Presentation of Multi-Skill Workforce Scheduling Model and Solving the Model Using Meta-Heuristic Algorithms

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1 Modern Appled Scence; Vol. 10, No. 2; 2016 ISSN E-ISSN Publshed by Canadan Center of Scence and Educaton Presentaton of Mult-Skll Workforce Scheduln Model and Solvn the Model Usn Meta-Heurstc Alorthms Iman Fozveh 1, Hooman Saleh 1 & Kamran Moharehabed 2 1 Department of Industral Enneern, Faculty of Enneern, Najafabad Branch, Islamc Azad Unversty, Najafabad, Iran 2 Department of Industral and Informaton Enneern, Faculty of Enneern Manaement, Poltecnco d Mlano, Mlan, Italy Correspondence: Iman Fozveh, Department of Industral Enneern, Faculty of Enneern, Najafabad Islamc Azad Unversty, Najafabad, Esfahan, Iran. Tel: E-mal: Fozveh.au@mal.com Receved: November 09, 2015 Accepted: November 19, 2015 Onlne Publshed: January 12, 2016 do: /mas.v10n2p194 URL: Abstract In the present artcle, a mult-objectve mathematcal model for scheduln mult-sklled mult-objectve workforce has been proposed wth the ams of mnmzn the number of nht-shft enneers, mnmzn the total cost of workforce and maxmzn the number of enaed workforce. To solve the proposed model for scheduln workforce, bee colony optmzaton alorthm and DE alorthm have been employed, and n order to nvestate the effcency of these two alorthms, the results have been compared wth each other n terms of qualty, dsperson and unformty factors. In order to solve the model three sample problems (40, 70 and 280 workforce) were desned and then solved by the two mentoned alorthms. Bee alorthm s able to fnd hher-qualty answers. Also the results of the comparson of dsperson and unformty ndex ndcate that bee colony alorthm s able to fnd answers wth more dsperson and more homoeneous than DE alorthm. The comparson of soluton tme of both alorthms ndcate that bee colony alorthm s faster than DE alorthm and needs less tme to reach qualty, dspersed and homoenous answers. Keywords: scheduln, mult-skll workforce, buldn projects, mult-purpose optmzaton, meta-heurstc alorthm 1. Introducton In all oranzatons, servce centers, and producton centers, the ssue of assnment of sklled workforce has been extensvely rearded as lmted resources to actvtes. In some envronments, plannn of workforces s conducted n two staes (Ho and Leun, 2010): Frst phase: In ths stae, the number of hours requred for each skll and the number of courses are determned. Second stae: at ths stae, based on data from the prevous stae, workforces are assned to demands. In prevous studes, wth reard to workforce capacty to tasks, the ssue of workforce plannn has been dvded nto mult-sklled and snle-sklled roups. In the frst roup, each of workforces has a specfc skll, and n the second roup, people have dfferent sklls (Montoya, 2012). One of the major tasks of the oranzaton s plannn workforces. Workforce plannn s a process n whch oranzaton s ensured that t has needed workforces to acheve ts oranzatonal oal. Realzaton of ths oal for oranzatons that have multple oals, and conflctn oals sometmes, and most mportantly for oranzatons that have mprecse and ambuous data and nformaton for any reason, ths ssue emeres as serous problem needed (Shahnazar et al., 2012). Workforces plannn as a means to ensure the provson of workforces requred n order to the oranzaton's oals and ts mathematcal modeln s also mportant for planners. Therefore, varous mathematcal models have been proposed to formulate workforces plannn, but the queston has always been rased out s that what s the effcent mathematcal model n workforce plannn. Ths study provded a mathematcal model for sklled workforce scheduln. In ths case, employees were cateorzed based on ther expertse and skll where workers at hher skll levels have the ablty to work n lower skll levels. The purpose of ths scheduln s optmum assnment of employees to shfts n order to acheve the mnmum costs and maxmum use of resources n order to meet the objectves of employees and acheve the maxmum satsfacton of employees. The proposed 194

