Production Planning under Hierarchical Workforce Environment

Transcription

1 , July 2-4, 2014, London, U.K. Poduction Planning unde Hieachical Wokfoce Envionment Sujan Piya, Nas Al-Hinai Abstact In a make-to-ode system, odes ae scheduled fo poduction based on the ageed due date with custome and the stategy of company. Poduction planning of such system includes scheduling of odes to the poduction peiods and allocation of wokes at diffeent wok centes. Complexity in the system aises when the opeation to pefom next and its pocessing time is skilled dependent whee a highe qualified woke type can substitute a lowe qualified one, but not vicevesa. Unde such woking envionment, efficient scheduling of odes and allocation of wokes at diffeent wok cente play a majo ole to impove system pefomance. This pape develops a mathematical model fo make-to-ode flow shop system that helps to identify optimum schedule of odes and allocation of wokes, unde hieachical wokfoce envionment, with an objective of minimizing the weighted aveage ealiness and tadiness of odes. A heuistic method is also poposed to educe the complexity and solve the model efficiently. Index Tems Woke allocation, Scheduling, Hieachical wokfoce, Make-to-Ode, Flow shop. D I. INTRODUCTION ue to fiece competition in the maket thee is an inceasing tend of company moving fom Make-to- Stock (MTS) model of poduction system towads Make-to-Ode (MTO) system. The main eason fo the populaity of MTO system is that the fim can offe a poduct which is unique and customized in natue [1]. Also, due to the success of Dell Copoation, many fims seek to offe geate poduct vaiety at a low cost by eliminating finished goods inventoy using MTO poduction model [2]. In MTO system, company stats poduction only afte getting confimed odes fom the customes and each ode will be fo a unique poduct. The scheduling of odes fo poduction depends on the due date attached to the ode and the stategy of company. The complexity in the poduction of odes occu when the opeations to be pefomed depends on the level of skill that the wokes posses, whee a highe qualified woke type can substitute fo a lowe qualified one but not vice-vesa. This hieachical natue of wokfoce will affect the time at which ode once scheduled will be finished. It will have diect implication on when to stat poduction and who should be allocated to poduce it because neithe poducing Manuscipt eceived Mach13, This wok was suppoted in pat by Sultan Qaboos Univesity unde gant IG/ENG/MIED/14/05. S. Piya and N. Hinai ae with the Depatment of Mechanical and Industial Engineeing, Sultan Qaboos Univesity, Muscat, Sultanat of Oman. (Coesponding autho phone: ; sujan@ squ.edu.om; nhinai@squ.edu.om). ealie than the due date (holding cost of inventoy) no poducing late (tadiness cost) is desiable to the manufactue. We can find numbe of liteatues dedicated to poduction scheduling i.e., scheduling of job to the machines fo poduction. To name few of them ae [3], [4], [5], [6], [7], [8], [9], [10]. These liteatues have developed new appoaches/ heuistics to solve scheduling poblem eithe in a flow shop o job shop settings with diffeent objective functions such as minimizing makespan, mean flow time, weighted tadiness / ealiness, total ideal time and so on. In these liteatues, scheduling is done by consideing the numbe and capacity of available machines. But, if the industy is labo intensive, then only consideing the numbe and capacity of machines is not enough. In such situation, the capability of woke fo pefoming a specific job, athe than the capacity of available machine, will detemine the duation on which the job can be completed. Since each woke will have thei own capability, foming a hieachical natue of wokfoce, scheduling ode based on the capability of wokes is anothe challenging task fo such system. Theefoe, togethe with scheduling and sequencing of odes to the machines, pope allocation of woke is also impotant to optimize available esouces. Accoding to [11] thee ae vey few published papes that have dealt on hieachical wokfoce poblem and moe eseach is needed in this aea. Reseach on hieachical wokfoce using combinatoial method can be taced back to [12]. The pape developed an algoithm which geneate feasible schedule unde the assumption that each woke must have 2 off-days pe week. In the same line of eseach [13] discusses necessay and sufficient condition fo a labo mix to be feasible and pesented one-pass method that gives the least labo cost. [14] shows that intege pogamming appoach is well suited fo solving the poblem