as machine groupsrsquoidentification (MGI) follows the reversal of the steps in thefirst class Manufacturing cells (grouped machines) are firstcreated based on similarity in part routings and then theparts are allocated to cells The third class which is referredto as part familiesmachine grouping (PFMG) identifies thepart families and machine groups simultaneously As in thethird class the proposed model in this paper forms the partfamilies and machine groups simultaneously

The relationship between machines and part types isrepresented by machine part incidence matrix The machinepart incidence matrix has zero and one entries (119886

119894119895) A 1

entry in row 119894 and column 119895 (119886119894119895

= 1) of the matrixindicates that part 119895 has one or more operations on machine119894 whereas a 0 entry indicates that it does not [3] Eachsolution to a cell formation problem is displayed as a final partmachinematrix in standard block diagonal form In the blockdiagonal form matrix out-of-cell operations are referred toas ldquoexceptional elementsrdquo and the nonvisited machines arereferred to as ldquovoidsrdquo Table 1 presents an example of machinepart incidencematrix For instance part type 1 has operationson machine types 1 and 3 Two cells (clusters) are formed asshown in Table 2 Cell 1 consists of machine types 2 and 4 andproduces part types 5 and 2 Cell 2 consists of machine types 1and 3 and produces part types 3 1 and 4 Part type 3 needs tobe processed onmachine types 1 and 3 in cell 2However parttype 3 also needs to be processed on machine type 2 whichhas been assigned to cell 1 Part type 3 has an exceptionaloperation so that it requires an intercell move In Table 2the 0 entry represents a void in cell 2 A void indicates that amachine assigned to a cell is not required for the processing of

Table 2 Solution matrix of cell formation


2 1 1 14 1 11 1 1 13 1 1 0

a particular part in that cell In Table 2 it can be observed thatpart type 4 has no operation to be performed on the machinetype 3

One way to increase the performance of any CF solutionmatrix is to minimize the number of exceptional elementsObviously this can be achieved by minimizing the numberof intercell movements For this purpose common solutionsinclude replicating the bottleneck machinesfacilities [15 16]considering alternative process plans for part types [17 18]and subcontracting the operations or part types [13 19]

As mentioned earlier numerous algorithms heuristicand nonheuristic methods have been developed for the CFproblem However most of these methods may be criticizeddue to their narrow scope and simplistic view of the problemFor instance it is often neglected to consider the operationssequence of each part (ie the order in which the operationsare performed on each part) An accurate count of intercellmovements must be based on the number of cell visitationsand thus the need to include the sequence of operations intothe analysis [20] However there is little analytical work thattakes into account the operations sequence of the parts inevaluating the intercell movements

From the literature it can be observed that several inter-related issues are involved in the cell formation problemwhile most of the existing studies only discuss a fractionof these issues Therefore development of comprehensivecell formation methodologyprocedures would seem to benecessary in order to simultaneously address these issues

Consequently in this research a comprehensive math-ematical model is proposed to solve CF problem Thismathematical programming model provides a large coverageof attributes such as part demand intercell traffic movementcost with focusing on sequence of operations differenttravel distances between cells cell size limits machine costmachine capacity and multiple identical machines Thismodel is also formulated in such a way as to ensure aunidirectional flow of material between cells

In fact it is widely accepted in the literature that thewholeproblemof designing aCMS taking into accounts the numer-ous criteria involved belongs to the class of NP-completeproblems [21] Furthermore the additional features of theproposed model increase its complexity and combinatorialnature For instance one feature of the proposed model is todetermine the number and types of machines to assign foreach part type or cell Logendran et al [22] have shown thatthe problem involved with this feature is NP-hardThereforethe proposed mathematical model in this paper is NP-hardtoo because of attributes model

Journal of Applied Mathematics 3

Cell 1 Cell 2 Cell 119899middot middot middot

Figure 1 A Schematic view of the cells in the proposed plan [20]

3 Mathematical Model Development

31 Model Development Consider a manufacturing systemconsisting of a number of machines to process different parttypes Each part typemay require some or all of themachinesfor processing Demands for different part types are assumedto be known fromwork orders or from forecastTheproposedmodel of this study therefore determines how to form themanufacturing cells and how to group the part types intopart families This model is developed through consideringmultiple identical machines (machine replication) machinescapacity cell size limit machines costs and intercell move-ments costs In order to copewith the possible cell revisitationand the resulting intercell backtracking traffic a simple planincluding certain positioning of the cells is considered Thisplan divides the underlying manufacturing system into 119899

cells as in Dahel [20] In this regard each cell is designedand positioned on the material flow pattern in such a wayto achieve a unidirectional flow of intercell traffic Figure 1illustrates a schematic view of the cells in this plan wherethe cells are numbered to reflect their relative position onthe material flow pattern For example cell 2 follows cell 1immediately in turn cell 3 follows cell 2 and so on Howeverin the case of existence of any exceptional part type in acell the exceptional part type(s) move to the cell imme-diately downstream for further processing Accordingly anoperation 119895 of part 119894 may be assigned to say cell 119897 onlyif the preceding operation (119895 minus 1) on the partrsquos routingsequence is assigned either to cell 119897 or to any cell upstream ofcell 119897 Since cell design is based on the operations sequenceexceptional parts (which should be processed in multiplecells) can process so without resorting to cell revisitationThis results in eliminating intercell backtracking and thussimplifies material handling A general description of theproposed planwith flow line cells and its advantages over ldquojobshoprdquo cells can be found in Dahel [20]

In this paper a mixed integer nonlinear programmingmodel is developed to solve the above-mentioned problemNotations including indices coefficients and parameters aredescribed in the following section

32 Notations

Indices

119894 Part type index 119894 = 1 119868

119895 Index of operations of part type 119894 119895 = 1 119869119894

119897 Cell index 119897 = 1 119871

119896 Machine index 119896 = 1 119870

Coefficients and Parameters

119863119894 Known demand of part type 119894

119872119896 Unit machine operating cost for machine type 119896

119865119894[119895119896]

