Improved approximation for non-preemptive single machine ow-time scheduling with an availability constraint

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1 Improved approximation for non-preemptive single machine ow-time scheduling with an availability constraint Joachim Breit Department of Information and Technology Management, Saarland University, Saarbrücken, Germany Abstract We study the problem of scheduling n non-preemptable jobs on a single machine which is not available for processing during a given time period. The objective is to minimize the sum of the job completion times. The best known approximation algorithm for this NP-hard problem has a relative worst-case error bound of 17.6%. We present a parametric O(n log n)-algorithm H with which better worst-case error bounds can be obtained. The best error bound calculated for the algorithm in the paper is 7.4%. In a computational experiment we test the algorithm with the performance guarantee set to 10.2%. It turns out that randomly generated instances with up to 1000 jobs can be solved with a mean (maximum) error of 0.31% (3.18%) and a mean (maximum) computation time of 0.8 (9.7) seconds. Keywords: Single machine scheduling; Approximation algorithms; Availability constraints. 1 Introduction In classical scheduling models it is usually assumed that machines are continuously available for processing throughout the planning period. This assumption is not justied in situations Address: Saarland University, Department of Information and Technology Management, P.O. Box , D Saarbrücken, Germany. address: jb@itm.uni-sb.de (J. Breit). Tel.: +49(681) Fax: +49(681)

2 where maintenance requirements, rest periods or breakdowns have to be considered. This motivates the investigation of generalized scheduling models in which machine availability may be restricted. In the past years such models have received considerable attention from researchers. Schmidt [7], Lee [3] and Braun et al. [2] provide surveys of this area of research. In the problem studied in this paper we are given a set N = {J 1,..., J n } of jobs to be processed on a single machine. The processing time of job J j, j = 1,..., n, is denoted by p j. The machine cannot process more than one job at a time and is assumed not to be available for processing during a given time interval [s, t] where = t s > 0. For convenience we will call this interval of non-availability the hole in the following. Job preemption is not allowed. A solution for the problem is a schedule S which assigns a completion time C j,s and a starting time C j,s p j to each job J j. The sum of the job completion times (or ow-time) of schedule S is denoted by C(S) = j C j,s. An optimal schedule S has a minimal ow-time of C. It is not dicult to see that jobs with zero processing time (p j = 0) can be optimally scheduled at time 0. We therefore assume throughout the paper that processing times are strictly positive. The objective function value obtained from an algorithm H is C(H). We say that algorithm H has a relative worst-case error bound of α if ρ(h) = (C(H) C )/C α holds for any instance of the problem. Adiri et al. [1] show that our problem is NP-hard. Lee and Liman [4] provide a shorter NP-hardness proof and show that the shortest processing time rule (SPT) which sequences the jobs in non-decreasing order of their processing times has a tight worst-case error bound of 2/7. In [5] Sad et al. propose a pseudo-polynomial dynamic programming algorithm with time complexity O(ns) to solve the problem exactly. The eciency of the algorithm is improved by a xation of variables technique. An improved approximation algorithm MSPT (modied SPT) with a worst-case error bound of 3/17 is provided by Sad et al. in [6]. We adopt the notation introduced in the aforementioned papers and denote by B (A) the set of jobs scheduled before (after) the hole in the SPT schedule S SP T. The MSPT algorithm rst generates schedule S SP T. Then it tries to improve the result by enumerating schedules which can be obtained by a single exchange of one job from set B with one job from set A. In each schedule jobs before and after the hole, respectively, appear in nondecreasing order of their processing times. Clearly, the maximum number of tentative schedules that the MSPT algorithm has to examine is B A (n/2) 2 which would result in a time complexity of O(n 2 ). However, Sad et al. showed that it is possible to decrease 2

