Two-Stage Flowshop Scheduling with Outsourcing Allowed. and Technology, Shanghai, Abstract

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1 , pp Two-Stage Flowshop Schedulig with Outsourcig Allowed Li Li 1, Juzheg Lua 2 ad Yiche Qiu 1 1 School of Busiess, East Chia Uiversity of Sciece ad Techology, Shaghai, School of Mechaical ad Power Egieerig, East Chia Uiversity of Sciece ad Techology, Shaghai, lili1220@ecust.edu.c Abstract I this paper we study the productio-schedulig problem for a two-stage flow shop with outsourcig optios. The maiteace of i-house machie ad job-related discout for outsourcig cost are cosidered i model, i which total cost ad makespa are regarded as bi-objective for the schedulig problem. Based o the aalysis of model, we itroduce a cocept-packagig to reduce the complexity of problem ad the develop heuristic algorithm to solve the problem. Computatioal experimets usig differet groups of data are coducted to test the algorithm. Keywords: Productio-Schedulig, Outsourcig, Maiteace, Cost discout, Heuristic Algorithm 1. Itroductio Nowadays, drive by the tred of ecoomic globalizatio, a large umber of moder compaies are seekig for trasformatios. I particularly, productio outsourcig is becomig commo pheomeo durig this trasformatio. The beefits icurred by outsourcig are quite sigificat. May maufacturers outsource some jobs to subcotractors so as to focus o their core busiess ad gai profits as much as possible. Outsourcig ca relive the burdes of their ow productio factory, spare more time for them to cocetrate o the core busiess-sellig or marketig for example-rather tha the traditioal productio activities. O the other sides, with the subcotractor ivolved i the productio activities, subsequet costs icurred by outsourcig are ievitable, which cotais both processig costs ad trasportatio costs. Moreover, as we all kow, the profit for maufacture is largely depeds o the efficiecy of its productio-schedulig pla. With the outsourcig resources, the optimal productio schedulig is eve more complicated, but also prospective ad valuable. It is the reaso why may researchers have delved ito this problem uder differet outsourcig models. Durig the outsourcig models, some maufacturer outsourced part of productio operatios of jobs to the subcotractor, but the other oe that part of jobs etirely are outsourced to subcotractor is more commo i practice, especially for raw products. Ulike the certai outsourcig cost assumed i previous models, the outsourcig cost is more likely to be aouced ad charged accordig to the umber of jobs ad related processig time. Furthermore, the objective of schedulig problem is usually regarded as the productio time (e.g., makespa). However, the schedulig problems with outsourcig allowed i this paper do ot oly cosider the time, but also balace the productio time ad productio cost, Correspodig Author ISSN: IJUNESST Copyright c 2016 SERSC

2 which is the key objective to make two decisios: whether adopt outsourcig ad which jobs are outsourced. I the rest of this paper, we will describe the problem i detail, followed by the literature review. After itroducig the basic mathematic model ad related assumptios, we will study the relatios betwee parameters ad fid out a objective fuctio to coordiate those ew elemets. The by developig a heuristic algorithm, we will coduct related computatioal experimets ad compare their results respectively, the to reach a coclusio i the practical level. 2. Literature Review Outsourcig uder machie schedulig models has become a hot issue i maufacturig fields. At the begiig, most such researches are focused o the sigle-stage problems. Che ad Li (2008) [2], ad Lee ad Sug (2008a,b) [5-6] have studied the schedulig problem with outsourcig optios uder the assumptio that the subcotractor has virtually ulimited capacity. Therefore they do ot eed to schedule the outsourced jobs. Qi (2008) [14] cosidered a sigle-machie schedulig problem that allows some jobs to be outsourced to a sigle outsourcig machie, with a trasportatio delay ad a trasportatio cost. Based o it, Qi(2009) [15] cosidered