Processing time of operation 119895 of part 119894 onmachine type 119896

119862119896 Capacity of one machine of type 119896 for one time

period

1198771198971198971015840 Cost of moving a unit of part type from cell 119897 to

cell 1198971015840

119871119861119897 Minimum number of machines in cell 119897

119880119861119897 Maximum number of machines in cell 119897

Binary Decision Variables

120575119894[119895119896]119897

=

1 if operation 119895 of part-type 119894 tobe processed by machine type 119896

is done in cell 1198970 otherwise

(1)

Subscripts 119894[119895119896] of variable 120575119894[119895119896]119897

indicate that machine119896 is required to process operation 119895 of part type 119894 Thisinformation is known from the given part process plan

Integer Decision Variables

119899119896119897 Number of machine type 119896 assigned to cell 119897

33 Mathematical Formulation Consider the following

minimize 119885 =

119870

sum

119896=1

119872119896

119871

sum

119897=1

119899119896119897

+

119868

sum

119894=1

119863119894

119895119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198971015840=1

1198771198971198971015840120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840

(2)

subject to

119871

sum

119897=1

120575119894[119895119896]119897

= 1 119894 = 1 119868 119895 = 1 119869119894 119896 = 1 119870

119895119894

(3)

119868

sum

119894=1

119869119894

sum

119895=1

120575119894[119895119896]119897

ge 119899119896119897 119894 = 1 119868 119895 = 1 119869

119894 119896 = 1 119870

119897 = 1 119871

(4)


119868

sum

119894=1

119869119894

sum

119895=1

119863119894 119865119894[119895119896]

120575119894[119895119896]119897

le 119862119896 119899119896119897 119896 = 1 119870 119897 = 1 119871

(5)

120575119894[119895119896]119897

le

119897

sum

119905=1

120575119894[(119895minus1)119896]119905

119894 = 1 119868 119895 = 2 119869119894

119896 = 1 119870 119897 = 1 119871

(6)

119871119861119897le

119870

sum

119896=1

119899119896119897

le 119880119861119897 119897 = 1 119871 (7)

119899119896119897

ge 0 and integer 120575119894[119895119896]119897

= 0 1 forall119894 119895 119897 1198971015840 119896 1198961015840 (8)

The objective function of the proposed model is given by(2) The objective function is the sum of machine cost andintercell material handling costThe first term of the objectivefunction is the machine operating cost It is assumed that themachines can be included when they are needed and can beremoved from the system when they are not required Thesecond term of the objective function is the intercell materialhandling cost This cost function is nonlinear since it hasbeen assumed that the distances between each pair of cellsare different (part type 119894 after completion of its operation 119895

by machine 119896 in cell 119897 moves to machine 1198961015840 for the next

operation 119895 + 1 in cell 1198971015840) It is further assumed that the

specifications of different part types (for example size orvolume of different part types) do not influence the materialhandling cost

Constraints of the model consist of (3) to (8) Constraintset (3) assigns each partrsquos operation to exactly one cellConstraint set (4) ensures that once machine 119896 is assignedto cell 119897 then the operations of part types may be assignedto that machine Constraint set (5) ensures that sufficientmachine capacity is assigned to each cell Furthermore itdetermines the number of machine types which will berequired Constraint set (6) states that an operation 119895 of parttype 119894 by machine type 119896 may be performed in cell 119897 only ifthe preceding operation (119897 minus 1) is performed either in cell 119897 orin any cell upstream of cell 119897 Constraint sets (7) specifies theminimum number (119871119861) and the maximum number (119880119861) ofmachines in cell 119897 Constraint sets (8) impose nonnegativityand integrality The model as shown previously has beenadapted and modified from Atmani et al [23] Dahel [20]and Chen [24]

34 Linearization of the Proposed Model The objective func-tion in the model is a nonlinear function due to its secondterm for material handling cost This term can be linearizedusing a procedure as follows First consider the second termof the objective function as follows

119868

sum

119894=1

119863119894

119895119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198971015840=1

1198771198971198971015840120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840 (9)

It can be modified as follows119868

sum

119894=1

119869119894

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198711015840=1

1198771198971198971015840 [120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840] (10)

In order to linearize the previous expression that assume

119884119894[119895119896]119897119896

10158401198971015840= 120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840 (11)

These variables imply that

119884119894[119895119896]119897119896

10158401198971015840 =

1 if part type 119894 moves to machine 1198961015840

in cell 1198971015840 to perform operation(119895 + 1) after performing operation 119895

on machine 119896 in cell 1198970 otherwise

(12)

Finally the second term of objective function can be replacedby the following linear expression

119868

sum

119894=1

119869119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198711015840=1

1198771198971198971015840119884119894[119895119896]119897119896

10158401198971015840 (13)

Moreover the following constraints should be added to thismodel

120575119894[119895119896]119897

+ 120575119894[(119895+1)119896

1015840]1198971015840 minus 2119884

119894[119895119896]11989711989610158401198971015840 ge 0 (14)

120575119894[119895119896]119897

+ 120575119894[(119895+1)119896

1015840]1198971015840 minus 119884119894[119895119896]119897119896

10158401198971015840 le 1 (15)

Constraints sets (14) and (15) imply that 119884119894[119895119896]119897119896

10158401198971015840(119905) is equal

to 1 if one unit of part type 119894 is moved to machine 1198961015840 in cell 1198971015840

for operation (119895+1) after performing operation 119895 onmachine119896 in cell 119897

4 Numerical Examples

Two numerical examples with different structures fromexisting literature are presented in this sectionThe exampleshave been solved using LINGO 120 a commercially availableoptimization software on a personal computer with IntelCore2 Duo T6400 200GHz processor and 4GB RAM

Example 1 Data set of this example has been adapted fromDahel [20] with slight modifications In this example 11part types and 7 machine types are considered A three-cellpartition is sought and the minimum number of machinesallowed per cell is set to two and four respectively Detailedproduction demand for each part type machine operatingcosts and machine capacity for each machine type arepresented in Table 3 Meanwhile Table 4 represents part-machine requirements sequence of operations for each parttype and processing time needed for each operation FromTable 4 it can be observed that for example part type 4requires three operations with the first operation onmachine6 the second on machine 3 and the third on machine 4 InTable 4 the lower number in each cell represents the process-ing time requirement corresponding to its relative operation


