AGGREGATED MODELS TECHNIQUE FOR INTEGRATING PLANNING AND SCHEDULING OF PRODUCTION TASKS

Transcription

1 Internatonal Journal of Modern Manufacturng Technologes ISSN , Vol III, No 1 / AGGREGATED MODELS TECHNIQUE FOR INTEGRATING PLANNING AND SCHEDULING OF PRODUCTION TASKS Ştefan Dumbravă Gheorghe Asach Techncal Unversty of Ias-Romana, Department of Automatc Control and Appled Informatcs Profdrdoc Dmtre Mangeron Street, No 27, , Ias, Romana Correspondng author: Ştefan Dumbravă, sdumbrav@actuasro Abstract: Effectve schedule of producton operatons that conducts to optmzed realstc and feasble solutons s a complex task that has been ntensvely studed due to ts economc mportance Startng from the graph of the detaled problem, a method of optmzaton the task schedule s emphaszed In order to obtan a mnmum tme for the project executon together the fulflment of resource constrants, the detaled model s aggregated, optmzed from the pont of vew of the graph heght and cardnalty and fnally the soluton s dsaggregated The method s appled n the case study of a Flexble Manufacturng System for an assembly project that s used for educatonal purposes and the results are emphaszed Key words: schedulng algorthms, flexble manufacturng system, executon tmes, graphs tmes based on hstory, the resultng producton plans are sometmes unrealstc or unfeasble, (Vollmann et al, 1997) The drawbacks of these methods may be accomplshed by usng complex models, able to characterze dfferent aspects of the manufacturng plannng problem One of the key problems of the producton management s that of fndng the optmum confguraton and explotaton of the manufacturng facltes It has been admtted that the soluton of the problem depends on the consdered tme horzon Consequently, t has been dvded nto three herarchcal plannng levels, Fg 1 1 INTRODUCTION Producton control vares greatly from one company/plant to the other There are stuatons where effectve producton control s possble and there are stuatons where the manufacturng challenge s so large that t s almost mpossble to generate any knd of feasble plan Due to the market fluctuatons, a good strategy s to work wth small sze nventores and a flexble control system that enables quckly react to the market changng demand, (Kenneth et al, 2004) For ths purpose, the development of mathematcal concepts or varous tools and ads for the producton control solvng problems has been a contnuous research preoccupaton Technques for plannng the mnmum safety stock, forecast of producton volume, a better schedulng n the detaled tmetables of the jobs or the computaton of the lead tme for quotng orders are some examples of research topcs n ths feld The goal of all these technques s to mprove the producton effcency wthn modern manufacturng concepts lke MRP/CRP systems Presently many companes use materal/capacty requrements plannng systems for medum and long term producton plannng In spte of MRP benefts, because of the assumptons of nfnte resource capacty, the dffculty to model resource constrants, fxed lead tmes and the heurstc estmaton of lead Fg 1 Levels of the plannng herarchy The levels of decson makng are called strategc (or long-term), tactcal (medum-term), and operatonal (short-term), (Kovacs, 2005) The level wth the longest tme horzon s the strategc one At ths level decson about fnal products, producton facltes, captal, resources and poltcs are taken In medum term plannng the decson taken mply the resources and to some degree, polces and facltes

2 40 Operatonal level s the shortest tme horzon and the decsons regard the way n whch avalable resources are used as effcent as possble n order to meet certan goals, lke cost mnmzaton and fllng customer orders Producton schedulng on the operatonal level develop the frst segments from producton plan n detaled resource assgnments and operaton sequences, (Kovacs, 2005) The last module n the plannng herarchy s the real-tme executon control It ensures a feedback about the status of the shop-floor tasks flow compared to the plannng The man objectve of the paper s to emphasze an aggregate modellng method n solvng the producton plannng problem that enables the soluton to be further developed nto detaled feasble schedules 2 METHOD DESCRIPTION