Research Article Multitasking Scheduling Problems with Deterioration Effect

Transcription

1 Hdaw Mathematcal Problems Egeerg Volume 017, Artcle ID , 10 pages Research Artcle Multtaskg Schedulg Problems wth Deterorato Effect Zhaguo Zhu, 1, Jl L, 3 ad Chegb Chu 4 1 College of Ecoomcs ad Maagemet, Najg Agrcultural Uversty, Najg 10095, Cha Laboratore d Iformatque, Bologe Itégratve et Systèmes Complexes (IBISC), EA 456, Uverstéd EvryVald Essoe, 9100 Evry Cedex, Frace 3 Resource-Coservg & Evromet-Fredly Socety ad Ecologcal Cvlzato 011 Collaboratve Iovato Ceter of Hua Provce, School of Busess, Cetral South Uversty, Chagsha , Cha 4 Laboratore Gée Idustrel, Ecole Cetrale Pars, Grade Voe des Vges, 995 Châteay-Malabry Cedex, Frace Correspodece should be addressed to Jl L; ljl@csu.edu.c Receved 5 September 016; Revsed 1 February 017; Accepted 1 March 017; Publshed 1 Aprl 017 Academc Edtor: Imaculada T. Castro Copyrght 017 Zhaguo Zhu et al. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Multtaskg schedulg problems wth a deterorato effect curred by coexstg behavoral pheomea huma-related schedulg systems cludg deteroratg task processg tmes ad deteroratg rate-modfyg actvty (DRMA) of huma operators are addressed. Uder the assumpto of ths problem, the processg of a selected task suffers from the jot effect of avalable but ufshed watg tasks, the posto-depedet deterorato of task processg tme, ad the DRMA of huma operators. Tradtoally, these ssues have bee cosdered separately; here, we address ther tegrato. We propose optmal algorthms to solve the problems to mmze makespa ad the total absolute dffereces completo tme, respectvely. Based o the aalyss, some specal cases ad extesos are also dscussed. 1. Itroducto Durg the past decade, multtaskg, as a atural respose to a growg umber of competg actvtes the workplace, has become a symbol for productvty ad has attracted growg terest the felds of behavoral psychology, cogtve egeerg, ad operatos maagemet [1, ]. Uder multtaskg, huma operators frequetly perform multple tasks by swtchg from oe task to aother, whch demad ther tme ad atteto the workplace. For example, health care, 1% of hosptal employees workg tme s spet o more tha oe actvty [3] whle, formato cosultg, formato workers usually egage about 1 workg spheres per day ad the cotuous egagemet wth each workg sphere before swtchg lasts oly 10.5 mutes o average [4]. As poted out by Rose [5], multtaskg mghthavesgfcateffectsotheecoomy:tsestmated that extreme multtaskg formato overload costs the US ecoomy $650 bllo per year owg to the loss of productvty. Although may multtaskers state that multtaskg has made them more productve [6 8], may studes have dcated that multtaskg may lower productvty [9, 10]. Thus, recet multtaskg-related lterature, vestgato has bee made about the effects of multtaskg o the productvty from dfferet perspectves [, 11 13]. Yet, cosderg multtaskg schedulg systems remas relatvely uexplored except for the works of Hall et al. [14], Sumetal.[15],Sumetal.[16],adZhuetal.[17].Hallet al. [14] tated schedulg problems wth multtaskg by proposg schedulg models a admstrato schedulg system. The, Sum et al. [15], Sum et al. [16], ad Zhu et al. [17] exteded ther work based o the basc settg of a multtaskg schedulg model, where the processg of a selected task suffers from terrupto by other tasks that are avalable but ufshed. Therearealsotwoothermportathumabehavorsthat affect productvty a huma-based schedulg system: the fatgue effect (agg effect) of huma operators ad ratemodfyg actvty. The fatgue effect here meas that productvty deterorates over tme because of the tredess of huma operators ad curs the durato of task processg whch becomes loger tha expected. Thus, the task processg tme

