Minimizing Makespan in Flow Shop Scheduling Using a Network Approach

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1 Minimizing Makespan in Flow Shop Scheuling Using a Network Approach Amin Sahraeian Department of Inustrial Engineering, Payame Noor University, Asaluyeh, Iran 1 Introuction Prouction systems can be ivie into three main categories, job shop, flow shop an fixe site Cellular technology an flexible manufacturing system are the subsystems of job shop Prouction scheuling for fixe site are categorize uner the title of project planning, so we can conclue that the prouction scheuling in general are three forms: a) job shop prouction scheuling b) flow shop prouction scheuling an c) project prouction scheuling In the traitional flow shop scheuling problem, it is assume that there is only one machine at each stage to execute passing jobs With the evelopment of harware, software, an theory in parallel computing, the traitional moel of flow shop scheuling is becoming somewhat unrealistic Define to capture the essence of parallel computing is the so-calle hybri flow shop moel, in which each job has to go through multiple stages with parallel machines instea of a single machine (Havill & Mao, 2002) Scheuling is one of the most important ecisions in prouction control systems Every prouction system shoul have a kin of prouction scheuling, no matter whether it is manage an organize traitionally or have a systematic an scientific approach to the planning in the prouction system If a scientific approach to prouction planning is organize, we can be sure that a better usage of the resources especially the machinery an the manpower are consiere an a better situation for competition are forme in the market In this chapter we try to use a mathematical optimization moel for oing this job The systems which we are concerne with are two subsystems of the flow shop, which are simple flow shop an hybri flow shop The goal is to minimize the total completion time of all the activities an the approach which are use is a linear programming which ominates the heuristic moels which mostly use for the NP-Har problems to o this, first of all we convert the prouction system into the network form, then we fin the critical activities which affects the total completion time (makespan), then we assign some buget to the activities to crash them, by assigning some buget to some of the operations, (Hojjati & Sahraeyan, 200) the operation time of these activities reuces an affects the total completion time of all the operations an because of the shortage of the buget, the problem is solve an etermines which activities are better to absorb the limite buget to minimize the makespan wwwintechopencom

2 Prouction Scheuling 11 Definition of prouction systems 111 Project prouction system A type of non-continuous systems, in which the final goos are complete in a fixe place an the machinery an manpower are move towar the finishe goos For planning the prouction in this situation we use a project program like CPM(Critical Path Metho), PERT(Program Evaluation an Review Technique) or GERT(Graphical Evaluation & Review Technique) The most characteristic of this system is that, either the movement of the prouct is impossible (like briges an roas an ) or it is very ifficult (like aircrafts an large ships) 112 Job shop prouction system A job shop is a type of manufacturing process structure where small batches of a variety of custom proucts are mae In the job shop process flow, most of the proucts prouce require a unique set-up an sequencing of processing steps Similar equipment or functions are groupe together, such as all rill presses in one area an grining machines in another in a process layout The layout is esigne to minimize material hanling, cost, an work in process inventories Job shops use general purpose equipment rather than specialty, eicate prouct-specific equipment Digital numerically controlle equipment is often use to give job shops the flexibility to change set-ups on the various machines very quickly Job shops compete on quality, spee of prouct elivery, customization, an new prouct introuction, but are unlikely to compete on price as few scale economies exist When an orer arrives in the job shop, the part being worke on travels throughout the various areas accoring to a sequence of operations Not all jobs will use every machine in the plant Jobs often travel in a jumble routing an may return to the same machine for processing several times This type of layout is also seen in services like epartment stores or hospitals, where areas are eicate to one particular prouct or one type of service A job is characterize by its route, its processing requirements, an its priority In a job shop the mix of proucts is a key issue in eciing how an when to scheule jobs Jobs may not be complete base on their arrival pattern in orer to minimize costly machine set-ups an change-overs Work may also be scheule base on the shortest processing time Capacity is ifficult to measure in the job shop an epens on lot sizes, the complexity of jobs, the mix of jobs alreay scheule, the ability to scheule work well, the number of machines an their conition, the quantity an quality of labor input, an any process improvements 11 Group technology an flexible manufacturing systems Group technology an flexible manufacturing system are the subsystems of job shop In cellular group technology, in which the layout is cellular, factory will be broken to units calle cells In these units family parts base on its characteristics, such as parts of geometry, size or similar process are forme An machinery that have the task of processing on the parts to be allocate cells Machinery mentione as possible locate close to each other an machinery group is forme Cellular prouction is the combination of the two job shop an project prouction systems In other wors, cellular system has avantages of job shop wwwintechopencom

