Adaptive subsequence adjustment with evolutionary ... - CiteSeerX

Journal of Systems Engineering and Electronics Vol. 25, No. 5, October 2014, pp.800–810

Adaptive subsequence adjustment with evolutionary asymmetric path-relinking for TDRSS scheduling Peng Lin1,3 , Linling Kuang2 , Xiang Chen2,* , Jian Yan2 , Jianhua Lu1,2 , and Xiaojuan Wang3 1. Department of Electronic Engineering, Tsinghua University, Beijing 100084, China; 2. Tsinghua Space Center, Tsinghua University, Beijing 100084, China; 3. China Electronic Equipment of System Engineering Institute, Beijing 100141, China

Abstract: Due to the limited transmission resources for data relay in the tracking and data relay satellite system (TDRSS), there are many job requirements in busy days which will be discarded in the conventional job scheduling model. Therefore, the improvement of scheduling efficiency in the TDRSS can not only help to increase the resource utilities, but also to reduce the scheduling failure ratio. A model of nonhomogeneous parallel machines scheduling problems with time window (NPM-TW) is firstly built up for the TDRSS, considering the distinct features of the variable preparation time and the nonhomogeneous transmission rates for different types of antennas on each tracking and data relay satellite (TDRS). Then, an adaptive subsequence adjustment (ASA) framework with evolutionary asymmetric path-relinking (EvAPR) is proposed to solve this problem, in which an asymmetric progressive crossover operation is involved to overcome the local optima by the conventional job inserting methods. The numerical results show that, compared with the classical greedy randomized adaptive search procedure (GRASP) algorithm, the scheduling failure ratio of jobs can be reduced over 11% on average by the proposed ASA with EvAPR. Keywords: nonhomogeneous parallel machines scheduling problem with time window (NPM-TW), adaptive subsequence adjustment (ASA), asymmetric path-relinking (APR), evolutionary asymmetric path-relinking (EvAPR). DOI: 10.1109/JSEE.2014.00093

1. Introduction In recent 30 years, eleven tracking and data relay satellites (TDRSs) have been launched by NASA [1], seven of which are still available to serve the spacecraft as a relay platform seamlessly all over the world. In last five years, China has launched three TDRSs to build up Chinese TDRS system Manuscript received September 18, 2013. *Corresponding author. This work was supported by the National Natural Science Foundation of China (61132002; 91338101; 91338108), the National S&T Major Project (2011ZX03004-001-01), the Research Fund of Tsinghua University (2011Z05117), and the Co-innovation Laboratory of Aerospace Broadband Network Technology.

(TDRSS) with global coverage, which can provide not only space-based telemetering and telecontrol service, but also data relay for space aircraft. As a matter of fact, the TDRSS has played a very important role in the space network (SN), space communications and navigation constellation integration project (SCaN), etc [2,3]. However, due to the dramatical growth of space aircraft, more and more transmission jobs will be undertaken by the TDRSS. Taking a busy-day model provided by NASA as an instance [4], there are more than 400 jobs of 20 aircraft should be served by two TDRSs. In such a heavy service scenario of the TDRSS, there are nearly 12% jobs will be discarded by current scheduling schemes [4]. Therefore, job scheduling efficiency becomes significantly important for the on-orbit TDRSs with a few antennas. In order to automatically solve the scheduling problem with high efficiency for the TDRSS, a parallel machines scheduling problem with time window (PM-TW) [5] was modeled for this problem. However, the difference of transmission rates for different types of antennas on each TDRS is not considered, which means the processing duration time of each job is treated homogeneous for any type of antennas. On each TDRS, at least there are two types of antennas for serving users, which are single access antennas (S or Ka bands) and multiple access antennas (S band). Each antenna (beam) of one TDRS is treated as a machine, so the antenna (beam) and the machine are not particularly distinguished in this paper. Then, a relevant non-identical parallel machines scheduling problem with multiple time windows (NIPM-MTW) was modeled for the earth observation case by low earth orbit (LEO) satellites [6], in which the switching time between targets and the observing time for different targets are both set as constants. However, these settings are not consistent with the real scenarios for the TDRSs, where the antenna preparation time of the TDRSs is variable with the rotation an-