2 Mathematcal Problems Egeerg s descrbed as a odecreasg fucto depedet o the fatgue of huma operators. The fucto form s smlar totheaggeffectthelterature.theyarebothcerta types of deterorato. For coveece troducg the problem, we use the term deterorato effect to deote both of them hereafter. For detals o such deteroratos, the readers ca refer to Gawejowcz [18], Mosheov [19], Zhao ad Tag [0], Jaak ad Rudek [1], S.-J. Yag ad D.-L. Yag[],Rudek[3],Yagetal.[4],A.RudekadR.Rudek [5],Rudek[6],adJetal.[7].Arate-modfygactvty (RMA) refers to huma operators regularly egagg rest breaks durg work shfts to recover some of the egatve effects of fatgue [8]. Lee ad Leo [9] frst vestgated schedulgwthrmabymodelgtasaspecaltypeof classcal mache mateace actvty durg whch o tasks are processed [30]. The work the was exteded two aspects.mostfocusedothemacheperspectve(e.g.,[31 36]). However, some work focused o the huma behavor perspectve(e.g.,[17,37,38]).however,bothsetsofwork utlze the same assumpto that the durato of the RMA s fxed. I fact, the later a RMA starts, the loger the durato of the RMA usually becomes [, 39, 40]. For example, the laterahumaoperatorhasabreak,thelogerthetmeheor she takes for recoverg to susta a acceptable producto rate. Here, we call ths a deteroratg rate-modfyg actvty (DRMA). As commo behavoral pheomea, multtaskg, the deterorato effect, ad DRMA play cocurretly mportat roles realstc huma-based schedulg systems by affectg the productvty; however, most pror research has cocetrated o them separately. Although Hall et al. [14] provded a practcal admstratve plag scearo llustratg schedulg wth multtaskg, the deterorato effect ad DRMA of huma operators were ot cosdered. Whe huma operators are multtaskg, the deterorato effect ad DRMA chage the processg tmes remarkably, whch may cur dfferet schedulg results. Lodree Jr. ad Geger [37] dscussed huma-lke characterstcs of fatgue (deterorato effect) ad recovery (RMA) huma task sequecg, but ot a multtaskg evromet. Zhu et al. [17] vestgated the multtaskg schedulg problem wth RMA, yet they gored the job deterorato effect ad the varablty (deterorato) of RMA from huma fatgue. Extedg the work of Hall et al. [14], Lodree Jr. ad Geger [37], ad Zhu et al. [17], we jotly cosder the above ssues huma-based schedulg systems to pursue more practcal results. We refer to the proposed problem as multtaskg schedulg problems wth deterorato effect. The rest of the paper s orgazed as follows. After presetg the problem formulato ad otato Secto, we provde the results for the cosdered problem Secto 3. The, we dscuss some extesos Secto 4. The paper s cocludedsecto5.. Problem Formulato ad Notato I ths secto, we formulate multtaskg schedulg problems wth DRMA ad a deterorato effect based o the multtaskg settg Hall et al. [14] ad the RMA ad fatgue effect Lodree Jr. ad Geger [37] ad S.-J. Yag add.-l.yag[].theschedulgevrometcabe preseted as follows: suppose that a set of tasks eeds to be processedbyahumaoperator.whlethehumaoperator s processg a selected task, other avalable but ufshed tasks uavodably terrupt the operator. Moreover, the productvty of the huma operator deterorates over tme due to the fatgue effect, whch creases the actual task processg tmes. The huma operator may regularly take rest breaks (DRMA) to recover to a certa level of productvty, yet the durato of the RMA s ot fxed ad t also deterorates over tme, whch meas that the later the DRMA s performed the loger ts durato s. For coveece descrbg the schedulg model, jobs ad maches are used to deote tasks ad huma operators, respectvely. Cosder a set J = J 1,J,...,J } of jobs avalable at tme 0 to be processed o a mache that ca process at most oe job at a tme. Each job has a ormal processg tme p j. As Hall et al. [14], whle job J j s beg scheduled as aprmaryjob,theavalableadufshedjobsarecalled watg jobs of prmary job J j,adweuses j Ntodeote the set of such watg jobs. Thus, for each prmary job J j, the actual job processg tme also cossts of terrupto tme curred by terruptg job J j from the watg jobs J k, k S j,adswtchg tme for hadlg the terruptg jobs durg whch o useful work s performed, addto to ormal processg tme. The prmary job J j s opreemptve except for the terruptos by the watg jobs J k, k S j ; thats,aprmaryjobsalwayscompletedbeforeaotherjob sscheduledasaprmaryoe.letp k deote the remag processg tme of job J k whe job J j starts to be scheduled as a prmary job. Let V k (l), 0 l 1, deote the remag processg tme of job J k after t has terrupted l prmary jobs. The multtaskg fucto s defed as f( S j ) k Sj g k (p k ) based o emprcal evdece the operatos maagemet lterature [41, 4]. Aga, lke Hall et al. [14], we use f( S j ) ad k Sj g k (p k ) to deote, respectvely, the swtchg tme depedet oly o the umber of watg jobs ad the terrupto tme, whch s the sum of the amout of tme for all the watg jobs J k S j terruptg job J j. Followg ther assumpto too, ths paper, g k (p k )=Dp k ad f( S j ) = β S j,where0 < D < 1, 0 < β.thus,the multtaskg fucto s β S j k Sj Dp k. There are two types of deterorato: the deterorato effect of RMA ad the deterorato effect of job processg tmes. Just lke the form of DRMA ad the agg effect S.