BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. 57, No. 3, 2009 Algoriths for parallel processor scheduling with distinct due windows and unit-tie obs A. JANIAK 1, W.A. JANIAK 2, and R. JANUSZKIEWICZ 1 1 Institute of Coputer Engineering, Control and Robotics, Wrocław University of Technology, 11/17 Janiszewskiego St., 50-372 Wrocław, Poland 2 Institute of Industrial Engineering and Manageent, Wrocław University of Technology, 25 Soluchowskiego St., 50-372 Wrocław, Poland Abstract. We have studied probles of scheduling n unit-tie obs on identical parallel processors, in which for each ob a distinct due window is given in advance. If a ob is copleted within its due window, then it incurs no penalty. Otherwise, it incurs a ob-dependent earliness or tardiness cost. The obective is to find a ob schedule such that the total weighted earliness and tardiness, axiu weighted earliness and tardiness or total weighted nuber of early and tardy obs is iniized. Properties of optial solutions of these probles are established. We proved that optial solutions for these probles can be found in On 5 ) tie in case of iniization of the total weighted earliness and tardiness and the total weighted nuber of early and tardy obs and in On 4 nlog n) tie in case of iniization of the axiu weighted earliness and tardiness. The established solution ethods are extended to solve the probles with arbitrary integer release dates. A dedicated algorith with tie coplexity On 3 ) is provided for the special case of the proble of iniizing total weighted nuber of early and tardy obs with agreeable earliness-tardiness weights. Key words: scheduling algoriths, parallel processor, earliness/tardiness, distinct due windows, unit-tie obs, integer release dates. 1. Introduction Probles of scheduling unit-tie obs with distinct due windows on identical parallel processors are studied. A due window is a tie interval associated with a ob. There are no restrictions iposed on due windows. A ob incurs no scheduling cost if it is copleted within its due window. Otherwise, an earliness or tardiness cost is incurred. Scheduling probles with distinct due windows odel any real-life probles that occur in the production of perishable goods. Consider for exaple a process in which one cheical is cobined with another to produce the final product and one of the cheicals deteriorates rapidly. If the deteriorating cheical is produced before the second cheical is ready it ay becoe useless. On the other hand, if this cheical is produced late, delay in production of the final product ay prove costly. Another exaple coes fro a branch of industry, in which a anufacturer agrees with his clients to deliver products in certain tie intervals. If the products are anufactured before the earliest acceptable tie, they ust be kept in warehouses, thus incurring an additional cost. On the contrary, if they are anufactured after the latest acceptable tie, the anufacturer ust pay for an express delivery of the products. Scheduling probles with distinct due windows were studied only in a single processor environent. Sidney [1] was aong the pioneers, who introduced such probles into scheduling. He studied a special case of a single processor proble with iniization of axiu cost associated with earliness and tardiness, in which any due window is not allowed to contain another due window as a proper subset. He proved that this proble is solvable in polynoial tie. However, in general a single processor scheduling proble with distinct due windows and iniization of the total weighted earliness and tardiness is strongly NP-hard, since its special case, i.e. iniization of total weighted tardiness or earliness), is already strongly NP-hard see [2]). Soe heuristic algoriths for this proble were analyzed in [3] and [4]. Later on in [5] a general case of single processor scheduling proble with distinct due windows and iniization of the weighted nuber of early and tardy obs was investigated and the proble was shown to be strongly NP-hard. Soe additional