Peng Lin et al.: Adaptive subsequence adjustment with evolutionary asymmetric path-relinking for TDRSS scheduling

gles for target switching, and the time window constraint for each transmission job is also independent. The above time differences for different antennas on TDRSs make the serving capabilities of such parallel machines (i.e., multiple antennas on TDRSs) for one job no longer the same, but nonhomogeneous. Moreover, the problem model of job scheduling on one TDRS is typically a knapsack loading problem (KLP) in the time domain, which has been proved to be NP-hard [4]. When multiple TDRSs are involved to serve more jobs, the complexity is increasing along with the number of jobs n, serving antennas m and TDRSs s, which is much larger than O((m∗s)n ). Therefore, some random searching based intelligent algorithms are often used to solve this problem, such as heuristic algorithms [7–10] and genetic algorithms (GA) [11–13]. In heuristic algorithms, Reddy and Brown [7] firstly developed a dynamic-programming (DP) heuristic scheme for the TDRSS. Then, Rojanasoonthon and Bard proposed a greedy randomized adaptive search procedure (GRASP) [4,8], which outperforms the DP scheme in scheduled jobs. However, due to the lack of job replacement and removement, the converged solutions of the GRASP will be easily influenced by the initial steps. Therefore, in general the multi-start iterative process [9] is needed to get the best solution in the GRASP. Reference [10] further combined the branch-and-price (B&P) algorithm with the GRASP to solve the scheduling problem in the presence of service priorities, but the scale of the problem is strictly limited by 100 jobs. For GA-type algorithms, the evolutionary strategy [11– 13] is used to solve the combinational programming problem with resource constraints in the TDRSS. The low efficiency of crossover and mutation operators of the GA-type algorithms makes the scheduled jobs generally no more than 15 [11,12], which is not applicable in the busy-day scenario with 418 jobs [4]. Except for heuristic and GA-type algorithms, there are some other researches [6,14,15] related to the job scheduling problem with the time window. However, according to the special features such as nonhomogeneous transmission rates of different antennas and variable antenna preparation time for target switching, the above methods cannot be generally extended to the TDRSS. In this paper, regarding the distinct features of the variable preparation time and different transmission rates for different antennas on one TDRS, a novel nonhomogeneous parallel machines scheduling problem with time window (NPM-TW) in the TDRSS is modeled. Then, an adaptive subsequence adjustment (ASA) framework, including the asymmetric path-relinking (APR) and the evolutionary

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strategy, is proposed to solve the NPM-TW. Herein, the APR operation with crossover operators can help to partially avoid the local optimal solutions after the ASA. Finally, the evolutionary strategy compounds the APR to iteratively generate optimized solutions, by which the largescale NPM-TW in the TDRSS can be handled efficiently. The remainder of this paper is organized as follows. In Section 2, the NPM-TW in the TDRSS is modeled. The framework of ASA with evolutionary APR (EvAPR) for the NPM-TW is presented in Section 3, where the details of ASA, APR, and the related evolutionary strategy are included. Then, in Section 4, three testing cases are given to evaluate the proposed algorithm. Finally, Section 5 concludes this paper.

2. Problem modeling 2.1 Basic operation scenario of the TDRSS As Fig. 1 shows, the basic operation scenario of the TDRSS given by NASA [16] includes two TDRSs, which are TDRS east and TDRS west. Spacecraft can communicate with the TDRSs only when they are within the lineof-sight range of any TDRS.

Fig. 1

Basic operation scenario of TDRSS by NASA

As defined by NASA [7], the main attributes of each transmission job include the time window, the duration time and the job priority, which are shown in Fig. 2.

Fig. 2

Attributes of each job in TDRSS

Within the line-of-sight range, the time window is defined as a time constraint [aj , bj ], where aj is the earliest starting time and bj is the ending time of the time window, only in which the transmission link can be established. Given tj is a feasible starting time for Jobj , the

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first time window constraint can be formulated as a j  tj < b j .

(1)

Besides, the duration time, signed as dj , is defined as the block length of contiguous antenna time by specifying the antenna preference. Thus, another time window constraint dj  bj − tj + 1

(2)

should be satisfied, which means Jobj should be finished within the time window without any interruption. In addition, the priority ρj is used to distinguish the scheduled profile for Jobj . Different from PM-TW [4] and NIPM-MTW [6], two categories of antennas, treated as nonhomogeneous parallel machines with both nonhomogeneous preparation time and transmission rates, are analyzed and modeled in detail for the TDRSS. Generally, the directional beams of two single access (SA) antennas on each TDRS can be rotated mechanically, while the beams of the phased array multiple access (MA) antenna are steered by means of electronic control. Each of the above directional beams can serve one proper spacecraft at a time. When the serving jobs on one machine (i.e., one antenna on a TDRS) are switched from one to another, the waiting time includes the antenna preparation time, idle time and link establishment time as shown in Fig. 3.

