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1 Computes & Industial Engineeing xxx (2008) xxx xxx Contents lists available at ScienceDiect Computes & Industial Engineeing jounal homepage: Optimization of opeation and changeove time fo poduction planning and scheduling in a flexible manufactuing system Kanchan Das a, M.F. Baki b, *, iangyong Li b a Technology Systems Depatment, East Caolina Univesity, Geensville, NC 75858, USA b Odette School of Business, Univesity of Windso, 401 Sunset Avenue, Windso, Ont., Canada N9B 3P4 aticle info abstact Aticle histoy: Received 29 June 2006 Received in evised fom 6 May 2008 Accepted 3 June 2008 Available online xxxx Keywods: Pocess planning Sequencing Flexible manufactuing system Machining Intege pogamming Pat gouping Machine loading Tool allocation This pape deals with the poduction planning poblem of a flexible manufactuing system. It specifically addesses issues of machine loading, tool allocation, and pat type gouping with the intent of developing an opeation sequencing technique capable of optimizing opeation time, non-poductive tool change times, and oientation change times when pocessing a goup s design featues. A hieachical appoach has been adopted to detemine the pat goups depending on the opeation, tool change and oientation change times at the uppe level. At the next level, we sequence the opeations of the pat goups. Intege pogamming models ae fomulated to goup the pats and to addess the opeation-sequencing poblem. The model is illustated with an example elated to an auto engine cylinde head machining plant. Ó 2008 Elsevie Ltd. All ights eseved. 1. Intoduction Machines in moden, flexible, manufactuing systems ae capable of pefoming all planned opeations. The inheent efficiency of a flexible manufactuing system (FMS), combined with additional capabilities, can be hanassed by developing a suitable poduction plan. As Stecke (1983) mentions, in ode to best utilize an FMS s capabilities, a caeful system set up is equied pio to poduction. This pape consides the specific poduction planning poblem of complex pats, simila to an automobile engine cylinde head, which combines design featues (DFUs) equivalent to pat types, and unit manufactuing featues (UMFs) equivalent to opeations. The DFUs ae located at diffeent locations and, in some situations, at diffeent faces. Although the tool change time is minimal in moden FMSs, the movement of tool heads fom one DFU to the next, combined with the time needed fo tool etacting and positioning, makes each tool change consideably time consuming. In addition to the tool changes, oientation changes equied to each the DFUs at diffeent faces of the cylinde heads make the system moe complex. We follow the geneal famewok intoduced by Stecke (1983) as a guideline, but esot to a system-dependent planning appoach moe suitable to ou poblem. * Coesponding autho. Tel.: ; fax: addess: fbaki@uwindso.ca (M.F. Baki). Among the FMS planning models, the machine loading one is most fequently cited. Pat gouping is the next most studied model, as in Hwang and Shogun (1989), Kulkani and Kiang (1995), Liang and Dutta (1993a, 1993b), Rajagopalan (1986), Sawik (1990), and Stecke and Kim (1988, 1991). Mohammed, Kuma, and Motwani (1999), Mukhopadhyay, Maiti, and Gag (1991), Rajagopalan (1986), Sodhi, Askin, and Sen (1994) conside an appoach that combines pat gouping and machine loading in conjunction with tool loading. Pat gouping, machine loading, and tool povisioning fo pat goups ae, in essence, linked. This pape takes a joint appoach, collectively addessing these thee poblem aeas fo a moe compehensive solution. To implement the above-mentioned joint appoach, this pape intoduces a methodology fo consideing opeations, tool change and oientation change times that addesses the FMS planning poblems of pat gouping, machine loading, and tool allocation. Assigning machines to a pat goup equipped with the equied numbe of tools is a cucial pat of both the machine loading and tool povisioning phases. As discussed, moden FMS machines ae capable of pefoming all opeations. Howeve, a diffeent set of tools gives a machine a diffeent capability as equied by the planning model to pocess a planned pat, mix, o goup. The output is then taken fom this planning model, and a detailed opeation schedule is geneated in the late stage. Section 2 eviews the elevant liteatue. Section 3 pesents the mathematical pogamming models fo the pat gouping, machine /$ - see font matte Ó 2008 Elsevie Ltd. All ights eseved. doi: /j.cie Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