Table 3 Machine cost machine capacity and part processing demand for Example 1

Machine type1 2 3 4 5 6 7

Machine cost 15 10 20 15 12 15 16Machine capacity 800 700 500 600 700 600 800

Part type1 2 3 4 5 6 7 8 9 10 11

Part processing demand 22 20 18 12 23 24 22 30 19 25 28

Table 4 Partial input data for Example 1

Part type1 2 3 4 5 6 7 8 9 10 11

Machinenumber Part operation requirement (upper number) and processing time requirement (lower number)

1 2 1 2 1109 89 218 20

2 1 1 2 2127 210 146 111

3 2 2 1 1 2125 125 200 184 71

4 3 2 2 2100 78 40 48

5 2 1117 191

6 3 1 3133 50 64

7 1 1 1278 80 128

Table 5 Intercell material handling cost for Example 1

Cell No 1 2 31 00 10 142 10 00 123 14 12 00

For instance 5 time units of processing on machine type 6are required in order to produce one unit of part type 4 Inother words part type 4 requires 5 time units of the capacityofmachine type 6 In addition the intercell material handlingcosts are shown in Table 5

Considering the part-operation requirements in Table 4in order to reduce the number of variables and constraintsthe variables which can be fixed to zero were removed fromthe model using sparse set membership filtering technique ofLINGO [25] After fixing these variables some constraintsbecame redundant and were subsequently removed TheLINGO solver defined this model as mixed integer linear

program (MILP) and used the branch and bound (B-and-B) method to solve it This problem consists of 222 variables(including 96 integer variables) and 284 constraints Theglobal optimal solution was achieved after 13 seconds of thesolver running and the objective value (ie the total cost) ofthe problem was 2828 Table 6 presents a solution matrix forthis example where bracketed numbers show the number ofcorresponding machine assigned to each cell

This solution has four exceptional part type includingparts 4 6 9 and 11 For instance part type 4 performs itsfirst two operations in cell 2 and then moves downstreamto cell 3 for its third operation Accordingly intercell trafficfollows a unidirectional pattern from cell 2 toward cell 3 andno backtracking is happened

From Table 6 it can be observed that multiple units of thesame machine can be used in different cells or even in eachcell For instance two machines of type 3 have been assignedto cell 1 since the capacity provided by a single machine forthese twomachine types is not sufficient to satisfy the capacityrequirements of the parts assigned to this cell Similarly two


Table 6 A solution matrix for Example 1

Cell number Machine number Part type1 2 6 11 3 4 7 9 5 8 10

11 2 12 1 1

3 [2] 2 1 2

2

1 1 23 2 15 2 16 3 3 1

32 2 24 3 2 2 2

7 [2] 1 1 1


Machine type1 2 3 4 5 6 7 8

Machine cost 70 160 160 180 165 65 110 185Machine capacity 18000 18000 20000 20000 22000 18400 18000 20000

Part type1 2 3 4 5 6 7 8 9 10

Part processing demand 50 150 500 75 500 1200 1500 750 5000 1300Part type (continued)

11 12 13 14 15 16 17 18 19 20Part processing demand 1239 575 1239 1500 14000 39 900 339 390 304

machines of type 7 are included in cell 3 and two machinesof type 1 are assigned to cells 1 and 2 Two machines types of2 are assigned to cells 1 and 3 too

Example 2 In this example the data for part-machineincidence matrix have been adapted from Chandrasekharanand Rajagopalan [26] 3 cells 20 part types and 8 machinetypes are considered in this example The minimum andmaximum numbers of machines in each cell are 2 and 5respectively Machine cost and machine capacity for eachmachine type and detailed production demands for each parttype are presented in Table 7 Part-machine requirementsand processing time needed for each operation are shown inTable 8 For example the 3rd column of Table 8 shows thatthere are 2 operations for processing part 2 It also indicatesthat machines 1 and 3 are required to perform operations 1to 2 respectively for part type 1 Furthermore the intercellmaterial handling costs are considered to be the same as thosein the first example

The LINGO code of this example was written similarto that of the first example The LINGO solver detectedthis example as a mixed integer linear program (MILP) andused the branch and bound (B-and-B) method to solve itThe linear model of this example consists of 576 variables(including 207 integer variables) and 731 constraints Thesolution was achieved after 2 minutes and 44 seconds of thesolver runningThe total cost of this problem (which appears

as objective value in the LINGO solution report) was 46128Table 9 shows the solution matrix (machines and part typesgroups) for this example Recall that the bracketed numbersshow the number of correspondingmachine assigned to eachcell

It is observed that in this solution there are four excep-tional part types (including parts 3 11 18 and 20) with12 exceptional elements Through a careful considerationof all the factors mentioned before it can be seen thatthe constraints (5) and (7) have significant influence onthe solution and the solver runtime For instance if themaximum number of machines in each cell is increasedfrom 5 to 6 the solver runtime will be reduced by 54through generating second remarkable results where there isno exceptional part type in the solution (Table 10)

5 Discussion

This section discusses about conditions and advantages ofthe proposed mathematical model The performance of theproposed model is compared with those of some existingwell-known methodologies

As matter of fact it seems that the proposed model hassome advantages over other research models For instanceAskin et al [6] Chan and Milner [8] are solved CF model bybinarymodels and they do not consider substantial attributes



Part type1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


1 1 1 1 1 1 1 1 1 1 1055 359 24 04 3 04 18 124 1 35

2 2 2 1 1 2 1 1284 059 383 203 37 05 24

3 2 2 2 2 2 3 2 3 2201 18 09 06 35 26 38 15 2

4 3 1 3 2 2 2 2201 28 032 352 24 06 08

5 1 2 2 1 1 1 2334 396 085 29 04 12 03

6 2 1 3 3 3 2 3218 178 04 1 02 17 03

7 4 3 4 4 3 2 3 5113 18 131 26 15 35 39 25

8 5 4 5 3 4 4217 016 03 03 06 14


Cell number Machine number Part type1 2 3 5 8 9 11 13 14 16 17 18 19 20 10 12 15 4 6 7

