VOL. 1, NO. 1, December International Journal of Economics, Finance and Management All rights reserved.

Transcription

1 Two Hybrd Algorthms for Solvng the Mult Objectve Batch Schedulng Problem on a Sngle Machne Based on Smulated Annealng and Clusterng Methods Hamdreza Haddad, 2 Payam Ghanbar,2 Iranan unversty of scence and technology, Narma, Tehran, Iran ABSTRACT Sngle machne batchng schedulng s one of the most mportant problems n the manufacturng area whch has appled many applcatons especally n feld of supply chan management. In the real world ndustry, the manufacturers requre a sutable plan so as to delver the fnshed tems to ther customers wth mnmum delvery and tardness cost. Ths paper studes the present problem wth objectve of mnmzng the total tardness and maxmzng the job values on a sngle machne when the deterorated jobs are delvered to each customer n varous sze batches and proposed a mathematcal model for ths target. Furthermore, all jobs are not ready for process at tme zero and each job s ready based on a predefned release date. In order to solve the proposed model, two hybrd algorthms, based on smulated annealng and clusterng methods are offered and the results are compared wth the global optmums that are generated by Lngo 0 software. Furthermore, based on the effectve factors of the problem, a number of senstvty analyses are also mplemented. Computatonal study demonstrates that usng clusterng methods leads to specfed mprovements n batchng process. Keywords: Batch schedulng; sngle machne; deteroraton; job values; clusterng; release date. INTRODUCTION The assumpton of Batchng s one of the most mportant assumptons that approach the problem of sngle machne schedulng n the real ndustres. By usng ths assumpton, the schedulng models could be developed as a supply chan wth several stages. Generally, two types of batchng problem have been studed n the lterature of schedulng. The frst model s related to the producton systems n whch jobs are grouped nto separate batches before beng processed on the machne and the generated batches are processed. Ths problem can be consdered as a parallel or seral batchng problem. In parallel batchng problem, the completon tme of each batch s equal to the maxmum processng tme of jobs that belong to that batch. On the other hand, n seral batchng problem, the completon tme of each batch s consdered as the completon tme of the last job assgned to t. Ng et al. (2002) studed the problem of seral batch schedulng on a sngle machne wth consderaton of precedence constrant and solved t by an polynomal tme algorthm. Ish et al. (200) presented a Lagrange relaxaton n order to solve the problem of sngle machne batch schedulng wth three objectves nvolvng maespan, maxmum flow tme and the satsfacton level. In the proposed model, the sze of each batch was consdered as a fuzzy number. Cheng et al. (200 b) studed the problem of sngle machne batch schedulng wth resource dependent setup wth objectve of mnmzng total weghed resource consumpton. The proposed model was solved n polynomal run tme. Cheng and Kovalyov (2000) accomplshed an extensve research about dfferent cost functons n sngle machne batch schedulng, ncludng maxmum lateness, number of late jobs, total tardness, total weghted completon tme and total weghted tardness. They also solved some of the proposed objectves by usng a dynamc programmng approach. Nong et al. (2008) studed the mpact of famly setups and release date n the problem of sngle machne parallel batchng wth objectve of mnmzng maespan and a polynomal approxmaton scheme was developed by some smplfcatons. Other researches that are related to the sngle machne batch schedulng are Ng et al. (2004 c), Coffman (990), Albers and Bruer (993) and Baptce (2000). The second model s related to the dstrbuton systems and develops the schedulng problems n a supply chan. In such problems that enttles batch delvery schedulng, jobs are processed ndependently on the machne and after completon are grouped nto some batches and generated batches are sent to the costumers. Each completed job may be held for fnshng the completon of next jobs or sent ndvdually. Most of these researches consder the delvery cost n addton to a schedulng functon for the problem. 38