example, 15 s). Therefore, the feature of different preparation time for different antennas on TDRSs is distinct from the case considered in the NIPM-MTW [6], which is obviously more reasonable for the real operation scenario in the TDRSS. Furthermore, the nonhomogeneous transmission rates of parallel machines (serving antennas) depend on the different beam gains of different antennas. The beam of the SA antenna in general has a higher gain [17] than that of the MA antenna, so the transmission rate of the former is naturally higher than that of the later, which means that a job served by an SA antenna will require less duration time than that by an MA antenna as shown in Fig. 4.

Fig. 4 Diagram of nonhomogeneous parallel machines with nonhomogeneous transmission rates in the TDRSS

This feature is so different from the homogeneous transmission rates considered in the PM-TW [4], where the job duration time is identical on different antennas. Therefore, in this paper the transmission rates of the two type antennas are treated nonhomogeneously and formulated as MA dSA j < dj

Fig. 3

Waiting time of machine during job switching in TDRSS

Herein, the antenna preparation time, signed as pij , is the beam switching time from Jobi to Jobj . Link establishment time, signed as ej , is the necessary link establishment time for Jobj , while the idle time means the left interval before link establishment. Therefore, the constraint during job switching can be formulated as wij  pij + ej

(3)

where wij represents the waiting time between Jobi and Jobj , which should be larger than the related necessary time. For SA antennas, the preparation time depends on the rotated angles required by the mechanical table, which is strictly limited by the stability of the TDRS attitude. Its typical value is between 15 s and 5 min [16]. For the MA antenna, the preparation time is often set as a constant (for

(4)

where dSA represents the duration time on SA antennas j represents the duration time on MA antennas, and and dMA j the former one is generally much smaller than the latter. 2.2 Model of the NPM-TW Based on the definitions of nonhomogeneous antenna preparation time and transmission rates above, the NPMTW in this paper can be modeled as follows. Assuming that there are n jobs in the set J = {Job1 , Job2 , . . . , Jobn } and m machines can be scheduled independently. Each Jobi has a single time window twi = [ai , bi ], where ai is the earliest starting time and bi is the latest starting time for this job. Each Jobi has a set of Ki available machines N P Mi that can serve this job, i.e., N P Mi = {Mik |k = 1, 2, . . . , Ki }

(5)

where dik is the duration time of the Jobi within twi by the machine Mik .

Peng Lin et al.: Adaptive subsequence adjustment with evolutionary asymmetric path-relinking for TDRSS scheduling

There are ρ priority sets, and ωρi of each Jobi has contribution to the scheduled resolution. The set of the total jobs within the higher q priorities is denoted as Uq , i.e., q 

Uq =

Jp .

j∈J 0 \{i} Mjk ∈N P Mj

where J 0 = J ∪ {0}, and {0} represents the starting and the ending of the job sequence on machine k. yjk : binary variable, where yjk = 1 if the machine k is scheduled to serve Jobj , otherwise yjk = 0. It means  yjk  1. (8) Mjk ∈N P Mj

sijk : setup time variable, which is constituted by the antenna preparation time and the link establishment time. In general, sijk  0. (9) wijk : waiting time variable, which is the necessary waiting time during Jobi switching to Jobj on the machine k. Also, wijk  0. (10) tik : starting time variable, which is the scheduled starting time of Jobi on the machine k. Different from previous works, in this model dik can be variable with the different machine k according to the nonhomogeneous transmission rates. Besides, the sijk is proportional to the antenna preparation time and can be variable with the beam switching between different targets. When each job requires to be processed exactly once without interruption by one antenna, a mixed integer linear programming formulation is modeled for the NPM-TW as    ωρi ( xijk ) (11) max

s.t. 



j∈J 0 \{i} Mjk ∈N P Mj



j∈J 0 \{i} Mjk ∈N P Mj



j∈J 0 \{i} Mjk ∈N P Mj

xijk =

xijk  1, ∀i ∈ J 0  Mjk ∈N P Mj



(12)

yjk , ∀i ∈ J 0 (13)



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xjik = 0

i∈J 0 \{j} Mik ∈N P Mi

ai yik  tik , ∀i ∈ J

0

(14) (15)

Mik ∈N P Mi



Besides, some variables are further defined to specify the job switching progress in the NPM-TW of the TDRSS as follows. xijk : flow variable, where xijk = 1 if the machine k is scheduled to serve Jobj after Jobi , otherwise xijk = 0. The related constraint is expressed as   xijk  1 (7)

xijk −

j∈J 0 \{i} Mjk ∈N P Mj

(6)

p=1

i∈J





(tjk − wijk − dik )yik  bi , ∀i, j ∈ J 0 (16)

Mik ∈N P Mi



(wijk − sijk )yjk  0, ∀i, j ∈ J 0 .

