- Email: [email protected]
Note on Shim and Kim’s lower bounds for scheduling on identical parallel machines to minimize total tardiness
European Journal of Operational Research 197 (2009) 422–426
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Short Communication
Note on Shim and Kim’s lower bounds for scheduling on identical parallel machines to minimize total tardiness Jeffrey Schaller * Department of Business Administration, Eastern Connecticut State University, 83 Windham Street, Willimantic, CT 06226-2295, United States
a r t i c l e
i n f o
Article history: Received 20 July 2007 Accepted 4 July 2008 Available online 17 July 2008 Keywords: Scheduling Branch-and-bound Lower bound Parallel machines
a b s t r a c t This note introduces a new lower bound for the problem of scheduling on parallel identical machines to minimize total tardiness that is based on the concepts used in the two lower bounds developed by Shim and Kim [Shim, S.O., Kim, Y.D., 2007. Scheduling on parallel identical machines to minimize total tardiness. European Journal of Operational Research 177, 135–146]. The note shows that the new lower bound dominates the three lower bounds used in Shim and Kim’s branch-and-bound algorithm and can be used in place of these lower bounds to lower the enumeration required. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction This note introduces a new lower bound that is based on the concepts used in lower bounds developed by Shim and Kim (2007) for assigning and sequencing a set of jobs on identical parallel machines that will minimize the total tardiness of the jobs. The note also shows that the new lower bound requires no more time to calculate than the lower bounds developed by Shim and Kim (2007). To help complete jobs in a timely manner there may be two or more identical machines of a type. These machines are referred to as identical parallel machines and the scheduling of jobs on the machines is referred to as identical parallel machine scheduling. The tardiness of a job is defined as the completion time of the job minus the due date for the job if the job is completed after the due date and the tardiness is equal to zero if the job is completed before the due date. Suppose there is a set of n jobs to be processed on m machines. Let pj, Cj, and dj represent the processing time, the completion time, and the due date of job j (j = 1, . . . , n), respectively. The tardiness of job j, Tj is P defined as: Tj = max{Cj dj, 0}, for j = 1, . . . , n. The objective function, Z, can be expressed as: Z ¼ nj¼1 T j . Using the notation of Graham P Tj. et al. (1979) this problem is referred to as Pm Three papers have addressed the problem of scheduling jobs on identical parallel machines to minimize total tardiness using branchand-bound algorithms that incorporate lower bounds. Azizoglu and Kirca (1998) developed several dominance properties and a lower bound that includes jobs that are not yet in a partial schedule. Their lower bound is based on a relaxed problem that allows jobs to be simultaneously processed on more than one machine. Yalaoui and Chu (2002) developed a branch-and-bound algorithm that included additional dominance properties and a new lower bound. Shim and Kim (2007) developed a lower bound that improves the lower bound developed by Azizoglu and Kirca (1998). Section 2 presents the lower bounds used by Shim and Kim (2007). Section 3 presents the new lower bound, Section 4 presents a computational test and Section 5 concludes the note. 2. Lower bounds This section first presents the three lower bounds used by Shim and Kim (2007). These lower bounds are referred to as LB1, LB2 and LB3. The section concludes by showing that one of the lower bounds (LB2) is dominated by another lower bound (LB3). To facilitate the discussion in this section and the next it is assumed that jobs are indexed to correspond to their completion times (C1 6 C2 6 6 Cn). The following notation is used in the lower bounds. Let r represent a partial schedule. S(r) is the set of jobs in the
* Tel.: +1 860 465 5226; fax: +1 860 465 4469. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.07.005
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J. Schaller / European Journal of Operational Research 197 (2009) 422–426
partial schedule r. Ck (r) is the completion time of the last job scheduled on machine k in the partial schedule r. Let U (r) equal mink¼1;...;m Ck (r), which is the earliest time a machine becomes available. Let cl (r) denote the lth smallest value of the Ck (r)’s. Let r0 be the set of jobs not included in r. Let n0 be equal to the number of jobs included in r0 . Let dEDD[j] (r0 ) equal the due date of the job sequenced in the jth position if the jobs in r0 are sorted in earliest due date order (dEDD[j] 6 dEDD[k] if j < k). Let pSPT[j](r0 ) equal the processing time of the job in the jth position of a sequence formed by sorting the jobs in r0 in shortest processing time order (pSPT[a](r0 ) 6 pSPT[b](r0 ) if a < b). 2.1. Lower bounds used in Shim and Kim (2007) Three lower bounds are used in Shim and Kim’s (2007) branch-and-bound algorithm to obtain a lower bound on the tardiness of the jobs not included in a partial schedule r. The maximum of these lower bounds is added to the total tardiness of the jobs included in a partial schedule r to obtain a lower bound on the total tardiness of the completion of the schedule. The resulting lower bound is then compared against the best solution found so far to see if the partial schedule r can be eliminated from further consideration. These lower bounds are referred to as LB1, LB2, and LB3. The first lower bound (LB1) is a job based lower bound that compares the earliest possible completion time for a job to its due date to obtain a lower bound on the job’s tardiness. The earliest possible completion time for each job is found by adding the job’s processing time P to the completion time of the first available machine (U (r)). Therefore, the lower bound is computed as: LB1 ¼ j2r0 maxfUðrÞ þ pj dj ; 0g. To calculate LB1 first U(r) must be found which requires O(m) time then U (r) + pj dj must be calculated for each job j 2 r0 which requires O(n0 ) = O(n) time. Therefore, the complexity of LB1 is O(m) + O(n). The second lower bound (LB2) was developed by Azizoglu and Kirca (1998) and is obtained by relaxing the problem to a single-machine tardiness problem in which the processing time of each job j is set to pj/m. Therefore, a lower bound on the completion time of the job comP pleted jth among the jobs in r0 can be computed as UðrÞ þ ji¼1 pspt½i ðr0 Þ=m. The lower bound uses the following lemma from Kim (1995). Pn 0 P 0 Lemma 1. If C1 6 C2 6 6 Cn then maxðC j dedd½j ðr0 Þ; 0Þ 6 nj¼1 maxðC j dj ; 0Þ. The second lower bound LB2 is computed as h . j¼1 i Pn 0 Pj LB2 ¼ j¼1 max UðrÞ þ i¼1 pspt½i ðr0 Þ m dedd½j ðr0 Þ; 0 . To calculate LB2 first U(r) must be found and the processing times (pj) and due dates . P (dj) must be sorted in non-descending order then UðrÞ þ ji¼1 pspt½i ðr0 Þ m dedd½j ðr0 Þ must be calculated for each job j 2 r0 . Finding U (r) requires O(m) time. Each of the two sorts requires O(nln (n)) time and the final calculation requires O(n) time. The complexity of LB2 is O(m) + O(2nln (n)) + O(n). Shim and Kim (2007) use the following lemma to develop a third lower bound. The lemma provides a lower bound on the completion time of the job that is completed jth among the jobs in r0 . Lemma 2. In the identical parallel machine scheduling problem, the completion time of the job that is completed jth among those in r0 is no less than
minl6k6minðj;mÞ
"( k X
cl ðrÞ þ
j X
), # k :
pspt½i
i¼1
l¼1
Proof. This proof is provided in Shim and Kim (2007).
