Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tcim20 Download by: [WU Vienna University Library], [Mohammadmehdi Hakimifar] Date: 09 June 2016, At: 04:20 International Journal of Computer Integrated Manufacturing ISSN: 0951-192X (Print) 1362-3052 (Online) Journal homepage: http://www.tandfonline.com/loi/tcim20 Robust and Fuzzy Optimisation Models for a Flow shop Scheduling Problem with Sequence Dependent Setup Times: A real case study on a PCB assembly company Seyed Mohammad Gholami-Zanjani, Mohammadmehdi Hakimifar, Najmesadat Nazemi & Fariborz Jolai To cite this article: Seyed Mohammad Gholami-Zanjani, Mohammadmehdi Hakimifar, Najmesadat Nazemi & Fariborz Jolai (2016): Robust and Fuzzy Optimisation Models for a Flow shop Scheduling Problem with Sequence Dependent Setup Times: A real case study on a PCB assembly company, International Journal of Computer Integrated Manufacturing, DOI: 10.1080/0951192X.2016.1187293 To link to this article: http://dx.doi.org/10.1080/0951192X.2016.1187293 Published online: 08 Jun 2016. Submit your article to this journal View related articles View Crossmark data
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Full Terms & Conditions of access and use can be found athttp://www.tandfonline.com/action/journalInformation?journalCode=tcim20
Download by: [WU Vienna University Library], [Mohammadmehdi Hakimifar] Date: 09 June 2016, At: 04:20
International Journal of Computer IntegratedManufacturing
Robust and Fuzzy Optimisation Models for aFlow shop Scheduling Problem with SequenceDependent Setup Times: A real case study on aPCB assembly company
Seyed Mohammad Gholami-Zanjani, Mohammadmehdi Hakimifar,Najmesadat Nazemi & Fariborz Jolai
To cite this article: Seyed Mohammad Gholami-Zanjani, Mohammadmehdi Hakimifar,Najmesadat Nazemi & Fariborz Jolai (2016): Robust and Fuzzy Optimisation Models for a Flowshop Scheduling Problem with Sequence Dependent Setup Times: A real case study on aPCB assembly company, International Journal of Computer Integrated Manufacturing, DOI:10.1080/0951192X.2016.1187293
To link to this article: http://dx.doi.org/10.1080/0951192X.2016.1187293
Robust and Fuzzy Optimisation Models for a Flow shop Scheduling Problem with SequenceDependent Setup Times: A real case study on a PCB assembly company
Seyed Mohammad Gholami-Zanjani, Mohammadmehdi Hakimifar, Najmesadat Nazemi and Fariborz Jolai*
University of Tehran, School of Industrial Engineering, Tehran, Islamic Republic of Iran
(Received 11 May 2014; accepted 28 January 2016)
In this paper, authors considered a flow shop scheduling problem with sequence dependent set-up times in an uncertainenvironment. Its objective function is to minimise weighted mean completion time. As for uncertainty, set-up andprocessing times are considered not to be deterministic. Authors propose two different approaches to deal with uncertaintyof input data: robust optimisation (RO) and fuzzy optimisation. First, a deterministic mixed-integer linear programmingmodel is presented for the general problem. Then, its robust counterpart of the proposed model is dealt with. Afterwards, thefuzzy flow shop model is developed. Moreover, a real case study on Tehran-Madar Company which is a producer of printedcircuit board and OEMs is studied. Finally, a considerable discussion is held on comparison of all three approaches ofnamely deterministic, fuzzy and ROs based on some generated numerical examples.
In many manufacturing and assembly facilities, each jobhas to undergo a series of operations. Often, these opera-tions have to be done on all jobs in the same order,implying that the jobs have to follow the same route.The machines are then assumed to be set-up in seriesand the environment is referred to as a flow shop. Thereis a set of jobs that must be processed by a given numberof machines. All machines are disposed in series and jobsvisit machine 1 first, then machine 2 and so on until thelast machine (Pinedo 1995).
