A Graphical Approach for Solving Single Machine Scheduling Problems Approximately

Transcription

1 A Graphica Approach for Soving Singe Machine Scheduing Probems Approximatey Evgeny R Gafarov Aexandre Dogui Aexander A Lazarev Frank Werner Institute of Contro Sciences of the Russian Academy of Sciences, Profsoyuznaya st 65, Moscow, Russia (e-mai: axe73@mairu, azarev@ipuru) Ecoe Nationae Superieure des Mines, FAYOL-EMSE, CNRS:UMR6158, LIMOS, F Saint-Etienne, France (e-mai: dogui@emsefr) Fakutät für Mathematik, Otto-von-Guericke-Universität Magdeburg, PSF 4120, Magdeburg, Germany (e-mai: frankwerner@mathematikuni-magdeburgde) Abstract: For five singe machine tota tardiness probems a fuy poynomia-time approximation scheme (FPTAS) based on a graphica agorithm is presented The FPTAS has the best running time among the known approximation schemes for these probems Keywords: Scheduing agorithms, Dynamic programming, FPTAS 1 INTRODUCTION We consider severa singe machine tota tardiness probems which can be formuated as foows We are given asetn = {1, 2,,n} of n independent jobs that must be processed on a singe machine Job preemption is not aowed The machine can hande ony one job at a time A jobs are assumed to be avaiabe for processing at time 0 For each job j N, a processing time p j > 0, a weight w j > 0 and a due date d j > 0aregiven A feasibe soution is described by a permutation π = (j 1,j 2,,j n ) of the jobs of the set N from which the corresponding schedue can be uniquey determined by starting each job as eary as possibe Let C jk (π) = k =1 p j be the competion time of job j k in the schedue resuting from the sequence π IfC j (π) >d j,thenjobj is tardy If C j (π) d j,thenjobj is on-time Moreover, et T j (π) =max{0,c j (π) d j } be the tardiness of job j in the schedue resuting from sequence π and et GT j (π) = min{max{0,c j (π) d j },p j } In the weighted tota tardiness minimization probem the objective is to find an optima job sequence π that minimizes weighted tota tardiness, ie, F (π) = n w jt j (π) Simiary, for the tota tardiness probem the objective function F (π) = n T j(π) and for the tota ate work probem the objective function F (π) = n GT j(π) have to be minimized The foowing specia cases of the probems are aso considered: Partiay supported by RFBR (Russian Foundation for Basic Research): , The authors are aso gratefu to Prof V Strusevich for his idea to use the GrA as a base for an FPTAS - minimizing weighted tota tardiness when a due dates are equa, ie, d j = d, j = 1, 2,,nThis probem is denoted by 1 d j = d w j T j ; - the specia case B 1 of the tota tardiness probem 1 T j,wherep 1 p 2 p n, d 1 d 2 d n and d n d 1 p n ; - the specia case B 1G of the probem 1 T j with d max d min p min,whered max = max j N {d j }, d min =min j N {d j } and p min =min j N {p j } For the NP-hard probem of maximizing weighted tota tardiness 1(no-ide) max w j T j, the objective is to find an optima job sequence that maximizes weighted tota tardiness, where each feasibe schedue starts at time 0 and there is no ide time between the processing of jobs Such probems with the maximum criterion have practica interpretations and appications, but their investigation is aso a theoreticay significant task A the probems and specia cases mentioned above are NP-hard in the ordinary sense (Gafarov et a (2012)) For the specia case B 1, an FPTAS with a running time of O(n 3 og n+n 3 /ε) is known, see Kouamas (2010) An FPTAS for the tota weighted ate work probem with the same running time was presented in Kovayov et a (1994) An FPTAS for the probem 1 d j = d w j T j with a running time of O((n 6 og w j )/ε 3 ) was presented in Keerer et a (2006) These probems can be considered as particuar cases when a compex function Ψ(t)+F (π, t) has to be minimized The function F (π, t) corresponds to one of the above mentioned functions F (π) if the jobs are processed not from time 0, but from time t Asan aternative, we have to partition the t-axis into intervas with the same optima schedue For exampe, the singe machine probem of minimizing the number of ate jobs,