1

1 1 1 1 1 1 1 1 1 1 12 2 1 13 2 2 2 2 2 3 2 3 25 1 2 26 2 1 3

2

4 2 25 1 1 1

6 [2] 3 3 3 27 3 2

3

2 2 2 1 14 3 2 1 3 25 27 4 3 5 3 4 48 5 4 4 4 5 3

such as demand time or cost but the proposedmodel is cov-ered many attributes Other examples such as Defersha andChen [27] and Zolfaghari and Liang [11] are comprehensivemodels to solve CF problem and consider many attributessuch as considering multiple copies of identical machinenevertheless in these models part type flow between cells isnot unidirectional but the proposed model has this attributeor the proposed mixed integer linear programming model issolved by branch and bound algorithm which can be found

global optimal solution but the previous two researchesmodels are found local optimum solution

On the other hand based on the examination of the lit-erature in order to quantify the performance of the differentmethodologiessolutions ldquoperformance measuresrdquo are usedFor this purpose several different performance measureshave been suggested in the literature A comprehensivereview of various performance measures of CF solutionswas presented by Sarker [28] Through various performance


Table 10 A secondary solution for Example 2 after applying a newlimit for maximum number machines in each cell

Cell number Machine number Part number1 2 4 5 6 7 8 1 5 6 7 10 12 202 1 3 5 6 [2]lowast 9 13 15 173 1 2 3 4 7 8 2 3 4 8 11 14 16 18 19lowastBracketed numbers show the number of corresponding machine assignedto each cell

Table 11 CF results of the proposedmodel versus the results of othermethods

Example number Model 119890119900

119890 GCI

1ROC 5 25 80Proposed model 4 25 84ROC2 15 61 754

2

HPH 9 61 852

Proposed model 12 61 803Secondary results ofproposed model 0 61 100

measures group capability index (GCI) which has beenproposed by Hsu [29] is used here The reason for selectingthe GCI as the performance measure in this study is dueto its simultaneous consideration of production volume andprocessing time of operations GCI excludes voids (ldquozerordquoentries) from the calculation of goodness and it can beformulated as follow

GCI = 1 minus

1198900

119890

(16)

where 1198900is the number of exceptional elements in the

machine partmatrix and 119890 is the total number of ldquoonerdquo entriesin the machine component matrix

The results of the first example fromTable 6 are comparedwith the results of rank order clustering (ROC) model [30]by means of GCI performance measure Similarly the resultsof second example (Tables 9 and 10) are compared with theresults of ROC2 and Hamiltonian Path Heuristics (HPH)models [6] by means of GCI performance measure Thesecomparisons are presented in Table 11 For the sake of concisepresentation those steps required to solve the examples withabove-mentioned models are not described in this paperFrom Table 10 it can be observed that the proposed modelrepresents improvements in GCI in comparison with theothermodels Furthermore the proposedmodel of this studyconsiders practical features such as operation sequence andmachine capacity while the ROC ROC2 and HPH modelsare not able to take these features into consideration

6 Conclusions

In this paper a mathematical model was developed in orderto solve the cell formation (CF) problem in the cellularmanufacturing (CM) systems This mathematical modeltakes into account the features such as production volume ofeach part operations sequence of each part intercell traffic

movement cost with focusing on different travel distancebetween cells machine capacity and machine replicationThe model designs cells with the flexibility of choosing thenumber of cells and specifying limits on the number ofmachines per cell The model also formulated in a way soas to ensure a unidirectional flow of material between cellsThis feature results in eliminating the intercell backtrackingwhich simplifies the material handling In order to examinethe proposed model two numerical examples with differentspecifications from existing literature were considered Theexamples were solved by means of LINGO optimization soft-ware In order to evaluate the performance of the proposedmodel it was compared with some existing well-knownmodels including rank order clustering (ROC) ROC2 andHamilton path heuristic (HPH) bymeans of group capabilityindex (GCI) performance measure In conclusion we can saythat the proposed CF model in this paper has a high andacceptable performance and it is reliable for the design andanalysis of the cellular manufacturing (CM) systems

Conflict of Interests

The authors certify that there is no conflict of interests(considering both financial and nonfinancial gains) with anyorganization regarding the material discussed in the paper

References

[1] H M Selim R G Askin and A J Vakharia ldquoCell formation ingroup technology review evaluation and directions for futureresearchrdquo Computers and Industrial Engineering vol 34 no 1pp 3ndash20 1998

[2] G Papaioannou and J M Wilson ldquoThe evolution of cell for-mation problem methodologies based on recent studies (1997ndash2008) review and directions for future researchrdquo EuropeanJournal of Operational Research vol 206 no 3 pp 509ndash5212010

[3] A Mungwattana Design of Cellular Manufacturing Systems forDynamic and Uncertain Production Requirements with Presenceof Routing Flexibility Virginia Polytechnic Institute and StateUniversity Blacksburg Va USA 2000

[4] U Wemmerlov and N L Hyer ldquoProcedures for the partfamilymachine group identification problem in cellular manu-facturingrdquo Journal of Operations Management vol 6 no 2 pp125ndash147 1986

[5] J CWei and N Gaither ldquoA capacity constrainedmultiobjectivecell formation methodrdquo Journal of Manufacturing Systems vol9 no 3 pp 222ndash232 1990

[6] R G Askin S H Cresswell J B Goldberg and A J VakharialdquoHamiltonian path approach to reordering the part-machinematrix for cellular manufacturingrdquo International Journal ofProduction Research vol 29 no 6 pp 1081ndash1100 1991

[7] S J Chen and C S Cheng ldquoA neural network-based cellformation algorithm in cellular manufacturingrdquo InternationalJournal of Production Research vol 33 no 2 pp 293ndash318 1995

[8] H M Chan and D A Milner ldquoDirect clustering algorithm forgroup formation in cellular manufacturerdquo Journal of Manufac-turing Systems vol 1 no 1 pp 65ndash75 1982

[9] S Zolfaghari ldquoDesign and plannng for cellular manufac-turing application of neural networks and advanced search


techniquesrdquo in Mechanical Engineering Universty of OttawaOttawa Canada 1997

[10] R Logendran ldquoA biary integer programming approach forsimultaneousmachine-part grouping in cellular manufacturingsystemsrdquo Computers and Industrial Engineering vol 24 no 3pp 329ndash336 1993

[11] S Zolfaghari and M Liang ldquoComprehensive machine cellpartfamily formation using genetic algorithmsrdquo Journal of Manu-facturing Technology Management vol 15 no 6 pp 433ndash4442004