2 mathematcal model and eneralzed meta-heurstc alorthms are applcable n other manufacturn and servce envronments wth slht chane. To solve the proposed model to schedule workforce, bee colony optmzaton alorthm was used, and n order to nvestate the effcency of ths alorthm, ts results were compared wth results obtaned from DE alorthm. 2. Lterature Revew The scheduln of workforce s assnment of qualfed workforce to meet the demand depends on tme varous for products or servces, whle leslaton and work areements have been consdered preferences of employees as far as possble. The specfc needs of manufacturn and servce ndustres to workforce have led to provdn the varous scheduln models and very dfferent technques. Costa et al. (2006) studed the ssue of the scheduln of rest days n condtons that demand for workforce wll vary from day to day and the total number of workn days for each workforce s known beforehand. Then, they proposed polynomal tme alorthm to solve the problem. Lusa et al. (2008) proposed a scenaro-based optmzaton approach to deal wth uncertan demand and presented a lnear prorammn model that wll be able to et a practcal answer for each scenaro and optmzn the expected value of total fractonal capacty. Hemerl and Kolsch (2010) offered a mxed nteer lnear prorammn model to solve the scheduln mult-project problem by consdern the mult-sklled workforce wth statc and heteroeneous effcency. L and Womer (2008) presented a combned alorthm based on mxed nteer lnear modeln and plannn of restrctons to solve the scheduln problem wth mult-sklled personnel, consdern the capacty of workload for each worker. In ther model, a worker s able to offer multple sklls, whle he s able to provde only one skll at the same tme. Ho and Leun (2010) formulated the ssue of the scheduln of workforce n provdn food and other supples for arlnes. Food and other supplers are delvered to arplanes on the runway before ther flht. Arplanes are servced by a team wthn a specfed tme. Wth reard to the servces that must be provded and the schedule of workers for each shft, the problem ncludes formn teams, assnn teams and start tmes to provde servces so that a lare number of flhts to be servced. Addtonally, the workload balance should be consdered between the teams. They used meta-heurstc methods of smulated annealn were used to solve the problem. Frat and Hurkens (2011) studed the scheduln problem of complex tasks n a project wth a heteroeneous set of resources. In ther model, resources ncluded techncans wth multple skll levels, whch n addton to the scheduln of project actvtes on a workn day, a combned set of techncans should be assned to the project actvtes consdern the resource constrants. They provded a mxed nteer prorammn for ther consdered model. Ths model examnes the scheduln of project actvtes n every workn day (perod) and assnment of workforce sklls to actvtes. They also proved that project scheduln wth mult-sklled workforce s one of the NP-HARD problems. M Hallah and Alkhabbaz (2013) nvestated the ssue of workforce scheduln n health care unt of Kuwat. They presented a mxed nteer mathematcal model for ther studed problem n whch the mnmum and maxmum lmts of permtted workn hours per day for each nurse and the lmtaton of number of consecutve nht shfts (maxmum two consecutve nht shfts allowed) were consdered. Tontarsk (2015) examned the mathematcal modeln and scheduln problem analyss nurses usn smulaton n a paper. He has provded a methodoloy to optmum assnment of nurses to work shfts n a planned perod. Hs mathematcal model has 2 shfts per day, n whch nurses have ther own specal condtons to be assned for shfts. He used smulaton technque to mplement the model both n the form of statc and dynamc. Bonutt et al (2014) started to model and solve the scheduln problem of workforce by consdern the rest tme lunch tme of employees. They scheduled workforce n workn shfts and they used smulated annealn alorthm to solve the consdered problem. 3. Mathematcal Modeln As mentoned, ths artcle provdes mult-skll workforce scheduln model. In the studed problem, a workn day s dvded nto two 12-hour shfts of mornn and nht, and the scheduln horzon s 28 days (4 weeks) n whch every week has 7 workn days. Employees are dvded nto 3 professonal roups (electrcal, mechancal and operaton) and they are placed at three skll levels (enneers, techncans and workers). The workforce who has the hher level of skll can perform a lower-level task n same expertse, but a lower level workforce cannot perform a hher-level task. Employees are allowed to work only one shft per day. Any workforce who works n nht shft, he cannot be assned to work n mornn shft of follown day. No workforce can work n 4 consecutve nht shfts. Every workforce should rest at least one day per week. The maxmum and mnmum workn hours of employees should be predetermned and ncluded n the model. Every workforce should work n two nht shfts per week. The number of forces from each skll and expertse and each shft has mnmum and maxmum workforce. Ths model has three objectves. In ths secton, the 195