2 when the starting time of the machine is variabe, was consid ered in Hoogeveen et a (2012) The same situation arises when it is known that some jobs have to be schedued one by one in a batch from an unknown time point t [t 1,t 2 ], eg, a set N contains two subsets N 1, N 2 and an optima job sequence can be represented as a concatenation (π 1,π 2,π 3 ), where {π 1 } {π 3 } = N 1 and {π 2 } = N 2 The jobs from N 2 have to be schedued according to one of the functions mentioned above The graphica and approximation agorithms presented in this paper can be used both for the initia probems and for the probems with variabe starting time Since the main topic of this paper is that of an anaysis of approximation agorithms, we reca some reevant definitions For the scheduing probem of minimizing a function F (π), a poynomia-time agorithm that finds a feasibe soution π such that F (π ) is at most ρ 1 times the optima vaue F (π ) is caed a ρ-approximation agorithm; the vaue of ρ is caed a worst-case ratio bound If a probem admits a ρ-approximation agorithm, it is said to be approximabe within a factor ρ A famiy of ρ- approximation agorithms is caed a fuy poynomia-time approximation scheme, or an FPTAS, if ρ =1+ε for any ε>0 and the running time is poynomia with respect to both the ength of the probem input and 1/ε For a practica reaization of some pseudo-poynomia agorithms with Booean variabes (eg a job can be ontime or tardy) one can use a modification caed a graphica agorithm (GrA) which we present for the probem 1 d j = d w j T j as we as an FPTAS based on this GrA with a running time of O(n 3 /ε) Modifications of these GrA and FPTAS for the cases B 1, B 1G and the probems 1 GT j and 1(no-ide) max w j T j are aso presented 2 DYNAMIC PROGRAMMING FOR THE PROBLEM 1 D J = D W J T J Lemma 1 There exists an optima job sequence π for probem 1 d j = d w j T j that can be represented as a concatenation (G, x, H), where a jobs j H are tardy and S j d for a j H, and a jobs i G are on-time A jobs from set G are processed in non-increasing order of the vaues pj w j and a jobs from set H are processed in non-decreasing order of the vaues pj w j Thejobx starts before time d and is competed no earier than time d The job x is caed stradding Assume that the jobs are numbered as foows: p2 w 2 p3 w 3 pn w n, (*) where the job with number 1 is the stradding job As a coroary from Lemma 1, there is a stradding job x N, to which the number 1 wi be assigned, such that for each {1, 2,,n}, there exists an optima schedue in which a jobs j {1, 2,,} are processed from time t one by one, and there is no job i {+1,+2,,n} which is processed between these jobs Thus, we can present a dynamic programming agorithm (DPA) based on Lemma 1 For each x N, we number the jobs from the set N \{x} according to (*) and perform Agorithm 1 Then we choose a best schedue among the n constructed schedues At each stage, 1 n, of Agorithm 1, we construct a best partia sequence π (t) for the set of jobs {1, 2,,} and for each possibe starting time t of the first job (which represents a possibe state in the DPA) F (t) denotes the weighted tota tardiness vaue for the job sequence π (t) Φ 1 (t) andφ 2 (t) are temporary functions, which are used to compute F (t) Agorithm 1 1 Number the jobs according to order (*); 2 FOR t := 0 TO n j=2 p j DO π 1 (t) :=(1),F 1 (t) :=w 1 max{0,p 1 + t d}; 3 FOR := 2 TO n DO FOR t := 0 TO n j=+1 p j DO π 1 := (, π 1 (t + p )), π 2 := (π 1 (t),); Φ 1 (t) :=w max{0,p + t d} + F 1 (t + p ); Φ 2 (t) :=F 1 (t)+w max{0, p