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3) If demand cannot be fully satisfied, then first recourse to other suppliers.
After fulfillment, if other suppliers still have inventory left, then they offer help.
If other suppliers dont have enough inventories, either, shortage penalty occurs.
3.2 Mathematical Formulation
With the demand being stochastic, we first make the model discrete, then build
one-objective and bi-objective models, respectively.
Monte Carlo Generating Stochastic Environment.
D
stands for demand
and stochastic demand function is written as
f
(
D
). In this problem, decision
variables are
r,Q
, so ultimate cost are decided upon these three variables
D,r,Q
. For every (
r,D
) , the expected cost has such form
E
(
C
)=
p
(
r,Q,D
)
f
(
D
)
dD
However, it is hard to implement a continuous function. We here use Monte
Carlo sampling method [16] to reach a precise estimation of this expectation.
Monte Carlo is to obtain many stochastic realizations under the condition that
D
follows its distribution function
f
(
D
). And then, function value will be calculated
under every realization. Through weighted combining, we get the estimation we
wanted. In this problem, we use
M
realizations, so that each has a probability
of 1
/M
. Eventually, the estimation can be written as
E
(
C
)=
p
(
r,Q,D
)
f
(
D
)
dD
M
m
=1
1
≈
Mp
(
r,Q,D
m
)
.
Multi-objective Model.
The parameters and decision variables are listed be-
low.
IOP
:
Inventory order position
Shortage
:
Shortage quantity
Holdinv
:
Holding inventory
Recourse
:
Borrowed quantity
Install
:
Fixed cost
Profit
:
P t
Demand
:
Demanded quantity
Sellout
:
Shortage quantity
S
:
Sold quantity
H
:
Cost for holding
R
:
Cost for borrowing
P
:
Price for selling
N
:
Stage number
K
:
Supply number
r
:
Supply number
Q
:
Ordering quantity
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