Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol66.Issue2.Jun2000:
Int. J. Production Economics 66 (2000) 149}158
On optimal two-stage lot sizing and inventory batching policies
Roger M. Hill*
School of Mathematical Sciences, University of Exeter, Laver Building, North Park Road, Exeter EX4 4QE, UK
Received 13 January 1999; accepted 13 August 1999
Abstract
A recent paper considered the two-stage lot-sizing problem with "nite production rates at both stages. The problem
was classi"ed according to: whether the production rate at the "rst stage is greater or less than that at the second stage,
whether the production batch size at the "rst stage is greater or less than that at the second stage and whether the transfer
from the "rst stage to the second stage is continuous or in batches. The objective of this note is to o!er an alternative
(more direct and intuitive) way to derive and present essentially similar results and also to extend the analysis by relaxing
one of the assumptions. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Inventory batching policies; Two-stage lot sizing
1. Introduction
2. De5nitions and assumptions
In an interesting recent paper [1] Kim carried
out a thorough analysis of the two-stage lot-sizing
problem with "nite production rates at both
stages. By considering the various ways of classifying such a model an optimal solution procedure
was derived and presented. The objective of this
note is to suggest an alternative way of performing
the analysis which we believe to be more intuitive
and concise and therefore possibly easier to
understand. In addition, by relaxing one of the
key assumptions it is shown how the policy
space can be extended and lower cost policies
derived.
For convenience we shall follow most of the
de"nitions and assumptions in [1] which we set out
again here. In addition we introduce one or two
further terms.
At the "rst production stage (stage 2) a raw
material is manufactured at a "nite rate, in batches,
to produce an intermediate product which we shall
call process stock. Stock is transferred, from
stage 2 to 1, either continuously (&continuous transfer') or as soon as a stage 2 batch has been "nished
(&batch transfer'). We note, in passing, that
another possible rule for stock transfer is to transfer
to stage 1 whatever process stock is required
to manufacture a batch at stage 1 so that this
stock arrives just at the time when the manufacture
of the stage 1 batch is due to start. This may
be applicable if the process stock holding cost increases after the transfer is made (see, for example,
[2,3]).
* Tel.: 44-1392-264-475; fax: 44-1392-264-460.
E-mail address: [email protected] (R.M. Hill).
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 2 2 - X
150
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
At stage 1 the process stock undergoes further
manufacturing at a "nite rate, in batches, to produce a "nished good (1 unit of process stock making 1 unit of "nished goods stock). Batch set up
costs are incurred at both stages and stockholding
costs are incurred on both process stock and "nished goods stock. There is a constant external demand for "nished goods which has to be met.
Everything about the system is assumed to be deterministic and all the stock variables are assumed
to be continuous in nature.
According to the policy structure considered by
Kim production at stage 2 is done in equal-sized
batches of Q and at stage 1 in equal-sized batches
2
of Q . Whichever of Q or Q is the greater deter1
1
2
mines the basic repeating production cycle. For
example, if Q *Q then the repeating cycle (of
2
1
duration Q /D) starts when the production of
2
a batch at stage 2 starts and "nishes when the
production of the next batch at stage 2 starts.
Within this cycle there are k batches of "nished
goods produced. Because Q will not in general be
2
an integral multiple of Q the actual pattern of
1
"nished goods production consists of (k!1)
batches of size Q and a "nal batch of size
1
dQ (0(dQ)Q ), determined by Q "(k!1)
1
2
Q #dQ. Similar comments apply if Q )Q .
1
2
1
Therefore it is not, in general, the case that all batch
sizes at a given stage are equal. In this note we
consider how the analysis might be extended if all
the batch sizes at a given stage are allowed to di!er.
2.1. Dexnitions
i
D
P
i
Q
i
F
i
c
i
AS
i
ATI
The stage/inventory index (i"2 for stage 2/
process stock and i"1 for stage 1/"nished
goods stock).
The constant (continuous) demand rate for
"nished goods (at stage 1).
The production rate at stage i (P 'D for
i
the problem to be non-trivial).
The (target) production batch size at stage i.
The set up cost at stage i.
The stockholding cost per unit per unit time
at stage i (we assume that c )c ).
2
1
The average stock level of inventory i.
The average total inventory in the system
("AS #AS ).
1
2
TC
The average total cost per unit time of set up
and stockholding.
Two general observations can be made:
(i) For some parameter combinations it will be
easier to compute AS and ATI and then de1
duce AS (as ATI!AS ) than to compute
2
1
AS directly. Similarly, it may sometimes be
2
more convenient to express the total stockholding cost per unit time as c ATI#
2
(c !c )AS (rather than c AS #c AS ).
