STABILITY OF JOHNSON S SCHEDULE WITH LIMITED MACHINE AVAILABILITY

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1 MOSIM 01 du 25 au 27 avril 2001 Troyes (France) STABILITY OF JOHNSON S SCHEDULE WITH LIMITED MACHINE AVAILABILITY Oliver BRAUN, Günter SCHMIDT Department of Information and Technology Management Saarland University, Postfach , D Saarbrücken, Germany ob@itm.uni-sb.de, gs@itm.uni-sb.de Yuri N. SOTSKOV Institute of Engineering Cybernetics Surganova St. 6, Minsk, Belarus Universite de Technologie Troyes, 12, Rue Marie-Curie, BP 2060, Troyes, France sotskov@newman.bas-net.by ABSTRACT: We investigate the two-machine n-job flow-shop scheduling problem with the objective of minimizing the makespan and given non-availability intervals on each of the two machines. It was proven recently that this problem is NP-hard even if there is only one non-availability interval either on the first machine or on the second machine. If there are no non-availability intervals on any machine, then the two-machine flow-shop problem may be solved using Johnson s schedule constructed in O(n log n) time. Our results demonstrate that the optimality of Johnson s schedule is not violated if there exist only sufficiently small non-availability intervals. The instrument we use is stability analysis which answers the question how stable the property of a schedule to be optimal is if there are independent changes in the processing times of the jobs. KEYWORDS: flow-shop, limited machine availability, stability analysis 1. INTRODUCTION A number of recent papers have been devoted to the flowshop problem with two machines which have intervals of non-availability, see (Schmidt, 2000) for a survey. Lee (Lee, 1997) has shown that this problem is NP-hard even if there is a single non-availability either machine 1 or machine 2. Lee also provided a heuristic with a makespan which is at most (3/2) times larger than minimal makespan if the non-availability interval is on machine 1; and (4/3) times larger than minimal makespan if the nonavailability interval is on machine 2. In (Cheng and Wang, 2000), it was shown that the error bound of (3=2) for the situation with a non-availability machine 1 is tight. Cheng and Wang developed a heuristic with an error bound of (4=3). In (Cheng and Wang, 1999), it was studied the problem with availability constraints imposed on each machine when the non-availability one machine is followed immediately by the non-availability the other machine. Cheng and Wang provided a heuristic with a worst-case error bound of (5=3) for the case when a job that is preempted by a non-availability interval has to restart after the machine becomes available again. In (Kubiak, Blazewicz, Formanowicz, Breit and Schmidt, 2000) it was shown (by experiments) the nearly-optimal behaviour of Johnson s schedule for the case when there are more than one non-availability interval. The aim of this paper is to use the stability analysis of optimal schedule for the two-machine flow-shop scheduling with limited machine availability. The non-availability of a machine may increase the time for processing jobs on the machine. The influence of possible changes of the processing times on the optimality of a schedule was investigated in (Sotskov, 1991) where the notion of stability radius of optimal schedule was introduced (stability radius denotes the largest quantity of independent variations of the processing times such that the given optimal schedule remains optimal). In (Kravchenko, Sotskov and Werner, 1995), necessary and sufficient conditions for a schedule in a job-shop to have an infinitely large stability radius were proven which means that a schedule remains optimal even if the processing times may vary arbitrarily (but remain non-negative). To test these conditions and to construct a schedule with infinitely large stability radius takes O(q 2 ) time where q means the number of operations which have to be processed. It was also shown that there are no schedules with infinitely large stability radii for the flow-shop and open-shop problems when n>2 or m>2 with n being the number of jobs and m the number of machines. In (Sotskov, Sotskova and Werner, 1997), it was shown (by experiments) that makespan optimal schedule is usually stable with respect to changes in the processing times. Let us assume that the non-availability intervals are known in advance (notation: offline, NC off ) and a job that cannot be completed before the non-availability interval of a machine can be continued after the machine is available again (notation: resumable, pmtn-res). Using threefield notation, the problem under consideration may be denoted as F 2;NC off jpmtn resjc max. Let ff be a sequence of jobs in Johnson s order which is optimal when all machines are continuously available during the planning horizon. We study the question whether this order remains optimal for the case of offline known non-availability intervals. The main idea is to consider non-availability intervals on a machine as parts of the operations that

