Optimal two-stage lot sizing and inventory batching policies

*Corresponding author. Tel.: #1 414 288 3699; fax: #1 414288 5754.

Int. J. Production Economics 58 (1999) 221—234

Optimal two-stage lot sizing and inventory batching policies

DaeSoo Kim*

Marquette University, Department of Management, College of Business Administration, P.O. Box 1881, Straz Hall, Milwaukee,WI 53201-1881, USA

Received 18 November 1996; accepted 1 November 1997

Abstract

This paper extends previous studies of two-stage lot sizing problems with finite production rates. We develop variouslot sizing and inventory batching (i.e., operation—unit batching (OUB) and unit—unit batching (UUB)) models underdifferent system characteristics and lot sizing and inventory policies. The analysis of the optimality of the lot size ratiobetween the two stages reveals (1) that both non-increasing and non-decreasing lot sizing policies can be optimal in bothOUB and UUB, (2) that a non-integer lot size ratio can be optimal in OUB, and (3) that an integer lot size ratio is alwaysoptimal in UUB. We present a simple, easy-to-implement, optimal solution approach to the two-stage lot sizing andinventory batching problem, along with examples. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Two-stage lot sizing; Inventory batching policies; Operation—unit batching; Unit—unit batching

1. Introduction

In this study, we consider two-stage lot sizingproblems with a constant demand rate and finiteproduction rates. The two-stage lot sizing problemcan be classified into various versions dependingupon production system characteristics such as lotor batch transfer and production rates betweenstages and lot sizing policies. The lot or batchtransfer from the initial (process inventory) stage2 to the final (finished goods inventory) stage 1and from the final stage 1 to a final destination

(warehouses or customers) can be divided into op-eration batching, transfer batching, and unit batch-ing. In the operation batching (OB), the entire lot istransferred to the next stage upon the completionof production. In the transfer batching (TB), thepartial lot is transferable to the next stage beforethe entire lot is processed. In the unit batching(UB), each unit is continuously transferred upon itscompletion. With regard to production systemcharacteristics, the two-stage system can involvenon-increasing and non-decreasing productionrates between the stages. The lot sizing policy canbe divided into that of non-increasing and non-decreasing lot sizes between the initial (process in-ventory) and final (finished goods) stages. And thelot size ratio between the two stages can be eitherinteger or non-integer.

0925-5273/99/$ — see front matter ( 1999 Elsevier Science B.V. All rights reservedPII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 0 5 7 - 7

In the two- or multi-stage lot sizing problemliterature, especially two studies [1,2] are directlyrelated to our study among many other two- ormulti-stage constant-demand lot sizing studies forserial and assembly systems [3—12]. Szendrovits[1] presented two-stage lot sizing and inventorymodels based on the operation—operation batching(OOB; OB from stage 2 to 1 and OB from stage 1 tocustomers) with different lot sizing policies andproduction system characteristics. He showed thatthe non-increasing lot sizing policy and the integerlot size ratio between the two stages are not neces-sarily optimal. Karimi [2] developed two-stage lotsizing and inventory models based on the unit—unitbatching (UUB; UB from stage 2 to 1 and UB fromstage 1 to customers). He simultaneously con-sidered different lot sizing policies and system char-acteristics given the same cycle time for each cyclefor every stage. He found that the non-integersplit/merge lot sizing policy (i.e., non-increasing ornon-decreasing lot sizing policy with a non-integerlot size ratio) might be optimal in some instances.For other related studies, readers are referred toreview articles by Gelders and Van Wassenhove[13] and Bahl et al. [14] and the references therein.

In this paper, we extend previous studies of thetwo-stage lot sizing and inventory batching prob-lem. We develop lot sizing and inventory batchingmodels based on the operation—unit batching (OUB;OB from stage 2 to 1 and UB from stage 1 tocustomers) and the unrestricted unit—unit batching(UUB; UB from stage 2 to 1 and from stage 1 tocustomers without the restriction of same cycletime) under different lot sizing policies and systemcharacteristics. Our OUB and UUB models differfrom Szendrovits’ [1] OOB models in that the lotor batch transfer from stage 1 to customers orgenerally from stage i (producing, supplying stage)to i!1 (using stage) is based on unit batching as inthe classic economic production quantity (EPQ)model. In addition, our UUB models differ fromKarimi’s [2] in that the cycle time for each stage isnot restricted to be the same for each cycle. Inpractice, these types of batching can be frequentlyfound in, for example, assembly lines, transfer lines,and flexible assembly systems [15]. For different lotsizing policies, we analyze optimality of the lot sizeratio between the two stages. Given the optimal lot

size ratio, we present a simple, easy-to-implement,optimal solution approach to the two-stage lotsizing problem. The rest of the paper is organized asthe description of problems and assumptions,batching models, solution approach, and numericalexamples, followed by conclusions.

