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This branching process results in the formation of a branchandbound tree (we will discuss the bounding next). Each node in this tree represents a linear programme consisting of the original linear programme and the extra branches added. Eventually all the leaf nodes in the tree will contain solutions where all the integer variables have integer values (an integer solution) and no further branching is needed. All these values can be compared and the best one is the solution to the original integer programme.  
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< <  Note For mixed integer programmes of any reasonable size this tree could be huge, in fact it grows exponentially as the number of integer variables increases.  
> >  Note For mixed integer programmes of any reasonable size this tree could be huge; in fact it grows exponentially as the number of integer variables increases.  
The bounding process allows sections of the branchandbound tree to be removed from consideration before all the leaf nodes have integer solutions. It relies on the following optimisation principle: Adding constraints to a mathematical programme will result in a deterioration of the optimal objective value. 
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Integer Programming with OR Software 
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Integer Programming with OR Software 
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Integer Programming with OR Software  
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To see integer programming in action, check out some of the integer programming case studies: 
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Integer Programming Topics  
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Integer Programming with OR Software 
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Integer Programming Topics
 
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Integer Programming with OR Software

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< Ready to Review  done  Lauren>
Integer Programming  
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< <  Integer programmes are almost identical to linear programmes with one very important exception. Some of the decision variables in integer programmes can only have integer values. The variables are known as integer variables. Since most integer programmes contain a mix of real variables (i.e., that can have any real value) and integer variables they are often known as mixed integer programmes. While the change from a linear programming formulation is a minor one, the effect on the solution process is enormous. Integer programmes can be very difficult problems to solve and currently a lot of research is focussing on finding "good"...  Lauren there is a lot of current research finding "good" ways to solve integer programmes.  
> >  Integer programmes are almost identical to linear programmes with one very important exception. Some of the decision variables in integer programmes can only have integer values. The variables are known as integer variables. Since most integer programmes contain a mix of real variables (i.e., that can have any real value) and integer variables they are often known as mixed integer programmes. While the change from a linear programming formulation is a minor one, the effect on the solution process is enormous. Integer programmes can be very difficult problems to solve and currently a lot of research is focussing on finding "good" ways to solve integer programmes.  
Integer programming, the process of solving a (mixed) integer programme, was originally done using the branchandbound process. The branch part of the process eliminated noninteger values for integer variables in the following way:
 
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The current best integer solution is called the incumbent. After solving a linear programme at a leaf node of the branchandbound tree one of the following conditions holds:
 
