Minimizing Makespan Subject to Budget Limitation in Hybrid Flow Shop

Seyed Mohammad Hossein Hojjati', Amin Sahraeyan"'Assistant Professor of Shiraz Islamic Azad University (Hojjaty mh@iaushiraz.ac.ir)

2M.Sc. in I.E. ofNajafabad Islamic Azad University (Aminsahraeyan2003@yahoo.com)

ABSTRACT

Flow shop production lines have been practiced for many years in manufacturing systems. Makespan is the totalcompletion time of all the activities and is known as one of the most criteria in manufacturing systems. Minimizing thiscriteria causes more efficiently use of the resources specially machinery and manpower. By assigning some budget to someof the operations the operation time of these activities will be reduced and will affect the total completion time of all theoperations (which is called makespan). In this paper this issue is practiced in hybrid flow shops. At first we convert hybridflow shop to a network model and by using a linear programming approach it is identified in order to minimize makespan (the total completion time of the network) which activities are better to absorb the predetermined and limited budget.Minimizing the total completion time ofall the activities in the network is equivalent to minimizing makespan in productionscheduling. The problem is solved by lindo and the result helps the decision maker to assign the budget to the desiredoperations.

Keywords: hybrid flow shop, makespan, linear programming, budget

1- INTRODUCTION

A kind of production scheduling is flow shopscheduling, Flow Shop is a production system inwhich the sequence of performing operations on themachines is fix throughout the production line. Whenwe use different production line in parallel form thisis called hybrid flow shop (H.F.S.). In real situationmost of the flow shops are in the form of hybrid, forwhich the execution of each job must go throughmultiple stages in one specific order and at each stagethere are parallel machines availab lc to process thejobs that have entered the stage. Most of the researchworks for scheduling problems are done in the simpleflow shop. In these production systems there is onlyone machine in each stage for processing thedifferent jobs.

In simple flow shop when there are two or threestages, Johnson [4] suggests a good solution tominimize make span. For more than 3 stagesCampbell, Dudeck and Smith [1] suggest theirsolution (called CDS). Hojjati and et al [2] also havesolution to minimize make span in job shop and flowshop systems. It seems that one of the recent papers issuggested by Hojjati and et al [3] which minimizesmake span in simple flow shop by assigning somebudget for crashing the activities with the goal ofminimizing Makespan. In their paper they haveconverted the simple flow shop to network model andby using linear programming approach the problem is

978-1-4244-4136-5/09/$25.00©2009IEEE

solved. The main difference between the suggestedpaper and that of Hojjati is the difference betweensimple flow shop and hybrid flow shop.

In this article a hybrid flow shop is practiced andit is tried to crash the operations by assigning somebudget, which results minimizing the totalcompletion time ofall the operations. Here at first theterminology of the approach is presented, then thegeneral formula for n jobs with m machines in eachstage and k stages are modeled, it is followed by anumerical example and fmally the problem is solvedusing a linear programming algorithm. At the end asensitivity analysis is presented to show the effect ofdifferent budgets to the termination time of theproject. General approach is to convert the hybridflow shop to network model, then by the use of linearprogramming ( L.P. ) it is tried to minimize thecompletion time of the last node. For solving theproblem the lindo software is used.

2-NOMENCLATURE

The following terminology is used for modelingthe problem:

M Number ofmachinesN Number ofjobsr Job numberm Machine numberS Stage numberT: Starting time ofnode jJ: An activity number

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Jrms: Job r on machine m in stage sij: Activity from node i to jDij: Normal duration time ofactivity from node i tonodejD: Minimum crashing time of activity i to jd:Crashed duration time ofactivity i to jCij: Slope ofcrashing cost for activity i to jB: Predetermined budget

3-CONVERTING HYBRID FLOW SHOP INTONETWORK MODEL

We can illustrate a general form ofH.F.S. with njobs mmachines, and s stages as in Fig.1

"

\\

\\

\\

\\

\ \

\ \ \\ \ \ \

"~\ """J2 11 "J212\ 7 9 ----,- ~ jm 'l3'\ 8 ... \,~" " <.,,' \\ "