h
The third lower bound (LB3) can then be computed as: LB3 ¼
Pn0
j¼1
h hnP o. i i Pj k 0 max minl6k6minðj;mÞ k dedd½j ðr0 Þ; 0 . l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ
The computational requirement to calculate LB3 is very similar to that of LB2. The same two sorts as required by LB2 are required by LB3 and a final calculation is required that has the same requirement as that of LB2. To calculate LB3 the machine completion times must be sorted (cl (r)) which requires O(mln (m)) time. The complexity of LB3 is O(mln (m)) + O(2nln (n)) + O(n). 2.2. LB2 is dominated by LB3 As shown by the following theorem for any partial schedule r the third lower bound (LB3) dominates the second lower bound (LB2). Theorem 1. For any given partial schedule r LB2 6 LB3. Proof. In both lower bounds dedd[j] is subtracted from a lower bound on the completion time of the job completed jth among the jobs in r0 therefore for any given partial schedule r to prove LB2 6 LB3 it must be shown
UðrÞ þ
j X
, pspt½i ðr0 Þ
m6
i¼1
Since k 6 m,
Pj
i¼1 pspt½i ð
6 minl6k6minðj;mÞ
hnP
.
r0 Þ m 6
min
"( k X
l6k6minðj;mÞ
Pj
i¼1 pspt½i ð
cl ðrÞ þ
l¼1
j X
), # pspt½i ðr0 Þ
k for j ¼ 1; . . . ; n0 :
i¼1
.
r0 Þ k. U (r) = c1 (r) 6 cl (r) for l = 2, . . . , k therefore UðrÞ 6
o. Pj k 0 k for j = 1, . . . , n0 and LB2 6 LB3. h l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ
Pk
.
c rÞ k ) UðrÞþ
l¼1 l ð
Pj
r0 Þ=m
i¼1 pspt½i ð
The next section presents a new lower bound. 3. A new lower bound The concepts used to form LB1 and LB3 can be combined to form a new lower bound referred to as LB4. This section presents this lower bound. It is then shown that LB4 dominates LB1 and LB3. The section concludes by showing that the complexity of LB4 is the same as LB1 and LB3.
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J. Schaller / European Journal of Operational Research 197 (2009) 422–426
3.1. LB4 0
To develop the new lower bound for the problem modified due dates dj ðr0 Þ are created for each job j 2 r0 as follows r0 Þ ¼ maxfdj ; UðrÞ þ pj g. The due dates used in LB3 are replaced by the modified due dates and the result is added to LB1. Formally LB4 is computed as
0 dj ð
LB4 ¼ LB1 þ
n0 X j¼1
" max
"( min
l6k6minðj;mÞ
k X l¼1
cl ðrÞ þ
j X
), # pspt½i ðr0 Þ
# 0
k dedd½j ðr0 Þ; 0 :
i¼1
Theorem 2. For any given partial schedule r LB4 is a lower bound on the sum of the tardiness of the jobs in the set r0 for any completion of the partial schedule. P Proof. The sum of tardiness of the jobs in the set r0 for any completion of the partial schedule r ¼ j2r0 maxfC j dj ; 0g. This theorem will P P 0 be proved in two parts: (1) we will first show that j2r0 maxfC j dj ; 0g can be rewritten as LB1 þ j2r0 maxfC j dj ðr0 Þ; 0g, Since LB1 P 0 h h n o . i i Pj P Pk P0 0 0 0 k dedd½j ðr0 Þ; 0 6 j2r0 maxfC j dj ðr0 Þ; 0g to complete we will then show 2) that nj¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ the proof. h P
P 0 0 0 maxfC j dj ; 0g ¼ LB1 þ j2r0 maxfC j dj ðr0 Þ; 0g. MaxfC j dj ; 0g ¼ maxfðC j dj ðr0 ÞÞ þ ðdj ðr0 Þ dj Þ; 0g for j 2 r0 . Since 0 0 0 0 0 Cj P U(r) + pj, dj ðr Þ ¼ dj if dj P Cj and maxfC j dj ; 0g ¼ maxfðC j dj ðr ÞÞ; 0g ¼ 0. If Cj > dj then C j dj ¼ ðC j dj P 0 0 0 0 0 0 0 0 ðr ÞÞ þ ðdj ðr Þ dj Þ > 0. Therefore, fmax C j dj ; 0g ¼ maxfC j dj ðr Þ; 0g þ dj ðr Þ dj for j 2 r so 0g ¼ j2r0 maxfC j dj ;P P 0 0 0 0 0 r0 Þ ¼ maxfUðrÞ þ pj ; dj g ) d0j ðr0Þ dj ¼ maxfUðrÞ þ pj dj ; 0g therefore j2r0 maxfC j dj ðr Þ; 0g þ dj ðr Þ dj Þ. dj ðP j2r0 0 0 0 ðmaxfC j dj ðr0 Þ; 0g þ dj ðr0 Þ dj Þ ¼ LB1 þ j2r0 maxfC j dj ðr0Þ; 0g and this completes the first part of the proof. h hnP o. i i P Pn0 Pj 0 0 k 0 (2) k dedd½j ðr0 Þ; 0 6 j2r0 maxfC j dj ðr0 Þ; 0g. It is shown by Shim and Kim j¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ hnP o . i Pj k (2007) that minl6k6minðj;mÞ k is a lower bound on the completion time of the job completed l¼1 cl ðrÞ þ i¼1 pspt½i h hnP o. i i P0 P Pj 0 0 k 0 k dedd½j ðr0 Þ; 0 6 nj¼1 maxfC j jth among the jobs in r0 (Cj) therefore nj¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ
(1)
j2r0
P0 P 0 0 0 dedd½j ðr0 Þ; 0g. To complete the proof it will be shown nj¼1 maxfC j dedd½j ðr0 Þ; 0g 6 j2r0 maxfC j dj ðr0 Þ; 0g. Suppose we have a sequence of modified due dates that are not in EDD order. It can be shown that the sequence can be converted to a sequence of modified due dates that is in EDD order by a series of exchanges of pairs of modified due dates. To prove P P 0 0 0 0 0 0 0 0 j2r0 maxfC j dedd½j ðr Þ; 0g 6 j2r0 maxfC j dj ðr Þ; 0g it is sufficient to prove if dj ðr Þ > dk ðr Þ and j < k then exchanging the two due dates will result in an objective value when compared to the completion times that is no greater than the resulting 0 0 0 0 objective value before the exchange. Let dj (r0 ) and dk (r0 ) be two modified due dates such that j < k and dj ðr0 Þ > dk ðr0 Þ. Let 00 0 d be modified due dates that are the same as the modified due dates d except the modified due dates for j and k are exchanged. 