Some Optimization Approaches for Transfer Lines with Blocks of Operations

SOME

OPTlMIZATlO~APPROACHES

FOR TRA:\SFER L1:\ES WITH BLOCKS OF OPERAT101\'S

A. Dolgui*, B. Finel **, 1\'. Guschinsky ***, G. Levin ***, and F. Vernadat **

* Industrial 51 'Stems Optimi::ation Lab. Unil'erSifl' of Technolof5\' oj'Troyes 12, me Marie Curie, B.P. 2060 - 10010 TrOl'es Cedex, France Ph.. +33 (0)3 2571 5629 Fax. +33 (0)325715649 E-mail: dolgui@uttF

** LaboratonIor Industrial and Mechanical Engineering ENIM, lie du Saulcv 57045 Met:: Cedex, France Ph.' 33 (0)3 87344269 Fax.' +33 (0)3 87346935 El11aif: {fine!,l'ernadat/@enil11..fr

*** Operations Research Lab., Institute ofEngineering Cybernetics Surganov St. 6, 220012 Minsk, Belanls Ph.. +375 (/7) 284 2152 Fax.' +375 (17) 231 84 03 E-mail: {f5\!shin.levin/@nelVl11an.bas-net.by

Abstract: In this paper some algorithms which can be used to minimise the line cost of lines where operations can be executed simultaneously (for example by a spindle head) on workstations in series are presented: the operations can be grouped in blocks of parallel operations. The duration of the longest operation of a block is its processing time; the workstation time is the sum of processing times of its blocks. Two types of methods are proposed: the first one is based on a MIP model, and the second one on random search heuristic. Some comparisons are given. Copyright@ 20031FAC Keywords: Manufacturing Optimisation.

systems

design,

Transfer

line,

Load

balancing.

equipment (tools and spindle heads) to create the corresponding production line. Such a line represents a big cost and a long exploitation period. To find a good (and if possible the best) design solution is a crucial step.

I. INTRODUCTIO Transfer lines are widely used in mechanical industry for mass production of a single type of product (GrooveI', 1987: Hitomi, 1996). Designing such lines is a "ery complex problem due to manufacturing and design constraints and to the large number of possible decisions. At the preliminary design stage studied in this paper. all the operations required to manufacture a product are known. It is necessary to define the workstations (machines) and the pieces of

To minimize the number of workstations and pieces of equipment (line eosl) as well as the occupied area for considered lines. the operations are grouped into blocks at every workstation (station). All operations of the same block are executed simultaneously by

261

one spindle head. The case when there are no intermediate buffers is considered: all blocks (spindle heads) of the samc station arc executed sequentially. if there arc severa L all the stations are linearly ordered and are activated simultaneously. So, at the preliminary design stage, it is necessary to assign each opcration required to manufacture a product, to one block and each block to one station. The assignment of blocks to a station defines in the same time the order of their activation at the station.

For this problem, two exact optimization methods (Dolgui et aI., 1999, 2000, 2001, 2002a) are proposed. Thc first method is based on a mixed integcr programming (MIP) approach and the second one on a graph optimization technique. The exact optimization methods can be used only for small and medium problems. For large-scale problems, some heuristic approaches are needed. Such an approach is proposcd in (Dolgui et aI., 2002b). It is based on the COMSOAL technique.

In this paper. it is assumed that the operation time of each block is equal to the longest operation time (for all operations of this block). The station time is equal to the sum of block times (for all blocks assigned to this station). The line cyclc timc is the maximal station time. The line cost is estimated by the following exprcssion:

In this paper. the MIP models and a COMSOALbascd heuristic are discussed. The paper is organized as follows. Section 2 introduces the basic MIP model. Section 3 deals with the proposed heuristic approach. In Section 4, numerical examples, tests and comparisons for the MIP model and the heuristic model are given. The Section 5 presents an improved MIP model. The conclusion remarks are presented in Section 6.

(1) where is the number of stations. Q is the total number of blocks, Cl and C, are the relative costs of one station and of one block, respectively.

111

2. MIP MODEL The considered line balancing problem is to assign all the operations of a given set to stations (machines) and blocks (spindle heads) in such a way that: - the line cost (1) is as small as possible; - the obtained line cycle time is not greater than the given line cycle; -the precedcnce and compatibility constraints are satisfied.