An ntegrated approach of the producton herarchcal plannng levels, namely producton plannng and producton schedulng, s proposed It enables based on an aggregate formulaton of the producton plannng, a feasble detaled schedule of the actvtes and tasks Startng from the detaled problem, ts decomposed soluton s obtaned by reformulatng the aggregate problem from the detaled one, obtanng the aggregate soluton and fnally decomposng t The steps followed n applyng the method are emphaszed n fg 2 Dsaggregate Level problems solutons problem problems solutons Level Aggregate Aggregate problem Aggregate soluton Fg 2 The actvty dagram of the method steps n plannng problem solvng The soluton of the detaled problem s obtaned n three steps: The detaled model of the ntal plannng problem s aggregated, replacng the varables and the set of ther constrans n the detaled model wth one aggregate varable wth an aggregate constran; The aggregate model s solved by an adequate algorthm; The soluton of the aggregate problem wll be decomposed nto the detaled soluton durng dsaggregaton The aggregaton/dsaggregaton procedure must mantan the temporal requrements and those of capacty The aggregaton model s a less complex one that permts to fnd easer an optmal feasble soluton 21 Modellng and representaton of the Producton Plannng Problem The detaled problem s modelled usng the classcal resource-constrant project schedulng problem (RCPSP) model, (Brucker, 2004) Its formulaton consders the set of projects P that must be overcome durng an aprory known tme horzon Each of the projects p P has defned the earlest start tme t s, p and the fnal executon tme t f, p Each of the projects p contans a set of taskst p Each task t T p lasts a fx tme d and requres a porton of the renewable cumulatve resource rt R The resource capacty s denoted by q ( r t ) The tasks belongng to the same project may be connected by precedence constrants t t j, meanng that the executon of task t j can start only after the executon of task t s fnshed The RCPSP model of the detaled producton plannng problem s a tree whose nodes are the tasks the project and ts edges represent the precedence relatons between tasks Runnng the project starts from the leaves and fnshes wth the root A sample project s presented n fg 3 22 Model aggregaton The aggregaton procedure s based on the tree parttonng n connected sub-trees composed of combned tasks that belong to an aggregate actvty The throughput of ths operaton on the project model s a new tree havng n the nodes the actvtes (composed of the tasks) of the project The model s an extenson of the RCPSP one, called resourceconstraned project schedulng problem wth varablentensty actvtes (RCPSVP), (Kovacs et al, 2004; Rogers et al, 1991) based on the followng formalsm The aggregate problem comprses a set of projects P, a set of actvtes A that buld the projects, a set of contnuously renewable and dvsble resources R and an acyclc graph G, whch descrbes the end-to-start precedence constrants The tme horzon of the project s assumed known and t s devsed nto dscrete tme unts of the aggregate plannng The length of s chosen as an nteger sub

3 multply of the project tme horzon In every tme unt of the dscrete tme unt, a part of actvty A A s executed Ths s denoted by x and s called the ntensty of the actvty A n tme unt Every actvty may requre the smultaneous use of some resources, proportonal wth ther ntensty The total work of actvty A usng resource r s denoted by: A r d (1) t ta : rt r Durng the tme unt, the A actvty uses A A r x unts of resource r The soluton of the aggregate problem conssts n the calculaton of the ntenstes for every actvty n each tme unt so that the precedence and temporal constrants to be fulflled whle the maxmum capacty of the resources s not exceeded and the total cost s mnmzed The partton of the project tree s called the aggregate project model If two tasks connected by a precedence constrant of the project tree are nserted nto the same actvty then the constrant s omtted, whle t s mantaned f the tasks belong to two dfferent actvtes The actvtes precedence graph s also a tree The actvty A for whch d ( t) s ta called complete, whle those for whch d ( t) ta are called broken actvtes In