the problem that oly stage-oe operatio is the outsourced stage ad stage-two operatios is started i-house. He developed a optimal algorithm ad a heuristic algorithm to solve the problem. I additio to trasportatio time delay ad cost, Mokhtari(2013) [11] believe the outsourcig cost is also the importat factor affectig the schedulig. He develop a iteger programmig formulatio to solve the joit schedulig of both i-house ad outsourced jobs simultaeously with the objective of miimizatio for sum of the total weighted completio time ad total outsourcig cost. Sice Johso s semial paper of 1954, which aalyzed the makespa miimizatio i a two-machie flow shop, may researchers have studied flow shop problems. Hou ad Hoogevee (2003) [9] ad Choi et. al., (2007)[3] cosidered a three-machie flow shop ad showed that the problem is NP-hard i the ordiary sese. For surveys of flow shop schedulig research, the reader is referred to Hejazi ad Saghafia (2005)[8] ad Cheg et. al., (2000)[1]. It has bee show that miimizig the makespa i a flow shop is strogly NP-hard, eve for the threemachie case. Ruiz-Torres et. al., (2008) [13] preseted a bi-objective model to deal with the problem of fidig outsourcig strategies ad cosidered trade- offs betwee the followig measures. Similar bi-objective job shop schedulig problem was studied by Guo(2014) [10], i which the total tardiess ad the outsourcig cost are cosidered. The lexico graphic approach is used to hadle these objectives simultaeously ad a effective two-phase eighbourhood search (TPNS) is preseted to solve this complicated problem. I terms of objective i schedulig problem with outsourcig allowed, may previous researches cosidered the weighted sum of time ad cost (Qi (2011)) [16]. Lee ad Choi (2011) [7] cosidered a two-stage productio problem i which objective is to miimize sum of makespa C max (s ) associated with sequece ad total outsourcig cost, where weight of processig time for differet operatios are give. I the two-machie ordered flow shop problem i Chug (2013) [4], a schedule is costructed ad its performace is measured by the makespa for ihouse jobs. Jobs processed by subcotractors require payig a outsourcig cost. The objective is to miimize the sum of the makespa ad the total outsourcig cost. Sice this problem is NP-hard, a approximatio algorithm is preseted ad three special cases were aalyzed. Moreover, Morteza(2013) [12] addressed the productio ad delivery schedulig itegratio problem, i which the objective is to 246 Copyright c 2016 SERSC

3 miimize the sum of the total weighted umber of tardy jobs ad the delivery costs. Ad special cases of the two-machie flow shop problem are ivestigated ad used to set up a ew brach ad boud algorithm. 3. Problem Descriptio I this paper, the two-stage flowshop schedulig problem with both i-house machies ad outsourcig machies are studied (See Figure 1), i which secod operatio i-house is the bottleeck (loger processig time ad machie maiteace). A subcotractor, kilometers away from the i-house factory, serves as a outsourcig choice for the maufacturer. Part operatios of jobs are ot allowed to outsourced to subcotractor. Istead, jobs are outsourced with both operatios, that is, they are etirely outsource. Oce the outsourcig producig is completed, all the outsourced jobs are trasported toward the i-house factory i oe batch. I- Outsourced: Operatio 1 Operatio 1 Figure 1. Two-Stage Flowshop Schedulig Problem with Subcotractor With outsourcig icluded i the productio activities, the subcotractors supply their workig machies that are parallel to the i-house machies, which complicates the schedulig problem compared with productio situatio without outsourcig. Apart from trasportatio delay ad its costs which may previous researchers has already cosidered, several elemets from practice i real world are briefly outlied as follows: Differet processig time for jobs i-house ad outsourced Job-related discouts o outsourcig cost Bi-objective i terms of cost ad time I actual productio activities, the processig speed differs from job to job. Due to disparate machie qualities, machies