Page 131 - IJOCTA-15-1

Duality for robust multi-dimensional vector variational control problem under invexity
Table 1. Author’s contribution in this field of study

Author(s) Variational Robustness Multiobjective Multi-Dimensional
Mititelu 21 × × × ✓
Gulati and ✓ × ✓ ×
Mehndiratta 22
Sachdev et al. 23 ✓ × ✓ ×
Singh et al. 24 ✓ × ✓ ×
Dubey et al. 25 ✓ × ✓ ×
Jayswal et al. 26 × ✓ × ✓
Baranwal et al. 27 × ✓ × ✓
Antczak and
Treant¸˘a 28 ✓ × × ✓
Pokharna and ✓ × ✓ ×
Tripathi 29
Nguyen et al. 30 × ✓ × ✓
Saeed and × ✓ ✓ ✓
Treant¸˘a 31
Bagri et al. 32 ✓ ✓ × ✓
Jayswal et al. 33 ✓ ✓ × ✓
This paper ✓ ✓ ✓ ✓

and folllowing holds
!
Z Z
Z Z
ˆ
max Ψ 1 (Λ, a 1 )dν, ..., max Ψ p (Λ, a p )dν ψ(Λ, σ κ , r)dν − ψ(Λ, ˆσ κ , r)dν (>)≧
a
a
Ω 1 ∈A 1 Ω p∈A p Ω Ω
Z
ˆ
subject to η(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)ψ σ (Λ, ˆσ κ , r)dν
Ω
θ(Λ, σ κ (t), b)≦0, b∈B Z
ˆ
+ (Λ, ˆσ κ r)dν
β(Λ, σ κ (t), µ)=0, µ∈M D κ η(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)ψ σ κ
Ω
t∈Ω, σ(t 0 )=δ 0 , σ(t 1 )=δ 1 . Z
ˆ
+ ξ(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)ψ ω (Λ, ˆσ κ , r)dν,
Ω
∀(σ, ω) ∈K ×Υ.
Thus, the set of feasible solutions for (RUVP) will Z
be Definition 3. A functional ψ(Λ, σ κ , r)dν is
Ω
defined as (strictly) pseudo-invex at (ˆσ, ˆω)∈K ×
T ={(σ, ω)∈K ×Υ : θ(Λ, σ κ (t), b)≦0, Υ if there exist η and ξ with η =0 if σ(t)= ˆσ(t)
such that
β(Λ, σ κ (t), µ)=0, σ(t 0 )=δ 0 , σ(t 1 )=δ 1 , t∈Ω, Z
ˆ
b∈B, µ∈M }. η(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)ψ σ (Λ, ˆσ κ , r)dν
Ω
Definition 1. A feasible point (¯σ, ¯ω)∈T is de- Z
ˆ
fined as the robust weak efficient solution for + D κ η(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)ψ σ κ (Λ, ˆσ κ r)dν
Ω
(RUVP) iff ∄ any (σ, ω)∈T such that Z
ˆ
Z Z + ξ(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)ψ ω (Λ, ˆσ κ , r)dν(>)≧0,
¯
maxΨ(Λ, a)dν < maxΨ Λ, a dν. Ω Z Z
Ω a∈A Ω a∈A
ˆ
⇒ ψ(Λ, σ κ , r)dν − ψ(Λ, ˆσ κ , r)dν (>)≧0,
For ψ ∈R, η(t,σ,ˆσ,σ κ , ˆσ κ , ω, ˆω)∈R q and Ω Ω
r
ξ(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)∈R , some definitions are ∀(σ, ω)∈K ×Υ.
presented which are extended on the lines of 20 Z
over uncertain parameter r: Definition 4. A functional ψ(Λ, σ κ , r)dν is
Ω
defined as quasi-invex at (ˆσ, ˆω)∈K ×Υ if there
Z
Definition 2. A functional ψ(Λ, σ κ , r)dν is exists η and ξ with η =0 for σ(t)= ˆσ(t) and
Ω
Z Z
defined as (strictly) invex at (ˆσ, ˆω)∈K ×Υ if
ˆ
ψ(Λ, σ κ , r)dν − ψ(Λ, ˆσ κ , r)dν ≦0,
there exist η and ξ such that η =0 if σ(t)= ˆσ(t)
Ω Ω
125