studied by[13]. [15] pesented an optimal algoithm fo multiple shifts scheduling of hieachical employees on fou days o thee days wokweeks. [11] intoduced the concept of compessed wokweeks in the model of [14]. The wok of [11] is futhe extended by [16] by intoducing the concept of suitability of task assignment to individual employees. Recently, [17] extended the model of [14] and [11] by developing mathematical models unde the assumption that the wok is divisible. The models poposed in these liteatues give the optimum numbe of wokfoces with diffeent hieachies needed to satisfy custome ode. Howeve, in Make-to- Ode poduction system each ode will have cetain due date attached to it. Hence, it is necessay fo the company to identify the sequence on which ode to poduce fist and then which next togethe with identifying optimum numbes of wokes with diffeent hieachies to minimizing the total

2 , July 2-4, 2014, London, U.K. cost of poduction. Futhemoe, eviewed liteatues on hieachical wokfoce neglect the pecedence elationships that exist between the opeations in the ode, which is a citical issue when job needs to be pocessed at diffeent wok centes. Fom the liteatue eview it can be seen that thee ae many papes published on poduction scheduling and hieachical wokfoce planning sepaately. Howeve, many manufactuing companies have to deal with both of these factos simultaneously, especially when the company is labo intensive. This eseach aims at developing a mathematical model by integating hieachical wokfoce planning with ode scheduling poblem with an objective of minimizing weighted aveage ealiness and tadiness, which as fa as we know is the fist attempt in this aea of eseach. The est of the pape is stuctued as follows. Section 2 addesses the poblem desciption. Section 3 explains the poposed mathematical model in detail. Section 4 discusses a heuistic method developed to educed the complexity and solve the poposed mathematical model efficiently. Finally, concluding emaks and futue eseach diections ae highlighted in section 5. II. PROBLEM STATEMENT Scheduling of odes and allocation of wokes to the available esouces ae some of the most impotant and widely eseached issues in the field of poduction planning and contol as these issues have diect impact on the poductivity and/o cost of poduction. This pape has consideed the integation of afoementioned issues in a peiodic setting whee all of the odes that aived on a given peiod will be poduced in the next succeeding peiod. Once the ode aives with cetain due date attached to it, ode will emains in a pe-shop pool till the next peiod. Company will accumulate all the odes of a given peiod plus any backlog of past peiod to make schedule fo next peiod, depending on which the ode will go fo poduction. To minimize the weighted aveage ealiness and tadiness of odes, which is the objective function hee, it is necessay duing scheduling to conside the hieachical natue of wokfoce. Thee diffeent levels of wokes hieachy ae consideed, namely level 1, level 2 and level 3. Wokes in these levels diffe in tems of thei capability to opeate vaious machines and in the speed with which the woke can pefom a cetain opeation. Level 1 epesents the most efficient goup of wokes who can do all types of tasks that ae necessay to be pefomed in the ode o it is the goup of woke who can opeate all of the machines. Plus, this goup of wokes can pefom the opeation faste (less pocessing time) than any othe goups. On the othe hand Level 3 epesents the goup of wokes who can wok only on fewe machines and takes moe time to finish job. Capability of wokes that fall on Level 2 lies in between level 1 and level 3 in tems of numbe of machines they can opeate and the speed at which they can pefom the assigned opeation. Hee, the poblem is to identify the equied numbe of wokes of each level, allocate these wokes to diffeent wok centes and schedule the odes on the basis of capability of identified numbes of wokes of vaious levels. The pape assumes this poblem in a flow shop envionment. The shop basically consists of vaious wok centes. An ode consists of batch of one poduct type that will ente fom the initial wok cente and exit though the last wok cente. Each ode involves opeations at all the wok centes. The objective is to develop ode scheduling model by taking into account the hieachical natue of wok foce in such a way as to minimize the weighted aveage ealiness and tadiness. Following assumptions ae used fo developing mathematical model: - Same ode cannot be pocessed on moe than one machine at