The classical form of flow shop scheduling problem istoo theoretical and sometimes impractical. Consideringset-up times helps us to model real industry (Ciavotta,Minella, and Ruiz 2013). Machine set-up time is an impor-tant item for production scheduling in flow line, and itmay easily consume about one-fifth of accessiblemachine’s capacity if not well handled (Pinedo 1995).Set-up times are not considered as a part of the job’sprocessing time and include non-productive operations.Set-up times can be classified into two major groups.The first one is Sequence Independent Set-up Time(SIST), i.e. the set-up or changeover time for a machineonly depends on the job that is to be processed next. Thesecond group is the Sequence Dependent Set-up Time(SDST) case, where set-up time depends both on thecurrent job being processed and on the next job in thesequence. This second category is much more complexand includes the first one as a particular case, permitting itto describe several operational scenarios. Scheduling
problems with SDSTs are classified as the difficult sche-duling problems (Naderi et al. 2011).
In the related existing literature, a large number ofresearches have been conducted towards deterministicflow shop scheduling problems. There are lots of studiesin the literature about deterministic flow shop schedulingproblems in which all parameters are considered to beknown and constant (Ciavotta, Minella, and Ruiz 2013;Mehravaran and Logendran 2012; Kurz and Askin 2004;Ziaee and Sadjadi 2007).
However, in the real world conditions, industrial sys-tems have probabilistic behaviour. For example, processingand set-up times can be random variables, machines canbreakdown and access to the jobs can be unstable. Thesenondeterministic behaviours show that uncertain models aremore realistic than certain ones (Gu et al. 2010).
The three common approaches for modelling the sche-duling problems with uncertain parameters are stochasticprogramming, fuzziness and robust optimisation (RO). Ifuncertain parameters’ distribution functions are known orhave probabilistic description, then the problem becomes astochastic scheduling problem that can be modelled withstochastic programming. This approach has a long andrich history. Elmaghraby and Thoney (1999) consideredstochastic job durations with arbitrary distribution for atwo-machine flow shop problem with the objective ofminimising the expected makespan. Gourgand, Grangeonand Norre (2003) considered flow shop problem withstochastic processing times. They suggested a recursivealgorithm based on a Markov Chain to compute Cmax andevaluate the solution. They presented a discrete event
simulation model. Choi and Wang (2012) proposed anovel decomposition-based approach, which combinesboth the shortest processing time and the genetic algo-rithm to minimise the makespan of a flexible flow shop(FFS) with stochastic processing times. Liefooghe et al.(2012) also introduced several multi-objective methodsthat are able to handle any type of probability distribution.In their model, processing times are random variables. Kuand Niu (1986) studied a stochastic generalisation ofordering of the set of jobs to minimise the time until thesystem becomes empty, i.e. the makespan in which jobprocessing times are independent random variables. Theirmain result is a sufficient condition on the processing timedistributions that imply that the makespan becomes sto-chastically smaller when two adjacent jobs in a given jobsequence are interchanged. They also give an extension ofthe main result to job shops. Wang, Zhang and Zheng(2005) presented a hypothesis-test method, an effectivemethodology in statistics, employed and incorporatedinto a genetic algorithm to solve the stochastic flow shopscheduling problem and to avoid premature convergenceof the genetic algorithm.
Gholami, Zandieh and Alem-Tabriz (2009) dealt with thehybrid flow shop scheduling problems in which there areSDSTs, and machines which suffer from stochastic break-down to optimise objectives based on expected makespan.But in the real cases, there is not enough historical data aboutuncertain parameters, so the probabilistic distribution ofuncertain parameters can hardly be found. Especially inview of chance constraint planning, chance constraint canomit convexity feature and increase complexity of the mainproblem (Charnes and Cooper 1959).