discussion on this proble and heuristic algoriths were presented in [6]. The present paper deals with the scheduling probles in which the due windows are given in advance. In a scientific literature there have also been studied probles with due window assignent. For a literature review and soe results see [7]. In this paper we analyze probles, whose special cases, i.e. probles of scheduling unit-tie obs with arbitrary ob release dates and due dates, have already been studied in the scientific literature. The probles of iniizing the total weighted tardiness and the total weighted nuber of tardy obs can be reduced to the network-flow proble see [8 9]). The proble of iniizing the total weighted nuber of tardy obs can also be solved by a polynoial tie algorith constructed by Baptiste et al. [10] for a ore general proble with equal processing ties. The proble of ini- e-ail: ada.aniak@pwr.wroc.pl 209
A. Janiak, W.A. Janiak, and R. Januszkiewicz izing the axiu tardiness is solvable by scheduling obs in the order of non-decreasing due dates [9]. The paper is organized as follows. In Sec. 2, a study of the parallel processor scheduling proble with unit-tie obs and iniization of weighted earliness and tardiness is presented. In Sec. 3, a parallel processor scheduling proble with unit-tie obs and iniization of the weighted nuber of early and tardy obs is analyzed. Section 4 concludes the paper. 2. Proble forulation There are n non-preeptive obs to be scheduled on identical parallel processors. Each processor can handle at ost one ob at a tie and each ob can be copletely processed on any processor. All obs have unit processing requireents and for each ob 1,...,n there is given a due window [ê, ˆd ] ê ˆd ), where ê and ˆd are non-negative integer nubers. A schedule σ deterines the allocation of obs to processors and ob starting and copletion ties. Let S σ) and C σ) denote the integer start and copletion tie of ob in schedule σ, respectively. For convenience throughout the paper S and C are also used instead of S σ) and C σ), respectively, if there is no possible confusion as to the schedule we refer to. Given a schedule σ, the earliness and tardiness of ob are defined as E σ) ax0, ê C σ) and T σ) ax0, C σ) ˆd, respectively. The obective is to find a schedule σ for which one of the following criteria f su σ) α E σ) + β T σ)), f ax σ) ax 1 n α E σ),β T σ), f nu σ) is iniized, where V σ) U σ) α V σ) + β U σ)), 1 if C σ) < ê 1 if C σ) > ˆd and where α and β deterine strictly positive integer costs of earliness and tardiness, respectively. V σ) and U σ) are indicators that in the schedule σ ob is early or tardy, respectively. The defined probles will be denoted as P-su, P-ax and P-nu, where suffixes su, ax and nu deterine the obective criterion. 3. Properties of optial solutions In this section, we provide properties, which allow us to find optial polynoial tie algoriths for the probles forulated in the previous section.,, 3.1. Miniization of weighted earliness and tardiness. Lea 1. There exists an optial solution for probles P-su and P-ax for which the starting tie S of each ob eets the following condition: ax ê S ê + 1. Proof. Assue there is an optial solution σ, which does not coply with the thesis of this lea. Therefore, there exists at least one ob, which eets one of the following conditions: a) S σ) < ê n/, b) S σ) > ê + n/ 1. Observe that if condition a) is satisfied for ob then there exists tie t such that ê n/ t ê 1 and interval [t, t + 1] in which at least one processor is idle. Assue that schedule σ has been obtained fro σ by setting S i σ ) S i σ), for i, and S σ ) t. Then for σ the obective criterion value is not greater than for solution σ, since α ê C σ )) < α ê C σ)) and α i ê i C i σ )) α i ê i C i σ)), for i. Therefore σ is also optial. On the other hand, if condition b) is satisfied for ob then there exists tie t such that ê t ê + n/ 1 and interval [t, t+1] in which at least one processor is idle. As for the previous case, assue that schedule σ has been obtained fro σ by