-J. Yag ad D.-L. Yag [], both deterorato effects are posto-depedet, meag that the actual durato of DRMA ad job processg are both affected by ther actual scheduled postos. Therefore t s assumed that the durato of DRMA s a fucto of ts actual scheduled posto. We deote the posto of DRMA as f t s scheduled mmedately after the completo of the prmary job J [], where = 0,..., 1.TheactualduratoofDRMAs t = b t 0,wheret 0 s ts ormal durato ad b > 0 s the deterorato dex. Thus, by combg ths wth the deterorato of jobs, the actual processg tme of J j s

3 Mathematcal Problems Egeerg 3 p A [j] = θ [j] p [j] f [j](j) β S j k Sj θ [k] Dp [k] f [k](k), for j = 1,,...,,adp A [j] = p [j] f [j](j) β S j k Sj Dp [k] f [k](k), forj = 1,,...,,where0 < θ j 1 s the job-depedet modfyg rate ad f j (r) s a geeral job-depedet deterorato, a odecreasg fucto of job J j depedet o ts posto r a sequece (schedule). For example, oe specal case s f j (r) = r a j,wherea j > 0 s the job-depedet deterorato factor. For a gve job sequece ad posto of DRMA, the actual processg tmes for jobs J [j], j = 1,,...,, multtaskg schedulg wth a deterorato effect ad DRMA ca be expressed by ducto as p A [j] = (1 D)j 1 p [j] f [j] (j) β ( j) D(1 D) j 1 p [k] f [k] (k), p A [j] =θ [j] (1 D) j 1 p [j] f [j] (j) β ( j) D(1 D) j 1 θ [k] p [k] f [k] (k), for,...,, for j=1,...,. The completo tme of each job J j, C [j],for,,...,,s C [j] = C [j] = j ((1 D) j 1 p [j] f [j] (j) β ( j) D(1 D) j 1 p [k] f [k] (k)), ((1 D) j 1 p [j] f [j] (j) β ( j) D(1 D) j 1 j j=1 p [k] f [k] (k)) b t 0,,...,, (θ [j] (1 D) j 1 p [j] f [j] (j) β ( j) D(1 D) j 1 θ [k] p [k] f [k] (k)), j=,...,, where C [0] =0. The objectve s to fd the optmal schedule of jobs ad the optmal posto of DRMA, whch mmzes the (1) () followg objectve fuctos, respectvely: the makespa (C max = max,,..., C j })adthetotalabsolutedffereces completo tme (TADC = =1 j= C C j ). Followg the three-feld otato of Graham et al. [43], we deote the problems as 1 MT, DE, DRMA C max ad 1 MT, DE, DRMA TADC, where MT meas multtaskg, DE meas deterorato effect, ad DRMA meas deteroratg rate-modfyg actvty. 3. Results The ma problems cosdered ths secto are 1 MT, DE, DRMA C max ad 1 MT, DE, DRMA TADC, where jobs are subject to a posto-depedet deterorato effect ad deteroratg RMA whle the huma operator s carryg out multtaskg. Schedulg models ad optmal solutos are proposed to fd the optmal job sequece π ad posto of DRMA such that the makespa ad the total absolute dffereces the completo tme of the schedule are mmzed, respectvely. For each problem, we aalyze the ma problem frst ad the dscuss some mportat specal cases Makespa Mmzato. We ow aalyze the 1 MT, DE, DRMA C max problem. The objectve fucto ca be expressed as Z=C [] = b t 0 β( j)d(1 D) j 1 j=1 ((1 D) j 1 p [j] f [j] (j) p [k] f [k] (k)) (θ [j] (1 D) j 1 p [j] f [j] (j) β ( j) D(1 D) j 1 j=1 θ [k] p [k] f [k] (k)) = b t 0 p [j] f [j] (j) (1 (1 D) ) j=1 p [j] f [j] (j) (1 D) ( 1) θ [j] p [j] f [j] (j) β = b t 0 p [j] f [j] (j) j=1 (1 D) θ [j] )p [j] f [j] (j) β ((1 (1 D) ) ( 1), for 1 1. Tosolvethsproblem,weuseE jr to deote the cost curred by a prmary job scheduled posto r ad set x jr =1f (3)

