A note on the time complexity of machine scheduling with DeJong’s learning effect

Accepted Manuscript Technical Note A note on the time complexity of machine scheduling with DeJong’s learning effect Chuanli Zhao, Fang Ji, T.C.E. Cheng, Min Ji PII: DOI: Reference:

S0360-8352(17)30351-0 http://dx.doi.org/10.1016/j.cie.2017.08.010 CAIE 4857

To appear in:

Computers & Industrial Engineering

Received Date: Accepted Date:

28 July 2017 7 August 2017

Please cite this article as: Zhao, C., Ji, F., Cheng, T.C.E., Ji, M., A note on the time complexity of machine scheduling with DeJong’s learning effect, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie. 2017.08.010

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A note on the time complexity of machine scheduling with DeJong’s learning effect Chuanli Zhao1, Fang Ji2, T.C.E. Cheng3, Min Ji2* 1

School of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, PR China 2

School of Management and E-Business, Contemporary Business and Trade Research Center, Zhejiang Gongshang University, Hangzhou 310018, PR China 3

Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Kowloon, Hong Kong

A note on the time complexity of machine scheduling with DeJong’s learning effect

Abstract In a recent paper [Ji, M., Yao, D.L., Yang, Q.Y., Cheng, T.C.E. (2015) Machine scheduling with DeJong’s learning effect. Computers & Industrial Engineering 80, 195-200], the authors provided a fully polynomial-time approximation scheme (FPTAS) for the considered NP-hard problem. Extending this research, the authors in another paper [Ji, M., Tang, X.Y., Zhang, X., Cheng, T.C.E. (2016) Machine scheduling with deteriorating jobs and DeJong’s learning effect. Computers & *

Corresponding author. Tel./Fax: +86 571 28008303 E-mail addresses: [email protected] (C.L. Zhao), [email protected] (J. Fang), [email protected] (T.C.E. Cheng), [email protected] (M. Ji)

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Industrial Engineering, 91, 42-47] gave an FPTAS for a related NP-hard problem. In this note we show that the time complexity of the two FPTASes can be improved. Key words: scheduling; DeJong’s learning effect; FPTAS; time complexity

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1．Introduction We study the DeJong’s learning effect model in scheduling. Following the assumptions and notation given in Ji et al. (2015), we have a set of n non-preemptive jobs J   J1 , J 2 ,

, J n  to be processed on m parallel machines, each of which can

handle at most one job at a time. Based on DeJong’s learning model, the actual processing time of job J j scheduled in position r is p jr  p j  M  1  M  r a  , where p j is the normal processing time of job J j , a  0 is the common learning index, and M is the incompressible factor

 0  M  1 . The objective is to minimize

the makespan. Denoting the problem as Pm DLE Cmax , Ji et al. (2015) pointed out that the problem is NP-hard and devised a fully polynomial-time approximation scheme (FPTAS) to solve the problem in O  n2 m1L2 m1  2 m  time. Extending Ji et al. (2015), Ji et al. (2016) took job deterioration into account, whereby the actual processing time of job J j is p j  p jr   t , where   0 is the common job deterioration rate and t is the starting time of job J j in the schedule. Denoting the problem as Pm p j  p j  M  1  M  r a    t Cmax , they proved that



the problem is NP-hard and provided an FPTAS to solve it in O n3m1L2 m1  2 m



time. In this note we show that the time complexity of the two FPTASes in Ji et al. (2015, 2016) can be reduced. We give the detailed analysis in the next section and suggest some extensions of the analysis in Section 3.

2. The reduced time complexity of the FPTASes Ji et al. (2015, 2016) gave the following results. Theorem 1. (Theorem 4 in Ji et al. (2015)) Given a fixed number of machines m, algorithm

Am

finds

x0  X

for

problem

Pm DLE Cmax

such

that

f n  x0   1    f n  x*  in O  n2 m1L2 m1  2 m  time. Theorem 2. (Theorem 1 in Ji et al. (2016)) Given a fixed number of machines m, 3

Am

algorithm

x0  X

finds

Pm p j  p j  M  1  M  r a    t Cmax O  n3m1L2 m1  2 m  time.

such

for

problem

Q  x0   1    Q  x* 

that

in

While FPTASes in Ji et al. (2015, 2016) are correct, their time complexity can be improved. In the FPTASes, they used the following Procedure Partition

 A, e,   ,

first developed by Kovalyov and Kubiak (1998), in which A  X , where X is the set of all the vectors x   x1 , x2 ,

, xn  , e is a non-negative integer function on X , and

0    1.