3 the number of tentative schedules considerably. In the computational experiment reported in [5] an improved version of the MSPT algorithm with time complexity O(n log n) was used. This version of the algorithm required an average computation time of only about 3 seconds on a personal computer for problem instances with jobs. The objective of this research is to provide approximation algorithms with performance guarantee better than the MSPT algorithm. In order to present our results we introduce further notation: J [i],s - job scheduled at position i in schedule S C [i],s - completion time of job J [i],s p [i],s - processing time of job J [i],s J[i] - job scheduled at position i in schedule S C[i] - completion time of job J[i] Cj - completion time of job J j in schedule S p [i] - processing time of job J[i] X - set of the rst B jobs in schedule S Y - set of the last A jobs in schedule S We assume that jobs are generally started as early as possible. Under this assumption let δ (δ ) denote the idle time immediately before the hole in schedule S SP T (S ). In [4] it is shown that δ δ. By B (A ) we denote the set of jobs scheduled before (after) the hole in the optimal solution. In the following section we will present our approximation algorithm and analyze its worst-case performance. The section is divided into two subsections. In the rst subsection we discuss the properties of a schedule S k which is important for our algorithm. In the second subsection we state the approximation algorithm and derive its worst-case bound based on the results of the rst subsection. In Section 3 we report on the results of a computational experiment. Section 4 nishes the paper with a summary. 2 Approximation algorithm and worst-case analysis 2.1 Schedule S k and its properties Let Z k = { J[ B k+1], J [ B k+2],..., J [ B ]} be the set of the last k jobs processed before the hole in the optimal solution, k 1. We denote by S k a schedule with the following structure: 3

4 The jobs of set Z k are processed in SPT order immediately before the hole. The jobs of set N \ Z k are processed before the jobs of set Z k and after the hole. Proceeding according to the SPT-ordered list of these jobs we schedule as many of them as possible at the beginning of the schedule before job set Z k and the rest after the hole. In schedule S k the set of jobs processed before (after) the hole is denoted by B k (A k ). The idle time immediately before the hole in schedule S k is denoted by δ k. In the following we present four lemmas and a corollary which contain basic properties of schedule S k. Lemma 1. δ k δ. Proof. Consider schedule S k in which the last k jobs before the hole are the same as in schedule S. We distinguish between two cases. (i) Suppose rst that B k = B. As we use the SPT-ordered list of the jobs in set N \Z k to select the jobs appearing at positions 1,..., B k k in schedule S k it is clear that C [j],sk C[j] for j = 1,..., B k k. Jobs at positions B k k + 1,..., B k are the same in both schedules S k and S. Thus we have C [j],sk C[j] for j = B k k + 1,..., B k as well. The completion time of the jobs appearing at positions j = B k + 1,..., n in schedules S k and S are given by C [j],sk = B k k i=1 p [i],sk + i Z k p i + δ k + + j i= B k +1 p [i],sk, (1) C [j] = B k k i=1 p [i] + i Z k p i + δ + + j i= B k +1 Because in schedule S k jobs of set N \ Z k are scheduled using the SPT rule we have p [i]. B k k i=1 p [i],sk + j i= B k +1 p [i],sk B k k i=1 p [i] + j p [i] i= B k +1 and thus C [j],sk C [j] δ k δ j = B k + 1,..., n. (2) It follows that the lemma holds, for otherwise we have C(S k ) < C, a contradiction. 4

5 (ii) Now suppose that B k = B + l where l 1. Let us rst compare the ow-time of the rst B k jobs in schedules S k and S. Suppose that we must schedule the jobs of set Z k in SPT order immediately before the hole. For schedule S k we selected the shortest B k k jobs besides the jobs of set Z k which is obviously the best choice we can make. We can either schedule any of these jobs before the jobs of set Z k or after the hole. It is not dicult to see that scheduling all of them in SPT order before job set Z k is optimal. Hence, among all possibilities to schedule the jobs of set Z k and in addition B k k jobs from set N \ Z k under the restriction that job set Z k is placed immediately before the hole, schedule S k is the best. It follows that B k i=1 C [i],sk B k i=1 C [i]. Consider now the jobs at positions j = B k +1,..., n in schedules S k and S. Equation (1) still holds. The completion times of the last A k jobs in schedule S are given by C [j] = B k k l i=1 p [i] + i Z k p i + δ + + B k i= B k l+1 p [i] + j i= B k +1 Because in schedule S k jobs of set N \ Z k are scheduled using the SPT rule we have p [i]. B k k i=1 p [i],sk + j i= B k +1 p [i],sk B k k l i=1 p [i] + B k i= B k l+1 p [i] + j i= B k +1 p [i]. Inequality (2) holds thus for case (ii) as well. otherwise we have C(S k ) < C, a contradiction. It follows that the lemma holds for In the next lemma we derive a bound on the absolute error of schedule S k. The proof uses Lemma 1. Lemma 2. The absolute error of schedule S k is given by C(S k ) C ( A k k 1) (δ k δ ). (3) Proof. Because the last k jobs before the hole are the same in schedules S k and S, and δ k δ, each of these jobs nishes δ k δ time units later in schedule S than in schedule S k. The same is obviously true for the job preceding the rst job of set Z k. We may assume that this job exists because otherwise schedule S k is clearly optimal. Let us again distinguish between the two cases B k = B and B k > B. 5