j + t d}; IF Φ 1 (t) < Φ 2 (t) THENF (t) :=Φ 1 (t) and π (t) :=π 1, ELSE F (t) :=Φ 2 (t) andπ (t) :=π 2 ; 4 π n (0) is an optima job sequence for the chosen job x with the objective function vaue F n (0) Theorem 2 By using Agorithm 1 for each x N, an optima job sequence of the type described in Lemma 1 can be found in O(n 2 p j )time It is obvious that for some chosen job x N, in the job sequence π n (0), the job x cannot be stradding (ie, either S x d or C x <d) This means that there exists another job x N for which the vaue F n (0) wi be ess Agorithm 1 can be modified by considering for each =1, 2,,n, ony the interva [0,d p ] instead of the n interva [0, p i ] since for each t > d p, job j is i=+1 tardy in any partia sequence π (t) and the partia sequence π 2 := (π 1 (t),) is optima Thus, the time compexity of the modified Agorithm 1 is equa to O(nd), and an optima schedue can be found in O(n 2 d)time Let UB be an upper bound on the optima function vaue for the probem which is found by the 2-approximation agorithm of Fathi and Nutte, ie, UB 2F (π ) If for some t UB (, + ) wehavef (t UB )=UB,thenfor each t>t UB we have F (t) >UB(since t denotes the starting time of an optima schedue obtained from the job sequence π (t) forjobs1, 2,, and F (t) denotes the corresponding vaue of the monotonic objective function) So, the states t>t UB seem to be unpromising, ie, for any job sequence π constructed using these states, we wi have F (π) >UB, ie, π is not optima Thus, we need to consider different vaues F (t) onyfort [0,t UB ]and assume F (t) =+ for t (t UB, + ) If a parameters p j,w j for a j N and d are integer, then there are at most UB +2differentvauesF (t) In addition, if there is a point t (, + ) such that F (t )=0andF (t +1)> 0, then F (t) = 0 for a t t and F (t) <F (t+1) for a t t +1, since a the functions F (t) are monotonic Thus, we can modify Agorithm 1 as foows If we wi save at each stage instead of a states t [0,t ] ony one state t, then the number of saved states wi be restricted by UB, since for a saved states t we have F (t) <F (t + 1) The running time of

3 Fig 1 Function F (t) in the GrA and in Agorithm 1 the modified agorithm is O(n min{d, U B}) If we consider ony the states t [d n p j,d] instead of [0,d]ateach stage, =1, 2,,n, then we obtain an optima soution for the chosen stradding job for each possibe starting time t (,t UB n )ino(nub) time Let π be a job sequence, where a n jobs are processed in non-decreasing order of the vaues pj w j DenotebyF(π,d) the weighted tota tardiness for the job sequence π,where the processing of the jobs starts at time ditisobviousthat for this starting time the schedue π is optima Then, by using the modified Agorithm 1, we can obtain an optima soution for each possibe starting time t (, + ) in O(n 2 F (π,d)) time We note that the inequaity F (t) < F (t + 1) does not necessariy hod for a t>t in the probem 1 GT j, ie, the running time of Agorithm 1 is not restricted by UB for the probem 1 GT j 3 GRA FOR PROBLEM 1 D J = D W J T J The GrA is a modification of Agorithm 1, in which function F (t) is defined for any t (, + ) (not ony for integer t) However, we need to compute these vaues ony at the break points separating intervas in which function F (t) is a inear function of the form F (t) = F k(t) =uk (t tk 1 )+b k F (t) is a continuous piecewise inear function whose parameters can be described in a tabuar form Namey, in each step of the GrA, we store the information on function F (t) for a number of intervas (characterized by the same best partia sequence and by the same tota weight of tardy jobs) in a tabuar form as given in Tabe 1 Tabe 1: Function F (t) k 1 2 m +1 m +2 int (,t 1 ] (t1,t2 ] (tm,t m +1 ] (t m +1, ) b k 0 b 2 b m +1 u k 0 u 2 u m +1 0 π k π 1 π 2 π m +1 (1, 2,, ) In Tabe 1, k denotes the