The Theory and Practice of a Dual Criteria Assignment Model with a continuously distributed Value-of-Time

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Model with a continuously distributed Value-of-Time

Fabien Leurent

To cite this version:

Fabien Leurent. The Theory and Practice of a Dual Criteria Assignment Model with a continuously distributed Value-of-Time. ISTTT, Jul 1996, Lyon, France. pp. 455-477. �hal-00348537�

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< * < =

> θ_m

π_k #

∂L

∂θ_m =q P_k π_k

∂π_k

∂θ_m k

∂²L

∂θ_m∂θ_n = q P_k π_k

∂²π_k

∂θ_m∂θ_n − ¹ π_k

∂π_k

∂θ_m

∂π_k

∂θ_n k

2 π_m(

{ }

H_µ,σ r^a U_m L^m

v _m^m+n:= ^P

m+n−P^m T^m−T^m+n 2

∂

∂µπ_m = _∂µ^∂H_µ,σ(U_m)−_∂µ^∂H_µ,σ(L_m)

∀D ∈ _∂µ^∂ ;_∂σ^∂ ;_{∂µ ∂σ}^∂2 ; ^∂2

∂µ²; ^∂2

∂σ²

Dπ_m =DH_µ,σ(U_m)−DH_µ,σ(L_m)

r^a

' X_a,k = X_a,m(k)

∂

∂r_av _m^m+n = ^Xa,m+n− X_a,m

T_m−T_m+n

∂²

∂r_a∂r_bv _m^m+n =0

*

∂π_m

∂r_a = ∂H_µ,σ(U_m)

∂x

∂U_m

∂r_a −∂H_µ,σ(L_m)

∂x

∂L_m

∂r_a

∂²π_m

∂r_a∂r_b = ∂²H_µ,σ(U_m)

∂x²

∂U_m

∂r_a

∂U_m

∂r_b −∂²H_µ,σ(L_m)

∂x²

∂L_m

∂r_a

∂L_m

∂r_b

∂²π_m

∂r_a∂µ ⁼

∂²H_µ,σ(U_m)

∂µ∂x

∂U_m

∂r_a −∂²H_µ,σ(L_m)

∂µ∂x

∂L_m

∂r_a

∂²π_m

∂r_a∂σ ⁼

∂²H_µ,σ(U_m)

∂σ∂x

∂U_m

∂r_a −∂²H_µ,σ(L_m)

∂σ∂x

∂L_m

∂r_a

< *

&%

µ σ

H(x)= Φ(^ln(x)−µ_σ ) Φ

F5" : . > φ

φ(t)=exp(−t²/ 2) / 2π 5

ln(x)−µ

σ ^tx

#

H⁻¹(y)=exp

(

)

F(x)= ¹

H⁻¹(u)du 0

x

=exp(σ²

2 −µ)Φ Φ

(

)

∂ /∂x /∂µ /∂σ

H(x)= φ(t_x) σ

1 /x

−1

−t_x

∂²

/∂x² /∂x∂µ /∂x∂σ

× /∂µ² /∂µ∂σ

× × /∂σ²

H(x)= φ(t_x) σ²

−(σ+t_x) /x² t_x/ x −(1−t_x²) /x

× −t_x 1−t_x²

× × t_x.(2−t_x²)

< *

)

+

,

? ?.# " .! ∃u∈E_rs^m (P _rsⁿ−P _rs^m) /u>0 0<T _rs^m−T _rsⁿ >

v₀:=(P _rsⁿ−P _rs^m) / (T _rs^m−T _rsⁿ)# .! v₀ ∈[supE_rs^m; inf E_rsⁿ ] (

supE_rs^m = inf E_rsⁿ v∈]sup E_rs^m; inf E_rsⁿ [ E_rsⁱ

B B E_rs^m E_rsⁿ $

v _rs^m =sup E_rs^m =inf E_rsⁿ =(P _rsⁿ−P _rs^m) / (T _rs^m−T _rsⁿ)

? ?1# - 8@

Q_rsⁿ = _mefficient ≤nq_rs^m

= q_{rs E} dH_rs(v)

rs

mefficient ≤n m = q_rsH_rs(v _rsⁿ)

e(n)=n (

Q_rsⁿ =Q_rs^e(n) Q_rsⁿ =q_rsH_rs(v _rs^e(n))

!

? # q_rs^m ≥0 H^rs

& n v _rs^e(n) % _mq_rs^m =q_rs

Q_rs^{m rs} =q_rsH_rs(v _rs^{e(m rs)}) = q_rsH_rs(H_rs⁻¹(1)) =q_rs ( e(n)= e(n−1)

Q_rsⁿ⁻¹=Q_rsⁿ q_rsⁿ =Q_rsⁿ −Q_rsⁿ⁻¹ =0 (

q_{rs E} dH_rs(v)

rs

m =

q_rsH_rs(v _rsⁿ)−q_rsH_rs(v _rs^e(n−1)) =Q_rsⁿ −Q_rsⁿ⁻¹ = q_rsⁿ

? ?7# ?1

& >

) & #

F A#

! !

Q_j =Q_m v _rs^m =v _rs^m v _rs^m ∈E_rs^m

G _rs^m(v _rs^m) =min_nG _rsⁿ(v _rs^m) ≤G _rs^j (v _rs^m)= T _rs^j+P _rs^j /v _rs^j

Q_j =Q_m I _rs^m ≤ I _rs^j

(P _rsⁱ −P _rsⁱ⁺¹) / v _rsⁱ m<i< j P _rs^m < P _rs^j j <i<m P _rs^m > P _rs^j

? # -

[q_rs^m]_m Ω_rs > _v∈ Ω_rs#

Ω_rs ⊂ _m[v _rs^m−1;v _rs^m] ' 8@ ! v∈[v _rs^m−1;v _rs^m]

v _rs^m−1<v _rs^m > ! )

n ≤m I _rsⁿ −I _rs^m =T _rsⁿ−T _rs^m+ _i=n^m−1(P _rsⁱ −P _rsⁱ⁺¹) /v _rsⁱ % _n ≤i<m

v _rsⁱ ≤v _rs^m−1≤v (P _rsⁱ −P _rsⁱ⁺¹) /v _rsⁱ ≤(P _rsⁱ −P _rsⁱ⁺¹) /v 2

_i=n^m−1(P _rsⁱ −P _rsⁱ⁺¹) / v _rsⁱ ≤ _i=^m−1_n (P _rsⁱ −P _rsⁱ⁺¹) /v =(P _rsⁿ −P _rs^m) /v

q_rs^m >0 0≤I _rsⁿ −I _rs^m 0≤T _rsⁿ−T _rs^m+(P _rsⁿ −P _rs^m) /v

G _rs^m(v)≤ G _rsⁿ(v) ( _n ≥m v _rsⁱ ≥v _rs^m−1≥v

n ≥i≥m −(P _rsⁱ −P _rsⁱ⁺¹) /v _rsⁱ ≤ −(P _rsⁱ −P _rsⁱ⁺¹) /v ! .

− _i=^m−1_n (P _rsⁱ −P _rsⁱ⁺¹) /v _rsⁱ ≤ − _i=n^m−1(P _rsⁱ −P _rsⁱ⁺¹) /v =(P _rsⁿ −P _rs^m) /v "

I _rsⁿ −I _rs^m =T _rsⁿ−T _rs^m− _i=mⁿ⁻¹(P _rsⁱ −P _rsⁱ⁺¹) /v _rsⁱ

0≤I _rsⁿ −I _rs^m G _rs^m(v)≤ G _rsⁿ(v)