1
2
1
1 1
2 2
(ii) Since c )c we generally wish to have a pol2
1
icy which, other things being equal, minimises
"nished goods stock and this will generally be
achieved by manufacturing "nished goods in
equal batch sizes. In this context we will make
use of the following fairly well-known
result (which can be proved by the method of
Lagrange multipliers):
If K is a positive constant then +n a2
j/1 j
is minimised, subject to the constraint
+n a "K, when all the a are equal. (A)
j/1 j
j
2.2. The eight parameter settings
Kim used the term &policy' to describe the eight
di!erent structural combinations of model parameters and decision variables under consideration.
We shall use the term &setting' and reserve the term
&policy' to identify a particular speci"cation of lot
sizes. The eight settings are:
Setting 1 Q *Q , P *P , batch transfer,
2
1 2
1
Setting 2 Q )Q , P *P , batch transfer,
2
1 2
1
Setting 3 Q *Q , P )P , batch transfer,
2
1 2
1
Setting 4 Q )Q , P )P , batch transfer,
2
1 2
1
Setting 5 Q *Q , P *P , continuous transfer,
2
1 2
1
Setting 6 Q )Q , P *P , continuous transfer,
2
1 2
1
Setting 7 Q *Q , P )P , continuous transfer,
2
1 2
1
Setting 8 Q )Q , P )P , continuous transfer.
2
1 2
1
Typical patterns of stock against time are shown
in Figs. 1}8. In these "gures the "nished goods
inventory is always shown as a solid line. In addition, the wide dash line shows either the total inventory or the process inventory, whichever is more
helpful in a particular setting. The narrow dash
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
Fig. 1. Cycle inventories for Setting 1 (k"4 illustrated).
Fig. 2. Cycle inventories for Setting 2 (k"3 illustrated).
151
152
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
Fig. 3. Cycle inventories for Setting 3 (k"4 illustrated).
lines are present only to draw out the structure of
the inventory behaviour. Some "gures show what
patterns result from allowing batch sizes at one
stage to di!er in a fairly general way. Values written
next to lines indicate the rate at which stock is
increasing or decreasing.
Given k and Q , AS is minimised, subject to
2
1
(k!1)Q #dQ"Q , by setting Q "dQ"Q /k
1
2
1
2
(from result (A)). If all the "nished goods batch sizes
are allowed to di!er then result (A) shows that it is
still optimal to have equal sized "nished goods
batches in this setting. Therefore,
3. Analysis
(P !D)Q
2
AS " 1
1
2kP
1
and
Setting 1: Q *Q , P *P , batch transfer (il2
1 2
1
lustrated in Fig. 1)
To be optimal the completion of a process batch
must coincide with the start of a "nished goods
batch, from which we get
DQ
x" 2 ,
P
2
DQ
(P !D)Q
Q
D
2 " 2 1#
ATI" 2 # 2
P
2P
2
P
2
2
2
and
A
(P !D)
((k!1)Q2 #dQ2).
AS " 1
1
1
2P Q
1 2
(1)
B
(2)
(3)
(4)
D
TC"
MF #kF N
2
1
Q
2
Q
(c !c )
D
D
2 1!
# 2 c 1#
# 1
2
2
k
P
P
1
2
GA
B
A
BH
.
(5)
This is convex in Q given k and also in k given
2
Q . Therefore, it is not di$cult to "nd the optimal
2
policy. (Similar comments can be applied to the
other settings.)
Setting 2: Q )Q , P *P , batch transfer
2
1
2
1
(illustrated in Fig. 2)
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
153
Fig. 4. (a) Cycle inventories for Setting 4 } the optimal policy if di!erent batch sizes are allowed (k"4 illustrated). (b) Cycle inventories
for Setting 4.
From Fig. 2 we see that, for the optimal solution,
the manufacture of the "rst batch of process stock
must "nish at the time when the manufacture of the
"nished goods batch is due to start, subsequent
process batches must "nish at the same time as the
previous batch is used up and the last process batch
must last until the end of "nished goods production. Since P *P this can be achieved by process
2
1
154
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
Fig. 5. Cycle inventories for Setting 5 (k"4 illustrated).
Fig. 6. Cycle inventories for Setting 6 (k"3 illustrated).