3 Minimizing Makespan in Flow Shop Scheuling Using a Network Approach (prouction flexibility an iversity of parts) an continuous prouction (high rate of prouction) together Cellular system is one of the most important tools to achieve the lean prouction A flexible manufacturing system (FMS) is a manufacturing system in which there is some amount of flexibility that allows the system to react in the case of changes, whether preicte or unpreicte This flexibility is generally consiere to fall into two categories, which both contain numerous subcategories The first category, machine flexibility, covers the system's ability to be change to prouce new prouct types, an ability to change the orer of operations execute on a part The secon category is calle routing flexibility, which consists of the ability to use multiple machines to perform the same operation on a part, as well as the system's ability to absorb large-scale changes, such as in volume, capacity, or capability Most flexible manufacturing systems consist of three main characteristics The work machines which are often automate CNC(Computer Numerically Controlle) machines are connecte by a material hanling system which is calle AGV (Automate Guie Vehicle ) to optimize parts flow an the central control computer which controls material movements an machine flow The main avantage of an FMS is its high flexibility in managing manufacturing resources like time an effort in orer to manufacture a new prouct The best application of an FMS is foun in the prouction of small sets of proucts like those from a mass prouction The other avantages are Faster, Lower- cost/unit, greater labor prouctivity, greater machine efficiency, improve quality, increase system reliability, reuce parts inventories, aaptability to CAD/CAM(Computer-aie Design/Computer-aie Manufacturing) operations, shorter lea times 11 Flow shop prouction system Flow shop prouction system in turn is ivie to three main categories: a) simple flow shop, b) hybri flow shop an c) parallel flow shop The simple an hybri flow shop are stuie in this chapter an the methoology for parallel flow shop is suggeste for further research 111 Simple flow shop Much research works both in acaemic an practical fiels have stuie flow shop scheuling In a flow shop system all jobs are processe on machines in the same sequence However, the processing time of each operation might vary All jobs are assume to be reay to be processe at time zero It is further assume that there is sufficient physical buffer space between two successive machines without being concerne about the busy or ile status of that machine A general objective is to evelop a scheule that minimizes the makespan The general flow shop scheuling problem is a NP-Complete problem an a non polynomial time algorithm is expecte for these type of problems (French, 12) The evelopment of heuristic algorithms guarantees goo solutions, (Campbel et al, 10) especially for large size problems In simple flow shop system there are a set of m machines (processors) an a set of n jobs Each job comprises a set of m operations which must be wwwintechopencom

4 0 Prouction Scheuling one on ifferent machines All jobs have the same processing operation orer when passing through the machines There are no preceence constraints among operations of ifferent jobs Usually some assumptions are consiere when the theoretical approach is consiere For example operations cannot be interrupte an each machine can process only one operation at a time These assumptions are consiere more eeply in the future The problem is to fin the job sequences on the machines which minimize the makespan, (Khoaai, 2011) ie the maximum of the completion times of all operations (Sea, 200) As the objective function, mean flow time, completion time variance (Gowrishankar, 2001) an total tariness can also be use (Pan et al, 2002) The flow shop scheuling problem is usually solve by approximation or heuristic methos Successful heuristic methos inclue approaches base on simulate annealing, tabu search, an genetic algorithms (Al-Dulaimi & AAli, 200) 112 Hybri flow shop Hybri flow shop scheuling problems are quite common, especially in the process inustry where multiple servers (machines) are available at each stage (Brah & Hunsucker, 11) as well as in certain flexible manufacturing environments (Zijm & Nelissen, 10) It is an extension of two classical scheuling problems, the classical flow shop an ientical parallel-machine problems Further, when processing times at a given stage ominate those at other stages, it is natural to increase the system capacity by aing another machine at this stage A Hybri Flow Shop (HFS) consists of a series of prouction stages Each stage has several machines operating in parallel Some stages may have only one machine, but at least one stage must have multiple machines The flow of jobs through the shop is uniirectional Each job is processe by one machine in each stage an it must go through one or more stage Machines in each stage can be ientical, uniform or unrelate (Linn & Zhang, 1) The hybri flow shop scheuling problem is NP-har an it is usually solve by heuristic methos, that is base on simulate annealing, (Wang et al) tabu search, an genetic algorithms There has been a significant amount of research one on the HFS scheuling problem since its first attempt in 11 The main ifference between the proucts of this prouction system an the other two is that the proucts of flow shop cannot be isassemble In the other wors, we o not have any assembly activity in flow shop systems but in job shop an Project prouction the prouct is ivie to ifferent components We can see these components in the final prouct an finishe goos but, in flow shop this issue is not obvious So without visiting the prouction system, by observing only the final prouct, we can conclue whether this prouct is mae in the flow shop or not 2 Methoology Makespan is one of the most important criteria in every prouction systems; it is equal to the total completion time of all the activities Minimizing this criterion cause better usage of the resources specially machinery an manpower In both simple an hybri flow shop, the methoology is to convert the flow shop into a network form, then a linear programming moel with the objective of minimizing the total completion time of all the activities are constructe Minimizing total completion time of all the activities is equivalent wwwintechopencom