number of the current interva whose vaues range from 1 to m + 2 (where the number of intervas m + 2 is defined for each = 1, 2,,n), (t k 1,t k ]isthekth interva (where t0 =, tm +2 =+ = t UB ), b k, uk are the parameters of the inear and t m +1 function F k (t) defined in the kth interva, and πk is the best sequence of the first jobs if they are processed from time t (t k 1,t k ] These data mean the foowing For each above interva, we store the parameters b k and u k for describing function F (t) and the resuting best partia sequence if the first job starts in this interva For each starting time t (t k 1,t k ] (t 0 = ) of the first job, we have a best partia sequence π k of the jobs 1, 2,, with a tota weight of the tardy jobs u k and the function vaue F (t) =u k (t tk 1 )+b k (see Fig 1(a)) We have F (t) =0fort (t 0,t1 ] Reca that function F (t) is defined not ony for integers t, but aso for rea numbers t For simpicity of the foowing description, we consider the whoe t-axis, ie, t (, + ) This tabe describes a function F (t) which is a continuous, piecewise-inear function in the interva (,t UB ] The points t 1,t2,,tm +1 are caed break points, since there is a change from vaue u k 1 to u k (which means that the sope of the piecewise-inear function changes) Note that some of the break points t k can be non-integer To describe each inear segment, we store its sope u k and its function vaue b k = F (t) atthepointt = t k 1 So,in the tabe b 1 = b 2 <b 3 < < b m +1 <UBhods, since t 1 <t2 <<tm +1 In the GrA, the functions F (t) refect the same functiona equations as in Agorithm 1, ie, for each t Z [0, n j=2 p j], the function F (t) has the same vaue as in Agorithm 1 (see Fig 1), but these functions are now defined for any t (, + ) As a resut, often a arge number of integer states is combined into one interva (describing a new state in the resuting agorithm) with the same best partia sequence In Fig 1 (a), the function F (t) from the GrA is presented and in Fig 1 (b), the function F (t) from Agorithm 1 is dispayed The function Φ 1 (t) is obtained from F 1 (t) by the foowing operations We shift the diagram of function F 1 (t) to the eft by the vaue p and in the tabe for F 1 (t) and add a coumn which resuts from the new break point t = d p Ift s 1 p < t < t s+1 1 p, s m 1, then we have two new intervas of t in the tabe for Φ 1 (t): (t s 1 p,t ] and (t,t s+1 1 p ] (for s = m 1 +1, we have (t UB 1 p,t ]and(t, + )) This means that we first repace each interva (t k 1,tk+1 1 ]by(tk 1 p,t k+1 1 p ] in the tabe for Φ 1 (t), and then repace the coumn with the interva (t s 1 p,t s+1 1 p ] by two new coumns with the intervas (t s 1 p,t ]and(t,t s+1 1 p ] Moreover, we increase the vaues u s+1 1,us+2 1,,um by w, ie, the tota weight of tardy jobs (and thus the sope of the function) increases The corresponding partia sequences π 1 are obtained by adding job as the first job to each previous partia sequence The function Φ 2 (t) is obtained from function F 1 (t) by the foowing operations In the tabe for F 1 (t), we add a coumn which resuts from the new break point t = d i=1 p iift h 1 <t <t h+1 1, h m 1, then there are two new intervas of t in the tabe for Φ 2 (t): (t h 1,t ] and (t,t h+1 1 ](forh = m 1 +1,wehave(t UB 1,t ]and (t, + )) This means that we repace the coumn with the interva (t h 1,th+1 1 ] by two new coumns with the intervas (t h 1,t ]and(t,t h+1 1 ]