Fuzzy optimisation is another uncertainty detectedmethod. Zadeh (1965) proposed the concept of fuzzysets. This method is useful in modelling real environment.The fuzzy scheduling problems have been investigated bymany researchers (Lai and Wu 2011). Some researchersused fuzzy approach to deal with uncertainty in flow shopscheduling problems. Fuzzy processing times and flexibledue dates were discussed in Kilic’s research (Kilic 2007).In his study, ant colony optimisation meta-heuristicapproach is used to find schedules. Temİz and Erol(2004) proposed the fuzzy concept to flow shop schedul-ing problems. The branch-and-bound algorithm of Ignalland Schrage was modified and rewritten for three machineflow shop problems with fuzzy processing times. Fuzzyarithmetic on fuzzy numbers is usually used to determinethe minimum completion time (Cmax). Proposed algorithmgets a scheduling result with a membership function forthe final completion time. With this determined member-ship function, a wider point of view is provided for themanager about the optimal schedule. McCahon and Lee(1992) proposed the Campbell, Dudek and Smith jobsequencing algorithm and modified it to accept trapezoidalfuzzy processing times. Ishibuchi et al. (1994) formulated
a fuzzy flow shop scheduling problem where the due dateof each job was given as a fuzzy set. The membershipfunction of the fuzzy due date was corresponding to thegrade of satisfaction of a completion time. The objectivefunction of the formulated problem is to maximise theminimum grade of satisfaction over given jobs.Tsujimura et al. (1993) showed that fuzzy set theory canbe useful in modelling and solving flow shop schedulingproblems with uncertain processing times. They also illu-strated a methodology for solving a job sequencing pro-blem, in which opinions of experts about processing timeswere totally different. Triangular fuzzy numbers (TFNs)are used to represent the processing times of experts. Zareand Fakhrzad (2011) presented an efficient algorithm tosolve FFS problems using fuzzy approach. Their goal wasto minimise the total job tardiness. Yao and Lin (2002)investigated an approach on incorporating statistics withfuzzy sets in the flow shop scheduling problem. This workwas based on the assumption that the precise value for theprocessing time of each job was unknown, but somesample data are available. Their work intended to extendthe crisp flow shop sequencing problem into a generalisedfuzzy model that would be useful in practical situations.
The next approach is ROwhich is a more recent and novelmethod to face by uncertainty in optimisation in which uncer-tainty model is neither stochastic nor fuzzy, but rather set-based. The RO method can provide a feasible solution underall possible realisations of uncertain parameters within theirbounds (known as a robust feasible solution) with the bestobjective function value between all the robust feasible solu-tions. In other words, this approach tries to find solutionswhich are less sensitive towards changing uncertain para-meters (Baohua and Shiwei 2009). The first step in developingthe RO theory was taken by Soyster (1973) who investigatedexplicit formulations for RO. Two decades later, Ben-Tal andNemirovski (1998, 2000) and El Ghaoui, Oustry and Lebret(1998) put a significant step forward in RO theory by propos-ing models for uncertain linear problems with ellipsoidaluncertainties. They also solved the counterparts of the nominalproblem in the form of conic quadratic problems (Pishvaee,Rabbani, and Torabi 2011).
Scheduling problems are subjected to volatility and uncer-tainty due to their random nature, measurement errors or manyother reasons. Two key sources of uncertainty in any schedul-ing system in real world problems are the processing time andset-up time, which might be caused by human factors oroperating faults. These parameters are assumed to be uncertainin this research. Often, in some cases, it is not appropriate orpossible to extract exact distribution functions for uncertaindata. Being mindful of the fact that the proposed problem isoperational, in many real cases, (1) there is not enough histor-ical data for the uncertain parameters, (2) the collected datawould be inadequate and (3) the collected data would beunreliable because of any errors. Therefore, in case of thesedrawbacks, which are common in scheduling environments,
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the stochastic programming becomes inapplicable. ROaddresses the problem of data uncertainty by guaranteeingthe feasibility and optimality of the solution for the worstinstances of the parameters. To the best of our knowledge,there is no research work applying the RO approach in thecontext of flow shop scheduling problem.Moreover, applyingdifferent uncertain conditions can give an insight to decisionmaker to make a more practical decision.
The remainder of this paper is organised as follows: Theconcerned problem is descripted and formulated in Section2. The robust counterpart and fuzzy mixed integer program-ming models are developed and presented in Sections 3 and4, respectively. Numerical analysis is reported in Section 5and finally, the conclusion is followed in Section 6.
2. Problem description
There are I jobs ready to be processed at time zero. Eachjob has K operations. The first operation is performed bythe first machine followed by the sequence of operationson machines to the last machine. A job once started on anymachine must be completed on it without interruption (i.e.no pre-emption is allowed).
In this model, assumptions are considered as follows:
● The objective function is to minimise the weightedmean flow-time.
● Machine efficiency is 100%.● All machines and jobs are available at zero in the
usage time.● Each operation can process only one operation in a
point of time and each operation in a point of timecan be done by just one machine.
● No pre-emption of jobs is allowed (once the proces-sing of an operation on a machine has started, itcannot be interrupted on that machine).