2 2 K. Das et al. / Computes & Industial Engineeing xxx (2008) xxx xxx loading, tool allocation, and scheduling, and Section 4 povides examples that illustate the planning and scheduling model. Conclusions ae given in Section Liteatue eview The FMS poduction planning poblem has attacted numeous authos ove the past two decades, due to its inheent potential fo flexibility, quality, and high poductivity all of which ae vital fo poviding a quick esponse to maket needs. Stecke (1983) outlines a geneal famewok fo the implementation of system set-up decisions pio to poduction in a FMS. The famewok divides the FMS poduction planning poblem into seveal hieachically stuctued sub poblems: pat type selection, machine gouping, poduction atio, esouce allocation, machine loading, and scheduling. The pat type selection and machine gouping poblems ae aimed at educing set up times and inceasing the thoughput of machining centes by taking the tool allocation poblem and othe tool specific constaints into account. Sain and Chen (1987) addess machine loading and tool allocation poblems. Thei poposed MIP model consides tool life, tool slot availability, and tool-specific space equiements, with an objective to detemine the pat outing in an effot to minimize the total machining cost. Modi and Shanka (1994) solve the opeational assignment of tools and machines by minimizing pat movement between the machines. Thei appoach attempts to maximize the numbe of opeations a tool and a machine ae capable of caying out. Thei study fomulates the model as a quadatic pogamming poblem and lineaizes it, following the appoach of Balas (1964) and Glove and Woolsey (1974). Mohammed et al. s (1999) eseach focuses on the FMS patgouping poblem to optimize makespan developing an MIP model to solve the pat gouping, machine loading, and tool allocation poblems. Thei model minimizes the deviation between the makespan s evaluation, whee, initially, pat gouping is consideed, to whee it is late ignoed. To decide the taget makespan time, they use a model that ignoes pat gouping. Thei study detemines the numbe of pat goups by solving a fomulation that depends on the compatibility between opeations, tools and machines late used as input fo the pat-gouping model. Thei eseach compaes the model output by taking examples fom liteatue ganeed though expeimental analysis and concludes that the model efficiently geneates lowe makespan and enhanced outing flexibility. Pesi, Ukovich, Pesenti, and Nicolich (1999) popose a two-level FMS poduction-planning model with the intention of impoving a eal industial poblem elated to machine utilization expeienced by Gandi Motoi in Italy. Fist, pat gouping, tool allocation, and machine loading models ae addessed. The focus then switches to the scheduling poblem at the factoy level, depending on the outcomes of the fist level models. Pesi et al. popose an MIP model fo the machine loading and batching model, poviding a 10-pat illustation. Fo the sequencing poblem, they test local dispatching ules such as FIFO, LIFO, SRPT, EDD, and othes. Vaious pefomance measues, including WIP, ae used to evaluate the suitability of the scheduling model. Siniech, Rubinovitz, Milo, and Nakbily (2001) develop a 0 1 intege pogamming model to detemine a job sequencing plan with the objective of minimizing unpoductive tool change time by educing the numbe of dissimila tools needed by adjacent jobs that ae subject to the availability of the fixtuing devices. This study concludes with a heuistic pocedue to solve the model. The solution quality of the heuistic is tested using a hi-tech company s pactical data, and is found to be satisfactoy when compaed to an optimal solution of the model. Gamila and Motavalli (2003) develop an integated model to addess the FMS poduction-planning poblem. Thei fist step solves an integated model fo machine loading and tool loading. Then the opeation schedule is obtained accoding to the outcomes of the fist step. A 0 1 MIP model is poposed to minimize the summation of maximum completion time, mateial handling time, and total pocessing time. Bad (1988) fomulates a non-linea intege-pogamming model to solve the FMS job-sequencing poblem while seeking to minimize the numbe of tool switches. The tool switching and job-sequencing poblems ultimately esult in a eduction in makespan. Tool switches ae instances when the tool is loaded in the magazine fom local stoage (as needed) to pocess a job in its scheduled position. The study assumes that each tool change time is identical and uses a dual-based Lagangian elaxation heuistic to solve the poblem. Koo and Tanchoco (1999) analyze the tool and opeation selection poblem of single-stage multifunctional machining systems (SSMS), whee tools ae dynamically shaed between machines. Avci and Aktuk (1996) popose an appoach to solve FMS tool magazine aangement and opeation sequencing poblems. The objective of this study is to minimize the total manufactuing cost by using an efficient tool-shaing concept. The model accounts fo a pecedence constaint, tool magazine capacity, tool availability, and tool life in developing solutions. Gieco, Semeao, Tolio, and Toma (1995) conduct a tool management study with the objective of evaluating the impact of a eduction of tools on the oveall pefomance of the FMS. This study investigates the possibility of educing investment in tools by allowing machines to shae them in a simulated envionment. Hetz, Lapote, Mittaz, and Stecke (1998), Cama, Colen, Oelemans, and Spieksma (1994) and Tang and Denado (1988a, 1988b) addess the tool-switching poblem. All tools equied fo pocessing must be installed befoe a pat is pocessed. Howeve, the installation of tools in the tool magazine is time consuming. These studies focus on sequencing jobs that minimize the numbe of tool switches needed. Ecke and Gupta (2005) look at the poblem of tool change time when scheduling FMS tasks hoping to optimize the total tool change time by consideing pecedence equiements. This pape intoduces the poblem of moving a tool head fom one position to anothe duing the intechange of tools, in contast to only tool change o tool switch instants. The machines unde consideation in this study have automatic tool changes that cay out tool changeove within a vey shot time span. While tool changeove time is insignificant compaed to opeational time in UMFs, tool movement time has been found to be quite substantial in the case of jobs involving engine cylinde heads and engine blocks. In addition, this pape also consides a change in oientation in educing cycle time and in developing an efficient schedule. 