2 have to be processed on that machine. So in a concrete schedule we have enlargements of processing times that are caused by non-availability intervals. In Section 3, we compute the stability radius ρ j of the processing times of the operations on machine j which is the minimum of all maximal possible enlargements r ij of jobs i on machine j such that Johnson s order is not changed. Then we compute the enlargement radius ffi j of the processing times of the operations on machine j which is the maximum of all maximal possible enlargements d ij of jobs i on machine j caused by non-availability intervals on machine j. It is easy to show that ff remains optimal if d ij» r ij for all i 2f1; 2;:::;ng and j 2f1; 2g. We show that the stability radius can be computed in O(n) time and that the enlargement radius can be computed in O(K 2 + n log n) time with K being the total number of non-availability intervals. Before providing the algorithms in Section 3., we give the notations and demonstrate Johnson s schedule in Section NOTATIONS AND EXAMPLE Let n jobs have to be processed in a two-machine flowshop: first on machine 1 and then on machine 2. We use the notations from Table 1 where j 2 f1; 2g and i 2f1; 2;:::;ng. Let machine j be not available from time s(n kj ) until time f (N kj )=s(n kj )+h(n kj );k2f1; 2;:::;K j g;j2 f1; 2g: The operation started before time s(n kj ) but not finished at time s(n kj ) is suspended during time h(n kj ) starting from time s(n kj ) and is allowed to resume its processing from time f (N kj )=s(n kj )+h(n kj ) on the same machine j. p ij K j K = K 1 + K 2 N kj s(n kj ) f (N kj ) h(n kj ) r ij ρ j s j f j d ij ffi j processing time of job i on number of non-availability intervals on total number of non-availability intervals, k th non-availability start time of the k th nonavailability finish time of the k th nonavailability length of the k th nonavailability maximal possible enlargement of processing time of job i such that Johnson s order is not changed, stability radius of the processing times of the operations on earliest possible start time of a job on machine j latest possible finish time of a job on maximal possible enlargement of processing of job i forced by non-availability intervals on enlargement radius of the processing times of the operations on machine j. Example: We consider a two-machine flow-shop problem with non-availability intervals on the two machines as shown in Fig. 1. The non-availability intervals of machines 1 and 2 are shaded in all figures. In what follows, this example will be used to illustrate the calculations. Figure 1. Example Table 2 presents the given processing times p ij of job i on machine j: In Section 3., we show how to compute values r ij ;ρ j ;s j ;f j and d ij (see Table 1) for i 2 f1; 2;:::;ng and j 2f1; 2g. For details of optimal schedule for the two-machine flowshop we refer to Johnson (Johnson, 1954). The optimal Table 1. Main notations schedule can be constructed as follows. Partition the jobs f1; 2;:::;ng into two subsets S 1 and S 2 with S 1 containing all the jobs with p i1» p i2 and S 2 all the jobs with p i1 >p i2. In the optimal schedule, the jobs from set S 1 are processed first, and they are processed in the increasing order of p i1 (SPT: shortest processing time priority). The jobs from set S 2 follow after jobs S 1 in the decreasing order of p i2 (LPT: longest processing time priority). Ties may be broken arbitrarily. If the n jobs are ordered in this way on each of the two machines, we say of Johnson s schedule or Johnson s order. In (Johnson, 1954), it has been shown that this schedule is optimal for the case where all machines are continuously available throughout the planning horizon. Example (continued): Figure 2 shows the schedule with the jobs following Johnson s order applied to the situation where the machines are not continuously available.

3 Machine 1 Machine 2 p 11 =3 r 11 d 11 p 12 =5 r 12 d 12 p 21 =4 r 21 d 21 p 22 =1 r 22 d 22 p 31 =7 r 31 d 31 p 32 =2 r 32 d 32 ρ 1 ffi 1 ρ 2 ffi 2 Table 2. The given processing times Figure 2. Jobs are scheduled in the order given by Johnson s rule In what follows, we show that this schedule remains optimal for the input data given in Fig. 1 and Table STABILITY AND ENLARGEMENT RADII Let ff be Johnson s schedule (which is optimal when both machines are continuously available during the planning horizon). Our aim is to answer the question whether this order remains optimal for the case of offline known nonavailability intervals. To this end, we consider non-availability intervals on a machine as parts of the operations that have to be processed on that machine, i.e. in a concrete schedule enlargements of processing times of some jobs are caused by non-availability intervals. We compute the stability radius ρ j of the processing times of the operations on machine j which is the minimum of all maximal possible enlargements r ij of jobs i on machine j such that Johnson s order is not changed. Then we compute the enlargement radius ffi j of the processing times of the operations on machine j which is the maximum of all maximal possible enlargements d ij of jobs i on machine j caused by non-availability intervals on machine j. It is shown that schedule ff remains optimal if d ij» r ij for all i 2f1; 2;:::;ng and j 2f1; 2g. For definition of the stability radius in general case we refer to (Sotskov, 1991). Here we need a slightly modified version of the stability radius since non-availability intervals cannot decrease the operation processing times. Definition 1 The stability radius ρ j of optimal order of operations on machine j is defined as the minimum of all maximal possible enlargements r ij of operations i on machine j such that this order is not changed. Next, we prove the following lemma. Lemma 1 The stability radii ρ j ; j 2f1; 2g; can be computed in O(n) time. Proof: We assume that we are given a sequence ff of