![Page 1: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/1.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Unbounded Knapsack Problemand the
Generalized Cordel Property
Lisa Schreiber
Friedrich-Schiller-Universitat Jena,Institut fur Angewandte Mathematik
November the 26th, 2010
![Page 2: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/2.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 3: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/3.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 4: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/4.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 5: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/5.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 6: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/6.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 7: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/7.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 8: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/8.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.
Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
![Page 9: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/9.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).
Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
![Page 10: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/10.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
![Page 11: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/11.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
![Page 12: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/12.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack Problem
G = N: Unbounded Knapsack Problem
![Page 13: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/13.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Knapsack Problem
Given n items and a knapsack with capacity C.Every item has a value v(i) and a weight w(i).Question: Which items shall be packed into the knapsacksuch that the capacity C is not injured and the total value isas large as possible?
Definition: Knapsack Problem
maxx∈Gn
f (x) :=n
∑i=1
v(i)xi subject ton
∑i=1
w(i)xi ≤ C
G = {0,1}: 0-1 Knapsack ProblemG = N: Unbounded Knapsack Problem
![Page 14: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/14.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 15: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/15.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.
two Criteria:1 An alternative should be good with respect to the objective
function.2 An alternative should not be too similar to the optimal
solution.
One good method: the Penalty Methodexamined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
![Page 16: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/16.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.two Criteria:
1 An alternative should be good with respect to the objectivefunction.
2 An alternative should not be too similar to the optimalsolution.
One good method: the Penalty Methodexamined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
![Page 17: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/17.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.two Criteria:
1 An alternative should be good with respect to the objectivefunction.
2 An alternative should not be too similar to the optimalsolution.
One good method: the Penalty Methodexamined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
![Page 18: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/18.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.two Criteria:
1 An alternative should be good with respect to the objectivefunction.
2 An alternative should not be too similar to the optimalsolution.
One good method: the Penalty Method
examined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
![Page 19: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/19.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method
Aim: Compute not only an optimal solution but also one (ormore) alternative solutions.two Criteria:
1 An alternative should be good with respect to the objectivefunction.
2 An alternative should not be too similar to the optimalsolution.
One good method: the Penalty Methodexamined by Schwarz (2003), Sameith (2005) andDornfelder (2009)
![Page 20: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/20.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
Important properties:
ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
![Page 21: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/21.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
Important properties:
ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
![Page 22: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/22.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) =
{v (i) , if B0(i) = 0v (i) · [1− ε · B0(i)] , if B0(i) > 0
Important properties:
ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
![Page 23: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/23.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) =
{v (i) , if B0(i) = 0v (i) · [1− ε · B0(i)] , if B0(i) > 0
Important properties:
ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
![Page 24: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/24.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) =
{v (i) , if B0(i) = 0v (i) · [1− ε · B0(i)] , if B0(i) > 0
Important properties:ε increases→ punishment gets higher
B0(i) > B0(j)→ punishment of item i is higher
![Page 25: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/25.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) =
{v (i) , if B0(i) = 0v (i) · [1− ε · B0(i)] , if B0(i) > 0
Important properties:ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
![Page 26: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/26.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) = v (i) · [1− ε · B0(i)]
Important properties:ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
![Page 27: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/27.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
![Page 28: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/28.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
![Page 29: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/29.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
![Page 30: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/30.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
![Page 31: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/31.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
![Page 32: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/32.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
![Page 33: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/33.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
A first example
C = 13values v =[6,8,3,1]
weights w=[5,7,3,6]
optimal solution: B0 = (2,0,1,0) with f (B0) = 15
e.g. ε = 0.7. This leads to the following punished values:
vε(2) = v(2) = 8 andvε(4) = v(4) = 1
vε(1) = v (1) · [1− ε · B0(1)] = 6 · [1− 0.7 · 2] = −2.4
vε(3) = v (3) · [1− ε · B0(3)] = 3 · [1− 0.7 · 1] = 0.9
C, vε = [−2.4, 8, 0.9, 1] and w provide Bε = (0,1,2,0) asbest solution and penalty alternative
![Page 34: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/34.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere else
P1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
![Page 35: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/35.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
![Page 36: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/36.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I2
2 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
![Page 37: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/37.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon
3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
![Page 38: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/38.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
![Page 39: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/39.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
![Page 40: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/40.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
Properties of Penalty Alternatives (Schwarz 2003)B penalty-alternative⇒ B is optimal for all parameters ε inan optimality interval IB = [ε1, ε2] and nowhere elseP1,P2 penalty alternatives with optimality intervals I1 and I2⇒ three possible cases
1 I1 = I22 I1 ∩ I2 = {ε}, intersection contains only a single epsilon3 I1 ∩ I2 = ∅, empty intersection
∞0ε0 ε1 ε2
P0 = B0 P1 P2 . . .