00 0 00 0 00 0 0 00 dp ðr0 Þ ¼ dp (r0 ) if p – j or k, dj ðr0 Þ ¼ dk (r0 ), and dk ðr0 Þ ¼ dj ðr0Þ. Let Z p ¼ maxfC p dp ðr0Þ; 0g and Z 0p ¼ maxfC p dp ðr0 Þ; 0g. To P0 P0 show nj¼1 Z 0p 6 nj¼1 Z p it must be shown Z 0j þ Z 0k 6 Z j þ Z k since Z 0p ¼ Z p if p – j or k. The following two cases cover all possibil00 00 ities: 1) C k dk ðr0 Þ P 0 and 2) C k dk ðr0 Þ < 0. 00 00 00 0 0 0 0 0 (Case 1) C k dk ðr Þ P 0. C k dk ðr0Þ P 0 ) Z 0k ¼ C k dk ðr0Þ ¼ C k dj ðr0 Þ. Since dj ðr0 Þ > dk (r0 ) and C k dj ðr0 Þ P 0 then 0 0 0 0 0 0 00 00 0 0 0 0 0 0 0 0 0 0 Z k ¼ C k dk ðr Þ ¼ C k dj ðr Þ þ ðdj ðr Þ dk ðr ÞÞ ¼ Z k þ dj ðr Þ dk ðr Þ. Z j ¼ maxfC j dj ðr Þ; 0g. Since dj ðr0 Þ ¼ dk ðr0 Þ and 0 0 0 0 0 0 0 0 0 0 0 dj ðr Þ >0 dk ðr Þ then Z j 6 Z0 j þ dj ðr Þ dk ðr Þ and Z j þ Z k 6 Z j þ Z k for this case. 0 0 00 0 0 (Case 2) C k dk ðr0 Þ < 0. C k dk ðr0 Þ < 0 ) Z 0k ¼ 0. Since Ck > Cj and dk ðr0 Þ ¼ dj ðr0 Þ then Zj = 0. Z k ¼ maxfC k dk ðr0Þ; 0g. 00 0 0 0 0 0 0 0 Z j ¼ maxfC j dj ðr Þ; 0g ¼ maxfC j dk ðr Þ; 0g. Since C k > C j Z k P Z j therefore Z j þ Z k 6 Z j þ Z k for this case. This completes the second part of the proof. h
3.2. LB4 dominates LB1 and LB3 The proofs of the following two theorems show that for any given partial schedule r the lower bounds LB1 and LB3 are dominated by the lower bound LB4. Theorem 3. For any given partial schedule r LB1 6 LB4. h hnP o. i i h hnP Pj Pn0 P0 0 k k 0 k dedd½j ðr0 Þ; 0 . Since Proof. LB4 ¼ LB1 þ nj¼1 max minl6k6minðj;mÞ j¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ Pj 0 0 0 h i¼1 pspt½i ðr Þg=k dedd½j ðr Þ; 0 P 0 LB1 6 LB4. Theorem 4. For any given partial schedule r LB3 6 LB4. h hnP o. i i P0 Pj 0 0 k 0 k dedd½j ðr0 Þ; 0 . LB4 = LB1 + LB30 . Since dedd½j ðr0 Þ P dedd½j ðr0 Þ for Proof. Let LB30 ¼ nj¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ P P P 0 0 0 0 0 n 0 0 0 0 LB3 LB30 6 nj¼1 ðdedd½j ðr0 Þ dedd½j ðr0ÞÞ ¼ LB1 therefore LB4 P j = 1, . . . , n0 j¼1 ðdedd½j ðr Þ dedd½j ðr ÞÞ ¼ j2r0 ðdj ðr Þ dj ðr ÞÞ ¼ LB1. LB3. h
425
J. Schaller / European Journal of Operational Research 197 (2009) 422–426 Table 1 Average reduction in seconds when m = 2 m=2
Number of jobs
Set 1 2 3 4 5 6
10 0.17 0.11 0.83 0.25 1.71 0.11
12 2.63 3.66 21.88 9.65 72.18 1.00
14 44.98 198.51 749.44 120.38 2278.33 30.92
12 11.60 27.78 28.27 8.30 21.71 2.96
14 886.64 949.89 1561.84 420.46 1507.85 19.19
Table 2 Average reduction in seconds when m = 4 m=4
Number of jobs
Set 1 2 3 4 5 6
10 0.39 0.54 0.58 0.26 0.42 0.06
3.3. Comparison of the complexity of LB4 with the lower bounds of Shim and Kim (2007) 0
To calculate LB4 LB1 must be calculated, the modified due dates must be created, processing times (pj) and modified due dates (dj (r0 )) h hnP k must be sorted in non-descending order, the machine completion times must be sorted (cl(r)), and max minl6k6minðj;mÞ l¼1 cl ðrÞ þ Pj 0 0 0 0 i¼1 pspt½i ðr Þg=k dedd½j ðr Þ; 0 must be calculated for each job j 2 r . The modified due dates can be created during the calculation of LB1 and U (r) (used to calculate LB1 and create the modified due dates) can be found during the sort of the machine completion times (cl (r)) so the complexity of LB4 is O(mln(m)) + O(2nln(n)) + 2 O(n). Shim and Kim (2007) calculate each of the lower bounds (LB1, LB2, and LB3) and then used the maximum value found when determining whether to prune a branch in their branch-and-bound algorithm. In this note, it was shown that LB2 is dominated by LB3 so the calculations required for LB2 can be eliminated without reducing the bound found. Comparing the computational requirement to calculate LB4 with that of calculating LB1 and LB3 it is found that LB4 requires slightly less time than calculating both LB1 and LB3 therefore using LB4 instead of the maximum of LB1 and LB3 will not hurt the performance of a branch-and-bound algorithm and offers the opportunity to significantly improve it. 4. Computational test To test whether using the new lower bound (LB4) in place of the two lower bounds developed by Shim and Kim (2007) (LB1 and LB3) increases the efficiency of a branch-and-bound algorithm the following test was used. A branch-and–bound algorithm was coded. Nodes in the branch-and-bound algorithm correspond to a partial schedule. A depth-first branching rule is used so the node with the most jobs in the corresponding partial schedule is selected for branching. Ties are broken in favor of the node with the minimum lower bound. The branch-and-bound algorithm was first tested on thirty six problem sets using Shim and Kim’s (2007) lower bounds and the average time to solve problems within each problem set was recorded. Then the branch-and-bound algorithm was tested using the new lower bound instead of Shim and Kim (2007)’s lower bounds. The branch-and-bound algorithm was coded in Turbo Pascal and was tested on a HP LP1965 2.4 GHz PC. Each problem set had 10 problems. The processing times of jobs for all of the problems were generated using a uniform distribution over the integers 1–50. All problems within a problem set have the same number of jobs, machines and due dates are drawn from the same distribution. The numbers of jobs tested were n = 10, 12 and 14. The number of machines tested were m = 2 and 4. The due dates for the jobs were randomly generated using a uniform distribution over the integers .AP (1 r R/2) and AP (1 r + R/2), where AP is Pn the total processing time for all n jobs divided by the number of machines AP ¼ j¼1 pj m , and R and r are two parameters called due date range and tardiness factors. Six sets of these parameters were used for each n and m combination: R = 0.5 and r = 0.75 (set 1), R = 1.00 and r = 0.75 (set 2), R = 0.5 and r = 0.5 (set 3), R = 1.00 and r = 0.5 (set 4), R = 0.5 and r = 0.25 (set 5), R = 1.00 and r = 0.25 (set 6). Tables 1 and 2 show the average reduction per problem in computation time when the new lower bound is used in place of the lower bounds developed by Shim and Kim (2007). Table 1 shows the results for problems sets with two machines and Table 2 shows the results for problem sets with four machines. The results show that using the new lower bound in place of the lower bounds developed by Shim and Kim (2007) reduced the time required to solve problems for all 36 problem sets. The results also show that as the number of jobs increases the computation savings increase rapidly for both the problem sets with two machines and four machines. 5. Conclusion This note showed that one of the lower bounds for identical parallel machine scheduling to minimize total tardiness developed by Shim and Kim (2007) (LB3) dominates a lower bound developed by Azizoglu and Kirca (1998) (LB2). The note also introduced a new lower bound
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J. Schaller / European Journal of Operational Research 197 (2009) 422–426
for the problem (LB4) that utilizes concepts based on the two lower bounds developed by Shim and Kim (2007) (LB1 and LB3). LB4 was shown to dominate LB1 and LB3 and since LB3 dominates LB2 LB4 also dominates LB2. The lower bound LB4 can be used in a branchand-bound algorithm in place of LB1, LB2 and LB3 to reduce the enumeration that is required. It was also shown that the computational requirement to calculate LB4 is less than that of calculating LB1, LB2 and LB3 so using the new lower bound improves the efficiency of a branch-and-bound algorithm. This improvement in efficiency was also demonstrated in a computational test. References Azizoglu, M., Kirca, O., 1998. Tardiness minimization on parallel machines. International Journal of Production Economics 55, 163–168. Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinooy Kan, A.H.G., 1979. Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics 5, 287–326. Kim, Y.D., 1995. Minimizing total tardiness in permutation flowshops. European Journal of Operational Research 85, 541–555. Shim, S.O., Kim, Y.D., 2007. Scheduling on parallel identical machines to minimize total tardiness. European Journal of Operational Research 177, 135–146. Yalaoui, F., Chu, C., 2002. Parallel machine scheduling to minimize total tardiness. International Journal of Production Economics 76, 265–279.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Short Communication
Note on Shim and Kim’s lower bounds for scheduling on identical parallel machines to minimize total tardiness Jeffrey Schaller * Department of Business Administration, Eastern Connecticut State University, 83 Windham Street, Willimantic, CT 06226-2295, United States
a r t i c l e
i n f o
Article history: Received 20 July 2007 Accepted 4 July 2008 Available online 17 July 2008 Keywords: Scheduling Branch-and-bound Lower bound Parallel machines
a b s t r a c t This note introduces a new lower bound for the problem of scheduling on parallel identical machines to minimize total tardiness that is based on the concepts used in the two lower bounds developed by Shim and Kim [Shim, S.O., Kim, Y.D., 2007. Scheduling on parallel identical machines to minimize total tardiness. European Journal of Operational Research 177, 135–146]. The note shows that the new lower bound dominates the three lower bounds used in Shim and Kim’s branch-and-bound algorithm and can be used in place of these lower bounds to lower the enumeration required. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction This note introduces a new lower bound that is based on the concepts used in lower bounds developed by Shim and Kim (2007) for assigning and sequencing a set of jobs on identical parallel machines that will minimize the total tardiness of the jobs. The note also shows that the new lower bound requires no more time to calculate than the lower bounds developed by Shim and Kim (2007). To help complete jobs in a timely manner there may be two or more identical machines of a type. These machines are referred to as identical parallel machines and the scheduling of jobs on the machines is referred to as identical parallel machine scheduling. The tardiness of a job is defined as the completion time of the job minus the due date for the job if the job is completed after the due date and the tardiness is equal to zero if the job is completed before the due date. Suppose there is a set of n jobs to be processed on m machines. Let pj, Cj, and dj represent the processing time, the completion time, and the due date of job j (j = 1, . . . , n), respectively. The tardiness of job j, Tj is P defined as: Tj = max{Cj dj, 0}, for j = 