(17)

Mjk ∈N P Mj

The objective function (11) is designed to maximize the total profit of weighted jobs. Constraint (12) limits that the scheduled jobs in the feasible solutions are sequentialized and each job should be served less than once. Constraint (13) ensures that each job must be served by one of the available antennas. Constraint (14) ensures that both the predecessor and the successor of each job should be processed on the same antenna. Constraints (15) and (16) show that the scheduled starting time of each job should lie in the duration, which is later than the earliest starting time, but earlier than the latest starting time of the time window. Finally, constraint (17) limits the waiting time between adjacent jobs, which is not less than the summation of the related antenna preparation and link establishment time. According to the three-field notation standard [18], the NPM-TW in the TDRSS can be rewritten as fol lows: nRm|twj , N P Mj , djk , sijk , wijk | ωρj Uj where nRm denotes the nonhomogeneous unrelated pa rallel machines, and ωρj Uj specifies the weighted objective function with Uj = 1 for scheduled Jobj and Uj = 0 for unscheduled Jobj . If the above related constraints of each job are considered, job confliction will be led by the scheduling sequence variation during the solution construction progress. Consequently, different scheduled results should be further constructed. Regarding the sequence expression of n jobs in the scheduling problem, the number of sequentialized results is simply to be n!. Moreover, each job may have m nonhomogeneous parallel machines to choose. Thus the complexity of such scheduling problem can reach up to O(mn! ), which is typically an NP-hard problem [19]. Then, in the next section, we will try to propose a novel framework to solve this NP-hard problem with high scheduling efficiency.

3. ASA with EvAPR framework 3.1 Proposed framework In our proposal, the framework is based on a novel ASA with EvAPR, as shown in Fig. 5.

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Note Fjk  0 means that Jobj cannot be shifted forward without violating the time window constraints of the subsequence (α, . . . , i, j, . . . , β)k . 3.2.2 Backward time slack The backward time slack for Jobi scheduled on the machine k, denoted by Bik , is the maximum value that the starting time of Jobi can be shifted backward without producing extra waiting time between Jobi and Jobj in the subsequence (α, . . . , i, j, . . . , β)k . The backward time slack of the job i can be defined as

Fig. 5 Framework of ASA with EvAPR algorithm

This framework can be divided into three phases. First of all, an ASA is basically proposed to realize job inserting operation adaptively, in which an initial solution can be enhanced to a guiding solution. Secondly, a novel APR with ASA is presented to generate an elite solution from the neighborhood of the former two solutions. After repeating the above two phases with independent initializations, a group of elite solutions can be generated. Finally, a hybrid evolutionary strategy with APR is involved to accelerate the convergence of searching in the elite solutions. 3.2 ASA based job insertion The ASA based job insertion starts with an arbitrary initial solution and randomly generates a sequential set of unscheduled jobs as the related neighborhood space. Each job in the neighborhood space can be inserted into the initial solution one by one, constructing a guiding solution better than the initial one. In the ASA based job insertion, three major characters, including forward time slack, back time slack and ASA, will be described as follows. 3.2.1 Forward time slack Let (α, . . . , i, j, . . . , β)k be a partial sequence of a feasible solution on the machine k. The forward time slack of Jobj , denoted by Fjk , is the maximum time that Jobj can be shifted forward without violating the time window of any job in the subsequence (α, . . . , i, j, . . . , β)k . The forward time slack of the Jobj can be defined as Fjk = min{tjk − aj , Fik }.

(18)

Bik = min{bi − tik , Bjk }.

