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ABACUS
(Mathematics
Science
Series) Vol. 45,
No
1,
Aug.
2018
FIXED
LIFETIME
INVENTORY
SYSTEM
WITTH
TWO
ORDERs
AND
STOCHASTIC
DEMAND.
Izevbizua.
O
and
Apanapudor.
S.J
Department
of
mathematics
University
of
Benin
Department
of
Mathematics
Delta
State
University
Abraka
maorobo@vahoo.com
Abstract.
Attempts
to
reduce the amount
of
items
outdating from the fixed lifetime inventory system has lead
to
an increase
in
the shortages
in
the
system.
To
address this issue
of
shortages,
we
introduce the
two
order
model, derive the total cost function
of
the model
and
gave
numerical
example
using
a
product
whose
lifetime
is
four
periods.
Key
words:
inventory, outdates, shortages, system, policy.
Introduction.
One
of
the major
problem
associated with the fixe
etime
inventory
system
is the outdating
of
products not
used
to
meet
demand before their exp.ra
ion
date.
Many
authors [eg
Nahmias
and
Pierskallla
(1973),
Chiu
(1995),
Liu
and
Lian
(1Sey),
Nahmias
(2011),
Tripathi
and
Sheweta
(2013),
Harshal
(2015),
Shen-Chin
el
ta
(2016),
Izevbizua
and
Omosigho
(2017)]
in
the literature based
the
decision
to order
new
products
on
the quantity
of
items on hand.
Policy such as
"order
this amount, when the quantity on
hand
drops to this
amount"
is
popular
among
researchers and inventory managers. However, Izevbizua and Omosigho
(2017) introduced a new fixed lifetime inventory model where the decision
to
order
new
items
is
based
on
the
number
of
useful lifetime remaining
on
the items
on
handrather than the
quantity
of
items
on
hand.
The
ordering policy
of
the
new
model reads "order this amount
when the remaining useful lifetime
of
the items on hand is this". The model was applied to
two
fixed lifetime products, egg and bread.
One
advantage
the
useful lifetime model
have
over the quantity model is that the
amount
of
items
outdating
is less.
However,
the useful lifetime
model
has
its
own
draw back.
The
number
of
shortages
resulting
from its
application
is
high.
This
is
because
only
one
order is placed at a time.
To
eliminate
or
reduce
the shortage quantities res':lting
from
placing
one order at a time,
we
introduce a
new
model
where
there are
two
orders,
one
period
apart in
arri
ving
into
the
inventory system.
The
second
order
is to
mop
up
excess
demand
that
cannot
be satisfied from the first order.
The
two
orders
form a set
of
order.In
this work,
we
shall
derive the total
cost
function
of
the
model.
Assumption
and
notation
of
the
model.
Assumption
1)
There
are
two
orders
y,
and
y,2,
one
period
apart
in
arriving
into
inventory.
if
y
arrives
in
period
i,
y2
arrives in
period
i+1.The
two
orders form a
set
of
order.
2)
We
place
order
for
new
products
when
the
useful-lifetime
remaining
on
the
items
from
the
first order
of
a
set
is
one
period.
The
issuing
policy
is
First
in
First
out
(FIF0)
Only
products
from the
previous
set
can
outdate
in the
current
cycle.
3)
4)
66
ABACUS
(Mathematics
Science
Series)
Vol. 45,
No
1,
Aug.
2018
Any
product(s)
not
used
to
meet
demand
by
the
end
of
period
m,
outdates
and
is
discarded.
5)
6)
7)
The
age
of
the
products
arriving
into
inventory
is
zero.
The lead time
is
zero
Notation.
Jirst
order
second
order
=
order
i
of
set
j
m =
lifetime
of
products
x=
the
reorder point
wrt
order
yi
x,
i =
1,2
reorder
po
int
for
y,,
i =
1,2
k =
ordering
cost
per
unit
6 outdate
cost
per
unit
V
shortage
cost
per
unit
h
holding
cosi
per
unit
d,, i =
1,2..
m,...demand
in
period i
t=
realization
of
demand
S)=
deamnd
densit
Description
of
the
model.
Tablel describe
the
model
for
the
case where
the
useful-lifetime
is
mperiods.
Table
1:Double
order
inventory
for
a
product
with
m
useful
-lifetime
T
T2
d--
age
0|
age
1.
age
m.
(outdates)
(outcates)
SET1
age
2.
age
m-1.
x-
if
>d,
|ifx>d
r-d
e 0 age 1
age
m-1
age
m
-2
age m
SET2
2 -
+-2
-d
-d
- -
age
age
l.
age
m.
(outdates)
(outdates)
age
m
-.
-
>a|rsd.
ge
age
m-I.
age
m.
age
m-2
Ta
SET3
-
age
0
age
age
m-
age
Y
age
m-2
ABACUS
(Mathematics
Science
Series) Vol. 45,
No
1, Aug.
2018
the
There
are
three
sets
of
orders
in Tablel,
each
set
consists
of
two
orders
y,
and
v.