● There are no precedence constraints among the jobs.● The set-up times are sequence dependent.● Each stage has only one machine.
The objective function is to minimise the weightedmean flow-time or weighted mean completion time;these two objectives are equal because of availability ofall jobs at time zero.
Constraint (2) ensures that a job is not allowed to beprocessed on more than one machine simultaneously. Itmeans that a job could not start being processed on amachine unless it has been released from the previousmachine. Constraint (3) ensures that a machine is notallowed to process more than one job at each time orstates that a job could start being processed on a machinewhen its predecessor has completed processing on thatmachine.
Since authors assume all jobs to be available at timezero, constraint (4) ensures that the first job on the firstmachine should start at zero time. Constraints (5) and (6)ensure that each sequence position is filled with only onejob on each machine, and each job is assigned to onlyone position in the job sequence. Constraint (7) demon-strates that if the binary variable related to each jobpositioned in each sequence on each machine xi;j;k
� �is
set to zero, its completion time has to be zero in that
I Number of jobsK Number of stages/machines/operationsi Denotes job i; i = 1 . . . Ik Denotes machine k; k = 1 . . . Kj Denotes order j; j = 1 . . . Itik The processing time of job i on machine kcijk The completion time of job i in order j on machine kxijk Binary variable taking value 1 if job i on machine k is
processed in order jyi0;i;j0;j;k Binary variable taking value 1 if xi;j;k � xi0;j0;k ¼ 1sii0k Set-up time dependent, if job i′ is processed just after job
iwi Weight of job i
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order. Constraints (8) and (9) are suggested here toreduce the complexity of the mathematical model causedby multiplication of xi;j;k by xi0;j0;k, which makes themodel non-linear.
The resulting model is a MIP model with IðK � 1Þj j þKðI � 1Þj j þ KIj j þ KJj j þ IJKj j þ 2 I2J 2K
�� �� number ofconstraints, IJKj j number of binary variables and IJKj jnumber of continuous variables.
3. RO
Considering a typical optimisation problem where thereare different input parameters and when one or limitednumbers of input parameters are uncertain, we can usesensitivity analysis to analyse optimal solutions. It is atraditional way but when number of uncertain para-meters increases, this method is not efficient. RO is animportant methodology, computationally tractable fordealing with data uncertainty in optimisation problems.And also, it does not require any information about theuncertain data distribution. The robust method hasrecently become popular among practitioners.Generally, RO is defined as the method for guaranteeingthe feasibility and optimality of the solution for theworst cases of the parameters. First in this method, adeterministic data set is defined within the uncertainspace and then the best solution which is feasible forany realisation of the data uncertainty in the given setthat called robust counterpart optimisation is obtained.Ben-Tal and Nemirovski (1998, 2000) show that a smallperturbation on the input data for some benchmark pro-blems may result in solutions are not only optimal butalso may be infeasible. In RO, it is assumed that thebounds of the uncertain parameters are known, and theaim is to find the solutions that are guaranteed to remainfeasible for all parameters values.
In set-induced RO, the uncertain data are assumed tobe varying in a given uncertainty set. The aim is tochoose the best solution among those ‘immunised’against data uncertainty, i.e. candidate solutions thatremain feasible for all realisations of the data from theuncertainty set.
To illustrate this method, consider the following sim-ple linear mathematical model in which parameters c; d;Aand b belong to uncertain interval.
min cxþ ds:t:Ax � b
According to Ben-Tal and Nemirovski (1998, 2000),the related linear mathematical model considering uncer-tain parameters can be defined as following model:
min cxþ ds:t:Ax � b;c; d;A; b 2 U
(11)
In this paper, the applied RO based on ‘worst-case-oriented’ philosophy is used. Ben-Tal and Nemirovski(1998, 2000) defined robust counterpart as follows:
min c Xð Þ ¼ supc;d;A;b2U cxþ d½ � : Ax� b "c;d;A;b 2 U� �
(12)
The best possible robust feasible solution of problem(12) is equal to an optimal solution to problem (11). Sucha solution satisfies the constraints for all possible realisa-tions of the data and guarantees an optimal objectivefunction value not worse than cðx�Þ.