of the subcotractor process jobs a little faster tha those of i-house machies. Uder the moder i-batch productio actualities, the more jobs maufacturers outsources to the subcotractors, the more discouts i processig costs that maufacturers ca ejoy. The subcotractors providig packagig service for example, you ca fid their service price o their website. The more aked products the maufacturers outsourced to it, the lower of its price of package service. Ulike the objective i most previous research, we believe that productio time ad costs caot be added together for their differet uits, eve though they are both the idex to schedulig problem. 4. Mathematic Modelig Operatio 2 Operatio 2 Custome rs Trasportatio Based o the problem defiitio, we suppose that jobs are to be processed by a two-stage flow shop, which has oly oe machie at each stage. At ay time, oe job ca oly be processed by oe machie, ad oe machie ca oly process oe job. All jobs are available at time zero, ad there is o job preemptio. Notatios are as follows: Copyright c 2016 SERSC 247

4 Parameters: C max-j: the maximum completio time of i-house ad outsourced jobs, j=1 suggests operatios are processed i-house ad j=2 for outsourced jobs; C max-3: the maximum completio time if all jobs are processed i-house without outsourcig; : the time of trasportatio for the fiished outsourced jobs; M: the deadlie for jobs, defied as M = max{c max-1,c max-2 + l} ; : the processig time for the secod operatio of job i o machie j; l q ij "i, j = 1 suggests that operatios are processed outsourced; j=2 for the i-house; p ij : the processig time for the first operatio of job i o machie j. "i, j = 1 suggests that operatios are processed outsourced; j=2 for the i-house; : the job umbers for maiteace t period of i-house stage-two machie; : the trasportatio costs; a m e j : the uit time cost; j=1,2 for the i-house ad outsourced uit time cost, e 1 < e 2. Decisios variables ( O j : the job sets. O 1 is the outsourced job-set, O 2 is for the i-house; X i: the job allocatio. X i =1 otes that job i is outsourced, otherwise, is i-house processed; S i : the startig time for the first operatio of job i; T i : the startig time for the secod operatio of job i. Other parameters ( i ): i ): F i: the completio time for the first operatio of job i; C i : the completio time for the secod operatio of job i; j i : the job set processed before the job i; d i: the job set ext to the job i. Cosequetly, job i begis its first-stage ad secod-stage operatio satisfyig F i = S i + p i1 (1- X i )+ p i2 X i C i = T i + q i1 (1- X i ) X i The job set before job i is deoted as j i = { k (S k < S i )Ç(X i - X k = 0),"i,k }, ad j i = Æ suggests that job i is the first job to be processed i-house or outsourced, i { }. which case, F di = 0 ; otherwise, d i = l S l < S k,l Îj i,"k Îj i Due to the maiteace for the i-house machie of stage-two operatio, the startig time of the secod operatio is ì é T i = max F i,c d i + (1- x i ) 1- sg(mod( j i ê ë a )) ù íî ú û t ü ýþ With the outsourcig, the productio might be saved compared to the case with bottleeck machie i house ad o outsourcig resources allowed. The saved time ca be viewed as reduced cost, which is related to the avoidace of puishmet for delayed productio. Thus, without losig geerality ad accuracy, we cosider two idexes: h 1 = M C max-3, h 2 = O 1 O 1 + O Copyright c 2016 SERSC

5 Defie the cost-reductio percetage brought by saved time as fuctio g1, by customer s awards as fuctio g2, by subcotractor as g3. Fuctio g1 ad g2 are solely related to the variable h 1, while h 2 is the oly variable to fuctio g3. Cosequetly, the objective fuctio the whole costs is described as follows: (1) s.t. s i ³ F di (2) The time ad costs are coordiated i formula (1) with the three fuctios of g 1,g 2,ad g 3. I actual productio activities, maufacturers ca roughly calculate the costs of each order accordig to the cotracts ad determie whether a schedulig pla is take or ot. Formula (2) is aother occupatio limit for the operatios of stage-oe, that is, oe machie oly process oe job at the