3. Poblem statement We intend to develop a pocess-planning model fo machining cylinde heads in a typical engine manufactuing plant. The pocess-planning poblem includes an extensive numbe of issues that must be addessed in an industy setting. This pape attempts to addess impotant issues that, if implemented, would make the pocess-planning system moe efficient. We conside jobs simila to auto engine cylinde heads. In a typical cylinde head thee ae seveal design featues, DFUs, (equivalent to pat types), while each featue pocesses one o moe UMF, o (opeations) fo its completion. The DFUs ae located at vaious faces of the head and, as such, ae pocessed on flexible manufactuing machines by changing thei oientation though a tilting fixtue. Each UMF Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

3 K. Das et al. / Computes & Industial Engineeing xxx (2008) xxx xxx 3 equies its own specific type of tool, which is placed in the limited capacity tool magazine, to begin pocessing. The capability of a machine is diffeent when allocated with a diffeent set of tools. We explain the poblem fom the pespective of optimizing makespan by minimizing non-poductive times. Tool changeove time, time fo moving the tool heads, oientation change and the time taken by a pat in visiting many wok stations ae the few non-poductive time items. To minimize the non-poductive tool movement and oientation time, we intoduce the following concepts: - If a tool can pocess moe than one UMF (opeation, o) ona numbe of DFUs, then these DFUs and UMFs can be gouped saving tool change time, minimizing tool head tavel distance, and optimizing the usage of tools and thei magazines. Thus, tool change time can be optimized oveall by developing goups of DFUs and UMFs that use simila tools. Fig. 1 explains these fist two advantages. - The DFUs in a cylinde head ae located at diffeent faces. Consequently, tool heads need to visit diffeent faces to pefom the equied opeations, usually by changing thei oientation, i.e., tilting the fixtue at an angle. While an oientation time cannot be avoided, we can goup DFUs at diffeent oientations so that oientation changes ae minimized, and kept as low as possible. Also, it is possible to educe oientation time by planning the stating and destination faces, depending on the angle of tilt. - In addition, the pocessing times of UMFs and the efixtuing times of DFUs ae consideed when gouping to optimize the makespan of the goup. The objective of the study is (i) to plan fo goups g of DFUs, allocate tools on machine m when pocessing all UMFs o of the goup g and (ii) to sequence the UMFs especting pecedence equiements so that the non-poductive tool eplacement times and oientation change times ae optimized. As the hieachical pocess-planning model shows in Fig. 2, ou focus is on the planning and sequencing poblems. The following assumptions ae made in developing the model: Assumptions: 1. An opeation (UMF) type is uniquely associated with one type of tool, and tools ae identified by tool numbes. 2. Tool head movement, tool positioning, and tool etacting times of all UMFs ae known. 3. Tools needed fo each opeation, including the numbe of slots applicable to a tool, ae known. 4. Tool life is adequate fo pocessing all UMFs assigned to it. Thus, duplicate tools, the eloading of tools, and tool life issues ae not addessed. 5. Oientation times equied fo the changing of faces is known. 6. All machines ae identical and ae capable of pefoming all opeations (UMFs). Featue Location 1 Featue Location 3 Tool change Featue Location 2 Featue Location 4 Fig. 1. Tavel distances with and without tool change. 7. No limit exists on the numbe of fixtues and on thei capability. The fixtues ae unique and capable of tilting to any degee, as pe the equiements of the pocessing. 8. Each machine is capable of pocessing multiple UMFs; effective capacity, as well as tool magazine capacity, is known fo each machine. Each machine has the same capacity and capability. 9. The model consides Euclidean distance duing the movement of tool heads fom one featue to the next. 10. The fist level of the poblem addesses: a. Gouping of design featues in goups (see Fig. 3). b. Allocating the UMFs of the design featues to the machines. c. Allocating tools to the machines. d. Selecting machines with appopiate tools to pocess UMFs elated to the design featues of a goup. e. Detemining the makespan when pecedence constaints ae not consideed The planning model The following notations ae used fo the fomulation of the poblem: Indices f 2 (1,2,...,F) face g 2 (1,2,...,G) goup 2 (1,2,...,R) design featue unit (DFU) s 2 (1,2,...,R) position of the DFU in sequence of DFUs. o 2 (1,2,...,O) unit machining featue (UMF, opeation) to be pefomed l 2 (1,2,...,L) tool Paametes O numbe of UMFs in DFU O fist fist UMF of DFU, i.e., UMF 1 þ P 1 0 ¼1 O0 O last (s) f l o Extact Design Featues, Unit Manufactuing Featues andpecedences fom the Job PlanGouping ofdesign Featues and Allocate Cutting Tools to Machines byoptimizing Opeation and Changeove Time Sequence Unit Manufactuing Featues fo eachgoup of Design Featues Geneate Numeical Contol (NC) Machine PogamstoAutomate the Machining Pocess Focus on Planning and Sequencing Fig. 2. The hieachical machining pocess-planning model. last UMF of DFU, i.e., UMF P 0 ¼1 O0 ¼ðOfist þ O 1Þ the DFU assigned to position s of sequence of DFUs. face at which DFU is located tool fo pocessing o (continued on next page) Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