4 Moreover, we increase the vaues u h+1 1,uh+2 1,,um by w, ie, the tota weight of tardy jobs increases The corresponding partia sequences π 2 are obtained by adding job at the end to each previous partia sequence We construct a tabe that corresponds to the function F (t) = min{φ 1 (t), Φ 2 (t)} as foows We consider functions Φ 1 (t) and Φ 2 (t) on a resuting intervas from both tabes and search for intersection points of the diagrams of these functions To be more precise, we construct a ist t 1,t 2,,t e, t 1 < t 2 < < t e, of a break points t from the tabes for Φ 1 (t) andφ 2 (t), which are eft / right boundary points of the intervas given in these tabes Then we consider each interva (t i,t i+1 ],i=1, 2,,e 1, and compare the two functions Φ 1 (t) andφ 2 (t) overthisintervaletthe interva (t i,t i+1 ] be contained in (t z 1,t z ] from the tabe for Φ 1 (t) and in (t y 1,t y ] from the tabe for Φ 2 (t) Then Φ 1 (t) =(t t z 1 ) u z + b z and Φ 2 (t) =(t t y 1 ) u y + b y Choose the coumn from both tabes corresponding to the maximum of the two functions in (t i,t i+1 ]andinsert this coumn into the tabe for F (t) If there exists an intersection point t of Φ 1 (t) andφ 2 (t) inthisinterva, then insert two coumns with the intervas (t i,t ] and (t,t i+1 ] This step requires O(m 1 )operations Let m be the number of coumns in the resuting tabe of F (t) and for k, 1 k < m, the inequaity b k < UB b k+1 hods From the equaity UB = (t UB t k 1 )u k + b k, we compute the vaue tub Inthecoumn with the interva (t k 1,t k ](coumnk), assign tk = t UB and substitute a the coumns k+1,k+2,,m,byonecoumn with the interva (t UB, + ), b k+1 =+, u k+1 =0and π k+1 =(1, 2,,) So, in the resuting tabe, there wi be no more then UB + 2 coumns if a the parameters of the probem are integer In the tabe corresponding to function F n (t) we determine the coumn (t k n,t k+1 n ], where t k n < 0 t k+1 n Thenwehave an optima sequence π = πn k+1 for the chosen job x and the optima function vaue F (π )=b k+1 n +(0 t k n) u k+1 n Theorem 3 By using the GrA for each x N, an optima job sequence of the type described in Lemma 1 can be found in O(n 2 F ) time, where F is the optima objective function vaue 4 ADVANTAGES OF GRA FOR 1 D J = D W J T J In each step = 1, 2,,n of the GrA, we do not consider a points t [0,d], t Z, but ony points from the interva in which the optima partia soution changes or where the resuting functiona equation of the objective function changes So, the main difference is that we operate not with independent vaues F in each of the points t, but with functions which are transformed in each step anayticay (according to their anaytica form), which can have obvious advantages For exampe, et us minimize a function Ψ(t) + F (π, t), where the function F (π, t) corresponds to a function F (π) when the jobs are processed not starting from time 0, but from time t Ifthe function F (t) is presented anayticay (not in a tabuar form (t, F )) and the function Ψ(t) is presented anayticay as we, then the search for the minimum of Ψ(t)+F (π, t) canbemadeinshortertime Moreover, such an approach has the foowing advantages when compared with Agorithm 1 (DPA): 1 GrA can sove instances, where (some of) the parameters p j,w j,, 2,,n or/and d are not in Z 2 The running time of the GrA for two instances with the parameters {p j,w j,d} and {p j 10 const ± 1,w j 10 const ±1,d 10 const ±1},const>1, is the same whie the running time of the DPA wi be 10 const times arger in the second case Thus, one can usuay sove consideraby arger instances with the GrA 3 Properties of an optima soution can be taken into account, and sometimes the GrA has even a poynomia time compexity, or we can at east essentiay reduce the compexity of the standard DPA 4 Unike DPA, it is possibe to construct a FPTAS based on GrA easiy which is presented in Section 5 Let us consider another type of a DPA This agorithm generates iterativey some sets of states In every iteration, = n, n 1,,1, a set of states is generated Each state can be represented by a string of the form (t, F ), where t is the competion time of the ast known job schedued in the beginning of a schedue and F is the vaue of the function, provided that the eary jobs start at time 0 and the ast