5 Minimizing Makespan in Flow Shop Scheuling Using a Network Approach 1 to minimizing makespan in the prouction system The result is that the sequencing an scheuling of all the activities are etermine In the next step, some buget is assigne to crash the possible activities It is possible that we can only assign buget to some of the activities to ecrease their times The amount of buget is always limite, so one question arises, an it is: what are the best activities to absorb this limite buget to minimize the total completion time of all the activities This causes to a some more constraints to the problem an the result is that the critical activities are etermine The output of the problem is that how much buget shoul be assigne to which activities to get the best result The best result is minimizing the total completion time of all the activities in the network which is equivalent to minimizing makespan Minimizing makespan in simple flow shop scheuling using a network approach 1 Assumptions 11 Assumptions regaring the jobs The sequencing moel consists of a set of n jobs which are simultaneously available (at time zero, static environment) Job r has a preetermine operation times on machine m m= 1,, M Set-up time is inepenent of sequence an is inclue in the processing time Jobs are inepenent of each other One unit of prouction for each job is consiere 12 Assumptions regaring the machines Each machine in the shop operates inepenently Machines can be kept ile All machines are continuously available No machine breakown is allowable Each machine can process only one operation at a time point 1 Assumption regaring the process Processing time for each job on each machine is eterministic an inepenent of the orer in which jobs are to be processe Transportation times of a job an set-up times are inclue in the processing time Problem methoology 1 Problem efinition In this chapter we are face with a solve n/m/f/c max problem By using heuristics such as (Campbell et al10, Nawaz et al 1) To reminimize the makespan, processing times of the operations can be reuce by proviing aitional resources, which are available at a wwwintechopencom

6 2 Prouction Scheuling cost The processing time of some of the operations can be reuce by assigning a cost We woul like to obtain minimum makespan by reucing processing time of some of these operations The problem is to fin these operations with a pre specifie buget 2 Converting flow shop scheuling into a network Since we are intereste in reucing makespan by employing aitional resources crashing the project we evelop a metho to convert a flow shop scheuling into the project network Let us efine the following notations to convert flow shop scheuling problems into a network Nomenclature The following terminology is use for moeling the problem: N: number of jobs M: number of machines J: an activity number m: machine number r: job number J rm : job r on machine m i,j: activity from noe i to noe j Tj: starting time of noe j Di,j: normal uration time of activity from noe i to noe j Df(i,j): minimum crashing time of activity i to j i,j : crashe uration time of activity i to j C i,j : slope of crashing cost for activity i to j B: preetermine buget Network requirements Certainly when flow shop scheuling problem is converte into a network, it is important to prepare network requirement Thus, let us efine network requirement as follows: Noes: Each noe represents an event of start or finish operation(s) on machines Activity: Activities are the operations to be one on specific machine an have uration equal to processing time Preecessors: Activity representing the previous operation for the same job constitutes a preceing activity to the operation Further the activity corresponing to the operation of the job on the same machine which is before this job in the sequence also constitutes preceing activity Duration time: "processing time" is the uration of the activity Resources: Machines are the resources Suppose we have a flow shop system with n jobs an m machines The ata is escribe in table 1 wwwintechopencom