order to characterse the aggregate model there are ntroduced the followng propertes: ( A ) s called the weght of actvty A and represent an estmaton of the requred tme necessary to execute all the tasks ncluded n that actvty The cardnalty of the aggregate model s denoted by c (P) The heght of the P tree s denoted by h(p) and t s gven by the longest path from the leave to the root The aggregate model wll dmnsh the computatonal complexty of the plannng problem but two long actvtes may affect the feasblty of the short-term schedules, (Toye, et al, 1990) A good compromse s related to the assgnment of the actvtes weght equal to the dscrete unt of tme that results from the dscretzaton of the project tme horzon However, the model aggregaton s not a relaxaton of the detaled model snce t ntroduces new constrants resultng from the tme horzon aggregaton n dscrete tme unts Thus, two coupled actvtes mght execute n dfferent dscrete tme unts Startng from a detaled model there are more than one soluton for obtanng the aggregate model A mnmal heght representaton conducts to smaller executon tmes and an ncrease of parallelsm n actvtes Based on the above remarks, an aggregate optmal model of a project s defned as follows 41 An actvty A s a connected component of the project tree f the estmated requred tme of ts executon, expressed by ( A ), fts nto an aggregate tme unt The optmal aggregate model P of a tree T means fndng a partton so that both h (P) and c (P) are mnmal 23 Optmal form of the aggregate model Gven the tree project T ( V, E, r), where V, E and r are the vertces, edges and the root of the tree then P st1,, st q s a parttonng f and only f each component st s a sub tree (connected sub graph) of T, the st components are dsjonted and the unon of the vertex-sets V ( st ) of the st equals V The root component of P s the one contanng r, and wll be denoted by RC (P) A component weght functon : st R on the sub-trees of T and a real postve constant W are defned The parttonng P st1,, st q of T s admssble f and only f ( st ) W, for every st P We assume that ( v ) W for each v V, whch mples that the trees have always admssble parttons Furthermore, we ntroduce the notaton of r( P) ( RC( P)) for the weght of the root component of P The functon s sad to be monotonous f for two sub trees st 1 and st 2 wth st1 st2 then ( st1 ) ( st2 ) It s denoted by S(v) the set of the sons of vertce v and T (u) a subtree of T rooted at u S(v) The level of a vertce u T (u) s defned as the heght of an optmal parttonng of a sub tree T (u) wth u S(v) Consderng P v and P u the parttonng of T (v) and T (u) respectvely, the maxmum heght of a partton P wth u S(v) s h max ( h( P )) and u max us ( v) K u u S( v) h( Pu ) hmax Based on the above notatons the followng parttons are defned: Pu u S( v), ) P u S( v), ) P1 v comb( K (2) P2 v comb( u (3) The algorthm runs nto two steps In the frst step the ntalzaton P v v s done The second step s an teratve one, durng whch one vertce v wth processed sons s chosen The optmal parttonng s found based on the optmal parttonng of the sub trees T (u) Ths step s repeated untl the root r s found The comb operator s appled to the optmal parttonng P u wth u S(v), obtanng the parttons P 1 and P 2 havng the heghts h( P v h and v v 1 ) u max

4 42 h( P2 v ) hmax 1 If P 1 v s admssble then the algorthm assgns Pv P1 v Else the equalty Pv P2 v takes place P 2 v s always admssble because t conssts of sub trees belongng to an admssble parttonng P u In order to decde f P 1 v s admssble J v x x T ( u) u K If ( J ) W then P 1 v t s admssble and Pv P1 v s the parttonng of mnmal heght 24 Dsaggregaton Dsaggregaton of the producton plan nvolves orderng each task of the detaled model nto one aggregate tme unt (Kovacs, et al, 2005) The dsaggregaton of the producton plan s complete wth solvng the detaled schedulng problems correspondng to the aggregate tme unts The representaton of the programmng problem based on constrants s formalsed as follows A sequence of operatons based on constrants s defned by the {X, D, C, O}, where X {x } s a fnte set of varables, each of them x may take values n ts doman D takng nto account a defned set of constrants C The soluton represents a sequence S S of