known ate job competes exacty at time n p This agorithm can be described as foows: Aternative DPA 1 Number the jobs according to order (*); 2 Put the state (0, 0) into the set of states V n+1 3 FOR := n TO 1 DO FOR each state (t, F ) from the set of states V +1 DO Put a state [t+p,f+ w max{0,t+ p d}]; Put a state [t, F + w max{0, n p j ( 1 p j t) d}]; 4 Find F =min{f (t, F ) V 1 } We need to consider ony states, where t d and F UB If in a ist V, there are two states with the same objective function vaue (t 1,F )and(t 2,F )andt 2 >t 1, then the state (t 2,F ) can be removed from consideration So, the running time of the Aternative DPA is O(n min{d, F }) which corresponds to the running time of the GrA (note that the GrA can be easiy modified to consider ony points t [0,d]) However, in the GrA some of the possibe but unpromising states are not considered and, in contrast to the aternative DPA, the GrA finds a optima schedues for a t [,t UB n ]ino(nf ) time So, the aternative DPA is not effective for the probems of minimizing a compex function Ψ(t)+F (π, t) 5 AN FPTAS FOR PROBLEM 1 D J = D W J T J εub 2n The idea of the FPTAS is as foows Let δ = To reduce the time compexity of the GrA, we have to diminish the number of coumns considered, which is the number of different objective function vaues 0 = b 1 = b 2,b3,,bm +1 = UB If we consider not the origina vaues b k but the vaues b k which are rounded up or down to the nearest mutipe of δ vaues b k, there are no more

5 than UB δ = 2n ε different vaues bk Then we wi be abe to convert the tabe F (t) into a simiar tabe with no more than 4 n ε coumns Furthermore, for such a modified tabe (function) F (t), we wi have F (t) F (t) <δ εf (π ) n If we do the rounding and modification after each stage of the GrA, then the cumuative error wi be no more than nδ εf (π ), and the tota running time of n runs of the GrA wi be O( n3 ε ), ie, an FPTAS is obtained By transforming the GrA, we save the approximated functions F A (t) in the same tabuar form but without the ast row describing an optima partia job sequence π k FPTAS (as a modification of the GrA) Assume that we have obtained a tabe for a function F (t) after stage with the objective function vaues b 1,b2,,bm +1 Ifm > 4 n ε, then do the foowing Round a the vaues b k from Tabe 1 to the nearest mutipe of δ Letthevauesb 1, b2,,bm +1 be obtained, where b 1 b 2 b m +1 We modify the tabe F A (t) as foows Assume that, for k 1 <k 2,wehaveb k1 < b k1+1 = = b k2 < b k2+1 We substitute the coumns which correspond to the vaues b k1,,b k2 1 for the two coumns presented in Tabe 2 Tabe 2: Substitution of coumns int k (t k1 1,t k1 ] (t k1,t k2 1 ] (t UB, ) b k b k1 u k u = bk 2 b k 1 t k 1 t k 1 1 b k2 0 0 In fact, in the modification we substitute some inear fragments which foow one by one and are cose (in terms of the absoute error which is δ/2) to the same ine by this ine Let F (t) be the function obtained at stage in the modified agorithm before rounding and et F A (t) describe the rounded function Lemma 4 For a t (t k1 1 (t) <δ/2 F,t k2 1 ], we have F A (t) Before the next stage + 1, we assign F A (t) := F (t) Assume that functions F (t), =1, 2,,n,are exact and constructed by the origina GrA Simiary, F A (t), = 1, 2,,n, are approximated functions constructed by the modified agorithm Lemma 5 For each, = 1, 2,, n, and a t (, + ), we have F A (t) F (t) δ/2 Let π be an optima job sequence for the chosen stradding job x Lemma 6 Inequaity F (π ) F (π ) n δ n 2ε F (π ) 2n hods Theorem 7 By using the modified GrA for each x N, a job sequence π of the type described in Lemma 1 wi be found in O( n3 ε ) time, where F (π ) (1 + ε)f (π ) The running time O( n3 ε ) is obtained as foows The modified GrA is used n times for each job x N The running time of the modified GrA depends on n and