This provides an algorithm, how to compute all penaltyalternatives P0,P1, . . . ,Pk !
![Page 41: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/41.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
vε (i) = v (i) · [1− ε · B0(i)]
⇒ fε(B) =n
∑i=1
vε(i) · B(i) =n
∑i=1
v(i) · [1− ε · B0(i)] · B(i)
=n
∑i=1
v(i) · B(i)︸ ︷︷ ︸=f (B)
−εn
∑i=1
v(i) · B0(i) · B(i)︸ ︷︷ ︸=:p(B)
![Page 42: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/42.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
vε (i) = v (i) · [1− ε · B0(i)]
⇒ fε(B) =n
∑i=1
vε(i) · B(i) =n
∑i=1
v(i) · [1− ε · B0(i)] · B(i)
=n
∑i=1
v(i) · B(i)︸ ︷︷ ︸=f (B)
−εn
∑i=1
v(i) · B0(i) · B(i)︸ ︷︷ ︸=:p(B)
· · · ∞0ε0 ε1 ε2 εk
threshold parameters
f (P0) ≥ f (P1) > . . . > f (Pk)p (P0) ≥ p (P1) > . . . > p (Pk)
![Page 43: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/43.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Properties of Penalty Alternatives
vε (i) = v (i) · [1− ε · B0(i)]
⇒ fε(B) =n
∑i=1
vε(i) · B(i) =n
∑i=1
v(i) · [1− ε · B0(i)] · B(i)
=n
∑i=1
v(i) · B(i)︸ ︷︷ ︸=f (B)
−εn
∑i=1
v(i) · B0(i) · B(i)︸ ︷︷ ︸=:p(B)
· · · ∞0ε0 ε1 ε2 εk
threshold parameters
f (P0) ≥ f (P1) > . . . > f (Pk)p (P0) ≥ p (P1) > . . . > p (Pk)
![Page 44: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/44.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Threshold Parameters
Let ε be the threshold parameter between the optimalityintervals of the two penalty alternatives Bl and Br .e.g.: Bl = Pi and Br = Pi+1Then we can compute ε the following way.
fε (Bl) = fε (Br )
⇔ f (Bl)− ε · p (Bl) = f (Br )− ε · p (Br )
⇔ ε =f (Bl)− f (Br )
p (Bl)− p (Br )
![Page 45: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/45.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
An Algorithm for Computing all Penalty Alternatives(Schwarz)
1 Initialization: Compute B0 (optimal solution) and B∞.Go to step 2 with [Bl = B0,Br = B∞].
2 Compute the possible threshold parameter ε between Bland Br and the penalty alternative Bε. Then we considerthe following two cases:fε (Bε) = fε (Bl) = fε (Br ):
bagagagsadgsagsadgdsaglabla
No further branching. ε is the real thresholdparameter between Bl and Br .
fε (Bε) 6= fε (Bl) = fε (Br ):
bagagagsadgsagsadgdsaglabla
With Bε we found a new penalty alternative,so we have to branch. Go to step 2 with[Bl = Bl ,Br = Bε] and [Bl = Bε,Br = Br ]
![Page 46: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/46.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 47: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/47.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
From Chess history
Siegbert Tarrasch(1862−1934)In every chess positionthere exists exactly onebest move!
