Formulation of suppliers' individual average profit
Differential equation for the supplier in Figure 2 in [0,T] is given by
with boundary conditions
and
. Solving the differential equation with the boundary condition, we have
(3)
.
Figure 2 Inventory level of supplier.
The total idle cost =
, purchase cost =
, selling price =
, and ordering cost =
.
(4)
Formulation of manufacturer individual average profit
Inventory level of manufacturer in Figure 3 in [0,T] is given by
(5)
with boundary conditions
and
.
Figure 3 Inventory level of manufacturer.
.
.
The total idle cost =
, purchase cost =
, selling price =
, and ordering cost =
.
Case 1
(When
)
.
.
.
(6)
Case 2
(When
)
(7)
Formulation of retailer individual average profit
Inventory level of retailer in Figure 4 in [0,T] is given by
(8)
with boundary conditions
and
.
Figure 4 Inventory level of retailer.
.
The total idle cost =
, purchase cost =
, selling price =
, and ordering cost =
.
Case 1
(When
)
Interest earned by the retailers for (
) cycle is given by
.
,
.
,
.
(9)
Case 2
(When
)
Interest earned by the retailers for (
) cycle is given by
Interest payable by the retailers for the first n cycle is given by
(10)
(11)
Crisp environment
Case 1
(
)
(12)
where
,
(13)
(14)
and
) (15)
(16)
(17)
Therefore,
is concave if
. (18)
Case 2
(
)
(19)
where A and B are given in (16) and (17), respectively and
F=[hmn2+n−2r−22+hr2n+12]TRDR+(n+1)cr1IeDcM22
+(nidm−ids−idr)TR+idsTs−(As+Am+Ar) (20)
(21)
(22)
Therefore,
is concave if
(23)
Proposed inventory model in uncertain environment
Let us consider
,
,
, and
as zigzag uncertain variables where
,
,
, and
. Then, the objective is reduce to the following:
- For Case 1 ( )
(24)
- For Case 2 ()
(25)
The equivalent crisp model
Using Lemma 1 and applying Theorem 2, the expected total average profit is given by the following:
- For Case 1
(26)
- For Case 2 (T′ ≤ M ≤ T R )
(27)