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Fully Polynomial Time Approximation Scheme for the Two-Parallel Capacitated Machines Scheduling Problem Under Unavailability Constraint
Fully Polynomial Time Approximation Scheme for the Two-Parallel Capacitated Machines Scheduling Problem Under Unavailability Constraint Imed Kacem ∗
Universite Paul Verlaine - Metz. Address: LITA, UFR M.I.M, Ile du Saulcy 57000, Metz. (e-mail: [email protected])
Abstract: In [Discrete Applied Mathematics, 41, 211-222] Lee and Liman studied the n-job twoparallel machines scheduling problem with the aim of minimizing the total flow-time. In this problem, instead of allowing both machines to be continuously available as is often assumed in the literature, we consider that one of the machines is available for a specified period of time after which it can no longer process any job. They proposed a heuristic that has a worst case error bound of 1/2. In this paper, we establish the existence of a Fully Polynomial Time Approximation Scheme (FPTAS) for the above problem. The complexity of the proposed FPTAS is strongly polynomial. Keywords: Scheduling, analytic approximations, heuristics, algorithms. 1. INTRODUCTION In [Discrete Applied Mathematics, 41, 211-222] Lee and Liman studied the n-job two-parallel machines scheduling problem with the aim of minimizing the total flow-time. In this problem, instead of allowing both machines to be continuously available as is often assumed in the literature, we consider that one of the machines is available for a specified period of time after which it can no longer process any job. They proposed a heuristic that has a worst case error bound of 1/2. In this paper, we establish the existence of a Fully Polynomial Time Approximation Scheme (FPTAS) for the above problem. The complexity of the proposed FPTAS is strongly polynomial. For self-consistency, we recall some necessary definitions related to the approximation area. A ρ-approximation algorithm for a problem of minimizing an objective function ϕ is an algorithm such that for every instance π of the problem it gives a solution Sπ verifying ϕ (Sπ ) /ϕ (OP Tπ ) ≤ ρ where OP Tπ is the optimal solution of π. Also, ρ is called the worst-case bound of the above algorithm. The approximation is tight if ρ is the best possible (i.e., the smallest value we can obtain by the algorithm for all the instances of the problem). A class of (1 + ε)-approximation algorithms is an FPTAS, if its running time is bounded by a polynomial function in 1/ε and the instance size for every ε > 0. A class of (1 + ε)approximation algorithms is a PTAS (Polynomial-Time Approximation Scheme), if its running time is an arbitrary function in 1/ε and the instance size for every ε > 0. Scheduling problems with unavailability constraints have attracted numerous researchers from all the world. This
has been motivated by practical and real industrial problems. It is noteworthy that during the last decade numerous problems of this class have been addressed in the literature (for more details, see the state-of-the-art paper by Schmidt 2000). However, to the best of our knowledge, the studied problem has only been addressed by Lee and Liman. That is why this paper is a good attempt to solve this problem by using a more efficient approximation scheme. The paper is organized as follows. Section 2 describes the problem. Then, a dynamic programming is presented in Section 3. Finally, we prove the existence of an FPTAS for the total completion time minimization problem in Section 4. Finally, Section 5 concludes the paper.
2. FLOW-TIME MINIMIZATION ON CAPACITATED TWO-PARALLEL MACHINES The problem is to schedule n jobs on two-parallel machines, with the aim of minimizing the total (weighted or unweighted) completion time. Every job i has a processing time pi . The first machine is available for a specified period of time [0, T1 ] (i.e., after T1 it can no longer process any job). Every machine can process at most one job at a time. Without loss of generality, we consider that all data are integers and that jobs are indexed according to the SP T rule: p1 ≤ p2 ≤ ... ≤ pn (1)
Due to the dominance of the SP T order, an optimal solution is composed of two sequences (one sequence for