The problem is to find the best assignment of all the operations to blocks, and of the blocks to stations, minimizing the estimated line cost, under a given cycle time (to obtain a given productivity of the line) under technological (precedence and and compatibility) constraints. This problem is close to Simple Assembly Line Balancing Problem (Baybars, 1986; Scholl, 1999).

The following constraints are taken into account:

For Simple Assembly Line Balancing (SALB) problem, two main approaches are used: Branch and Bound algorithms, see for example (Scholl and Klein, 1998) and heuristics and meta-heuristics (Helgenson and Bimie, 1961; Arcus, 1966; Rekiek et aI., 2000). A recent state-of-art is presented in (Rekiek et aI., 2002).

a partial order relation over the set of all operations to be assigned, which defines a set of possible operations sequences;

ii)

the necessity to perform some groups of operations in one block or at the same station (e.g., because of a required machining tolerance);

iii) the impossibility to perform some groups of operations in one block or at the same station (e.g., due to spindle head and station constructions or because of manufacturing incompatibility of operations).

The known methods for SALB optimization cannot be used directly for the considered problem because: i)

i)

the operations of each block are executed simultaneously (the block time is equal to the maximum of operation times and not the sum of operations times);

The following notation will be used: :" is the given set of operations to be assigned;

To is the required line cycle time;

ii) the blocks are not known in advance and must be built during optimization;

1110

is the maximal number of available stations;

iii) the line cost not only depends on the number of stations but also on the number of blocks;

/lo is the maximal number of available blocks at one

i\') some complementary constraints on compatibility of grouping operations in the blocks and at the stations exist (to obtain realizable manufacturing and designing decisions).

ti

Then. the considered balancing problem.

problem

is

a new

station; is the processing time of the operation j

E

1'\;

PredOJ is the set of direct predecessors of the operation j E:"; j(e) is some fixed operation from a set e~:";

load

k is the station number. k

262

=

I. ... , 1110:

qo = mol1 o is the maximal number of blocks; q is the block number. q = I. .... qo;

X. = { I, if the operation i is assigned to the block q, Iq 0, otherwise

5(k) = {(k-I )110+ I. .. .,kno I is the set of block numbers of the station k;

and real variables:

E5 is the collection of operation sets representing

Fq

the exclusion (impossibility) constraints for stations. Operations of the same set of the collection cannot be assigned to the same station all together;

Zk

0 , to count the number of blocks, Yq

=

I if the

~

0, to count the number of stations, Zk

=

I if the

station k is created. Then, the problem can be formulated as the following MIP model (Dolgui et 01, 2000):

£5 is the collection of operation sets representing the inclusion (necessity) constraints for the stations. All operations of the same set of the collection must be assigned to the same station.

qo

L

X jq = I , for each operation)

E

N,

(2)

q=1

i.e. each operation must be assigned to one block exactly.

Notes: I) The sets Pred(.) define not strict precedence constraints, i.e. iEPred(j), for example, means that operation ) cannot be executed before operation i (operation i is executed before) or i and) are executed simultaneously). This type of precedence constraints takes place together with strict precedence constraints under designing the considered type of transfer lines. To present strict partial order relation between i and) (i r strictly before i), then both an arc (ij) in G and

LXiq , ~ Xjq I Pred(j)1

L iEPred(j)

' for

q'~q

each operation}

E

Nand q = 1,2, ... , qo,

(3)

i.e. for each operation and each block, the precedence constraints must be respected. IAI is the cardinality of a set A.

the pair {ij} in £B should be used.