2 Mahdav Mazdeh et al. (20 a) presented a smulated annealng for solvng the problem of mnmzng the number of weghted tardy jobs wth delvery cost so as to fnd the near optmal soluton. rejecton. The objectve functon was mnmzng the maespan and total cost of rejected jobs. A polynomal run tme method and an approxmaton algorthm were presented for solvng the problem. Tan et al. (2007) studed the problem of mnmzng the sum of total weghted flow tmes wth delvery cost and proposed an optmal algorthm for solvng t. Mahdav Mazdeh et al. (20 c) solved the problem of mnmzng weghted sum of flow tmes wth delvery cost analytcally usnga branch and bound approach. Another research related to Mahdav Mazdeh et al. (2007 b) presented a branch and bound to mnmze flow tme and delvery cost smultaneously. Hamdna et al. (202) studed a sngle batch delvery system to mnmze total tardness and earlness smultaneously. They proposed a mathematcal model for descrbng the problem and presented a genetc algorthm for solvng t. Deteroraton s one of the other assumptons whch approach the schedulng problems n the real world ndustres and means that f processng a job becomes delayed, the processng tme wll be ncreased by a specal functon. Mosheov (994) s nown as the poneer of presentng the deteroraton n sngle machne problems. He nvestgated several functons nvolvng mnmzng the maespan, total completon tme, total weghted completon tme, total weghted watng tme, and total tardness, number of tardy jobs, maxmum lateness and maxmum tardness by consderng a smple lnear deteroraton model. Wang et al. (2009) studed the sngle machne schedulng problem wth learnng effect and deteroratng jobs smultaneously so as to mnmze the total weghted completon tme and the maxmum lateness. The other researches n the feld of the applcaton of deteroraton n sngle machne problems could be mentoned as Wang and Wang (200), Husang et al. (200) and Cheng and J (200). In the classc models of sngle machne, t was assumed that all jobs are ready for process before the startng of schedulng. But n real ndustres especally n supply chans, the orders enter to the system perodcally. Hence, for any job a predefned release date can be consdered that determnes the tme whch that job s ready to be processed. Although the assumpton of release date has been studed n several sngle machne problems, there are few papers that consder t n batchng problem. Lu et al. (200) consdered the problem of sngle machne parallel batch schedulng wth release date and L and Yuan (2006) studed the sngle machne parallel batchng wth release date and three objectve functons ncludng mnmzng maespan, mnmzng machne occupaton tme and mnmzng stocng cost. The problem was solved by an tme dynamc programmng algorthm. Rarou et al. (202) extended a new Mehta heurstc based on genetc local search and recoverng beam search for solvng the problem of mnmzng the total completon tme. One of the closest researches to our study s related to KarmNasab et al. (202) n whch the problem of sngle machne batch schedulng wth deteroraton and precedence constrants was studed. The objectve functon was mnmzng the total tardness and maxmzng the job values n maespan and a smulated annealng was used for fndng near optmal solutons ap. The man dfference of our research here s usng the clusterng algorthms wth SA as two hybrds n order to mprove the effcency of objectve functon. The other dfference s related to the release date assumpton that s consdered n ths study. In ths paper, we consder the ssue of seral batch schedulng problem wth release date and deteroraton, where the objectve s to mnmze the total tardness and delvery costs and maxmze the job values smultaneously, whch, based on our nowledge s not consdered n the lterature. MahdavMazdeh et al. (20 a) mentoned that the problems that combne a schedulng functon wth delvery costs are rather complex and solvng them by smple optmzaton methods s not commercal. On the other hand, Lawler (997) showed that problem of sngle machne schedulng wth total tardness functon s strongly NPhard. Therefore, the present problem that consders the objectves of total tardness, delvery costs and job values smultaneously s NPHard too. Two hybrd algorthms, based on smulated annealng and clusterng algorthms, are offered for fndng near optmal solutons. In order to chec the verfcaton of SA, some data sets of problem are solved optmally, usng Lngo software, and the results are compared wth each other. Some senstvty analyses are also done based on mportant factors of the problem. The remanng parts of ths paper are as follows: n secton 2 the proposed model s presented and the varables and parameters are ntroduced. In secton 3 the soluton 39