4 4 K. Das et al. / Computes & Industial Engineeing xxx (2008) xxx xxx Design Featues DFU 1 DFU 2 DFU 3 DFU... DFU... DFU... DF... Goup 1 Goup 2 Goup 3 PoductionLine Fig. 3. Allocation of design featue units (DFUs) to goups of DFUs. b maximum available time (cycle time) on each machine D total numbe of design featue to be pocessed (demand) h l numbe of tool slots occupied by tool l H tool magazine size on each machine MAG g maximum numbe of machines allowed in goup g NM an uppe limit on the total numbe of machines allowed ROPN lg maximum numbe of opeations possible by tool l in goup g TO o time fo pocessing UMF o RT o efixtuing o eloading time fo pocessing UMF o TFC 0 time fo oientation change to visit face f 0 fo pefoming an opeation on design featue 0 fom face f afte completing an opeation on featue TTC oo 0 time fo eplacing tool l o afte pefoming an opeation o by tool l o 0 while pefoming an opeation o 0 Decision vaiables C lg 1 if tool l is loaded on goup g, 0 othewise EOR 0 g 1 if oientation is changed to face f 0 fo pefoming an opeation on design featue 0 afte pefoming an opeation on design featue located at face f when design featue and 0 ae in goup g, 0 othewise (lineaization vaiable) Y oo0 g 1 if tool is changed to l o 0 fo pocessing UMF o 0 afte pocessing UMF o by tool l o when the UMFs ae in goup g, 0 othewise (lineaization vaiable) NM g intege, numbe of machines in goup g og 1 if UMF o is pocessed in goup g, 0 othewise Z g 1 if featue is pocessed in goup g, 0 othewise Constaints G g¼1 A design featue is pocessed in one goup only. Z g ¼ 1 8: ð1þ Many design featues may be assigned to a goup g. If no design featue is assigned to a goup, the goup is actually not fomed. To allow a vaiable numbe of actual goups, the following is an implicit constaint: R ¼1 Z g P 0 8g: ð2þ When goup g fo design featue is selected, each UMF o of the featue is pocessed in goup g: og ¼ Z g 8; g; O fist 6 o 6 O last : ð3þ If UMF o is to be pocessed in goup g, then the equied tool must be chosen and loaded onto the goup; M is a constant indicating an uppe limit on the numbe of UMFs assigned to tool l fo:l o¼lg og 6 M C lg 8l; g: ð4þ Total tool slots occupied in a machine s magazine should not exceed its tool magazine capacity L l¼1 h l C lg 6 H 8g: ð5þ A machine cannot be oveloaded beyond its capacity. (The tems used in constaint 5 ae simila to the tems used in the objective function. Please see the discussion on objective function fo a discussion on the tems.) R ¼1 Olast o¼o fist þ R ¼1 D ðto o þ RT o ÞZ g þ R TTC oðoþ1þ Z g þ R O last 1 o¼o fist R ¼1 0 ¼1 R ¼1 0 ¼1 TFC 0EOR 0 g TTC O last ;O fist 0 EOR 0 g 6 bnm g 8g: ð6þ The numbe of machines to be used in goup g cannot exceed the maximum numbe of machines allowed in the goup NM g 6 MAG g 8g: ð7þ The total numbe of machines cannot exceed the maximum numbe of machines allowed G g¼1 NM g 6 NM: Objective function The objective function minimizes the total time consisting of opeation and efixtuing time (OPT), tool change time (TLCT), and oientation change time (ORCT). The opeations and efixtuing times, OPT fo pefoming UMF o, on DFU ove the entie demand D may be expessed as OPT ¼ R ¼1 Olast o¼o fist D ½TO o þ RT o Š: Since OPT is a constant, independent of gouping, the objective function includes TLCT and ORCT. Minimize: TLCT + ORCT ð8þ ð9þ Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

5 K. Das et al. / Computes & Industial Engineeing xxx (2008) xxx xxx 5 ORCT computes the oientation change time taken by a tool head to move to a diffeent face f 0 to pefom the next scheduled opeation on a DFU 0 afte pefoming one opeation on a DFU located at face f of the job. Sequencing is optimized in the sequencing model. In the cuent planning model, we assume that the sequence of DFUs is known and that all UMFs of a DFU ae pocessed contiguously. Hence, an oientation change time TFC(s)(s 0 ) fo some s 0 P (s + 1) is incued if DFUs (s) and (s 0 ) ae assigned to the same goup and if no othe DFU k, fo (s +1)6 k 6 (s 0 1) is assigned to that goup. Hence, ORCT ¼ R 1 R G s¼1 s 0 ¼sþ1 g¼1 TFC ðsþðs 0 ÞZ ðsþg Z ðs 0 Þg Ys 0 1 k¼sþ1 ð1 Z ðkþg Þ: ð10þ The above non-linea tem is fist lineaized following Balas (1964). Accoding to the pocedue, we fist convet Z ðsþg Z ðs 0 Þg Ys 0 1 k¼sþ1 ð1 Z ðkþg Þ to a zeo-one vaiable EOR ðsþðs 0 Þg o EOR 0 g: Afte lineaization, the function is conveted to ORCT ¼ R 1 o ORCT ¼ R R G s¼1 s 0 ¼sþ1 g¼1 R G ¼1 0 ¼1 g¼1 TFC ðsþðs 0 ÞEOR ðsþðs 0 Þg TFC 0EOR 0g : ð11þ ð12þ We need the following two equations fo this lineaization pocess: Z ðsþg þ Z ðs 0 Þg s0 1 Z ðkþg ðs 0 s þ 1ÞEOR ðsþðs 0 Þg and k¼sþ1 þðs 0 s 1Þ P 0 8g; s 0 P ðs þ 1Þ; s 2f1; 2;...; R 1g ð13þ Z ðsþg þ Z ðs 0 Þg s0 1 Z ðkþg EOR ðsþðs 0 Þg 6 1 k¼sþ1 P ðs þ 1Þ; s 2f1; 2;...; R 1g: 8g; s0 ð14þ Following the pocedue