the number of coumns in the tabes which describe functions F A (t) The number of coumns does not exceed O( n ε ) 6 ALGORITHMS FOR THREE SPECIAL CASES Lemma 8 There exists an optima job sequence π for the specia case B 1G that can be represented as a concatenation (G, x, H), where a jobs j H are tardy and a jobs i G are on-time A jobs from set G are processed in non-increasing order of the vaues p j (LPT order) and a jobs from set H are processed in nondecreasing order of the vaues p j (SPT order) The job x is caed stradding Lemma 9 (Gafarov et a (2012)) There exists an optima job sequence π for the specia case B 1 that can be represented as a concatenation (G, H) A jobs from the set G are processed in LPT order and a jobs from the set H are processed in SPT order Assume that for the case B 1G, the jobs are numbered as foows: p 1 p 2 p n Lemma 10 For the specia cases B 1 and B 1G andajobsequenceπ SPT =(n, n 1,,1), inequaity F (π SPT ) 3F (π )hods Without oss of generaity, we wi consider ony cases whereinajobsequenceπ SPT at east two jobs are tardy Lemma 11 (Gafarov et a (2012)) There exists an optima job sequence π for the probem 1 GT j that can be represented as a concatenation (G, H), where a jobs j H are tardy and GT j (π) =p j For a jobs i G, we have 0 GT i (π) <p i A jobs from the set G are processed in EDD (eariest due date) order and a jobs from the set H are processed in LDD (ast due date) order Let for the probem 1 GT j thejobsbenumberedas foows: d 1 d 2 d n and et π EDD =(1, 2,,n) Denote by T the maxima tardiness of a job in the sequence π EDD, ie, T =max j N {GT j (π EDD )} Lemma 12 For probem 1 GT j, the foowing inequaity hods: F (π EDD ) nf (π ) Lemma 13 (Gafarov et a (2012)) There exists an optima job sequence π for the probem 1(no-ide) max w j T j that can be represented as a concatenation (G, H), where a jobs j H are tardy and a jobs i G are on-time A jobs from the set G are processed in non-increasing order of the vaues wj p j and a jobs from the set H are processed in non-decreasing order of the vaues wj p j For probem 1(no-ide) max w j T j, the vaue LB maxt T =max j N (w j ( n i=1 p i d j )) is a ower bound Then UB maxt T = nlb maxt T is an upper bound on the optima objective function vaue To sove these probems, Agorithms 1, GrA and FPTAS can be modified as foows For the specia case B 1G: - DPA Use the fact described in Lemma 8 In Agorithm 1, we number the jobs according to the order p 2 p 3 p n Assume that F 1 (t) := max{0,p 1 + t d 1 },Φ 1 (t) :=max{0,p + t d } + F 1 (t+p )andφ 2 (t) :=F 1 (t)+max{0, p j +t

6 d } A other steps of the agorithm remain the same Remember that w j =1foraj N for the probem 1 T j The running time of the modified Agorithm 1 is O(nd max ) Since it is necessary to consider n stradding jobs x N, an optima job sequence can be found in O(n 2 d max )timebyusingthemodified Agorithm 1; - GrA The GrA remains the same The parameters u k denote the number of tardy jobs which is equa to the tota weight of the tardy jobs, since w j =1fora j N Inaddition,assumethatUB = F (π SPT ) By using the GrA, an optima schedue can be found in O(n 2 min{d max,f })time; εf (πsp T ) 3n - FPTAS In the FPTAS, assume that δ = Since the GrA is without changes, the time compexity of the FPTAS based on the GrA for the specia case B 1G has a running time of O(n 3 /ε), which is ess than the running time O(n 7 /ε) ofthefptasfor the genera case presented by Lawer For the specia case B 1: - DPA Use the fact described in Lemma 9 Agorithm 1 is modified as for the specia case B 1G The running time of the modified Agorithm 1 is O(nd max ) Since there is no stradding job, an optima job sequence can be found in O(nd max )timebyusing the modified Agorithm 1 ony once; - GrA The GrA remains the same as for the specia case B 1G By the GrA, an optima