each machine) of jobs scheduled in nondecreasing order of their indexes (Smith 1956). In the remainder of the paper, (Q) denotes the studied problem, F ∗ (Q) denotes the minimal weighted sum of the completion times for problem Q and FS (Q) is the sum of the completion times of schedule S for problem Q. Proposition 1. If ni=1 pi ≤ 2T1 , then problem (Q) has an FPTAS. Proof. We relax the unavailability constraint (i.e., we assume that the first machine is continuously available). Then, the relaxed problem has an FPTAS according to Sahni (1976). Let σ1 be the obtained schedule by applying the FPTAS by Sahni for the relaxed problem, B1′ be the completion time of the last job scheduled on the first machine and B2′ denote the completion time of the last job scheduled on the second machine. By assumption, B1′ + B2′ ≤ 2T1 . Hence, either B1′ ≤ T1 or B2′ ≤ T1 must hold. If B1′ ≤ T1 , then σ 1 is also (1+ε)−approximation for the original problem (Q). If B2′ ≤ T1 , then by exchanging the role of the two machines, we obtain a new (1 + ε)−approximation schedule σ ′1 that is also feasible for the original problem (Q) and this completes the proof. Based on the result of Proposition 1, we only consider the case where n pi > 2T1 (2) i=1
3. DYNAMIC PROGRAMMING ALGORITHM The problem can be optimally solved by applying the following standard dynamic programming algorithm B. This algorithm generates iteratively some sets of states. At every iteration k, a set Uk composed of states is generated (0 ≤ k ≤ n). Each state [t, f ] in Uk can be associated to a feasible schedule for the first k jobs. Variable t denotes the completion time of the last job scheduled on the first machine before T1 and f is the total flow-time of the corresponding schedule. This algorithm can be described as follows: Algorithm B (i). Set U0 = {[0, 0]}. (ii). For k ∈ {1, 2, ..., n}, For every [t, f ] in Uk−1 state : k 1) Put t, f + p − t in Uk i=1 i 2) Put [t + pk , f + (t + pk )] in Uk if t + pk ≤ T1 Remove Uk−1 (iii). F ∗ (Q) = min[t,f ]∈Un {f}.
the second step of our FPTAS, we modify the execution of algorithm B in order to reduce the running time. The main idea is to remove a special part of the states generated by the algorithm. Therefore, the modified algorithm becomes faster and yields an approximate solution instead of the optimal schedule. Hence, we have to take care when removing such states so that the approximation will be of a good quality. The approach of modifying the execution of an exact algorithm to design FPTAS, was initially proposed by Ibarra and Kim (1975) for solving the knapsack problem. It is noteworthy that during the last decades numerous scheduling problems have been addressed by applying such an approach. A sample of these papers includes Gen and Levner (1981), Kacem (2009-2010), Sahni (1976), Kovalyov and Kubiak (1999), Kellerer and Strusevich (2006) and Woeginger (2005). Given an arbitrary ε > 0, we define 2FH (Q) LB = , (3) 3 3n , (4) q1 = ε 2 n q2 = , (5) ε FH (Q) δ1 = (6) q1 and T1 δ2 = . q2 We split the interval [0, FH (Q)] into q1 equal subintervals Ir1 = [(r − 1)δ 1 , rδ 1 ]1≤r≤q1 of length δ 1 . We also split the interval [0, T1 ] into q2 equal subintervals Is2 = [(s − 1)δ 2 , sδ 2 ]1≤s≤q2 of length δ 2 . Our algorithm Bε′ generates reduced sets Uk# instead of sets Uk . The algorithm can be described as follows: Algorithm Bε′ (i). Set U1# = {[0, 0]}. (ii). For k ∈ {1, 2, 3, ..., n}, # For every state [t, f] in Uk−1 : k 1) Put t, f + ( i=1 pi − t) in Uk#
2) Put [t + pk , f + (t + pk )] in Uk# if t + pk ≤ T1 # Remove Uk−1 Let [t, f ]r,s be the state in Uk# such that f ∈ Ir1 and t ∈ Is2 with the smallest possible t (ties are broken by choosing the state of the smallest
f). # Set U = [t, f ] |1 ≤ r ≤ q , 1 ≤ s ≤ q 1 2 . The analysis of this dynamic algorithm shows that its k r,s complexity can be reduced to O (nT1 ). (iii). FBε′ (Q) = min[t,f ]∈Un# {f }. 4. FPTAS
Here, we are interested in the existence of an FPTAS for this problem. We present a new algorithm for this problem based on trimming technique. The algorithm starts by applying Lee’s heuristic H to obtain a feasible solution. In
The worst-case analysis of this algorithm and the comparison of the execution of algorithms B and Bε′ allow us to deduce the following theorem. The details of the proof will be presented at the conference.
Theorem 2. Algorithm Bε′ is an FPTAS and it can be implemented in O(n3 /ε2 ) time.