Procedure Partition Step 1.

 A, e,  

1 2 Arrange the vectors x  A in the order x  , x  ,

    

0  e x   e x 1

Step 2.

2

 

 e x  . 1

1 2 Assign the vectors x  , x  ,

 

 

e x 1   1    e x  i

1

A , x  such that

i , x 1  to set A1e until i1 is found such that



and e x

 i1 1

  1    e  x   . If such 1

i1 does

not exist, then take Akee  A1e  A , and stop. i 1 i 2 Assign the vectors x 1  , x 1  ,

     1    e  x  

that e x

i1 1

i2

i , x 2  to set A2e until i2 is found such



and e x

 i2 1

  1    e  x   . If such i1 1

i2 does not exist, then take Akee  A2e  A  A1e , and stop. Continue the above construction until x

 A

is included in Akee for some

ke . We first give a lemma, which is crucial to improving the time complexity of the FPTASes.





Lemma 1. Given a finite set E  e  x  x  A , if the number of elements in E is less than or equal to N (i.e., | E | N ), then Ke  N . Proof. Let v j be the minimum value of e  x  over x  Aej , j  1,2, be the maximum value of e  x  when x  Aej , j  1,2, 4

, ke , and u j

, ke . The method of

constructing

subsets

Aej

in

Partition

 A, e,  

Procedure

v j  u j  v j 1     v j 1 . The worst case is v j  u j , j  1,2, for the partition. That is, the vectors in Partition Procedure

ensures

that

, ke , when   0

 A, e,  

are divided into

different subsets according to the different values of e  x  . Therefore, the number of

 A, e,  

subsets generated by Partition Procedure

is less than or equal to N.

□

The following results help reduce the time complexity of the FPTASes in Ji et al. (2015, 2016). Theorem 3. (Improvement of Theorem 1) Given a fixed number of machines m ,

Am

algorithm

x0  X

finds

for

Pm DLE Cmax

problem

such

that

f n  x0   1    f n  x*  in O  n2 m +1Lm1  m  time. Proof. We follow the FPTAS for problem Pm DLE Cmax in Ji et al. (2015). In Section 4.1.2 of Ji et al. (2015), the partial recursive functions on X, which is the set of all the vectors x   x1 , x2 ,

, xn  with x j  k , j  1,2,

, n , and k  1,2,

,m ,

are given below:

r0i  x   0 , i  1, 2,

, m,

rjk  x   rjk1  x   1 , for x j  k , rji  x   rji1  x  , for x j  k , i  k. Through derivation, we can easily obtain

E ij  0,1, 2,



Thus, the number of E ij , i  1, 2, 1, we have krji  n  1, i  1,2, By

Property





, n for E ij  rji  x  x  A , i  1, 2,

3

, m is no more than  n  1 . Due to Lemma

,m. of

Ji

et



krji  min  n  1 ,  2  n  1 log  n    2  , i  1,2, krji  n  1, i  1,2,

,m.

al.

, m.

(2015),

we

have

It

follows

that

, m , which is an improvement over the analysis in Ji et al.

(2015). So we obtain the improved time complexity of O  n2 m +1Lm1  m  for the FPTAS.

□

Theorem 4. (Improvement of Theorem 2) Given a fixed number of machines m, 5

Am

algorithm

x0  X

finds

Pm p j  p j  M  1  M  r a    t Cmax O  n3m +1Lm1  m  time.

such

that

for

problem

Q  x0   1    Q  x* 

in

Proof. Similarly, we follow all the assumptions and notation, and the algorithm in Ji et

al.

(2016).