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Only the last condition requires more branching, all the other conditions result in the node becoming fathomed and no more branching is required from that node.  
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Integer Programming Topics  
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Integer Programming with OR Software  
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To see integer programming in action, check out some of the integer programming case studies: 
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> >  < Ready to Review  done  Lauren>  
Integer Programming 
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Note For mixed integer programmes of any reasonable size this tree could be huge, in fact it grows exponentially as the number of integer variables increases.  
Changed:  
< <  The bounding process allows sections of the branchandbound tree to be removed from consideration before all the leaf nodes have integer solutions. It relies on the following optimization principle: Adding constraints to a mathematical programming will result in a deterioration of the optimal objective value.  
> >  The bounding process allows sections of the branchandbound tree to be removed from consideration before all the leaf nodes have integer solutions. It relies on the following optimisation principle: Adding constraints to a mathematical programme will result in a deterioration of the optimal objective value.  
This means that adding the branching constraints to the linear programmes at the branchandbound tree nodes will mean the resulting nodes will have an optimal objective function value that is equal to or worse than the optimal objective function value of the original linear programme. Thus the objective function values get worse the deeper into the tree you look. Since we are finding the integer solution in the branchandbound tree with the best objective value, we can use any integer solutions to bound the tree.  
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Integer Programming Topics  
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Integer Programming with OR Software  
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To see integer programming in action, check out some of the integer programming case studies: 
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To see integer programming in action, check out some of the integer programming case studies:  
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< <  Results from OpsRes web retrieved at 13:00 (GMT)</twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> Number of topics: 10 </patternSearchResultCount>  
> >  Results from OpsRes web retrieved at 13:00 (GMT)</twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> \usepackage{amsmath} Case Study: Submitted: Operations Research Topics: Application Areas: Contents Problem Description Problem Description Return... </twikiSummary> </twikiBottomRow> </patternSearchResult> Number of topics: 10 </patternSearchResultCount>  
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Only the last condition requires more branching, all the other conditions result in the node becoming fathomed and no more branching is required from that node. 
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< <  < Under Construction >  
Integer ProgrammingInteger programmes are almost identical to linear programmes with one very important exception. Some of the decision variables in integer programmes can only have integer values. The variables are known as integer variables. Since most integer programmes contain a mix of real variables (i.e., that can have any real value) and integer variables they are often known as mixed integer programmes. While the change from a linear programming formulation is a minor one, the effect on the solution process is enormous. Integer programmes can be very difficult problems to solve and there is a lot of current research finding "good" ways to solve integer programmes. 
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Integer Programming with OR Software  
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To see integer programming in action, check out some of the integer programming case studies: 
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< <  Integer Programming with OR Software ??? Note to Mike > parent these from the guides to the OR Software ???  
> >  Integer Programming with OR Software  
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< Under Construction >  
Integer ProgrammingInteger programmes are almost identical to linear programmes with one very important exception. Some of the decision variables in integer programmes can only have integer values. The variables are known as integer variables. Since most integer programmes contain a mix of real variables (i.e., that can have any real value) and integer variables they are often known as mixed integer programmes. While the change from a linear programming formulation is a minor one, the effect on the solution process is enormous. Integer programmes can be very difficult problems to solve and there is a lot of current research finding "good" ways to solve integer programmes. 
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Integer ProgrammingInteger programmes are almost identical to linear programmes with one very important exception. Some of the decision variables in integer programmes can only have integer values. The variables are known as integer variables. Since most integer programmes contain a mix of real variables (i.e., that can have any real value) and integer variables they are often known as mixed integer programmes. While the change from a linear programming formulation is a minor one, the effect on the solution process is enormous. Integer programmes can be very difficult problems to solve and there is a lot of current research finding "good" ways to solve integer programmes. 
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Integer Programming  
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< <  Advanced Integer Programming Topics  
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Integer Programming  
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< <  Integer Programming with OR Software  
> >  Integer Programming with OR Software ??? Note to Mike > parent these from the guides to the OR Software ???  
To see integer programming in action, check out some of the integer programming case studies: 
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Integer Programming  
Changed:  
< <  Integer programmes are almost identical to linear programmes with one very important exception. Some of the decision variables in integer programmes may need to have only integer values. The variables are known as integer variables. Since most integer programmes contain a mix of real variables and integer variables they are often known as mixed integer programmes. While the change from linear programming is a minor one, the effect on the solution process is enormous. Integer programmes can be very difficult problems to solve and there is a lot of current research finding “good” ways to solve integer programmes.  
> >  Integer programmes are almost identical to linear programmes with one very important exception. Some of the decision variables in integer programmes can only have integer values. The variables are known as integer variables. Since most integer programmes contain a mix of real variables (i.e., that can have any real value) and integer variables they are often known as mixed integer programmes. While the change from a linear programming formulation is a minor one, the effect on the solution process is enormous. Integer programmes can be very difficult problems to solve and there is a lot of current research finding "good" ways to solve integer programmes.
Integer programming, the process of solving a (mixed) integer programme, was originally done using the branchandbound process. The branch part of the process eliminated noninteger values for integer variables in the following way:
This branching process results in the formation of a branchandbound tree (we will discuss the bounding next). Each node in this tree represents a linear programme consisting of the original linear programme and the extra branches added. Eventually all the leaf nodes in the tree will contain solutions where all the integer variables have integer values (an integer solution) and no further branching is needed. All these values can be compared and the best one is the solution to the original integer programme. Note For mixed integer programmes of any reasonable size this tree could be huge, in fact it grows exponentially as the number of integer variables increases. The bounding process allows sections of the branchandbound tree to be removed from consideration before all the leaf nodes have integer solutions. It relies on the following optimization principle: Adding constraints to a mathematical programming will result in a deterioration of the optimal objective value.  
Deleted:  
< <  Integer programming (the process of solving a (mixed) integer programme) was originally done using the branchandbound process. The branch part of the process eliminated noninteger values for integer variables in the following way:
This branching process results in the formation of a branchandbound tree (we will discuss the bounding next). Each node in this tree represents a linear programme consisting of the original linear programme and the extra branches added. Eventually all the leaf nodes in the tree will contain solutions where all the integer variables have integer values and no further branching is needed. All these values can be compared and the best one is the solution to the original integer programme. The bounding process allows sections of the branchandbound tree to be removed from consideration before all the leaf nodes have integer solutions. It relies on the following optimization principle: Adding constraints to a mathematical programming will result in a deterioration of the optimal objective value.  
This means that adding the branching constraints to the linear programmes at the branchandbound tree nodes will mean the resulting nodes will have an optimal objective function value that is equal to or worse than the optimal objective function value of the original linear programme. Thus the objective function values get worse the deeper into the tree you look. Since we are finding the integer solution in the branchandbound tree with the best objective value, we can use any integer solutions to bound the tree.  
Changed:  
< <  The current best integer solution is called the incumbent. After solving a linear programme at a leaf node of the branchandbound tree one of the following conditions holds:
 