3 mathematcal model s presented. Accordnly, ndces, parameters and varables are descrbed frstly. Then, lmtatons and objectve functons are desned. 3.1 Indces of Model : ndex for employees ( = 1,..., I) d: ndex for days durn a week (d = 1,..., 6) h: ndex for days durn the scheduln horzon (h = 1,..., 4) p: ndex for shfts (1 = mornn, 2 = nht) e: ndex for expertse, n ths paper 3 expertse of mechancs, electroncs, and and effcency are used (e1, e2, e3) l: ndex for skll levels n each expertse, enneers and techncans and workers n ths study (l1, l2, l3). 3.2 Parameters of Model t_pd: lenth of shfts per day (hours) Tmn: mnmum hours requred for the work force durn schedule Tmax: maxmum allowable workn hours for each worker durn schedule Ndpel: Maxmum total employees requred of skll level of I from expertse e n shft of p of day d Mdpel: Mnmum total employees requred of skll level of I from expertse e n shft of p of day d Cpdel: the wae of workforce of n shft p rel: Ths parameter s equal to 1, f workforce of I wth expertse of e can be assned to work n lower skll level, otherwse s equal to Decson Varables of Model Ths model ncludes has one decson varable as follows: Xpdhel: Ths s a bnary varable equals to 1, f the workforce of I wth the expertse of e of skll I n day d s assned to shft p, otherwse t s equal to Lmtatons and Objectve Functons A) Objectve functons 1. Mnmzn the number of enneers n nht shft Mn X,,,,, (1) 2. Mnmzn total costs of workforce Mn X C (2) 3. Maxmzn the number of workforce employed Max X,,,,, + X,,,,, (3) B) Lmtatons _e X_pdhel 1,, d, h, (4) B) Lmtatons: 4. Each workforce n each day can be assned to one shft to work n one skll level of an expertse. X.t T,, e, l (5) 5. Maxmum and mnmum workn hours of each workforce can be assned to work wthn a scheduln perod. X M, e, l, p, d, h (6) 6. Total number of employees requred at each skll level of expertse n every shft of every day. X r,, e, l, p, d, h (7) 7. Each workforce n each expertse can be assned to work at hs real skll or level lower than hs real skll n day. X 1,, e, l, d, h (8) 8. Employees are allowed to work only n one shft of day. 196

4 X,,,,, + X,,,,, 1,, e, l, h (9) 9. Workforce assned to evenn shft should not be assned to mornn shft. X,,,,, + X,,,,, +X,,,,, +X,,,,, 3,, e, l, h (10) 10. No workforce can work n 4 consecutve nht shfts X 6,, e, l, h (11) 11. each workforce should rest at least one day per week X,,,,, 2,, e, l, h (12) 12. each workforce should work n 2 day shfts per week 13. each workforce should work at least n two nht shfts per week X,,,,, 2,, e, l, h (13) 4. Alorthm Structure of Soluton To solve the presented model to schedule the workforce, bee colony optmzaton alorthm and DE alorthm were used. 4.1 The Structure of Proposed DE Alorthm To solve the studed problem, DE was used that ts proposed structure s as follows: {Intalzaton: Intalze feasble unt populaton by mult start varable nehborhood search (VNS). Intalze pareto archve. For counter=1 to max teraton For each soluton n populaton Create a mutant vector for each soluton n current populaton by eq.(14) Apply crossover operator and select elements of soluton by eq.(15) Apply repar method to constructon a feasble soluton. End for Update pareto archve. Select N solutons as the best from current solutons. End for } A possble soluton to consdered problem s a two-dmensonal matrx that ts rows represent the number of employees and ts columns represent the multplcaton the number of days of scheduln perod and number of shfts per day. The vales boxes of matrx reflect the skll level of the workforce n day and the shft assned. For example, for the studed problem, f the value of the box (3.11) s equal to 2, t means that thrd workforce s assned to work nn 6 th day, n the mornn shft wth expertse of The Constructon Manner of Intal Solutons To construct hh-qualty ntal solutons, mult start varable nehborhood search was used n ths study. The developed varable nehborhood search method n ths study ncluded three nehborhood search structures (NSS). Each of these structures has local search operator that these operators are descrbed below. The method starts wth an ntal feasble soluton. The manner of constructon of feasble soluton s as follows: 1. For each day of, the frst shft for each expertse and each skll s selected randomly amon the employees that can work, as requred. 1.1 Scheduln of other days and shfts, except frst shft of day I, the frst shft of the day, s done as follows: For each expertse and skll, that employer s selected that has been assned to work less than others, so that demands to be met. In addton, when selectn an employer for a skll, nvestaton starts from those employees that ther skll 197

