Table 2 Description of symbols

CA

Cloud provider agent.

PA

Cloud provider agent. iPA denotes the isolated Cloud providers, hPA is the home Cloud provider in the federation and fPA is the foreign Cloud provider in the federation.

ℑ

Interrelations between two classes of agents.

\(\mathbf{\mathfrak{L}}\)

The set of interrelations between hPA and fPA.

U

The universe of discourse for the entire distributed fuzzy constraint network (DFCN).

X^k

The tuple of non-recurring objects of the k^th agent.

C^k

The set of fuzzy constraints of the k^th agent.

\({\mathbf{\mathfrak{N}}}^k\)

The fuzzy Constraint Network (FCN) of the k^th agent.

\({\prod}_{{\mathbf{\mathfrak{N}}}^k}\)

The intention of a fuzzy constraint network \({\mathbf{\mathfrak{N}}}^k\), which is an n-ary possibility distribution.

\(\overline{{\mathrm{C}}_i^k}\)

The cylindrical extension of \({\mathbf{C}}_i^k\) in the space X^k.

\({{}_{\alpha}\prod}_{{\mathbf{\mathfrak{N}}}^k}\)

The α − level cut of \({\prod}_{{\mathbf{\mathfrak{N}}}^k}\), which can be viewed as a set of solutions satisfying all constraints that are greater than or equal to an acceptable threshold α.

Ψ^k(S)

The aggregated satisfaction value (ASV) of the solution S.

F_l(S)

The fuzzy membership function about the l^th issue of the solution S.

ρ, δ, r, λ

The concession according to the own satisfaction degree, the responsive information of the opponent, the time constraint and the market factor, respectively.

ε, Δε, ε^∗

The threshold, concession value and new threshold, respectively.

D(A, B)

The distance measure between offer A and counter-offer B.

G( )

The distance measure function of two fuzzy sets.

P

The set of feasible solutions.

T( )

The appropriate measure function between a solution and the counteroffer.

S^∗

The prospective solution.

A^∗

The new offer, which is the marginal particularized possibility distribution in the space X^k of the k^th agent.

\({\mathbf{X}}^{k^{\hbox{'}}}\)

The tuple of non-recurring objects of the k^th agent in the second tier negotiation.

\({\mathbf{C}}^{k^{\hbox{'}}}\)

The set of fuzzy constraints of the k^th agent in the second tier negotiation.

ρ^', δ^', r^', λ^'

The satisfaction, response, time and market factor of the second tier, respectively.

Δε^'

The concession value of the second tier.

P^'

The set of second tier feasible solutions.