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
155
Fig. 7. (a) Cycle inventories for Setting 7 (k"4 illustrated). (b) Cycle inventories for the optimal policy for setting 7 when c "c (k"3
1
2
illustrated).
batches of the required form. Exactly the same
reasoning as that applied to Setting 1 shows that
it is optimal to have equal-sized process batches
(and again the argument can be extended to show
that it will always be optimal to have equal-sized
batches in this setting). The average inventories are
therefore
Q (P !D)
AS " 1 1
1
2P
1
(6)
156
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
Fig. 8. Cycle inventories for Setting 8.
and
A
B
1
Q D 1
#
AS " 1
.
2
P
2k P
1
2
(7)
Setting 3: Q *Q , P )P , batch transfer
2
1
2
1
(illustrated in Fig. 3)
The analysis of this setting is identical to that of
Setting 1.
Setting 4: Q )Q , P )P , batch transfer
2
1
2
1
(illustrated in Fig. 4a)
From Fig. 4a we can see directly that
(P !D)Q
1.
AS " 1
1
2P
1
(8)
We can also see that the next batch of process
stock must be available by the time the previous
batch has been used up. We have the same constraint on process batch sizes as we had in Setting
2 but this is now much more binding since
P )P . Again, the batching policy for process
2
1
stock, given Q and k, does not in#uence the other
1
costs involved. The optimal batching policy, from
Fig. 4a, is to manufacture continuously, transferring batches to stage 1 as they are required. We can
interpret this either as manufacturing a single batch
per cycle at stage 2 and making k transfers to stage
1 or, as is the case in Kim's context, that k batches
are manufactured per cycle with each batch starting
as soon as the previous one is "nished.
Average process stock is therefore minimised
when process manufacture is "nished as late as
possible in the "nished goods production cycle.
This in turn is achieved by making the last batch
(transfer) of process stock as small as possible subject to the constraint of maintaining continuous
production at stage 2.
If all process batch sizes are allowed to di!er then
the optimal policy is for the "rst process batch to
"nish when the "nished goods batch is due to start
and for the time to manufacture each subsequent
process batch to be the same as the time for the
previous process batch to be used up at stage 1.
This requires successive process batches to decrease
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
by the factor P /P . From this we can determine all
2 1
the process batch sizes and hence, after a few lines
of algebra, show that
DQ (1!(P /P ))(1#(P P )k)
2 1
2 1 .
AS " 1
(9)
2
2P
(1!(P P )k)
2
2 1
AS is convex in k and hence the total cost is
2
convex in both Q and k and therefore the optimal
1
policy can be determined.
To "nd the optimal &equal' batch size policy we
require the time to use up the penultimate process
batch at stage 1, at rate P , to be the same as the
1
time to manufacture the last process batch at rate
P . It follows that dQ"(P /P )Q and hence
2
2 1 2
Q "Q ((k!1)#P /P )"Q kH, where kH"(k!1)
1
2
2 1
2
#P /P . It is now probably easier to consider
2 1
total inventory. This is illustrated in Fig. 4b. Total
inventory reaches a maximum of (P !D)Q /P
1
1 1
#(DdQ/P ) and falls (at a uniform rate) to a
1
minimum
of
(P !D)Q /P #D dQ/P
1
1 1
1
!(P !D)Q /P . Therefore,
2
1 2
D dQ (P !D)Q
(P !D)Q
1
1#
! 2
(10)
ATI" 1
2P
P
P
2
1
1
and hence,
(P !D)Q
D dQ (P !D)Q
1#
1
AS " 1
! 2
2
2P
2P
P
1
2
1
1
D dQ DQ
1
# 1
!
"
2 P
P
P
1
1
2
1
DQ
1
DP Q
!
" 2 1# 1
P
2
P
P2 kH
2
1
1
DQ 2 P 2
P
2 # 1! 2 .
" 1
2P kH P
P
2
1
1
This gives
A
B
A
G A B A
B
BH
(11a)
(11b)
(11c)
(11d)
G A
B H
G AG A B A BHB
A BH
D
P
TC"
F # kH#1! 2 F
2
1
P
Q
1
1
P
Q c D 2 P 2
2 # 1! 2
# 1 2
P
kH P
2 P
1
1
2
D
#c 1!
1
P
1
which is convex in kH and hence in k.
(12)
157
Setting 5: Q *Q , P *P , continuous transfer
2
1 2
1
(illustrated in Fig. 5)
This setting is essentially similar to Setting 1. To
be optimal we require the start of a process batch to
coincide with the start of a "nished goods batch
and minimising "nished goods stock given Q and
2
k requires the "nished goods batches to be of equal
size. This gives
Q (P !D)
,
AS " 2 1
1
2kP
1
Q (P !D)
ATI" 2 2
2P
2
and
(13)
(14)
AS "ATI!AS .