3 approach s offered and n secton 4 the computatonal studes are presented. 2. PROBLEM FORMULATION Consder that there s a machne that processes n jobs wth no preempton and each job s avalable for processng based on a predefned release date. The completed jobs can be delvered to the customer mmedately after completon or be awated next jobs to be delvered as a batch. Accordng to the number of jobs, N batches are consdered and the jobs are assgned to the batches. Clearly, any batch that does not have any jobs wll be omtted and then the sequence of batches s determned so that the proposed objectve functon s mnmzed. As mentoned, the seral batchng assumpton s consdered n ths paper, therefore, the processng tme of each batch s calculated as sum of all the processng jobs located on t. The due date and release date of each batch s also the maxmum due date and release date of jobs whch are assgned to t respectvely. The decson varables and parameters of the problem are lsted below: N N b Number of jobs ready to be scheduled Number of batches contanng at least one job p d r The normal processng tme of job The due date of th job The release date of th job p bat () The processng tme of batch D bat () The due date of batch R bat () The release date of batch n Number of jobs assgned to batch t j The tardness of batch where scheduled on jth poston c j The completon tme of job where scheduled on jth poston The rate of deteroraton The decreasng rate of job value s y x j a The shppng cost of batch If job s assgned to batch Otherwse If batch s located on jth poston n the sequence Otherwse If there s at least one job assgned to batch Otherwse And the proposed model s presented as follows: mn St: z s.a t e j c x } j c 0 0 j.( p bat ( ). j ) max{ x j. R bat ( ), c j p ) y. p bat β.c j j=,2,,n b (2) ( K=,2,,N b (4) R ) max y. r bat ( (5) n K=,2,,N b (6) n y n K=,2,,N b (7) a n ( n ).( t j max 0, c j D bat ( ). x n j ) () (3) j=,2,,n b (8) 40

4 D ( ) mn y. d y 0, K=,2,,N b (9) j x j y bat x =,2,,N b (0) j x j=,2,,n b () j y =,2,,N b (2) 0 or 0 or Equaton () ntroduces the objectve functon where the frst term corresponds to the delvery cost, the second term corresponds to the sum of tardness and the thrd term s related to the sum of job values. Constrant (2) states that how the completon tme of each batch s calculated. Constrant (3) mentons that the machne s avalable at tme zero. Constrant (4) mentons that the release date of each batch s equal to the maxmum release date of jobs assgned to t. Constrant (5) declares that the processng tme of each batch s equal to the sum of processng tmes of jobs assgned to t. Constrant (6) determnes how many jobs are located on each batch. Constrant (7) states that whether batch s empty or not. Constrant (8) clarfes that the tardness of each batch s equal to the gap between the tme that the processng of that batch has been completed and ts due date. Clearly, the value of tardness must be postve. Equaton (9) shows how the due date of each batch s calculated. Constrant (0) states that each batch s processed only once at each sequence and constrant () determnes that each poston can be assgned to just one batch. Constrant (2) mentons that each job can be assgned to exactly one batch to be processed and constrant (3), (4) shows that x and y are bnary varables. 3. SOLUTION APPROACH As mentoned, the consdered problem s NPhard; hence, t s not reasonable to use ordnary optmzaton methods. In ths case, two hybrd algorthms, based on smulated annealng meta heurstc (SA) and some clusterng algorthms, n order to reach near Optmal solutons n sutable run tmes, are presented. The reason for usng SA s that t can be consdered as a Marcov chan (Van Laarhoven and Aarts 988), so ts senstvty to the ntal soluton s much less than the other metaheurstcs and has a great ablty to avod gettng n local optmum solutons. 