jobs following Johnson s order. W.l.o.g. we assume that ff = (1; 2;:::;n). Moreover, we assume that the first k jobs belong to set S 1 and the remaining n k jobs belong to set S 2, i.e. S 1 = f1; 2;:::;kg and S 2 = fk +1;k + 2;:::;ng. We compute the numbers r ij which represent the maximal possible values one can add to the processing times p ij such that Johnson s order remains the same. We distinguish the following two cases: i 2 S 1 or i 2 S 2. In the case when i 2 S 1, inequality p i1» p i2 holds, and maximal possible enlargements r i1 are the minimum of two values a i1 and b i1 where a i1 = p i2 p i1 represents the maximum amount of time one can add to p i1 such that job i remains in set S 1. The value ρ pi+1;1 p b i1 ; i =1; 2;:::;k 1; i1 = 1; i = k; represents the maximum amount of time one can add to p i1 such that the SPT ordering within the set S 1 is preserved. Similar arguments are valid for the case i 2 S 2 when p i1 > p i2. Every computation of an r ij can be done in constant time. So the stability radii ρ j ; j 2f1; 2g, can be computed in O(n) time. 2 Example (continued): In Table 3, the results of calculation of the stability radii for machines 1 and 2 are shown. Machine 1 p 11 =3 r 11 =minf2; 1g =2 p 21 =4 r 21 = 1 p 31 =7 r 31 = 1 ρ 1 =2 Machine 2 p 12 =5 r 12 = 1 p 22 =1 r 22 =minf3; 1g =1 p 32 =2 r 32 =minf5; 1g =5 ρ 2 =1 Table 3. Stability radii Thus we have shown that, if no processing time on machine 1 is enlarged by more than 2 units and if no processing time on machine 2 is enlarged by more than 1 unit, then Johnson s schedule ff remains optimal for the given non-availability intervals on the machines. Next, independently of any concrete schedule we want to determine the maximal possible enlargements of operations on a machine j 2f1; 2g caused by non-availability intervals on machine j. Definition 2 The enlargement radius ffi j of the processing times of the operations on machine j is defined as the maximum of all maximal possible enlargements of operations on machine j caused by non-availability intervals on machine j.

4 Thus, the enlargement radius gives the maximum amount of time that the finishing of an operation on a machine may be delayed by one or more non-availability intervals of that machine. So, in a sense, the processing time of the operation is enlarged by non-availability intervals. Next, we prove the following lemma. Lemma 2 The enlargement radii ffi j ; j 2f1; 2g; can be computed in O(K 2 + n log n) time. Proof: For machine 1 we do the following. If an operation on machine 1 starts or is resumed immediately after a non-availability interval, we add the length of this nonavailability interval to the processing time of this operation. First, we make a pre-computation of sums of lengths of non-availability intervals: D i = K1+1 i max a=1 ( a+i 1 X k=a h(n k1 ) Similarly we compute sums of availability intervals A k1 with s(a k1 )=f(n k1 ) and f (A k1 )=s(n k+1;1 ). We set E 0 =0and E i = K 1+1 i min a=1 ( a+i 1 X k=a h(a k1 ) These computations need O(K 2 ) time. Thus, all operations on machine 1 with E k 1» p i1 <E k have a maximal enlargement caused by non-availability intervals with length of D k. Similar arguments are valid for machine 2. The concrete determination of the enlargement of each job takes O(K + n log n) time, so we need O(K 2 + n log n) time for the computation of the enlargement radii ffi j ; j 2f1; 2g. 2 Example (continued): In Table 4, the enlargement radii for machines 1 and 2 are presented. Machine 1 Machine 2 p 11 =3 d 11 =2 p 12 =5 d 12 =2 p 21 =4 d 21 =3 p 22 =1 d 22 =1 p 31 =7 d 31 =3 p 32 =2 d 32 =1 ffi 1 =3 ffi 2 =2 Table 4. Enlargement radii Thus we shown that any processing time on machine 1 (machine 2) cannot be enlarged by more than 3 (by more than 2) units if one sticks non-availability intervals with jobs. Figure 3 presents the polytope of the enlargement values of operations on machine 1. Point A is determined by the given processing times p i1 of job i 2f1; 2; 3g on machine 1. Using Lemma 1 and Lemma 2 we can prove the following theorem. ) ) : : Figure 3. Enlargement polytope of operations on machine 1 Theorem 1 Johnson s order defined for problem F 2jjC max remains optimal for problem F 2;NC off jpmtn resjc max with the same processing times if d ij» r ij for all i 2 f1; 2;:::; ng and j 2 f1; 2g. To test the latter conditions takes O(K 2 + nlogn) time. In order to reduce the number of non-availability intervals that have to be considered, one can use Lemma 3. Lemma 3 The earliest possible start times s j and the latest possible finish times f j of any job on machine j 2 f1; 2g can be computed in O(n + K) time. Proof: On machine 1, the earliest start time of any job is equal to ρ f (N11 ); if s(n s 11 )=0; 1 =max 0; otherwise. The latest finish time of any job on machine 1 is equal to f 1 = p 1 + a + b with p 1 = P p i1 and a = P k i=1 h(n i1) with s(n k1 ) <p 1 for maximal k. Let b initially be 0, and k be taken from the former computation. The value b is increased by h(n k+1;1 ) and k is increased by 1 as long as s(n k+1;1 ) < p 1 + a + b. The computation of p 1 takes O(n) time and the computation of a and b takes O(K) time. So we need O(n + K) time to compute the latest finish time of any job on machine 1. Let p min (1) be the smallest processing time of an operation on machine 1. On machine 2, the earliest possible start time of any job is the finish time of the first operation (which is assumed to be the shortest one) on machine 1.

