Quality Control in Food Supply Chain Incorporating Government ...

of quality effort under the given requirement is
K M , and
KM
is an increasing convex function,
2
kei
e
which means i , KM '( ei) > 0, KM ''( ei) > 0 . The function of quality effort is
K
M
=
2 (3),
and the
quality effort cost parameter k>0. Similar cost function of quality effort is also used by Gurnani and
Erkoc(2008),Kaya and Ozer (2009).
e
H
The company’s profit is
2
⎧ kei
⎪ DM − − [ θϕd + θ (1 − ϕ)0 ],
ei
< e
⎪
2
2
⎪ kei
Π
M
= ⎨ DM − , eL ≤ ei
≤ eH
⎪ 2
2
⎪ kei
⎪ DM − +R
G,
ei
> eH
2
⎩ (4)
α
As can be seen from above in either case, company face reflected function is ( p-c)
ei
=
k
Here, on
e ∗ i
case discussion,we know that
⎧ M + kR M + M −k Me − R −ke
⎪ e < e < < e e < e <
⎪ k k k k k
∗ ⎪αM αM αM+ 2kRG
αM
αM+
2kθϕd
αM+ 2 k( θϕd+
RG
)
ei = ⎨ , 当 < eH且 eH > ; 若 < eL, eL > , eH
>
⎪ k k k k k k
⎪ αM+ α M − (2kR + 2 kαMe − k e ) αM+
2kθϕd
⎪ < < <
k k k
2 2 2
αM
α 2
G αM
αM+
2kθϕd
α α (2α
L
2
G L
)
H, 当
H
; 若
L, H
,
H
2 2 2 2
αM
G H H
eL 当 eL , eL
⎪⎩ , ,
α M
eL
k
<
( 一 )If the
(A)The government quality level line is in the range
2 2 2
αM + 2kθϕ d αM + α M −k(2αMeL −2 RG −keL
)
< , eH
<
k
k
Enterprise will choose eH
as the quality level
kG
∏
G
= ln − θ( l+ αeH) m+ r( eH −eL)
At this time 1−
θ
The government in order to achieve maximum profits, θ on both sides to be derivative
1
θ = 1 − ( l + αe ) m
H
2
keH
∏
M
= ( l+ αeH)
M − + RG
∏
G
= ln kG( l+ αeH) m− ( l+ αeH) m+ 1 + r( eH −eL)
2
,
(7)
αM + 2kθϕ d αM + 2 k( θϕd + RG
)
eL
> , eH
>
(B) From the condition, we can get
k
k
M
e
∗ = α
Enterprise will choose i
k
kG
αM αM
∏
G
= ln − θ( l+ α ) m+ r( − eL)
+ d
1− θ k k
(8)
The government in order to achieve maximum profits, θ on both sides to be derivative
1 k
θ = − ( 2
kl + α M ) m
Put (7) and (9) in (8)
2 2
α M ( kl+
α M)
m αM
G G L
∏ = ln k ( l+ ) m− + r( − e ) + d
k k k ,
2 2 2
α M ( kl+ α M) m−1
∏
M
= lM + −
ϕd
2
2 k ( kl+
α M)
m
L
(5)
(6)
(9)
(10)
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