Alsa
dh
quantity
of
items
on
hand
when
new
orders
are
placed
is
represented
by
X,
and
x,
.For
set
1,
the
first
order
arrived
at
the
start
of
period
1.
At
the
start
of
period
2,
the
first
orderic
depleted
by
demand
in
period1
and
the
second
order
arrives.
The
periodic demand continues
until
the
useful
lifetime
left
on
items
from the first
order
of
setl
is
one
period, which
is
the
reorder
period
for
new
items.
Items from
the
first
order
not
used
to
meet
demand
at
the end of
.period m
outdate
in
period
m+1,
while
items
from
the
second
order
not
used
to
meet
demand
at
the
end
of
period
m+l
outdate
in
period
m
+2.
The
process
continues
for
the
next
set
of
orders.
Next
,we
look
at
the
amount
ofitems
on
hand and their
age
categories.
This
enable
us
keep
track
of
items
from a
particular
order.
age
{0)
T,
=(-d)+y,age
{l,0
T=
(-d
-d,)+(-d,),
age
12,1
T.=-4,)+(-4,)+.ai
-1,m-2.0)
7,
=-4)
+%-4,)+0f-4)+i.
age
tm,
m -
1,1,09)
-
T..=
(-4,)+0f-d,
-d,,)+(i-d4,),
age{m,2.1}
i
T (-4,)+0-4,-4,)+.agetm-1,2.0)
The
component
of
the total cost function are holding cost, ordering cost, outdate cost and
shortage
cost
Holding Cost.
The holding cost refer
to
the cost
for
holding items in inventory. The cost
is
normally charged
at
the
end of a period after demand
and
outdating
has
taking place. The quantity of items
on
hand is giving as
On
hand inventory +
+x
+x,
-1)St)dt
(1)
With
holding
cost
of
h
per
unit
held
in
inventory,
we
have
our
holding
cost
as
Holding
cost= h
JO++
X
+x,
-t)f(t)dt
(2)
Shortage cost
One of the advantage
of
the
two
order fixed lifetime inventory system is that shortage
is
highly minimized.
The
second order
of
each set act has
backup
for
the
first order.
The
68
ABACUS
(Mathematics
Science
Series)
Vol.
45,
No
1,
Aug.
2018
incoming second order takes care
of
any demand that cannot be satisfied by the first order.
It's however
not
completely
safe
to
assume,
there
are
no
shortages
in
the
system,
as
there exists
(no
matter
how
small)
the
probability
of
shortage
occurring
when demand is very
high.
Shortage
occur
when
demand
is
greater
than
the
items
on
hand.
The
shortage
quantity
is
given
as
Shortage
quantity=
-,+y,
+x, +x,)f)dt
(3)
*y2+*+*2
With shortage cost
of
v for each demand that cannot be satisfied, our shortage cost
is
Shortage
cost=y
--(y, + y2
+x,
+)S(t)dt
(4)
*y2
+X
tX2
Outdate
quantity
Products outdate
if
they not used to meet demand at the end
of
their useful-lifetime in
inventory. Since
we
have
two orders whose items are one period different in age they will
outdate
one
period
apart,
if
not
used
to
meet
demand
at
the
end
of
their
useful
life.
While
products
from
will outdate
at
the
end
of period m, products
from
will
outdate
at
the
end
of
period m
+1.
So
our attention
will
be
focus
on
periods m
and
m+1
for
each
set
of orders.
Casel:period
m.
Items on hand at the end
of
period
m,
from Tablel include
(-4)(-2,)+-d,)+»
of
the four age categories,
only
products
from (
-d,)
will outdate
if
and
only
if
x
>d,.
Case2:
period
m+1
Items
on
hand
at
the
end
of
period m
+1,
from
Tablel include
-d)+(-d,
-4,)+(-4,)
Again only products
from
(x
-d,.)
will
outdate
if
and only
if
x >
d,
Thisjustify our assumption that only products from the previous set will outdate in the
current cycle. Therefore our outdate quantity per set will be
outdate quantity= Jx-)/tydt+ Ja
-1)fMdt
5)
For
convince
the
outdate
quantity
can
represented
by
*
outdate
quantity
=(x,
+x2
-1)f)dt
6)
0
outdate
cost
per
unit
is
6,
so
outdate
cost
=0
+x,
-1)f)dt
(7
0
Ordering
cost
There
is a
fixed
ordering
cost
k
per
unit
ordered.
Our
ordering
cost
per
setis
Ordering
cost
= k
(y,
+y2)
8)
69
ABACUS
(Mathematics
Science
Series) Vol. 45, No
1,
Aug.
2018
Adding
equations
(2), (4),
(¬)
and
(8)
we
obtain
our
total
cost
function
as
y,+t
'1
C(yyX,,X,)
=
k(y,+y,)+h
++x
+*^
-1)f)dt
+
v
-(,+
+x,
+*,))S)dt
(9)
+0
,+
-1)SU)dt
0
Equation
(9)
is the Total
cost
function
per
set
for the
two
order
model.