To extend the robust counterpart of the proposedmodel, flow shop with SDSTs, processing time and set-up time are considered as uncertain parameters. Each ofthese uncertain parameters is assumed to vary in a speci-fied closed bounded box (Ben-Tal et al. 2005; Ben-Tal,Ghoui, and Nemirovski, 2009). The general form of thisbox can be specified as follows:
� �where ��t is the nominal value of the �t as the parameter of
vector � (n-dimension vector) and the positive number Gt
represents ‘uncertainty scale’ and ρ � 0 is the ‘uncertaintylevel’. A particular case of interest is Gt ¼ ��t, which corre-sponds to a simple case where the box contains ��t whoserelative deviation from the nominal data is of size up to ρ.
Ben-Tal et al. (2005) have showed that in this case(closed bounded box), the robust counterpart problem canbe converted to attractable equivalent model where UBOX
is replaced by a finite set Uext consisting of the extremepoints of UBOX. To represent the tractable form of therobust model, Equations (2), (3) and (5) should be con-verted to their equivalent tractable ones. As an examplefor Equation (2), we have
XIj¼1
ci;j;kþ1 �XI
i0¼1;i0�i
si0i;kþ1xi;j�1;kþ1
!xi;j;kþ1
" #
�XIj¼1
ci;j;kxi;j;k� � � ti;kþ1
"s2UsBOX Us
BOX¼ s2<ns : st��stj j�ρsGts; t¼1;2;...;nf g��
(13)
"T2UTBOX UT
BOX¼ T2<nT : Tt��Ttj j�ρTGtT ; t¼1;2;...;n
� ���(14)
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So, we have
XIj¼1
ci;j;k �XI
i0¼1;i0�i
si0;i;kþ1yi0;i;j�1;j;kþ1 þ ηsi0;i;kþ1
" #
�XIj¼1
ti;kþ1xi;j;kþ1 þ ηti;kþ1
� ��XIj¼1
ci;j;k "i;"k < K
(15)
ρsGsi0;i;kþ1yi0;i;j�1;j;kþ1 � ηsi0;i;kþ1 "i0; i; j; k
ρsGsi0;i;kþ1yi0;i;j�1;j;kþ1 � �ηsi0;i;kþ1 "i0; i; j; k
Similarly, for all constraints that include uncertainparameters, the same is done. Finally, with respect to(13), (14) and (15), the tractable form of the robustmodel can be presented as follows:
Z ¼ minXIi¼1
XIj¼1
ci;j;kwi
XIj¼1
ci;j;k �XI
i0¼1; i0�i
si0;i;kþ1yi0;i;j�1;j;kþ1 þ ηsi0;i;kþ1
" #
�XIj¼1
ti;kþ1xi;j;kþ1 þ ηti;kþ1
� ��XIj¼1
ci;j;k "i;"k < K
ρsGsi0;i;kþ1yi0;i;j�1;j;kþ1 � ηsi0;i;kþ1 "i0; i; j; k
ρsGsi0;i;kþ1yi0;i;j�1;j;kþ1 � �ηsi0;i;kþ1 "i0; i; j; k
A fuzzy set number is a normal convex fuzzy subset of thereal line with an upper semi-continuous membership func-tion. Assuming ~c as a fuzzy number, its membershipfunction is as follows:
μ~cðxÞ ¼fcðxÞ if a1 � x � a21 if a2 � x � a3gcðxÞ if a3 � x � a40 if x � a1 or x � a4
8>><>>:
where fcðxÞ and gcðxÞ are considered as left and right handside of the membership function.
Since many operators distort the shape of TFNs inscheduling problems such as flow shop scheduling(Hong and Wang 2000), ~c is considered to be TFN here.Thus, the membership function can be defined as follows:
μ~cðxÞ ¼fcðxÞ ¼ x�a1
a2�a1if a1 � x � a2
1 if x ¼ a2 ¼ a3gcðxÞ ¼ a4�x
a4�a2if a2 � x � a4
0 if x � a1 or x � a4
8>><>>:
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According to Jiménez et al. (2007), the expected inter-val (EI) and expected value (EV) of TFN, ~c, are defined asfollows:
EIð~cÞ ¼ Ec1;E
c2
¼ ð10f �1c ðxÞdx;
ð10g�1c ðxÞdx
� �
¼ 1
2a1 þ a2ð Þ; 1
2a2 þ a4ð Þ
� �
EVð~cÞ ¼ Ec1 þ Ec
2
2¼ a1 þ 2a2 þ a4
4
Moreover, regarding the ranking method presented byJiménez et al. (2007), for any pairs of fuzzy numbers ~a
and ~b, the degree in which ~a is greater than ~b is defined asfollows:
μM ð~a; ~bÞ ¼0 if Ea
2 � Eb1 0
Ea2�Eb
1
Ea2�Eb
1� Ea1�Eb
2ð Þ if 0 2 Ea1 � Eb
2 ;Ea2 � Eb
1
1 if Ea
1 � Eb2 � 0
8><>:
(16)
When μM ð~a; ~bÞ � α, it says that ~a is greater than, or
equal to, ~b at least in degree α and it represents as ~a�α~b.