4 4 Mathematcal Problems Egeerg job J j s scheduled as a prmary job posto r; otherwse, x jr =0,for,,...,ad r=1,,...,.thus, x jr = 1, f job J j s scheduled as a prmary job posto r, for,,...,, r=1,,...,, 0, otherwse, for,,...,, r=1,,...,. E jr = p j f j (r), r=1,...,, ((1 (1 D) )(1 D) θ j )p j f j (r), r=1,...,. (4) (5) Thus, the makespa mmzato ca be represeted as the followg lear programmg problem: m subject to Z= E jr x jr b t 0 β r=1 r=1 x jr =1, x jr =1,,,...,, r=1,,...,, ( 1) x jr =1or 0,,,...,, r=1,,...,. (LP 1 ) Note that the above objectve fucto cossts of three terms. The last term s costat, ad the secod oe s also fxed for each gve posto of DRMA. Twosetsofcostratsare used to guaratee that each job s scheduled as a prmary job exactly oce ad each posto s take by oly oe prmary job. Therefore, gve the posto of DRMA, addressg the above lear programmg problem (LP 1 ) s equvalet to solvg the followg classcal assgmet problem (AP 1 ): m subject to E jr x jr r=1 r=1 x jr =1, x jr =1,,,...,, r=1,,...,, x jr =1or 0,,,...,, r=1,,...,. (AP 1 ) The way to fd the optmal job sequece π ad the posto of DRMA to mmze the makespa for problem 1 MT, DE, DRMA C max s formally descrbed as the followg optmzato algorthm based o Hall et al. [14] ad Zhu et al. [17]. Theorem 1. For the 1 MT, DE, DRMA C max problem, fdg the optmal schedule of jobs ad the posto of DRMA ca be doe O( 4 ). Proof. As dscussed Algorthm 1, we frst preprocess the jobs to obta the remag processg tmes of all jobs after they have terrupted l prmary jobs, for l = 1,..., 1, whch requres O() tme.the,foreachgvepostoof DRMA, the1 MT, DE, DRMA C max problem s coverted to a classcal assgmet problem that ca be solved O( 3 ) (see [44]). Thus, the assgmet problem s executed 1 tmes. Cosequetly, the tme complexty of Algorthm 1 s O( 4 ). Further, we dscuss three specal cases of the ma problem 1 MT, DE, DRMA C max : the oe wthout DRMA, deoted as 1 MT, DE C max ; the oe whch the deterorato effect s job-depedet (f(r), a odecreasg fucto depedet o posto r; e.g., oe of ts specal cases s f(r) = r a,wherea>0s a job-depedet deterorato factor) ad DRMA s ot allowed, deoted as 1 MT, DE d C max ; ad the oe whch the deterorato of RMA s ot allowed, deoted as 1 MT, DE, RMA C max. If the DRMA s ot allowed, the ma problem s reduced to a multtaskg schedulg problem wth a deterorato effect to mmze the makespa, the 1 MT, DE C max problem, whch has the followg results. Property. For the 1 MT, DE C max problem, fdg the optmal schedule of jobs ca be doe O( 3 ). Proof. The makespa of all jobs the 1 MT, DE C max problem s gve by C max = D(1 D) j 1 β ((1 D) j 1 p [j] f [j] (j) β ( j) ( 1). p [k] f [k] (k)) = p [j] f [j] (j) It ca be mmzed O( 3 ) tme by creatg a assgmet problem smlar to the above ma problem. If the DRMA s ot allowed ad the deterorato effect s job-depedet, the ma problem s reduced to a multtaskg schedulg problem wth a job-depedet deterorato (6)

5 Mathematcal Problems Egeerg 5 Step 1.Forkfrom 1 to do Step 1.1. V k (0) fl p k. Step 1..Forlfrom 1 to 1do V k (l) fl V k (l 1) DV k (l 1). Step.Forfrom 0 to 1do Step.1. Compute every cost coeffcet E jr wth Equatos. (5) for j, r = 1,,...,. Step.. Obta the local optmal sequece (π ) ad the correspodg makespa related cost by solvg the assgmet problem (AP 1 ). Step 3. The optmal soluto s the sequece π ad the posto of the DRMA that leads to the lowest makespa-related cost. Step 4. Schedule the DRMA the posto of ad the jobs wth multtaskg as the sequece π. Algorthm 1 effect to mmze the makespa, the 1 MT, DE d C max problem, whch has the followg results. Property 3. For the 1 MT, DE d C max problem, fdg the optmal schedule of jobs ca be doe O( log ). Proof. The makespa of all jobs the 1 MT, DE d C max problem s gve by C max = D(1 D) j 1 ( 1). ((1 D) j 1 p [j] f(j)β( j) p [k] f (k)) = p [j] f (j) β Accordg to a well-kow result o two vectors proposed by Hardy et al. [45], the makespa ca be mmzed O( log ) tmebyassggthejobwththelargestp [j] to the posto wth the smallest value of f(j), thejobwththe secod largest p [j] to the posto wth the secod smallest value of f(j), adsoo.thustheoptmaljobsequececa be foud O( log ) tme. If the deterorato of RMA s ot allowed, the ma problem s reduced to a multtaskg schedulg problem wth a job-depedet deterorato effect ad RMA to mmze the makespa, the 1 MT, DE, RMA C max problem, whch has the followg results. We use Z 1, Z,adZ 3 to deote the objectve values for the 1 MT, DE, RMA C max problem whe the RMA s located precedg, wth, ad followg the