Asymmetric Path-Relinking Based Heuristics for ...

Asymmetric Path-Relinking Based Heuristics for Large-scale Job Scheduling Problem in TDRSS Lin Peng1,3 , Kuang Linling2 , Chen Xiang2,∗ , Yan Jian2 , Lu Jianhua1,2 , Wang Xiaojuan3 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 Email: {chenxiang}@tsinghua.edu.cn Abstract—In the busy-day scheduling scenario for tracking and data relay satellite system (TDRSS) provided by NASA, there are generally more than 400 jobs to be scheduled over two TDRSs, which builds up a large-scale job scheduling problem (LJSP) on multiple parallel machines. Since the global optima of such a LJSP can only be obtained with Non-deterministic Polynomial (NP) complexity, some classical heuristic algorithms, e.g., the GRASP, are often used to approach local optima with lower efficiency. In this paper, by fully exploiting the timedomain flexibility and statistical property of jobs’ slack time windows, a heuristic job scheduling framework is proposed to speed up the time convergence. The asymmetric path-relinking (APR) is firstly involved as a basic progressive operator, followed by the positive and negative adaptive subsequence adjustment (p/n-ASA) strategies, which are two adaptive construction and replacement mechanisms by maximizing usage the slack of jobs’ time windows. The key-path APR and hybrid evolutionary are further presented to accelerate the convergence. Simulation results show that, compared with the GRASP, our proposal can shorten the convergence time by almost 9 times, meanwhile with an improvement on the number of scheduled jobs.

I.

I NTRODUCTION

As a space-based transmission platform on the synchronous geostationary orbit, tracking and data relay satellite system (TDRSS) not only undertakes the reservation jobs, but also serves the emergency jobs burst all around the world. The burst jobs will disrupt the pre-scheduled plan inevitably, and a new plan must be rescheduled as quick as possible. With the dramatically increasing of space-based transmission demands, the time convergence is more and more significant for the largescale job scheduling problem (LJSP) in TDRSS nowadays. However, the LJSP in TDRSS has been proved as an NPhard problem [1], which is one of the principal unsolved problems today. Assuming that each one of the three servable antennas on each TDRS [2] can be scheduled as an independent machine, so, the computational complexity is proportional to the number of machines m and jobs n, which will be much greater than O((m)n ) under the multidimensional constraints, such as time-domain, spacial positions and antennas capability constraints. Take the classical busy-day scenario for an example, the LJSP provided by NASA [1], the number of the jobs is 418, the number of antennas is 6, and all the multidimensional constraints should be satisfied. The exhaustive of all the feasible combinations is 6418! , and almost 44000 seconds is necessary to find as good as the given target value by the greedy random

adaptive search procedure (GRASP) [3]. Therefore, when the pre-scheduled plan is disrupted frequently, the rescheduling time of the LJSP can not be tolerated for TDRSS. As the best of our knowledge, two heuristic algorithms are used to solve the LJSP provided by NASA, the best performance of the GRASP [4] just reaches 87.8% job-completion probability, and performs better than 84.93% of the ReddyBrown heuristic algorithm [5]. However, the GRASP lacks of jobs deletion and replacement, and the solutions are partly determined during the initial construction steps. Besides, there are some relevant algorithms for TDRSS scheduling, such as the genetic algorithm (GA) [6, 7] and the branch-and-price algorithm [8]. However, the reference GA algorithms are only feasible to solve the instance with only 15 jobs, and the number of jobs scheduled by the branch-and-price algorithm just reaches 100. Different from the traditional algorithms, in which only the greedy construction methods are discussed [4, 5, 8], and both the time convergence and the scheduling efficiency are not sufficient. Therefore, a heuristic job scheduling framework based on the asymmetric path-relinking (APR) is proposed in this paper, in which both the advantages of progressive and evolutionary strategies are adopted. Besides, the timedomain flexibility and statistic property of the jobs with slack time windows are further utilized to optimize the scheduling efficiency and accelerate the time convergence. The paper is organized as follows. First, the model for TDRSS scheduling is represented in Section II. Then, the heuristic framework and the related algorithms are proposed in Section III. Further, simulation results are demonstrated in Section IV. Finally, this paper is concluded in Section V. II.

