01.sol

01.sol - Industrial Engineering Fall Integer Programming So...

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Unformatted text preview: Industrial Engineering Fall Integer Programming So utions to Homework Prob em . De ne y i = if box i is used, otherwise for all i = , . . . , m ; x ji = if item j is put into box i , otherwise for all i = , . . . , m and j = , . . . , n . Consider the integer program maximize subject to m i = b i y i Q , n j = a j x ji b i y i for all i = , . . . , m , m i = x ji = for all j = , . . . , n , x ji { , } for all i = , . . . , m and j = , . . . , n , y i { , } for all i = , . . . , m . e above integer program is feasible if and only if the move is possible. Prob em . For j = , . . . , n , let a j , . . . , a j k ( j ) denote the support points of the functions { f i j i = , . . . , m } ; i.e., the functions { f i j i = , . . . , m } are linear on the intervals {[ a j , a j + ] = , . . . , k ( j ) - } . It follows that k ( j ) ( m + )( - ) + . en, the following integer program is valid: minimize n j = k ( j ) = j l f , j ( a j ) subject to n j = k ( j ) = j l f i , j ( a j ) b i for i = , . . . , m , k ( j ) = j = for all j = , . . . , n , j y j for all j = , . . . , n , j y j - + y j for all j = , . . . , n and = , . . . , k ( j ) - , j k ( j ) y j k ( j )- for all j = , . . . , n , k ( j )- = y j = for all j = , . . . , n , j for all j = , . . . , n and = , . . . , k ( j ) , y j { , } for all j = , . . . , n and = , . . . , k ( j ) - . Prob em . a. Let e be a vector of ones,...
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01.sol - Industrial Engineering Fall Integer Programming So...

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