Group Theory in the General Nm Job Shop Problem
Group Theory in the General Nm Job Shop Problem
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Derivation of the Group Problem In this section we will follow the strategy suggested by the group theoretic 3pproach to integer programming. Beginning with the optimal LP solution to (1), (2) and (3) which has some variables at non-integer values, we derive a group problem whose solution set corresponds to the set of "corrections" to (1), (2) and (3). A correction is a set of non- negative integer values for non-basic variables which force all basic 4 Gomory, op. Cit. Shapiro, op. Cit.
11 vari...ables to integer values. In determining a correction via the group problem we ignore the non-negativity constraints on the basic variables so the solution determined may still be infeasible. The group problem is determined by mapping all non-basic columns and the right hand side of the optimal LP tableau into their fractional parts.
We start by writing (1), (2), and (3) without the objective function in partitioned matrix form as follows: (^^> Vb^^21^R^Wf ^^22^2R =^2 W (III) 83^X3 + R3^T^ + A3Y^ -IS3^ = b3 av) Wb^^i'^r +Vb -^^b = ^ (^^ ^51^3-^ Sl^R -^Vr =N Where (I) represents equations (1), (II) equations (3) without the objective, (III) equations (2) in which 3.
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To quickly assess the difficulty of the text, read a short excerpt:
Derivation of the Group Problem In this section we will follow the strategy suggested by the group theoretic 3pproach to integer programming. Beginning with the optimal LP solution to (1), (2) and (3) which has some variables at non-integer values, we derive a group problem whose solution set corresponds to the set of "corrections" to (1), (2) and (3). A correction is a set of non- negative integer values for non-basic variables which force all basic 4 Gomory, op. Cit. Shapiro, op. Cit.
11 vari...ables to integer values. In determining a correction via the group problem we ignore the non-negativity constraints on the basic variables so the solution determined may still be infeasible. The group problem is determined by mapping all non-basic columns and the right hand side of the optimal LP tableau into their fractional parts.
We start by writing (1), (2), and (3) without the objective function in partitioned matrix form as follows: (^^> Vb^^21^R^Wf ^^22^2R =^2 W (III) 83^X3 + R3^T^ + A3Y^ -IS3^ = b3 av) Wb^^i'^r +Vb -^^b = ^ (^^ ^51^3-^ Sl^R -^Vr =N Where (I) represents equations (1), (II) equations (3) without the objective, (III) equations (2) in which 3.
What to read after Group Theory in the General Nm Job Shop Problem?
You can find similar books in the "Read Also" column, or choose other free books by Marshall L Fisher to read onlineMoreLess
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