Tight bounds for periodicity theorems on the unbounded Knapsack problem

► Developed three new periodicity bounds for UKP. ► Proved each new bound is tight. ► Showed two new bounds subsume known results. Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of the best i...

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Vydané v:	European journal of operational research Ročník 215; číslo 2; s. 319 - 324
Hlavní autori:	Huang, Ping H., Lawley, Mark, Morin, Thomas
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Amsterdam Elsevier B.V 01.12.2011 Elsevier Elsevier Sequoia S.A
Edícia:	European Journal of Operational Research
Predmet:	Applied sciences Combinatorial optimization Combinatorial optimization Integer programming Knapsack problem Number theory Periodicity Empirical analysis Exact sciences and technology Flows in networks. Combinatorial problems Integer programming Knapsack problem Mathematical programming Number theory Operational research Operational research and scientific management Operational research. Management science Optimization Optimization techniques Periodicity Studies Theorems Integer programming Combinatorial optimization Number theory Periodicity Knapsack problem Empirical method Profit
ISSN:	0377-2217, 1872-6860
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Abstract	► Developed three new periodicity bounds for UKP. ► Proved each new bound is tight. ► Showed two new bounds subsume known results. Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of the best item (the one with the highest profit-to-weight ratio). The successive applications of periodicity theorems can drastically reduce the size of the Knapsack problem under analysis, theoretical or empirical. We prove that each new bound is tight in the sense that no smaller bound exists under the given condition.
AbstractList	Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of the best item (the one with the highest profit-to-weight ratio). The successive applications of periodicity theorems can drastically reduce the size of the Knapsack problem under analysis, theoretical or empirical. We prove that each new bound is tight in the sense that no smaller bound exists under the given condition. [PUBLICATION ABSTRACT] Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of the best item (the one with the highest profit-to-weight ratio). The successive applications of periodicity theorems can drastically reduce the size of the Knapsack problem under analysis, theoretical or empirical. We prove that each new bound is tight in the sense that no smaller bound exists under the given condition. ► Developed three new periodicity bounds for UKP. ► Proved each new bound is tight. ► Showed two new bounds subsume known results. Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of the best item (the one with the highest profit-to-weight ratio). The successive applications of periodicity theorems can drastically reduce the size of the Knapsack problem under analysis, theoretical or empirical. We prove that each new bound is tight in the sense that no smaller bound exists under the given condition.
Author	Huang, Ping H. Lawley, Mark Morin, Thomas
Author_xml	– sequence: 1 givenname: Ping H. surname: Huang fullname: Huang, Ping H. email: huang74@purdue.edu organization: Regenstrief Center for Healthcare Engineering, Purdue University, West Lafayette, Indiana 47907, United States – sequence: 2 givenname: Mark surname: Lawley fullname: Lawley, Mark organization: School of Biomedical Engineering, Purdue University, West Lafayette, Indiana 47907, United States – sequence: 3 givenname: Thomas surname: Morin fullname: Morin, Thomas organization: School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907, United States