5. CONCLUSION In this paper we deal with a scheduling problem and proposes an FPTAS for solving it. Such a problem consists in sequencing n-job two-parallel machines scheduling problem with the aim of minimizing the total flow-time. In this problem, instead of allowing both machines to be continuously available as is often assumed in the literature, we consider that one of the machines is available for a specified period of time after which it can no longer process any job. In this paper, we establish the existence of an FPTAS for the weighted version of the above problem. The complexity of the proposed FPTAS is strongly polynomial. Moreover, our approach can be used for Min/Max criteria (like the makespan, see Kacem (2009)) where the other approach cannot handle this type of criteria. In our future works, we hope to extend these results to other variants of this problem. The development of better approximation algorithms is also a challenging subject.
ACKNOWLEDGEMENTS This work has been carried out when the author was with Troyes University of Technology (Institut Charles Delaunay, OSI Team). Sponsor and financial support acknowledgment goes here to the Regional Council of Champagne Ardenne, France (Project OCIDI from 2006 to 2009).
REFERENCES Gens, G.V., Levner, E.V., (1981) Fast approximation algorithms for job sequencing with deadlines. Discrete Applied Mathematics 3, 313—318. Ibarra, O., Kim, C.E., (1975) Fast approximation algorithms for the knapsack and sum of subset problems. Journal of the ACM 22, 463—468. Kacem, I. (2009) Approximation algorithms for the makespan minimization with positive tails on a single machine with a fixed non-availability interval. Journal of Combinatorial Optimization 2009, 17:2, 117-133. Kacem, I. (2010) Fully Polynomial-Time Approximation Scheme for the Weighted Total Tardiness Minimization with a Common Due Date. Discrete Applied Mathematics, 2010, doi: 10.1016/j.dam.2010.01.013 Kellerer, H., Strusevich, V.A., (2006) A fully polynomial approximation scheme for the single machine weighted total tardiness problem with a common due date. Theoretical Computer Science 369: 230-238. Kovalyov, M.Y., Kubiak, W., (1999) A fully polynomial approximation scheme for weighted earliness-tardiness problem. Operations Research 47: 757-761.
Lee, C.Y., Liman, S.D., (1993) Capacitated two-parallel machines sceduling to minimize sum of job completion times. Discrete Applied Mathematics 41, 211-222. Sahni, S., (1976) Algorithms for scheduling independent tasks. Journal of the ACM 23, 116—127. Schmidt, G., (2000) Scheduling with limited machine availability. European Journal of Operational Research 121, 1-15. Smith, W.E., (1956) Various optimizers for single stage production. Naval Research Logistics Quarterly 3, 5966. Woeginger, G.J., (2005) A comment on scheduling two machines with capacity constraints. Discrete Optimization 2, 269—272.
Universite Paul Verlaine - Metz. Address: LITA, UFR M.I.M, Ile du Saulcy 57000, Metz. (e-mail: [email protected])
Abstract: In [Discrete Applied Mathematics, 41, 211-222] Lee and Liman studied the n-job twoparallel machines scheduling problem with the aim of minimizing the total flow-time. In this problem, instead of allowing both machines to be continuously available as is often assumed in the literature, we consider that one of the machines is available for a specified period of time after which it can no longer process any job. They proposed a heuristic that has a worst case error bound of 1/2. In this paper, we establish the existence of a Fully Polynomial Time Approximation Scheme (FPTAS) for the above problem. The complexity of the proposed FPTAS is strongly polynomial. Keywords: Scheduling, analytic approximations, heuristics, algorithms. 1. INTRODUCTION In [Discrete Applied Mathematics, 41, 211-222] Lee and Liman studied the n-job two-parallel machines scheduling problem with the aim of minimizing the total flow-time. In this problem, instead of allowing both machines to be continuously available as is often assumed in the literature, we consider that one of the machines is available for a specified period of time after which it can no longer process any job. They proposed a heuristic that has a worst case error bound of 1/2. In this paper, we establish the existence of a Fully Polynomial Time Approximation Scheme (FPTAS) for the above problem. The complexity of the proposed FPTAS is strongly polynomial. For self-consistency, we recall some necessary definitions related to the approximation area. A ρ-approximation algorithm for a problem of minimizing an objective function ϕ is an algorithm such that for every instance π of the problem it gives a solution Sπ verifying ϕ (Sπ ) /ϕ (OP Tπ ) ≤ ρ where OP Tπ is the optimal solution of π. Also, ρ is called the worst-case bound of the above algorithm. The approximation is tight if ρ is the best possible (i.e., the smallest value we can obtain by the algorithm for all the instances of the problem). A class of (1 + ε)-approximation algorithms is an FPTAS, if its running time is bounded by a polynomial function in 1/ε and the instance size for every ε > 0. A class of (1 + ε)approximation algorithms is a PTAS (Polynomial-Time Approximation Scheme), if its running time is an arbitrary function in 1/ε and the instance size for every ε > 0. Scheduling problems with unavailability constraints have attracted numerous researchers from all the world. This
has been motivated by practical and real industrial problems. It is noteworthy that during the last decade numerous problems of this class have been addressed in the literature (for more details, see the state-of-the-art paper by Schmidt 2000). However, to the best of our knowledge, the studied problem has only been addressed by Lee and Liman. That is why this paper is a good attempt to solve this problem by using a more efficient approximation scheme. The paper is organized as follows. Section 2 describes the problem. Then, a dynamic programming is presented in Section 3. Finally, we prove the existence of an FPTAS for the total completion time minimization problem in Section 4. Finally, Section 5 concludes the paper.