a time. - The pocessing time of opeation of an ode on a given machine by the given level of woke is pedefined. - Peemption of ode is not allowed. - Each machine can pocess at most one opeation at any time. - Due date of odes ae known and fixed. - Allocation of wokes in the wok cente fo the given ode emains fixed once the wok is stated. - Allocation of wokes in the wok cente fo diffeent ode may be diffeent. - Set up time is consideed negligible. - All odes ae available fo pocessing at the beginning of peiod. III. MATHEMATICAL MODEL This section discusses the poposed mathematical model developed to solve the poblem descibed in Section 2. All indexes, paametes, and decision vaiables used in the model ae listed below. Index i Wok cente, i= {1,.., m} o Custome ode, o= {1,, O} k Woke level, k= {1,..,K} f Unit in the ode, f={1,..,f} v Position of ode in the sequence, v={1,, V) Paamete q o Total numbe of units in ode o d o Due date of ode o t o Time at which poduction of ode o can be stated oi Waiting time fo opeation in wok cente i of ode o p iok Pocessing time woke level k takes to pocess one unit of ode o in wok cente i Cap Capacity of wok cente in a peiod CT iok Completion time of a batch of ode o in wok cente i by woke level k ST iof Stating time of f unit of ode o at wok cente i w Ealiness penalty of ode o fo each time unit of ealiness w Tadiness penalty of ode o fo each time unit of tadiness Decision Vaiable CT mo Completion time of ode o at wok cente m E o Ealiness time of ode o T o Tadiness time of ode o X iov 1 if at machine i ode o is assigned to v th position in the sequence; 0 othewise

3 , July 2-4, 2014, London, U.K. Y iok 1 if in wok cente i ode o is pocessed by woke level k; 0 othewise The Model: ' '' wo Eo woto min z O (1) o1 Subject to: Eo do CTmo (2) To CTmo do (3) CTmok CT( m 1) ok pmok o 1,..., O; k 1,......, K (4) CTiok STiof piok i 1,..., m 1; o 1,...,O ; k 1,...., K ; f 1 (5) K Y iok 1 i 1,......, m ; k 1 o 1,..., O (6) O X iov 1 i 1,......, m (7) o1 Cap i q o p iok Yiok o 1 i 1,..., m ; k (8) ST CT i( o 1) f iok o 1,....., O ; k 1,......, K ; f 1 ST( i 1) of CTiok o 1,....., O ; k 1,......, K ; f 1 CTmo, Eo, To 0 o 1,....., O ; (11) Y iok, X io 0,1 i 1,..., m ; o 1,...,O ; k 1,...., K The objective function, shown in Equation (1), is to minimize the weighted aveage ealiness and tadiness. While developing the cuent model it is assumed that all the odes that aive in pevious peiod can be completed within the next peiod with the egula poduction time of wokes. Equation (2) calculates the ealiness of an ode o which is geate than o equal to the diffeence between its due date and completion time at the last wok cente m. Similaly, Equation (3) calculates the tadiness of an ode. Completion time of an ode at the last wok cente will detemine the ealiness and tadiness of an ode. This can be calculated by Equation (4). Howeve, completion time of ode at the last wok cente depends on its completion time at the peceding wok cente, which is calculated by Equation (5). It indicates the completion time of an ode in a given wok cente by woke level k. Equation (6) is a binay vaiable to ensues that in each wok cente i ode o will be pocessed by one woke level k. The value of Y iok equals 1 means that woke level k is assigned to wok cente i to pocess ode o. Othewise, the value of it will be zeo. Similaly, equation (7) is also a binay vaiable to ensue that each ode is assigned only once in the given machine. The capacity constaint is epesented by Equation (8) and indicates that the total pocessing time fo odes with a given woke type must be less than o equal to the available capacity within a peiod. Constaint that a machine can opeate only one opeation of an ode at a time is indicated by Equation (9). Equation (10) is a pecedence constaint. Equation (11) is non-negative constaint and Equation (12) is binay vaiable. The stating time in the above equations can be calculated by using iteative method as follows. When i = 1 and f = 1 ST iof = max [t o, io ] (13) When i = 1 and f = 2, 3,., F ST iof = ST io(f-1) + p iok (14) When i = 2, 3,., m and f = 1 ST iof = max [ST (i-1)of + p (i-1)ok, io ] (15) When i = 2, 3,., m and f = 2, 3,., F ST iof = max [ST io(f-1) + p iok, ST (i-1)of + p (i-1)ok ] (16) IV. PROPOSED HEURISTIC Flow shop scheduling poblem with the objective function consideed in this pape is NP-had [18]. The complexity of the poblem is exacebated by hieachical natue of wokfoce that we have consideed hee. Theefoe, in ode to solve industial size poblem a heuistic method is poposed