5 levels s closer to nvestated skll. 2. Per, one deree s added and we proceed to frst step. Developed structures to search nehborhood of soluton are as follows: The frst nehborhood search operator: The operator acts n the way that one employer s selected amon all employees randomly and t s ted the assnments of ths employer to be cancelled n some days (wth lmtatons), and the assnment to be ven for employer who s allowed. The second nehborhood search operator: The operator acts n the way that one employer s selected amon all employees and n all shfts f that employer s assned to skll level lower than hs own level, t s tred that he s assned to skll closer to hs real skll level. If possble, the assnment of ths employer s cancelled and t s ven to qualfed employer. The thrd nehborhood search operator: In ths operator, the ndex of one day and one employer are constructed randomly and assnment the consdered employer (f any) s cancelled n that day wth consdern the condtons, and t s ven the employer who has qualfed and has hher desrablty. Each of the operators uses specfc structure known as nehborhood search structure (NSS). Developed nehborhood search structure n ths study s as follows: NSS type k, {for =1 to nom do n_s=local search k th(current soluton) Current soluton=acceptance procedure End for return current soluton. } Ths structure s same for all three local search operators. Here, the admsson crteron has been developed usn non-equatons. Now, f both solutons are at the same level n terms of the qualty, that soluton s selected that has hhest Eucldean dstance from the best answer found so far. Three nehborhood search structures have been mxed each other n the form of nehborhood search of varable that ths mx has been represented n fure below. The pseudo-code of our VNS s as follows: {For each nput soluton K=1 Whle stoppn crteron s meet do New soluton=apply NSS type k If new soluton s better then K=1 Else K=k+1 If k=4 then K=1 End f End f End whle } As sad before, mult start varable nehborhood search (mult start VNS) was used construct ntal solutons. VNS procedure starts wth an ntal random soluton and to construct n number of solutons, t s mplemented n number of tmes. To mplement, t starts wth random soluton frstly, whle t uses outputs of prevous 198

6 mplementaton as ntal soluton n follown mplementatons. The condton to accept the soluton n each teraton s that the selected soluton must have the hhest qualty and has the hhest Eucldean dstance from the found soluton by alorthm. Then, t s added to soluton populaton, and t s used as nput for follown mplementaton of VNS The Mutaton Operator One of the components of DE alorthm s mutaton operator that ts performance has been descrbed n Formula 14. In ths paper, the performance of mutaton operator s exactly as Formula 14. M, j = C + A rand C C ) (14), j, j ( 1, j 2, j In formula above, factor A s a postve number reater than or equal to 1 rate that controls the rate of populaton development. Rand s a random number n the nterval of (0.1). The eneral procedure used n ths paper s as follows: 0. For each soluton n populaton such as s (each structure or matrx) 1. Two ndces of j and k that should not be equal to are constructed randomly and one soluton s obtaned from dfferenced of these elements, that s, (s j -s k ). Next, by usn follown formula for each structure of th soluton, one new vector s constructed based on Formula f s smaller than N or equal to t, o to step 1, otherwse, o to step End For each of the structures, Formula 14 was used and accordn to each of them, one mutaton vector as M s constructed Intersecton Operator The used ntersecton operator n ths paper s based on Formula 15. In fact, one mutaton vector wll be constructed for each of current solutons n ths alorthm. Then, the elements of ntersecton vector elements wll be determned amon the mutaton vector elements and ts soluton by comparn based on factor cr. In the follown stae, these elements are repared and feasble soluton s constructed. In other words, for each soluton n populaton, we have one ntersecton vector as T that has been constructed that becomes a feasble soluton for next eneraton. The follown formula shows the performance of ntersecton operator n ths alorthm. T, j M, j f ( r, j cr orj = = C, j f otherwse jr) (15) In the formula above, cr s the ntersecton factor that ntersecton operator determnes the elements of vectort. It should be noted that the value of r, j s also n nterval of (0..1) Repar Operator After applyn the ntersecton operator, the selected operator on the resultn vector s appled and feasble soluton s constructed. The process of repar s done as follows. In