3. Smulated Annealng Smulated annealng (SA), whch has been appled wdely to solve many combnatoral optmzaton problems, s a class of optmzaton Meta heurstcs and (3) (4) performs a stochastc neghborhood search through the soluton space. In ths paper, the algorthm starts from an ntal temperature termed as T 0, n whch two random strngs are generated as below Fg : The codng scheme of proposed SA The frst strng corresponds to assgnng the jobs to batches and the other s related to fndng a sutable schedulng of generated batches. Based on ths fact, consder that there are fve avalable jobs for processng on the machne. The frst strng shows that the frst and the second jobs are assgned to batch 2, the thrd job s located on batch and the fourth and ffth jobs are assgned to batches 3 and 4 respectvely. Snce number 5 s not mentoned n ths strng, therefore the ffth batch has remaned empty and wll be removed. Afterwards, n second strng, a random schedule of all the batches that contan at least one job s generated. Ths procedure contnues to the pont that the equlbrum condton for ths temperature occurs. In ths problem, the equlbrum condton s met when the gaps between proposed objectve functons n consecutve teratons n a certan temperature are as less as possble. Ths condton can be demonstrated by: Z Z Z (4) j Where Z + s the value of objectve functon n +th teraton of algorthm, j s the number of objectves that are consdered to calculate the total gap and s a very small number. 4

5 By reachng the equlbrum n the temperature, the temperature decreases and the procedure starts from the lower temperature and contnues untl reachng the next equlbrum. Neghborhood search s also mplemented by swappng two randomly selected postons n the second strng (batch strng) n order to meet other nodes of soluton space. 3.2 Clusterng Clusterng s the assgnment of a set of observatons to subsets (called clusters) so that the observatons n the same cluster are smlar n some sense and the smlarty of generated clusters s lttle. Clusterng s a method of unsupervsed learnng, and a common technque for statstcal data analyss whch has been appled n many felds, ncludng machne learnng, data mnng, pattern recognton, mage analyss, nformaton retreval, and bonformatcs. Clusterng contans several algorthms and methods. For more detals about clusterng see (Jan et al. 999 and Kttaneh 202) In ths paper the Herarchcal clusterng and K means clusterng are consdered n order to assgn the jobs to a batch based on some smlartes. a. Herarchcal Clusterng (HC) Herarchcal Clusterng, whch s presented by Johnson (967), contans agglomeratve and dvsve schemes. In agglomeratve clusterng, at frst, N clusters are consdered, n whch N s equal to the number of observatons. Each one of the N clusters s assgned to only one batch.. In each step, two tems whch have more smlartes are merged wth each other and a unt batch s generated. Ths procedure contnues untl the number of clusters s equal to the predefned number. On the other hand, dvsve scheme starts the assgnment procedure by puttng all the tems n one batch. In each step, an tem that has the least smlarty wth others s removed from the batch and assgned to another batch. Ths procedure contnues untl the number of clusters becomes equal to the desred number. The dentfcaton of smlartes between the tems can be done n dfferent ways, whch s what dstngushes sngle lnage from complete lnage and average lnage clusterng. In sngle lnage clusterng, also called the connectedness or mnmum method, the dstance between two clusters s consdered equal to the shortest dstance from any member of one cluster to any member of the other cluster. In complete lnage clusterng, also called the dameter or maxmum method, the dstance between two clusters s consdered equal to the greatest dstance from any member of one cluster to any member of the other cluster. In average lnage clusterng, the dstance between two clusters s consdered equal to the average dstance from any member of one cluster to any member of the other cluster. Generally, dvsve method s a more complex method n comparson to the agglomeratve scheme and requres more run tmes to assgn the tems nto separate clusters. b. Kmeans clusterng (KC) The basc concept of Kmeans clusterng (MacQueen, 967) s very smlar to the Herarchcal clusterng. However, the Kmeans method consders the assgnments as an optmzaton problem. In ths regard, frst a number s defned as the desred number of clusters. Then each batch s occuped wth a random tem. These tems are correspondences to the center of ther clusters and are termed as Kcentrods. The next step s related to assgnng the remanng tems n the clusters accordng to the maxmum smlarty between the remanng tems and the centerods. By addng a new tem n each cluster, the centerod values are recalculated and ths procedure contnues untl the centrod values no longer change. Although t could be proved that the procedure wll always termnate, the