6 (i) Suppose that B k = B. We have C [j],sk C [j] 0 j = 1,..., B k k 1, (4) C [j],sk C [j] = (δ k δ ) j = B k k,..., B k. (5) Integrating (2), (4) and (5) we obtain (3). (ii) Suppose now that B k = B + l, l 1. In this case we have C [j],sk C [j] 0 j = 1,..., B k k l 1, (6) C [j],sk C [j+k+1] 0 j = B k k l,..., B k k 1, (7) C [j],sk C [j l] = (δ k δ ) j = B k k,..., B k. (8) Integrating (2), (6), (7), (8) we obtain (3). The following lemma gives a lower bound on the minimal ow-time based on Lemma 1 and the structure of schedule S k. Let z be an integer and, for convenience, dene the function ω(z) = z(z + 1)/2. Lemma 3. If schedules S SP T and S k are not optimal then a lower bound on the minimal ow-time is given by C (ω( A k ) + 4) (δ k δ ). (9) Proof. All jobs processed after the hole in schedule S k must have a processing time greater than the idle time before the hole, i.e. p j > δ k δ k δ, J j A k. It follows that j A k C j > ω( A k ) (δ k δ ). It is not dicult to see that the SPT order for the jobs in sets A and B is dominant. Assume that the last job before the hole in schedule S belongs to set B. Then, due to the dominance of the SPT order for set B, all jobs in set B must belong to set B and all jobs of set A are processed after the hole in the optimal schedule. This implies that schedule S SP T is optimal. Hence, if schedule S SP T is not optimal, then the job processed immediately before the hole in schedules S and S k must belong to set A. For the same reason we may assume that the rst job after the hole in schedules S S k belongs to set B. Then we have p [ Bk ],S k p [ Bk +1],S k > δ k δ and, using Lemma 1, t > C [ B ] = C [ B k ],S k +(δ k δ ) p [ Bk ],S k +(δ k δ ) > 2(δ k δ ). Notice that by denition and 6

7 we have J [ B ] / A k. Under the condition of the lemma, nally, we may assume that one job of set B \ Z k is processed after the hole in schedule S, i.e. J l B k \ Z k : C l > 2(δ k δ ). It follows that inequality (9) holds. Our next lemma is a generalization of two results in [4] and [6]. Lemma 4. If B = B + l, l 0, then ρ(spt) ( A 1 l) (δ δ ) (ω( A ) + 1) (δ δ ) = W C1( A, l). (10) Proof. Lee and Liman [4] proved that δ δ and C [j],ssp T C [j] δ δ, j = B +1,..., n. We focus now on the rst B jobs in schedules S SP T and S. If l jobs of set X are processed after the hole in schedule S then the last job scheduled before the hole in schedule S is job J[ B l]. We have C [ B l] C [ B ],S SP T = δ δ. As C[ B l+i] > C[ B l], i = 1,..., l, and C [ B i],ssp T < C [ B ],SSP T, i = 1,..., l, we have C [j],ssp T C [j] (δ δ ), j = B l,..., B. Also, due to the SPT order in schedule S SP T we have C [j],ssp T C [j] 0, j = 1,..., B l 1. It follows that the maximum absolute error of schedule S SP T is given by C(S SP T ) C ( A 1 l) (δ δ ). In [4] it is shown that C (ω( A )+1) (δ δ ). Integrating the two bounds we obtain (10). The following corollary simply combines lemmas 2 and 3. Corollary 1. If schedules S SP T and S k are not optimal then the relative worst-case error bound of schedule S k is given by C(S k ) C C A k k 1 ω( A k ) + 4 = W C2( A k, k). (11) 2.2 Algorithm H and its performance guarantee In order to state our approximation algorithm we have to introduce two parameters a max (k) and l max (k) both of which depend on k only and can be computed a priori. Parameter a max (k): For a given value k we can easily compute the maximum value that function W C2( A k, k) can attain. In rows 2 and 3 of Table 1 we have listed these maximum values for k = 1,..., 5 as fractions and as decimal numbers, respectively. Row 4 shows for which value A k the maximum is attained. For example, if k = 3 the function W C2( A k, 3) attains its maximum value of 5/ for A k = 9. Following [4] we have ρ(spt) ( A 1)/(ω( A ) + 1) = W C3( A ) where W C3( A ) 0 as A. For 7