Emanuel Lasker(1868−1941)In every chess positionthere exist as manyappropriate moves asthere are different players!
Oskar Cordel’s (1843−1913) “Three moves law”In every chess position there exists either
exactly one best moveor at least three equally best moves.
This assumption does not hold in every chess position!
![Page 48: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/48.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
From Chess history
Siegbert Tarrasch(1862−1934)In every chess positionthere exists exactly onebest move!
Emanuel Lasker(1868−1941)In every chess positionthere exist as manyappropriate moves asthere are different players!
Oskar Cordel’s (1843−1913) “Three moves law”In every chess position there exists either
exactly one best moveor at least three equally best moves.
This assumption does not hold in every chess position!
![Page 49: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/49.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
From Chess history
Siegbert Tarrasch(1862−1934)In every chess positionthere exists exactly onebest move!
Emanuel Lasker(1868−1941)In every chess positionthere exist as manyappropriate moves asthere are different players!
Oskar Cordel’s (1843−1913) “Three moves law”In every chess position there exists either
exactly one best moveor at least three equally best moves.
This assumption does not hold in every chess position!
![Page 50: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/50.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
From Chess history
Siegbert Tarrasch(1862−1934)In every chess positionthere exists exactly onebest move!
Emanuel Lasker(1868−1941)In every chess positionthere exist as manyappropriate moves asthere are different players!
Oskar Cordel’s (1843−1913) “Three moves law”In every chess position there exists either
exactly one best moveor at least three equally best moves.
This assumption does not hold in every chess position!
![Page 51: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/51.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidates
moves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)
a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
![Page 52: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/52.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)
a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
![Page 53: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/53.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)
a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
![Page 54: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/54.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)a1,a2, and a3 solutions of a given maximization problem
f (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
![Page 55: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/55.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
![Page 56: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/56.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
![Page 57: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/57.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Generalized Cordel Property (GeCoP)
Generalizing Cordel: examine differences betweencandidatesmoves a1 and a2 are equally good, if |f (a1)− f (a2)| isvery small
Definition: Generalized Cordel Property (GeCoP)a1,a2, and a3 solutions of a given maximization problemf (a1) ≥ f (a2) ≥ f (a3)
define differences di := f (ai)− f (ai+1)
We say that a1,a2, and a3 fulfill the Generalized CordelProperty (GeCoP), iff the following holds:
d1 ≥ d2 (GeCoP)
![Page 58: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/58.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)
⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
![Page 59: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/59.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)
⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
![Page 60: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/60.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)
⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
![Page 61: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/61.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)⇒ d1 = f (a1)− f (a2) will be very small
⇒ d2 = f (a2)− f (a3) has to be very small too,because of GeCoP (d1 ≥ d2)
⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
![Page 62: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/62.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)
⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
![Page 63: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/63.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
![Page 64: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/64.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Connection between Cordel and GeCoP
as a reminder: Cordel said there are never exactly twobest moves!
best moves in chess position: a1,a2 and a3
assume a1 and a2 are equally good (two best moves)⇒ d1 = f (a1)− f (a2) will be very small⇒ d2 = f (a2)− f (a3) has to be very small too,
because of GeCoP (d1 ≥ d2)⇒ at least three best moves!
My thesis: analyzing the validity of GeCoP for severalkinds of optimization problems andkinds of solutions (e.g. three best solutions,penalty-alternatives)
![Page 65: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/65.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
GeCoP for Penalty Alternatives
0ε0 ε1 ε2P0 P1 P2 . . .