of Beale and Tomlin (1972), the 0 1 assignment poblem also equies the constaint that each design featue unit (DFU) be pefomed in exactly one goup: G g¼1 Z g ¼ 1 8: ð15þ This equiement has aleady been addessed by constaint 1, pesented above. TLCT computes the tool change time fo changing tool l o to l o 0 fo pefoming UMF o 0 (opeation) afte pefoming UMF o. In geneal, TLCT ¼ O O G o¼1 o 0 ¼1 g¼1 TTC oo 0Y oo 0 g: ð16þ Howeve, in ou planning model, we compute the tool change time using Z g and EOR 0 g. Notice that tool changes take place within a featue o between featues. Within a featue, a tool change is equied fo pocessing opeation o 0 =(o + 1) afte opeation o. In this case, o, (o + 1), and all belong to the same goup. Hence, Z g = 1 fo some g. Between featues and 0, a tool change is equied fo pocessing the fist opeation of DFU 0 afte pocessing the last opeation of DFU. In this case, EOR 0 g ¼ 1 fo someg. Hence, TLCT is defined as follows: 0 1 TLCT ¼ G R O last 1 TTC oðoþ1þ Z g þ R TTC O last EOR ;O fist 0 ga: g¼1 ¼1 o¼o fist Integality constaint: ¼1 0 ¼1 0 ð17þ ½ og ; Y oo 0 g; Z g ; EOR 0g ; C lg Š2f0; 1g 8g; o; ; o 0 ; 0 ; l: ð18þ Estimation of efixtuing time RT o is the efixtuing time fo each UMF. We ae poposing the following pocedue fo detemining an appoximate value fo RT o. {Tool positioning time fo UMF o of featue }+{Tool etacting time fom peceding }+{Rapid time (time elapsed fo tool head movement fom the peceding position to the next position theeby pocessing the same UMF type on a diffeent design featue when no tool eplacement is necessay)}. Tool etacting time and apid time ae needed when tools go fom one featue to the next, eithe afte a tool is changed to pefom a diffeent opeation, o when the same tool is used to pefom the same opeation type. If an aveage time estimate is available fo the tool positioning and the tool etacting steps of each opeation o UMF, then the tool changing step will only add the exta time equied to cove the distance fom the peceding position to the tool magazine and finally, fom the tool magazine to the next machining position, plus tool eplacement time. Depending on tool positioning, the etacting time, and the apid time between two UMFs, an aveage estimation of RT o is possible. Estimating RT in this way educes needless complexity in evaluating RT as well as the tool change time, TTC, as discussed peviously. A tool can pefom moe than one opeation (the same o a simila type of opeation) on one DFU o on a numbe of DFUs in a goup. This constaint is unusual fo the standad assignment poblem, but is often found in manufactuing situations and in the distibution of a single poduct in moe than one depot. Since ou aim is to minimize the numbe of tool changes, we define the constaint as educing tool movement in this manne: og ¼ ROPN lg 8l; g: ð19þ fo:l o¼lg Notice that theefoe, we can substitute M by ROPN lg in constaint Eq. (4) The sequencing model We addess the sequencing poblem of the UMFs in the second level. The sequencing poblem is solved fo each goup of UMFs sepaately (see Fig. 4). Since the machine and the goup to which the machine belongs have aleady been decided in the fist planning model, the two vaiables ae consideed fixed fo the sequencing phase. Assumptions fo the sequencing poblem: 1. All gouping poblems of the design featues ae completed with the objective of optimizing the makespan of the planned lot of DFUs (pats). 2. All machines, mateial handling systems, o any othe poduction suppot systems shall un without need fo sevice. 3. All tools ae new in the initial stage, and no tool beakage shall occu duing the pocessing of the planned lot. Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

6 6 K. Das et al. / Computes & Industial Engineeing xxx (2008) xxx xxx Notations fo the sequencing poblem: The following notations and all pevious notations ae applicable to this sequencing model: p 2ð1; 2;...; P ¼ P O o¼1 ogþ positions in the pocessing sequence fo pefoming UMFs. Decision vaiables U op 1 if UMF o is pocessed at position p, 0 othewise The sequencing model The sequencing model s objective function also minimizes the makespan time components. As it is in the planning model, the opeation and efixtuing time is a constant. Theefoe, the objective function has two tems only: one fo tool change time and the othe fo oientation change time. As discussed ealie, the diffeences in the sequencing model include the pocessing step positions based on pecedence equiements while single goup DFUs ae pocessed on a single selected machine type based on the outcomes of the planning model. Fo each goup g, Minimize h g, whee h g ¼ P 1 o: og ¼1 o 0 : o 0 g ¼1 p¼1 O last o¼o fist Olast 0 TTC oo 0U op U o 0 ðpþ1þ þ P 1 :Z g ¼1 0 :Z 0 g ¼1 p¼1 o 0 ¼O fist 0 TFC 0U opu o0ðpþ1þ: ð20þ subject to: Only one UMF o is assigned to any position p U op ¼ 1 8p: ð21þ o: og ¼1 Each UMF o will be assigned only once in the sequence P p¼1 U op ¼ 1 8o 3 og ¼ 1: ð22þ The succeeding opeation o + 1 cannot be pefomed without fulfilling the pecedent of the UMFs fo each design featue : p U op 0 P p U ðoþ1þp 0 p 0 ¼1 p 0 ¼1 8p; 3 Z g¼1 ; O fist 6 o 6 ðo last 1Þ: ð23þ The objective function fo the scheduling model is also non-linea. It can be lineaized by following a pocedue simila to the one used by the planning model. We mention the lineaized objective and constaints below: h g ¼ P 1 o: og ¼1 