(15)
2
1
Setting 6: Q )Q , P *P , continuous transfer
2
1 2
1
(illustrated in Fig. 6)
It is again straightforward to show that we need
equal process batch sizes, with
Q (P !D)
AS " 1 1
1
2P
1
and
(16)
Q (P !P )D
1 .
AS " 1 2
(17)
2
2kP P
1 2
Setting 7: Q *Q , P )P , continuous transfer
2
1 2
1
(illustrated in Fig. 7a)
The structure in this setting is similar to that
described in [3] for which a full analysis (allowing
the "nished goods batch sizes to vary in a quite
general way) is quite complex. In [3] no production
takes place at stage 1 (or production at this stage
takes zero time), and there is a "xed cost associated
with the transfer of stock from stage 2 to 1, but the
structure of the optimal policy is essentially the
same as that in this setting. It is possible to show
that for a policy to be optimal in this setting the
sequence of "nished goods batch sizes within a process production cycle must be non-decreasing.
Therefore, a policy of equal-sized "nished goods
batches must have a lower cost than one for which
the last batch is smaller than the previous batches.
If all the "nished goods batch sizes are allowed to
vary then, in the limit as c /c P0, the optimal
1 2
158
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
policy tends to one with equal-sized "nished goods
batches, since, given Q and k, the primary objec2
tive is to minimise AS .
1
In the opposite limit, when c "c , we need not
1
2
distinguish between process stock and "nished
goods stock and the primary objective is to minimise ATI. This is achieved when the total stock in
the system at the start of the process stock production cycle (x in Fig. 7a) is minimised and the means
of achieving this is shown diagrammatically in Fig.
7b. For this policy the size of the "rst "nished goods
batch is P P x/D(P !P ) and successive "nished
1 2
1
2
goods batches increase in the ratio (P !D)
1
P /D(P !P ). Using the fact that the sizes of the
2
1
2
"nished goods batches must sum to Q we can
2
determine x, and hence ATI and AS , in terms of
1
Q and k and hence "nd the optimal solution.
2
The optimal policy for relative values of c and
1
c between the two extremes mentioned above can
2
be shown to be a sequence of "nished goods batch
sizes which increase in the ratio (P !D)P /
1
2
D(P !P ) followed by a number of equal-sized
1
2
batches.
Returning to Kim's equal "nished goods batchsize approach, the time from the beginning of the
process production batch to the completion of the
"rst "nished goods batch in the cycle is the time to
consume x at rate D plus the time to manufacture
Q /k at rate P , which is x/D#(Q /kP ). The
2
1
2
1
quantity of process stock manufactured during this
time, at rate P , must be exactly Q /k. This gives
2
2
Q (P !P )D
2 ,
(18)
x" 2 1
kP P
1 2
Q (P !P )D Q (P !D)
2 # 2 2
(19)
ATI" 2 1
2P
kP P
2
1 2
and
Q (P !D)
AS " 2 1
.
1
2kP
1
(20)
Setting 8: Q )Q , P )P , continuous transfer
2
1 2
1
(illustrated in Fig. 8)
Since P )P it is clearly not worth manufac2
1
turing more than one batch of process stock per
cycle and so k"1 and Q "Q . This gives
2
1
Q (P !D)
AS " 1 1
(21)
1
2P
1
and
A
B
Q D 1
1
.
AS " 1
!
2
2 P
P
2
1
(22)
4. Concluding observations
An algorithm procedure for tackling a particular
problem is fairly fully described by Kim [1].
It is important to note that if all the batch sizes at
a given stage are required to be equal then it is quite
possible that better solutions may be achieved by
considering schedules for which nQ "mQ , where
2
1
m and n have no common factors (other than 1) and
both m and n are greater than 1. In this case the
cycle time is nQ /D ("mQ /D) which is neither
2
1
Q /D nor Q /D. However, such solutions would be
2
1
di$cult to "nd and even more di$cult to sell to
management and implement in practice.
References
[1] D. Kim, Optimal two-stage lot sizing and inventory batching policies, International Journal of Production Economics 58 (3) (1999) 221}234.
[2] R.M. Hill, The single-vendor single-buyer integrated production}inventory model with a generalised policy, European Journal of Operational Research 97 (3) (1997) 493}499.
[3] R.M. Hill, An analysis of the single-vendor single-buyer
integrated production}inventory problem, International
Journal of Production Research 37 (11) (1999) 2463}2475.