7 Minimizing Makespan in Flow Shop Scheuling Using a Network Approach Activity Preecessors Duration Time Machine 1,1 i,j r,m (i,j-1),(i-1,j) (r,m-1),(r-1,m) D 11 D ij D rm M 1 M j M m Table 1 Data for general moel Linear programming application to fin minimum makespan subject to buget limitation Accoring to objective that is to select operations to be crashe for fining minimum makespan subject to buget limitation, after converting flow shop scheuling problem into the network, now it is possible to crash the network to fin minimum makespan Problem formulation The problem can be formulate as follows: MinZ = T T ST n 1 Cij, ( Dij, ij, ) B (1) T T (2) j i i, j D D () f(,) i j i, j i, j T, T, = integer i j i, j Constraint (1) is relate to buget limitation, that aitional cost for crashing coul not be greater than pre specifie buget Constraint (2) states that the start time of event j shoul be at least equal to start time of i an crash uration of activity i-j Constraint () is relate to the lower an upper bouns on crash uration It is obvious that all t i 's an ij must be non negative an also integer However, ue to structure of the problem, it can be solve as a linear programming problem Converting the flow shop problems into a network In next section we have shown how to convert flow shop problems to project network We can calculate the earliest start, latest start, floats for all the activities using CPM metho Earliest finishing time of the project is equal to makespan for the flow shop problem After that we can use the linear programming moel for crashing the network wwwintechopencom

8 Prouction Scheuling Numerical example Consier that we have a flow shop problem with four machines an five jobs Accoring to Campbell et al (10), the sequence has been obtaine as A-B-C-E-D with corresponing processing times as given in table 2 JOB MACHINE Cutting (min) Pressing (min) Drilling (min) Weling (min) Part A Part B Part C Part D 2 Part E Table 2 Processing time of a flow shop with machines an jobs Conversion of this problem to a CPM network with information of prerequisite is given in table an the corresponing network is shown in figure 1 Noe (i,j) 1,2 2,,,, 2,, 10,11 12,1 1,1,1 1,1 1,20 21,22 2,2 1,2 2,2 2,2 2,2 2,2 Activity (J rm ) J A1 J B1 J C1 J E1 J D1 J A2 J B2 J C2 J E2 J D2 J A J B J C J E J D J A J B J C J E J D Preecessor J A1 J B1 J C1 J E1 J A1 J B1,J A2 J C1,J B2 J E1,J C2 J D1,J E2 J A2 J B2,J A J C2,J B J E2,J C J D2,J E J A J B,J A J C,J B J E,J C J D,J E Duration time (D i,j ) 2 Table The project network ata for corresponing example wwwintechopencom

9 Minimizing Makespan in Flow Shop Scheuling Using a Network Approach J A1 J B1 J C1 J E1 J D1 1 2 J A2 J B2 J C2 J E2 J D J A 1 Fig 1 Project network for given example J B J C J E J D J A JB JC JE : Dummy operation is use to consier the technological constraints Now we shoul assign some buget to some activities (operation) for which their time can be reuce These are shown in table J D 2 2 Noe (i,j) 1,2 2,,,, 2,, 10,11 12,1 1,1,1 1,1 1,20 21,22 2,2 1,2 2,2 2,2 2,2 2,2 Activity (J rm ) J A1 J B1 J C1 J E1 J D1 J A2 J B2 J C2 J E2 J D2 J A J B J C J E J D J A J B J C J E J D Preecessor J A1 J B1 J C1 J E1 J A1 J B1,J A2 J C1,J B2 J E1,J C2 J D1,J E2 J A2 J B2,J A J C2,J B J E2,J C J D2,J E J A J B,J A J C,J B J E,J C J D,J E Duration time (D i,j ) 2 Minimum Duration time (D f(i,j) ) 2 2 Cost Slope ($) Table Cost of reuce times wwwintechopencom

10 Prouction Scheuling 1 Problem solution Consiering the information given for the problem in tables an an Fig 1 the objective function an the constraints can be written as follows: MinZ = T T Sto : 2 1 ( 12 ) ( 2 ) ( ) ( 2 ) ( 1 ) ( ) ( ) T2 T1 12 T T2 2 T T T T T T T T2 2 T T T11 T T1 T T1 T1 11 T1 T 1 T1 T1 11 T20 T1 120 T22 T T2 T2 22 T2 T1 12 T21 T20 T21 T1 T2 T22 T2 T1 T2 T1 T2 T20 T2 T22 T2 T = T2 T2 22 T2 T2 22 T2 T2 22 T2 T2 22 T T T10 T T T T10 T T12 T11 T12 T T1 T1 T1 T T1 T T1 T1 T1 T1 T1 T11 = = 1011 = 121 = 11 = 11 = 2122 = 22 = 2 12 = 22 = 22 = 22 = Ti, Tj, i, j = int eger wwwintechopencom