(19)

Note Bik  0 means that Jobi cannot be shifted backward any further without violating the time window constraints of the subsequence (α, . . . , i, j, . . . , β)k . 3.2.3 ASA Different from traditional fixed or adjusted subsequences with only forward time slack [16] for job insertion, in our proposal both the forward time slack and the backward time slack are considered to adaptively adjust the subsequence (so-called the ASA), then, one unscheduled job can be inserted in the feasible solution. By the ASA operation, the candidate positions for unscheduled job insertion along with the adjusted subsequences can be sufficiently increased, which can help to generate better guiding solutions. Here, the best insertion point that maximizes the slack of the scheduling result can be used to determine where an unscheduled job should be inserted in an adjusted subsequence. The ASA based job insertion is shown in Fig. 6, where more than six positions of the feasible solution can be attempted. The steps of the ASA based job insertion can be summarized as follows: Step 1 A sequential set Useq of unscheduled jobs should be generated randomly, which will be tried to be inserted into the initial solution one by one.

Fig. 6 Diagram of the ASA based job insertion

Peng Lin et al.: Adaptive subsequence adjustment with evolutionary asymmetric path-relinking for TDRSS scheduling

Step 2 Select one unscheduled job from the set Useq randomly, then, its time widow can be obtained. Step 3 Under the constraint of the time window for one unscheduled job, by both forward and backward time slack calculations as (19) and (20), an adjusted subsequence of the scheduled jobs (α, . . . , β)k on the machine k can be obtained. Thus the each interval between the adjacent Jobi and Jobj , (i, j)k ∈ (α, . . . , β)k can be obtained, which will be further attempted as one candidate insertion position. Step 4 Pick the best pair of the adjacent jobs (i∗ , j ∗ )k for such job insertion to maximize the flexibility function fk (i, u, j), i.e., (i∗ , j ∗ )k = argmax{fk (i, u, j) : ∀(i, j)k ∈ (α, . . . , β)k } (20) where fk (i, u, j) = Fuk + Buk

(21)

is a flexibility function to measure how many slacks a scheduled solution has after inserting an unscheduled job u ∈ U in (i, j)k . Then, return to Step 2 to select the next unscheduled job until all the candidates in Useq are attempted. 3.3 APR Path-relinking (PR) was firstly introduced in the context of tabu search [20], as an intensification strategy to explore trajectories connecting elite solutions obtained by heuristic methods. In the problems with solutions represented by binary vectors, such as knapsack problems, probabilistic scores are successfully provided to evaluate the results [21]. PR in general consists of trajectories interconnecting high-quality solutions, which are the paths starting from an initial solution towards another neighbour one. Therefore, PR can be viewed as a symmetric combination method [22]. However, in some cases in the NPM-TW scheduling problem of the TDRSS, the scheduled jobs cannot be crossovered between two feasible solutions by PR symmetrically due to the different starting time or duration time. For example, assuming the starting time of Jobi in two feasible solutions is different, crossovering Jobi between the two solutions will result in job conflict with other Jobj (j = i). On the other hand, when the duration time of two jobs is different, i.e., dik = djk for i = j, crossovering Jobi and Jobj will interrupt one of the jobs with a longer duration time. Furthermore, three remarks about the usability problems of PR in the NPM-TW are given as follows.

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Remark 1 Since one job could be scheduled at different starting time both in the initial solution and the guiding solution, the job may be arranged twice in one neighborhood solution inherited from the above two solutions when applying PR directly. Remark 2 The time constraints during job switching (as shown in Fig. 3) should be satisfied in both the initial solution and the guiding solution. However, in the neighborhood solutions produced by PR, the time constraints of jobs may be no longer satisfied due to the variation of antenna preparation time following the predecessor job. Remark 3 As the procedure of PR will not insert any unscheduled jobs, the performance of the neighborhood solutions by PR may be even worse than the initial and guiding solutions. In order to overcome the shortcomings of the traditional PR, the APR with a bi-directional (forward and backward) asymmetric crossover operation is proposed in this paper. Firstly, in the APR only the ending time point of each scheduled job is adopted as the crossover point, which means the asymmetric crossover operation of the APR is just applied to all the end time points in both the initial and the guiding solutions. Then, in the forward (backward) direction of APR, only the conflicted job of the guiding solution (the initial solution) will be reserved, while the other one job of the initial solution (the guiding solution) will be removed. The detailed description of the APR is given with a simple example in Fig. 7. Given two solutions si and sg on the machine k, where si represents an initial solution, and sg represents a guiding solution. Each path is built by leading from si to sg or by leading from sg to si . Denote the neighborhood solutions produced by the APR between two solutions si and sg as N S(si , sg ), where si = s s {Jobs1i , . . . , Jobs|sii | } and sg = {Job1g , . . . , Job|sgg | }. Let the asymmetric difference between si and sg be defined as s

s

δsi →sg = {j = 1, ..., |sg | | Jobsj i = Jobj g or tsjki = tjkg } (22) s

s

δsg →si = {j = 1, ..., |si | | Jobj g = Jobsj i or tsjki = tjkg } (23) s

where tsjki = tjkg denotes that the sequential scheduled jobs do not start at the same time. Further, the bi-directional difference can be calculated as |δsi ↔sg | = |δsi →sg | + |δsg →si | = |sg | + |si |.