On optimal two-stage lot sizing and inventory batching policies
Roger M. Hill*
School of Mathematical Sciences, University of Exeter, Laver Building, North Park Road, Exeter EX4 4QE, UK
Received 13 January 1999; accepted 13 August 1999
Abstract
A recent paper considered the two-stage lot-sizing problem with "nite production rates at both stages. The problem
was classi"ed according to: whether the production rate at the "rst stage is greater or less than that at the second stage,
whether the production batch size at the "rst stage is greater or less than that at the second stage and whether the transfer
from the "rst stage to the second stage is continuous or in batches. The objective of this note is to o!er an alternative
(more direct and intuitive) way to derive and present essentially similar results and also to extend the analysis by relaxing
one of the assumptions. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Inventory batching policies; Two-stage lot sizing
1. Introduction
2. De5nitions and assumptions
In an interesting recent paper [1] Kim carried
out a thorough analysis of the two-stage lot-sizing
problem with "nite production rates at both
stages. By considering the various ways of classifying such a model an optimal solution procedure
was derived and presented. The objective of this
note is to suggest an alternative way of performing
the analysis which we believe to be more intuitive
and concise and therefore possibly easier to
understand. In addition, by relaxing one of the
key assumptions it is shown how the policy
space can be extended and lower cost policies
derived.
For convenience we shall follow most of the
de"nitions and assumptions in [1] which we set out
again here. In addition we introduce one or two
further terms.
At the "rst production stage (stage 2) a raw
material is manufactured at a "nite rate, in batches,
to produce an intermediate product which we shall
call process stock. Stock is transferred, from
stage 2 to 1, either continuously (&continuous transfer') or as soon as a stage 2 batch has been "nished
(&batch transfer'). We note, in passing, that
another possible rule for stock transfer is to transfer
to stage 1 whatever process stock is required
to manufacture a batch at stage 1 so that this
stock arrives just at the time when the manufacture
of the stage 1 batch is due to start. This may
be applicable if the process stock holding cost increases after the transfer is made (see, for example,
[2,3]).
* Tel.: 44-1392-264-475; fax: 44-1392-264-460.
E-mail address: [email protected] (R.M. Hill).
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 2 2 - X
150
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
At stage 1 the process stock undergoes further
manufacturing at a "nite rate, in batches, to produce a "nished good (1 unit of process stock making 1 unit of "nished goods stock). Batch set up
costs are incurred at both stages and stockholding
costs are incurred on both process stock and "nished goods stock. There is a constant external demand for "nished goods which has to be met.
Everything about the system is assumed to be deterministic and all the stock variables are assumed
to be continuous in nature.
According to the policy structure considered by
Kim production at stage 2 is done in equal-sized
batches of Q and at stage 1 in equal-sized batches
2
of Q . Whichever of Q or Q is the greater deter1
1
2
mines the basic repeating production cycle. For
example, if Q *Q then the repeating cycle (of
2
1
duration Q /D) starts when the production of
2
a batch at stage 2 starts and "nishes when the
production of the next batch at stage 2 starts.
Within this cycle there are k batches of "nished
goods produced. Because Q will not in general be
2
an integral multiple of Q the actual pattern of
1
"nished goods production consists of (k!1)
batches of size Q and a "nal batch of size
1
dQ (0(dQ)Q ), determined by Q "(k!1)
1
2
Q #dQ. Similar comments apply if Q )Q .
1
2
1
Therefore it is not, in general, the case that all batch
sizes at a given stage are equal. In this note we
consider how the analysis might be extended if all
the batch sizes at a given stage are allowed to di!er.
2.1. Dexnitions
i
D
P
i
Q
i
F
i
c
i
AS
i
ATI
The stage/inventory index (i"2 for stage 2/
process stock and i"1 for stage 1/"nished
goods stock).
The constant (continuous) demand rate for
"nished goods (at stage 1).
The production rate at stage i (P 'D for
i
the problem to be non-trivial).
The (target) production batch size at stage i.
The set up cost at stage i.
The stockholding cost per unit per unit time
at stage i (we assume that c )c ).
2
1
The average stock level of inventory i.
The average total inventory in the system
("AS #AS ).
1
2
TC
The average total cost per unit time of set up
and stockholding.
Two general observations can be made:
(i) For some parameter combinations it will be
easier to compute AS and ATI and then de1
duce AS (as ATI!AS ) than to compute
2
1
AS directly. Similarly, it may sometimes be
2
more convenient to express the total stockholding cost per unit time as c ATI#
2
(c !c )AS (rather than c AS #c AS ).