[12] R Raminfar N Zulkifli M R Vasili and T Sai Hong ldquoAnintegratedmodel for production planning and cell formation incellular manufacturing systemsrdquo Journal of Applied Mathemat-ics vol 2013 Article ID 487694 10 pages 2013

[13] F M Defersha and M Chen ldquoA comprehensive mathematicalmodel for the design of cellular manufacturing systemsrdquo Inter-national Journal of Production Economics vol 103 no 2 pp767ndash783 2006

[14] N Singh ldquoDesign of cellular manufacturing systems an invitedreviewrdquo European Journal of Operational Research vol 69 no 3pp 284ndash291 1993

[15] H Seifoddini ldquoDuplication process in machine cells formationin group technologyrdquo IIE Transactions vol 21 no 4 pp 382ndash388 1989

[16] S Viswanathan ldquoA new approach for solving the P-medianproblem in group technologyrdquo International Journal of Produc-tion Research vol 34 no 10 pp 2691ndash2700 1996

[17] G K Adil D Rajamani and D Strong ldquoCell formation con-sidering alternate routeingsrdquo International Journal of ProductionResearch vol 34 no 5 pp 1361ndash1380 1996

[18] S Sofianopoulou ldquoManufacturing cells design with alternativeprocess plans andor replicate machinesrdquo International Journalof Production Research vol 37 no 3 pp 707ndash720 1999

[19] N Safaei and R Tavakkoli-Moghaddam ldquoIntegrated multi-period cell formation and subcontracting production planningin dynamic cellular manufacturing systemsrdquo International Jour-nal of Production Economics vol 120 no 2 pp 301ndash314 2009

[20] N E Dahel ldquoDesign of cellular manufacturing systems intandem configurationrdquo International Journal of ProductionResearch vol 33 no 8 pp 2079ndash2095 1995

[21] J R King and V Nakornchai ldquoMachine-component groupformation in group technology review and extensionrdquo Interna-tional Journal of Production Research vol 20 no 2 pp 117ndash1331982

[22] R Logendran P Ramakrishna and C Sriskandarajah ldquoTabusearch-based heuristics for cellular manufacturing systems inthe presence of alternative process plansrdquo International Journalof Production Research vol 32 no 2 pp 273ndash297 1994

[23] A Atmani R S Lashkari and R J Caron ldquoA mathematicalprogramming approach to joint cell formation and operationallocation in cellular manufacturingrdquo International Journal ofProduction vol 33 no 1 pp 1ndash15 1995

[24] M Chen ldquoAmodel for integrated production planning in cellu-lar manufacturing systemsrdquo Integrated Manufacturing Systemsvol 12 no 4 pp 275ndash284 2001

[25] LINDO Systems LINGO Userrsquos Guide LINDO SystemChicago Ill USA 2010

[26] M P Chandrasekharan andR Rajagopalan ldquoAn ideal seed non-hierarchical clustering algorithm for cellular manufacturingrdquoInternational Journal of Production Research vol 24 no 2 pp451ndash463 1986

[27] F M Defersha and M Chen ldquoA parallel multiple Markovchain simulated annealing for multi-period manufacturingcell formation problemsrdquo International Journal of AdvancedManufacturing Technology vol 37 no 1-2 pp 140ndash156 2008

[28] B R Sarker ldquoGrouping efficiency measures in cellular manu-facturing a survey and critical reviewrdquo International Journal ofProduction Research vol 37 no 2 pp 285ndash314 1999

[29] C P Hsu The Similarity Coefficient Approaches of Machine-Component Cell Formation in Cellular Manufacturing A Com-parative Study University of Wisconsin-Milwaukee Milwau-kee Wis USA 1990

[30] M P Groover Automation Production Systems and Computer-Integrated Manufacturing Prentice Hall Press 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 Machine-part matrix


1 1 1 12 1 1 13 1 14 1 1

Section 4 to illustrate the proposed model Discussions toverify the model and conclusions are presented in Sections5 and 6 respectively

2 An Overview of CellFormation (CF) Problem

A variety of methods have been proposed by researchers forsolving the CF problem General overviews on the methodsof solving the CF problem can be found in Singh [14] Selim etal [1] and Papaioannou and Wilson [2] These methods canbe classified based on the proceduresformulations employedto form manufacturing cells and also the part families Selimet al [1] identified three solution strategies The first classwhich is referred to as part families identification (PFI)begins the cell formation process by identifying the familiesof parts first and then allocates machines to the familiesThe second class which is referred to as machine groupsrsquoidentification (MGI) follows the reversal of the steps in thefirst class Manufacturing cells (grouped machines) are firstcreated based on similarity in part routings and then theparts are allocated to cells The third class which is referredto as part familiesmachine grouping (PFMG) identifies thepart families and machine groups simultaneously As in thethird class the proposed model in this paper forms the partfamilies and machine groups simultaneously

The relationship between machines and part types isrepresented by machine part incidence matrix The machinepart incidence matrix has zero and one entries (119886

119894119895) A 1

entry in row 119894 and column 119895 (119886119894119895

= 1) of the matrixindicates that part 119895 has one or more operations on machine119894 whereas a 0 entry indicates that it does not [3] Eachsolution to a cell formation problem is displayed as a final partmachinematrix in standard block diagonal form In the blockdiagonal form matrix out-of-cell operations are referred toas ldquoexceptional elementsrdquo and the nonvisited machines arereferred to as ldquovoidsrdquo Table 1 presents an example of machinepart incidencematrix For instance part type 1 has operationson machine types 1 and 3 Two cells (clusters) are formed asshown in Table 2 Cell 1 consists of machine types 2 and 4 andproduces part types 5 and 2 Cell 2 consists of machine types 1and 3 and produces part types 3 1 and 4 Part type 3 needs tobe processed onmachine types 1 and 3 in cell 2However parttype 3 also needs to be processed on machine type 2 whichhas been assigned to cell 1 Part type 3 has an exceptionaloperation so that it requires an intercell move In Table 2the 0 entry represents a void in cell 2 A void indicates that amachine assigned to a cell is not required for the processing of

Table 2 Solution matrix of cell formation


2 1 1 14 1 11 1 1 13 1 1 0

a particular part in that cell In Table 2 it can be observed thatpart type 4 has no operation to be performed on the machinetype 3