setting S i σ ) S i σ), for i, and S σ ) t, for which the criterion value is not greater than for the solution σ, 0, C σ ) ˆd 0, C σ) ˆd since β ax β ax and β i ax 0, C i σ ) ˆd i β i ax 0, C i σ) ˆd i, for i. Therefore σ is optial. Repeating the above arguent, if necessary, proves the result. Lea 2. Let the set of all obs J be partitioned into two subsets J and J, such that J k < n and J n k and ax J ê + k/ in J ê n k)/. There exists an optial solution to probles P-su and P- ax in which only obs fro J are scheduled in interval [ax 0, in J ê k/,ax J ê + k/ ]. Proof. According to Lea 1 ) k k J ax 0, ê S ê + 1, and J ) n k n k ax 0, ê S ê + 1. Therefore, and J S, C S + 1 k [ ax 0, in ê J [ ax 0, in ê J, ax J ê + J S, C S + 1 k, ax J ê + ] k, k ]. 210 Bull. Pol. Ac.: Tech. 573) 2009
Algoriths for parallel processor scheduling with distinct due windows and unit-tie obs Since ax J ê + k/ in J ê n k)/, then the intervals in which the obs fro subsets J and J ust be scheduled are disoint. The following algorith PARTITION based on Lea 2 partitions the set of obs into disoint subsets. With each subset there is associated an interval in which the obs fro the subset should be scheduled according to Lea 1). The intervals associated with different subsets can contain at ost one coon nuber borders of the intervals. Algorith PARTITION input: J the set of obs) 1. Let J 1 Ø, J 2 J. 2. Let i be the ob such that ê i in J2 ê. 3. Set J 1 J 1 i, J 2 J 2 \ i. 4. If J 2 Ø then stop. 5. If ax J1 ê + J 1 / in J2 ê J 2 /, then J 1 is one of the subsets and to find the other subsets apply PARTITION to J 2, else go to step 2. Lea 3. If the set of all obs is partitioned using algorith PARTITION into k n disoint subsets J 1,..., J k, then the length of intervals associated with the subsets in which the obs should be scheduled is at ost O n J i /), for i 1,...,k. Proof. Suppose the algorith PARTITION is applied to the set of all obs. At the beginning there are two subsets, the first one, say J, contains one ob and the other one, say J contains the rest of obs. The length of the interval associated with J equals 2 J 1 / 2 and in the worst case ax ê + J J 1 in J ê J Hence, the distance between the upper bound of the interval associated with J and in ê is J / +1 n/ + J 1 O n/) and the ob with the sallest ê fro J have to be added to J and reoved fro J. If the described procedure is repeated until ax J ê + J / in J ê J / it is clear that the length of the interval associated with J is at ost J O n/)+2 J / O n J /). If the algorith PARTITION is next applied to J the sae reasoning leads to the conclusion that the length of the associated interval is at ost O n J /) and the sae for the rest of subsets and the associated intervals. Lea 4. If the set J of all obs is partitioned using algorith PARTITION into k n disoint subsets J 1,...,J k, such that the obs fro the subset J l should be scheduled in interval [t l1, t l2 ], then an optial allocation of obs to the processors and deterination of starting ties for probles P-su and P-ax can be obtained by solving a proper insu or in-ax assignent proble for each subset J l and an associated interval [t l1, t l2 ]. Proof. According to Lea 2, it is possible to partition the set J into disoint subsets, and obs that belong to each such subset J l can be scheduled in different interval. Therefore,. the obective criterion for probles P-su and P-ax can be rewritten as follows: f su σ) α E σ) + β T σ)) and J i α E σ) + β T σ)) f ax σ) ax α E σ), β T σ) 1 n ax ax α E σ), β T σ) 1 i k J i respectively, under assuption that k J i J and J i J Ø, for i, 1,...,n, i. Thus, the iniization of the global obective criterion value can be obtained by the iniization of the criterion value for each disoint subset of obs and an associated interval. For each subset J l and an associated interval [t l1, t l2 ] the iniization of the criterion value for the proble P-su can be forulated as follows: in : i l c i x i, 1) for i l t l2 t l1 ) and l J l, where α ê t l1 + i)) if t l1 + i < ê c i+qtl2 t l1 )) β t l1 + i ˆd ) if t l1 + i > ˆd 2) for 1 i t l2 t l1 and 0 q 1, and where is the -th ob in J l, subect to x i 1, 3) 1 i il 1 l i l x i 1. 4) In case of the proble P-ax, the iniization of the obective value can be forulated as follows: in : ax 1 i i l,1 l c i x i 5) for i l t l2 t l1 ) and l J l, subect to constraints 3) 4), where c i is given by 2). The value x i should be interpreted as follows: x i 1 i -th ob in J l is scheduled on processor t l2 t l1 S t l1 + i 1) od t l2 t l1 ) 6) for 1 i i l and 1 l. Bull. Pol. Ac.: Tech. 573) 2009 211