1
2
1
1 1
2 2
(ii) Since c )c we generally wish to have a pol2
1
icy which, other things being equal, minimises
"nished goods stock and this will generally be
achieved by manufacturing "nished goods in
equal batch sizes. In this context we will make
use of the following fairly well-known
result (which can be proved by the method of
Lagrange multipliers):
If K is a positive constant then +n a2
j/1 j
is minimised, subject to the constraint
+n a "K, when all the a are equal. (A)
j/1 j
j
2.2. The eight parameter settings
Kim used the term &policy' to describe the eight
di!erent structural combinations of model parameters and decision variables under consideration.
We shall use the term &setting' and reserve the term
&policy' to identify a particular speci"cation of lot
sizes. The eight settings are:
Setting 1 Q *Q , P *P , batch transfer,
2
1 2
1
Setting 2 Q )Q , P *P , batch transfer,
2
1 2
1
Setting 3 Q *Q , P )P , batch transfer,
2
1 2
1
Setting 4 Q )Q , P )P , batch transfer,
2
1 2
1
Setting 5 Q *Q , P *P , continuous transfer,
2
1 2
1
Setting 6 Q )Q , P *P , continuous transfer,
2
1 2
1
Setting 7 Q *Q , P )P , continuous transfer,
2
1 2
1
Setting 8 Q )Q , P )P , continuous transfer.
2
1 2
1
Typical patterns of stock against time are shown
in Figs. 1}8. In these "gures the "nished goods
inventory is always shown as a solid line. In addition, the wide dash line shows either the total inventory or the process inventory, whichever is more
helpful in a particular setting. The narrow dash
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
Fig. 1. Cycle inventories for Setting 1 (k"4 illustrated).
Fig. 2. Cycle inventories for Setting 2 (k"3 illustrated).
151
152
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
Fig. 3. Cycle inventories for Setting 3 (k"4 illustrated).
lines are present only to draw out the structure of
the inventory behaviour. Some "gures show what
patterns result from allowing batch sizes at one
stage to di!er in a fairly general way. Values written
next to lines indicate the rate at which stock is
increasing or decreasing.
Given k and Q , AS is minimised, subject to
2
1
(k!1)Q #dQ"Q , by setting Q "dQ"Q /k
1
2
1
2
(from result (A)). If all the "nished goods batch sizes
are allowed to di!er then result (A) shows that it is
still optimal to have equal sized "nished goods
batches in this setting. Therefore,
3. Analysis
(P !D)Q
2
AS " 1
1
2kP
1
and
Setting 1: Q *Q , P *P , batch transfer (il2
1 2
1
lustrated in Fig. 1)
To be optimal the completion of a process batch
must coincide with the start of a "nished goods
batch, from which we get
DQ
x" 2 ,
P
2
DQ
(P !D)Q
Q
D
2 " 2 1#
ATI" 2 # 2
P
2P
2
P
2
2
2
and
A
(P !D)
((k!1)Q2 #dQ2).
AS " 1
1
1
2P Q
1 2
(1)
B
(2)
(3)
(4)
D
TC"
MF #kF N
2
1
Q
2
Q
(c !c )
D
D
2 1!
# 2 c 1#
# 1
2
2
k
P
P
1
2
GA
B
A
BH
.
(5)
This is convex in Q given k and also in k given
2
Q . Therefore, it is not di$cult to "nd the optimal
2
policy. (Similar comments can be applied to the
other settings.)
Setting 2: Q )Q , P *P , batch transfer
2
1
2
1
(illustrated in Fig. 2)
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
153
Fig. 4. (a) Cycle inventories for Setting 4 } the optimal policy if di!erent batch sizes are allowed (k"4 illustrated). (b) Cycle inventories
for Setting 4.
From Fig. 2 we see that, for the optimal solution,
the manufacture of the "rst batch of process stock
must "nish at the time when the manufacture of the
"nished goods batch is due to start, subsequent
process batches must "nish at the same time as the
previous batch is used up and the last process batch
must last until the end of "nished goods production. Since P *P this can be achieved by process
2
1
154
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
Fig. 5. Cycle inventories for Setting 5 (k"4 illustrated).
Fig. 6. Cycle inventories for Setting 6 (k"3 illustrated).