job sequece, respectvely. These ca be expressed by ducto as follows: Z 1 = Z =t 0 p [j] f [j] (j) θ [j] β p [j] f [j] (j) ( 1) t 0, for =0, (7) Z 3 = (1 D) β j=1 (1 θ [j] )p [j] f [j] (j) ( 1), for 1 1, p [j] f [j] (j) β ( 1), for =. If we let Q = (1 θ [j])p [j] f [j] (j), the the followg lemma holds. Lemma 4. For the 1 MT, DE, RMA C max problem, f Q<t 0, the optmal job sequece ca be obtaed oly by creatg a assgmet problem wthout the RMA. Otherwse, the optmal job sequece ca be obtaed by schedulg the RMA frst ad the creatg a assgmet problem to sequece the jobs. Proof. We prove the case of Q<t 0 ;theproofforthecaseof Q t 0 s smlar. Z 3 Z 1 = p [j] f [j] (j) =Q t 0 <0. Z 3 Z = (1 D) (1 D) j=1 p [j] f [j] (j) θ [j] t 0 (1 θ [j] )p [j] f [j] (j) t 0 (1 θ [j] )p [j] f [j] (j) t 0 (1 θ [j] )p [j] f [j] (j) t 0 =Q t 0 <0. Therefore, for the 1 MT, DE d, RMA C max problem, the stuato of =(.e., whch the objectve fucto s Z 3 )s the optmal stuato, whch meas the optmal job sequece cabeobtaedbyschedulgorma.thetheoptmaljob sequece ca be obtaed by covertg the mmzato of Z 3 to a assgmet problem, whch takes O( 3 ) tme. (8) (9)

6 6 Mathematcal Problems Egeerg Property 5. For the 1 MT, DE, RMA C max problem, fdg the optmal schedule of jobs ad the posto of RMA ca be doe O( 3 ). Proof. Accordg to Lemma 4, a optmal sequece ca be obtaed by schedulg the RMA at tme zero ad the sequecg the jobs by solvg a assgmet problem after the RMA for the case of Q t or sequecg the jobs by solvg a assgmet problem wthout the RMA for the case of Q<t. Therefore Lemma 4 dcates that a optmal sequece for problem 1 MT, RMA C max ca be obtaed O( 3 ) tme. Property 5 holds. 3.. Total Absolute Dffereces Completo Tme Mmzato. We ow aalyze the 1 MT, DE, DRMA TADC problem. The objectve fucto s where TADC = ψ j p [j] f [j] (j) j=1 (ρ j θ [j] λ)p [j] f [j] (j) ( j1)β( j)( ) b t 0, ψ j =(j 1)( j1)(1 D) j 1 j 1 (k 1)( k1) D (1 D) k 1, ρ j =(j 1)( j1)(1 D) j 1 λ= j k= D (1 D) k ( k) k, D (1 D) k 1 (k 1)( k1). (10) (11) Smlarly, the rght sde of the above equato ca be mmzed by solvg the followg lear programmg problem: m subject to Ω jr y jr r=1 r=1 ( ) b t 0 y jr =1, y jr =1, ( j1)β( j),,...,, r=1,,...,, y jr =1or 0,,,...,, r=1,,...,, (1) where Ω jr = ψ r p j f j (r), r=1,...,, (13) (ρ r θ j λ)p j f j (r), r=1,...,. Property 6. For the 1 MT, DE, DRMA TADC problem, fdg the optmal schedule of jobs ad the posto of DRMA ca be doe O( 4 ). Proof. The proof s smlar to that of Theorem 1. Nowwedscussthreespecalcasesofthemaproblem 1 MT, DE, DRMA TADC: the oe wthout DRMA, deoted as 1 MT, DE TADC, the oe whch the deterorato effect s job-depedet (f(r), a odecreasg fucto depedet o posto r; e.g., oe specal case s f(r) = r a, where a>0s the job-depedet deterorato factor) ad DRMA s ot allowed, deoted as 1 MT, DE d TADC, ad the oe whch the deterorato of RMA s ot allowed, deoted as 1 MT, DE, RMA TADC. If the schedule of DRMA s ot allowed, the ma problem s reduced to a multtaskg schedulg problem wth a job-depedet deterorato effect ad RMA to mmze the total absolute dffereces completo tme, the 1 MT, DE TADC problem, whch has the followg results. Property 7. For the 1 MT, DE TADC problem, fdg the optmal schedule of jobs ca be doe O( 3 ). Proof. The objectve fucto of the 1 MT, DE TADC problem s gve by TADC = (j 1) ( j 1) p A [j] = 1)((1 D) j 1 p [j] f [j] (j) β ( j) D(1 D) j 1 = j 1 p [k] f [k] (k)) ((j 1)( j1)(1 D) j 1 (j 1) ( j (k 1)( k1) D (1 D) k 1 )p [j] f [j] (j) (j 1)( j1)β( j). (14) It ca be mmzed O( 3 ) tme by creatg a assgmet problem wth the coeffcet ((j 1)( j 1)(1 D) j 1 j 1 (k 1)( k 1)D(1 D)k 1 )p [j] f [j] (j). If the DRMA s ot allowed ad the deterorato effect s job-depedet, the ma problem s reduced to a multtaskg schedulg problem wth a job-depedet deterorato