A. Janiak, W.A. Janiak, and R. Januszkiewicz Theore 1. Optial solutions for probles P-su and P-ax can be found in O n 5) and O n 4 n log n ) tie, respectively. Proof. At first the set J of all n obs is partitioned using algorith PARTITION into k n disoint subsets J 1,...,J k. If the obs in J are sorted according to the non-decreasing order of ê that takes O n log n) tie) then coputation in algorith PARTITION takes O n) tie. Thus, the operations of sorting and partitioning the obs take O n log n) tie. By Lea 3, with subset J i there is associated an interval of length at ost O n J i /), for i 1,...,k. The data of the instance of the assignent proble, which by Lea 4 ust be solved for each subset J i and the associated interval, can be represented as a graph with a nuber O n J i ) of vertices and O n 2 J i ) of edges. The in-su assignent proble with a nuber V of vertices ) and E of edges is solvable in O V 2 log V + V E tie see [11]) and the inax assignent proble is solvable in O E ) V log V tie as a special case of the bottleneck transportation proble with unit edge capacities, see [12]). Therefore, the optial solution for proble P-su can be found in ) O n J i ) 2 log n J i ) + n J i n 2 J i O n J i ) 2 log n 2 + n 3 J i 2) O n J i ) 2 log n + n 3 J i 2) ) O n 2 J i 2 log n + O n 3 J i 2)) O n 3) k J i 2 O n 5) tie, and for proble P-ax in O n 2 J i ) n J i log n J i ) O n 2 J i ) n J i log n 2 O n 2 ) n log n J i 1.5 O n 2 ) n log n J i 2 O n 4 ) n log n tie, since k J i 2 k 2 i ) J J 2 n 2. The solution procedure for probles P-su and P-ax established by Theore 1 can be extended to the case when for each ob there is given an arbitrary integer release date r r ê ). Notice, that S σ) r ust hold for each ob in any feasible schedule σ. If the set of all obs J is partitioned into disoint subsets using algorith PARTITION, then for each subset J l and ob J l S [t l1, t l2 ], where t l1 ax 0, in i Jl ê i J l / and t l2 ax i Jl ê i + J l /. Observe that r ax i Jl ê i for each J l. Since all the obs fro subset J l can be scheduled in an even narrower interval [ax i Jl ê i,t l2 ], then they can also be scheduled in [t l1, t l2 ]. Thus, it is only necessary to ensure that the starting tie of each ob is equal to or greater than its release date. To do this, the cost atrix c in 2) has to be replaced with c defined as follows: c i+qt l2 t l1 )) M if t l1 + i < r α ê t l1 + i)) if t l1 + i < ê and t l1 + i r β t l1 + i ˆd ) if t l1 + i > ˆd, 7) where M is an arbitrary large nuber, e.g. M n n α + β ). Therefore, the following theore can be stated. Theore 2. Optial solutions for probles P-su and P-ax with arbitrary integer ob release dates r r ê ) can be found in O n 5) and O n 4 n logn ) tie, respectively. 3.2. Miniization of weighted nuber of early and tardy obs. In the following part of the paper we analyze probles of iniizing the total weighted nuber of early and tardy obs. The proble with arbitrary ob weights and the proble with agreeable earliness-tardiness weights are studied separately. We provide properties of optial solutions and optial polynoial tie algoriths to solve the above entioned probles. Lea 5. There exists an optial solution for proble P- nu such that the starting tie S of any ob eets the following condition: ax ê S ê + 1. Lea 6. Let the set of all obs J be partitioned into two subsets J and J, such that J k < n and J n k and ax J ê + k/ in J ê n k)/. There exists an optial solution to proble P-nu in which only obs fro J are scheduled in interval [ax 0, in J ê k/,ax J ê + k/ ]. Proofs of the above provided leas are siilar to ones of Leas 1 and 2 for probles P-su and P-ax. Lea 7. If the set J of all n obs is partitioned using algorith PARTITION into k n disoint subsets J 1,..., J k, such that the obs fro subset J l should be scheduled in interval [t l1, t l2 ], then an optial allocation of obs to the processors and deterination of starting ties for proble P-nu can be 212 Bull. Pol. Ac.: Tech. 573) 2009