2. FLOW-TIME MINIMIZATION ON CAPACITATED TWO-PARALLEL MACHINES The problem is to schedule n jobs on two-parallel machines, with the aim of minimizing the total (weighted or unweighted) completion time. Every job i has a processing time pi . The first machine is available for a specified period of time [0, T1 ] (i.e., after T1 it can no longer process any job). Every machine can process at most one job at a time. Without loss of generality, we consider that all data are integers and that jobs are indexed according to the SP T rule: p1 ≤ p2 ≤ ... ≤ pn (1)
Due to the dominance of the SP T order, an optimal solution is composed of two sequences (one sequence for
each machine) of jobs scheduled in nondecreasing order of their indexes (Smith 1956). In the remainder of the paper, (Q) denotes the studied problem, F ∗ (Q) denotes the minimal weighted sum of the completion times for problem Q and FS (Q) is the sum of the completion times of schedule S for problem Q. Proposition 1. If ni=1 pi ≤ 2T1 , then problem (Q) has an FPTAS. Proof. We relax the unavailability constraint (i.e., we assume that the first machine is continuously available). Then, the relaxed problem has an FPTAS according to Sahni (1976). Let σ1 be the obtained schedule by applying the FPTAS by Sahni for the relaxed problem, B1′ be the completion time of the last job scheduled on the first machine and B2′ denote the completion time of the last job scheduled on the second machine. By assumption, B1′ + B2′ ≤ 2T1 . Hence, either B1′ ≤ T1 or B2′ ≤ T1 must hold. If B1′ ≤ T1 , then σ 1 is also (1+ε)−approximation for the original problem (Q). If B2′ ≤ T1 , then by exchanging the role of the two machines, we obtain a new (1 + ε)−approximation schedule σ ′1 that is also feasible for the original problem (Q) and this completes the proof. Based on the result of Proposition 1, we only consider the case where n pi > 2T1 (2) i=1
3. DYNAMIC PROGRAMMING ALGORITHM The problem can be optimally solved by applying the following standard dynamic programming algorithm B. This algorithm generates iteratively some sets of states. At every iteration k, a set Uk composed of states is generated (0 ≤ k ≤ n). Each state [t, f ] in Uk can be associated to a feasible schedule for the first k jobs. Variable t denotes the completion time of the last job scheduled on the first machine before T1 and f is the total flow-time of the corresponding schedule. This algorithm can be described as follows: Algorithm B (i). Set U0 = {[0, 0]}. (ii). For k ∈ {1, 2, ..., n}, For every [t, f ] in Uk−1 state : k 1) Put t, f + p − t in Uk i=1 i 2) Put [t + pk , f + (t + pk )] in Uk if t + pk ≤ T1 Remove Uk−1 (iii). F ∗ (Q) = min[t,f ]∈Un {f}.