7 Mathematcal Problems Egeerg 7 effect to mmze the total absolute dffereces completo tme, the 1 MT, DE d TADC problem, whch has the followg results. Property 8. For the 1 MT, DE d TADC problem, fdg theoptmalscheduleofjobscabedoeo( log ). Proof. The objectve fucto of the 1 MT, DE d TADC = =1 j= C C j problem s gve by TADC = (j 1) ( j 1) p A [j] = 1)((1 D) j 1 p [j] f(j)β( j) D(1 D) j 1 = j 1 We set p [k] f (k)) ((j 1)( j1)(1 D) j 1 (j 1) ( j (k 1)( k1) D (1 D) k 1 )p [j] f(j) (j 1)( j1)β( j). φ j =((j 1)( j1)(1 D) j 1 j 1 (k 1)( k1) D (1 D) k 1 ) f (j). (15) (16) Accordg to a well-kow result o two vectors proposed by Hardy et al. [45], the absolute dffereces completo tme ca be mmzed O( log ) tme by assgg the job wth the largest p [j] to the posto wth the smallest value of φ j, the job wth the secod largest p [j] to the posto wth the secod smallest value of φ j,adsoo.thustheoptmaljob sequececabefoudo( log ) tme. If the deterorato of RMA s ot allowed, the ma problem s reduced to a multtaskg schedulg problem wth a job-depedet deterorato effect ad RMA to mmze the total absolute dffereces completo tme, the1 MT, DE, RMA TADC problem, whch has the followg results. Property 9. For the 1 MT, DE, RMA TADC problem, fdg the optmal schedule of jobs ad the posto of RMA ca be doe O( 4 ). Proof. Ths problem ca be coverted to a lear programmg smlar to that for the 1 MT, DE, DRMA TADC problem wth the dfferece the last tem of the objectve fucto. I ths problem t s ( ) b t 0 whle t s ( )t 0 that problem. 4. Exteso I the above problems, oly oe DRMA s cosdered. Now, we dscuss a further exteso whch multple DRMAs are allowed. Followg J ad Cheg [33] ad Zhu et al. [46], we assume that there exst at most h depedet DRMAs. We deote the postos of the uth DRMA as u f t s scheduled mmedately after the completo of the prmary job J [u ],where u = 0,..., 1.Theactualduratoof DRMA s t A u = b u t u,wheret u s ts ormal durato ad b > 0 s the deterorato dex. Thus, for ay job J j scheduled as a prmary job the posto mmedately after the u th DRMA, ts actual processg tme ca be expressed as p A [j] =θ [u][j] p [j] f [j](j) β S j k Sj θ [u][k] Dp [k] f [k](k), where 0 < θ uj 1 s a job-depedet modfyg rate, u = 1,...,h. The correspodg problems ca be deoted as 1 MT, DE, MDRMA C max ad 1 MT, DE, MDRMA TADC, respectvely, where MDRMA meas multple DRMAs. We aalyze the 1 MT, DE, MDRMA C max problem frst. The objectve fucto ca be expressed as Z=C [] = h 1 e=1 e1 k= e 1 j= h 1 ( p [j] f [j] (j) 1 e=1 e1 k= e 1 p [j] f [j] (j) D (1 D) k 1 D (1 D) k 1 θ e[j] (1 D) h θ h[j] ) 3 j= 1 ( D (1 D) k 1 D (1 D) k 1 θ e[j] (1 D) θ [j] ) j= 1 ( (1 D) θ 1[j] )p [j] f [j] (j) β( j) b u t u. h u=1 D (1 D) k 1 p [j] f [j] (j) (17) Thus, the makespa mmzato problem ca be represeted as the followg lear programmg problem: m Z = Λ jr y jr b u t u r=1 h u=1 β( j)

8 8 Mathematcal Problems Egeerg subject to r=1 y jr =1,,,...,, y jr =1or 0,,,...,, r=1,,...,, (LP ) y jr =1, r=1,,...,, where p j f j (r) r=1,...,, ( D (1 D) k 1 (1 D) θ 1j )p j f j (r), r = 1,...,, Λ jr = (. D (1 D) k 1 1 e1 e=1 k= e 1 D (1 D) k 1 θ ej (1 D) θ j )p j f j (r), r = 1,..., 3, (18) ( D (1 D) k 1 h 1 e1 e=1 k= e 1 D (1 D) k 1 θ ej (1 D) h θ hj )p j f j (r), r = h 1,...,. Note that the above objectve fucto cossts of three terms. The last term s costat. The secod oe s also fxed ad addressg the above lear programmg problem (LP ) s equvalet to solvg the followg classcal assgmet problem whle the postos of MDRMA,,..., h are gve. Ths meas that, to obta the optmal multtaskg job sequece (π 1, ), the assgmet problem s executed at most h tmes. Therefore, the followg theorem,..., h holds. Theorem 10. For the 1 MT, DE, MDRMA C max problem, fdg the optmal schedule of jobs ad the postos of MDRMA ca be doe O( h3 ). Smlarly, we also have the followg property. Property 11. For the 1 MT, DE, MDRMA TADC problem, fdg the optmal schedule of jobs ad the posto of DRMAcabedoeO( h3 ). Proof. The 1 MT, DE, MDRMA TADC problem ca be coverted to solvg a lear programmg problem, the objectve fucto of whch ca be expressed as where Z= Γ jr y jr u ( u ) b u t u r=1 h u=1 β(j 1)( j1)( j), (19) Γ jr r 1 ( (k 1)( k1) D (1 D) k 1 )p j f j (r), r=1,...,, r ( k ( k) D (1 D) k θ 1j k= = r ( k ( k) D (1 D) k θ j k=. (k 1)( k1) D (1 D) k 1 )p j f j (r), r = 1,...,, 1 e1 e=1 k= e 1 (k 1)( k1) D (1 D) k 1 θ ej (k 1)( k1) D (1 D) k 1 )p j f j (r), r = 1,..., 3, (0) r ( h 1 k ( k) D (1 D) k θ hj k= h e=1 e1 k= e 1 (k 1)( k1) D (1 D) k 1 θ ej (k 1)( k1) D (1 D) k 1 )p j f j (r), r = h 1,...,.