(24)

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APR are not better than the guiding solution in the aspect of scheduling efficiency. Thus the ASA is usually combined with the APR to further optimize the neighborhood solutions. 3.4 EAPR

Fig. 7 Asymmetric crossover operation by APR

The difference value |δsi ↔sg | is linear growth with the amount of the scheduled jobs. The bi-directional paths of solutions are produced as follows. |δsi →sg |

si → sg : si = s1f , s2f , . . . , sf

|δsg →si |

sg → si : sg = s1b , s2b , . . . , sb

= sg

(25)

= si .

(26)

Following the above APR operation, a necessary correction step should further be used to solve the jobs confliction. On one hand, since each job should be served less than once, the duplicated job arranged twice in the neighborhood solutions can be deleted according to the direction of asymmetric crossover operation. Taking Fig. 7 for an instance, Job1 is arranged twice in s1b , which should be deleted, so as Job3 in s2b . On the other hand, when the time constraints during job switching are not satisfied, the successor job can also be deleted in this correction step. Also as shown in Fig. 7, the time constraints of Job3 after switching from Job1 in Sf2 are no longer satisfied, so the successor Job3 should be deleted from Sf2 . Thereby, a group of new feasible solutions is generated by the bi-directional APR, which can be considered as the neighborhood solutions N S of the initial and guiding solutions. The forward neighborhood solutions F N S = {sjf , j = 1, . . . , |sg |} are generated in the forward APR si → sg , and the backward neighborhood solutions BN S = {sjb , j = 1, . . . , |si |} are generated in the backward APR sg → si . Besides, from the above description on the APR we can see that there is no novel unscheduled job added to the neighborhood solutions, so the generated solutions by the

In order to speed up the convergence of searching in the neighbourhood solutions by the APR with the ASA, an evolutionary strategy can be further involved with the APR (so-called the EvAPR) for the NPM-TW, the initial idea of which comes from GA-type algorithms [11–13]. In the EvAPR, a group of elite solutions, treated as initial populations, should be firstly generated by multiple independent repetitions, in which the APR with the ASA is repeated with independent initializations. After drawing on the evolutionary procedure of GA, all the populations are randomly combined in pairs and the ASA with the APR is applied to each pair to generate new populations of the next generation. The above evolutionary progress will not stop until meeting a proper constraint, i.e., computing time cost, iteration number. In this paper, a constraint of maximal iteration number (e.g., three is set as the maximal generation of populations) is adopted as the evolutionary stop criterion. To sum up, the pseudo-code of the ASA with EvAPR algorithm for the NPM-TW is presented in Algorithm 1 in Fig. 8. The loop in line 1 ensures that the jobs with higher priorities must be scheduled firstly. The line 2 defines the initial solution for the current iteration loop, which is the scheduled result of the jobs with higher priorities in the last iteration loop. When p = 1, {optimal solutionk,0 , k = 1, . . . , m} denote the empty solutions. The loop in line 3 presents that the nonhomogeneous parallel machines should be scheduled one by one. The loop in line 4 presents the multiple independent repetitions to generate a group of independent elite solutions, which is used as the initial population for the EvAPR. Lines 5–7 present the bidirectional APR process, followed by the ASA based job insertion in lines 8–10 to optimize the related neighborhood solutions. In line 11, the best solution with the largest scheduled profit in the above optimized neighborhood solutions is picked up and denoted as the elite solution. The loop in line 13 presents the number of elite solutions in pairs, i.e. M axRepetitions/2, in which each elite solution should be selected once. Line 14 ensures that each pair of two elite solutions is selected randomly. In lines 15 and 16, the forward APR and backward APR are implemented, respectively. Thereafter, the ASA based job insertion is further used to optimize F N S and BN S in lines 17–24. If

Peng Lin et al.: Adaptive subsequence adjustment with evolutionary asymmetric path-relinking for TDRSS scheduling

the forward and backward elite solutions are better than the former two solutions, the updated solution pair will replace the old one as new populations of the next gene-

Fig. 8

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ration in lines 25–29. Finally, in line 30 the optimal solution obtained in the current generation is updated.