same time. Formula (3) suggests that oe job which cosists of two processig operatios must be processed i-house or outsourced etirely, operatio-divisio ot allowed. Formula (4) ad (5) limit the value for the startig time ito a reasoable rage. With the assumptio that processig speed i subcotractor s shop goes faster tha that i i-house factory ad uit time costs also lower, it is well aticipated that whe the trasportatio costs ca be egligible whe the umber of jobs is large, productio activities with outsourcig are must ecoomic that those of ooutsourcig. The, we will have a deep view o the quality of our objective fuctio. Defie the fuctio of the costs of o-outsourcig as f f ' = e 2 ( å p i2 + å q i2 ) (6) Machie of stage-oe/two i the subcotractor processes the first/secod operatios for the same job faster that those of i-house, that is, p i1 < p i2,q i1 < q i2. It { } Î[0,1], m e 2 ( å p i2 is obvious that 1-[1- g 1 )] [1- g 2 )] Df = f '- f = e 2 ( å p i2 ) ). Thus, O 2 O é 1 ù -[1- g 1 )] [1- g 2 )] êm + e 2 å (p i2 ) +[1- g 3 (h 2 )]ie 1 i å(p i1 + q i1 ) ú > 0 ë û Defie k 1 = O 1 å O 1 å (p i1 + q i1 ) (p i2 ),k 2 = e 1 e 2,l = O 1 å å (3) (4) (5) (p i2 ). m = [1- g 1 )][1- g 2 )], sigifies the (p i2 ) cost-discout icurred by reduced makespa. = k 1 k 2 l[1- g 3 (h 2 )]. I most situatios, (7) Copyright c 2016 SERSC 249

6 the processig speed for each job of i-house machies is almost a costat ratio to that of subcotractor s machies, so that the idex k 1 ad k 2 ca be viwed as costat. The idex l ad fuctio Cosequetly, the idex sigifies the effects of the quatity of outsourced jobs o the evetual costs. g 3 (h 2 ) has the same effect o the costs-value. The, Df = [1- m(1- l + )]e 2 å(p i2 )- mm» [1- m(1- l + )]e 2 å (p i2 ) (8) Geerally speakig, the idex l ad has the same effect o the costs-value, but from formula (8) it is hard to say the costs-value must be fewer if the maufacturer outsources more jobs to the subcotractor i that o the oe had, there is some balace betwee the idex l ad ad o the other had, more jobs that are outsourced to the subcotractor may also raise the whole makespa, resultig i the icreasig of the costs-value. It is well aticipated that the optimal result may lie i the situatio that the quatity of outsourced jobs is ear to half of the umber of the whole jobs ad the makespa of i-house factory is ear to that of the subcotractor s shop. 5. Algorithm Desig It is obvious that the problem i this paper is a NP hard problem, with those ew elemets attached o the basic F2 P: F2 model ad there is o certai precise algorithm for this improved problem i this paper, so approximatio algorithm is our choice for this problem. I order to get a productio pla with less programruig time ad fewer costs, we coceive a heuristic algorithm to achieve this goal Package Method ad Time Complexity Give the fact that the jobs i a order may be eormous i productio practice, there is o eed ad ecessity for the maufacturers to carry out his productio activities uder a optimal productio-schedulig schema. Otherwise, time complexity will soar i the way of expoetial growth as the umber of jobs icrease. I other words, computer quality ad program-ruig time are two mai limitatios for such searchig. I order to get a relatively optimal schedulig pla with less program-ruig time, we develop a heuristic algorithm ad itroduce a ew approach for this problem---package. Packagig is to iclude a certai umber of jobs ito a group as a pack. Istead of dealig with jobs oe by oe, package is the uit for sequecig ad outsourcig, i which way the time complexity ad program-ruig time is largely reduced as the umber of jobs i a pack icrease. However, as the program-ruig time lesses, the results uder this package-algorithm will be further from the optimal results. Cosequetly, we eed to maage to strike a balace betwee the productio costs ad the program-ruig time, which are two equally importat aspects for the makig of schedulig pla. Furthermore, we defie a idex D max here as the max differece that we cotrol i the program betwee the pack umbers of i-house ad outsourced. There are two reasos for this defiitio: The applicatio of D max ca further reduce the program-ruig time whe the umber of jobs is great ad pack-umbers are large. As formula (8), the whole makespa ad outsourced-job-scale are ifluetial o the costs so that cotrollig the pack-umber differece betwee the i-house ad outsourced packages ito a rage ca esure that the whole makespa is ot 250 Copyright c 2016 SERSC

