Table 1 Notations of this paper

T = {t₁, t₂, …, t_m, …, t_M}

A set of M subtasks

S = {S₁, S₂, …, S_m, …, S_M}

A set of M  candidate service sets

\({S}_m=\left\{{s}_{m,1},{s}_{m,1},\dots, {s}_{m,{n}^{\prime }},\dots, {s}_{m,{N}^{\prime }}\right\}\)

A set of N^′ candidate services

\({S}^{\prime }=\left\{{\textrm{s}}_{1,2},{\textrm{s}}_{2,2},\dots, {s}_{m,{n}^{\prime }},\dots, {s}_{M,{N}^{\prime }}\right\}\)

A composite service

P = {P₁, P₂, …, P_n, …, P_N}

A set of N individual

\({P}_n=\left\{{\textrm{s}}_{1,2},{\textrm{s}}_{2,2},\dots, {s}_{m,{n}^{\prime }},\dots, {s}_{M,{N}^{\prime }}\right\}\)

Mapping of services and individual

\({\prod}_{m=1}^M{N}^{\prime }\)

The composite service path

T

Task

s_m

Candidate service

\({s}_{m,{n}^{\prime }}\)

The n^′-th service of service set S_m

t_m

Subtask

\({q}_{Te}^{S_m}\), \({q}_{Ct}^{S_m}\), \({q}_{Rl}^{S_m}\), \({q}_{Rp}^{S_m}\)

Time, cost, reliability and reputation of candidate service

q₁,  q₂, q₃, q₄

Time, cost, reliability and reputation of aggregation values

\({\textrm{Q}}_{\textrm{k}}^{+}\), \({\textrm{Q}}_{\textrm{k}}^{-}\)

Normalized positive and negative attribute indicators

q_k

The QoS attribute

minq_k, maxq_k

The minimum and maximum aggregated values of the QoS attribute

ω_k

The weight of the attribute

w₁, w₂, w₃, w₄

The weight of time, cost, reliability and reputation

R_max

Maximum iterations

P

The population

P_n

The individual

\({P}_{T^{\prime }, new}^i\), \({P}_{T^{\prime }, old}^i\)

The new and old position

P_mean

The mean position

P_teacher

The teacher (the best individual at current iteration)

T_F

The teaching factor

r, r₁, r_j , r₂, c₁, c₂

The random numbers

\({P}_{c, old}^{l_1}\), \({P}_{c, old}^{l_2}\)

The old position of the l₁-th and l₂-th individual in teaching-phase

\({P}_{c, old}^{l_1,d}\), \({P}_{c, old}^{l_2,d}\)

The old dimension of teaching-phase

\({P}_{c, new}^{l_1,d}\), \({P}_{c, new}^{l_2,d}\)

The new dimension of teaching-phase

\({P}_{L, new}^j\), \({P}_{L, old}^j\), \({P}_{L, old}^k\)

The new and old position of learning-phase

\({P}_{L, new}^{j,h}\), \({P}_{L, old}^{j,h}\), \({P}_{L, old}^{k,h}\)

The new and old dimension of learning-phase