by assuming that at least one level of woke (Level 1) can wok on all the machines and they ae the most expeienced goup of wokes thus esulting into less woking time as well in all the wok centes. Let So be the set of odes to be poduced in any given peiod. Poposed heuistic woks as follow: Step 1: Geneate sub-sets of ode by selecting any two odes fom the set (So). The total numbe of sub-sets (z) n 1 will be equal to ( n a) whee, n is the total odes in the a1 set So. Fo eg.: Let, So =(o1,..,o6) and assume that these odes have to be pocessed in 6 diffeent machines (m1,, m6). Also, let s assume that thee ae thee levels of wokes (l1, l2, l3) out of which level 1 can wok on all the machines, level 2 on machines m3, m4, m5 and m6. Howeve, level 3 can wok only on machines m5 and m6.fo this example, the sub-sets ae (o1, o2), (o1, o3),, (o5, o6). Step 2: Select any sub-set and define the possible sequence. Hee, since each sub-set will have two odes, the total numbe of possible sequences will be equal to 2z. Fo sub-set (o2, o6) the possible sequences will be (o2-o6) and (o6-o2). Step 3: Fo any possible sequence of odes allocate Level 1 woke on all the machines as it has been assumed that woke that falls in this level can wok on the entie machines. Then, 3(a): Identify the numbe of diffeent levels of wokes who can wok on same sets of machines and allocate the woke on those machines accodingly. This esults into 3y diffeent possibilities on woke allocation whee, y is the numbe of diffeent levels of wokes that can wok

4 , July 2-4, 2014, London, U.K. on same sets of machines and is the numbe of machines on which y can wok. Fo ou example, in m1 and m2 only woke with level 1 can wok. So, fo any sequence of ode in these machines, assign woke with level 1 and fix it. Then, fo machines m3 and m4, wokes with level 1 and 2 can wok. So, one possible allocations fo ode sequence (o2-o6) on these machines will be [o2(m3/l1);o2(m4/l1)- o6(m3/l1;o6(m4/l2)]. The total possibilities fo woke allocations hee will be equal to 12. 3(b): Repeat Step 3(a) fo the emaining sequence. This esults into 6y diffeent possibilities on woke allocation fo odes (o2, o6). 3(c): Fo each possibility, calculate the objective function and then select the one that gives the minimum value of objective function. Hee, it should be noted that while calculating the objective function, fo emaining machines m5 and m6, still woke belonging to level 1 will be allocated. Let the best possibilities fo ou example be [o2(m3/l1);o2(m4/l2)-o6(m3/l2;o6(m4/l1)]. 3(d): Similaly, epeat the steps fom step 3(a) fo emaining machines by fixing woke level on machines based on the esult of Step 3 (c). Fo the example, emaining machines fo woke allocation ae m5 and m6 and all level of wokes can wok on it. The total possible solution fo woke allocations hee will be equal to 27. These situations will be enumeated by fixing woke level 1 in m1, m2 fo o2 and o6. Also, woke level 1 at m3 and m4 fo o2 and o6, espectively. Howeve, at m4 and m3 of o2 and o6 espectively woke with level 2 will be fixed. Step 3 esults into 3y s s diffeent possibilities on ode ss sequencing and woke allocation fo a set of ode whee s = (1,., S) epesents set of woke level who can wok on specific set of machine. 2 sets of wokes fo the example ae (l1, l2) and (l1, l2, l3) who can wok on machine sets (m3, m4), (m5, m6) espectively. The outcome of Step 3 will be the best sequencing of odes and allocation of wokes at all the machines fo odes (o2, o6). Step 4: Repeat steps fom step 2 fo all the emaining subset of odes and select the sub-set with the sequence and allocation that esults into minimum value of objective function. This esults into 6 ys s ( z) possibilities of ode ss sequencing and woke allocations fo all the sub-sets geneated in step 1. The outcome of Step 4 will be the best sequencing of odes and allocation of wokes at all the machines fo the pai of ode that must be scheduled fist fo poduction. Step 5: Next, epeat steps fom step 1 fo the emaining odes in the set (So) until the set emains empty. While epeating the steps it should be noted that the ode pai that has been selected in Step 4 with thei sequence and woke allocation should be fixed and the sequencing of new ode pai will stat afte the selected ode pai. x Poposed heuistic esults into 3 ys s n 2a n 2a 1 ss ao possible solutions on ode sequencing and woke allocations fo all the odes in the set So instead of ss 3 y s s n!. V. CONCLUSION This pape developed a mathematical model to allocate hieachical natue of wokfoce to diffeent