the second step of our FPTAS, we modify the execution of algorithm B in order to reduce the running time. The main idea is to remove a special part of the states generated by the algorithm. Therefore, the modified algorithm becomes faster and yields an approximate solution instead of the optimal schedule. Hence, we have to take care when removing such states so that the approximation will be of a good quality. The approach of modifying the execution of an exact algorithm to design FPTAS, was initially proposed by Ibarra and Kim (1975) for solving the knapsack problem. It is noteworthy that during the last decades numerous scheduling problems have been addressed by applying such an approach. A sample of these papers includes Gen and Levner (1981), Kacem (2009-2010), Sahni (1976), Kovalyov and Kubiak (1999), Kellerer and Strusevich (2006) and Woeginger (2005). Given an arbitrary ε > 0, we define 2FH (Q) LB = , (3) 3 3n , (4) q1 = ε 2 n q2 = , (5) ε FH (Q) δ1 = (6) q1 and T1 δ2 = . q2 We split the interval [0, FH (Q)] into q1 equal subintervals Ir1 = [(r − 1)δ 1 , rδ 1 ]1≤r≤q1 of length δ 1 . We also split the interval [0, T1 ] into q2 equal subintervals Is2 = [(s − 1)δ 2 , sδ 2 ]1≤s≤q2 of length δ 2 . Our algorithm Bε′ generates reduced sets Uk# instead of sets Uk . The algorithm can be described as follows: Algorithm Bε′ (i). Set U1# = {[0, 0]}. (ii). For k ∈ {1, 2, 3, ..., n}, # For every state [t, f] in Uk−1 : k 1) Put t, f + ( i=1 pi − t) in Uk#
2) Put [t + pk , f + (t + pk )] in Uk# if t + pk ≤ T1 # Remove Uk−1 Let [t, f ]r,s be the state in Uk# such that f ∈ Ir1 and t ∈ Is2 with the smallest possible t (ties are broken by choosing the state of the smallest
f). # Set U = [t, f ] |1 ≤ r ≤ q , 1 ≤ s ≤ q 1 2 . The analysis of this dynamic algorithm shows that its k r,s complexity can be reduced to O (nT1 ). (iii). FBε′ (Q) = min[t,f ]∈Un# {f }. 4. FPTAS
Here, we are interested in the existence of an FPTAS for this problem. We present a new algorithm for this problem based on trimming technique. The algorithm starts by applying Lee’s heuristic H to obtain a feasible solution. In
The worst-case analysis of this algorithm and the comparison of the execution of algorithms B and Bε′ allow us to deduce the following theorem. The details of the proof will be presented at the conference.
Theorem 2. Algorithm Bε′ is an FPTAS and it can be implemented in O(n3 /ε2 ) time.
5. CONCLUSION In this paper we deal with a scheduling problem and proposes an FPTAS for solving it. Such a problem consists in sequencing n-job two-parallel machines scheduling problem with the aim of minimizing the total flow-time. In this problem, instead of allowing both machines to be continuously available as is often assumed in the literature, we consider that one of the machines is available for a specified period of time after which it can no longer process any job. In this paper, we establish the existence of an FPTAS for the weighted version of the above problem. The complexity of the proposed FPTAS is strongly polynomial. Moreover, our approach can be used for Min/Max criteria (like the makespan, see Kacem (2009)) where the other approach cannot handle this type of criteria. In our future works, we hope to extend these results to other variants of this problem. The development of better approximation algorithms is also a challenging subject.
ACKNOWLEDGEMENTS This work has been carried out when the author was with Troyes University of Technology (Institut Charles Delaunay, OSI Team). Sponsor and financial support acknowledgment goes here to the Regional Council of Champagne Ardenne, France (Project OCIDI from 2006 to 2009).
REFERENCES Gens, G.V., Levner, E.V., (1981) Fast approximation algorithms for job sequencing with deadlines. Discrete Applied Mathematics 3, 313—318. Ibarra, O., Kim, C.E., (1975) Fast approximation algorithms for the knapsack and sum of subset problems. Journal of the ACM 22, 463—468. Kacem, I. (2009) Approximation algorithms for the makespan minimization with positive tails on a single machine with a fixed non-availability interval. Journal of Combinatorial Optimization 2009, 17:2, 117-133. Kacem, I. (2010) Fully Polynomial-Time Approximation Scheme for the Weighted Total Tardiness Minimization with a Common Due Date. Discrete Applied Mathematics, 2010, doi: 10.1016/j.dam.2010.01.013 Kellerer, H., Strusevich, V.A., (2006) A fully polynomial approximation scheme for the single machine weighted total tardiness problem with a common due date. Theoretical Computer Science 369: 230-238. Kovalyov, M.Y., Kubiak, W., (1999) A fully polynomial approximation scheme for weighted earliness-tardiness problem. Operations Research 47: 757-761.
Lee, C.Y., Liman, S.D., (1993) Capacitated two-parallel machines sceduling to minimize sum of job completion times. Discrete Applied Mathematics 41, 211-222. Sahni, S., (1976) Algorithms for scheduling independent tasks. Journal of the ACM 23, 116—127. Schmidt, G., (2000) Scheduling with limited machine availability. European Journal of Operational Research 121, 1-15. Smith, W.E., (1956) Various optimizers for single stage production. Naval Research Logistics Quarterly 3, 5966. Woeginger, G.J., (2005) A comment on scheduling two machines with capacity constraints. Discrete Optimization 2, 269—272.