One way to increase the performance of any CF solutionmatrix is to minimize the number of exceptional elementsObviously this can be achieved by minimizing the numberof intercell movements For this purpose common solutionsinclude replicating the bottleneck machinesfacilities [15 16]considering alternative process plans for part types [17 18]and subcontracting the operations or part types [13 19]

As mentioned earlier numerous algorithms heuristicand nonheuristic methods have been developed for the CFproblem However most of these methods may be criticizeddue to their narrow scope and simplistic view of the problemFor instance it is often neglected to consider the operationssequence of each part (ie the order in which the operationsare performed on each part) An accurate count of intercellmovements must be based on the number of cell visitationsand thus the need to include the sequence of operations intothe analysis [20] However there is little analytical work thattakes into account the operations sequence of the parts inevaluating the intercell movements

From the literature it can be observed that several inter-related issues are involved in the cell formation problemwhile most of the existing studies only discuss a fractionof these issues Therefore development of comprehensivecell formation methodologyprocedures would seem to benecessary in order to simultaneously address these issues

Consequently in this research a comprehensive math-ematical model is proposed to solve CF problem Thismathematical programming model provides a large coverageof attributes such as part demand intercell traffic movementcost with focusing on sequence of operations differenttravel distances between cells cell size limits machine costmachine capacity and multiple identical machines Thismodel is also formulated in such a way as to ensure aunidirectional flow of material between cells

In fact it is widely accepted in the literature that thewholeproblemof designing aCMS taking into accounts the numer-ous criteria involved belongs to the class of NP-completeproblems [21] Furthermore the additional features of theproposed model increase its complexity and combinatorialnature For instance one feature of the proposed model is todetermine the number and types of machines to assign foreach part type or cell Logendran et al [22] have shown thatthe problem involved with this feature is NP-hardThereforethe proposed mathematical model in this paper is NP-hardtoo because of attributes model








32 Notations

Indices

119894 Part type index 119894 = 1 119868


119897 Cell index 119897 = 1 119871

119896 Machine index 119896 = 1 119870




119865119894[119895119896]



period


cell 1198971015840




120575119894[119895119896]119897

=



(1)






minimize 119885 =

119870

sum

119896=1

119872119896

119871

sum

119897=1

119899119896119897

+

119868

sum

119894=1

119863119894

119895119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198971015840=1

1198771198971198971015840120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840

(2)

subject to

119871

sum

119897=1

120575119894[119895119896]119897

= 1 119894 = 1 119868 119895 = 1 119869119894 119896 = 1 119870

119895119894

(3)

119868

sum

119894=1

119869119894

sum

119895=1

120575119894[119895119896]119897

ge 119899119896119897 119894 = 1 119868 119895 = 1 119869

119894 119896 = 1 119870

119897 = 1 119871

(4)


119868

sum

119894=1

119869119894

sum

119895=1

119863119894 119865119894[119895119896]

120575119894[119895119896]119897

le 119862119896 119899119896119897 119896 = 1 119870 119897 = 1 119871

(5)

120575119894[119895119896]119897

le

119897

sum

119905=1

120575119894[(119895minus1)119896]119905

119894 = 1 119868 119895 = 2 119869119894

119896 = 1 119870 119897 = 1 119871

(6)

119871119861119897le

119870

sum

119896=1

119899119896119897

le 119880119861119897 119897 = 1 119871 (7)

119899119896119897

ge 0 and integer 120575119894[119895119896]119897

= 0 1 forall119894 119895 119897 1198971015840 119896 1198961015840 (8)







119868

sum

119894=1

119863119894

119895119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198971015840=1

1198771198971198971015840120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840 (9)


sum

119894=1

119869119894

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198711015840=1

1198771198971198971015840 [120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840] (10)


119884119894[119895119896]119897119896

10158401198971015840= 120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840 (11)


119884119894[119895119896]119897119896

10158401198971015840 =




(12)


119868

sum

119894=1

119869119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198711015840=1

1198771198971198971015840119884119894[119895119896]119897119896

10158401198971015840 (13)


120575119894[119895119896]119897

+ 120575119894[(119895+1)119896

1015840]1198971015840 minus 2119884

119894[119895119896]11989711989610158401198971015840 ge 0 (14)

120575119894[119895119896]119897

+ 120575119894[(119895+1)119896

1015840]1198971015840 minus 119884119894[119895119896]119897119896

10158401198971015840 le 1 (15)


10158401198971015840(119905) is equal










Part type1 2 3 4 5 6 7 8 9 10 11



Part type1 2 3 4 5 6 7 8 9 10 11


1 2 1 2 1109 89 218 20

2 1 1 2 2127 210 146 111

3 2 2 1 1 2125 125 200 184 71

4 3 2 2 2100 78 40 48

5 2 1117 191

6 3 1 3133 50 64

7 1 1 1278 80 128


Cell No 1 2 31 00 10 142 10 00 123 14 12 00









11 2 12 1 1

3 [2] 2 1 2

2

1 1 23 2 15 2 16 3 3 1

32 2 24 3 2 2 2

7 [2] 1 1 1




Part type1 2 3 4 5 6 7 8 9 10








5 Discussion





Part type1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


1 1 1 1 1 1 1 1 1 1 1055 359 24 04 3 04 18 124 1 35

2 2 2 1 1 2 1 1284 059 383 203 37 05 24

3 2 2 2 2 2 3 2 3 2201 18 09 06 35 26 38 15 2

4 3 1 3 2 2 2 2201 28 032 352 24 06 08

5 1 2 2 1 1 1 2334 396 085 29 04 12 03

6 2 1 3 3 3 2 3218 178 04 1 02 17 03

7 4 3 4 4 3 2 3 5113 18 131 26 15 35 39 25

8 5 4 5 3 4 4217 016 03 03 06 14



1

1 1 1 1 1 1 1 1 1 1 12 2 1 13 2 2 2 2 2 3 2 3 25 1 2 26 2 1 3

2

4 2 25 1 1 1

6 [2] 3 3 3 27 3 2

3

2 2 2 1 14 3 2 1 3 25 27 4 3 5 3 4 48 5 4 4 4 5 3









119890 GCI


2

HPH 9 61 852



GCI = 1 minus

1198900

119890

(16)