o 0 : o 0 g ¼1 p¼1 O last 0 TTC oo 0EU oo 0 p þ P 1 :Z g ¼1 0 :Z 0 g ¼1 p¼1 Olast o¼o fist o 0 ¼O fist 0 TFC 0EU oo0p: ð24þ U op þ U o 0 ðpþ1þ EU oo 0 p 6 1 8p 6 P 1; o 3 og ¼ 1; o 0 3 o 0 g ¼ 1; ð25þ U op þ U o 0 ðpþ1þ 2EU oo0p P 0 8p 6 P 1; o 3 og ¼ 1; o 0 3 o 0 g ¼ 1; ð26þ whee 1 if UMF o is assigned to position p and UMF o0 EU oo 0 p ¼ is assigned to position ðp þ 1Þ; 0 othewise: ð27þ Finally, the integality constaint: ½U op ; EU oo 0 pš2f0; 1g 8; o; o 0 ; p: ð28þ 4. Numeical example This section intoduces a numeical example theeby illustating the model s applicability. The instance is solved sequentially in two phases, namely, the planning model and the sequencing model. In phase 1, the planning model goups DFUs, allocates Table 1 Pat demand and pocessing infomation DFU Numbe of DFUs Location on face Data type UMF Tool# TO (s) RT (s) UMF Tool# TO (s) RT (s) UMF Tool# TO (s) RT (s) UMF Tool# TO (s) RT (s) UMF Tool# 9 10 TO (s) RT (s) UMF Tool# TO (s) RT (s) UMF Tool# TO (s) RT (s) UMF Tool# TO RT UMF Tool# 3 4 TO RT UMF Tool# 1 2 TO RT UMF Tool# TO RT UMF Tool# TO RT TO, opeation time in seconds; RT, efixtuing time in seconds. Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

7 K. Das et al. / Computes & Industial Engineeing xxx (2008) xxx xxx 7 goups of DFUs to machines, and loads the machines with the appopiate tools by optimizing the makespan. The planning model is solved hieachically. We fist minimize the numbe of machines used, which gives an estimate of the paamete NM. Given the numbe of machines allowed, the tool change and oientation change times ae then minimized. The hieachical appoach acknowledges that minimizing the numbe of machines is key to educing costs as it is an expensive esouce. In phase 2, the opeation sequencing poblem fo each goup of DFUs is solved futhe optimizing the tool change and oientation change times in espect to the uling pecedent. Since ou model assumes a known sequence, we follow an iteative appoach in solving these two models, whee the sequence is modified at each iteation until the makespan stops impoving. At the fist iteation, we assume a DFU sequence (s) = s "1 6 s 6 R. In othe wods, at iteation 1, the DFU sequence is fist set to 1,2,3,...,R. At subsequent iteations, the DFU sequence is modified accoding to the fist occuence of a DFU in the solution geneated by the sequencing model. The solution pocedue is biefly descibed below: Phase 1: The planning model Step 1: Let NM = min{nmj (1), (3) (8), (13) (14), (18)}. Step 2: Let (s)=s "1 6 s 6 R. Step 3: Solve the poblem min{orct + TLCTj(1), (3) (8), (12) (14), (17) (18)}. Phase 2: The sequencing model Step 4: Fo each goup g, solve the poblem min {h g j(21) (27)}. Step 5: If ORCT + TLCT is impoved, update and go to Step 3, othewise stop and etun the best solution. The model example is solved using LINGO 9.0 on an INTEL Coe 2CPU, GHz, 2 GB RAM compute Input data Table 1 summaizes the numbe of DFUs (demands) to be machined, the UMFs (opeations) and thei sequence fo each DFU, the tool types fo each UMF, and the locations fo the DFUs on diffeent faces. Fo example, DFU type 1 is located at face 1 with a demand fo 16 DFUs, each equiing 4 UMFs (opeations) using tool numbes 1, 2, 3 and 4, espectively. Table 1 also shows that the opeation time (TO) and efixtuing time (RT) fo pefoming UMF 1 on DFU 1 ae 2.88 and 1.73 s, espectively. The UMFs in the example poblem equies 12 types of tools fo thei pocessing some of which (tools) can pocess simila UMF (opeations) on moe than one DFU. Fo example, tool #1 can pocess UMF1 of DFUs 1, 2, 3 and 10. The tool change times in seconds ae geneated as integes unifomly distibuted in [2, 10]. The tool change time matix is Howeve, due to limited page space, we epot only the patial data elated to DFUs 1 and 2 in Table 2. A tool change time is composed of tool movement and tool eplacement times. As discussed in the pevious section, efixtuing times account fo most of the movement times. Theefoe, a tool change time includes a tool eplacement time plus an exta tavel time that Table 3 Chaacteistics of tools and machines Tool no. Numbe of slots Tool magazine capacity slots (fo each machine) Table 2 Tool change times matix elated to DFUs 1 and 2 DFU type DFU type UMF UMF Tool # DFU Type DFU type UMF UMF Tool # indicates that such combinations ae not pemitted. Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

8 8 K. Das et al. / Computes & Industial Engineeing xxx (2008) xxx xxx Table 4 Oientation change time matix Face Face indicates that such combinations ae not pemitted. Table 5 Chaacteistics of the planning and sequencing models at iteation 1 Item Planning Sequencing model model Goup 1 Goup 2 Goup 3 Goup 4 Total numbe ,683 14,737 11,867 11,811 of vaiables Numbe of ,616 14,670 11,800 11,744 intege vaiables Numbe of constaints CPU time 18 min 49 s 18 min 39 s 1 min 19 s 2 s Table 6 Solution of the planning model at iteation 1 DFUs Goup 1 Goup 2 Goup 3 Goup 4 Total 4, 5, 7, 11 6, 9, 12 3, 8, 10 1, 2 Numbe of machines used Tool changes Tool change time (TLCT) Oientation change Oientation change time (ORCT) ORCT + TLCT Opeation and efixtuing time Total (makespan) coves tavel to the tool magazine followed by tavel fom the tool magazine to the next position. When a single tool pefoms two consecutive opeations, the tool change time is set to zeo. Table 3 povides infomation on the