means algorthm does not necessarly fnd the most optmal confguraton, whch s correspondng to the global objectve functon whch s mnmum. The algorthm s also sgnfcantly senstve to the ntally randomly selected cluster centers. The means algorthm can be run multple tmes to reduce ths effect. The presented clusterng algorthms are effcent for assgnng the tems to batches, but have some dsadvantages. One mportant dsadvantage of the presented clusterng algorthms s the ncapablty of them to calculate the optmal number of batches. Consequently, the batch numbers must be gven to them as an nput data. In order to overcome to ths problem these methods are used by SA as hybrd methods whch are presented n detals n followng sectons. 3.3 Descrpton of Hybrd Algorthms In ths secton, the performances of proposed hybrds are descrbed. In both hybrds, the procedure begns wth the proposed SA to determne the desred value for number of batches. Afterwards, one of the clusterng algorthms s used n order to assgn the jobs to sutable batches. Then the generated batches are scheduled by the second strng presented n SA secton. In computatonal study secton, t s shown that usng the 42

6 clusterng methods cause mmense mprovement va the objectve functon. a. SAHC hybrd Frst, the SA algorthm s run and the numbers of batches are determned. Then the jobs are assgned to batches usng Herarchcal clusterng. The lnage method s consdered as the average between tems and the dstance measure s defned as squared Eucldean. As mentoned, the smlarty of tems s calculated based on. Start wth the SA and determne the number of batches. ( N_fnal) 2. Consder N vrtual batches. (N s equal to the number of jobs) 3. Assgn each job to a batch randomly so that each batch occupes only one batch. 4. whle N < > N_fnal DO 5. Merge two batches that has the closest smlartes 6. N=N 7. loop 8. Enter the generated batches n the SA by a random schedule. (state ) 9. T=T 0 0. Whle T<T F DO. Do whle the equlbrum condton has not occurred 2. Generate a neghborhood ( state j) 3. Z=f(j)f() 4. If z <0 then =j 5. Else f random (0,) < exp( z/t) 6. K=j 7. loop 8. T= T.(coolng rate) 9. Loop 20. End Fg 2: The pseudocode of SAHC algorthm the processng tmes and due dates of them. It means that the jobs that have close values of processng tme and due date are joned to generate a unt batch. Afterwards, the generated batches are entered nto the SA agan and ther schedule s specfed. The performance of proposed hybrd s presented below n more detal: b. SAKC Hybrd The begnnng of SAKC hybrd s smlar to SA HC hybrd; SA algorthm s used at frst to determne the number of batches. Then the jobs, usng Kmeans clusterng, are assgned to batches. The smlarty of tems s calculated based on ther processng tmes and due dates. It means that the jobs that have near values of processng tme and due date are joned to generate a unt batch. Afterwards, the generated batches are entered nto the SA agan and ther schedule s specfed. The performance of proposed hybrd s presented below n more detal:. Start wth the SA and determne the number of batches. (N_fnal) 2. Assgn N jobs to each batch randomly as the center of them. (center()) 3. whle the center new (j) center (j) < >0 do 4. assgn the remanng jobs to the generated batches based on the maxmum smlarty Recalculate the center of each batch (center new (j)) 5. loop 6. Enter the generated batches n the SA by a random schedule. (state ) 7. T=T 0 8. Whle T<T F DO 9. Do whle the equlbrum condton has not occurred 0. Generate a neghborhood ( state j). Z=f(j)f() 43

7 2. If z <0 then =j 3. Else f random (0,) < exp( z/t) 4. K=j 5. loop 6. T= T.(coolng rate) 7. Loop 8. End Fg 3: The pseudocode of SAKC algorthm 3.4 Calbratng the SA Parameters In order to calbrate the proposed SA, a Taguch approach s presented n whch attempts to dentfy controllable factors (control factors) whch mnmze the effect of the nose factors, have been made. Durng the expermentaton, the nose factors are manpulated to force varablty to occur and then to fnd the optmal control factor settngs that mae the process or product robust or resstant to varatons from the nose factors. In ths paper, the S/N rato whch s consdered to be nomnal, s the best of the nd and s calculated by the followng equaton: (5) The effectve factors and ther levels are also descrbed n Table. Table : The Taguch experment nputs Factor symbol levels Type Intal temperature A 3 Number of objectves that are