5 Let d = p min (1) + c. The values c and k are initially set to 0. The value c is increased by h(n k+1;1 ) and k is increased by 1 as long as s(n k+1;1 ) <p min (1) + c. Then we have s 2 =max 8 >< >: f (N is ); if there is a non-availability interval N i2 with s i2» d and f i2 d; d; otherwise. This computation takes O(n + K) time. Let p max (2) be the largest processing time of an operation on machine 2. The latest possible finish time of any job on machine 2 is estimated by the upper bound f 2 = f 1 + P p i2 + d. The value k is the maximum index with s(n k2 )» f 1. The value d initially is set to be equal to f (N k2 ) f 1. The value d is increased by h(n k+1;2 ) and k is increased by 1 as long as s(n k+1;2 ) <f 1 + P p i2 +d. This computation takes O(n + K) time. 2 It is easy to see that all the above bounds with the exception of p max (2) are tight. 4. CONCLUSION In Section 3, sufficient conditions are given for Johnson s schedule to be optimal in the case of given non-availability intervals on both machines in the two-machine flow-shop problem. Due to Lemma 1 and Lemma 2 these conditions may be tested in polynomial time of the number of jobs and the number of non-availability intervals. Figure 4 shows the results of the computation of the stability and enlargement radii for machine 1 and machine 2. One can see that Johnson s order remains optimal for the non-availability case under consideration since d ij» r ij for all i 2f1; 2;:::;ng and j 2f1; 2g. Figure 4. Resulting schedule In this example, non-negative integer numbers are used as input data. Note that it was done only for simplicity of the presentation: all the above results are valid for nonnegative real input data. It should be noted also that the above stability analysis may be used not only for problem F 2;NC off jpmtn resjc max, but for other scheduling problems with limited machine availability if optimal schedule for the corresponding pure setting of the problem (i.e. when all machines are available during the whole planning horizon) may be constructed applying priority rule of jobs such as SPT, LPT and so on. Moreover, one can use the above results for some kind of on-line settings of scheduling problems when there is no prior information about exact location of the non-availability intervals on the time axis but values d ij or values ffi j ; i = f1; 2;:::; ng; j = f1; 2g; are given before scheduling. In a straightforward matter it will be possible to extend the results of this paper to a job-shop. To underline the practical interest of the results of this paper, it will be sensible to make experimental tests similar to those in (Kubiak, Blazewicz, Formanowicz, Breit and Schmidt, 2000). We would answer the question of how many schedules following Johnson s order remain optimal in spite of the non-availability intervals on the machines. ACKNOWLEDGMENTS This research was partially supported by INTAS (Project INTAS ). The third author was supported also by NATO and ISTC (Project B ). REFERENCES Cheng, T.C.E. and G. Wang, Two-machine flowshop scheduling with consecutive availability constraints. Information Processing Letters, 71, p Cheng, T.C.E. and G. Wang, An improved heuristic for two-machine flowshop scheduling with an availability constraint. Operations Research Letters, 26, Johnson, S.M., Optimal two- and three-stage production schedules with setup times included. Naval Research Logistics Quarterly, 1, Kravchenko, S.A., Y. N. Sotskov and F. Werner, Optimal schedules with infinitely large stability radius. Optimization, 33, Kubiak, W., J. Blazewicz, P. Formanowicz, J. Breit and G. Schmidt. Two-machine flow shops with limited machine availability, European Journal of Operational Research, accepted for publication. Lee, C.Y., Minimizing the makespan in the twomachine flowshop scheduling problem with an availability constraint. Operations Research Letters, 20, Schmidt, G., Scheduling with limited machine availability. European Journal of Operational Research, 121, Sotskov, Y.N., Stability of an optimal schedule. European Journal of Operational Research, 55, Sotskov, Y.N., N. Y. Sotskova and F. Werner, Stability of an optimal schedule in a job shop. Omega, International Journal of Management Sciences, 25,