7 very log ad the costs-discout from the subcotractor is ot very small. Cosequetly, the evetual costs are relatively satisfactory. Combied with Johso s rule which is to provide a optimal sequece for jobs cosistig of two idepedet operatios i oe process factory, we devise the program procedures show i Sectio Program Procedures Step 1: Sequece the jobs to be processed i accordace with the Johso s rule. Step 2: Package the sequeced jobs every percetage, makig these packs as a array A. If 100 x ÏN +, the left umber of jobs is the they are also treated as oe pack. ì é - x 1 ùü íî ê ë x ú ýþ û, which is less that x 100 Step 3: Determie the pack i array A oe by oe that whether it should be processed i-house or outsourced ad the wipe it off from array A. If array A is empty, calculate the evetual costs accordig to the objective fuctio Mi f. Whe the program is ruig,defie the dyamic jobs set which is processed i-house/outsourced as B 2/B 1, at the ed of the program, B 1=O 1, B 2=O 2. Step 4: Calculate the value of d = B 1 - B 2. If i-house factory or the subcotractor s shop, the go back to step 3. If to step5. d < D max, distribute that pack ito the d = D max Step 5: If d = D max, it sigifies that there are too may outsourced packs to be processed, the distribute that pack to be processed i-house; if d = -D max, the ihouse packs are too may, the distribute that pack to the subcotractor shop. Go to step3. Step 6: Output the jobs i O 1 ad O 2 i order respectively ad calculate the value of the objective fuctio, compare them ad take the miimum result ad its correspodig pla of pack-allocatio. 6. Computatioal Results ad Compariso We will firstly aim at the costs chage with differet D max ; therefore, the objective fuctio Mi f is a fuctio of two variables: Mi f 1 = f 1 (x,d max ). The we will compare the effects of disparate rules o the costs with same idex of D max, i which case, Mi f is also a two-variables fuctio: Mi f 2 = f 2 (x,rule). After determiig D max ad the sequecig rule accordig to the previous results, we move forward to study whether the idex a, the job umber betwee twice maiteaces, does effects o our coclusio. Uder this case, Mi f ca be sigifies as Mi f 3 = f 3 (x,a). ;,go 6.1. Variable D max Suppose that the sequecig rule is i accordace with the Johso s Rule ad the idex is valued 1, 2, 3 respectively. The simulatio-results are show i Table 1 ad Figure 2. Copyright c 2016 SERSC 251

8 Jobs umber i each package Table 1. Results ad Ruig Time Uder Differet D max x % max differece=3 max differece=2 max differece=1 fuctio value ruig time fuctio value ruig time fuctio value ruig time 15 3% % % % % % % % % % % % % % % % Figure 2. Fuctio Value Uder Differet D max From the Figure 2, ad Table 1, we ca have a clear reorgaizatio that both costs- value ad figure-fluctuatio whe D max =1 are iferior to the cases of D max =2,3. For the cases of D max =2 ad D max =3, there is ot large differece betwee the costsvalue but the program-ruig time of D max =2 is always less tha that of D max =3 whe the percetage x is the same. Ad it is withi our aticipatio that the costs - value will ot chage much as icreases other tha the ruig time soarig mayfold. Cosequetly, we choose D max =2 as a costat for the followig experimets. 252 Copyright c 2016 SERSC