consdered to calculate the total gap n each temperature B 3 A()=500 A(2)=500 A(3)=3000 B()=3 B(2)=4 B(3)=5 The assocated degree of freedom for these two factors s equal to 8. Accordng to Taguch standard table of orthogonal array, the L 9 whch fulfls all the mnmum necessary requrements should be selected. In order to analyze the Taguch outputs, three mportant measures are consdered as the S/N rato (as robust measure), the average responses for each combnaton of control factors and the varablty n the responses due to the nose (standard devaton). The results are depcted n below fgures. M a n E f f e c t s P l o t f o r S N r a t o s D a t a M e a n s M e a n o f S N r a t o s A S g n a l t o n o s e : N o m n a l s b e s t ( 0 * L o g 0 ( Y b a r * * 2 / s * * 2 ) ) B 2 3 Fg 4: the results for responses based on S/N rato 44

8 M a n E f f e c t s P l o t f o r M e a n s D a t a M e a n s A B M e a n o f M e a n s Fg 5: the results for responses based on means M a n E f f e c t s P l o t f o r S t D e v s D a t a M e a n s A B M e a n o f S t D e v s A measure of robustness s used to dentfy control factors that reduce varablty n a product or process by mnmzng the effects of uncontrollable factors. Fg. 4 ndcates the robustness of each combnaton of factors. Clearly, t s desred to select a par of factors that generate the maxmum robustness. Therefore, based on ths fgure, A(3) and B(2) are selected. Fg 6: the results for responses based on standard devatons proposed factors are A(3) and B(2) whch better satsfes the response values. 4. COMPUTATIONAL EXPERIMENT All the nstances for ths problem were coded by Vsual Basc 6 and were run on personal computer wth CORE I 7 processor and 4 GB of RAM. The requred data was generated randomly based on below scheme: Fg. 5 shows the average responses for each combnaton of control factors. Snce the objectve of the functon s mnmzaton, the mnmum value for ths measure s desred, so A (2) and B (3) are selected. Fnally, Fg. 6 shows the varablty n the responses due to the nose whch s desred to be mnmum, hence A(3) and B(2) are selected. Based on mentoned measures, the most effcent combnaton of the Processng tme of jobs from unform dstrbuton [00] Due date of jobs from unform dstrbuton [ α P] where α s consdered equal to 0.2 manually. The nstances are also solved usng Lngo 0 software to determne the effcency and capablty of proposed SA to reach the global optmum. To mae the senstvty analyss, some mportant factors, ncludng problem dmenson (number of 45

9 consdered batches) and rate of deteroraton, have been consdered. 4. Senstvty Analyss Based On Problem Dmenson In order to mplement ths analyss frst, some nstances wth small dmensons are consdered and the results are compared to equvalent Lngo results. But Lngo s ncapable of solvng the problem for greater than 5 jobs by vrtue of mmense complexty of the problem. Rate of deteroraton and rate of job values are also consdered as 0.2 and respectvely n ths secton. Table 2 depcts the comparson of the results of SA, hybrds and Lngo based on the problem dmenson. Table 2: Senstvty analyss based on small problem dmensons Numb er of tass Number of batches Smple SA tme %GAP SAKC SAHC Lngo tme %GAP tme %GAP tme The second column ndcates the number of batches and the thrd to ffth columns show the performance of sngle SA, ncludng (Value of The results gap between SA and global optmum s calculated by: (6) Objectve Functon), run tme n seconds and the percentage of gap between them and global optmum. For each nstance, the SA was run 5 tmes and the best obtaned soluton was consdered as the best. The other columns also present the results of proposed hybrds and global optmums by Lngo.Based on the results from Table 2, usng both clusterng algorthms leads to decrease the requred run tme for solvng the problem. On the other hand, the value of objectve functon s mproved drastcally. As mentoned n the ntroducton secton, the proposed model s NPhard and solvng t optmally s not commercal n reasonable run tme. In ths regards, the Lngo solver s ncapable to solve the problem wth ncreasng the dmenson. Therefore, the comparson of gaps between the results and optmum value objectve functon s not possble for larger scales. Accordng to the results of table 2, by usng the clusterng methods not only the value of objectve functon s mproved, but also the run tme s decreased. For small dmensons of problem, both hybrds perform very effcently and offer the solutons wth low error n a reasonable tme. By ncreasng the number of jobs, does not seem that the hybrd of SAKC performs faster, but n feld of objectve functon, generally, there s no domnance. The comparson of SA and proposed hybrds for solvng the problem n larger scales are presented n table below. 46