8 Table 1: Worst-case bounds of schedule S k depending on k. and parameter values a max (k) and l max (k) k max Ak W C2( A k, k) 4/25 1/8 5/49 3/35 7/95 max Ak W C2( A k, k) A k a max (k) l max (k) any given value k we can thus determine a minimum value a max (k) such that ρ(spt ) W C3( A ) W C2( A k, k) holds if A > a max (k). In Table 1 the values a max (k) are listed in row 5. As an example, the value a max (3) = 17 indicates that the SPT-rule guarantees a worst-case bound of 10.21% or better if A > 17. In cases where A > a max (k) holds schedule S SP T has a worst-case performance guarantee at least as good as schedule S k. Parameter l max (k): A second special case of the problem in which schedule S SP T has a better worst-case performance guarantee than schedule S k can be deduced from Lemma 4. Let again l = B B be the number of jobs in set X which are processed after the hole in schedule S. For any given value k we can compute a minimum value l max (k) such that ρ(spt) W C1( A, l) W C2( A k, k) holds if l > l max (k). In Table 1 the values l max (k) are listed in row 6. The value l max (3) = 3 for example indicates that if k = 3 and l = B B > 3 then the SPT-rule guarantees a worst-case bound of 10.21% or better. In cases where l = B B > l max (k) holds schedule S SP T has a worst-case performance guarantee at least as good as schedule S k. We will now explain the basic idea of our approximation algorithm H. The algorithm is called with a specic accuracy parameter value k as its input. The aspired performance guarantee then is max Ak W C2( A k, k); see Table 1 for concrete values. As we have seen at the beginning of this subsection there are cases where the SPT rule guarantees a worst-case bound at least as good as max Ak W C2( A k, k). The algorithm therefore rst generates schedule S SP T. The remaining part of the algorithm only has to deal with cases in which the SPT schedule may have a performance worse than max Ak W C2( A k, k). In this part of the algorithm we try to nd schedules S 1,..., S k by full enumeration. Schedule S k will guarantee the desired worst-case bound of max Ak W C2( A k, k) for the case B > k. For the case B k one of the schedules S 1,..., S k 1 will be an optimal schedule. 8

9 We are now ready to state the algorithm and dene Q = { A { J [ B lmax(k) k+1],s SP T,..., J [ B ],SSP T }}, the set of the last A + l max (k) + k jobs in schedule S SP T. Algorithm H. Input: Accuracy parameter k. Output: Schedule S H. Step 1. Generate schedule S SP T. If A > a max (k) then go to step 3. Step 2. For q := 1 to k do begin For all subsets Z of set Q with Z = q do begin Generate a tentative schedule with the following structure: (1) Jobs of set Z are processed in SPT order immediately before the hole. (2) Jobs of set N \ Z are processed before the jobs of set Z and after the hole. Proceeding according to the SPT-ordered list of these jobs we schedule as many of them as possible at the beginning of the schedule before job set Z and the rest after the hole. end. end. Step 3. Find the best among all generated feasible schedules and call this schedule S H. end. The structure of schedules S SP T, S k and S is depicted in Figure 1. Figure 1: Structure of schedules S SP T, S k and S 9