9 Mathematcal Problems Egeerg 9 Thus, smlar to the proof of Theorem 10, Property 11 holds. 5. Coclusos I addto to multtaskg, there exst other mportat behavoral pheomea related to huma operators that also affect productvty huma-based schedulg systems. I ths study, we addressed the tegrato of these ssues by studyg multtaskg schedulg problems wth a deterorato effect. We showed that all the cosdered cases are polyomally solvable, ad we proved the tme complexty. Some of the results dffer from those obtaed wthout multtaskg, deterorato effects, or DRMA. The results are ot lmted to huma operator schedulg but may also be applcable to mache-based schedulg problems. Further studes may vestgate dfferet objectve fuctos, for example, the total weghted completo tme, uder the cotext of multtaskg. Coflcts of Iterest The authors declare that they have o coflcts of terest. Ackowledgmets Ths work s supported part by NSF of Cha uder Grats ad , the Qg La Project, ad a project fuded by the Prorty Academc Program Developmet of Jagsu Hgher Educato Isttutos (PAPD). Refereces [1] V. Jez, Searchg for the meag of multtaskg, Norsk Koferase for Orgasasjoers Bruk av Iformasjostekolog, pp ,UverstetetTromsø,Tromsø,Norway,November 011. [] K.C.DwasSgh, Doesmulttaskgmproveperformace? evdece from the emergecy departmet, Maufacturg ad Servce Operatos Maagemet,vol.16,o.,pp ,014. [3] K.J.O Leary,D.M.Lebovtz,adD.W.Baker, Howhosptalsts sped ther tme: sghts o effcecy ad safety, Joural of Hosptal Medce,vol.1,o.,pp.88 93,006. [4]V.M.Goza lezadg.mark, Maaggcurretsofwork: mult-taskg amog multple collaboratos, Proceedgs of the 9th Europea Coferece o Computer-Supported Cooperatve Work (ECSCW 05), H. Gellerse, K. Schmdt, M. Beaudou-Lafo, ad W. Mackay, Eds., pp , Sprger, Pars, Frace, 005. [5] C. Rose, The myth of multtaskg, The New Atlats, vol. 0, pp , 008. [6] R. Peebaker, The medocre multtasker, New York Tmes, August 009, [7]E.Ophr,C.Nass,adA.D.Wager, Cogtvecotrol meda multtaskers, Proceedgs of the Natoal Academy of Sceces of the Uted States of Amerca, vol.106,o.37,pp , 009. [8] V. Vega, K. McCracke, ad C. Nass, Multtaskg effects o vsual workg memory, workg memory ad executve cotrol, Proceedgs of the Aual Meetg of the Iteratoal Commucato Assocato,Motreal,Caada,May008. [9] E. M. Hallowell, Overloaded crcuts: why smart people uderperform, Harvard Busess Revew, vol. 83, o. 1, pp. 54 6, 005. [10] R. E. Mayer ad R. Moreo, Ne ways to reduce cogtve load multmeda learg, Educatoal Psychologst,vol.38,o.1, pp.43 5,003. [11] D. Covello, A. Icho, ad N. Persco, Tme allocato ad task jugglg, Amerca Ecoomc Revew, vol. 104, o., pp , 014. [1] D. M. Sabomatsu, D. L. Strayer, N. Mederos-Ward, ad J. M. Watso, Who mult-tasks ad why? Mult-taskg ablty, perceved mult-taskg ablty, mpulsvty, ad sesato seekg, PLoS ONE,vol.8,o.1,ArtcleIDe5440,013. [13] Realzato, The effects of multtaskg o orgazatos, 014, Orgazatos.pdf. [14] N. G. Hall, J. Y.-T. Leug, ad C.-L. L, The effects of multtaskg o operatos schedulg, Producto ad Operatos Maagemet,vol.4,o.8,pp ,015. [15] J. Sum, C. S. Leug, ad K. I.-J. Ho, Itractablty of operato schedulg the presece of multtaskg, Operatos Research, I submsso. [16] J. Sum ad K. Ho, Aalyss o the effect of multtaskg, Proceedgs of the IEEE Iteratoal Coferece o Systems, Ma, ad Cyberetcs (SMC 15), pp , IEEE, Hog Kog, October 015. [17] Z. Zhu, F. Zheg, ad C. Chu, Multtaskg schedulg problems wth a rate-modfyg actvty, Iteratoal Joural of Producto Research, vol. 55, o. 1, pp , 017. [18] S. a. Gawejowcz, A ote o schedulg o a sgle processor wth speed depedet o a umber of executed jobs, Iformato Processg Letters,vol.57,o.6,pp ,1996. [19] G. Mosheov, Parallel mache schedulg wth a learg effect, Joural of the Operatoal Research Socety, vol.5,o. 10, pp , 001. [0] C.-L. Zhao ad H.-Y. Tag, Sgle-mache schedulg problems wth a agg effect, Joural of Appled Mathematcs ad Computg,vol.5,o.1-,pp ,007. [1] A. Jaak ad R. Rudek, Schedulg jobs uder a agg effect, Joural of the Operatoal Research Socety, vol.61,o.6,pp , 010. [] S.-J. Yag ad D.-L. Yag, Mmzg the makespa o sglemache schedulg wth agg effect ad varable mateace actvtes, Omega,vol.38,o.6,pp ,010. [3] R. Rudek, Schedulg problems wth posto depedet job processg tmes: computatoal complexty results, Aals of Operatos Research,vol.196,pp ,01. [4] D.L.Yag,T.C.E.Cheg,S.J.Yag,adC.J.Hsu, Urelated parallel-mache schedulg wth agg effects ad multmateace actvtes, Computers & Operatos Research,vol. 39,o.7,pp ,01. [5] A. Rudek ad R. Rudek, O flowshop schedulg problems wth the agg effect ad resource allocato, The Iteratoal JouralofAdvacedMaufacturgTechology,vol.6,o.1 4, pp ,01. [6] R. Rudek, The computatoal complexty aalyss of the twoprocessor flowshop problems wth posto depedet job