every shft of every day and for every skll that can be assned amon employees (n terms of feasblty), that employer s chosen ts number s smaller (n matrces produced n prevous stae, each of boxes has one number constructed durn the mutaton and ntersecton, and we have them to prortze, here). If n a shft f one day for one skll, no qualfed employer, no employer s assned to that skll and other assnments follow ths process. Fnally, ths soluton s evaluated as not feasble soluton and f t s determned that t s not feasble, all lmtatons of model are appled on t so that t can chane to feasble soluton f possble. The repar process s done for all vectors and new justfable solutons are constructed Updatn Pareto Archve As there s no snle soluton n whch all objectves are optmzed n solvn the mult-purpose problems due to contradcton amon the objectves, a set of optmum solutons are provded( close to optmum). Here that we 199

7 used a method based on Pareto archve to solve the problem, the qualty of solutons n the archve s very mportant. In ths paper, the archve wll be updated at each teraton of DE alorthm. All solutons n the Pareto archve and new constructed solutons are poured n a pool of soluton and they are classfed. Then, frst level solutons are selected as new Pareto archve solutons. In each teraton, alorthm needs to populaton of solutons. In ths paper, to select the next teraton populaton, the current solutons n the populaton of that teraton and new constructed solutons are poured n a pool of soluton by alorthm. After the classfcaton the crowdn dstance crteron for each soluton and based on level of that soluton, the n number of soluton that has the hhest qualty and dstrbuton s selected as the next teraton populaton of alorthm, usn the Deb et al (2002) Selecton of Populaton for the Next Iteraton In each teraton, alorthm needs to populaton of solutons. In ths paper, to select the next teraton populaton, the current solutons n the populaton of that teraton and new constructed solutons are poured n a pool of soluton by alorthm. After the classfcaton the crowdn dstance crteron for each soluton and based on level of that soluton, the n number of soluton that has the hhest qualty and dstrbuton s selected as the next teraton populaton of alorthm, usn the Deb e (2002). 4.2 The Proposed Structure of the Bee Colony Alorthm The pseudo-code of the proposed structure of the bee colony alorthm s as follows: Purposed Bee colony optmzaton alorthm {Intalzaton: Intalze the alorthm parameter. Generate N feasble soluton as ntal populaton. Whle crteron s meet Calculate the ftness for each soluton n current populaton. Select the best bees and ther locaton as p1 set. Select the other bees and ther locaton as p2 set. Apply nehborhood search operator on p1 set, Assn some bees to obtaned solutons and calculate ther ftness. Apply random nehborhood search operator on p2. Calculate ther ftness. Select the N best bees of each locaton. Update pareto archve. Select N soluton as populaton of next eneraton. End whle Return the best soluton.} In fact, n the bee colony alorthm, there s a populaton of bees that each of them s on a food source. Bees wll be dvded nto two cateores of P1 and P2 n whch P1 cateory moves to better food sources usn the nehborhood search. P2 cateory s also randomly lookn for food. Ths wll contnue untl optmum food sources (questons) are found. In ths alorthm, the way of dsplayn soluton,, the way of constructn ntal solutons, the Pareto archve updatn and choce of next-eneraton solutons have been developed as DE alorthm Calculaton of Ftness Functon To calculate the ftness functon, solutons are frstly cateorzed usn Deb et al (2002), and crowdn dstance crteron s calculated for each soluton accordn to the level of soluton. Then, CS crteron s calculated for each soluton that ndcates the ftness value of ftness. = (16) In the formula above, crowdn dstance represents the dstance of crowdn and rank represents the level of 200