M ODEL FOR TDRSS S CHEDULING

A nonhomogeneous parallel machine scheduling problem with time window (NPM-TW) is modeled for TDRSS [3]. Assuming that there are n jobs in the set J = {Job1 , Job2 , · · · , Jobn } and m parallel machines can be scheduled independently. Each Jobi has a time window twi = [ai , bi ], in which the service is possible [9], where ai is the earliest starting time and bi is the latest ending time. 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 }. (1) Some variables are further defined to specific the job switching progress in the NPM-TW of TDRSS in Table I.

TABLE I: Variables of the NPM-TW Variable

Definition flow variable, where xijk = 1 if the machine k is scheduled to serve Jobj after Jobi , otherwise xijk = 0. binary variable, where yjk = 1 if the machine k is scheduled to serve Jobj , otherwise yjk = 0. setup time variable, which represents the antenna preparation time of machine k. waiting time variable, which is the time interval during Jobi switching to Jobj on machine k. starting time variable, which is the scheduled starting time of Jobi on machine k. duration time variable, which is the duration time of Jobi by machine k.

xijk yjk sijk wijk tik dik

The variables as defined in Table I should satisfy the related constraints as follows. ∑ ∑ xijk ≤ 1 (2)

The objective function (6) is designed to maximize the total profit of weighted jobs. Constraint (7) limits that the scheduled jobs in the feasible solutions are sequentialized and each job should be served less than once. Constraint (8) ensures that each job must be served in one of the available machines. Constraint (9) ensures that both the predecessor and the successor of each job should be processed on the same machine. Constraint (10) limits that the scheduled starting time of each job should be later than the earliest starting time of its time window. Constraint (11) limits that the scheduled ending time of each job should be earlier than the latest ending time of its time window. Constraint (12) limits that the waiting time must be larger than the setup time. III.

A SYMMETRIC PATH -R ELINKING BASED H EURISTICS

A. Heuristic Job Scheduling Framework Utilizing both the progressive and evolutionary strategies, a heuristic job scheduling framework is proposed to solve the LJSP in TDRSS, and the framework is shown in Figure 1. 6WHS

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where J = J ∪ {0}, and {0} represents the starting and the ending of the jobs sequence on machine k. 0

In the NPM-TW of TDRSS, dik is variable with the different machines according to the different transmission rates [2]. Besides, sijk is variable with the rotation angle [1] by switching from one job to another. Further, there are ρ priority sets, and each Jobi has ωρi contribution to the scheduled resolution. Since each job requires less than once service, the NPM-TW can be formulated as ∑ ∑ ∑ max ωρi ( xijk ) (6) j∈J 0 \{i} Mjk ∈N P Mj

i∈J

∑

s.t.

j∈J 0 \{i}

∑

∑

j∈J 0 \{i}

∑

∑

xijk ≤ 1, ∀i ∈ J 0

Mjk ∈N P Mj

xijk =

Mjk ∈N P Mj

∑

∑

Mjk ∈N P Mj

xijk =

j∈J 0 \{i} Mjk ∈N P Mj

∑

∑

yjk , ∀i ∈ J 0 ∑

xjik

(7) (8) (9)

i∈J 0 \{j} Mik ∈N P Mi

ai yik ≤ tik , ∀i ∈ J 0

(10)

(tik + dik )yik ≥ bi , ∀i, j ∈ J 0

(11)

(wijk − sijk )yjk ≥ 0, ∀i, j ∈ J 0 .

(12)

Mik ∈N P Mi

∑

Mik ∈N P Mi

∑

Mjk ∈N P Mj

There are mainly four steps in this framework. First, a group of initial solutions should be generated randomly, which can be treated as a multi-start progress [8, 10]. And then, the positive adaptive subsequence adjustment (p-ASA) is applied to each initial solution independently, which can construct a relevant guiding solution. Further, under the guidance of the probability crossover by Key-path, the APR with the p-ASA is applied to each pair of the guiding solutions, which is designed as a progressive optimization and can generate a group of elite solutions. In addition, the evolutionary asymmetric pathrelinking (EvAPR) with the negative adaptive subsequence adjustment (n-ASA) is applied to the elite solutions, by which one best solution can be obtained. B. Asymmetric Path-Relinking Path-relinking (PR) was first represented in the context of Tabu Search [11], as an intensification strategy to explore trajectories connecting elite solutions obtained by heuristic methods [12]. The PR progress starts from an initial solution and symmetrically moves to a guiding solution, by which both the attributes of the two parent solutions are combined and inherited to generate a group of neighborhood solutions. Then, the symmetric movement that best improves the parent solutions will be picked up [13]. However, in the solutions of TDRSS scheduling, the sequential jobs of feasible solutions are asymmetrically arranged