1, . . . , n. The objective function, Z, can be expressed as: Z ¼ nj¼1 T j . Using the notation of Graham P Tj. et al. (1979) this problem is referred to as Pm Three papers have addressed the problem of scheduling jobs on identical parallel machines to minimize total tardiness using branchand-bound algorithms that incorporate lower bounds. Azizoglu and Kirca (1998) developed several dominance properties and a lower bound that includes jobs that are not yet in a partial schedule. Their lower bound is based on a relaxed problem that allows jobs to be simultaneously processed on more than one machine. Yalaoui and Chu (2002) developed a branch-and-bound algorithm that included additional dominance properties and a new lower bound. Shim and Kim (2007) developed a lower bound that improves the lower bound developed by Azizoglu and Kirca (1998). Section 2 presents the lower bounds used by Shim and Kim (2007). Section 3 presents the new lower bound, Section 4 presents a computational test and Section 5 concludes the note. 2. Lower bounds This section first presents the three lower bounds used by Shim and Kim (2007). These lower bounds are referred to as LB1, LB2 and LB3. The section concludes by showing that one of the lower bounds (LB2) is dominated by another lower bound (LB3). To facilitate the discussion in this section and the next it is assumed that jobs are indexed to correspond to their completion times (C1 6 C2 6 6 Cn). The following notation is used in the lower bounds. Let r represent a partial schedule. S(r) is the set of jobs in the
* Tel.: +1 860 465 5226; fax: +1 860 465 4469. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.07.005
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partial schedule r. Ck (r) is the completion time of the last job scheduled on machine k in the partial schedule r. Let U (r) equal mink¼1;...;m Ck (r), which is the earliest time a machine becomes available. Let cl (r) denote the lth smallest value of the Ck (r)’s. Let r0 be the set of jobs not included in r. Let n0 be equal to the number of jobs included in r0 . Let dEDD[j] (r0 ) equal the due date of the job sequenced in the jth position if the jobs in r0 are sorted in earliest due date order (dEDD[j] 6 dEDD[k] if j < k). Let pSPT[j](r0 ) equal the processing time of the job in the jth position of a sequence formed by sorting the jobs in r0 in shortest processing time order (pSPT[a](r0 ) 6 pSPT[b](r0 ) if a < b). 2.1. Lower bounds used in Shim and Kim (2007) Three lower bounds are used in Shim and Kim’s (2007) branch-and-bound algorithm to obtain a lower bound on the tardiness of the jobs not included in a partial schedule r. The maximum of these lower bounds is added to the total tardiness of the jobs included in a partial schedule r to obtain a lower bound on the total tardiness of the completion of the schedule. The resulting lower bound is then compared against the best solution found so far to see if the partial schedule r can be eliminated from further consideration. These lower bounds are referred to as LB1, LB2, and LB3. The first lower bound (LB1) is a job based lower bound that compares the earliest possible completion time for a job to its due date to obtain a lower bound on the job’s tardiness. The earliest possible completion time for each job is found by adding the job’s processing time P to the completion time of the first available machine (U (r)). Therefore, the lower bound is computed as: LB1 ¼ j2r0 maxfUðrÞ þ pj dj ; 0g. To calculate LB1 first U(r) must be found which requires O(m) time then U (r) + pj dj must be calculated for each job j 2 r0 which requires O(n0 ) = O(n) time. Therefore, the complexity of LB1 is O(m) + O(n). The second lower bound (LB2) was developed by Azizoglu and Kirca (1998) and is obtained by relaxing the problem to a single-machine tardiness problem in which the processing time of each job j is set to pj/m. Therefore, a lower bound on the completion time of the job comP pleted jth among the jobs in r0 can be computed as UðrÞ þ ji¼1 pspt½i ðr0 Þ=m. The lower bound uses the following lemma from Kim (1995). Pn 0 P 0 Lemma 1. If C1 6 C2 6 6 Cn then maxðC j dedd½j ðr0 Þ; 0Þ 6 nj¼1 maxðC j dj ; 0Þ. The second lower bound LB2 is computed as h . j¼1 i Pn 0 Pj LB2 ¼ j¼1 max UðrÞ þ i¼1 pspt½i ðr0 Þ m dedd½j ðr0 Þ; 0 . To calculate LB2 first U(r) must be found and the processing times (pj) and due dates . P (dj) must be sorted in non-descending order then UðrÞ þ ji¼1 pspt½i ðr0 Þ m dedd½j ðr0 Þ must be calculated for each job j 2 r0 . Finding U (r) requires O(m) time. Each of the two sorts requires O(nln (n)) time and the final calculation requires O(n) time. The complexity of LB2 is O(m) + O(2nln (n)) + O(n). Shim and Kim (2007) use the following lemma to develop a third lower bound. The lemma provides a lower bound on the completion time of the job that is completed jth among the jobs in r0 . Lemma 2. In the identical parallel machine scheduling problem, the completion time of the job that is completed jth among those in r0 is no less than
minl6k6minðj;mÞ
"( k X
cl ðrÞ þ
j X
), # k :
pspt½i
i¼1
l¼1
Proof. This proof is provided in Shim and Kim (2007).