Algorithm 1: the ASA with EvAPR for the NPM-TW

4. Numerical results

twelve 2.67 GHz processors and 63.9 GB of RAM.

4.1 Data sets and test environment

4.2 Computational results

To test the performance of the proposed algorithm, we use the same instances in [16], which are generated based on the characteristics of the original TDRSS data set provided by NASA. The planning horizon is 86 400 s (i.e., one day). There are two TDRSs treated as six nonhomogeneous parallel machines, as shown in Fig. 1. All the test algorithms are implemented by Matlab (verR Xeon R server with sion 2012a) and executed on an Intel

In this subsection, three cases are provided to evaluate the performance of our proposal and comparison algorithms, including one-priority data sets, two-priority data sets and time-to-target tests. 4.2.1 One-priority data sets In the first data sets with one-priority, fifteen independent instances are generated randomly. M axRepetitions is set

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as 10. Three algorithms are performed to schedule these instances, which are the GRASP [16], the ASA+APR, and the ASA+EvAPR, respectively. In order to compare the results by the above three algorithms, we firstly take the scheduling results of the GRASP as the baseline and define the reduction ratio of scheduling failure (RRSF) of jobs as the new measure to evaluate the scheduling performance of the proposals. The RRSF of jobs is denoted as: RRSF = 1−

number of unscheduled jobs by others . (27) number of unscheduled jobs by GRAP S

The RRSF performance by the ASA+APR and ASA+EvAPR are finally shown in Fig. 9. The statistic results are also shown in Table 1. Table 1 Data Set 1

Set 2

Fig. 9 RRSF of jobs by ASA+APR and ASA+EvAPR on onepriority instances

Computational results on the two data sets

Evaluation Average failure ratio Average RRSF of jobs Maximum RRSF of jobs Average failure ratio Average RRSF of jobs Maximum RRSF of jobs Average weighted failure ratio Average RRSF of weighted-jobs Maximum RRSF of weighted-jobs

From Fig. 9, it can be seen that the RRSF performance of the ASA+APR is about 3.36% by averaging for all instances, while the more obvious performance improvements (about 11.61% by averaging for all instances) are obtained by the ASA+EvAPR. It means that by the advantages of the evolutionary strategy combined with the ASA, the ASA+EvAPR can sufficiently help to optimize the elite solutions obtained by the ASA+ARP. Besides, the maximal RRSF by the ASA+EvAPR can reach 14.74% (for the instance 11), compared with the GRASP.

GRASP 22

%

ASA+APR

ASA+EvAPR

3.36 6.32

11.61 14.74

2.83 5.43

6.43 11.11

5.62 14.99

23.74 48.13

22.25

6.25

those in Fig. 9. The ASA+EvAPR outperforms the GRASP and the ASA+APR obviously. From Fig. 10, it can be found that the performance of the ASA+EvAPR is better than the GRASP with averaged RRSF about 6.43% and maximal RRSF about 11.11% (for the instance 3). For the weighted scheduled jobs in Fig. 11, the RRSF of the ASA+EvAPR compared with the GRASP is about 14.99% on average and about 48.13% at maximum (for the instance 11).

4.2.2 Two-priority data sets In the second data sets with two-priority, 15 instances are also randomly generated, in which each priority has 200 jobs. The lower-priority jobs are assigned a weight of 1, while the higher-priority jobs are given a weight of 201. Thus each job with higher-priority would never be replaced by even 200 jobs with lower-priority in this case. This scheme implies an upper bound on the objective function (1a) is 40 400. The RRSF performance of jobs and those of weighted jobs are shown in Fig. 10 and Fig. 11, respectively. The similar results are obtained in Fig. 10 and Fig. 11, as

Fig. 10 RRSF of jobs by ASA+APR and ASA+EvAPR on twopriority instances

Peng Lin et al.: Adaptive subsequence adjustment with evolutionary asymmetric path-relinking for TDRSS scheduling

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It is found in Fig. 12 that the theoretical probability distribution curves of time-to-target for three test algorithms are both close to the approximations of the experimental probability distribution curves. Regarding the performance of time-to-target, the ASA+EvAPR outperforms the GRASP and the ASA+APR. The maximum computing time by the ASA+EvAPR is only about 1/6 of the time by the GRASP, and is about half of the time by the ASA+APR.