in the time windows. Besides, the preparation time of antennas are variable with the rotation angles during jobs switching. Further, the PR progress lacks of job addition, which cannot improve the objective of the scheduled jobs number. So, the PR can not be directly implemented in the NPM-TW of TDRSS. Therefore, a bidirectional (backward and forward) APR is proposed as a basic crossover operator fundamentally, in which both the features of asymmetric jobs arrangement and antennas preparation time variation are utilized in the NPMTW. Given two parent solutions si and sg on machine k, where si represents an initial solution, and sg represents a guiding solution. Each path is an asymmetric movement both in leading from si to sg and in leading from sg to si . So, the asymmetric paths value |δsi ↔sg | is linear growth with the amount of the scheduled jobs in the parent solutions as follows |δsi ↔sg | = |δsi →sg | + |δsg →si | (13) s s δsi →sg = {j = 1, · · · , |sg | Jobsjki ̸= Jobjkg or tsjki ̸= tjkg } (14) s s δsg →si = {j = 1, · · · , |si | Jobsjki ̸= Jobjkg or tsjki ̸= tjkg }, (15) s where tsjki ̸= tjkg notes that the sequential scheduled jobs are arranged asymmetrically. Instead of illogical interrupting jobs within consequent binary vectors, the APR can produce neighborhood solutions linearly that inherit the scheduled jobs from the two parent solutions. The asymmetric crossover operator of the APR is implemented at job unit, which is an improved strategy of the crossover operator at binary vector. The bi-directional paths of APR can be represented as follows. Such as the solutions Sf = {sjf , j = 1, · · · , |δsi →sg |} are produced in the forward APR (si → sg ), the solutions Sb = {sjb , j = 1, · · · , |δsg →si |} are produced in the backward APR (sg → si ). |δsi →sg |

si → sg : si = s1f , s2f , · · · , sf

|δsg →si |

sg → si : sg = s1b , s2b , · · · , sb

= sg

(16)

= si

(17)

During the asymmetric crossover operation one job may conflict with another job at duration time or duplicate in its time window. The direction of the movement can be a basis to correct these problems, by which the job of the destiny solution will be reserved, otherwise will be deleted. Besides, when the constraint of antenna preparation time during jobs switching are not satisfied, the successor should be deleted. C. Positive Adaptive Subsequence Adjustment As an adaptive construction mechanism, the p-ASA is designed to insert unscheduled jobs into the feasible solutions one by one. During the insertion attempt in the p-ASA, one unscheduled job may have multiple feasible inserting positions and the best one should be picked up. Take Figure 2 for an example, in the time window twul of the unscheduled Jobu , several positions of the feasible solution on machine k should be evaluated. A flexibility criterion is represented to determine where is the best inserting position for the unscheduled job. Define

aul Machine k

[

twul twul dulk

bul

]

Jobu

Time

Machine k Feasible solution

Time

Fig. 2: Construction mechanism in the p-ASA

fk (Jobi , Jobu , Jobj ) as a flexibility function that measures ę ߚ ߚ how much slack a scheduleߚ has after inserting Jobu between ߚ the adjacent Jobi and Jobj on machine k. The objective is designed to maximize the flexibility function as follows. max : fk (Jobi , Job u , Jobj ) ߚ

fk (Jobi , Jobu , Jobj ) = Fuk + Buk ,

(18) (19)

where Fuk is the maximum time value that Jobu can be shifted forward on machine k without causing any of the time windows along the scheduled sequence to be violated, and Buk represents the maximum backward shift value that does not produce any violation in the scheduled sequence. { min{tuk −ߚ au , Fik }, if Fߚ ik ≥ siuk Fuk = (20) ߚ min{tuk − au − siuk + Fik , Fik }, ozws. { min{bu − tuk − duk , Bjk }, if Bjk ≥ sujk Buk = min{bu − tuk − duk − sujk + Bjk , Bjk }, ozws. (21) ę

