We can define the unconditional attainable set as: varPsi =left{left.left x,yright)in {mathrm{Re}}_^{p+q}right|x mathrm{can} mathrm{produce} yright}, (1 which according to Daraio and Simar (2005; 2007a) can also be characterized as: Ψ = {(x, y)|H X, Y(x, y) > 0} where {H}_{X,Y}left x,yright)=mathrm{Prob}left(Xle x,Yge yright).
More importantly, as the descendants of 19th century immigrants find it increasingly difficult to outdo their parents, new immigrants, starting from a more easily attainable set of goals, can achieve a "success" in more manageable terms.
Exterior trajectories executing ballistic lunar capture are formalized through the definition of special attainable sets.
The attainable sets are used in both cases to handle families of either coast arcs or low-thrust orbits.
Lakshmikantham and Mohapatra [13] presented a theorem to show under what conditions the attainable sets of FDIs are the level sets of a fuzzy map.
For any set (mathcal{M}subset X), let (mathcal {M}^ z)=C^ z capmathcal{M}-{z}), where (G z,t)={win X|Pi w,t)=z} ) is the attainable set of z at (tin {mathbb {R}}_).
With this scenario, Figure 1 displays the concepts of an SCF f, the set of outcomes P and the set of utilities U K. Let g Sk, s−k) be the attainable set of user k at s−k, i.e., the set of outcomes that user k can induce when the other users select s−k.
For any set (M subset X), let M^ z)=C^ z cap M-{z} quadmbox{and}quad M^ z)=G z cap M-{z}, where G z)=cupbigl{ G z,t)| tin R_bigr} quad mbox{and} quad G z,t)=bigl{ win X| Pi (w,t)=zbigr} is the attainable set of z at (tin R_).
Then the attainable set can be defined as ( {varPsi}_t^zsubseteq {varPsi}_tsubset {mathrm{Re}}_^{p+q} ), and its distribution can be characterized as: {H}_{left.X,Yright|Z}^tleft left.x,yright|zright)=mathrm{Prob}left left.Xle x,Yge yright|Z=z,T=tright). (4).
In using optimal control for scheduling stage, feedback adaptive control for execution stage, and attainable sets for analysis of the achievement of the planned performance in a real execution environment, we pro-vide a mathematically unified framework for SC scheduling and execution control.
