A manpower allocation problem wit layout considerations Luca Zeppetella, Elisa Gebennini, Andrea Grassi, Bianca Rimini Dipartimento di Scienze e Metodi dell Ingegneria, Università degli Studi di Modena e Reggio Emilia Via Amendola 2, 42122 Reggio Emilia, Italy Abstract In tis study we investigate te problem of assigning tass to operators in a facility caracterized by longitudinal parallel macines suc as in a sop floor served by an overead travelling crane. Given a master production scedule (MPS) te obective is to assign all te obs sceduled on te macines (i.e., te tass) to te operators in order to fill to capacity te available worforce minimizing te distance between operators and tass. In te model we assume tat one tas, i.e., a particular production ob processed by a particular macine, must be entirely completed by a single operator. Different levels of automation of te macines are considered, from manual macines tat require a permanent employee to igly-automated macines were a single operator can oversee several macines. During te setup time or repair time of a macine te operator is considered free to operate on te remaining tass assigned to im, if any. On te basis of te MPS te number of operators is pre-defined in te long-term planning orizon taing in consideration a fixed mean transfer time between te tass, tat are te different production obs on different macines. Tis value as a uge uncertainty because it is igly influenced by te tass allocation. In fact a simultaneous multiple allocation means a continuous bac and fort of te operator between is assigned macines. Te obective of te model is te maximization of te operators utilization troug minimizing te operator-tas distances. Te baclogged wor is not admitted, terefore eac day is independent of te oter days, so a daily staffing is modelled. Te study arises from a specific real-world problem but it could be easily extended to oter contexts in wic te operator-tas allocation is subect to spatial-layout considerations. In general, non-optimized operators travel times may result in production losses, i.e., macine blocing and wor in progress. Keywords: Personnel sceduling, Manpower allocation, Manufacturing, Production 1. Introduction Every morning in some enterprises, especially in small ones, te production planner decides te allocation of operators to te macines. A centralized operator allocation is suitable in contexts were obs are caracterized by a low level of creativity (see, e.g., Currivan, 1999). Several autors, among wic Gordon and Erut ( 2004), underline te importance of te maximization of te ob satisfaction, but in some manufacturing sectors wit single operation tass and intercangeable operators, te purpose of te ob satisfaction can t be simply acieved troug te ob allocation and intrinsic motivation. In addiction, te possibility of considering employees preference, as in Sanazari-Sarezaei et al. ( 2013), is not suitable because in tese contexts operators ave usually a uniform preference for te same obs due to te limited features of te tass. In some manufacturing contexts, te sceduling procedures first consider te macines sceduling, because te macines and teir setups represent te bottlenec of te processes. For tis reason a typical sceduling procedure in small enterprises primarily involves te creation of te master production scedule ( MPS) and ten te dimensioning of te total amount of te worforce needed. A sop floor wit unrelated parallel macines can present different levels of automation, terefore different degrees of overseeing are involved and some operators could be assigned to oversee more tan one macine at te same time. In a situation of daily allocation of operators, te production planner needs a support in order to tae into account all te variables involved in te decision. Oterwise, it could lead to a large inefficiency, in particular if te te combination of te facility layout and macines sceduling is not favourable. Nowadays tis is particularly true of small and medium enterprises tat are increasingly looing for applying lean practices ( see, e.g., Zeppetella et al., 2013) in order to eliminate wastes and sources of inefficiency. Te remainder of tis paper is organized as follows. Section 2 deals wit te related literature in te studied field. Section 3 develops te model and analyses its input parameters. Section 4 presents te formal matematical model, wile section 5 applies te formulation to a realworld problem. Finally, Section 6 concludes te paper and 132