ߚ

ߚ

Then, find the best pair of the adjacent jobs (Jobi , Jobj )∗k on machine k that maximizes fk (Jobi , Jobu , Jobj ); i.e., (Jobi , Jobj )∗k = argmax{fk (Jobi , Jobu , Jobj )}

(22)

D. Probability Crossover by Key-Path For each neighborhood solution generated by the APR, the p-ASA may be repeated |U | times, where U is the unscheduled jobs set. With the neighborhood solutions increasing, the implementation time of the p-ASA repetition could be unacceptable. Therefore, a probability crossover by Key-path method is further proposed to reduce the amount of the neighborhood solutions for the APR. Firstly, define the time-window tightness twtjk of the time window twj = [aj , bj ] of Jobj on machine k as follows. twtjk =

djk bj − aj + 1

(23)

The time-window tightness is an inherent feature of each time window, which can quantify the slack and directly reflect the time-domain flexibility of one job in its time window. Note that the slack of the parent solutions are restricted by some jobs, named as key-paths, which have less flexibility and are harder to shift both forward and backward. Obviously, the bigger twtjk is, the worse flexibility is. Then, the time windows of scheduled jobs are grouped into two categories by a threshold of time-window tightness. The bigger ones are classified as key-paths, which are picked up as the asymmetric crossover operation units to generate the neighborhood solutions, and the others are ignored. Therefore,

the neighborhood solutions of the parent solutions are reduced dramatically, so as theߚ implementationߚ time of the p-ASA repetition. ߚ

Given T is the threshold of time-window tightness, whether Jobj scheduled on machine k is a key-path is defined as flows. { 1 if twtjk ≥ T (24) Cjk = 0 otherwise, ę ߚ ߚ

where Cjk is 1 if twtjk is bigger than T , and then, Jobj scheduled on machine k is classified as a key-path in the APR, otherwise Cjk = 0, the path of Jobj can be ignored. ߚ

ߚ

Further, the threshold T can be chosen dynamically by the statistical property of the scheduled jobs in the parent ߚ solutions. Then, the probability of crossover operation by Keypath can be calculated as follows. Ĵ jk ≥ĴT ߚ|j ߚ∈ J} PT = P‫{׊‬twt

(25)

In LJSP in TDRSS, the probability of crossover operation by Key-path should use a smaller value to reduce the computational complexity. However, in the case of small-scale JSP in TDRSS, the bigger one can be adopted, which should be chosen according to the computing time can be tolerated. ߚ

ߚ

E. Evolutionary APR with Negativeߚ Adaptive Subsequence Adjustment By the evolutionary strategy, the elite solutions obtained in multi-start process can be treated as a group of initial populations. Then, with the method of probability crossover by Key-path, the APR can be implemented to each pair of randomly selected populations to generate better populations, which is the randomly evolutionary progress of the EvAPR. ߚ ߚ Besides, as an adaptive replacement mechanism, the n-ASA is proposed for the EvAPR compatibly, which is designed to ߚ replace one scheduled job by some unscheduled jobs during the EvAPR.

Once a scheduled job is removed from ‫ ׊‬Ĵ ‫ ߚ ؿ‬a ߚfeasible solution, more flexibility will be induced. Take Figure 3 for an instance, Jobβ is going to be deleted from the feasible solution, then, a time interval [aβ , bβ ] will be idle on machine k. Different Machine k Feasible solution

aߚ

Delete

bߚ

Jobߚ

Jobα

Jobγ

Time

Insert Machine k

[

Jobc

]

Time

Candidates: {Jobc}

Fig. 3: Optimization mechanism in the n-ASA from the p-ASA, only a smaller candidate jobs set {Jobc } will be picked up, which should satisfy the constraint as follows. ||[aβ , bβ ] ∩ twc || ≥ wαck + dck + wcγk ,