9 6.2. Variable a I order to certify that job-umber betwee twice maiteace dose little effect o the costs-value, we coduct this experimet with =1000, ad a =50,75,100,125 respectively. From the followig figure ad table, we ca see that four figure lies almost overlap together, that is, uder the same cases, the fluctuatio of a does little effect o the evetual costs-value. 7. Coclusios Figure 3. Results of the Ifluece of Differet α I this paper, we cosider a specific model with several ew practical elemets ivolved i, besides the trasportatio costs ad delay, such as costs discout, differet processig speed, ad periodical maiteace. We itroduce a cocept of package to deal with the eormous jobs to be processed ad choose a idex D max to balace the whole makespa ad the scale of outsourcig jobs. Havig mathematically attested that the outsourcig model ca help the maufacturer save much moey, we coduct several experimets uder differet sequecig rules ad through comparig the costs- value, we fially fid out a relatively satisfactory productio pla for the maufacturer i practice. However, there are may other outsourcig models ad upredictable but practical factors that are ot cosidered i this paper, such as policy, road circumstaces at the time of trasmittig ad the extra time delay icurred by that, etc., Nevertheless, our practical level results would provide importat guidace for the strategic decisios o the whole. Ackowledgmets This research was supported by the Natioal Natural Sciece Foudatio of Chia (NSFC) for the projects of Productio Strategy ad Schedulig Problem for Hybrid Productio System with Outsourcig Allowed (No ). Refereces [1] T. C. E. Cheg, J. N. D. Gupta ad G. Wag, A review of flowshop-schedulig research with set-up times, Productio ad Operatios Maagemet, vol. 9, (2000), pp [2] Z. L. Che ad C. L. Li, Schedulig with subcotractig optios, IIE Trasactios, vol. 40, (2008), pp Copyright c 2016 SERSC 253

10 [3] B. C. Choi, S. H. Yoo ad S. J. Chug, Miimizig maximum completio time i a proportioate flow shop with oe machie of differet speed, Europea Joural of Operatioal Research, vol. 176, (2007), pp [4] D. Y. Chug ad B. C. Choi, Outsourcig ad schedulig for two-machie ordered flow shop schedulig problems, Europea Joural of Operatioal Research, vol. 226, (2013), pp [5] I. S. Lee ad C. S. Sug, Miimizig due date related measures for a sigle machie schedulig problem with outsourcig allowed, Europea Joural of Operatioal Research, vol. 186, (2008), pp [6] I. S. Lee ad C. S. Sug, Sigle machie schedulig with outsourcig allowed, Iteratioal Joural of Productio Ecoomics, vol. 111, (2008), pp [7] K. Lee ad B. C. Choi, Two-stage productio schedulig with a outsourcig optio, Europea Joural of Operatioal Research, vol. 213, (2011), pp [8] S. R. Hejazi ad S. Saghafia, Flowshop-schedulig problems with makespa criterio: A review, Iteratioal Joural of Productio Research, vol. 43, (2005), pp [9] S. Hou ad H. Hoogevee, The three-machie proportioate flow shop with uequal machie speeds, Operatios Research Letters, vol. 31, (2003), pp [10] X. P. Guo ad D. M. Lei, Bi-objective job shop schedulig with outsourcig optios, Iteratioal Joural of Productio Research, vol. 52, vol. 13, (2014), pp [11] H. Mokhtari ad I. N. K. Abadi, Schedulig with a outsourcig optio o both maufacturer ad subcotractors, Computers & Operatios Research, vol. 40, (2013), pp [12] R. B. Morteza, R. H. Seyed ad M. M. Mohammad, A brach ad boud algorithm to miimize the total weighed umber of tardy jobs ad delivery costs, Applied Mathematical Modelig, vol. 37, (2013), pp [13] A. J. R. Torres, F. J. Lopez, J. C. Ho ad P. J. Wojciechowski, Miimizig the average tardiess: the case of outsource machies, Iteratioal Joural of Productio Research, vol. 46, o. 13, (2008), pp [14] X. Qi, Coordiated logistics schedulig for i-house productio ad outsourcig, IEEE Trasactios o Automatio Sciece ad Egieerig, vol. 5, (2008), pp [15] X. Qi, Two-stage productio schedulig with a optio of outsourcig from a remote supplier, Joural of Systems Sciece ad Systems Egieerig, vol. 18, (2009), pp [16] X. Qi, Outsourcig ad productio schedulig for a two-stage flow shop, Iteratioal Joural of Productio Ecoomics, vol. 129, (2011), pp Author Li Li is a associate professor i School of Busiess, East Chia Uiversity of Sciece ad Techology. She received her PhD i Maagemet Sciece ad Idustrial Egieerig from TogJi Uiversity i Her curret research iterests iclude Productio Schedulig Problem ad supply chai maagemet. 254 Copyright c 2016 SERSC