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Erratum to “Improved WLP and GWP lower bounds based on exact integer programming” [J. Statist. Plann. Inference 140 (2010) 1154–1161]
Journal of Statistical Planning and Inference 141 (2011) 2500–2501
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Erratum
Erratum to ‘‘Improved WLP and GWP lower bounds based on exact integer programming’’ [J. Statist. Plann. Inference 140 (2010) 1154–1161] Dursun A. Bulutoglu, David M. Kaziska Air Force Institute of Technology, WPAFB, OH 45433-7765, United States
a r t i c l e in f o Article history: Received 28 December 2010 Available online 19 January 2011
P. 1154 lines 16–18: In the abstract, the sentence ‘‘By using the constraints in our integer programs and algorithms in the literature we prove that f ð16l,2,4Þ r9 if l is odd where f ð2t l,2,tÞ is the maximum n for which an OA(N,n,2,t) exists.’’ should be replaced with ‘‘Our IP formulations enable us to find improved upper bounds on f ð16l,2,4Þ for several odd l where f ð2t l,2,tÞ is the maximum n for which an OA(N,n,2,t) exists.’’ P. 1156 lines 7–8: The sentence ‘‘Finally, by using algorithms in the literature and the constraints in our integer programs we prove that f ð16l,2,4Þ r9 if l is odd where f ð2t l,2,tÞ is the maximum n for which an OA(N,n,2,t) exists.’’ should be replaced with ‘‘Our IP formulations enable us to find improved upper bounds on f ð16l,2,4Þ for several odd l where f ð2t l,2,tÞ is the maximum n for which an OA(N,n,2,t) exists.’’ P. 1157 lines 27–28: The sentence ‘‘The proof is skipped as it is similar to the proof of Lemma 3 in Stufken and Tang (2007).’’ should be deleted. P. 1157 line 32. In Lemma 1 (3) ‘‘If l is odd and t is even then ml is odd if jljt 1 or 2 (mod 4) and even otherwise.’’ jlj1 should be replaced with ‘‘If l is odd, then ml is odd if ðjljt1 Þ 1 (mod 2) and even otherwise.’’ P. 1157 line 33. In Lemma 1, (4) should be deleted. P. 1157 line 34. The following proof of Lemma 1 should be placed here. Proof. By Lemma 3, in Stufken and Tang (2007) it suffices to show that jlj X r1 jlj 1 t r r ¼ tþ1
ðmod 2Þ:
This is shown by induction on |l| and applying the following identities: ! ! jlj X r jlj jlj jljq ¼2 , q q r r¼q
jlj r
¼
jlj1 r1
þ
jlj1 r
:
&
DOI of original article: 10.1016/j.jspi.2009.10.013
Corresponding author.
E-mail addresses: dursun.bulutoglu@afit.edu (D.A. Bulutoglu), [email protected] (D.M. Kaziska). 0378-3758/$ - see front matter & 2011 Published by Elsevier B.V. doi:10.1016/j.jspi.2011.01.003
D.A. Bulutoglu, D.M. Kaziska / Journal of Statistical Planning and Inference 141 (2011) 2500–2501
2501
Table 5 Improvements to f ð16l,2,4Þ upper bounds for odd l.