8 soluton Local Search (P1 Cateory of Bees) To solve the problem, a new method was developed whch s based on local search. The nput of ths method s the populaton of of P1 cateory solutons. Ths method s based on nehborhood search. In other words, ths method receves a set of solutons as nput and tres to acheve the ood nehborhood solutons by mprovn each of them. To develop the mentoned method n ths paper, varable nehborhood search was used. Varable nehborhood search s obtaned by specal combnaton of several nehborhood search operator n VNS 3. The overall structure of the VNS alorthm and structure of nehborhood search operators n t are descrbed n sectons related to DE alorthm Randomzed Nehborhood Search (P2 Cateory of Bees) Random search n the colony alorthm of bee s for second cateory of bees that they are randomly search for food sources (soluton). To mplement a random search n ths paper, a number between 1 and 3 s randomly enerated and the th local search operator s appled on the soluton and f the better soluton s acheved, t s replaced by the current soluton. 5. Computatonal Results In ths paper, a mathematcal model was presented for mult-objectve scheduln problem. Due to structure overnn on the problem and ts complexty (NP-Hard), two dfferent meta-heurstc alorthms were eneralzed for solvn the model. In order to evaluate the performance of the proposed model and alorthms, 3 sample problems wll be solved usn eneralzed alorthms. In ths paper, computatonal results obtaned from the model wll be dscussed and nvestated usn the eneralzed alorthms encoded n software envronment of R2009a MATLAB on a personal computer wth features of 4 GB of RAM and dual-core 2.27 GHz CPU. 5.1 Characterstcs of Sample Problems Snce mathematcal model presented for model s for solvn the mult-skll workforce scheduln problem and due to chane n the nature of the defned problem, the use of sample problems n the area of workforce scheduln does not help to compare and evaluate the performance of alorthms. Hence, 3 sample problems based on realty wll be used to nvestate the performance of lmts n model and evaluaton of eneralzed alorthms effcency. Tables 1, 2 and 3 show typcal characterstcs of these sample problems. Table 4 shows the common characterstcs of these samples. Table 1. Sample problem 1 (40 workforces) mornn nht All day Total nventory enneer e techncan e Worker e enneer e techncan e Worker e enneer e techncan e Worker e Total of one shft Table 2. Sample problem 1 (70 workforces) mornn nht All day Total nventory enneer e techncan e Worker e enneer e techncan e Worker e enneer e techncan e

9 Worker e Total of one shft Table 3. Sample problem 1 (280 workforces) mornn nht All day Total nventory enneer e techncan e Worker e enneer e techncan e Worker e enneer e techncan e Worker e Total of one shft Table 4. Common characterstcs of sample problems Scheduln horzon 28 days Number of shfts Shft 2 Number of expertse (Shft 1: mornn, Shft 2: Nht) The number of skll levels n each expertse 3 expertse The lenth each shft 3 skll levels The maxmum allowable workn hours for each worker durn the perod (Skll 1: Enneer, skll 2: techncans, and Skll 3: worker) Mnmum workforce hours requred for each scheduled 12 hours perod 5.2 Comparatve Indces To evaluate the effectveness of the proposed alorthm, three ndces of qualty, dstrbuton and unformty were used. Qualty Index - the ndex s equal to the number of Pareto solutons (non-domnated). Unformty ndexths crteron tests the dstrbuton of Pareto solutons on the border of solutons. Ths ndex s defned as follows: s N 1 = 1 = d mean ( N 1) d d mean In the equaton above, d ndcates the Eucldean dstance between two found nehbor non-defeated solutons and d mean represents the mean of values of d. Dstrbuton ndex - ths ndex s used to determne the number of non-defeated solutons on the optmum border. Dsperson ndex s defned as follows: D = N = max( 1 x t y t ) In the equaton above, x y t t shows the Eucldean dstance between two nehbor solutons of xt and y t on the optmum border. 5.3 Results of Soluton As sad before, bee colony alorthm and DE alorthm were used to solve the presented model n ths paper. In 202

10 ths secton, we present the results of problem solvn by these two alorthms. Sample problems were solved by two alorthms. Results of the objectve functon values, solvn tme, and values of comparatve ndces are shown n the follown tables. Table 5. The results of solvn sample problem 1 (40 workforces) alorthm comparatve ndces Solvn tme qualty dstrbuton unformty bee DE Table 6. The results of solvn sample problem 2 (70 workforces) alorthm comparatve ndces Solvn tme qualty dstrbuton unformty bee DE Table 7. The results of solvn sample problem 3 (280 workforces) alorthm comparatve ndces Solvn tme qualty dstrbuton unformty bee DE The results showed that n all three developed problems, bee alorthm s able to fnd solutons wth hher qualty. Addtonally, the results also show that the dstrbuton and unformty ndces of are able to fnd solutons wth hh dstrbuton and unformty than compared wth DE alorthm. Comparn the solvn tme two alorthms ndcates that bee colony alorthm s faster than DE alorthm, and t requres less tme to acheve hh-qualty, dstrbuted, and unform solutons. Comparn the results of two alorthms accordn to comparatve ndcators show that the rsk of dsablty n local optmzaton n bee alorthm s less DE alorthm, because bee alorthm can search more areas (due to dstrbuton ndex) of soluton space n more unform form and t s able to hher qualty solutons. 