11 Minimizing Makespan in Flow Shop Scheuling Using a Network Approach 2 Results The results are given using LINGO 0 optimally In table, optimum uration for each activity is given Accoring to buget limitation makespan coul be reuce from 1 (before crashing) to Noe (i,j) 1,2 2,,,, 2,, 10,11 12,1 1,1,1 1,1 1,20 21,22 2,2 1,2 2,2 2,2 2,2 2,2 Activity (J rm) J A1 J B1 J C1 J E1 J D1 J A2 J B2 J C2 J E2 J D2 J A J B J C J E J D J A J B J C J E J D Crashe Duration time ( i,j) 2 Table Optimal uration for each activity accoring to buget limitation Table emonstrates ifferent minimum makespan accoring to ifferent buget assignment Buget ($) Makespan Table Minimum makespan accoring to the buget Minimizing makespan in hybri flow shop scheuling using a network approach 1 Assumptions 11 Assumptions regaring the jobs The sequencing moel consists of a set of n jobs which are simultaneously available (at time zero, static environment) wwwintechopencom

12 Prouction Scheuling Job r has a preetermine operation times on machine m Set-up time is inepenent of sequence an is inclue in the processing time Jobs are inepenent of each other (Shortest Processing Time) SPT rule is use to assign the jobs to the machines One unit of prouction for each job is consiere 12 Assumptions regaring the machines Each machine in the shop operates inepenently Machines can be kept ile All machines are continuously available No machine breakown is allowable Each machine can process only one operation at a time point Each machine starts at its earliest starting time possible Interruption of the machines is not allowe (no repairing uring processing) 1 Assumption regaring the process Processing time for each job on each machine is eterministic an inepenent of the orer in which jobs are to be processe Transportation times of a job an set-up times are inclue in the processing time 2 Nomenclature The following terminology is use for moeling the problem: N: number of jobs M: number of machines J: an activity number m: machine number r: job number s: stage number J rms : job r on machine m in stage s i,j: activity from noe i to noe j Tj: starting time of noe j Di,j: normal uration time of activity from noe i to noe j Df(i,j): minimum crashing time of activity i to j i,j : crashe uration time of activity i to j C i,j : slope of crashing cost for activity i to j B: preetermine buget Converting HFS into a network moel We can illustrate a general form of HFS with n jobs m machines, an s stages as in Fig 2 Each operation has a preecessor which is shown in table There are two sets of preecessors, one, the operational constraint, for which every job shoul be processe in its earlier stage, an secon technological constraint for which each machine shoul operate the jobs in chronological orer wwwintechopencom

13 Minimizing Makespan in Flow Shop Scheuling Using a Network Approach Stage Stage Stage s J J J 121 Job 1 J 122 J 211 J J J Job 2 Job 1 J J Job r 1 J J 22 Fig 2 General moel of HFS Stage activity preecessor Duration time 1 J 1m1 J rm1 J Nm1 J rm1 m = 1or 2 or or M, r = 1 J rms m = 1,2,,M, r = 1, s = s-1 J rm1 m = 1or 2 or or M, r = 1,2,,r-1 J rms m = 1,2,,M, r = r, s = s-1 J rm1 m = 1or 2 or or M, r = 1,2,,N-1 J rms m = 1,2,,M, r = N, s = s-1 D 1m1 D rm1 D Nm1 J 1m2 J rm2 m = 1or 2 or or M, r = 1 J rms m = 1,2,,M, r = 1, s = s-1 D 1m2 2 J rm2 J Nm2 J rm2 m = 1or 2 or or M, r = 1,2,,r-1 J rms m = 1,2,,M, r = r, s = s-1 J rm2 m = 1or 2 or or M, r = 1,2,,N-1 J rms m = 1,2,,M, r = N, s = s-1 D rm2 D Nm2 Table Preecessors for general moel wwwintechopencom

14 0 Prouction Scheuling Problem formulation The problem can be formulate as follows: MinZ = T T ST j i i, j 1 C ( D ) B ij, ij, ij, f(,) i j i, j i, j T, T, = integer i j i, j n T T D D Numerical example The methoology is illustrate using a numerical example with jobs, stages an respectively,, an 2 machines in each stage The problem is solve using SPT (Shortest Processing Time) The sequence has been obtaine as A-C-B-D with corresponing processing times as given in table Job Part A Part B Part C Part D M Stage 1 M2 M M1 1 1 Stage 2 M2 M M M1 11 Stage M M Stage M1 M Table Processing time The network is illustrate in Fig J11 J112 J1 J J1 J22 J J J211 J222 J22 J Fig The network of numerical example 1 J21 J12 J1 J wwwintechopencom