l 11 13 15 17 19 21 23 25 27 29
Old upper bound
New upper bound
16 18 20 21 21 22 25 25 26 29
10 13 14 14 21 22 24 25 26 29
P. 1158 lines 12–13: ‘‘When t is even and j 4t,’’ should be replaced with ‘‘For j4t,’’ and in constraint (3)(b) ‘‘if jjjt 1 or j1 2 (mod 4),’’ should be replaced with ‘‘if ðjt1 Þ 1 (mod 2),’’ P. 1158 lines 15–17: In constraint (3), (c) should be deleted. P. 1159 line 4: ‘‘Constraints (2)(a), (3)(b) and (3)(c) follow from Lemma 1.’’ should be replaced with ‘‘Constraints (2)(a) and (3)(b) follow from Lemma 1.’’ P. 1160 line 11: ‘‘Using the constraints in (3) we next show that no OA ð16l,10,2,4Þ exists for odd l.’’ should be deleted. P. 1160–1161: Lemma 2, Lemma 3, Corollary 4, Corollary 5 and their proofs should be deleted. The paragraph after Corollary 5 should be replaced with ‘‘In Table 5 we present new upper bounds on f ð16l,2,4Þ obtained by solving our IP formulations side by side with the best-known upper bounds in the literature. In 5 out of 10 cases our new upper bound is an improvement on the old upper bound and a tie in the remaining cases.’’ P.1161: In the references section 3rd, 4th, and 12th references should be deleted.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Erratum
Erratum to ‘‘Improved WLP and GWP lower bounds based on exact integer programming’’ [J. Statist. Plann. Inference 140 (2010) 1154–1161] Dursun A. Bulutoglu, David M. Kaziska Air Force Institute of Technology, WPAFB, OH 45433-7765, United States
a r t i c l e in f o Article history: Received 28 December 2010 Available online 19 January 2011
P. 1154 lines 16–18: In the abstract, the sentence ‘‘By using the constraints in our integer programs and algorithms in the literature we prove that f ð16l,2,4Þ r9 if l is odd where f ð2t l,2,tÞ is the maximum n for which an OA(N,n,2,t) exists.’’ should be replaced with ‘‘Our IP formulations enable us to find improved upper bounds on f ð16l,2,4Þ for several odd l where f ð2t l,2,tÞ is the maximum n for which an OA(N,n,2,t) exists.’’ P. 1156 lines 7–8: The sentence ‘‘Finally, by using algorithms in the literature and the constraints in our integer programs we prove that f ð16l,2,4Þ r9 if l is odd where f ð2t l,2,tÞ is the maximum n for which an OA(N,n,2,t) exists.’’ should be replaced with ‘‘Our IP formulations enable us to find improved upper bounds on f ð16l,2,4Þ for several odd l where f ð2t l,2,tÞ is the maximum n for which an OA(N,n,2,t) exists.’’ P. 1157 lines 27–28: The sentence ‘‘The proof is skipped as it is similar to the proof of Lemma 3 in Stufken and Tang (2007).’’ should be deleted. P. 1157 line 32. In Lemma 1 (3) ‘‘If l is odd and t is even then ml is odd if jljt 1 or 2 (mod 4) and even otherwise.’’ jlj1 should be replaced with ‘‘If l is odd, then ml is odd if ðjljt1 Þ 1 (mod 2) and even otherwise.’’ P. 1157 line 33. In Lemma 1, (4) should be deleted. P. 1157 line 34. The following proof of Lemma 1 should be placed here. Proof. By Lemma 3, in Stufken and Tang (2007) it suffices to show that jlj X r1 jlj 1 t r r ¼ tþ1
ðmod 2Þ:
This is shown by induction on |l| and applying the following identities: ! ! jlj X r jlj jlj jljq ¼2 , q q r r¼q
jlj r
¼
jlj1 r1
þ
jlj1 r
:
&
DOI of original article: 10.1016/j.jspi.2009.10.013
Corresponding author.
E-mail addresses: dursun.bulutoglu@afit.edu (D.A. Bulutoglu), [email protected] (D.M. Kaziska). 0378-3758/$ - see front matter & 2011 Published by Elsevier B.V. doi:10.1016/j.jspi.2011.01.003
D.A. Bulutoglu, D.M. Kaziska / Journal of Statistical Planning and Inference 141 (2011) 2500–2501
2501
Table 5 Improvements to f ð16l,2,4Þ upper bounds for odd l.
l 11 13 15 17 19 21 23 25 27 29
Old upper bound
New upper bound
16 18 20 21 21 22 25 25 26 29
10 13 14 14 21 22 24 25 26 29
P. 1158 lines 12–13: ‘‘When t is even and j 4t,’’ should be replaced with ‘‘For j4t,’’ and in constraint (3)(b) ‘‘if jjjt 1 or j1 2 (mod 4),’’ should be replaced with ‘‘if ðjt1 Þ 1 (mod 2),’’ P. 1158 lines 15–17: In constraint (3), (c) should be deleted. P. 1159 line 4: ‘‘Constraints (2)(a), (3)(b) and (3)(c) follow from Lemma 1.’’ should be replaced with ‘‘Constraints (2)(a) and (3)(b) follow from Lemma 1.’’ P. 1160 line 11: ‘‘Using the constraints in (3) we next show that no OA ð16l,10,2,4Þ exists for odd l.’’ should be deleted. P. 1160–1161: Lemma 2, Lemma 3, Corollary 4, Corollary 5 and their proofs should be deleted. The paragraph after Corollary 5 should be replaced with ‘‘In Table 5 we present new upper bounds on f ð16l,2,4Þ obtained by solving our IP formulations side by side with the best-known upper bounds in the literature. In 5 out of 10 cases our new upper bound is an improvement on the old upper bound and a tie in the remaining cases.’’ P.1161: In the references section 3rd, 4th, and 12th references should be deleted.