outlines some of te potential extensions of our model. 2. Literature review As confirmed by Van den Berg et al. (2013) te personnel sceduling problem as been studied widely but at te same time is considerably evolved since te introduction by Dantzig (1954) and Edie (1954) in te 1950s. Several classifications of te problem ave been put forward. One of te first is te recurring classification of Abernaty et al. ( 1973). In is framewor, composed of planning, sceduling and allocation, our study can be included in te tird class. Our wor deals wit te allocation of te worers at te beginning of te worday considering available worers and actual requirements. A typical example of tis class of problem is te ospital nurse sceduling. Anoter classification was proposed by Baer (1976). According to Baer (1976), tree main groups can be distinguised: sift sceduling, days off sceduling and tour sceduling. Our model is a sift sceduling, tat is a sceduling across a daily planning orizon and wit te possibility of involving overlapping sift. A more recent classification, introduced by Ernst et al. ( 2004), focuses on sceduling and rostering metods. Teir classification presents te personnel sceduling process as a number of combinable modules. Witin tis framewor our study involves te modules of tas assignment and staff assignment. Tis classification also distinguises te application areas, in our case is te manufacturing one. Ernst et al. ( 2004) also consider te approac used: our approac is constraint programming. In our wor we don t consider employees training or ability, even if it is widely studied in literature, as in Brusco (2008), Corominas et al. (2010) and Nembard and Bentefouet ( 2012). Altoug many autors ave investigated te personnel sceduling, as far as we now, it is rarely investigated te correlation between sceduling and travel time of te operators between macines in a given facility layout. Eiselt and Marianov ( 2008) approac te tass assigning problem to a spatial point of view but tey focus on te equity of te solution acieved. Moreover tey minimize te employee-tas distances not taing in account te possibility of parallel assignment of obs to one operator at te same time. In tis paper we consider te terms ob and tas and te terms employee and operator as synonymous. Anoter example of travel time consideration is te specific case of te obs dispersion in te manpower allocation problem wit time window and obteaming constraints ( MAPTWTC) proposed by Li et al. ( 2005). Te correlation about sceduling and facility layout is usually taen in account for te new design or reengineering of facilities layout, and models for te layout optimization tat tae in account te manpower allocation are widely proposed in literature, for example Grao ( 1995) and Canen and Williamson ( 1998). In case of a macine failure our model can be used to react and reassign te operator tat was overseeing te macine. In tis way te worload is levelled among te available worforce. In employee sceduling researc te rolling adustment approac as been presented in literature, for example in te two-stage stocastic program proposed by Morton and Popova (2004) or in conunction wit integer program for reacting sceduling of nurses in a ospital by Bard and Purnomo ( 2005). Te reacting scedules ave te benefit to reduce in case of a macine failures te disparity of worload, tat influences te ob satisfaction ( Eiselt and Marianov ( 2008)). 3. Model description In tis section, we describe te model tat is formally presented in te next section. Te primary scope of te model is te maximization of te operators utilization troug te minimization of transfer times between teir assigned tass. In general, an operator oversees eiter a manual macine or several automated macines at te same time. In te latter case, te operator as to move from is assigned macines continually bac and fort, causing a reduction of te effective operator capacity. From ere, te need of eeping tese travel times to a bare minimum. We assume te following conditions: Te total number of available operators is fixed ( i.e., te operator dimensioning problem is out of te scope of tis paper). Te capacity of te worforce is supposed to be sufficient to complete all te sceduled tass. Te facility layout (number, types and locations of macines) is given. Te MPS is nown daily. Eac day, te capacity of te macines is saturated by te assigned tass and teir setups. For eac tas, resulting from te assignment of a ob to a macine, te exact personnel requirement is computed. Tis parameter is denoted by δ man, for any ob, in te following. Te personnel requirement of tas, δ man, is te fraction of te macine cycle time in wic te operator as to oversee te very macine (e.g., te time necessary to remove te