15 Minimizing Makespan in Flow Shop Scheuling Using a Network Approach 1 For this problem the preecessors are shown in table Noe (i,j) 2,,,,,,,10 10,11 12,1 1,1 1,1 1,1 1,1 1,1 1,20 20,21 Activity (J rms ) J 11 J 112 J 1 J 12 J 1 J 22 J J 2 J 211 J 222 J 22 J 21 J 21 J 12 J 1 J 1 Preecessor J 11 J 112 J 1 J 11 J 1, J 222 J 22, J 1 J, J 12 J 211 J 222 J 22 J 21, J 112 J 12 J 1, J 21 Duration time (D i,j ) Table Preecessor of the numerical example Now we shoul assign some buget to some activities (operation) for which their time can be reuce These are shown in table 10 Noe (i,j) 2,,,,,,,10 10,11 12,1 1,1 1,1 1,1 1,1 1,1 1,20 20,21 Activity (J rms ) J 11 J 112 J 1 J 12 J 1 J 22 J J 2 J 211 J 222 J 22 J 21 J 21 J 12 J 1 J 1 Preecessor J 11 J 112 J 1 J 11 J 1, J 222 J 22, J 1 J, J 12 J 211 J 222 J 22 J 21, J 112 J 12 J 1, J 21 Duration time (D i,j ) Minimum Duration Time (D f(i,j) ) Cost Slope ($) Table 10 Cost of reuce times wwwintechopencom

16 2 Prouction Scheuling 1 Problem solution Consiering the information given for the problem in tables an 10 an Fig, the objective function an the constraints can be written as follows: MinZ = T22 T1 Subject To : ( ) ( ) ( ) ( ) ( ) (1 ) T T2 2 T T T T T T T T T T T10 T 10 T11 T T1 T T1 T1 11 T1 T1 11 T1 T1 11 T1 T1 11 T1 T1 11 T20 T1 120 T21 T T2 T1 T12 T1 T1 T1 T T T T1 T T T10 T T1 T T20 T1 T22 T21 T22 T = 1 = 1 = 11 = = 10 = = = = i j i, j = 1 T, T, = integer 2 Results The problem that is formulate in section 1 is solve by LINDO software In table 11, optimum uration (crashe time) an buget use for each activity accoring to cost slop that is shown in table 10 is given So with buget limitation of 10000$, we can ecrease completion time of prouction Accoring to buget limitation makespan coul be reuce from 0 (before crashing) to wwwintechopencom

17 Minimizing Makespan in Flow Shop Scheuling Using a Network Approach Activity ( ij) Crashe Time Buget Use Objective 00 Table 11 The result of the numerical example By assigning ifferent bugets, ifferent results can be obtaine, this is calle the sensitivity analysis of the problem For example, if we assign 2000$ buget, completion time ecrease from 0 to, an so The result can be shown as in table 12 Buget Table 12 Sensitivity analysis of the numerical example Makespan 0 Conclusions This chapter reviewe literature on the flow shop scheuling to etermine the optimum completion time(minimum makespan) Flow shop system is involve three subsystems, which are simple flow shop, hybri flow shop an parallel flow shop In this chapter iscusse about only both simple an hybri flow shop systems In simple flow shop we use one machine in each stage with ientical process for all jobs, but in hybri flow shop at least in one stage there is more than one machine for processing Accoring to the literature review, it was foun that we have consiere the problem of fining minimum makespan for a given sequence of jobs in both simple an hybri flow shop by using network approach As a sequence of jobs on machines is known, the problem can be represente as a critical path network It is shown that both simple an hybri flow shop problems can be converte to a network moel with regar to preecessor relations an processing time wwwintechopencom