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
155
Fig. 7. (a) Cycle inventories for Setting 7 (k"4 illustrated). (b) Cycle inventories for the optimal policy for setting 7 when c "c (k"3
1
2
illustrated).
batches of the required form. Exactly the same
reasoning as that applied to Setting 1 shows that
it is optimal to have equal-sized process batches
(and again the argument can be extended to show
that it will always be optimal to have equal-sized
batches in this setting). The average inventories are
therefore
Q (P !D)
AS " 1 1
1
2P
1
(6)
156
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
Fig. 8. Cycle inventories for Setting 8.
and
A
B
1
Q D 1
#
AS " 1
.
2
P
2k P
1
2
(7)
Setting 3: Q *Q , P )P , batch transfer
2
1
2
1
(illustrated in Fig. 3)
The analysis of this setting is identical to that of
Setting 1.
Setting 4: Q )Q , P )P , batch transfer
2
1
2
1
(illustrated in Fig. 4a)
From Fig. 4a we can see directly that
(P !D)Q
1.
AS " 1
1
2P
1
(8)
We can also see that the next batch of process
stock must be available by the time the previous
batch has been used up. We have the same constraint on process batch sizes as we had in Setting
2 but this is now much more binding since
P )P . Again, the batching policy for process
2
1
stock, given Q and k, does not in#uence the other
1
costs involved. The optimal batching policy, from
Fig. 4a, is to manufacture continuously, transferring batches to stage 1 as they are required. We can
interpret this either as manufacturing a single batch
per cycle at stage 2 and making k transfers to stage
1 or, as is the case in Kim's context, that k batches
are manufactured per cycle with each batch starting
as soon as the previous one is "nished.
Average process stock is therefore minimised
when process manufacture is "nished as late as
possible in the "nished goods production cycle.
This in turn is achieved by making the last batch
(transfer) of process stock as small as possible subject to the constraint of maintaining continuous
production at stage 2.
If all process batch sizes are allowed to di!er then
the optimal policy is for the "rst process batch to
"nish when the "nished goods batch is due to start
and for the time to manufacture each subsequent
process batch to be the same as the time for the
previous process batch to be used up at stage 1.
This requires successive process batches to decrease
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
by the factor P /P . From this we can determine all
2 1
the process batch sizes and hence, after a few lines
of algebra, show that
DQ (1!(P /P ))(1#(P P )k)
2 1
2 1 .
AS " 1
(9)
2
2P
(1!(P P )k)
2
2 1
AS is convex in k and hence the total cost is
2
convex in both Q and k and therefore the optimal
1
policy can be determined.
To "nd the optimal &equal' batch size policy we
require the time to use up the penultimate process
batch at stage 1, at rate P , to be the same as the
1
time to manufacture the last process batch at rate
P . It follows that dQ"(P /P )Q and hence
2
2 1 2
Q "Q ((k!1)#P /P )"Q kH, where kH"(k!1)
1
2
2 1
2
#P /P . It is now probably easier to consider
2 1
total inventory. This is illustrated in Fig. 4b. Total
inventory reaches a maximum of (P !D)Q /P
1
1 1
#(DdQ/P ) and falls (at a uniform rate) to a
1
minimum
of
(P !D)Q /P #D dQ/P
1
1 1
1
!(P !D)Q /P . Therefore,
2
1 2
D dQ (P !D)Q
(P !D)Q
1
1#
! 2
(10)
ATI" 1
2P
P
P
2
1
1
and hence,
(P !D)Q
D dQ (P !D)Q
1#
1
AS " 1
! 2
2
2P
2P
P
1
2
1
1
D dQ DQ
1
# 1
!
"
2 P
P
P
1
1
2
1
DQ
1
DP Q
!
" 2 1# 1
P
2
P
P2 kH
2
1
1
DQ 2 P 2
P
2 # 1! 2 .
" 1
2P kH P
P
2
1
1
This gives
A
B
A
G A B A
B
BH
(11a)
(11b)
(11c)
(11d)
G A
B H
G AG A B A BHB
A BH
D
P
TC"
F # kH#1! 2 F
2
1
P
Q
1
1
P
Q c D 2 P 2
2 # 1! 2
# 1 2
P
kH P
2 P
1
1
2
D
#c 1!
1
P
1
which is convex in kH and hence in k.
(12)
157
Setting 5: Q *Q , P *P , continuous transfer
2
1 2
1
(illustrated in Fig. 5)
This setting is essentially similar to Setting 1. To
be optimal we require the start of a process batch to
coincide with the start of a "nished goods batch
and minimising "nished goods stock given Q and
2
k requires the "nished goods batches to be of equal
size. This gives
Q (P !D)
,
AS " 2 1
1
2kP
1
Q (P !D)
ATI" 2 2
2P
2
and
(13)
(14)
AS "ATI!AS .