product from te macine and put it in a storage buffer). Terefore, according to its definition, δ man is equal to 1 for a manual macine wic requires a permanent overseeing, wile less tan 1 for obs assigned to automated macines. Te operators wor is also influenced by te possibility of interrupting a ob. Tis results in a certain frequency of travels among te assigned tas, e.g., a ob could require an operator once every 3 minutes or every 30 minutes depending on te type of ob. In our model, no inventory of processed wor is allowed. Tis maes sense considering tat te operators only support te production, so tey could limit but not increase te production rate given by te macines. According to te model proposed by Wild and Scneewei 133
( 1993), assuming tat tere is no baclogging or inventories of processed wor, daily staffing decisions are independent of one anoter. Terefore, te considered orizon is te worday. In order to facilitate te matematical formulation, te worday is discretized into time bucets. Te lengt of te time bucets must be defined according to te specific application. In general, te time bucet sould last as te duration of te sortest operation of te day among woring operations, setup operations or maintenance operations. We assume tat te condition of any macine ( i.e., if it is woring or idle due to setup or maintenance operations) is nown at te beginning of eac time bucet. In te present formulation of te model, an approximation in te evaluation of te operator capacity is introduced. Te transfer time between te different time bucet is not considered as a reduction of te available capacity of te operator. Tis approximation is balanced by te obective function tat drives te solution so tat at eac tas tat last more tan one time bucet sould be assigned te same operator. In addiction, te approximation could be balanced wit a sligt reduction of te lengt of te time bucet. Te days off and breas sceduling is out of te scope of tis model, due to te fact tat are commonly fixed in small and medium enterprises. In our model, we don t distinguis between full-time operator and part-time or contingent operator, due to te fact tat te dimensioning of te staff is given as input and cost evaluations are not proposed. During te setup time or repair time, a macine is not assigned to te operators because we assume te presence of specific teams for setups and maintenances. Te model presents two obective functions. Te first obective function connects adacent time bucets and minimizes te canges of operators on te same ob. Te second obective function addresses te core of te problem minimizing te operator-tass distance in eac time bucet. Tis obective function is quadratic due to te fact tat te distances travelled by any operator depend on te set of is/er assigned tass. Te two obective functions are solved in lexicograpical order. Te cosen order advocates minimum canges of te operator rater tan a myopic distance minimization. 4. Matematical formulation of te model Tis section develops a formal matematical model based on te discussion in te previous section. In te model, we consider: tass : 1,...,; time bucets t: 1,...,T ; operators in period t: 1,...,. In every time bucet te number of available operator could be different, in order to mae te model suitable for several possible manpower dimensioning and breas policies. All time bucets ave te same duration tat is set to te value T cap. We indicate te fraction of time required for an operator to complete te ob wit te parameter δ man. According to te definition, its value is bounded as follows: 0 δ man 1 Te parameter τ t denote if te ob is performed in te time bucet t and it is defined as follows: { 1 if ob is sceduled in t, τ t = 0 oterwise. Wen a ob is performed in te time bucet, it is performed for te wole time bucet. According to tis assumption, te duration of te time bucets, T cap, was fixed as te minimum duration among te operations. According to te facility layout and te MPS, te matrix of te distances between any couple of tass can be obtained. We denote by d te distance, expressed in travel time, between tas and tas, tat is te distance between te macine associated to te ob and te one associated to ob. Witout loss of generality te matrix d is assumed to be symmetric. Given tat we now te frequency of operator travels associated to te ob, denoted by f, we can compute te frequency related to te time bucets duration, f T cap, as f T cap = f T cap. Ten, we define a not symmetric matrix in te form of { f T cap f T cap = if >, 0 oterwise. were f T cap is f T cap 4.1. Decision variable = max [ f T cap ; f T cap ]. Te variable x t defines te allocation of te operators to te obs sceduled by te MPS. { 1 if ob is allocated to operator in t, x t = 0 oterwise. 4.2. Constraints One operator can be assigned to a ob only if te ob is active in te time bucet. In a time bucet, one and only one operator can be assigned to a ob. Terefore, we can require tat x t = τ t,t Consider now te fixed duration of te time bucets. Te worload assigned to an operator is limited by te time 134