odes at vaious wok centes in a flow shop envionment. This allocation poblem is integated with ode scheduling poblem. The poblem that the pape dealt with is stongly NP-had. Theefoe, a heuistic method is also poposed to addess industial size poblem efficiently. The poposed heuistic method educes the numbe of possible combinations of ode sequence and woke allocation dastically. The developed model helps a company to identify equied numbe of wokes with diffeent level of hieachies, ode pocessing sequence, and allocation of diffeent wokes to the available wok centes. As fa as we know this is the fist attempt to integate ode schedule poblem with woke allocation unde the situation whee diffeent levels of wokes exists. In the nea futue extensive numeical analysis will be conducted to see the pefomance of poposed model and check its sensitivity. Even with the poposed heuistic, if the numbes of odes ae vey high, with many levels of woke hieachy, then it may be difficult to solve the poblem in a easonable timing. Theefoe, intoduction of intelligent method such as GA, PSO, TS can be consideed as an extension of pesent eseach. Also, the next avenue fo the eseach expansion may be to conside the constaint on woking hous of wokes fo eg. each woke woks only 8hs pe day and max 5 days a week with 2 day est peiod. ACKNOWLEDGMENT The paticipation of the autho in this confeence has been made possible though the patial contibution fom Sultan Qaboos Univesity. This suppot is gatefully acknowledged. REFERENCES [1] J. Gallien, Y.L. Tallec, and T. Schoenmey, A model fo make to ode evenue management, Sloan School of Management, Massachusetts Institute of Technology, Cambidge, MA, 2004 (Woking Pape). [2] K.L. Keame, J.Dedick, and S. Yamashio, Refining and extending the business model with infomation technology: Dell compute copoation, The Infomation Society, 2000, Vol. 16, pp [3] I.Lee, and M.J. Shaw, A neual net appoach to eal time flow shop sequencing, Computes and Industial Engineeing, 2000, Vol. 38, pp [4] I. Kacem, S. Hammadi, and P. Bone, Appoach by localization and multi-objective evolutionay optimization fo flexible job-shop scheduling poblems, IEEE Tansactions on Systems, Man and Cybenetics, 2002, Vol. 32 (1), pp [5] B. Toktas, M. Azizoglu, and S.K. Koksalan, Two machine flow shop scheduling with two citeia: Maximum ealiness and makespan, Euopean Jounal of Opeation Reseach, 2004, Vol. 64, pp [6] L. Tang, J. Liu, and W. Liu, A neual netwok model and algoithm fo the hybid flow shop scheduling poblem in a dynamic envionment, Jounal of Intelligent Manufactuing, 2005, Vol. 16, pp

5 , July 2-4, 2014, London, U.K. [7] L. Liu, H. Gu, and Y.Xi, Robust and stable scheduling of a single machine with andom machine beakdowns, Intenational Jounal of Advanced Manufactuing Technology, 2007, Vol. 31, pp [8] B. Yagamahan, and M. Yenise, Ant colony optimization fo multiobjective flow shop scheduling poblem, Computes and Industial Engineeing, 2008, Vol. 54(3), pp [9] Y. Xia, B. Chen, and J. Yue, Job Sequencing and due date assignment in a single machines shop with uncetain pocessing times, Euopean Jounal of Opeational Reseach, 2008, Vol. 184, pp [10] N. Al-Hinai, and T.Y. El Mekkawy, Robust and stable flexible job shop scheduling with andom machine beakdowns using a hybid genetic algoithm, Intenational Jounal of Poduction Economics, 2011, Vol. 132, pp [11] S.U. Seckine, H. Gokcen, and M. Kut, An intege pogamming model fo hieachical wokfoce scheduling poblem, Euopean Jounal of Opeational Reseach, 2007, Vol. 183, pp [12] H. Emmons, and R.N. Buns, Off-day scheduling with hieachical woke categoies, Opeations Reseach, 1991, Vol. 39, pp [13] R. Hung, Single-shift off-day scheduling of a hieachical wokfoce with vaiable demands, Euopean Jounal of Opeational Reseach, 1994, Vol. 78, pp [14] A. Billonnet, Intege pogamming to schedule a hieachical wokfoce with vaiable demands, Euopean Jounal of Opeational Reseach, 1999, Vol. 114, pp [15] R. Naashiman, An algoithm fo multiple shift scheduling of hieachical wokfoce on fou-day o thee-day wokweeks, INFOR, 2000, Vol. 38, pp [16] R. Pasto, and A. Coominas, A biciteia intege pogamming model fo the hieachical wokfoce scheduling poblem, Jounal of Modeling in Management, 2010, Vol. 5, pp [17] C. Ozguven, and B. Sungu, Intege pogamming models fo hieachical wokfoce scheduling poblems including excess off-days and idle labo times, Applied Mathematical Modelling, 2013, Vol. 37, pp [18] C. Koulamas, The total tadiness poblem: Review and extensions, Opeations Reseach, 1994, Vol. 42(6), pp