6 Conclusions





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















32 Notations

Indices

119894 Part type index 119894 = 1 119868


119897 Cell index 119897 = 1 119871

119896 Machine index 119896 = 1 119870




119865119894[119895119896]



period


cell 1198971015840




120575119894[119895119896]119897

=



(1)






minimize 119885 =

119870

sum

119896=1

119872119896

119871

sum

119897=1

119899119896119897

+

119868

sum

119894=1

119863119894

119895119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198971015840=1

1198771198971198971015840120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840

(2)

subject to

119871

sum

119897=1

120575119894[119895119896]119897

= 1 119894 = 1 119868 119895 = 1 119869119894 119896 = 1 119870

119895119894

(3)

119868

sum

119894=1

119869119894

sum

119895=1

120575119894[119895119896]119897

ge 119899119896119897 119894 = 1 119868 119895 = 1 119869

119894 119896 = 1 119870

119897 = 1 119871

(4)


119868

sum

119894=1

119869119894

sum

119895=1

119863119894 119865119894[119895119896]

120575119894[119895119896]119897

le 119862119896 119899119896119897 119896 = 1 119870 119897 = 1 119871

(5)

120575119894[119895119896]119897

le

119897

sum

119905=1

120575119894[(119895minus1)119896]119905

119894 = 1 119868 119895 = 2 119869119894

119896 = 1 119870 119897 = 1 119871

(6)

119871119861119897le

119870

sum

119896=1

119899119896119897

le 119880119861119897 119897 = 1 119871 (7)

119899119896119897

ge 0 and integer 120575119894[119895119896]119897

= 0 1 forall119894 119895 119897 1198971015840 119896 1198961015840 (8)







119868

sum

119894=1

119863119894

119895119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198971015840=1

1198771198971198971015840120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840 (9)


sum

119894=1

119869119894

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198711015840=1

1198771198971198971015840 [120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840] (10)


119884119894[119895119896]119897119896

10158401198971015840= 120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840 (11)


119884119894[119895119896]119897119896

10158401198971015840 =




(12)


119868

sum

119894=1

119869119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198711015840=1

1198771198971198971015840119884119894[119895119896]119897119896

10158401198971015840 (13)


120575119894[119895119896]119897

+ 120575119894[(119895+1)119896

1015840]1198971015840 minus 2119884

119894[119895119896]11989711989610158401198971015840 ge 0 (14)

120575119894[119895119896]119897

+ 120575119894[(119895+1)119896

1015840]1198971015840 minus 119884119894[119895119896]119897119896

10158401198971015840 le 1 (15)


10158401198971015840(119905) is equal










Part type1 2 3 4 5 6 7 8 9 10 11



Part type1 2 3 4 5 6 7 8 9 10 11


1 2 1 2 1109 89 218 20

2 1 1 2 2127 210 146 111

3 2 2 1 1 2125 125 200 184 71

4 3 2 2 2100 78 40 48

5 2 1117 191

6 3 1 3133 50 64

7 1 1 1278 80 128


Cell No 1 2 31 00 10 142 10 00 123 14 12 00









11 2 12 1 1

3 [2] 2 1 2

2

1 1 23 2 15 2 16 3 3 1

32 2 24 3 2 2 2

7 [2] 1 1 1




Part type1 2 3 4 5 6 7 8 9 10








5 Discussion





Part type1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


1 1 1 1 1 1 1 1 1 1 1055 359 24 04 3 04 18 124 1 35

2 2 2 1 1 2 1 1284 059 383 203 37 05 24

3 2 2 2 2 2 3 2 3 2201 18 09 06 35 26 38 15 2

4 3 1 3 2 2 2 2201 28 032 352 24 06 08

5 1 2 2 1 1 1 2334 396 085 29 04 12 03

6 2 1 3 3 3 2 3218 178 04 1 02 17 03

7 4 3 4 4 3 2 3 5113 18 131 26 15 35 39 25

8 5 4 5 3 4 4217 016 03 03 06 14



1

1 1 1 1 1 1 1 1 1 1 12 2 1 13 2 2 2 2 2 3 2 3 25 1 2 26 2 1 3

2

4 2 25 1 1 1

6 [2] 3 3 3 27 3 2

3

2 2 2 1 14 3 2 1 3 25 27 4 3 5 3 4 48 5 4 4 4 5 3









119890 GCI


2

HPH 9 61 852



GCI = 1 minus

1198900

119890

(16)




6 Conclusions





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119868

sum

119894=1

119869119894

sum

119895=1

119863119894 119865119894[119895119896]

120575119894[119895119896]119897

le 119862119896 119899119896119897 119896 = 1 119870 119897 = 1 119871

(5)

120575119894[119895119896]119897

le

119897

sum

119905=1

120575119894[(119895minus1)119896]119905

119894 = 1 119868 119895 = 2 119869119894

119896 = 1 119870 119897 = 1 119871

(6)

119871119861119897le

119870

sum

119896=1

119899119896119897

le 119880119861119897 119897 = 1 119871 (7)

119899119896119897

ge 0 and integer 120575119894[119895119896]119897

= 0 1 forall119894 119895 119897 1198971015840 119896 1198961015840 (8)







119868

sum

119894=1

119863119894

119895119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198971015840=1

1198771198971198971015840120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840 (9)


sum

119894=1

119869119894

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198711015840=1

1198771198971198971015840 [120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840] (10)


119884119894[119895119896]119897119896

10158401198971015840= 120575119894[119895119896]119897

120575119894[(119895+1)119896

1015840]1198971015840 (11)


119884119894[119895119896]119897119896

10158401198971015840 =




(12)


119868

sum

119894=1

119869119894minus1

sum

119895=1

119870

sum

119896=1

119870

sum

1198961015840=1

119871

sum

119897=1

119871

sum

1198711015840=1

1198771198971198971015840119884119894[119895119896]119897119896

10158401198971015840 (13)


120575119894[119895119896]119897

+ 120575119894[(119895+1)119896

1015840]1198971015840 minus 2119884

119894[119895119896]11989711989610158401198971015840 ge 0 (14)

120575119894[119895119896]119897

+ 120575119894[(119895+1)119896

1015840]1198971015840 minus 119884119894[119895119896]119897119896

10158401198971015840 le 1 (15)