bucet duration troug te constraint x t δ man T cap + x t x t d f T cap T cap,t. Denoting by γ te number of canges of operator for te ob, we can formulate ( T ) [ T γ = t t=1τ t=2 4.3. Obective functions =1 ] (x t 1 x t ) 1. In te previous section, we introduced te presence of two obective functions. Te first obective is about te minimization of te canges of operators on one ob among time bucets, tat is min γ Te second obective involves te minimization of te travel distances in case of multiple parallel allocation of obs to one operator in a time bucet min T t x t x t d f T cap Te obective functions are solved in lexicograpical order. Ten te problem can be formulated as follows: ( lex min γ, subect to T t ) x t x t d f T cap x t = τ t,t, (1) x t δ man T cap + (2) + x t x t d f T cap T cap,t, ( T ) ] γ = t (x t 1 x t ) 1 (3) t=1τ [ T t=2 =1 0 δ man 1, (4) x t [0,1],,t, (5) 5. Case study τ t [0,1],t, (6) γ 0. (7) Te model is based on a real-world problem and we test te model on real data taen from a small enterprise located in Reggio Emilia. Tis company produces plastic inection moulding products according to te specific design of te customers, so teir products are very customized and wit a low level of standardization. As a consequence, te production process is variable and igly influenced by te customers requirements. Te company employs about 20-30 full-time operators at te sop floor level and occasionally tey use interim worers. We loo at a subset of 15 employees. Tey are considered at te same training level and completely intercangeable. Te facility layout of tis company is particularly unfavourable for te employee sceduling because it is a straigt area, long and tigt, and te macines are parallel aligned. Te macines are eat presses and every macine requires an operator to supervise te process. Te degree of overseeing depends on te level of automation of te process. Te complexity of product and mould and te degree of automation of te macine determine te complexity of te overall process and consequently te adequate degree of overseeing. At one extreme, an operator oversees continuously a manual operation, suc as te insertion of a piece in te mould and te subsequent witdrawal of te product from te mould. On te oter, igly-automated macines ust require a manual intervention every alf an our. Given te specific facility layout of te firm te operator sceduling could ave a big impact on te productivity and te possibility of completing all te obs. A non-optimized scedule results in te accumulation of ig inventories of processed wor on te macines buffer. We considered 19 macines and 5 operators per sift, so in tis specific case te number of operators is fixed for te wole day, and te parameter reduces to K. Table 1 sows te distance between te macines. According to Table 1 and assuming an operator wal speed of 1.3m/s we computed te travel times between te tass. Te worday is composed of 3 sifts of 8 ours eac, terefore 48 time bucets of alf-our eac (T cap = 1800 seconds). We assume to now te MPS, i.e., te value of te parameter τ t for eac tas in time bucet t. Setups last from 1 to 6 time bucets. Table 2 and 3 present te values of δ man and f T cap for te tass involved in te solution presented in Table 4. We run te experiments using LocalSolver(TM), on a Intel(R) Core(TM) i7 PC 3.00 GHz, 12 GB RAM of memory. Te solver LocalSolver(TM) applies a ybrid neigbourood searc approac. After 1 our run te solution obtained suggested 230 operator canges and 53373.2 m, wile in 24 ours te solution was improved: 187 operator canges wit a 53666.2 m of total distance of all te operators. Due to space limits Table 4 presents te first 20 time bucets of te overall solution obtained in te 24- ours run. Analysing te solution, we can observe tat in te first time bucet operator 1 is assigned to tass 7, 8, 9 and 15. Te sum of te personnel requirements of tese four tass is 0.7 of te time bucet (1260 sec). Ten, te operator capacity is reduced by te total travel time ( 427.7 sec) associated wit te set of assigned tass. Tus, te utilization is 0.94 for operator 1 in te first time bucet. Te same consideration applies to te oter operators in te different time bucets. It can be noted tat tas 11 is man- 135