6. Concluson In ths paper, a new mathematcal model s presented for mult-objectve workforce scheduln problem. In ths problem, employees are cateorzed based on ther expertse and skll, and workforces at hher skll levels have the ablty to work n lower skll levels. Employers are tryn to assn the mnmum workforce at hher skll levels by an approprate plannn and scheduln, because t wll lead to hher costs. Ths model has three objectves, ncludn mnmzn the number of enneers on the nht shft, mnmzn the total cost of the workforce, and maxmzn the number of workn force. To solve the model presented for scheduln of workforce, the bee colony optmzaton alorthm and DE alorthms were used, and ther results were compared wth each other n terms of ndces of qualty, dstrbuton, and unformty to evaluate the effcency of these two alorthms. To solve the model, three sample problems (40, 70 and 280 workforces) were developed, and they were solved by two mentoned alorthms. In eneral, results of solvn the sample problem are as follows: Bee Alorthm s able to fnd hher qualty solutons. Addtonally, the comparson of the results of dstrbuton and unformty ndces suests bee colony alorthm s able to fnd solutons wth hh dstrbuton and unformty than DE alorthm. Comparn the results of two alorthms accordn to comparatve ndcators shows that the rsk of dsablty n local optmzaton n bee alorthm s less DE alorthm, because bee alorthm can search more areas (due to dstrbuton ndex) of soluton space n more unform form and t s able to hher qualty solutons. Fnally, t can be sad that the objectve of ths scheduln s optmzed assnment of employees to shfts n order to meet the objectves of employers and to acheve the maxmum satsfacton of employees. The presented mathematcal models and eneralzed meta-heurstc alorthms can be appled n other manufacturn and servce envronments wth slht chane n them. The man recommendatons for future research are as follows: 203

11 Consdern the educaton model and psycholocal factors for employees plannn and scheduln Consdern the absence, resnaton, and job leave durn the scheduled perod. Consdern the hrn and frn of employees Makn the scheduln dynamc, so that the plannn and scheduln of next perod can be done wth reard to scheduln of some last shfts or some last days of prevous perod. References Bonutt, A., De Cesco, F., Muslu, N., & Schaerf, A. (2014). Modeln and Solvn a Real-Lfe Mult-Skll Shft Desn Problem. 10 th Internatonal Conference of the Practce and Theory of Automated Tmetabln PATAT, York, Unted Kndom Auust. Casas, C. E. M. (2012). New methods for the mult-sklls project scheduln problem. Doctoral dssertaton. Ecole des Mnes de Nantes. Costa, M. C., Jarray, F., & Pcouleau, C. (2006). An acyclc days-off scheduln problem. 4OR, 4(1), Deb, K., Pratap, A., Aarwal, S., & Meyarvan, T. (2002). A fast and eltst multobjectve enetc alorthm: NSGA-II. IEEE Transacton on Evolutonary Computaton, 6(2), Fırat, M., & Hurkens, C. A. J. (2011). An mproved MIP-based approach for a mult-skll workforce scheduln problem. Journal of Scheduln, 15(3), Hemerl, C., & Kolsch, R. (2009). Scheduln and staffn multple projects wth a mult-sklled workforce. OR Spectrum, 32(2), Ho, S. C., & Leun, J. M. Y. (2010). Solvn a manpower scheduln problem for arlne catern usn metaheurstcs. European Journal of Operatonal Research, 202(3), L, H., & Womer, K. (2008). Scheduln projects wth mult-sklled personnel by a hybrd MILP/CP benders decomposton alorthm. Journal of Scheduln, 12(3), Lusa, A., Coromnas, A., & Muñoz, N. (2008). A multstae scenaro optmsaton procedure to plan annualsed workn hours under demand uncertanty. Internatonal Journal of Producton Economcs, 113(2), M Hallah, R., & Alkhabbaz, A. (2013). Scheduln of nurses: A case study of a Kuwat health care unt. Operatons Research for Health Care, 2(1-2), Shahnazar-Shahrezae, P., Tavakkol-Mohaddam, R., Azarksh, M., & Sadehnejad-Barkousarae, A. (2012). A dfferental evoluton alorthm developed for a nurse scheduln problem. The South Afrcan Journal of Industral Enneern, 23(3), Tontarsk, C. (2015). Modeln and Analyss of OR Nurse Scheduln Usn Mathematcal Prorammn and smulaton. MA Thess. Rochester Insttute of Technoloy. Copyrhts Copyrht for ths artcle s retaned by the author(s), wth frst publcaton rhts ranted to the journal. Ths s an open-access artcle dstrbuted under the terms and condtons of the Creatve Commons Attrbuton lcense ( 204