5. Conclusions Fig. 11 RRSF of weighted-jobs by ASA+APR and ASA+EvAPR on two-priority instances

4.2.3 Time-to-target To further evaluate the convergence performance of the proposed algorithm, the time-to-target method [23,24] is used to analyze the probability distribution of the computing time of the test algorithms. Without generality, we select an arbitrary instance from the first data sets with onepriority, and 200 independent runs are performed for each test algorithm. The stop criterion of the simulations for each algorithm is that the generated novel solution is better than or equal to a given target. Machine 1 is the first machine considered to be scheduled, and the initial solution is empty. The target value is set to be 88 on machine 1, which is the value of the best previously known solution for the instance. The experimental and theoretical probability distributions of time-to-target [24] are both plotted in Fig. 12.

Fig. 12 Experimental and theoretical probability distribution of time-to-target (machine 1)

Based on the analysis of the nonhomogeneous transmission rates and preparation time for different antennas on TDRSs, a novel NPM-TW scheduling model for the TDRSS is firstly proposed. Then, a framework of the ASA with EvAPR is developed to solve the NPM-TW. Herein, the forward time slack and backward time slack are both considered in the ASA to adaptively adjust the subsequence for potential further job insertion, which helps to generate the better guiding solutions. The evolutionary strategy combined with the APR is presented to speed up the convergence of searching in a large number of neighbourhood solutions. Numerical results verify the effectiveness of the proposed ASA+EvAPR, in both one-priority and two-priority data sets, compared with the GRASP. By the ASA+EvAPR, the scheduling efficiency in the aspects of job quantity and computing time can be obviously improved, which is guiding to the real operation of the TDRSS with large-scale jobs.

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Biographies Peng Lin was born in 1982. He recieved his B.E. degree from Xi’an Communication Institute, Xi’an, China, in 2004, and M.E. degree from Xi’an University of Technology, Xi’an, China, in 2007. Since April 2007, he has been with the Department of Software Testing Center, China Electronic Equipment of System Engineering Institute, Beijing. Since 2010, he has been pursuing a Ph.D. degree in the Department of Electronic Engineering, Tsinghua Univer-

sity, Beijing, China. His research interests include satellite communications and resource optimization. E-mail: [email protected] Linling Kuang was born in 1973. She received her B.S. and M.S. degrees from National University of Defense Technology, Changsha, China, in 1995 and 1998, respectively, and Ph.D. degree in electronic engineering from Tsinghua University, Beijing, China, in 2004. Since 2007, she has been with Tsinghua University, where she is currently an associate researcher in the Tsinghua Space Center. Her research interests include wireless broadband communications, signal processing, and satellite communications. E-mail: [email protected] Xiang Chen was born in 1980. He received his B.E. and Ph.D. degrees both from the Department of Electronic Engineering, Tsinghua University, Beijing, China, in 2002 and 2008, respectively. Since 2012, he has servd as an assistant researcher at Tsinghua Space Center, School of Aerospace, Tsinghua University, Beijing, China. His research interests mainly focus on signal processing, software radio, and wireless communications. E-mail: [email protected] Jian Yan was born in 1975. He received his B.S., M.S. and Ph.D. degrees in wireless communications from Tsinghua University, Beijing, China, in 1998, 2000, and 2010, respectively. From 2001 to 2011, he was with the Department of Electronic Engineering, Tsinghua University. Since 2011, he has been with the Tsinghua Space Center, Tsinghua University, and now serves as an associate researcher. His research interests are in the area of satellite communications. E-mail: yanjian [email protected] Jianhua Lu was born in 1963. He received his B.S. and M.S. degrees from Tsinghua University, Beijing, China, in 1986 and 1989, respectively, and Ph.D. degree in electrical and electronic engineering from the Hong Kong University of Science and Technology. Since 1989, he has been with the Department of Electronic Engineering, Tsinghua University Beijing, China, where he now serves as a professor. His current research interests include wireless multimedia communications, satellite communications, and wireless networking. E-mail: [email protected] Xiaojuan Wang was born in 1971. She recieved her B.E. degree from Second Artillery Engineering Institute, Xi’an, China, in 1994, and M.E. degree from Beijing Institute of Technology, Beijing, China, in 1999. Since July 2002, she is with the Department of Software Testing Center, China Electronic Equipment of System Engineering Institute, Beijing, China. Her research interests include communications system, and testing Technology. E-mail: [email protected]