(15)
2
1
Setting 6: Q )Q , P *P , continuous transfer
2
1 2
1
(illustrated in Fig. 6)
It is again straightforward to show that we need
equal process batch sizes, with
Q (P !D)
AS " 1 1
1
2P
1
and
(16)
Q (P !P )D
1 .
AS " 1 2
(17)
2
2kP P
1 2
Setting 7: Q *Q , P )P , continuous transfer
2
1 2
1
(illustrated in Fig. 7a)
The structure in this setting is similar to that
described in [3] for which a full analysis (allowing
the "nished goods batch sizes to vary in a quite
general way) is quite complex. In [3] no production
takes place at stage 1 (or production at this stage
takes zero time), and there is a "xed cost associated
with the transfer of stock from stage 2 to 1, but the
structure of the optimal policy is essentially the
same as that in this setting. It is possible to show
that for a policy to be optimal in this setting the
sequence of "nished goods batch sizes within a process production cycle must be non-decreasing.
Therefore, a policy of equal-sized "nished goods
batches must have a lower cost than one for which
the last batch is smaller than the previous batches.
If all the "nished goods batch sizes are allowed to
vary then, in the limit as c /c P0, the optimal
1 2
158
R.M. Hill / Int. J. Production Economics 66 (2000) 149}158
policy tends to one with equal-sized "nished goods
batches, since, given Q and k, the primary objec2
tive is to minimise AS .
1
In the opposite limit, when c "c , we need not
1
2
distinguish between process stock and "nished
goods stock and the primary objective is to minimise ATI. This is achieved when the total stock in
the system at the start of the process stock production cycle (x in Fig. 7a) is minimised and the means
of achieving this is shown diagrammatically in Fig.
7b. For this policy the size of the "rst "nished goods
batch is P P x/D(P !P ) and successive "nished
1 2
1
2
goods batches increase in the ratio (P !D)
1
P /D(P !P ). Using the fact that the sizes of the
2
1
2
"nished goods batches must sum to Q we can
2
determine x, and hence ATI and AS , in terms of
1
Q and k and hence "nd the optimal solution.
2
The optimal policy for relative values of c and
1
c between the two extremes mentioned above can
2
be shown to be a sequence of "nished goods batch
sizes which increase in the ratio (P !D)P /
1
2
D(P !P ) followed by a number of equal-sized
1
2
batches.
Returning to Kim's equal "nished goods batchsize approach, the time from the beginning of the
process production batch to the completion of the
"rst "nished goods batch in the cycle is the time to
consume x at rate D plus the time to manufacture
Q /k at rate P , which is x/D#(Q /kP ). The
2
1
2
1
quantity of process stock manufactured during this
time, at rate P , must be exactly Q /k. This gives
2
2
Q (P !P )D
2 ,
(18)
x" 2 1
kP P
1 2
Q (P !P )D Q (P !D)
2 # 2 2
(19)
ATI" 2 1
2P
kP P
2
1 2
and
Q (P !D)
AS " 2 1
.
1
2kP
1
(20)
Setting 8: Q )Q , P )P , continuous transfer
2
1 2
1
(illustrated in Fig. 8)
Since P )P it is clearly not worth manufac2
1
turing more than one batch of process stock per
cycle and so k"1 and Q "Q . This gives
2
1
Q (P !D)
AS " 1 1
(21)
1
2P
1
and
A
B
Q D 1
1
.
AS " 1
!
2
2 P
P
2
1
(22)
4. Concluding observations
An algorithm procedure for tackling a particular
problem is fairly fully described by Kim [1].
It is important to note that if all the batch sizes at
a given stage are required to be equal then it is quite
possible that better solutions may be achieved by
considering schedules for which nQ "mQ , where
2
1
m and n have no common factors (other than 1) and
both m and n are greater than 1. In this case the
cycle time is nQ /D ("mQ /D) which is neither
2
1
Q /D nor Q /D. However, such solutions would be
2
1
di$cult to "nd and even more di$cult to sell to
management and implement in practice.
References
[1] D. Kim, Optimal two-stage lot sizing and inventory batching policies, International Journal of Production Economics 58 (3) (1999) 221}234.
[2] R.M. Hill, The single-vendor single-buyer integrated production}inventory model with a generalised policy, European Journal of Operational Research 97 (3) (1997) 493}499.
[3] R.M. Hill, An analysis of the single-vendor single-buyer
integrated production}inventory problem, International
Journal of Production Research 37 (11) (1999) 2463}2475.