10 In the following we will analyze the worst-case performance of algorithm H. We want to prove the following theorem. Theorem 1. Algorithm H guarantees a worst-case error bound of ρ(s H ) A k k 1 ω( A k ) + 4 = W C2( A k, k). (12) Proof. We have to show that bound (12) is either guaranteed by schedule S SP T or by step 2 of algorithm H. By denition of parameters a max (k) and l max (k) schedule S SP T guarantees bound (12) if A > a max (k) or B B > l max (k). For the following analysis of step 2 we may thus restrict attention to those situations where the following two conditions hold: A a max (k), (13) B B l max (k). (14) Notice that bound (12) is identical with the bound presented in Corollary 1 for schedule S k. Hence, if we show that under conditions (13) and (14) schedule S k or a better schedule is found in step 2 we are done. Suppose rst that l max (k) + k B and thus Q = N. If B k then schedule S k exists. Because we obviously have Z k N, algorithm H nds set Z k in the k-th passage of the outer loop of step 2 and thus generates schedule S k. Using Corollary 1 we have that bound (12) holds. Now assume that B < k. In this case algorithm H nds set Z B = B in the passage q = B of the outer loop of step 2 and thus generates an optimal schedule. In the following we may assume that l max (k) + k < B, (15) and thus Q N. Suppose rst that B k and schedule S k exists. If we show that one of the tentative sets Z generated in the inner loop of step 2 is set Z k, then we have shown that schedule S k is found by the algorithm. This task is obviously accomplished if we show that Z k Q holds. Aiming for contradiction suppose that Z k Q does not hold. Then, at least one of the last k positions before the hole in schedule S is taken by a job of set N \ Q. We want to demonstrate that under this condition a minimum number of jobs in set X is processed after the hole in the optimal schedule. Thus, assume 10

11 that as many jobs of set N \ Q as possible are processed before the hole. This is the case if J[i] = J [i],ssp T, i = 1,..., B l max (k) k = B k + 1. This implies that B B = l max (k) + 1, a contradiction to (14). Finally, suppose that B < k. Using (15) this implies l max (k) + B < B, again a contradiction to (14). The time complexity of step 1 of algorithm H is O(n log n). Step 2 of algorithm H is only carried out if A a max (k). As parameters l max (k) and a max (k) are constant for a given k, the number of jobs in set Q is constant as well. Hence, the number of tentative schedules enumerated in step 2 is constant. Each tentative schedule can be generated in O(n log n) time. Thus, the running time of algorithm H for a given k is O(n log n). 3 Computational experiment The previous section dealt with the theoretical worst-case behaviour of algorithm H. In this section we want to quantify the error and computation time of algorithm H using experimental analysis and draw a comparison to the algorithms SPT and MSPT. The data set for the experiment was generated almost in the same way as the data set for the experiment presented in [6]. Job processing times were random integers from a uniform distribution [1, 100]. The number n of jobs in an instance ranged from 10 to The start time of the hole was set to a percentage Rperc of the sum of the processing times in an instance, i.e. we set s = Rperc j N p j where Rperc {0.1, 0.3, 0.5, 0.7, 0.9}. The hole length was set to the average processing time of the instance, i.e. = j N p j /n. For each pair of the parameters n and Rperc ten instances were generated. The algorithms were implemented in the programming language Tcl and the experiment was run on an AMD Duron/1200 MHz personal computer. Algorithm H was tested with the accuracy parameter value k = 3. According to Table 1 the performance guarantee with this setting is 10.21%. Table 2 lists the mean (ME) and worst (WE) relative errors of the algorithms SPT, MSPT and H depending on the number of jobs. All values are percent deviations from the minimal ow-time C. The minimal ow-time was computed using the dynamic programming algorithm presented in [5]. We see that the mean and worst errors of algorithm H are very low (0.31% and 3.18% deviation from C, respectively). Algorithm H improves the SPT results distinctly. On average the mean error of algorithm H is 50% of the mean SPT error (0.31% above C in comparison to 0.62% above C ). This result is astonishing if we 11