Also, according to the definition of fuzzy equations inParra et al. (2005), for any pair of fuzzy numbers ~a and~b, it says that ~a is equal to ~b in degree of α if the followingrelationships hold simultaneously:
~a�α=2~b; ~a�α=2
~b (17)
The above equations can be rewritten as follows:
α2� μM ð~a; ~bÞ � 1� α
2
Now, the corresponding fuzzy mathematical program-ming model in which all parameters are defined as TFNsis considered as follows:
min z ¼ ~ctx
s:t:
~ai x � ~bi i ¼ 1; . . . ; l
~aix ¼ ~bi i ¼ l þ 1; . . . ;m
x � 0
(18)
According to Jiménez et al. (2007), a decision vector,x 2 <n, is feasible in degree α if the following term is
correct: mini¼1;...;m μM ~aix; ~bi� �� � ¼ α.
According to (16) and (17), the equations ~aix � ~bi and
~aix ¼ ~bi are, respectively, equivalent to
Eaix2 � Ebi
1
Eaix2 � Eaix
1 þ Ebi2 � Ebi
1
� α; i ¼ 1; . . . ; l
α2� Eaix
2 � Ebi1
Eaix2 � Eaix
1 þ Ebi2 � Ebi
1
� 1� α2; i ¼ l þ 1; . . . ;m
Therefore, considering EI and EV of fuzzy numberdefinitions, the equivalent crisp α-parametric model ofmodel (18) can be written as follows:
min EVð~cÞxs:t:
ð1� αÞEai2 þ αEai
1
x � αEbi
2 þ 1� αð ÞEbi1
1� α2
� �Eai2 þ α
2Eai1
h ix � α
2Ebi2 þ 1� α
2
� �Ebi1
α2Eai2 þ 1� α
2
� �Eai1
h ix � 1� α
2
� �Ebi2 þ α
2Ebi1
x � 0
According to above descriptions, the equivalent aux-iliary crisp model of the SDST flow shop model can beformulated as follows:
To assess the performance of fuzzy and robust flow shopmodels with SDSTs, some numerical experiments are con-ducted, and the obtained results are reported in thissection.
To this aim, three test problems with different sizes areconsidered. These test problems are generated with regardto Taillard’s (1993) benchmark. Since the considered pro-blem is NP-hard, small-sized instances of the problem arerandomly generated so as to be solved in a reasonablecomputational time. Table 1 shows the distributions usedto generate random instances.
The experiments are performed under four differentuncertainty levels for both robust and fuzzy methods (i.e.ρ ¼ 0:2; 0:5; 0:7; 1 and α ¼ 0:2; 0:5; 0:7; 1). Uncertainty
bound is 10% of nominal data. All the models are solvedunder nominal data by GAMS 23.5 optimisation softwareand all tests are carried out on a Core i7 1.6 GHz computerwith 4 GB RAM.
The results of the experiments under uncertain proces-sing times and set-up times are reported in Table 2.
Table 2 shows results of three different approaches(deterministic, robust and fuzzy) for a given problem.Each part consists of problem characteristics such asobjective function value, run time, job sequences, numberof constraints and variables.
As the results show, although some information aboutmembership function and experts’ idea on uncertain para-meters are required, the optimum sequences gained fromdeterministic and fuzzy approaches for different levels ofα are exactly the same. However, in RO approach, somechanges are resulted in terms of job sequences to deal withuncertainty for some levels of ρ. It is noteworthy thatrobust approach requires less information about uncertainparameters in comparison with the fuzzy approach. Forinstance, for the problem with six jobs and two machines,the optimum sequence is changed to uncertainty levelρ � 0:5, in the same way for the problem 5 × 3 the changehappens in ρ ¼ 0:7 and ρ ¼ 1 for the third conductedexperiment. These changes of a robust approach in opti-mal sequences show that RO applied to this problem issensitive to variations of input data. In other words, itmakes the solution more immune to uncertainties butwith costs more than deterministic and fuzzy approaches,so it depends on the decision maker to consider a trade-offbetween the system’s reliability and cost.
