By Cai M.-C., Favaron O., Li H.
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Extra resources for (2,k)-Factor-Critical Graphs and Toughness
1 are properly changed. 5b) are substituted with the constraints below. For each component h of i, the non-negative variables lβhi are introduced, assuming that all variables χ are set to one: 8β ∈ B, 8i, j ∈ I=i < j, 8h ∈ Ci , 8k ∈ Cj Á 1À wβ0hi À wβ0kj ! lβhi þ lβkj À Dβ 1 À σ þ βhkij , 2 ð3:5aÞ 8β ∈ B, 8i, j ∈ I=i < j, 8h ∈ Ci , 8k ∈ Cj Á 1À wβ0kj À wβ0hi ! 1 Alternative Models 31 8ω ∈ Ω, 8β ∈ B, 8i ∈ I, 8h ∈ Ci lβhi ! Lωβhi ϑωi : The following (surrogate) objective function is defined: X lβhi : max ð3:6Þ ð3:7Þ β ∈ B, i ∈ I, h ∈ Ci For each component h of each item i, the terms lβhi represent (for the orientation ω assumed by i) the projections, on the axes wβ, of an enclosing parallelepiped, containing component h and centred with it.
One in particular (Sect. 3) is an actual alternative to the general MIP model, no longer restricted to the feasibility subproblem. It could also be utilized (at least partially) in the heuristics of Chap. 4. This aspect would definitely represent an interesting objective for future research. 1 General MIP Model First Linear Reformulation The rationale of the general MIP model reformulation presented hereinafter stresses the introduction of an ad hoc objective function. This aims at reducing the solution search region, as much as possible, in order to obtain any integer-feasible solution.
As a partially alternative version of this model reformulation, the constraints À 8 β ∈ B, 8 i, j ∈ I/i < j, 8 h ∈ Ci, 8 k ∈ Cj, dþ Dβ could also βhkij + dβhkij be added to tighten the feasibility region (creating in the subspace Sβ the two extreme points (Dβ, 0) and (0, Dβ), without excluding any solution. inequalities are X These À Á þ À Lωβhi ϑωi þ Lωβkj ϑωj , obviously tighter than the bounds d βhkij þ d βhkij 2Dβ À ω∈Ω XÀ Á þ when Dβ À Lωβhi ϑωi þ Lωβkj ϑωj ! 0. The conditions dÀ βhkij , dβhkij ∈ [0, Dβ], ω∈Ω moreover, if explicitly introduced in the model, can be of computational advantage, when the linear solver adopted treats the variable bounds independently (as in the case of simplex-based ones).
(2,k)-Factor-Critical Graphs and Toughness by Cai M.-C., Favaron O., Li H.