h
The third lower bound (LB3) can then be computed as: LB3 ¼
Pn0
j¼1
h hnP o. i i Pj k 0 max minl6k6minðj;mÞ k dedd½j ðr0 Þ; 0 . l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ
The computational requirement to calculate LB3 is very similar to that of LB2. The same two sorts as required by LB2 are required by LB3 and a final calculation is required that has the same requirement as that of LB2. To calculate LB3 the machine completion times must be sorted (cl (r)) which requires O(mln (m)) time. The complexity of LB3 is O(mln (m)) + O(2nln (n)) + O(n). 2.2. LB2 is dominated by LB3 As shown by the following theorem for any partial schedule r the third lower bound (LB3) dominates the second lower bound (LB2). Theorem 1. For any given partial schedule r LB2 6 LB3. Proof. In both lower bounds dedd[j] is subtracted from a lower bound on the completion time of the job completed jth among the jobs in r0 therefore for any given partial schedule r to prove LB2 6 LB3 it must be shown
UðrÞ þ
j X
, pspt½i ðr0 Þ
m6
i¼1
Since k 6 m,
Pj
i¼1 pspt½i ð
6 minl6k6minðj;mÞ
hnP
.
r0 Þ m 6
min
"( k X
l6k6minðj;mÞ
Pj
i¼1 pspt½i ð
cl ðrÞ þ
l¼1
j X
), # pspt½i ðr0 Þ
k for j ¼ 1; . . . ; n0 :
i¼1
.
r0 Þ k. U (r) = c1 (r) 6 cl (r) for l = 2, . . . , k therefore UðrÞ 6
o. Pj k 0 k for j = 1, . . . , n0 and LB2 6 LB3. h l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ
Pk
.
c rÞ k ) UðrÞþ
l¼1 l ð
Pj
r0 Þ=m
i¼1 pspt½i ð
The next section presents a new lower bound. 3. A new lower bound The concepts used to form LB1 and LB3 can be combined to form a new lower bound referred to as LB4. This section presents this lower bound. It is then shown that LB4 dominates LB1 and LB3. The section concludes by showing that the complexity of LB4 is the same as LB1 and LB3.
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3.1. LB4 0
To develop the new lower bound for the problem modified due dates dj ðr0 Þ are created for each job j 2 r0 as follows r0 Þ ¼ maxfdj ; UðrÞ þ pj g. The due dates used in LB3 are replaced by the modified due dates and the result is added to LB1. Formally LB4 is computed as
0 dj ð
LB4 ¼ LB1 þ
n0 X j¼1
" max
"( min
l6k6minðj;mÞ
k X l¼1
cl ðrÞ þ
j X
), # pspt½i ðr0 Þ
# 0
k dedd½j ðr0 Þ; 0 :
i¼1
Theorem 2. For any given partial schedule r LB4 is a lower bound on the sum of the tardiness of the jobs in the set r0 for any completion of the partial schedule. P Proof. The sum of tardiness of the jobs in the set r0 for any completion of the partial schedule r ¼ j2r0 maxfC j dj ; 0g. This theorem will P P 0 be proved in two parts: (1) we will first show that j2r0 maxfC j dj ; 0g can be rewritten as LB1 þ j2r0 maxfC j dj ðr0 Þ; 0g, Since LB1 P 0 h h n o . i i Pj P Pk P0 0 0 0 k dedd½j ðr0 Þ; 0 6 j2r0 maxfC j dj ðr0 Þ; 0g to complete we will then show 2) that nj¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ the proof. h P
P 0 0 0 maxfC j dj ; 0g ¼ LB1 þ j2r0 maxfC j dj ðr0 Þ; 0g. MaxfC j dj ; 0g ¼ maxfðC j dj ðr0 ÞÞ þ ðdj ðr0 Þ dj Þ; 0g for j 2 r0 . Since 0 0 0 0 0 Cj P U(r) + pj, dj ðr Þ ¼ dj if dj P Cj and maxfC j dj ; 0g ¼ maxfðC j dj ðr ÞÞ; 0g ¼ 0. If Cj > dj then C j dj ¼ ðC j dj P 0 0 0 0 0 0 0 0 ðr ÞÞ þ ðdj ðr Þ dj Þ > 0. Therefore, fmax C j dj ; 0g ¼ maxfC j dj ðr Þ; 0g þ dj ðr Þ dj for j 2 r so 0g ¼ j2r0 maxfC j dj ;P P 0 0 0 0 0 r0 Þ ¼ maxfUðrÞ þ pj ; dj g ) d0j ðr0Þ dj ¼ maxfUðrÞ þ pj dj ; 0g therefore j2r0 maxfC j dj ðr Þ; 0g þ dj ðr Þ dj Þ. dj ðP j2r0 0 0 0 ðmaxfC j dj ðr0 Þ; 0g þ dj ðr0 Þ dj Þ ¼ LB1 þ j2r0 maxfC j dj ðr0Þ; 0g and this completes the first part of the proof. h hnP o. i i P Pn0 Pj 0 0 k 0 (2) k dedd½j ðr0 Þ; 0 6 j2r0 maxfC j dj ðr0 Þ; 0g. It is shown by Shim and Kim j¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ hnP o . i Pj k (2007) that minl6k6minðj;mÞ k is a lower bound on the completion time of the job completed l¼1 cl ðrÞ þ i¼1 pspt½i h hnP o. i i P0 P Pj 0 0 k 0 k dedd½j ðr0 Þ; 0 6 nj¼1 maxfC j jth among the jobs in r0 (Cj) therefore nj¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ
(1)
j2r0