In this part, a remarkable sensitivity analysis is appliedto job processing times from another point of view. Theeffects of the processing times’ standard deviation areassessed on the flexibility of the system by using differentapproaches of uncertain models. For this purpose, nominalprocessing times were generated with different standarddeviations, and then the proposed models were run. Theresults are depicted in Figures 1 and 2. For differentscenarios of standard deviation, the robust and fuzzy mod-els were run in the range of [0, 1] for α and ρ with stepsof 0.1.
The results demonstrate that while the fuzzy model isnot affected by the standard deviation, and job sequencesare constant and correspond to deterministic model’s solu-tion for different levels of α, the robust model is totally
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sensitive. The less standard deviation of processing timesleads to more changes in job sequences, which means themore standard deviation, the less sensitive robust model.The robust model tries to change the sequences to keep thecosts at the least possible amount. For instance, thesequence changes for Sigma = 0 and each step of uncer-tainty level, but these changes are reduced by increasingthe Sigma level and finally for more than Sigma = 2; therobust model is no more sensitive to uncertainty level andthus, three approaches of deterministic, fuzzy and robustresult in the same sequences.
6. Case study
In this section, the proposed uncertain models areapplied to a printed circuit board (PCB) assembly com-pany. The company studied is Tehran-MadarManufacturing department (established in 2004). Themain productions of this manufacturing department arePCBs. Because of highly competitive situations, mini-mising the manufacturing times of PCBs and also deal-ing with uncertain data are crucial to electronicsindustries. Like any other industries, electronics indus-tries are forced to be robust confronting changes to keepcosts down. A typical PCB manufacturing process con-sists of Kitting, Insertion, Wave Solder, NodalImpedance Test, Burn-In and Functional Test.
The very common characteristics of these assemblysystems are presented below:
1. Multiple types of boards are assembled in the lineand each board follows a fixed route.
2. Boards are processed on a machine just once.3. The line consists of different types of machines.4. Processing time variation is not negligible.5. Orders are assumed to be ready at time zero.6. Set-up times are not too small compared to proces-sing times.
7. Machines are assumed to be reliable with availabil-ity of 100%.
The assembly centre seeks to schedule PCBs such that theweighted mean flow-time is minimised.
As shown in Table 3, it is supposed that seven jobs aregoing to be scheduled on six steps of a manufacturingprocess. Since the company owners were not enthusiasticto release the exact real data, the input data are scaled.Twenty per cent of uncertainty is controlled via a proposedRO method. As it was expected with reference to theprevious section, the sequences resulted from determinis-tic and fuzzy models are the same and therefore, it is notpresented in Table 3. Resulted sequences indicate that aminor change has happened in ρ ¼ 0:4 and a big changein ρ ¼ 1. In other words, if we face uncertain processingtimes, on level of 20%, the sequence 5-6-2-4-7-3-1 is theTa
ble2.
Resultsof
differentapproaches.
Problem
size
(|J|
*|K|)
Deterministic
Robust
Fuzzy
Objectiv
efunctio
nTim
e(s)
Sequence
No.
ofconstraints
No.
ofvariables
ρO.F.
Tim
e(s)
Sequence
No.
ofconstraints
No.
ofvariables
αO.F.
Tim
eSequence
No.
ofconstraints
No.