schedue can be found in O(n min{d max,f })time; - FPTAS The FPTAS remains the same as for the specia case B 1G Since there is no stradding job, the FPTAS for the specia case B 1 has a running time of O(n 2 /ε), which is ess than the running time of O(n 3 og n + n 3 /ε) of the FPTAS mentioned by Kouamas For the probem 1 GT j : - DPA Use the fact described in Lemma 11 In Agorithm 1, we number the jobs according to the order d 1 d 2 d n Assume that F 1 (t) := min{p 1, max{0,p 1 + t d 1 }}, Φ 1 (t) := min{p, max{0,p +t d }}+F 1 (t+p )andφ 2 (t) := F 1 (t)+min{p, max{0, p j +t d }} The running time of the modified Agorithm 1 is O(nd max ) Since there is no stradding job, an optima job sequence can be found in O(nd max )timebyusingthemodified Agorithm 1 ony once; - GrA The GrA remains amost the same In addition to the break points t and t,twonewbreak points τ = d and τ = d 1 p j are considered The sope u k of the function F (t) is changed according to the function min{p, max{0,p + t d }} By the GrA, an optima schedue can be found in O(n min{d max,nf }) time, since UB = F (π EDD ) nf (π )andthereareatmost2ub + 2 coumns in each tabe F (t) considered in the GrA; εf (πedd) - FPTAS Here, we assume that δ = n So,the 2 FPTAS has a running time of O(n 3 /ε) For the probem 1(no-ide) max w j T j : - DPA We use the fact described in Lemma 13 In Agorithm 1, we enumerate the jobs according to the order w1 p 1 w2 p 2 wn p n We assume that F 1 (t) := w 1 max{0,p 1 + t d 1 }, Φ 1 (t) := w max{0,p + t d } { + F 1 (t + p ) and Φ 2 (t) := F 1 (t) + w max 0, p i + t d } Since tota tardiness is i=1 maximized, we have F (t) := max{φ 1 (t), Φ 2 (t)} The running time of the modified Agorithm 1 is O(nd max ) Since there is no stradding job, an optima job sequence can be found in O(nd max )timebyusing the modified Agorithm 1 ony once; - GrA The GrA remains the same as for the probem 1 d j = d w j T j WehaveF (t) =max{φ 1 (t), Φ 2 (t)} It is known (Gafarov et a (2012)) that the functions F (t) represent continuous, piecewise-inear and convex functions By the GrA, an optima schedue can be found in O(n min{d max,nf, w j })time - FPTAS In the FPTAS, assume that δ = εubmaxt n 2 T So, the FPTAS has a running time of O(n 3 /ε) In Gafarov et a (2012), GrA for 8 scheduing probems and FPTAS for 6 scheduing probems are presented 7 CONCLUSION In this paper, an FPTAS was presented, which can be used with some simpe modifications for severa singe machine probems The FPTAS is based on a graphica approach The idea of such a modification of graphica agorithms enabes us to construct an FPTAS easiy The graphica approach can be appied to probems, where a pseudo-poynomia agorithm exists and Booean variabes are used in the sense that yes/no decisions have to be made (eg, a job may be competed on-time or not) For the singe machine probem of maximizing tota tardiness, the graphica agorithm improved the compexity from O(n p j )too(n 2 ) Thus, the graphica approach has not ony a practica but aso a theoretica importance REFERENCES ER Gafarov, A Dogui and F Werner Dynamic Programming Approach to Design FPTAS for Singe Machine Scheduing Probems CNRS Report, 26 pp (2012) C Kouamas The Singe-Machine Tota Tardiness Scheduing Probem: Review and Extensions European Journa of Operationa Research, 202, 1 7 (2010) M Y Kovayov, C N Potts and L N van Wassenhove A Fuy Poynomia Approximation Scheme for Scheduing a Singe Machine to Minimize Tota Weighted Late Work Mathematics of Operations Research, Vo 19, No 1, (1994) H Hoogeveen and V T Kindt Minimizing the Number of Late Jobs When the Starting Time of the Machine is Variabe Proceedings PMS, (2010) H Keerer and VA Strusevich A Fuy Poynomia Approximation Scheme for the Singe Machine Weighted Tota Tardiness Probem with a Common Due Date Theoretica Computer Science, 369, (2006)