Comparing to the p-ASA, a much smaller candidate jobs set can be generated basing on the n-ASA, which can shapely reduce the invalid attempts of job insertion, so as the computational complexity. Therefor, the n-ASA can further accelerate the time convergence dramatically during the EvAPR. F. Pseudo-codes of Heuristics The pseudo-codes of the APR with Key-path and p-ASA algorithm for the NPM-TW is represented as Algorithm 1. The loop in line 1 ensures that the jobs with higher priorities must be scheduled first. The line 2 defines the initial solution, which is the scheduled result of the jobs with higher priorities, where {best solutionk,0 , k = 1, · · · , m} denote the empty solutions. The loop in line 3 represents the parallel machines should be scheduled one by one. The loop in line 4 represents the multiple-start process, in which the M axIteration is the repetition times. The lines 5-6 represent the construction process of the guiding solution. The line 7 represents the progressive optimization by APR with Keypath that produces the Key-path neighborhood solutions. The loop in lines 8-10 represent that the p-ASA applies to the neighborhood solutions to find better solutions, denote by the neighborhood elite solutions. In line 11 the best solution with largest scheduled profit is picked up and denoted as elite solution, which represents the best result in current independent process. At last, in line 12 the best solution is picked up by comparing the elite solutions, which are obtained during the multiple-start process. The pseudo-codes of the EvAPR with Key-path and n-ASA algorithm for the NPM-TW is represented as Algorithm 2, which can be integrated into Algorithm 1 by replacing the line 12. The stop criterion in line 1 ensures that the EvAPR can stop adaptively, which can be determined by the computing time or iterative times. In this paper a constraint of iterative times is adopt as the stop criterion. If the best solution did not update for 3 times, the EvAPR procedure will stop and output the current best solution. The loop in line 2 represents the number of elite solutions in pairs, by which each elite solution should be selected once. The line 3 ensures that each pair of the two parent populations is selected randomly. Then, the forward and backward APR with Key-path are implemented in line 4 and 5 respectively. The loop in lines 6-8 represents the optimization application of the n-ASA in the forward Key-path neighborhood solutions. In line 9 the current best solution in the forward APR, defined as forward elite solution, can be obtained. If the forward elite solution is better than the guiding solution, the new one will replace the parent as a new population in lines 14-15, which will updated in line 18 for the next generation. Similarly, the lines 10-13 and 16-17 repeat the former procedure in the backward APR. At last, the best solution obtained in current generation is updated in line 19.

(26)

where ||[aβ , bβ ] ∩ twc || means the overlapped time between [aβ , bβ ] and time window twc of Jobc , which is the servable time on machine k and should be larger than the waiting time wαck + wcγk and the duration time dck .

IV.

S IMULATION R ESULTS

To evaluate the scheduling performance of the proposed heuristics, we use the instances developed by Rojanasoonthon [1], which are generated basing on the original data set

Algorithm 1: APR with Key-path and p-ASA algorithm for the NPM-TW Input: Set of n jobs by priority classes {J1 , · · · · · · , Jρ }; Set of p highest priority unscheduled jobs Up = ∪pi=1 Ji ; Weight of jobs of each priority ωρj ; Number of parallel machines m; Number of independent repetition M axIteration; Crossover probability of Key-path PT ; Output: Best solution of feasible schedule. 1 for p = 1 to ρ do 2 si = initial solutionk,p ← best solutionk,p−1 ; 3 for k = 1 to m do 4 for q = 1 to M axIteration do ′ 5 U ← randomly generate a sequence of unscheduled jobs Up on machine k; ′ 6 sg = guiding solutionk,p ← p-ASA apply to initial solutionk,p with U ; 7 KP N S = {Key-path neighborhood solutions} ← (si ↔ sg ) crossover by APR with the probability of PT ; 8 for l = 1 to |KP N S| do ′ 9 U ← randomly generate a sequence of unscheduled jobs Upl of KP N S(l); ′ 10 neighborhood elite solutionk,p (q, l) ← p-ASA apply to KP N S(l) with U ; 11 elite solutionk,p (q) ← M ax{neighborhood elite solutionk,p (q, l), l = 1, · · · , |KP N S|}; 12 best solutionk,p ← M ax{elite solutionk,p (q), q = 1, · · · , M axIteration}; 13

return {best solutionk,ρ , k = 1, · · · , m}.