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References	Serre (b0075) 1973 Toth (b0095) 2000; 125 Kohli, Krishnamurti, Mirchandani (b0050) 2004; 154 Nemhauser, Wolsey (b0065) 1988 Caccetta, Kulanoot (b0010) 2001; 47 Hardy, Wright (b0040) 1938 Hu (b0045) 1969 Mathur, Venkateshan (b0060) 2007; 178 Poirriez, Yanev, Andonov (b0070) 2009; 6 Kellerer, Pferschy, Pisinger (b0035) 2004 Shapiro (b0080) 1970; 18 Martello, Toth (b0055) 1990 Gilmore, Gomory (b0030) 1963; 11 Gilmore, Gomory (b0025) 1966; 14 Andonov, Poirriez, Rajopadhye (b0005) 2000; 123 Garey, Johnson (b0015) 1990 Srisuwannapa, Charnsethikul (b0085) 2007; 3 Garfinkel, Nemhauser (b0020) 1972 Tamir (b0090) 2009; 37 Kellerer (10.1016/j.ejor.2011.06.010_b0035) 2004 Caccetta (10.1016/j.ejor.2011.06.010_b0010) 2001; 47 Kohli (10.1016/j.ejor.2011.06.010_b0050) 2004; 154 Tamir (10.1016/j.ejor.2011.06.010_b0090) 2009; 37 Mathur (10.1016/j.ejor.2011.06.010_b0060) 2007; 178 Poirriez (10.1016/j.ejor.2011.06.010_b0070) 2009; 6 Toth (10.1016/j.ejor.2011.06.010_b0095) 2000; 125 Hu (10.1016/j.ejor.2011.06.010_b0045) 1969 Shapiro (10.1016/j.ejor.2011.06.010_b0080) 1970; 18 Andonov (10.1016/j.ejor.2011.06.010_b0005) 2000; 123 Srisuwannapa (10.1016/j.ejor.2011.06.010_b0085) 2007; 3 Gilmore (10.1016/j.ejor.2011.06.010_b0030) 1963; 11 Garfinkel (10.1016/j.ejor.2011.06.010_b0020) 1972 Nemhauser (10.1016/j.ejor.2011.06.010_b0065) 1988 Hardy (10.1016/j.ejor.2011.06.010_b0040) 1938 Serre (10.1016/j.ejor.2011.06.010_b0075) 1973 Martello (10.1016/j.ejor.2011.06.010_b0055) 1990 Garey (10.1016/j.ejor.2011.06.010_b0015) 1990 Gilmore (10.1016/j.ejor.2011.06.010_b0025) 1966; 14
References_xml	– year: 1990 ident: b0015 article-title: Computers and Intractability: A Guide to the Theory of NP Completeness – volume: 47 start-page: 5547 year: 2001 end-page: 5558 ident: b0010 article-title: Computational aspects of hard Knapsack problems publication-title: Nonlinear Analysis – volume: 123 start-page: 394 year: 2000 end-page: 407 ident: b0005 article-title: Unbounded Knapsack problem: dynamic programming revisited publication-title: European Journal of Operational Research – year: 2004 ident: b0035 article-title: Knapsack Problems – volume: 18 start-page: 432 year: 1970 end-page: 440 ident: b0080 article-title: Turnpike theorem for integer programming problem publication-title: Operations Research – year: 1969 ident: b0045 article-title: Integer Programming and Network Flows – year: 1972 ident: b0020 article-title: Integer Programming – volume: 14 start-page: 1045 year: 1966 end-page: 1074 ident: b0025 article-title: The theory and computation of Knapsack functions publication-title: Operations Research – year: 1988 ident: b0065 article-title: Integer and Combinatorial Optimization – volume: 6 start-page: 110 year: 2009 end-page: 124 ident: b0070 article-title: A hybrid algorithm for the unbounded Knapsack problem publication-title: Discrete Optimization – volume: 154 start-page: 36 year: 2004 end-page: 45 ident: b0050 article-title: Average performance of Greedy heuristics for the integer Knapsack problem publication-title: European Journal of Operational Research – volume: 3 start-page: 138 year: 2007 end-page: 143 ident: b0085 article-title: An exact algorithm for the unbounded Knapsack problem with minimizing maximum processing time publication-title: Journal of Computer Science – volume: 11 start-page: 863 year: 1963 end-page: 888 ident: b0030 article-title: A linear programming approach to the cutting stock problem – Part II publication-title: Operations Research – volume: 125 start-page: 222 year: 2000 end-page: 238 ident: b0095 article-title: Optimization engineering techniques for the exact solution of NP-hard combination optimization problems publication-title: European Journal of Operational Research – year: 1938 ident: b0040 article-title: An Introduction to the Theory of Numbers – year: 1973 ident: b0075 article-title: A Course in Arithmetic – volume: 178 start-page: 738 year: 2007 end-page: 754 ident: b0060 article-title: A new lower bound for the linear Knapsack problem with general integer variables publication-title: European Journal of Operational Research – year: 1990 ident: b0055 article-title: Knapsack Problems: Algorithms and Computer Implementation – volume: 37 start-page: 303 year: 2009 end-page: 306 ident: b0090 article-title: New pseudopolynomial complexity bounds for the bounded and other integer Knapsack related