the varables x so that x X : x v D, and the constrants are fulflled S S c C : c( v,, v ) true O - s the last component 1 N of the quadruple, denotes an objectve functon that assgns a real number to a soluton S The RCPS model n ths formalsm s composed of the varables representng the start moments of the tasks denoted by start t, where the tasks t T The task duraton d t s assumed to be an nteger number Consequently, the ntal doman of the varables s a set of nteger numbers These domans are further restrcted by the constrants that may be of temporal, precedence and resource capacty type The temporal constrants are condtonng the start and the end of the tasks, and belong to only one varable The precedence constrants are related to more than one task and mpose the order of the tasks executon The resource capacty constrant ensures that the total capacty requred does not exceed the avalable one A resource capacty constrant s responsble of a resource r and all the tasks that need that resource An optmsaton problem nvolves the mnmsaton of functon O, meanng the mnmsaton of the processng tme of all the tasks 3 JOB SCHEDULE IN THE FLEXIBLE MANUFACTURING SYSTEM For the purpose of the paper, a real world educatonal Flexble Manufacturng System (FMS) developed wthn the faculty department was consdered The layout of the system, composed of two flexble manufacturng cells, s shown n Fg 3 The frst manufacturng cell has the task of automatcally machnng of the raw parts beng developed around the robot #1 whch manpulates the parts from the raw part storages to the mllng machne and from the mllng machne to the part buffer, or to the conveyor The second cell s composed of robot #2, the vsual nspecton staton and the fnshed part storage The two cells are exchangng parts usng a crcular conveyor wth pallets Fg 3 A 3D vew of the FMS layout, [2] 1 - The part storage M1; 2 - The processed part buffer; 3 - ABB robot#1 (IRB 1400); 4 - The CNC Mllng machne; 5 - The Flex Lnk conveyor; 6 Assembly and vsual nspecton computer staton; 7 - ABB Robot#2 (IRB 2400); 8 - The part storage M2 The man objectve of the system s to produce a mxed type of parts, belongng to a group wth smlar features and to depost them on the ndcated pallets of fnshed parts n the programmed postons The fnshed parts are nspected by the vsual nspecton staton wthn the on-lne qualty system The manufacturng project conssts of assembly operatons of a fnal product whch are scheduled usng the aggregaton/ dsaggregaton method The resources of the system are defned as follows: r 1 - robot#1, r 2 - the CNC mllng machne, r 3 - the conveyor, r 4 - robot#2, r 5 - Vsual nspecton computer staton and assembly centre The tasks of the assembly project are denoted by t, 1, 12 : t 1 - postonng of the robot#1 at the raw part storage M1, t2 - a raw part s pcked and t s carred to the CNC mllng machne by robot#1; t 3 - the CNC mllng machne s setup; t 4 - CNC mllng machne s processng the part of type A; t 5 - The part A s unloaded from the mllng machne and s carred to the conveyor; t 6 - the conveyor s carryng the part A to the assembly centre; t 7 - robot#2 takes the part from the conveyor and places t to the vsual nspecton system; t 8 - robot#2 s postonng at storage M2; t 9 - a part B s carred from the storage M2 to the vsual nspecton and assembly staton by

5 robot#2; t10 - robot#2 fulfls the assembly task; t 11 - the assembly s nspected by the vsual nspecton system; t12 - the assembly part s carred to the storage M3 by robot #2 The detaled tree of the manufacturng project s bult, taken nto consderaton the sequence of the tasks, fg 4, wth the model parameters gven n table 1 43 In the second teratve step, an unprocessed task wth predecessor tasks processed s chosen There are P u t, t, t, t, t, t t obtaned the parttons , 7 and P u t, where u u Sv t u 2 8,t9 1, 2, v 10 1 t7 and u2 t8 Furthermore, the actvtes A1 t 1, t2, t3, t4, t5, t6, A2 t 7 and A3 t 8, t9 are bult n accordance to the constrant A On the obtaned parttons the comb operator s appled There are chosen K1 t7, K 2 t9 whch conducts to the parttons, fg 7 Fg 4 The project tree Table 1 The parameters of the detaled project model t d (t) (t) r 1 2 r r r r r r r r r r r r 4 Based on the detaled