10 Number of tass 25 Table 3: senstvty analyss based on medum and large problem dmensons Number of batches Smple SA tme SAKC tme SAHC tme Based on the results from table 3, the effcency of sngle SA declnes by ncreasng the dmenson of problem, but the hybrds offer better solutons n lower run tme. For larger scales of the problem, t can also be mentoned that the SAKC solves the nstances faster than other algorthms, though t s not possble to suggest any comparson for the value of objectve functon for the hybrd algorthms. 4.2 Senstvty analyss based on the rate of deteroraton The effect of deteroraton s evaluated by dmenson of 0 jobs and value of for the rate of job values. Table 4 llustrates the results of SA for several rates of deteroraton. Rate of deteroraton Table 4: senstvty analyss based on rate of deteroraton for N=0 Number of batches Smple SA SAKC SAHC tme tme tme Accordng to table 4, the rate of deteroraton s hghly effectve on the performance of SA and hybrds and clearly alters ther fnal solutons. On the other hand, does not seem to be any relatonshp between more rates of deteroraton and the run tmes. 4.3 Senstvty Analyss Based On The Rate Of Job Values The senstvty analyss also s conducted for the varous values of reducton rate of job values and the results are shown n table 4. In ths regard, rate of deteroraton s consdered equal to 0.2 and the number of jobs s 0. Rate of job values Table5: senstvty analyss based reducton rate of job values Number of batches Smple SA SAKC SAHC tme tme tme

11 Based on depcted plots t can be concluded that the proposed hybrds perform much better than smple SA for all the values of reducton rate and generally, there are no relaton between ths rate and the value of objectve functon. 4.4 The Comparson Of The Two Proposed Hybrd In all of the prevous sectons, the results of both proposed hybrds are much better than sngle SA, especally n large scale problems. In ths secton, t s desred to compare the results of hybrd algorthms n a far way. In order to do so, ther performance s calculated n equal tme ntervals. Each algorthm s run 30 tmes and the best, average and the worst solutons are recorded. Table 6 summarzes ths nformaton. Table 6: the comparson of hybrds performances tme Alg. Best Avg. Worst SD SD/Avg. SAHC SAKC SAHC SAKC SAHC SAKC SAHC SAKC Boxplot of HC, KC Boxplot of HC, KC Data 6000 Data HC KC HC KC Fg 7: performances of hybrds n 2.5 seconds Fg 8: performances of hybrds n 5 seconds 48

12 Boxplot of HC, KC Boxplot of HC, KC Data 5300 Data HC KC 4950 HC KC Fg 9: performances of hybrds n 7.5 seconds Fg 0: performances of hybrds n 0 seconds Based on the results of table 6 and depcted plots, the performance of HC s better than KC n the feld of objectve functon value, except n the case of 2.5 second run tme; n ths tme nterval KC performs better. On the other hand, the standard devatons of both algorthms decrease as tme passes. 5. CONCLUSION AND FUTURE RESEARCH Ths paper studed the problem of sngle machne batch schedulng wth release date and deteroraton. The objectve functon was mnmzng the total tardness and delvery costs and maxmzng the job values smultaneously; n whch a mathematcal model was presented. In order to solve the proposed model, two hybrd algorthms were proposed and ther performances was compared to global optmum for small dmensons of problem whch were generated usng Lngo software. Afterwards, a number of senstvty analyses were mplemented based on effectve factors of the problem, ncludng problem dmenson, rate of deteroraton and rate of job values n maespan. For future researches, the soluton method can be consdered as some exact algorthms le branch and bound, branch and cut and Lagrangan relaxaton n order to reach the global optmum even for medum and larger scales of jobs. Furthermore, the problem could be consdered n more nearest envronments n real ndustry le job shop or parallel machne nstead of sngle machne. REFERENCES [] Albers, S. and Brucer, P. (993). The complexty of onemachne batchng problems. Dscrete Appled Mathematcs 46, [2] Baptste, P. (2000). Batchng dentcal jobs. Mathematcal