The

partial

recursive

functions

on

X

for

problem

Pm p j  p j  M  1  M  r a    t Cmax are given below: r0i  x   0 , i  1, 2, , m, rjk  x   rjk1  x   1 , for x j  k , rji  x   rji1  x  , for x j  k , i  k , R0i  x   0 , i  1, 2,

R kj  x 

, m,

 r  x    r  x   1 a

k j 1

k j 1

a

, for x j  k ,

Rij  x   Rij 1  x  , for x j  k , i  k. Through derivation, we can easily obtain

E ij  0,1, 2,



T ji  0,1, 2a ,3a



Lemma 1, we obtain kRij  n  1, i  1,2, Property



3

in

,m.

, m is no more than  n  1 . Due to

,m.

Ji

et

al.

    2 , i  1, 2,

 kRi  min  n  1 ,  2C  n  1 log 1a j n  

kRij  n  1, i  1,2,

,m,

, na  for T ji  Rij  x  x  A , i  1, 2,

Therefore, the number of T ji , i  1, 2,

By



, n for E ij  rji  x  x  A , i  1, 2,

(2016),

, m.

It

we follows

have that

, m. Following the analysis in Ji et al. (2016), we obtain the

improved time complexity of O  n3m +1Lm1  m  for the FPTAS.

□

Based on the analysis above, the time complexity of the FPTAS for problem

Pm DLE Cmax is reduced from O  n2 m1L2 m1  2 m  to O  n2 m +1Lm1  m  and that of the FPTAS for problem Pm p j  p j  M  1  M  r a    t Cmax decreases from

O  n 3m1L2 m 1 2 m  to O  n3m +1Lm1  m  .

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3. Comments In this section we improve the properties of Partition

 A, e,   ,

which were first

derived by Kovalyov and Kubiak (1998), and were subsequently developed by Ji et al. (2016). Some related properties are as follows:

   2

Property 1. (Lemma 2 in Kovalyov and Kubiak (1998)) Ke  log e x

 

A

for

A 0    1 and e x  1.



 1    2 for i  1    e x 1  

Property 2. (Property 3 in Ji et al. (2016)) K e  C log 



0    1 and 0  e x 1

i 1



   1, where

e x

A

C  max 1 log 2,1 .

Based on these properties, we improve the properties of Partition

 A, e,  

by

Lemma 1 (proved in Section 2), where N is the number of e  x  values when x  A . The properties are given as follows:

     2, N for 0    1 and e  x   1.

Property 3. Ke  min log e x

Property



0  e x 1

i 1

A

A

4.

    1 K e  min C log    2, N   i  1    e x 1    



e x

   1, where A

for

0   1

and

C  max 1 log 2,1 .

The above properties are improvements over those given in Kovalyov and Kubiak (1998), and Ji et al. (2016). They can be used in all the algorithms that adopt Procedure Partition

 A, e,   . In essence, they help trim the number of the subsets in

the partitioning operation, leading to improvement in the time complexity of the corresponding FPTAS.

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References Ji, M., Tang, X.Y., Zhang, X., Cheng, T.C.E. (2016) Machine scheduling with deteriorating jobs and DeJong’s learning effect. Computers & Industrial Engineering 91, 42-47. Ji, M., Yao, D.L., Yang, Q.Y., Cheng, T.C.E. (2015) Machine scheduling with DeJong’s learning effect. Computers & Industrial Engineering 80, 195-200. Kovalyov, M.Y., Kubiak, W. (1998) A fully polynomial approximation scheme for minimizing makespan of deteriorating jobs. Journal of Heuristics 3(4), 287-297.

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Acknowledgements This research was supported in part by Zhejiang Provincial Natural Science Foundation of China (Grant No. LR15G010001 and Grant No. LQ15G010001), and the Contemporary Business and Trade Research Center of Zhejiang Gongshang University, which is a key Research Institute of Social Sciences and Humanities of the Ministry of Education. Cheng was supported in part by The Hong Kong Polytechnic University under the Fung Yiu King - Wing Hang Bank Endowed Professorship in Business Administration.

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Highlights  This note improves time complexity of two FPTASes provided in two papers published in CAIE.  Further general comments are also proposed in the last.

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