Algoriths for parallel processor scheduling with distinct due windows and unit-tie obs obtained by solving a proper in-su assignent proble for each set J l and an associated interval [t l1, t l2 ]. Proof. According to Lea 6, it is possible to partition set J into disoint subsets, and obs that belong to each such subset J l should be scheduled in different interval. Therefore, the obective criterion for proble P-nu can be rewritten as follows: f nu σ) α V σ) + β U σ)) α V σ) + β U σ)) J i under assuption that k J i J and J i J Ø, for i, 1,...,n, i. For each subset J l and an associated interval [t l1, t l2 ] the iniization of the criterion value for proble P-nu can be forulated as follows: i l in : c i x i 8) for i l t l2 t l1 ) and l J l, where α if t l1 + i < ê c i+qtl2 t l1 )) β if t l1 + i > ˆd 9) for 1 i t l2 t l1 and 0 q 1, and where is the -th ob in J l, subect to x i 1, 10) 1 i il 1 l i l x i 1. 11) Value x i should be interpreted as stated in Eq. 6). Theore 3. An optial solution to proble P-nu can be found in tie On 5 ). Proof. Analogous to the proof of Theore 1. As for probles P-su and P-ax, the solution ethod established for proble P-nu can be extended to the case when for each ob there is given an arbitrary integer release date r r ê ). To do this, the cost atrix c in 9) has to be replaced with c defined as follows: c i+qt l2 t l1 )) M if t l1 + i < r α if t l1 + i < ê and t l1 + i r 12) β if t l1 + i > ˆd, where M is an arbitrary large nuber, e.g. M n n α + β ). Thus, the following theore can be stated. Theore 4. An optial solution to proble P-nu with arbitrary integer ob release dates r r ê ) can be found in O n 5) tie. We now pass to consider the special case of proble P-nu with agreeable earliness-tardiness weights, i.e. α i α β i β. This siplified proble will be denoted as PA-nu. In the following part of the paper there are provided additional properties of probles P-nu and PA-nu which allow us to construct a dedicated algorith for proble PA-nu. Lea 8. For each optial solution to proble P-nu the following properties ust be satisfied: a) if α > β then ob cannot be early; b) there do not exist two obs i, for which S i ˆd i, ê 1 S i ˆd 1 and S < ê 1 or S ˆd ; c) there do not exist two obs i, for which S i < ê i 1, S < ê 1 and ê 1 S i ˆd 1. Proof. Case a) Assue that there is an optial schedule σ in which there exists ob such that α > β and S < ê 1. Assue that the schedule σ has been obtained fro σ by setting S σ ) t for any t ˆd such that at least one processor is idle in the interval [t, t+1]. Thus, ob is tardy in schedule σ. Observe that the obective criterion value for the schedule σ is saller than for σ by α β. This contradicts the optiality of the schedule σ. Case b) Assue that there is an optial schedule σ in which there exist two obs i and such that S i ˆd i, S < ê 1 or S ˆd, and ê 1 S i ˆd 1. Assue that the schedule σ has been obtained fro σ by setting S σ ) S i σ) and S i σ ) t for soe t S i σ) or t > S i σ) in case of a single processor). Observe that the penalty incurred by obs i and in the schedule σ is α + β i or β + β i, and in the schedule σ is at ost β i. Thus, σ cannot be optial. Case c) Assue that there is an optial schedule σ in which there exist two obs i and such that S i < ê i 1, S < ê 1 and ê 1 S i ˆd 1. Observe that S < S i in σ. Assue that the schedule σ has been obtained fro σ by setting S σ ) S i σ) and S i σ ) S σ). The penalty incurred by obs i and in the schedule σ is α i + α, and in the schedule σ is α i. This contradicts the optiality of σ. Lea 9. In each optial solution to proble P-nu if S < ê 1 for soe ob then for each ob i scheduled such that ê 1 S i ˆd 1 and ê i 1 S i ˆd i 1 we have α i α. Proof. Assue there is an optial schedule σ in which there exist two obs i and such that S < ê 1, ê 1 S i ˆd 1, ê i 1 S i ˆd i 1 and α i < α. Assue that the schedule σ has been obtained fro σ by setting S i σ ) S σ) and S σ ) S i σ). Observe that the penalty incurred by obs i and in the schedule σ is α, and in the schedule σ is at ost α i. Since α i < α, then σ cannot be optial. Bull. Pol. Ac.: Tech. 573) 2009 213