2. Problems and assumptions

The problem of interest is to minimize total in-ventory costs (i.e., setup and inventory holdingcosts) for a two-stage serial production and inven-tory system with a constant demand rate and finiteproduction rates over the infinite horizon. Themodels developed are based on the operation—unitbatching (OUB) and unit—unit batching (UUB)from stage 2 (initial, process inventory stage) tostage 1 (final, finished goods inventory stage) andfrom stage 1 to warehouses or customers. To facilit-ate discussion, we generally follow the notationused in [1].

i production stage index, i"2 (initial, pro-cess inventory), 1 (final, finished goods in-ventory),

D demand (consumption or usage) rate ofa final product (at stage 1),

Pi

production rate at i where Pi'D,

Fi

setup cost per lot at i,ci

unit inventory carrying cost per unit time ati (c

2)c

1),

CIi

inventory per cycle at i,Q

iproduction lot size or operation batch sizeper setup at i,

k QL/Q

S(k*1, real), lot size ratio between

the two stages, where QL, Q

S"larger,

smaller Q,kC, kB non-integer k rounded-up and rounded-

down to the nearest integer,K integer lot size ratio,TC

ptotal (setup and inventory holding) costs fora lot sizing and inventory batching policy p.

2.1. Assumptions

1. The two-stage production and inventory systeminvestigated has a serial structure which consists

222 D. Kim/Int. J. Production Economics 58 (1999) 221—234

of an initial (process inventory) stage (stage 2),a final (finished goods inventory) stage (stage 1),and a final destination (warehouses or cus-tomers), as found in assembly line, transfer line,or flexible assembly system environments.

2. The demand rate for an item to produce isknown and constant.

3. The shipping to meet the demand without short-ages or back orders occurs continuously, whichis represented by the unit batching as in theEPQ model.

4. The batch transfer is based on either the opera-tion batching (OB; all-unit entire lot transfer) orthe unit batching (UB; one-unit continuoustransfer) from stage 2 to 1 and on the unitbatching (UB) from stage 1 to customers.

5. The production rates of the system are finite, i.e.,non-instantaneous production or inventory re-plenishment. The production rate at each stageis greater than or equal to the demand rate.

6. The process and finished goods inventory hold-ing cost is calculated by the area under theinventory curves. The process inventory holdingcost is assumed to be the same before and afterthe transfer of a lot or batch to the next stageuntil the process inventory is used for finishedgoods production, as is typical in the literature.Further due to the constant demand rate, theinventory cycle will repeat over time with thecycle time of a larger lot size over demand rate(i.e., Q

L/D).

7. Capacity is assumed to be unlimited at eachstage with negligible setup and transportation ormaterial handling time.

8. The product is considered to be infinitely divis-ible, thereby not requiring integer lot sizes.

9. The lot size ratio between the two stages is notrestricted to integer values.

2.2. Policy notation for the models investigated

Policy 1 (5): OUB (UUB) [Q2*Q

1] for P

2*P

1Policy 2 (6): OUB (UUB) [Q

2)Q

1] for P

2*P


2*Q

1] for P

2)P


2)Q

1] for P

2)P

1

3. Operation–unit batching (OUB) models

In this section, we formulate lot sizing modelsbased on the operation batching (OB, i.e., the trans-fer of an entire lot or batch upon its completion)from stage 2 to 1 and the unit batching (UB, i.e.,the continuous unit transfer of a lot or batchduring its production) from stage 1 to customers.For each policy, we analyze optimality of a lot sizeratio between the two stages. Given the optimal lotsize ratio, we derive the optimal lot sizes and totalcost.