Table 1: Distances between te macines [m]. Macine 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 0 8 15 18 21 24 30 32 35 37 39 41 45 43 52 5 8 10 12 2 8 0 7 10 13 16 22 24 27 29 31 33 37 35 50 7 5 7 10 3 15 7 0 8 11 16 22 24 27 30 32 35 38 36 45 18 14 11 7 4 18 10 8 0 8 14 16 18 21 24 26 29 32 30 39 21 17 14 9 5 21 13 11 8 0 10 17 20 23 26 28 31 34 32 41 25 21 18 13 6 24 16 16 14 10 0 4 7 10 13 15 18 25 23 34 26 22 19 15 7 30 22 22 16 17 4 0 8 11 14 16 19 26 24 36 28 24 21 17 8 32 24 24 18 20 7 8 0 8 11 13 18 25 23 33 30 26 23 19 9 35 27 27 21 23 10 11 8 0 8 10 15 22 20 30 32 28 25 21 10 37 29 30 24 26 13 14 11 8 0 8 13 20 18 28 34 30 27 23 11 39 31 32 26 28 15 16 13 10 8 0 9 16 14 24 36 32 29 25 12 41 33 35 29 31 18 19 18 15 13 9 0 8 6 16 40 34 31 27 13 45 37 38 32 34 25 26 25 22 20 16 8 0 4 14 47 36 33 29 14 43 35 36 30 32 23 24 23 20 18 14 6 4 0 10 45 40 37 32 15 52 50 45 39 41 34 36 33 30 28 24 16 14 10 0 60 42 39 35 16 5 7 18 21 25 26 28 30 32 34 36 40 47 45 60 0 8 12 16 17 8 5 14 17 21 22 24 26 28 30 32 34 36 40 42 8 0 8 18 18 10 7 11 14 18 19 21 23 25 27 29 31 33 37 39 12 8 0 8 19 12 10 7 9 13 15 17 19 21 23 25 27 29 32 35 16 18 8 0 Table 2: Values of δ man for te obs. ob 1 2 3 4 5 6 7 8 9 10 11 δ man 0.2 0.2 0.1 0.2 0.2 0.3 0.1 0.2 0.3 0.2 1 ob 12 13 14 15 16 17 18 19 21 34 37 δ man 0.1 0.2 0.2 0.1 0.3 0.1 0.2 0.2 0.3 0.1 0.2 Table 3: Values of f Tcap for te obs. ob 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 34 37 f T cap 2 2 5 5 5 4 6 4 3 3 5 6 4 3 2 2 4 5 4 6 3 3 Table 4: Te first 20 time bucets of te solution obtained. (Te value in te cell represents te operator assigned to te ob in te time bucet) Macine Time b. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 ob 1 2 2 2 2 ob 2 5 5 5 5 5 5 3 3 3 3 3 3 3 2 2 2 1 1 1 5 3 ob 3 4 4 4 4 4 4 4 4 4 1 4 4 4 4 ob 4 5 5 5 5 5 5 2 1 1 5 5 2 2 2 2 2 1 1 1 1 5 ob 5 5 5 5 5 5 5 2 1 1 5 5 2 2 2 2 2 1 1 1 3 6 ob 6 2 2 2 2 2 3 3 3 3 5 5 2 2 5 5 5 5 5 5 3 7 ob 7 1 3 3 3 5 5 5 5 2 2 1 1 5 5 5 5 5 5 5 3 8 ob 8 1 3 3 3 4 4 4 4 4 4 4 4 4 3 3 3 3 5 5 5 9 ob 9 1 3 3 1 1 1 5 5 2 2 1 1 5 3 3 3 3 4 4 4 10 ob 10 3 3 3 3 3 3 3 3 3 3 3 2 2 2 11 ob 11 3 1 4 4 3 2 1 2 5 1 2 5 1 4 1 1 2 3 3 12 ob 12 4 4 1 1 1 1 5 5 2 2 1 1 5 5 5 5 5 4 4 4 13 ob 13 4 4 1 1 1 1 5 5 2 2 1 1 5 5 5 5 5 5 5 14 ob 14 2 2 1 1 1 1 15 ob 15 1 3 3 2 2 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 16 ob 16 4 4 2 2 2 3 2 1 1 3 3 3 3 1 4 4 4 2 2 2 17 ob 17 5 5 5 5 4 4 4 4 4 4 18 ob 18 3 4 4 4 4 4 4 4 4 4 2 2 2 1 1 1 1 19 ob 19 4 4 4 1 4 4 4 2 2 2 3 ob 21 1 1 ob 34 4 4 4 14 ob 37 5 136
ual so tat its assigned operator cannot be assigned to any oter tass until tas 11 is completed. 6. Conclusions and furter researc In tis paper we proposed a model for te allocation of tass to operators, in order to reduce te tassoperator distance and acieve an improved productivity. Te model as been designed to suit parallel macines sop floors wit different level of automation and terefore different manpower requirements for te obs. An optimized scedule of te operators results in minimized travel times and in well balanced worload among te operators. In order to gratify te employees in case of repetitive tass wit low levels of required training a balanced worload is mandatory, tis is one of te most powerful leverage to positively influence te ob satisfaction. Te model in its present form applies to te case in wic all tass must be performed and te obectives are to minimize te canges of te operator on a single ob and to minimize te tass-operator distance. A limitation of te model is te approximation introduced in te computation of te available capacity lacing te evaluation of te travel time among te time bucets. Te model as been solved by a numerical example. Computational results ave been obtained for a set of operators woring in a plastic inection moulding company. Te results sow tat te model provides support to daily staffing problem. As future researc, te introduction of training level considerations and employees medical prescription can be considered to expand te present model. Moreover, te development of training plans based on te routinization and te employees outloo can be added in order to acieve a more complete scedule. Acnowledgements Tis wor was supported by Unindustria Reggio Emilia. Tis support is gratefully acnowledged. We would also lie to express our tans to te firms involved in te proect for te data and te elpful discussion about te problem. References Abernaty, W.., Baloff, N., Hersey,.C., and Wandel, S. ( 1973). 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