LX jq :0;

The constraints concerning the necessity to execute some operations in the same block are omitted by grouping these operations in a macro-operation which will be considered as one operation (Dolgui et al. 2000).

jEe

le

1- I , for each eE EH

and (4)

q= 1,2, ... ,qo,

Constraint (4) defines the impossibility to include some groups of operations in the same block (exclusion constraint for the blocks), i.e. for each

For the constraints ES and EB, any proper

e E EH

subset of ec ES (ec EB) can be assigned to a

and for each block q, the number of

operations from e assigned to this block q must be smaller than the number of these operations in e (if the sum is equal to lel, all operations from e are

station (block). Only the whole set ec ES (ec EB) is forbidden. I.e. all ec ES and ec EB

included in q, then the constraint given by eE £H

are the minimal sets. For example, EB =< {3, 4, 6} > means that operations 3, 4 and 6 cannot be assigned simultaneously to the same block, but operations 3 and 4, 3 and 6, 4 and 6 can be assigned to the same block. 4)

0, to determine block time for block q,

block q is created,

the exclusion (impossibility) constraints for the blocks. Operations of the same set of the collection cannot be assigned to the same block together;

3)

~

Yq

£H is the collection of operation sets representing

2)

~

IS

not respected).

L

L X jq

:0;

Ie 1- I,

for each e E E5

jEe qES(k)

£5 is a collection of maximal sets, for example,

andk= 1,2, ... ,1110,

E5=<{1,2,5j,{3,4}> means that operations 1,2 and 5 must be assigned to the same workstation as well as operations 3 and 4. Of course, in £5 there are no intersecting sets (such sets should be united).

(5)

Constraint (5) defines the impossibility to include some groups of operations in the same station (exclusion constraint for the stations); for each eE £5

and for each station k, the number of

operations from e assigned at this station must be smaller than the number of these operations in e (but

Let us introduce the binary variables:

263

from the analysis of all the constraints (see in detail, Dolgui el aI., 2000).

a subset of e may be included, see definition of ES and EB ).

I

I

X

iq .

iEe\{j(e)} "ES(k)

for each e

E

(I e I-I) I

0=

"ES(k)

X

i(e)q , .

ES and b I,2, ... , lilO,

3. A COMSOAL-BASED HEURISTIC

(6)

In real design problems, the number of operations can exceed several hundreds, the number of workstations and the total number of the blocks may be equal to several tens. To solve such large scale problems in an acceptable time, special techniques need to be developed.

Constraint (6) indicates the necessity to execute some operations at the same station k (inclusion constraint) for any k and e E ES: if the operation j(e) is assigned to station k, then all other operations from e must be assigned at the same station k. I~ ~ t;

X Iq , for each block q

and for each operation}

E

=

In this paper, a random search approach based on ideas of the COMSOAL method (Arcus, 1966) is explained. A new algorithm is derived. This algorithm works with the following list: LC regroups all the operations which can be assigned to the current station.

1,2, ... , qo

N,

(7)

I.e. the time of a block q is the longest time of the operations which are assigned to this block.

LC contains an operation i if:

I

F" ~To,foreachbl,2,... ,mo.

(i,l) the processing time I I is not greater than the slack

(8)

time Tk of the current station k, i. e.:

"ES(k)

t;

i.e. the time of station k (equal to sum of corresponding block times) is not allowed to be greater than the given cycle time To.

E

N,

h

0=

To -

IF", qES(k)

(12) exclusion constraints ES are satisfied (taking also into account all the operations already assigned to the current station), (g,) all their predecessors are either assigned or in Le.

Yq ~jq , for each block q = 1,2, ... , qo and for each operation}

~

(9)

At each iteration of the algorithm, one operation from i.e. if J0q = I for any} then Yq must be equal to I (the existence of block q in the design decision). Zk~Yq,

LC is chosen in a random way to be assigned to the current block. If this operation has some predecessors in LC, then the predecessors are assigned first. When an operation must be assigned, all operations present with this operation in set eEES and in LC can be assigned if possible in the same loop of the algorithm.

foreachbl,2, ... ,lI1o

and for each qES(k),

(10)

if Yq =1 for any qES(kj, then Zk must be equal to I (the existence of workstation k in the design decision).

For each candidate operation, if this operation cannot be assigned to the current block, then a new block is created for the current station. If the slack

Then, the objective function is: m

C

I

I

o

time and ES prohibit creating a new block, then a new station is created. When a station is closed, the constraints ES are tested. If there is a subset of operations from a set eEES not assigned to this station with other operations from the same set e, then this is a non-realizable solution. When an operation is assigned, the LC is completed. An iteration of this algorithm gives a realizable solution for the line (or the conclusion that the solution of this iteration is not realizable).