numbe of slots fo each tool and tool magazine capacity fo each machine. Table 4 pesents the matix fo the oientation change times. Fo example, changing fom face 1 to face 2 takes 1.95 s, as shown in the table. DFU o UMF types do not influence oientation change times. Diffeences in oientation change times, as seen in the table, esult fom diffeent angles of tilt coveed by the fixtue while changing fom face to face. The oientation change times fo opeations TTC oo 0 can be deived fom Table 4. It is set as follows: the oientation change time TFC 0 fom DFU to 0 is given by the oientation change time fom face f to f 0 found in Table 4. Machine types with an effective capacity of 330 s each have been consideed as input infomation in the example. While these machines have simila capabilities fo pocessing each UMF, they ae loaded with diffeent types of tools. We assume a maximum of five machines allowed in each goup. The infomation in Tables 1 4 povide input data fo pat gouping, machine loading, and tool allocation, as well as fo sequencing the DFU opeations. All input infomation is geneated andomly in consideation of simila pactical data fom an engine manufactuing plant while pat gouping and machine loading model outcomes ae used as input fo the sequencing model Analysis of the model esult Table 5 summaizes the chaacteistics of the planning and the sequencing model. Fo each model, the following infomation is epoted fo iteation 1: the total numbe of vaiables, the numbe of intege vaiables, the numbe of constaints, and the computational time equied to obtain the global optimal solution. The pocedue teminated at iteation 3. The numbe of vaiables, constaints, and CPU times of iteations 2 and 3 ae simila to those of iteation 1. Tables 8a and 8b list the CPU times at iteations 2 and 3, espectively. The esults indicate that both planning and sequencing models have been solved efficiently. The computational esults of planning and sequencing models of the fist iteation ae epoted in Table 6. The esults indicate that the UMFs (opeations) of DFUs 4, 5, 7 and 11 ae assigned to Goup 1; Goup 2 is associated with the UMFs of DFUs 6, 9 and 12; the UMFs of DFUs 3, 8 and 10 ae assigned to Goup 3; finally, Goup 4 pocesses UMFs of DFUs 1 and 2. In each goup, only one machine is used. Table 6 also pesents a model of optimum opeation, efixtuing, tool change and oientation change times including the numbe of tool changes and oientation changes fo pocessing each goup of UMFs following this planning model. Evidently, when pat goups ae fomed and machines ae allocated to the goup, effective tool change and oientation change planning may be caied out accoding to the sequencing model. Tables 7a 7d epot computational esults of the fist iteation of sequencing models elevant to the UMFs in goups 1, 2, 3 and 4. The computational esults indicate that thee ae 12 UMFs in Goup 1, 10 UMFs in Goup 2, 8 UMFs in Goup 3, and 8 UMFs in Table 7a Sequencing model solution fo UMFs in goup 1 at iteation 1 Results Goup 1 Sequence shows [,o], whee = DFU, o = UMF Machining positions fo UMFs 1 * Sequence of UMFs [11,31] * [11,32] [11,33] [11,34] [4,12] [4,13] [4,14] [5,15] [5,16] [7,21] [7,22] [7,23] Tool sequence 11 * Tool changes 0 1 ** FACES 5 * Face changes 0 0 *** Tool change (s) 20 Oientation change (s) 5.95 Makespan (s) Note: * [11,31]: UMF 31 of DFU 11 is pocessed at position 1 (* at ow 3), using tool 11 (* at ow 5) at face 5 (* at ow 7). ** Tool change is 1 because diffeent tools 11 and 12 have been used in the successive positions 1 and 2. *** Face change equals 1 if the successive two opeations belong to diffeent faces; equals 0 othewise. Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

9 K. Das et al. / Computes & Industial Engineeing xxx (2008) xxx xxx 9 Table 7b Sequencing model solution fo UMFs in goup 2 at iteation 1 Results Goup 2 Sequence shows [,o], whee = DFU, o = UMF Machining positions fo UMFs Sequence of UMFs [12,35] [12,36] [6,17] [6,18] [12,37] [12,38] [6,19] [9,27] [9,28] [6,20] Tool sequence Tool changes FACES Face changes Tool change (s) 8 Oientation change (s) 7.5 Makespan (s) Table 7c Sequencing model solution fo UMFs in goup 3 at iteation 1 Results Goup 3 Sequence shows [,o], whee = DFU, o = UMF Machining positions fo UMFs Sequence of UMFs [8,24] [8,25] [10,29] [3,9] [3,10] [3,11] [8,26] [10,30] Tool sequence Tool changes FACES Face changes Tool change (s) 6 Oientation change (s) 12.9 Makespan (s) Table 7d Sequencing model solution fo UMFs in goup 4 at iteation 1 Results Goup 4 Sequence shows [,o], whee = DFU, o = UMF Machining positions fo UMFs Sequence of UMFs [1,1] [2,5] [2,6] [1,2] [1,3] [1,4] [2,7] [2,8] Tool sequence Tool changes FACES Face changes Tool change (s) 8 Oientation change (s) 0 Makespan (s) Goup 4. Consequently, the sequencing fo goups 1, 2, 3 and 4 is developed fo 12, 10, 8 and 8 positions, espectively. The optimal numbe of tool changes and face changes including times fo tool change, oientation (face) change and pocessing (as obtained fom the sequencing model) ae also descibed in Tables 7a 7d. Fo example, optimal tool changes and oientation (face) changes fo the UMFs in goup 1 ae 9 and 2 (Table 7a). Howeve, these figues fo the UMFs in goup 2, goup 3, and goup 4 ae {6 and 3}, {5 and 4} and {4 and 0}, espectively. The sequence fo the goup of UMFs is geneated with the aim of minimizing tool