A. Janiak, W.A. Janiak, and R. Januszkiewicz Lea 10. In each optial solution to proble P-nu if S ˆd 1 for soe ob then for each ob i scheduled such that ê 1 S i ˆd 1 and ê i 1 S i ˆd i 1 we have β i β. Proof. Analogous to the proof of Lea 9. Lea 11. In each optial solution to proble P-nu if S i < ê i 1, S ˆd, S ˆd i, and ê 1 S i ˆd 1 for soe obs i and then β i α i + β. Proof. Assue there is given an optial schedule σ in which exist two obs i and such that S i < ê i 1, S ˆd, S ˆd i, ê 1 S i ˆd 1 and β i < α i +β. Assue that the schedule σ has been obtained fro σ by setting S i σ ) S σ) and S σ ) S i σ). The penalty incurred by obs i and in the schedule σ is α i + β, and in schedule the σ is β i. Since β i < α i + β, then σ cannot be optial. On the basis of the above leas, we will construct an optial polynoial tie algorith for the proble PA-nu. Algorith SCHEDULE input: J the set of obs) 1. Partition J into disoint subsets using algorith PARTITION. 2. In each subset J l sort the obs in a non-increasing order of ax α, β and set L l ax 0, in i Jl ê i J l / and U l ax i Jl ê i + J l /. 3. For each subset J l until it is epty a) Choose fro J l ob with the greatest value of ax α, β and reove fro J l. b) If there exists an interval [t, t+1] such that ê 1 t in ˆd 1, U l in which soe processor is idle, then set S t and go to a). c) If for any interval [t, t+1] there exists ob i such that ê i 1 t ˆd i 1 which can be scheduled in a different interval in which soe processor is idle) without introducing an additional penalty, then set S t, i and go to b) Lea 9 and 10, and agreeable earliness-tardiness weights). d) If for any interval [t, t+1] such that ê 1 t in ˆd 1, U l there exists ob i which is tardy, then set S t, i and go to b) Lea 8, case b). e) If for any interval [t, t+1] such that ê 1 t in ˆd 1, U l there exists ob i which is early and there exists an interval [t, t +1] such that L l t t L l t < t in case of a single processor) in which soe processor is idle, then set S t, i and go to b) Lea 8, case c). f) If α β and there exists an interval [t, t+1] such that L l t < ê 1 in which soe processor is idle, then set S t and go to a) Lea 8, case a). g) Choose any interval [t, t+1] such that ˆd t U l in which soe processor is idle, set S t and go to a). Theore 5. Algorith SCHEDULE solves optially the proble PA-nu in O n 3) tie. Proof. Algorith SCHEDULE is based on the properties established by Lea 6, 8, 9 and 10, and it relies on the condition that α i α β i β for any obs i and, therefore the solutions obtained by the algorith are optial. Assue that the set J was partitioned into k n disoint subsets J 1, J 2,..., J k. The bound on the tie coplexity of the algorith is based on the assuptions that operations in step c) can be done in Oq) tie where q is the nuber of already scheduled obs and that checking if interval [t, t +1] exists in step e) can be done in O1) tie. To satisfy the first assuption, for each ob there ust be stored a nuber of possible allocations of ob to processors within its due window. The nuber has to be updated for each ob whenever soe ob is scheduled in the interval in which soe processor is idle. To fulfill the other assuption, for each t [ ax 0, in i J l ê i J l / ], ax ê i + J l / i J l there ust be stored a nuber of idle intervals for all processors within [ the interval ] ax 0, in ê i J l /, t. i J l The nuber has to be updated for each t whenever soe ob is scheduled in the interval in which soe