12 Table 2: Mean (ME) and worst (WE) percent deviations of algorithms SPT, MSPT and H from the minimal ow-time. n ME(SPT) WE(SPT) ME(MSPT) WE(MSPT) ME(H) WE(H) average take into account that for 52% of the test instances the abort condition A > a max (k) held, i.e. for these instances the algorithm stopped after generating the SPT schedule. The overall average computation time of algorithm H is only 0.8 seconds. For those instances where the abort condition did not hold and step 2 was carried out, the average computation time was 1.7 seconds. The maximum computation time was 9.7 seconds. Taking into account that we used a relatively slow interpreted programming language for the experiment and that our implementation of the algorithm was not optimized we conclude that algorithm H is quite fast and superior to the SPT algorithm unless extreme computation speed is of vital importance. The results in Table 2 also show that algorithm H (for k = 3) is outperformed by algorithm MSPT for instances with n 30 with respect to average and worst solution quality. For the smaller instances algorithm H achieves better results than algorithm MSPT. On average the mean error of the MSPT algorithm is only 11% of the mean SPT error (0.07% above C in comparison to 0.62% above C ). We refrain from reporting computation times for the MSPT algorithm and comparing them to those of algorithm H because our implementations of the algorithms were not optimized. Taking into account the computational experiments reported in [5] it is likely that algorithm MSPT is faster than algorithm H (for k = 3). We conclude that algorithm H (for k = 3) achieves a better 12

13 solution quality than algorithm MSPT for problem instances with n 20 while algorithm MSPT outperforms algorithm H for the larger problem instances. Sad et al. report in [6] that parameter Rperc has a distinct inuence on the performance of the MSPT algorithm. The mean errors increased as Rperc was increased from to 0.5. Our experiment conrms this result for Rperc values between 0.1 and 0.5. However, as Rperc is further increased (values 0.7 and 0.9) the mean error decreases again. The mean errors for algorithm H decreases from 0.66% to 0.002% as Rperc is increased from 0.1 to Summary We investigated the problem of minimizing the sum of the job completion times on a single machine which is not available during a given time interval. The best known approximation algorithm for this problem, the MSPT algorithm, has a relative worst-case error bound of 17.6%. We presented a parametric O(n log n)-algorithm H with which better worst-case error bounds can be obtained. The best error bound calculated for the algorithm in the paper is 7.4%. In a computational experiment algorithm H was tested exemplarily with a performance guarantee setting of 10.2%. It turned out that instances with up to 1000 jobs were solved with a mean (maximum) error of 0.31% (3.18%) and a mean (maximum) computation time of 0.8 (9.7) seconds. In the experiment algorithm H outperformed the MSPT algorithm with respect to mean and worst error for problem instances with at most 20 jobs. References [1] I. Adiri, J. Bruno, E. Frostig, A.H.G. Rinnooy Kan, Single machine ow-time scheduling with a single breakdown, Acta Informatica 26 (1989) [2] O. Braun, J. Breit, G. Schmidt, Deterministic machine scheduling with limited machine availability, discussion paper B0403 (2004), Department of Economics, University of Saarland. [3] C.-Y. Lee, Machine scheduling with availability constraints, in: J.Y.-T. Leung (Ed.), Handbook of Scheduling, CRC Press, 2004,

14 [4] C.-Y. Lee, S.D. Liman, Single machine ow-time scheduling with scheduled maintenance, Acta Informatica 29 (1992) [5] C. Sad, B. Penz, C. Rapine, A dynamic programming algorithm for the single machine total completion time scheduling problem with availability, 8th International Workshop on Project Management and Scheduling - PMS, Valencia, Spain, (2002) [6] C. Sad, B. Penz, C. Rapine, J. Bªaºewicz, P. Formanowicz, An improved approximation algorithm for the single machine total completion time scheduling problem with availability constraints, European Journal of Operational Research 161 (2005) 310. [7] G. Schmidt, Scheduling with limited machine availability, European Journal of Operational Research 121 (2000) 115. Acknowledgements The author would like to express his gratitude to Dr. Chérif Sad and two anonymous referees for many constructive comments and hints which helped improve the paper. 14