ofvariables
6×2
2130
321-5-4-2-3-6
5296
144
0.2
2282
96.3
1-5-4-2-3-6
5815
2815
0.2
2179
64.9
1-5-4-2-3-6
5299
2737
0.5
2444
72.9
4-1-5-2-3-6
0.5
2124
80.8
1-5-4-2-3-6
0.7
2536
110.9
4-1-5-2-3-6
0.7
2088
78.52
1-5-4-2-3-6
12674
72.6
4-1-5-2-3-6
12033
65.6
1-5-4-2-3-6
5×3
3528
124-2-3-1-5
3877
150
0.2
3600
23.94
4-3-2-1-5
4440
2111
0.2
3587
20.13
4-3-2-1-5
3880
2026
0.5
3756
17.7
4-3-2-1-5
0.5
3528
19.1
4-3-2-1-5
0.7
3870
18.54
4-2-1-5-3
0.7
3489
24.18
4-3-2-1-5
14002
28.24
4-2-1-5-3
13430
21.64
4-3-2-1-5
8×2
3728
4528
7-6-4-1-2-3-5-8
16,566
256
0.2
4018.1
23,415
7-6-4-1-2-3-5-8
17,737
8385
0.2
3799.9
8565
7-6-4-1-2-3-5-8
16,569
8449
0.5
4506.6
20,815
7-6-4-1-2-3-5-8
0.5
3713.8
18,107
7-6-4-1-2-3-5-8
0.7
4846
7898
7-6-4-1-2-3-5-8
0.7
3656.3
10,146
7-6-4-1-2-3-5-8
15415
10,488
6-4-7-1-2-3-5-8
13675
9016
7-6-4-1-2-3-5-8
8 S. M. Gholami-Zanjani et al.
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best with the objective function of 6789.5 and as a result,any other sequences will necessitate both more time andmore objective functions. It shows that the model is robustto the levels of uncertainty and is able to change thesequence to find the optimum solution.
As it was mentioned earlier, SDSTs impose consider-able complexity to the problem and makes it difficult todeal with. To show this complexity, especially in theproposed uncertain mathematical models of this paper, asensitivity analysis is presented in Figure 3. To this aim,applied to the considered case, set-up times are increased
from 0 to their nominal value in 10 steps uniformly. Thisis implemented for two scenarios of SDST and SIST.These adjustments are applied to each level of uncertaintyfor both robust and fuzzy counterparts. According toFigure 3, the mean value of computational times in therobust and fuzzy models is presented at each set-up levelof each scenario. To make it clear, in the SDST case, anincrease in set-up times will augment the complexitywhich, in turn, raises computational time dramatically.On the other hand, in the SIST case, since the set-uptime can be considered as the increase in the processing
0
2
4
6
8
10
12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cu
mu
lativ
e N
o.
of c
ha
ng
es
Uncertainty Level
sigma=.25
sigma=.50
sigma=.75
sigma=1
sigma=1.5
sigma=2
sigma=5
sigma=0
Figure 1. Comparison of processing time standard deviations with regard to uncertainty levels.
0
2
4
6
8
10
12
0 0.25 0.5 0.75 1 1.5 2 3 5
To
ta
l N
o.
of C
ha
ng
es
Sigma
Figure 2. Total numbers of changes regarding standard deviation.
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time, the imposed complexity is negligible; even in someproblem numbers, the computational time decreases. So,despite the fact that the proposed models are close to realproblems, a great part of complexity in these uncertainmodels are imposed by the SDSTs.
With respect to this paper’s main goal as analysis ofparameter’s uncertainty on a flow shop scheduling pro-blem and based on results obtained from both numericalexperiments and case study, it can be claimed that robustapproach performs much better than fuzzy approach inthis problem. To put it more clearly, the robust modelcan absorb fluctuations imposed by parameters’ variationsand it provides constraints feasibility to immunise themodel in dealing with uncertainty. However, implement-ing fuzzy approach does not gain any advantages.
7. Conclusion
In this paper, a flow shop scheduling problem that involvesSDSTs and minimises the weighted mean flow-time is inves-tigated. Initially, the corresponding mixed integer mathema-tical model is presented. Since facing uncertainty in realworld applications is inevitable, processing times and set-up times are assumed to be uncertain parameters. To deal
with uncertainty, two uncertainty approaches, fuzzy and ROmethods, are applied. Afterwards, the three approaches,deterministic, fuzzy and ROs, are compared with eachother in terms of objective functions and the resultedsequences. The obtained results show that RO providesmore flexible sequences at different levels of uncertainty.However, fuzzy optimisation generates sequences the sameas the corresponding deterministic model.
The results can help decision makers to possessbroader views of scheduling and analyse better. As forfuture research, since the proposed problem is NP-hard,meta-heuristic algorithms can be developed for large-sizedproblems to obtain results in a less amount of time.
Disclosure statementNo potential conflict of interest was reported by the authors.
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0
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