> >  The current best integer solution is called the incumbent. After solving a linear programme at a leaf node of the branchandbound tree one of the following conditions holds:
Only the last condition requires more branching, all the other conditions result in the node becoming fathomed and no more branching is required from that node.  
Changed:  
< <  Only the last condition requires more branching, all the other conditions result in the node becoming fathomed and no more branching is required from that node.  
> >  Example of a branchandbound tree  
Changed:  
< <  Example  
> > 
Advanced Integer Programming Topics
Integer Programming with OR SoftwareTo see integer programming in action, check out some of the integer programming case studies:  
Results from OpsRes web retrieved at 13:00 (GMT)</twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> Number of topics: 10 </patternSearchResultCount>  TWikiAdminGroup  20 Feb 2008  
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> >  Integer ProgrammingInteger programmes are almost identical to linear programmes with one very important exception. Some of the decision variables in integer programmes may need to have only integer values. The variables are known as integer variables. Since most integer programmes contain a mix of real variables and integer variables they are often known as mixed integer programmes. While the change from linear programming is a minor one, the effect on the solution process is enormous. Integer programmes can be very difficult problems to solve and there is a lot of current research finding “good” ways to solve integer programmes. Integer programming (the process of solving a (mixed) integer programme) was originally done using the branchandbound process. The branch part of the process eliminated noninteger values for integer variables in the following way:
This branching process results in the formation of a branchandbound tree (we will discuss the bounding next). Each node in this tree represents a linear programme consisting of the original linear programme and the extra branches added. Eventually all the leaf nodes in the tree will contain solutions where all the integer variables have integer values and no further branching is needed. All these values can be compared and the best one is the solution to the original integer programme. The bounding process allows sections of the branchandbound tree to be removed from consideration before all the leaf nodes have integer solutions. It relies on the following optimization principle: Adding constraints to a mathematical programming will result in a deterioration of the optimal objective value. This means that adding the branching constraints to the linear programmes at the branchandbound tree nodes will mean the resulting nodes will have an optimal objective function value that is equal to or worse than the optimal objective function value of the original linear programme. Thus the objective function values get worse the deeper into the tree you look. Since we are finding the integer solution in the branchandbound tree with the best objective value, we can use any integer solutions to bound the tree. The current best integer solution is called the incumbent. After solving a linear programme at a leaf node of the branchandbound tree one of the following conditions holds:
Only the last condition requires more branching, all the other conditions result in the node becoming fathomed and no more branching is required from that node.
Example
Results from OpsRes web retrieved at 13:00 (GMT)</twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> </twikiTopRow> </twikiSummary> </twikiBottomRow> </patternSearchResult> Number of topics: 10 </patternSearchResultCount>  
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