processor is idle. Based on the assuptions, the ost tie deanding operation in step 3) is: a) finding an interval [t, t+1] which can be done in O n J l ) tie, since the length of the interval in which the obs fro subset J l ust be scheduled is O n J l /) and there are processors, b) updating for each t the nuber of idle intervals for all processors which can be done in O n J l ) tie since the nuber for t is easily calculated based on the nuber for t 1). Assignent of a ob to the processor ay require a change of the assignent for at ost one ob. Observe that the change of the assignent for an already scheduled ob is possible only in cases c), d) and e) of step 3). If the operations of case c) are perfored then the reassignent ust be done in case b) of step 3) in the next iteration of the algorith, hence without further reassignent. On the other hand, if the operations of case d) or e) are perfored then the ob that ust be reassigned is tardy or early, respectively. Thus, the reassignent of this ob ust be perfored is case f) or g) of step 3), hence also without further reassignent. The nuber of operations required to schedule obs in the subset J l can therefore be estiated by O n J l 2). The total coputation tie of algorith SCHEDULE is O n J l 2) O n) J l 2 l1 l1 2 O n) J l ) O n 3). l1 214 Bull. Pol. Ac.: Tech. 573) 2009
Algoriths for parallel processor scheduling with distinct due windows and unit-tie obs 4. Conclusions Probles P-su, P-ax and P-nu of scheduling n unit-tie obs on identical parallel processors have been studied, in which with each ob is associated a distinct due window and cost of earliness and tardiness. The optial solutions to probles P-su and P-nu can be found in On 5 ) tie and to proble P-ax in O n 4 n log n ) tie by solving a polynoial nuber of instances of in-su or in-ax assignent proble. The solution ethods have been extended to the case of obs arbitrary integer release dates. For the special case of proble P-nu with agreeable earliness-tardiness weights there has been presented a dedicated optial algorith with tie coplexity O n 3). Further research ight be focused on developent of heuristic solution algoriths for scheduling probles with distinct due windows and arbitrary processing ties in the parallel processor environent. These algoriths ight be based on the algoriths presented in this paper. REFERENCES [1] J. Sidney, Optial single-achine scheduling with earliness and tardiness penalties, Operations Research 25 1), 62 69 1977). [2] E.L. Lawler, A pseudopolynoial algorith for sequence obs to iniizing total tardiness, Annals of Discrete Matheatics 1, 331 342 1977). [3] C. Koulaas, Single-achine scheduling with tie windows and earliness/tardiness penalties, Eur. J. Operational Research 96, 190 202 1996). [4] G. Wan and B.P.-C. Yen, Tabu search for single achine scheduling with distinct due windows and weighted earliness/tardiness penalties, Eur. J. Operational Research 142, 271 281 2002). [5] C. Koulaas, Maxiizing the weighted nuber of on-tie obs in single achine scheduling with tie windows, Matheatical and Coputer Modelling 25 10), 57 62 1997). [6] W.-S. Yoo and L.A. Martin-Vega, Scheduling single-achine probles for on-tie delivery, Coputers & Indusrial Engineering 39, 371 392 2001). [7] A. Janiak, W.A. Janiak, and M. Marek, Single processor scheduling probles with various odels of a due window, Bull. Pol. Ac.: Tech. 57 1), 95 101 2009). [8] J. Blazewicz, K.H. Ecker, E. Pesch, G. Schidt, and J. Weglarz, Handbook on Scheduling. Fro Theory to Applications, Springer, Berlin, 2007. [9] Peter Brucker, Scheduling Algoriths, Springer, Berlin, 2007. [10] P. Baptiste, P. Brucker, S. Knust, and V. Tikovsky, Ten notes on equal-execution-tie scheduling, 4OR 2, 111 127 2004). [11] M.L. Fredan, R.E. Taran, Fibonacci heaps and their uses in iproved network optiization algoriths, J. ACM 34 3), 596 615 1987). [12] A.P. Punnen and R. Zhang, Bottleneck flows in unit capacity networks, Inforation Processing Letters 109, 334 338 2009). Bull. Pol. Ac.: Tech. 573) 2009 215