![Page 66: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/66.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
GeCoP for Penalty Alternatives
0ε0 ε1 ε2P0 P1 P2 . . .
d1 := f(P0)− f(P1) d2 := f(P1)− f(P2)
![Page 67: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/67.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
GeCoP for Penalty Alternatives
0ε0 ε1 ε2P0 P1 P2 . . .
d1 := f(P0)− f(P1) d2 := f(P1)− f(P2)≥?
![Page 68: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/68.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 69: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/69.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]
C = 0.5 ·∑ni=1 w(i)
Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
![Page 70: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/70.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)
Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
![Page 71: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/71.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
![Page 72: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/72.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
![Page 73: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/73.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
![Page 74: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/74.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Random instances
according to Martello, Toth: “Knapsack Problems” (1990)two parameters v , r , e.g. v = 1000, r = 100w(i) uniformly randomly generated in the int range [1, v ]C = 0.5 ·∑n
i=1 w(i)Three different types of instances:
Uncorrelated: v(i) uniformly randomly generatedin the range [1, v ].
Weakly correlated: v(i) uniformly randomly generated inthe range [w(i)− r ,w(i) + r ]with respect to v(i) > 0.
Strongly correlated: v(i) = w(i) + r .
![Page 75: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/75.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Comparison of P (d1 ≥ d2) and P (d2 ≥ d3)
w(i), v(i) ∈ [1,100], v(i) ∈ [w(i)− 10,w(i) + 10], v(i) = w(i) + 10
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uncorrelatedweakly correlatedstrongly correlated
![Page 76: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/76.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Comparison of P (d1 ≥ d2) and P (d2 ≥ d3)
w(i), v(i) ∈ [1,1 000], v(i) ∈ [w(i)− 100,w(i) + 100], v(i) = w(i) + 100
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uncorrelatedweakly correlatedstrongly correlated
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uncorrelatedweakly correlatedstrongly correlated
![Page 77: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/77.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Comparison of P (d1 ≥ d2) and P (d2 ≥ d3)
w(i), v(i) ∈ [1,100 000], v(i) ∈ [w(i)− 10 000,w(i) + 10 000], v(i) = w(i) + 10 000
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5 10 20 50 100 200 500
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uncorrelatedweakly correlatedstrongly correlated
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uncorrelatedweakly correlatedstrongly correlated
![Page 78: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/78.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Results
1 Prob (d1 ≥ d2) �Prob (d2 ≥ d3)
2 The behavior changes heavily with the rangesize v .
The first phenomenon is unique according to our wider studieson other optimization problems!
![Page 79: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/79.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Results
1 Prob (d1 ≥ d2) �Prob (d2 ≥ d3)
2 The behavior changes heavily with the rangesize v .
The first phenomenon is unique according to our wider studieson other optimization problems!
![Page 80: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/80.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Results
1 Prob (d1 ≥ d2) �Prob (d2 ≥ d3)
2 The behavior changes heavily with the rangesize v .
The first phenomenon is unique according to our wider studieson other optimization problems!
![Page 81: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/81.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Average number of penalty alternatives
w(i), v(i) ∈ [1,100], v(i) ∈ [w(i)− 10,w(i) + 10], v(i) = w(i) + 10
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uncorrelatedweakly correlatedstrongly correlated
![Page 82: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/82.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Average number of penalty alternatives
w(i), v(i) ∈ [1,1 000], v(i) ∈ [w(i)− 100,w(i) + 100], v(i) = w(i) + 100
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![Page 83: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/83.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Average number of penalty alternatives
w(i), v(i) ∈ [1,100 000], v(i) ∈ [w(i)− 10 000,w(i) + 10 000], v(i) = w(i) + 10 000
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uncorrelatedweakly correlatedstrongly correlated
![Page 84: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/84.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Outline
1 Knapsack Problems
2 A Penalty Method for the Unbounded Knapsack Problem
3 The Generalized Cordel Property (GeCoP)
4 Experimental Results
5 Open Questions
![Page 85: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/85.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Open Questions
1 Do these phenomenona also occur at other optimizationproblems?
2 Is there a better method to generate alternatives for theunbounded knapsack problem?Is there a good method, that generates more than only 6alternatives in average?