problems publication-title: Operations Research Letters – volume: 11 start-page: 863 year: 1963 ident: 10.1016/j.ejor.2011.06.010_b0030 article-title: A linear programming approach to the cutting stock problem – Part II publication-title: Operations Research doi: 10.1287/opre.11.6.863 – volume: 47 start-page: 5547 year: 2001 ident: 10.1016/j.ejor.2011.06.010_b0010 article-title: Computational aspects of hard Knapsack problems publication-title: Nonlinear Analysis doi: 10.1016/S0362-546X(01)00658-7 – year: 2004 ident: 10.1016/j.ejor.2011.06.010_b0035 – volume: 3 start-page: 138 year: 2007 ident: 10.1016/j.ejor.2011.06.010_b0085 article-title: An exact algorithm for the unbounded Knapsack problem with minimizing maximum processing time publication-title: Journal of Computer Science doi: 10.3844/jcssp.2007.138.143 – volume: 125 start-page: 222 year: 2000 ident: 10.1016/j.ejor.2011.06.010_b0095 article-title: Optimization engineering techniques for the exact solution of NP-hard combination optimization problems publication-title: European Journal of Operational Research doi: 10.1016/S0377-2217(99)00453-1 – volume: 37 start-page: 303 year: 2009 ident: 10.1016/j.ejor.2011.06.010_b0090 article-title: New pseudopolynomial complexity bounds for the bounded and other integer Knapsack related problems publication-title: Operations Research Letters doi: 10.1016/j.orl.2009.05.003 – volume: 123 start-page: 394 year: 2000 ident: 10.1016/j.ejor.2011.06.010_b0005 article-title: Unbounded Knapsack problem: dynamic programming revisited publication-title: European Journal of Operational Research doi: 10.1016/S0377-2217(99)00265-9 – volume: 14 start-page: 1045 year: 1966 ident: 10.1016/j.ejor.2011.06.010_b0025 article-title: The theory and computation of Knapsack functions publication-title: Operations Research doi: 10.1287/opre.14.6.1045 – year: 1990 ident: 10.1016/j.ejor.2011.06.010_b0015 – volume: 6 start-page: 110 year: 2009 ident: 10.1016/j.ejor.2011.06.010_b0070 article-title: A hybrid algorithm for the unbounded Knapsack problem publication-title: Discrete Optimization doi: 10.1016/j.disopt.2008.09.004 – volume: 178 start-page: 738 year: 2007 ident: 10.1016/j.ejor.2011.06.010_b0060 article-title: A new lower bound for the linear Knapsack problem with general integer variables publication-title: European Journal of Operational Research doi: 10.1016/j.ejor.2006.02.018 – year: 1988 ident: 10.1016/j.ejor.2011.06.010_b0065 – year: 1938 ident: 10.1016/j.ejor.2011.06.010_b0040 – volume: 18 start-page: 432 year: 1970 ident: 10.1016/j.ejor.2011.06.010_b0080 article-title: Turnpike theorem for integer programming problem publication-title: Operations Research doi: 10.1287/opre.18.3.432 – year: 1972 ident: 10.1016/j.ejor.2011.06.010_b0020 – year: 1969 ident: 10.1016/j.ejor.2011.06.010_b0045 – year: 1973 ident: 10.1016/j.ejor.2011.06.010_b0075 – volume: 154 start-page: 36 year: 2004 ident: 10.1016/j.ejor.2011.06.010_b0050 article-title: Average performance of Greedy heuristics for the integer Knapsack problem publication-title: European Journal of Operational Research doi: 10.1016/S0377-2217(02)00810-X – year: 1990 ident: 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Snippet	► Developed three new periodicity bounds for UKP. ► Proved each new bound is tight. ► Showed two new bounds subsume known results. Three new bounds for... Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of...
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SubjectTerms	Applied sciences Combinatorial optimization Combinatorial optimization Integer programming Knapsack problem Number theory Periodicity Empirical analysis Exact sciences and technology Flows in networks. Combinatorial problems Integer programming Knapsack problem Mathematical programming Number theory Operational research Operational research and scientific management Operational research. Management science Optimization Optimization techniques Periodicity Studies Theorems
Title	Tight bounds for periodicity theorems on the unbounded Knapsack problem
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