10158401198971015840(119905) is equal










Part type1 2 3 4 5 6 7 8 9 10 11



Part type1 2 3 4 5 6 7 8 9 10 11


1 2 1 2 1109 89 218 20

2 1 1 2 2127 210 146 111

3 2 2 1 1 2125 125 200 184 71

4 3 2 2 2100 78 40 48

5 2 1117 191

6 3 1 3133 50 64

7 1 1 1278 80 128


Cell No 1 2 31 00 10 142 10 00 123 14 12 00









11 2 12 1 1

3 [2] 2 1 2

2

1 1 23 2 15 2 16 3 3 1

32 2 24 3 2 2 2

7 [2] 1 1 1




Part type1 2 3 4 5 6 7 8 9 10








5 Discussion





Part type1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


1 1 1 1 1 1 1 1 1 1 1055 359 24 04 3 04 18 124 1 35

2 2 2 1 1 2 1 1284 059 383 203 37 05 24

3 2 2 2 2 2 3 2 3 2201 18 09 06 35 26 38 15 2

4 3 1 3 2 2 2 2201 28 032 352 24 06 08

5 1 2 2 1 1 1 2334 396 085 29 04 12 03

6 2 1 3 3 3 2 3218 178 04 1 02 17 03

7 4 3 4 4 3 2 3 5113 18 131 26 15 35 39 25

8 5 4 5 3 4 4217 016 03 03 06 14



1

1 1 1 1 1 1 1 1 1 1 12 2 1 13 2 2 2 2 2 3 2 3 25 1 2 26 2 1 3

2

4 2 25 1 1 1

6 [2] 3 3 3 27 3 2

3

2 2 2 1 14 3 2 1 3 25 27 4 3 5 3 4 48 5 4 4 4 5 3









119890 GCI


2

HPH 9 61 852



GCI = 1 minus

1198900

119890

(16)




6 Conclusions





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Part type1 2 3 4 5 6 7 8 9 10 11



Part type1 2 3 4 5 6 7 8 9 10 11


1 2 1 2 1109 89 218 20

2 1 1 2 2127 210 146 111

3 2 2 1 1 2125 125 200 184 71

4 3 2 2 2100 78 40 48

5 2 1117 191

6 3 1 3133 50 64

7 1 1 1278 80 128


Cell No 1 2 31 00 10 142 10 00 123 14 12 00









11 2 12 1 1

3 [2] 2 1 2

2

1 1 23 2 15 2 16 3 3 1

32 2 24 3 2 2 2

7 [2] 1 1 1




Part type1 2 3 4 5 6 7 8 9 10








5 Discussion





Part type1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


1 1 1 1 1 1 1 1 1 1 1055 359 24 04 3 04 18 124 1 35

2 2 2 1 1 2 1 1284 059 383 203 37 05 24

3 2 2 2 2 2 3 2 3 2201 18 09 06 35 26 38 15 2

4 3 1 3 2 2 2 2201 28 032 352 24 06 08

5 1 2 2 1 1 1 2334 396 085 29 04 12 03

6 2 1 3 3 3 2 3218 178 04 1 02 17 03

7 4 3 4 4 3 2 3 5113 18 131 26 15 35 39 25

8 5 4 5 3 4 4217 016 03 03 06 14



1

1 1 1 1 1 1 1 1 1 1 12 2 1 13 2 2 2 2 2 3 2 3 25 1 2 26 2 1 3

2

4 2 25 1 1 1

6 [2] 3 3 3 27 3 2

3

2 2 2 1 14 3 2 1 3 25 27 4 3 5 3 4 48 5 4 4 4 5 3









119890 GCI


2

HPH 9 61 852



GCI = 1 minus

1198900

119890

(16)




6 Conclusions





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11 2 12 1 1

3 [2] 2 1 2

2

1 1 23 2 15 2 16 3 3 1

32 2 24 3 2 2 2

7 [2] 1 1 1




Part type1 2 3 4 5 6 7 8 9 10








5 Discussion





Part type1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


1 1 1 1 1 1 1 1 1 1 1055 359 24 04 3 04 18 124 1 35

2 2 2 1 1 2 1 1284 059 383 203 37 05 24

3 2 2 2 2 2 3 2 3 2201 18 09 06 35 26 38 15 2

4 3 1 3 2 2 2 2201 28 032 352 24 06 08

5 1 2 2 1 1 1 2334 396 085 29 04 12 03

6 2 1 3 3 3 2 3218 178 04 1 02 17 03

7 4 3 4 4 3 2 3 5113 18 131 26 15 35 39 25

8 5 4 5 3 4 4217 016 03 03 06 14



1

1 1 1 1 1 1 1 1 1 1 12 2 1 13 2 2 2 2 2 3 2 3 25 1 2 26 2 1 3

2

4 2 25 1 1 1

6 [2] 3 3 3 27 3 2

3

2 2 2 1 14 3 2 1 3 25 27 4 3 5 3 4 48 5 4 4 4 5 3









119890 GCI


2

HPH 9 61 852



GCI = 1 minus

1198900

119890

(16)




6 Conclusions





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Part type1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


1 1 1 1 1 1 1 1 1 1 1055 359 24 04 3 04 18 124 1 35

2 2 2 1 1 2 1 1284 059 383 203 37 05 24

3 2 2 2 2 2 3 2 3 2201 18 09 06 35 26 38 15 2

4 3 1 3 2 2 2 2201 28 032 352 24 06 08

5 1 2 2 1 1 1 2334 396 085 29 04 12 03

6 2 1 3 3 3 2 3218 178 04 1 02 17 03

7 4 3 4 4 3 2 3 5113 18 131 26 15 35 39 25

8 5 4 5 3 4 4217 016 03 03 06 14



1

1 1 1 1 1 1 1 1 1 1 12 2 1 13 2 2 2 2 2 3 2 3 25 1 2 26 2 1 3

2

4 2 25 1 1 1

6 [2] 3 3 3 27 3 2

3

2 2 2 1 14 3 2 1 3 25 27 4 3 5 3 4 48 5 4 4 4 5 3









119890 GCI


2

HPH 9 61 852



GCI = 1 minus

1198900

119890

(16)




6 Conclusions





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119890 GCI


2

HPH 9 61 852



GCI = 1 minus

1198900

119890

(16)




6 Conclusions





References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Mathematical Programming Model for Cell