graph a partton that s not optmsed nto actvtes s obtaned as n fg 5 Fg 7 The partton obtaned after the comb operator was appled The admssblty of ths partton t s verfed, checkng the condton 10 7 ( ) 15 wth 15 In the next step the fnal optmal partton s obtaned as: P v t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12 It results an optmal model composed of 3 aggregated actvtes: A1 t 1, t2, t3, t4, t5, t6, A2 t 7, t8, t9, t10, A3 t 11, t12 wth mnmal heght h ( P v ) 2 and cardnalty c ( ) 5, fg 8 P v t Fg 5 The project parttonng wthout optmsaton In the second step of the algorthm, the aggregaton model of mnmal heght s generated In the frst step t s obtaned the P v t 1, t3, t8 partton represented n fg 6 Fg 6 The partton obtaned after the frst step of the algorthm Fg 8 The aggregated model of mnmum heght Table 2 The mnmal heght aggregated model parameters A A A A A A r 1 r 2 r 3 r 4 r 5 r A A A JOB SCHEDULE GENERATION The dsaggregated schedule of the obtaned model conssts n transformng the 3 actvtes nto three tmetables, each of them representng the task

6 44 schedule on resources durng every tme unt W and establshng for every task the start tme and the fnsh tme The parameters of the dsaggregated model are gven n table 3 The tmetables are bult usng constrant-based solver and are represented n fgs 9-11 Table 3 The parameters of the dsaggregated model t d (t) (t) r t s, p t f, p 1 2 r r r r r r r r r r r r r 1 r 2 r 3 r 4 Fg 9 The correspondng A 1 actvty tmetable of the r 5 aggregated model r 1 r 2 r 3 r 4 r 5 Fg 10 The correspondng A 2 actvty tmetable of the aggregated model r 1 r 2 r 3 r 4 r 5 Fg 11 The correspondng A 3 actvty tmetable of the aggregated model 5 CONCLUSIONS At the operatonal level n producton plannng herarchy, the producton schedulng of tasks s a complex operaton that must conduct to realstc and feasble tmetable of operatons Consequently, a model of the system that enables to emphasze the shop constrants and the fnte capacty of resources together the sequence constrants s requred Due to the complexty of such models an optmzed soluton s dffcult to be obtaned A method to solve the problem s the aggregaton of operaton that mantans the ntal constrants and conduct to smpler models The optmzaton of the aggregated models refers to the heght of the model graph and ts cardnalty Algorthmcally, the optmzed aggregated model s obtaned and the tmetable of the operaton s detaled by dsaggregaton of the optmzed model In ths paper, the man notaton together the steps of the algorthm are presented consderng a case study The man objectve of the paper s to emphasze the aggregate/dsaggregate method appled n the case of an educatonal Flexble Manufacturng System Startng from an assembly project the actvtes, the task and the resource occupaton are fnally obtaned 6 REFERENCES 1 Brucker, A Drexl, R Mohrng, K Neumann & E Pesch (2004) Resource-constraned project schedulng: Notaton, classfcaton, models, and methods European Journal of Operatonal Research, 112(1): 3 41, ISSN: Kenneth N McKay & Vncent CS Wers (2004) Practcal Producton Control a survval gude for planners and schedulers, Co-publshed wth APICS, pp27-45, ISBN: , Prnted and bound n the USA 3 Kovacs (2005) Novel Models and Algorthms for Integrated Producton Plannng and Schedulng, Ph D Thess, Budapest Unversty of Technology and Economcs 4 Kovacs & T Ks (2004) Parttonng of trees for mnmzng heght and cardnalty Informaton Processng Letters, 89(4): , ISSN: Markus, J Vancza, T Ks & A Kovacs (2003) Project schedulng approach to producton plannng CIRP Annals Manufacturng Technology, 52(1): , ISSN: Rogers, RD Plante, RT Wong & JR Evans (1991) Aggregaton and dsaggregaton technques and methodology n optmzaton Operatons Research, 39(4): , ISSN: X 7 Toye, C A (1990) Let's Update Capacty Requrements Plannng Logc Proceedngs of the Amercan Producton and Inventory Control Conference, Atlanta, Georga: Amercan Producton and Inventory Control Socety 8 T E Vollmann, Berry WL, Jacobs FR & Whybark DC (2004) Manufacturng Plannng and Control Systems, 5 th ed McGraw-Hll, ISBN 10: X, New York Receved: December 20, 2010 / Accepted: May 30, 2011 / Paper publshed onlne: June 10, 2011 Internatonal Journal of Modern Manufacturng Technologes