10 10 Mathematcal Problems Egeerg processg tmes, Appled Mathematcs ad Computato, vol. 1, pp , 013. [7]M.J,D.Yao,Q.Yag,adT.C.E.Cheg, Sgle-mache commo flow allowace schedulg wth agg effect, resource allocato, ad a rate-modfyg actvty, Iteratoal Trasactos Operatoal Research,vol.,o.6,pp ,015. [8] S. Elo, O a mechastc approach to fatgue ad rest perods, Iteratoal Joural of Producto Research,vol.3,o. 4, pp , [9] C.-Y. Lee ad V. J. Leo, Mache schedulg wth a ratemodfyg actvty, Europea Joural of Operatoal Research, vol. 18, o. 1, pp , 001. [30]Y.Ma,C.Chu,adC.Zuo, Asurveyofschedulgwth determstc mache avalablty costrats, Computers ad Idustral Egeerg,vol.58,o.,pp ,010. [31] G. Mosheov ad D. Oro, Due-date assgmet ad mateace actvty schedulg problem, Mathematcal ad Computer Modellg,vol.44,o.11-1,pp ,006. [3]V.S.GordoadA.A.Tarasevch, Aote:commodue date assgmet for a sgle mache schedulg wth the ratemodfyg actvty, Computers ad Operatos Research,vol.36, o., pp , 009. [33] M.JadT.C.E.Cheg, Schedulgwthjob-depedetlearg effects ad multple rate-modfyg actvtes, Iformato Processg Letters,vol.110,o.11,pp ,010. [34] Y. Ozturkoglu ad R. L. Bulf, A uque teger mathematcal model for schedulg deteroratg jobs wth ratemodfyg actvtes o a sgle mache, Iteratoal Joural of Advaced Maufacturg Techology,vol.57,o.5 8,pp , 011. [35] K. Rustog ad V. A. Strusevch, Combg tme ad posto depedet effects o a sgle mache subject to rate-modfyg actvtes, Omega,vol.4,o.1,pp ,014. [36] Y. Y, T. C. E. Cheg, C.-C. Wu, ad S.-R. Cheg, Sglemache batch delvery schedulg ad commo due-date assgmet wth a rate-modfyg actvty, Iteratoal Joural of Producto Research,vol.5,o.19,pp ,014. [37] E. J. Lodree Jr. ad C. D. Geger, A ote o the optmal sequece posto for a rate-modfyg actvty uder smple lear deterorato, Europea Joural of Operatoal Research, vol. 01, o., pp , 010. [38] Y. Y. Öztürkoğlu ad R. L. Bulf, Schedulg jobs to cosder physologcal factors, Huma Factors ad Ergoomcs Maufacturg & Servce Idustres,vol.,o.,pp ,01. [39] T. C. E. Cheg, S.-J. Yag, ad D.-L. Yag, Commo duewdow assgmet ad schedulg of lear tme-depedet deteroratg jobs ad a deteroratg mateace actvty, Iteratoal Joural of Producto Ecoomcs,vol.135,o.1,pp , 01. [40] B. Mor ad G. Mosheov, Schedulg a deteroratg mateace actvty ad due-wdow assgmet, Computers ad Operatos Research,vol.57,pp.33 40,015. [41] G. Dobso, T. Tezca, ad V. Tlso, Optmal workflow decsos for vestgators systems wth terruptos, Maagemet Scece,vol.59,o.5,pp ,013. [4] S. Seshadr ad Z. Shapra, Maageral allocato of tme ad effort: the effects of terruptos, Maagemet Scece, vol. 47, o. 5, pp , 001. [43]R.L.Graham,E.L.Lawler,J.K.Lestra,adA.H.Ka, Optmzato ad approxmato determstc sequecg ad schedulg: a survey, Aals of Dscrete Mathematcs, vol. 5, pp , [44] P. Brucker, Schedulg Algorthms, Sprger, 007. [45] G. H. Hardy, J. E. Lttlewood, ad G. Polya, Iequaltes, Cambrdge Uversty Press, Lodo, UK, [46] Z. Zhu, M. Lu, C. Chu, ad J. L, Multtaskg schedulg wth multple rate-modfyg actvtes, Iteratoal Trasactos Operatoal Research,017.

11 Advaces Operatos Research Hdaw Publshg Corporato Advaces Decso Sceces Hdaw Publshg Corporato Joural of Appled Mathematcs Algebra Hdaw Publshg Corporato Hdaw Publshg Corporato Joural of Probablty ad Statstcs The Scetfc World Joural Hdaw Publshg Corporato Hdaw Publshg Corporato Iteratoal Joural of Dfferetal Equatos Hdaw Publshg Corporato Submt your mauscrpts at Iteratoal Joural of Advaces Combatorcs Hdaw Publshg Corporato Mathematcal Physcs Hdaw Publshg Corporato Joural of Complex Aalyss Hdaw Publshg Corporato Iteratoal Joural of Mathematcs ad Mathematcal Sceces Mathematcal Problems Egeerg Joural of Mathematcs Hdaw Publshg Corporato Hdaw Publshg Corporato Hdaw Publshg Corporato Joural of Volume 01 Hdaw Publshg Corporato Dscrete Dyamcs Nature ad Socety Joural of Fucto Spaces Hdaw Publshg Corporato Abstract ad Appled Aalyss Hdaw Publshg Corporato Hdaw Publshg Corporato Iteratoal Joural of Joural of Stochastc Aalyss Optmzato Hdaw Publshg Corporato Hdaw Publshg Corporato