Algorithm 2: EvAPR with Key-path and n-ASA algorithm for the NPM-TW Input: {elite solutionk,p (q), q = 1, · · · , M axIteration}; Output: best solutionk,p . 1 while StopCriterion do 2 for h = 1 to M axIteration/2 do 3 (si , sg ) ← randomly select two populations from {elite solutionk,p (q), q = 1, · · · , M axIteration}; 4 F KP N S = {f orward Key-path neighborhood solutions} ← forward APR (si → sg ) with PT ; 5 BKP N S = {backward Key-path neighborhood solutions} ← backward APR (si ← sg ) with PT ; 6 for l = 1 to |F KP N S| do 7 {[ai , bi ], i = 1, · · · }l ← get the idle time intervals during forward APR with Key-path; 8 f orward neighborhood elite solutionk,p (l) ← n-ASA apply to F KP N S(l) with {[ai , bi ], i = 1, · · · }l ; 9 10 11 12 13 14 15 16 17 18 19 20

f orward elite solution ← M ax{f orward neighborhood elite solutionk,p (l), l = 1, · · · , |F KP N S|}; for l = 1 to |BKP N S| do {[ai , bi ], i = 1, · · · }l ← get the idle time intervals during backward APR with Key-path; backward neighborhood elite solutionk,p (l) ← n-ASA apply to BKP N S(l) with {[ai , bi ], i = 1, · · · }l ; backward elite solution ← M ax{backward neighborhood elite solutionk,p (l), l = 1, · · · , |BKP N S|}; if f orward elite solution > sg then sg ← f orward elite solution; if backward elite solution > si then si ← backward elite solution; Update the two populations by si and sg ; best solutionk,p ← M ax{elite solutionk,p (q), q = 1, · · · , M axIteration}; return best solutionk,p .

provided by NASA’ TDRSS. The planning horizon is 86, 400 seconds (1 day). There are 6 antennas of 2 TDRSs treated as 6 parallel machines, and each instance contains 400 jobs with same priority. Then, based on the planning horizon, the parallel machines, and the jobs, the average duration time of each job is 1296 ≈ 86400×6 400 , by which all the available time is used up. There are four APR based heuristics implemented in Matlab (version 2012a) and executed on an Intel-Xeon with 12 processors of 2.67 GHz and 63.9 GB of RAM. Twenty independent instances are randomly generated, and then, the APR based heuristics are used to solve these instances and compared with the GRASP [4]. A smaller value 10 of multi-start process is selected, and the probability crossover of Key-path is 0.5.

TABLE II: Statistics of convergence Heuristic GRASP APR + p-ASA APR + Key-path + p-ASA EvAPR + n-ASA EvAPR + Key-path + n-ASA

380 GRASP APR + p-ASA APR + Key-path + p-ASA EvAPR + n-ASA EvAPR + Key-path + n-ASA

375

Target 88

Maximum value (sec) 44010.19 18933.65 14966.67 12411.00 4623.54

scheduling results of the twenty random instances are shown in Figure 5.

A. Convergence Evaluation

370 365 Scheduled jobs

The time-to-target method [14] is used to analyze the probability distribution of the computing time of the proposed heuristics. For one instance of the twenty instances, each heuristic runs 200 times [15] independently. Each run stops when a solution is generated better than or equal to a given target. The target value is set to 88 on the machine 1, which is a better solution less than the best one, previously known as 92. The experimental probability distribution of the time-to-target on the machine 1 is plotted in Figure 4.

Minimum value (sec) 314.77 261.04 59.98 108.74 83.60

360 355 350 345 340 335 330

1 2

3

4

5

6

1

7

8

9 10 11 12 13 14 15 16 17 18 19 20 Instance ID

Fig. 5: Evaluation by the one-priority instances. 0.8

Cumulative probability

The number of the scheduled jobs by the proposed heuristics perform much better than the GRASP, in which the EvAPR with Key-path and n-ASA performs the best. The statistical results of these random instances are shown in Table III.

0.6

0.4

GRASP empirical APR + p-ASA empirical APR + Key-path + p-ASA empirical

0.2

EvAPR + n-ASA empirical EvAKPR + Key-path + n-ASA empirical 0 0

5000

10000 15000 20000 25000 30000 35000 40000 45000 Time to target solution (sec)

Fig. 4: Empirical probability distribution of time-to-target.

The time-to-target results demonstrate that the EvAPR with Key-path and n-ASA could probability find the target solution in least computing time than the others, in which the maximum computing time is almost 19 of the GRASP and about 12 of the EvAPR with n-ASA as shown in Figure 4. Besides, the variation range of the computing time between the maximum value (4624.54 seconds) and minimum value (83.60 seconds) of time-to-target by the EvAPR with Key-path and n-ASA is smallest than the others too. More statistics are shown in Table II.