3 How can the results presented be used in practice?Is Cordel more than a Gedanken-Experiment?
![Page 86: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/86.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Open Questions
1 Do these phenomenona also occur at other optimizationproblems?
2 Is there a better method to generate alternatives for theunbounded knapsack problem?Is there a good method, that generates more than only 6alternatives in average?
3 How can the results presented be used in practice?Is Cordel more than a Gedanken-Experiment?
![Page 87: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/87.jpg)
Knapsack Problems Penalty Method GeCoP Experimental Results Open Questions
Open Questions
1 Do these phenomenona also occur at other optimizationproblems?
2 Is there a better method to generate alternatives for theunbounded knapsack problem?Is there a good method, that generates more than only 6alternatives in average?
3 How can the results presented be used in practice?Is Cordel more than a Gedanken-Experiment?
![Page 88: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/88.jpg)
additional slides
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from the optimal solution B0.have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) = v (i) · [1− ε · B0(i)]
Important properties:ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
![Page 89: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/89.jpg)
additional slides
The Penalty Method for the Knapsack Problem
The goal is to compute a second solution, which shalldiffer from a given reference solution Br .have a good function value.
Main idea: Punish the items used in the optimal solutionby reducing their values.
vε (i) = v (i) · [1− ε · Br (i)]
Important properties:ε increases→ punishment gets higherB0(i) > B0(j)→ punishment of item i is higher
![Page 90: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/90.jpg)
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.
2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13
optimal solution: B0 = (2,0,1,0)
![Page 91: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/91.jpg)
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13vw ≈ [1.2, 1.14, 1, 0.17]→ order: 1,2,3,4
(2,0,0,0)→ (2,0,0,0)→ (2,0,1,0)
![Page 92: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/92.jpg)
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13vw ≈ [1.2, 1.14, 1, 0.17]→ order: 1,2,3,4(2,0,0,0),C ′ = 3
![Page 93: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/93.jpg)
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13vw ≈ [1.2, 1.14, 1, 0.17]→ order: 1,2,3,4(2,0,0,0),C ′ = 3→ (2,0,0,0),C ′ = 3
![Page 94: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/94.jpg)
additional slides
Reference Solutions
Possible reference solutions are:1 An optimal solution.2 A Greedy solution
sort items in decreasing order of value per unit of weightput every item in sequence with maximal frequency in theknapsack such that the capacity is not exceeded
Example: v = [6,8,3,1], w = [5,7,3,6], C = 13vw ≈ [1.2, 1.14, 1, 0.17]→ order: 1,2,3,4(2,0,0,0),C ′ = 3→ (2,0,0,0),C ′ = 3→ (2,0,1,0),C ′ = 0
![Page 95: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/95.jpg)
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How often is the Greedy Solution Optimal?
w(i), v(i) ∈ [1,1 000], v(i) ∈ [w(i)− 100,w(i) + 100], v(i) = w(i) + 100
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Optimal Solution as Reference Solution
w(i), v(i) ∈ [1,1 000], v(i) ∈ [w(i)− 100,w(i) + 100], v(i) = w(i) + 100
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![Page 97: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/97.jpg)
additional slides
Greedy Solution as Reference Solution
w(i), v(i) ∈ [1,1 000], v(i) ∈ [w(i)− 100,w(i) + 100], v(i) = w(i) + 100
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![Page 98: The Unbounded Knapsack Problem and the Generalized Cordel ... · The Unbounded Knapsack Problem and the Generalized Cordel Property Lisa Schreiber Friedrich-Schiller-Universitat Jena,¨](https://reader034.vdocuments.us/reader034/viewer/2022050714/5d6778ff88c993d5408b9bf8/html5/thumbnails/98.jpg)
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Other Studied Optimization-Problems
Shortest Path ProblemBinary Knapsack ProblemMinimal Spanning TreesAssignment Problemsome theoretical problem types