"0

Zk + C 2

k=l

I

Y" ~ min

(11 )

,,=]

This MIP model is based on the constraints given by -

-

the collections ES, ES, EB and by the sets Pred(j), for all}

E

X The main decision variables are (Xjq , )

E

:", q = 1,2, ... , qv) and the complementary variables Y,(, q = 1,2, ... , qv; Zk, k = 1,2, ... , 1110. The

complementary variables decision variables.

depend

on

the

main 4. TESTS AND COMPARlSONS

To reduce the number of decision variables and constraints, intervals of possible \alues of block and station number for each operation can be obtained

About 30 different test examples have been used. Some results of twelve of them are presented here. The models ha\e been tested in the same conditions.

264

Parameters of 12 studied test examples are given in Table I. where n £B is number of subsets in £B,

the criterion value and computation time obtained with MIP using Cplex. The fourth and five give the criterion value and computation time obtained with the heuristic based on COMSOAL.

n £S is number of subsets in £S. n£S is number of subsets in £5.

Table 3 Correspondence between heuristic computation time and Rap

Table 2 propose some others data for these examples. ;\IBo and mBo (NSo and II1S0) represent the maXllllum and the minimum number of blocks (respectively stations) in which the operations can be located. The MoyBo and MoySo columns give the average number of blocks and stations allowed for each operation. The MoyPredo column shows the average number of predecessors for each operation for the sallle set of examples.

N 13 18 23 29 35 38 45

Table 1 Some studied examples

lOO Ex.

N

1

4

2

6

3 4

13 18 23 29 35 30 45 100 150 120 35 17

5 6

7 8 9 10 11 12 13 14

nEH 2 2 4 3 4 3

4 4 5

6 26 14

4 4

nES

nES

1 I

3 3

4 3

2 3 3

3

4 4

6 4

5 4

7 13 7 3 3

5 9

8 7 3

4 3 4

3 4 4

5

3

4 6 5

4 3 4

7 7 7 8 7 4 43

3 3 3 3 3 5

3

150 120

9 7

* .N /

73 116 160 164 388 1078 1672

-----,1-

1400·--------------1--1200 . '000 BOO 600 . 400 200 .

o· .......--.2

6

11

15

20

43

55

344

638

Figure 1: Correspondence between heuristic computation time and Rap Table 4 Computation times comparison Ex

Moy Moy So Predo 3.50 1.25 1.83 3.00 3.15 4.15 3.56 6.06 2.70 7.43 7.41 4.93 4.31 11.60 5.53 12.21 6.42 11.49 3.47 28.86 5.44 44.43 5.51 49.00

1 3 4 5 6 8 13 14

obj 26 33 40 40 38 40 40 36

CPLEX tps 30" 1'29" 23" 5'1 " lh9'4" 26" 18h22'15" 45"

COMSOAL tps I" 26 I" 33 0.2" 40 6" 40 0.6" 38 5" 40 0.39" 40 3.2" 38

obj

For small and medium examples, all the methods give the same optimal results (for the criterion value point of view). For large example, only the heuristic is possible, but, it do not guarantee the optimum.

Table 3 show that there is a relation between the computation time for heuristic based on COMSOAL approach and Rap:

Rap = MoyPredo

18 23 43

1800 '600 - - -

11 15 70 80 70 20 100 35 80 27 300 12

Table 2 Some other computed results for studied examples

Moy Ex MHo mHo MSo 11/S0 Ho I 12 4 10.25 9 3 5.33 3 2 6 5 3 3 16 3 11 3 12.38 4 12 4.72 4 9 3 8 5 15 2 3.43 4 4 5.55 6 18 12 6 4 7 12 1 4.77 3 8 20 14 7 16.16 5 9 2J 7 6 693 16 10 15 6 5 2 10.06 14.00 4 11 21 10 7 12 21 13 7 6.15 5

Rap

Heuristic time 2" 6" 8" 11" 16" 20" 43" 55" 344" 638"

5. MODIFIED MIP MODEL Two following modifications can improve the MIP model. The first consists in the following: a) All the variables Zk, k = 1.2, ... , 1110, are replaced by one new variable Z ::: 0 that represents the number of stations: b) The constraints (10) are changed:

,WinBo

The MIP model is resolved with Cplex solver. In Table 4. comparison between the results of two methods is given. In the first column the example number is given. The second and third colulllns give

Z :::kr,!. k=!.!.!..!.!.!.+ I. .... 11I0. qES(k).