~\ \,", ............~" "8 J

221 11 ~ - - - - - 12 h22 14"

3 " 5 "

"~\ "~\\ J \ J\ 311 17 \ 19 312 21\ 15 -----," 7 \ "<, ,,'\ 7 \

\ \ ... -:., \ \

\ \"" ...... , \ "~J321 .... 'J8< ~~ '~ J 3 22 .... '22' ',~\\~\\

"~\ "~\"J4 11 "J4 12\ 23 25 - - - - - ""\ 27 29 ...\ 8 ... " 8 ........\ ' ... " \ .............t®\ ;', \ 31

\ " ... , \ "h J421 _ C"""(' ..._~ J422 _~ ,,'

~------~'

Fig.1 - General Model ofH.F.S.

In figure 1, performing operations of the machinesare indicated by solid lines and the dash lines areused to show the dummy activities or thetechnological constraints [5]. By technologicalconstraint we mean each machine can operate jobnumber 2 when it is finished job number one in onehand and in the other hand job number 2 can beoperated when the job number one has beenprocessed.

3-1 ASSUMPTIONS

The following assumptions are considered:l-(Shortest Processing Time) SPT rule is used toassign the jobs to the machines.2- Each machine starts at its earliest starting possibletime.

3- The set up time is included in the processing time.4- One unit of item for each job is considered.5- Interruption of the machines is not allowed (norepairing during processing).6- Each machine can process only one job at a timepoint.

Each operation has a predecessor which is shownin table 1. There are two sets of predecessors, one,the operational constraint, for which every job shouldbe processed in its earlier stage, and secondtechnological constraint for which each machineshould operate the jobs in chronological order. Thisissue is defined more below the figure one.

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Table1- Predecessors for General Model

Jrm1 m = lor 2 or ... or M , r = 1Jrms m = 1,2,... ,M , r = 1 , s = s-

1

Jrm1 m = lor 2 or ... or M , r =1,2, ,r-1

Jrm1 Jrms m = 1,2, ,M , r = r , S = s-l

The problem can be formulated as follows:

MinZ = TN - T 1

ST.

~~C .. (D .. -d .. )~BL..J L..J I,} I,} I,}

D f(i,j) ~ d i,j ~ D i,j

(1)

(2)

(3)

(4)

2Jrm2

Jrm1 m = lor 2 or ... or M , r =1,2,... ,N-1

Jrms m = 1,2,...,M , r = N , s = s-l

Jrm2 m = lor 2 or ... or M , r = 1Jrms m = 1,2,... ,M , r = 1 , s = s-

1

Jrm2 m = lor 2 or ... or M , r =1,2, ,r-1

Jrms m = 1,2, ,M , r = r , s = s-l

Jrm2 m = lor 2 or ... or M , r =1,2, ,N-1

Jrms m=1,2, ,M,r=N,s=s-1

Drm2

Ti,Tj,di,j = int eger

In the above model the objective function is tominimize the difference between the final nodecompletion time and the initial node starting time,this will cause the total completion time of all thenetwork to be minimized.The first constraint equation (2) is the budgetconstraint. The term D-d is the amount of crashingtime if it is multiplied by the unit crash time cost, thetotal cost for crashing the network is determined.This amount should be less than the total budgetassigned to the programming which is equal to B.The second constraint equation (3) is thetechnological constraint which is discussed earlierand the third constraint equation (4) is the lower andupper bound ofthe times for each operation.