P0 P 0 0 0 dedd½j ðr0 Þ; 0g. To complete the proof it will be shown nj¼1 maxfC j dedd½j ðr0 Þ; 0g 6 j2r0 maxfC j dj ðr0 Þ; 0g. Suppose we have a sequence of modified due dates that are not in EDD order. It can be shown that the sequence can be converted to a sequence of modified due dates that is in EDD order by a series of exchanges of pairs of modified due dates. To prove P P 0 0 0 0 0 0 0 0 j2r0 maxfC j dedd½j ðr Þ; 0g 6 j2r0 maxfC j dj ðr Þ; 0g it is sufficient to prove if dj ðr Þ > dk ðr Þ and j < k then exchanging the two due dates will result in an objective value when compared to the completion times that is no greater than the resulting 0 0 0 0 objective value before the exchange. Let dj (r0 ) and dk (r0 ) be two modified due dates such that j < k and dj ðr0 Þ > dk ðr0 Þ. Let 00 0 d be modified due dates that are the same as the modified due dates d except the modified due dates for j and k are exchanged. 00 0 00 0 00 0 0 00 dp ðr0 Þ ¼ dp (r0 ) if p – j or k, dj ðr0 Þ ¼ dk (r0 ), and dk ðr0 Þ ¼ dj ðr0Þ. Let Z p ¼ maxfC p dp ðr0Þ; 0g and Z 0p ¼ maxfC p dp ðr0 Þ; 0g. To P0 P0 show nj¼1 Z 0p 6 nj¼1 Z p it must be shown Z 0j þ Z 0k 6 Z j þ Z k since Z 0p ¼ Z p if p – j or k. The following two cases cover all possibil00 00 ities: 1) C k dk ðr0 Þ P 0 and 2) C k dk ðr0 Þ < 0. 00 00 00 0 0 0 0 0 (Case 1) C k dk ðr Þ P 0. C k dk ðr0Þ P 0 ) Z 0k ¼ C k dk ðr0Þ ¼ C k dj ðr0 Þ. Since dj ðr0 Þ > dk (r0 ) and C k dj ðr0 Þ P 0 then 0 0 0 0 0 0 00 00 0 0 0 0 0 0 0 0 0 0 Z k ¼ C k dk ðr Þ ¼ C k dj ðr Þ þ ðdj ðr Þ dk ðr ÞÞ ¼ Z k þ dj ðr Þ dk ðr Þ. Z j ¼ maxfC j dj ðr Þ; 0g. Since dj ðr0 Þ ¼ dk ðr0 Þ and 0 0 0 0 0 0 0 0 0 0 0 dj ðr Þ >0 dk ðr Þ then Z j 6 Z0 j þ dj ðr Þ dk ðr Þ and Z j þ Z k 6 Z j þ Z k for this case. 0 0 00 0 0 (Case 2) C k dk ðr0 Þ < 0. C k dk ðr0 Þ < 0 ) Z 0k ¼ 0. Since Ck > Cj and dk ðr0 Þ ¼ dj ðr0 Þ then Zj = 0. Z k ¼ maxfC k dk ðr0Þ; 0g. 00 0 0 0 0 0 0 0 Z j ¼ maxfC j dj ðr Þ; 0g ¼ maxfC j dk ðr Þ; 0g. Since C k > C j Z k P Z j therefore Z j þ Z k 6 Z j þ Z k for this case. This completes the second part of the proof. h
3.2. LB4 dominates LB1 and LB3 The proofs of the following two theorems show that for any given partial schedule r the lower bounds LB1 and LB3 are dominated by the lower bound LB4. Theorem 3. For any given partial schedule r LB1 6 LB4. h hnP o. i i h hnP Pj Pn0 P0 0 k k 0 k dedd½j ðr0 Þ; 0 . Since Proof. LB4 ¼ LB1 þ nj¼1 max minl6k6minðj;mÞ j¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ Pj 0 0 0 h i¼1 pspt½i ðr Þg=k dedd½j ðr Þ; 0 P 0 LB1 6 LB4. Theorem 4. For any given partial schedule r LB3 6 LB4. h hnP o. i i P0 Pj 0 0 k 0 k dedd½j ðr0 Þ; 0 . LB4 = LB1 + LB30 . Since dedd½j ðr0 Þ P dedd½j ðr0 Þ for Proof. Let LB30 ¼ nj¼1 max minl6k6minðj;mÞ l¼1 cl ðrÞ þ i¼1 pspt½i ðr Þ P P P 0 0 0 0 0 n 0 0 0 0 LB3 LB30 6 nj¼1 ðdedd½j ðr0 Þ dedd½j ðr0ÞÞ ¼ LB1 therefore LB4 P j = 1, . . . , n0 j¼1 ðdedd½j ðr Þ dedd½j ðr ÞÞ ¼ j2r0 ðdj ðr Þ dj ðr ÞÞ ¼ LB1. LB3. h
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J. Schaller / European Journal of Operational Research 197 (2009) 422–426 Table 1 Average reduction in seconds when m = 2 m=2
Number of jobs
Set 1 2 3 4 5 6
10 0.17 0.11 0.83 0.25 1.71 0.11
12 2.63 3.66 21.88 9.65 72.18 1.00
14 44.98 198.51 749.44 120.38 2278.33 30.92
12 11.60 27.78 28.27 8.30 21.71 2.96
14 886.64 949.89 1561.84 420.46 1507.85 19.19
Table 2 Average reduction in seconds when m = 4 m=4
Number of jobs
Set 1 2 3 4 5 6
10 0.39 0.54 0.58 0.26 0.42 0.06
3.3. Comparison of the complexity of LB4 with the lower bounds of Shim and Kim (2007) 0
To calculate LB4 LB1 must be calculated, the modified due dates must be created, processing times (pj) and modified due dates (dj (r0 )) h hnP k must be sorted in non-descending order, the machine completion times must be sorted (cl(r)), and max minl6k6minðj;mÞ l¼1 cl ðrÞ þ Pj 0 0 0 0 i¼1 pspt½i ðr Þg=k dedd½j ðr Þ; 0 must be calculated for each job j 2 r . The modified due dates can be created during the calculation of LB1 and U (r) (used to calculate LB1 and create the modified due dates) can be found during the sort of the machine completion times (cl (r)) so the complexity of LB4 is O(mln(m)) + O(2nln(n)) + 2 O(n). Shim and Kim (2007) calculate each of the lower bounds (LB1, LB2, and LB3) and then used the maximum value found when determining whether to prune a branch in their branch-and-bound algorithm. In this note, it was shown that LB2 is dominated by LB3 so the calculations required for LB2 can be eliminated without reducing the bound found. Comparing the computational requirement to calculate LB4 with that of calculating LB1 and LB3 it is found that LB4 requires slightly less time than calculating both LB1 and LB3 therefore using LB4 instead of the