265

(10')

ACKNOWLEDGMENT where !!1 is a lower bound of the number of the stations at the line; c) The objective function (11) is replaced by: qo

lft)

C,Z+C 2 IYq lJ~O

where

1;,

IqYq

+1;,

~min,

This work IS partially supported by EU INTAS project 00-217.

(11')

REFERE CES

lJ~O

Arcus, A.L. (1966) COMSOAL: A computer method of sequencing operations for assembly lines. International Journal of Production Research 4, 259-277. Baybars, I. (1986) A survey of exact algorithms for the simple assembly line balancing. Management Science, 32, 909-932. Dolgui, A., Guschinsky, N. and Levin, G. (1999) On Problem of Optimal Design of Transfer Lines with Parallel and Sequential Operations. Proc. of the 7th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA'99), October 18-21, 1999, Barcelona, Spain. J.M. Fuertes (Ed.), Vol. 1, pp. 329-334. Dolgui, A., Guschinsky. N. and Levin, G. (2000) Approaches to balancing of transfer line with block of parallel operations, Institute of Engineering Cybernetics/University of Technology of Troyes, Minsk, Preprint No. 8, 42 pages. Dolgui, A., Guschinsky, N., Levin, G. and Harrath, Y. (2001) Optimal Design of A Class of Transfer Lines with Parallel Operations. In: Manufacturing, Modeling, Management and Control: A Proceedings Volume of the IFAC Symposium, P. Groumpos and A. Tzes (Eds.), Elsevier Science, pp. 36-41. Dolgui, A., Guschinsky, N. , Harrath, Y. and Levin, G. (2002a) Une approche de programmation lineaire pour la conception des lignes de transfert. Journal Europeen des Systemes Automatises, 36 (1),11-33. Dolgui, A., Finel, B., Guschinsky, N., Levin, G. and Vernadat, F. (2002b) A Transfer Line Design and Balancing Approach. Proceedings of the Digital Enterprise Technology (DET'02) Seminar, Durham, UK, pp. 207-210. Groover, M.P. (1987) Automation, Production Systems and Computer Integrated Manufacturing, Prcntice Hall, Englewood Cliffs, New Jersey. K. (1996) Manufacturing Systems Hitomi, Engineering, Taylor & Francis. Rekiek, B., Dolgui, A., Delchambre, A., and Bratcu. A. (2002) State of art of assembly lines design optimisation. Annual Reviews in Control 26(2), ]63-174. Schol\. A. and Klein, R. (1998) Balancing assembly lines effectively: a computational comparison. European Journal of Operational Research 114, 51-60. Schol1. A. (1999) Balancing and sequencing of assembly lines. Heidelbcrg Physica.

is a sufficiently small nonnegative value.

The constraint (10') and the objective function (11') allow to obtain problem decisions with the consecutive numeration of stations and blocks at each station (there are no "empty" blocks nor stations). However these decisions have as rule a non balanced block distribution between the stations. The blocks will be concentrated at the first stations. This drawback can be eliminated by to introduce another objective function: lJ u

lJ u

C,Z + C 2 IYq + 1;2 q~O

Iq * YlJ

~ min,

(I 1")

lJ~O

where q*=q- Jq/no[no. Ja[ is the integer part of the number a, and 1;2 is also a sufficiently small nonnegative value. The objective function (11") gives the maximal balance of blocks into stations under the same value of cost criterion. This property of the model may be important for some transfer line.

6. CONCLUSION In this paper. a balancing problem for transfer lines in mechanical industry is presented. The problem is to assign the given set of operations to the workstations and to the blocks satisfying given technological constraints. The generation of optimal or "good" design decision for such a type of line is a very complex optimisation problem. In comparison with the classical assembly line balancing problem, this problem has a lot of additional properties and constraints. The most important difference is the possibility to group operations in blocks. All the operations of the same block are executed simultaneously. In this paper. one exact and one heuristic methods are presented. The exact MIP method gives optimal solutions for small and medium problems. For large problems the heuristic proposed based on COMSOAL give good solutions and the computation time is not too big.

266