Numerical
Example.
We
applied
the
model
to
a
fixed
lifetime
product
with
four
useful
lifetime
and
zero
lead
time.
The result is shown in Table2 and Table3. First the ordering
policy
was
used
to
generate the
orders
of
the
sets while equation
(9)
was
used
to
generate the associated cost for each
model.
Table2:
orders,
demand,
shortage
and
outdates
for
a
produet
with
4useful-lifetime
and
zero
lead
time.
Day
Y2
demand
shortage
Outdates
0 20
= 70
50 40
25
10
25
32
=20
20
3
29
38
32 30
=50
6 50 15
=25
37
25
17
= 30
3
20
25
15 5
2
25
9
30
40
=10
20
10 15 20
31
10
=10
11
10 4 38
yi
=
40
10
12 26
30
45
41
13
50
=
45
14
36 0
28
=
45
70
ABACUS
(Mathematics
Science
Series)
Vol.
45,
No
1,
Aug.
2018
15
8
39
45 30
16
39
23
25
35
35
17
37
15
=37
35
18
22
18 4
37
=15
19
37
35
48
20
15
20
24
15
44
20
20
21
15
20
25
'=35
22
35
10
40
35
23
35
35
=30
24 30
30
=25
25 |25
40
L
40
26
35
y=45
27
40
38
5 y
30
37
28 45
29
10 30 32
30
30
30
40
y35
At the end
of
30 days
of
applying the ordering policy
of
the model, the number
of
shortages
was
zero
while the
outdates
was
9.
The
problem
of
shortages
have
been
addressed
by
the
model
as
the
second
order
mops
up
excess
demand.
From
Table
2,
we
have
Tabie
3 showing
the
cost
associated
with
each
set
of
orders.
The
ordering
cost,
shortage
cost,
holding
cost
outdate
coSt
per
units
are
given
as
k=
150,
v=
10,
h 20, 6
=0.005.
71
ABACUS
(Mathematics Science Series)
Vol.
45, No 1, Aug.
2018
Table
3:
double
orders
and
associated
cost
for
m=4
and
l = 0
outdates demand
Costs
set
X
70
10
95
22970.87
20
38
58
6356.89
57114.95
33510
1633.41
50
25
37
25
5
70
30
20
30
20
50
10 0
10
20
40
45 0
85
12191.67
45
23 90
32979.26
45
39
35 37
35
70 70638.19
37
15
37
15
52
36215.02
10
20
20
15 10 40 11596.96
35
35
5 70
12340.79
30
25
0
55
6695.45
13
40 45 2 85
13061.83
14
45
30 30
10
8 75 18025.77
15
35
Conclusion.
The
fixed lifetime
inventory
model
with
two
orders
have
been
considered.
Unlike the
systemn
with
one
order,
where
shortage
is
fairly
high,
the
shortages
associated
with
the
two
order
model
is
low
(sometimes
no
shortages
as
shown
in the
example)
this
is
because
the second
order
of
each set
mop
up any
excess
demand
that
cannot
be
satisfied
from
the
first
order.
This
result will
help
inventory
managers
overcome
the
problem
of
shortages
which
can
lead
to
loss
of
customers
goodwill.
References.
[1]
Chiu,HN
(1995).
An
apprOximetion
t
the
continuous
review
inventory
model
with
perishable
items and
lead
times.
European
Journal
of
Operational
Research.
Vol
87,
pp
93 -108.
2]
Nahmias,
S.
(2011).
Perishable
inventory
systems.
International
series in
operations
research
and
management science.
Vol
160.
[31
Izevbizua,
O and
Omosigho,
S.E
(2017).
A
review
of
the
fixed
lifetime
inventory
system.
The
journal
of
the
Mathematical
Association
of
Nigeria.
Vol
2
,pp
188-198.
14
Shen-
Chih
Chen,
Jie
Min,
Jinn-Tsair
Teng
and
Fuan
Li
(2016);
Inventory
and
Shelf
-Space
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for
fresh
produce
with
expiration
date
under
freshness-
and
-stock-
dependent
demand
rate.
Journal
of
the
Operational
Research
Society.
(S]
Harshal
.L,
Rahul.
N
and
Ravichandran.
N
(2015);
analysis
of
an
order-up-to-level
policy
for
perishable
with
random
issuing.
Journal
of
the
operations
research
society.
Vol67,
Pp
483-505.
6]
Tripathi,
R.P
and
Shweta.S.T
(2013);
optimal
order
policy
for
time-dependent
Deteriorating
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in
response
to
temporary
price
discount
linked
to
order
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Journal
of
applied
mathematical
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Vol7,
pp
2869-2878.
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Liu,
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Lian,
Z.
(1999);
(s,S)
Continuous
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pp.
150-158.
[8J
Nahmias,
S.
and
Pierskalla,
W.
P.
(1973);
Optimal
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Policies
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Product
thar
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two
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rishes
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20