18 Prouction Scheuling Then it is estimate the cost of crashing time for each activity, which is possible (cost slope) By using a linear programming formulation the critical activities are etermine Assigning some buget to activities that can be crashe by time, causes to reuce the completion time of all the project or makespan, this by itself causes better use of the resources specially machinery an manpower, which by itself increase prouctivity In aition to above, the important subject in this research is ability to sensitivity analysis So that, by assigning ifferent bugets, it can be obtaine ifferent completion time Thereupon we can select the optimum completion time consiering to corresponing buget For further research it is suggeste to apply linear programming technique to etermine makespan subject to buget limitation in some other systems like parallel flow shop Acknowlegement The author is very grateful to Dr Hojjati who reviewe this chapter an gave some helpful comments 10 References Al-Dulaomi, B & AAli, H (200) A Novel Genetic Algorithm Approach for Solving Flow Shop Problem International Journal of Computer Science an Network Security, Vol, No, pp 22-2 Brah, SA & Hunsucker, JH (11) Branch an Boun Metho for the Flow Shop with Multiple Processors European Journal of Operational Research, No1, pp -1 Campbell, HG; Dueck, RA & Smith, ML (10) A Heuristic Algorithm for the n Job, m Machine Sequencing Problem Management Science, No1, pp 0- French, S (12) Sequencing an Scheuling: An Introuction to the Mathematics of the Job Shop Harwoo, Chi Chester Gowrishankar, K (2001) Flow Shop Scheuling Algorithms for Minimizing the Completion Time Variance an the Sum of Squares of Completion Time Deviation from a Common Due Date European Journal of Operational Research, Vol 12, pp - Havill, J & Mao, W (2002) On-line Algorithms for hybri Flow Shop Scheuling Hojjati, SMH & Sahraeyan, A (200) Minimizing Makespan Subject to Buget Limitation in Hybri Flow Shop Proceeings of International Conference on Computer an Inustrial Engineering, Troyes, France, July -, 200 Khoaai, A (2011) Solving Constraine Flow-Shop Scheuling Problem with Three Machines International Journal of Acaemic Research, Vol, No 1, January, 2011, pp -0 Linn, R & Zhang, W (1) Hybri Flow Shop Scheuling: A Survey Proceeings of International Conference on Computer an Inustrial Engineering, pp -1 Pan, JCH; Chen, JS & Chao, CM (2002) Minimizing Tariness in a Two-Machine Flow Shop Computer an Operations Research, Vol 2, pp - Sea, M (200) Mathematical Moels of Flow Shop an Job Shop Scheuling Problems Worl Acaemy of Science, Engineering an Technology, pp Wang, H; Chow, F & Wu, F (n) A Simulate Annealing for Hybri Flow Shop Scheuling with Tasks to Minimize Makespan International Journal of Avance Manufacturing Technology, Vol, No, pp1- Zijm, WHM & Nelissen, E (10) Scheuling a Flexible Machining Centre Proceeings of Conference on Engineering Costs an Prouction Economics 1, pp 2-2 wwwintechopencom

19 Prouction Scheuling Eite by Prof Rorigo Righi ISBN Har cover, 22 pages Publisher InTech Publishe online 11, January, 2012 Publishe in print eition January, 2012 Generally speaking, scheuling is the proceure of mapping a set of tasks or jobs (stuie objects) to a set of target resources efficiently More specifically, as a part of a larger planning an scheuling process, prouction scheuling is essential for the proper functioning of a manufacturing enterprise This book presents ten chapters ivie into five sections Section 1 iscusses rescheuling strategies, policies, an methos for prouction scheuling Section 2 presents two chapters about flow shop scheuling Section escribes heuristic an metaheuristic methos for treating the scheuling problem in an efficient manner In aition, two test cases are presente in Section The first uses simulation, while the secon shows a real implementation of a prouction scheuling system Finally, Section presents some moeling strategies for builing prouction scheuling systems This book will be of interest to those working in the ecision-making branches of prouction, in various operational research areas, as well as computational methos esign People from a iverse backgroun ranging from acaemia an research to those working in inustry, can take avantage of this volume How to reference In orer to correctly reference this scholarly work, feel free to copy an paste the following: Amin Sahraeian (2012) Minimizing Makespan in Flow Shop Scheuling Using a Network Approach, Prouction Scheuling, Prof Rorigo Righi (E), ISBN: --0--, InTech, Available from: InTech Europe University Campus STeP Ri Slavka Krautzeka /A 1000 Rijeka, Croatia Phone: + (1) 0 Fax: + (1) 1 wwwintechopencom InTech China Unit 0, Office Block, Hotel Equatorial Shanghai No, Yan An Roa (West), Shanghai, 20000, China Phone: Fax: