*Tel.: 44-1392-264-475; fax: 44-1392-264-460. E-mail address:[email protected] (R.M. Hill).
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
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 classify-ing 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.
2. De5nitions and assumptions
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
trans-fer') 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 in-creases after the transfer is made (see, for example, [2,3]).
At stage 1 the process stock undergoes further
manufacturing at a "nite rate, in batches, to
pro-duce a"nished good (1 unit of process stock
mak-ing 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"
nish-ed goods stock. There is a constant external
de-mand for "nished goods which has to be met.
Everything about the system is assumed to be de-terministic 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 ofQ
2and at stage 1 in equal-sized batches
ofQ
1. Whichever ofQ1orQ2is the greater
deter-mines the basic repeating production cycle. For
example, if Q
2*Q1 then the repeating cycle (of
duration Q
2/D) starts when the production of
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. BecauseQ
2will not in general be
an integral multiple of Q
1 the actual pattern of
"nished goods production consists of (k!1)
batches of size Q1 and a "nal batch of size
dQ(0(dQ)Q
1), determined by Q2"(k!1)
Q
1#dQ. Similar comments apply if Q2)Q1.
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 The stage/inventory index (i"2 for stage 2/
process stock andi"1 for stage 1/"nished
goods stock).
D The constant (continuous) demand rate for
"nished goods (at stage 1). P
i The production rate at stagethe problem to be non-trivial).i (Pi'D for
Q
i The (target) production batch size at stagei.
F
i The set up cost at stagei.
c
i The stockholding cost per unit per unit timeat stagei(we assume thatc
2)c1). AS
i The average stock level of inventoryi.
ATI The average total inventory in the system
("AS
1#AS2).
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
1 and ATI and then
de-duce AS
2 (as ATI!AS1) than to compute
AS
2 directly. Similarly, it may sometimes be
more convenient to express the total
stock-holding cost per unit time as c
2ATI#
(c
1!c2)AS1 (rather thanc1AS1#c2AS2).
(ii) Sincec
2)c1 we generally wish to have a
pol-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 +nj/1a2
j
is minimised, subject to the constraint
+nj/1a
j"K, when all theaj are equal. (A)
2.2. The eight parameter settings
Kim used the term &policy' to describe the eight
di!erent structural combinations of model
para-meters 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
2*Q1,P2*P1, batch transfer,
Setting 2 Q
2)Q1,P2*P1, batch transfer,
Setting 3 Q
2*Q1,P2)P1, batch transfer,
Setting 4 Q
2)Q1,P2)P1, batch transfer,
Setting 5 Q
2*Q1,P2*P1, continuous transfer,
Setting 6 Q2)Q1,P2*P1, continuous transfer, Setting 7 Q
2*Q1,P2)P1, continuous transfer,
Setting 8 Q
2)Q1,P2)P1, continuous transfer.
Typical patterns of stock against time are shown
in Figs. 1}8. In these "gures the "nished goods
Fig. 1. Cycle inventories for Setting 1 (k"4 illustrated).
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.
3. Analysis
Setting 1:Q
2*Q1, P2*P1, batch transfer
(il-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
x"DQ2
1 is minimised, subject to
(k!1)Q
1#dQ"Q2, by setting Q1"dQ"Q2/k
(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,
AS
This is convex inQ
2 givenkand also inkgiven
Q
2. Therefore, it is not di$cult to"nd the optimal
policy. (Similar comments can be applied to the other settings.)
Setting 2:Q
2)Q1, P2*P1, batch transfer
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
produc-tion. SinceP
Fig. 5. Cycle inventories for Setting 5 (k"4 illustrated).
Fig. 7. (a) Cycle inventories for Setting 7 (k"4 illustrated). (b) Cycle inventories for the optimal policy for setting 7 whenc
1"c2(k"3
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
AS 1"
Q
1(P1!D)
2P
1
Fig. 8. Cycle inventories for Setting 8. (illustrated in Fig. 3)
The analysis of this setting is identical to that of Setting 1.
Setting 4:Q
2)Q1, P2)P1, batch transfer
(illustrated in Fig. 4a)
From Fig. 4a we can see directly that
AS
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 con-straint on process batch sizes as we had in Setting 2 but this is now much more binding since P
2)P1. Again, the batching policy for process
stock, givenQ
1andk, does not in#uence the other
costs involved. The optimal batching policy, from Fig. 4a, is to manufacture continuously, transfer-ring batches to stage 1 as they are required. We can interpret this either as manufacturing a single batch
per cycle at stage 2 and makingktransfers to stage
1 or, as is the case in Kim's context, thatkbatches
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 sub-ject 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
by the factorP
2/P1. From this we can determine all
the process batch sizes and hence, after a few lines of algebra, show that
AS
2 is convex in k and hence the total cost is
convex in bothQ
1andkand therefore the optimal
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
1, to be the same as the
time to manufacture the last process batch at rate P
total inventory. This is illustrated in Fig. 4b. Total
inventory reaches a maximum of (P
1!D)Q1/P1
(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 givenQ
2and
krequires the"nished goods batches to be of equal
size. This gives
AS
(illustrated in Fig. 6)
It is again straightforward to show that we need equal process batch sizes, with
AS
(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
pro-cess 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
policy tends to one with equal-sized"nished goods
batches, since, given Q
2 andk, the primary
objec-tive is to minimise AS 1.
In the opposite limit, whenc
1"c2, we need not
distinguish between process stock and "nished
goods stock and the primary objective is to mini-mise ATI. This is achieved when the total stock in the system at the start of the process stock
produc-tion cycle (xin 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 isP
1P2x/D(P1!P2) and successive"nished
goods batches increase in the ratio (P
1!D)
P
2/D(P1!P2). Using the fact that the sizes of the
"nished goods batches must sum to Q
2 we can
determine x, and hence ATI and AS
1, in terms of
Q
2 andkand hence"nd the optimal solution.
The optimal policy for relative values of c
1 and
c
2between the two extremes mentioned above can
be shown to be a sequence of"nished goods batch
sizes which increase in the ratio (P
1!D)P2/
D(P
1!P2) followed by a number of equal-sized
batches.
Returning to Kim's equal"nished goods
batch-size 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
consumexat rateDplus the time to manufacture
Q
2/k at rate P1, which is x/D#(Q2/kP1). The
quantity of process stock manufactured during this
time, at rateP
(illustrated in Fig. 8)
SinceP
2)P1 it is clearly not worth
manufac-turing more than one batch of process stock per
cycle and sok"1 andQ
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 whichnQ
2"mQ1, where
mandnhave no common factors (other than 1) and
bothm andn are greater than 1. In this case the
cycle time is nQ
2/D ("mQ1/D) which is neither
Q
2/DnorQ1/D. However, such solutions would be
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 batch-ing policies, International Journal of Production Econ-omics 58 (3) (1999) 221}234.