TABLE III: Statistics of efficiency

Heuristic GRASP APR+p-ASA APR+Key-path+p-ASA EvAPR+n-ASA EvAPR+Key-path+n-ASA

Avg. scheduled jobs 342.95 347.15 348.1 355.15 355.25

Avg. jobcompletion probability 85.74% 86.79% 87.03% 88.79% 88.81%

B. Efficiency Evaluation

The average number of the scheduled jobs by the EvAPR with n-ASA is about 355.15, which is about 12.2 jobs higher than the GRASP, while the APR with p-ASA is about 347.15 and about 4.2 jobs higher. Besides, with the method of probability crossover by Key-path, the EvAPR with Key-path and n-ASA reaches 355.25 jobs, which is about 12.3 jobs higher than the GRASP, while the APR with Key-path and p-ASA is about 348.1 and about 5.15 jobs higher.

Each instance of the twenty random instance is independently scheduled by the relevant heuristics respectively, and the

Therefore, the best performance of the average jobcompletion probability obtained by the proposed heuristics

reaches 88.81% is beyond 3% higher than 85.74% of the GRASP. Moreover, the statistics show that the probability crossover by Key-path method can not improve the scheduling efficiency of the job-completion probability apparently, but to reduce computational complexity and accelerate the time convergence. V.

C ONCLUSION

To speed up the time convergence of the LJSP in TDRSS, a heuristic job scheduling framework based on the APR is proposed in this paper. Since the APR is just a basic progressive operator, the p-ASA and the n-ASA are proposed respectively as two optimization mechanisms, by which the time-domain flexibility of jobs’ slack time windows is maximum utilized. Then, the adaptive solution construction of the p-ASA and the adaptive job replacement of the n-ASA are realized following the APR. Further, to accelerate the time convergence, the probability crossover by Key-path method is represented. The jobs with slack time windows are differentiated treated based on the statistical property of the time windows, by which the computational complexity of APR can be shapely reduced. Finally, a hybrid evolutionary strategy with the APR is proposed to further accelerate the time convergence, by which a group of solutions are iteratively crossed to generate a better solution randomly. Two verification methods are adopted to evaluate the performance of the proposed heuristics. The simulation results demonstrate that the time convergence of the EvAPR with Key-path and n-ASA is shorten almost 9 times than the GRASP, so as the smaller variation range of the computing time, and better scheduling efficiency can be systematically obtained. ACKNOWLEDGMENT This work was partially supported by the National Natural Science Fundation of China (61132002, 91338101, 91338108), Co-innovation Laboratory of Aerospace Broadband Network Technology, National S&T Major Project (2011ZX03004-00101) and Research Fund of Tsinghua University (2011Z05117). R EFERENCES [1] Rojanasoonthon, S. Parallel machine scheduling with time windows. Ph.D. thesis, Graduate Program in Operations Research and Industrial Engineering, University of Texas, Austin, TX. 2003. [2] https://www.spacecomm.nasa.gov/spacecomm/programs/tdrss/default.cfm [3] Lin Peng, Kuang Linling, Chen Xiang and etc. Adaptive Subsequence Adjustment with Evolutionary Asymmetric Path-relinking for TDRSS Scheduling. Journal of Systems Engineering and Electronics, submitted, 2013. [4] Siwate Rojanasoonthon, Jonathan Bard. A GRASP for Parallel Machine Scheduling with Time Windows. INFORMS Journal on Computing, 2005, 17(1): 32–51. [5] S Rojanasoonthon, JF Bard, and SD Reddy. Algorithms for parallel machine scheduling: a case study of the tracking and date delay satellite system. Journal of the Operational Research Society, 2003, 54:806–821. [6] Yan-Shen Fang; Ying-Wu Chen. Constraint Programming Model of Tdrss Single Access Link Scheduling Problem [C]. International Conference on Machine Learning and Cybernetics, 2006: 948–951. [7] Yan-Shen Fang, Ying-Wu Chen, Zhong-Xun GU. CSP Model of the Relay Satellite Scheduling. Journal of National University of Defence Technology, 2005, 27: 6–10.

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