Methods of Operatons Research 52, [3] Cheng, TCE. and Kovalyov, MY. (2000). Sngle machne batch schedulng wth sequental job processng. IIE Transactons 33 (5), , DOI: 0.023/A: [4] Cheng, TCE. Jana, A. and Kovalyov, M. (200). Sngle machne batch schedulng wth resource dependent setup and processng tmes. European Journal of Operatonal Research 35 (), [5] Coffman, E.G. Yannaas, M. Magazne, MJ. and Santos, C. (990). Batch szng and sequencng on a sngle machne. Annals of Operatons Research 26, [6] Hamdna, A. Khahabmamaghan, S. Mahdav Mazdeh, M and Jafar, M. (202).A genetc algorthm for mnmzng total tardness/earlness of weghted jobs n a batched delvery system. Computers and ndustral engneerng 62 (), [7] Husang, H. Wang, JB. and Wang, XR. (200). A generalzaton for snglemachne schedulng wth deteroratng jobs to mnmze earlness penaltes. Int J Adv Manuf Technol 47, [8] Ish, H. Xuesong, L. and Masuda, T. (200). Sngle machne batch schedulng problem wth fuzzy batch sze. 40th Internatonal Conference on Computers and Industral Engneerng. Awaj, 5 49

13 [9] Jan, A.K. Murty, M.N. and Flynn, P.J. (999) Data clusterng: A revew. ACM Computng Surveys 3, 3. [0] Johnson, S. (967). Herarchcal Clusterng Schemes. Psychometra, 2, mnmzng weghted flow tmes and delvery costs. Appled Mathematcal Modellng 35, [20] Mosheov, G. (994). Schedulng jobs under smple lnear deteroraton, Computers and Operatons Research 2, [] KarmNasab M, Haddad H.R, Ghanbar P. A smulated annealng for the Sngle machne batch schedulng wth deteroraton and precedence constrants. Asan Journal of ndustral engneerng 4(202) 6. [2] Kttaneh R, Abdullah S, Abuhamdah A. Iteratve smulated annealng form medcal clusterng problems. Trends n appled scences research 7 (2): 03, 7, 202. [3] Lawler EL. A pseudo polynomal algorthm for sequencng jobs to mnmze total tardness. Annals of Dscrete Mathematcs (997) [4] L W, Yuan J. Sngle machne parallel batch schedulng problem wth release dates and three herarchcal crtera to mnmze maespan, machne occupaton tme and stocng cost. Internatonal Journal of Producton Economcs 02 (2006) [5] Lu S, Feng H, L X. Mnmzng the maespan on a sngle parallel batchng machne. Theoretcal Computer Scence 4 (200) [6] MacQueen, J. (967). Some Methods for classfcaton and Analyss of Multvarate observatons, Proceedngs of 5th Bereley Symposum on Mathematcal Statstcs and Probablty. Bereley, Unversty of Calforna Press, [7] Mahdav Mazdeh, M. Hamdna, A. and Karamouzan, A. (20). A mathematcal model for weghted tardy jobs schedulng problem wth a batched delvery system. Internatonal Journal of Industral Engneerng Computatons 2. [8] Mahdav Mazdeh, M. Sarhad, M. and Hnd, KS. (2007). A branchandbound algorthm for snglemachne schedulng wth batch delvery mnmzng flow tmes and delvery costs. European Journal of Operatonal Research 83, [9] Mahdav Mazdeh, M. Shashaan. S. Ashour, A. and Hnd, KS. (20), Sngle machne batch schedulng [2] Ng, CT. Cheng, TCE. and Kovalyov, M. (2004) Sngle machne batch schedulng wth jontly compressble setup and processng tmes. European Journal of Operatonal Research 53 (), 229. [22] Ng, CT. Cheng, TCE. and Yuan, JJ. (2002). A note on the sngle machne seral batchng schedulng problem to mnmze maxmum lateness wth precedence constrants. Operatons Research Letters 30, [23] Nong, Q. Ng, CT. and Cheng, TCE. (2008). The bounded snglemachne parallelbatchng schedulng problem wth famly jobs and release dates to mnmze maespan. Operatons Research Letters 36, [24] Rarou M.A, Ladhar T, T ndt V. Couplng Genetc Local Search and Recoverng Beam Search algorthms for mnmzng the total completon tme n the sngle machne schedulng problem subject to release dates. Computers & Operatons Research 39 (202) [25] Tan, J. Fu, R. and Yuan, J. (2007). On lne schedulng wth delvery tme on a sngle batch machne. Theoretcal Computer Scence 374, [26] Van Laarhoven, P.J.M. and Aarts, E.H. (988). Smulated Annealng: Theory and Applcatons, Kluwer Academc Publshers, Dordrecht. [27] Wang, D. and Wang, JB. (200). Sngle machne schedulng wth smple lnear deteroraton to mnmze earlness penaltes, Int J Adv Manuf Technol 46, , DOI 0.007/s [28] Wang, JB. Huang, X. Wang, X. Yn, N. and Wang L. (2009). Learnng effect and deteroratng jobs n the sngle machne schedulng problems, Appled Mathematcal Modelng 33, [29] Cheng, TCE. and J M., (200). Batch schedulng of smple lnear deteroratng jobs on a sngle machne to mnmze maespan. European Journal of Operatonal Research 202,