maximum of LB1 and LB3 will not hurt the performance of a branch-and-bound algorithm and offers the opportunity to significantly improve it. 4. Computational test To test whether using the new lower bound (LB4) in place of the two lower bounds developed by Shim and Kim (2007) (LB1 and LB3) increases the efficiency of a branch-and-bound algorithm the following test was used. A branch-and–bound algorithm was coded. Nodes in the branch-and-bound algorithm correspond to a partial schedule. A depth-first branching rule is used so the node with the most jobs in the corresponding partial schedule is selected for branching. Ties are broken in favor of the node with the minimum lower bound. The branch-and-bound algorithm was first tested on thirty six problem sets using Shim and Kim’s (2007) lower bounds and the average time to solve problems within each problem set was recorded. Then the branch-and-bound algorithm was tested using the new lower bound instead of Shim and Kim (2007)’s lower bounds. The branch-and-bound algorithm was coded in Turbo Pascal and was tested on a HP LP1965 2.4 GHz PC. Each problem set had 10 problems. The processing times of jobs for all of the problems were generated using a uniform distribution over the integers 1–50. All problems within a problem set have the same number of jobs, machines and due dates are drawn from the same distribution. The numbers of jobs tested were n = 10, 12 and 14. The number of machines tested were m = 2 and 4. The due dates for the jobs were randomly generated using a uniform distribution over the integers .AP (1 r R/2) and AP (1 r + R/2), where AP is Pn the total processing time for all n jobs divided by the number of machines AP ¼ j¼1 pj m , and R and r are two parameters called due date range and tardiness factors. Six sets of these parameters were used for each n and m combination: R = 0.5 and r = 0.75 (set 1), R = 1.00 and r = 0.75 (set 2), R = 0.5 and r = 0.5 (set 3), R = 1.00 and r = 0.5 (set 4), R = 0.5 and r = 0.25 (set 5), R = 1.00 and r = 0.25 (set 6). Tables 1 and 2 show the average reduction per problem in computation time when the new lower bound is used in place of the lower bounds developed by Shim and Kim (2007). Table 1 shows the results for problems sets with two machines and Table 2 shows the results for problem sets with four machines. The results show that using the new lower bound in place of the lower bounds developed by Shim and Kim (2007) reduced the time required to solve problems for all 36 problem sets. The results also show that as the number of jobs increases the computation savings increase rapidly for both the problem sets with two machines and four machines. 5. Conclusion This note showed that one of the lower bounds for identical parallel machine scheduling to minimize total tardiness developed by Shim and Kim (2007) (LB3) dominates a lower bound developed by Azizoglu and Kirca (1998) (LB2). The note also introduced a new lower bound
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for the problem (LB4) that utilizes concepts based on the two lower bounds developed by Shim and Kim (2007) (LB1 and LB3). LB4 was shown to dominate LB1 and LB3 and since LB3 dominates LB2 LB4 also dominates LB2. The lower bound LB4 can be used in a branchand-bound algorithm in place of LB1, LB2 and LB3 to reduce the enumeration that is required. It was also shown that the computational requirement to calculate LB4 is less than that of calculating LB1, LB2 and LB3 so using the new lower bound improves the efficiency of a branch-and-bound algorithm. This improvement in efficiency was also demonstrated in a computational test. References Azizoglu, M., Kirca, O., 1998. Tardiness minimization on parallel machines. International Journal of Production Economics 55, 163–168. Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinooy Kan, A.H.G., 1979. Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics 5, 287–326. Kim, Y.D., 1995. Minimizing total tardiness in permutation flowshops. European Journal of Operational Research 85, 541–555. Shim, S.O., Kim, Y.D., 2007. Scheduling on parallel identical machines to minimize total tardiness. European Journal of Operational Research 177, 135–146. Yalaoui, F., Chu, C., 2002. Parallel machine scheduling to minimize total tardiness. International Journal of Production Economics 76, 265–279.