change times and oientation change times while especting pecedence equiements. The model geneates the tool change and oientation change sequences necessay fo pocessing the UMFs accoding to the sequence. The numbe of tool changes and oientation changes and, consequently, the times fo these changes ae diffeent in the sequencing model than the figues in the highe level planning model because the planning model does not optimize sequence. Diffeences in this example may be obseved by compaing esults in Table 6 with those in Tables 7a 7d. The sequencing model impoves the makespan of Goup 1 to s fom the s given by the planning model. This indicates a 2.35% impovement. Similaly, impovements ae 3.96%, 1.57%, 2.49% and 2.58%, espectively, fo Goups 2, 3, 4 and oveall. The esults of iteations 2 and 3 ae shown in bief in Tables 8a and 8b. As esults indicate, ORCT + TLCT has impoved at iteation 2. As shown in Table 6, ORCT + TLCT is 99.9 at the end of iteation 1. The impovement by iteation 2 is shown in Table 8a. The planning model impoves ORCT + TLCT to The sequencing model futhe impoves ORCT + TLCT to The esult does not impove at iteation 3. As shown in Table 8b, the ORCT + TLCT obtained by the planning model at iteation 3 is 85.35, which is moe than the ORCT + TLCT obtained by the sequencing model at iteation 2. Recall that the planning model assumes that all UMFs of a DFU ae pocessed contiguously. Theefoe, the solution geneated by the sequencing model is not necessaily a feasible solution of the planning model. The sequencing model elaxes the assumption that all UMFs of a DFU ae pocessed contiguously. At iteation 3, the sequencing model detemines an optimal sequence with ORCT + TLCT equal to 66.3, lowe than that of iteation 2. Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

10 10 K. Das et al. / Computes & Industial Engineeing xxx (2008) xxx xxx Table 8a Computational esults at iteation 2 Planning model DFU sequence 11, 4, 5, 7, 12, 6, 9, 8, 10, 3, 1, 2 (input fom the esult of iteation 2) CPU time 14 min 9 s Goups Total DFUs 3, 8, 10 5, 6, 7, 9 4, 11, 12 1, 2 ORCT TLCT ORCT + TLCT OPT Total (makespan) Sequencing model Goups CPU time 38 s 10 min 30 s 3 min 48 s 9 s Sequence [8,1], [8,2], [10,1], [3,1], [3,2], [3,3], [8,3], [10,2] [6,1], [5,1], [5,2],[6,2], [7,1], [7,2], [6,3], [9,1], [9,2], [7,3], [6,4] [12,1], [12,2], [12,3], [12,4], [11,1], [11,2], [11,3], [11,4], [4,1], [4,2], [4,3] [1,1], [2,1], [2,2], [1,2], [1,3], [1,4], [2,3], [2,4] ORCT TLCT ORCT + TLCT OPT Total (makespan) Table 8b Computational esults at iteation 3 Planning model DFU sequence 8, 10, 3, 6, 5, 7, 9, 12, 11, 4, 1, 2 (input fom the esult of iteation 1) CPU time 9 min 52 s Goups Total DFUs 1, 2 5, 7, 11, 12 4, 6, 9 3, 8, 10 ORCT TLCT ORCT + TLCT OPT Total (makespan) Sequencing model Goups CPU time 7 s 28 min 7 s 35 s 55 s Sequence [1,1], [2,1], [2,2], [1,2], [1,3], [1,4], [2,3], [2,4] [12,1], [12,2], [5,1], [5,2], [7,1], [12,3], [12,4], [11,1], [7,2], [6,1], [6,2], [6,3], [9,1], [6,4], [9,2], [4,1], [4,2], [4,3] [8,1], [8,2], [10,1], [3,1], [3,2], [3,3], [8,3], [10,2] [7,3], [11,2], [11,3], [11,4] ORCT TLCT ORCT + TLCT OPT Total (makespan) Oveall, the makespan is at the second iteation. Howeve, as shown in Table 8b, the makespan inceases to by the thid iteation. Theefoe, the iteative pocedue stops afte the thid iteation. The final solution is shown in Table 8a. 5. Conclusion Ou model intoduces a new appoach to pat gouping, tool allocation, and machine loading components fo FMS planning poblems poviding a way to optimize the tool change and oientation change times. Ou appoach will impove the oveall pefomance of use oganization by educing non-poductive times, such as the time taken to make tool and oientation changes. This study solves FMS planning poblems at highe levels and detemines the optimal sequencing of UMFs fo each goup in the second level. Whee only a limited numbe of UMFs (as patitioned by the pat gouping model) ae involved in the sequencing pat of the model, the computational effot equied fo the opeational level (sequencing poblem) poblem is also significantly educed. A numeical example is povided to demonstate the applicability of the poposed planning and sequencing models. This model may be consideed computationally feasible and may be solved fo poblem sizes simila to the example by commecially available solves, like LINGO 9.0. Ou eseach acknowledges that not expeimenting with eal, industial-sized poblems ceates limitations and, as such, futhe exploation in this espect may be consideed fo futue study. Acknowledgements This eseach is patially suppoted by a gant fom the Natual Sciences and Engineeing Reseach Council (NSERC) awaded to M.F. Baki. The authos thank two anonymous efeees fo thei helpful comments. Thanks ae also due to M. Kenneth Teague and M. Mike Pape fo thei contibutions to fuitful discussion. Refeences Avci, S., & Aktuk, M. S. (1996). Tool magazine aangement and opeations sequencing on CNC machines. Computes and Opeations Reseach, 23(11), Balas, E., Extension de l algoithme additif á la pogammation en nombes enties et á la pogammation non linéaie. Pais: C.R. Acad. Sci. Please cite this aticle in pess as: Das, K. et al., Optimization of opeation and changeove time fo poduction planning..., Computes & Industial Engineeing (2008), doi: /j.cie

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