4- NUMERICAL EXAMPLE

The methodology is illustrated using a numericalexample with 4 jobs, 2 stages and 2 machines in eachstage. The problem is solved using SPT rule. The

network is illustrated in Fig. 2

3-2 PROBLEM FORMULATION

Fig. 2- Network ofthe Numerical Example

J112

J 11 14

1

21 J3213

20

Table 2- Processing Time Information Table 4- The Assumptions of the Example For Th eCosts

4-1 PROBLEM SOLUTION

Table 3- Predecessor ofthe Numerical Example

For this problem the predecessors are shown In

table 3

8(4-d12) +15(3-dI4) +20(3-d46) +(7-d3S)+12(8-d78) <=64 (6)

Activity Crashed time Budget used(d. )

dl2 4 0

d23 4 0

dl4 2 15

d46 3 0

d45 5 0

d67 3 0

d78 5 36

d38 5 12

obiective 13 63

Table 5- The Result ofthe Example

By assigning different budgets , different results canbe obtained, this is cal1ed the sensitivity analysis ofthe problem, the result can be shown as in table 6.

The problem is solved by lindo and the result isshown in table 5

--- I;l ~ "".£~

s:: ...: 0.

2 '" .9 (1) --- :::l 0

oS § "" .~ [;i

"" o .... '@ .5 6' 0"""0 ""0

u....., "0 :::l ~ "-' ~o "'inZ <: "-' 2 0 ~

0c, u

1,2 1111 --- 4 3 82,3 1112 1111 4 4 ---1,4 1221 --- 3 1 154'5 Jz22 1221 5 5 ---4,6 1321 1221 3 2 203,8 1312 1112• 132 1 7 5 6

6'7 1421 1321 3 3 ---7,8 1422 1222 1421 8 5 12

(5)

(8)1<=dI4<=3 2<=d46<=35<=d78<=8 d2F4

d6F3 Tt, Tj = integer

T3-T2>=d32 T4-Tl >=d41 (7)T5-T4>=d54 T7-T6>=d76T3-T6>=O Ts-T7>=ds7

Subject To:

Min. Z= 'rs-Tt

3<=d12<=45<=d3S<=7D4s=5

job Stag e I Stage 2MC I MC2 MC I MC2

I 4 3 4 52 4 3 4 53 7 3 7 84 8 3 8 8

Considering the information given for the problem intables 2 and 3 and Fig. 2 the objective function andthe con straints can be written as fol1ows:

T2-Tl >=d21T6-T4>=d46T7-T5>=OTs-T3>=ds3

activity Duration(i,j) node predecessor time(lrm, ) (DLi)

1,2 1111 --- 42,3 1112 1111 41,4 1221 --- 34,5 1222 1221 54,6 1321 1221 33,8 h12 1112 1321 76,7 1421 1321 37,8 1422 1222 1421 8

Now we should assign some budget to someactivities (operation) for which their time can bereduced. These are sho wn in table 4

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Table 6- Sensitivity Analysis of the NumericalExample

budget Make span

0 17

12 16

24 15

42 14

64 13

5- CONCLUSION AND FURTHER RESEARCH

It is shown that hybrid flow shop problems can beconverted to network model and by the use ofa linearprogramming formulation the critical activities aredetermined. Assigning some budget to activities thatcan be crashed by time, causes to reduce thecompletion time ofall the project or make span, thisby itselfcauses better use of the resources speciallymachinery and manpower, which by itself increaseproductivity.

For further research it is suggested to apply themethodology to some other systems like parallel flowshops

REFERENCES

[I] Campbell, H.G., R.A. Dudeck, M. L. Smith" Aheuristic Algorithm for the n job m machinesequencing problem" Management Science vol. 16,PP 630-637 june 1970

[2] Hojjati, S.M.H.,M.T. Taghavifard "Schedulingnjobs on m machines using Branch and BoundTechnique" Moddaress journal of Research, No.11,Spring 2003

[3] Hojjati, S.M.H., M. Bagherpour, "MinimizingMake Span Subject to Budget Limitation in FlowShop Scheduling". The 35th CIE Conference onComputer & Industrial Engineering. June 19-22,2005 Istanbul, Turkey

[4] Johnson S.M. "Optimal two-and three-stageproduction schedules with setup times included".Nav. Res. Log. PP61-68, Qeuarterly 1

[5] Baker, Kenneth R. " Inrtoduction to sequencingand shceduling " john wiley and sons ,PP 181-187,1974

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