3.1. Policy 1: OºB [Q2*Q

1("Q

2/k)] for

P2*P

1

Fig. 1 shows the inventory curves (with respectto time) given in this policy. In this OUB policy theentire lot Q

2of process inventory produced at

stage 2 is transferred to the next finished goodsinventory stage (stage 1) upon its completion anddepleted at a rate of P

1during only each pro-

duction run period at stage 1 (i.e., Q1/P

1for the first

kB finished goods batches and (k!kB)Q1/P

1for the

last kCth batch). Therefore, the cycle inventoryat stage 2 (CI

2) is depicted by the area (surrounded

by the thicker lines) of the polygon with its sideslopes of P

2and kC (!P

1)’s and the CI at stage

1 (CI1) is the area of the kC triangles each with its

side slopes of P1!D and !D, which are cal-

culated as

CI2"(1

2)[k2Q2

1/P

2#kBQ2

1/P

1

#(k!kB)2Q21/P

1#kB(kB!1)Q2

1/D

#2kB(k!kB)Q21/D]

"(12)[(1/P

2#1/P

1)#(1/D!1/P

1)2kB/k

!(1/D!1/P1)(kB2#kB)/k2]Q2

2(1)

and

CI1"(1

2)[kB(Q2

1/D!Q2

1/P

1)

#(k!kB)2(Q21/D!Q2

1/P

1)]

"(12)[(1/D!1/P

1)!(1/D!1/P

1)2kB/k

#(1/D!1/P1)(kB2#kB)/k2]Q2

2. (2)

D. Kim/Int. J. Production Economics 58 (1999) 221—234 223

Fig. 1. Inventory curves for policy 1.


Then, the total cost (i.e., sum of setup and inventoryholding costs) as a function of k and Q

2is

TC(k, Q2)"[(F

2#c

2CI

2)#(F

1kC#c

1CI

1)](D/Q

2)

"D(F2#F

1kC)/Q

2#(1

2)D(c

2w2#c

1w

1)Q

2,

(3)

where

(w2, w

1)"([(1/P

2#1/P

1)#(1/D!1/P

1)2kB/k

!(1/D!1/P1)(kB2#kB)/k2],

[(1/D!1/P1)!(1/D!1/P

1)2kB/k

#(1/D!1/P1)(kB2#kB)/k2])

from Eqs. (1) and (2).From LTC(k, Q

2)/Q

2"0, we obtain Q

2and TC

as a function of k:

Q2(k)"[2(F

2#F

1kC)/(c

2w

2#c

1w1)]0.5

and

TC(k)"D[2(F2#F

1kC)]0.5(c

2w2#c

1w

1)]0.5. (4)

From the analysis of optimality of a lot size ratiobetween the two stages, we find that the optimalratio is integer (see Appendix A). Thus, we obtainthe following TC* and Q*

2given an optimal integer

lot size ratio K ("k"kC"kB) by substituting k,kC, and kB with K into Eq. (3):

TC*(K)"D[2(F2#F

1K)]0.5[c

2(1/D#1/P

2)

#(c1!c

2)(1/D!1/P

1)/K]0.5

"D20.5[F2(c

1!c

2)(1/D!1/P

1)/K

#F1c2(1/D#1/P

2)K

#F1(c

1!c

2)(1/D!1/P

1)

#F2c2(1/D#1/P

2)]0.5

and

Q*2(K)"[2(F

2#F

1K)/Mc

2(1/D#1/P

2)

#(c1!c

2)(1/D!1/P

1)/KN]0.5. (5)

3.2. Policy 2: OºB [Q2

("Q1/k))Q

1] for

P2*P

1

As seen from Fig. 2, in this policy the productionfinish time of each of the kC batches of processinventory is synchronized with the finish time of thedepletion of its immediately preceding batch ata rate of P

1, in order to minimize the process

inventory while ensuring the continuous produc-tion of finished goods Q

1. Thus, we obtain the cycle

inventory at stage 2 (CI2) as the area of the kC

triangles each with its side slopes of P2

and !P1

and the CI at stage 1 (CI1) as the area of the big

triangle with its side slopes of P1!D and !D.

(See Table 1 for the summary of CI2

and CI1,

TC(k, Q2) or TC(k, Q

2), Q

2(k) or Q

1(k), and TC(k),

since the derivation of total costs and lot sizes areessentially the same as that in Policy 1, along with


Table 1Summary of the cycle inventory, total cost, and lot size functions

Cycle inventory functions

Policy 1: see those in the text

Policy 2

CI2"(1

2)[kB(Q2

2/P

2#Q2

2/P

1)#(k!kB)2(Q2

2/P

2#Q2

2/P

1)]

"(12)[(1/P

2#1/P

1)(1!2kB/k#(kB2#kB)/k2)]Q2

1CI

1"(1

2)[Q2

1/D!Q2

1/P

1]"(1

2)[1/D!1/P

1]Q2

1

Policy 3: same as those in policy 1

Policy 4a

CI2"(1

2)[kBQ

2M(kBQ

2/P

2!(kB!1)Q

2/P

1)#Q

2/P

1N#(k!kB)2Q2

2(1/P

2#1/P

1)]

"(12)[(1/P

2#1/P

1)!(1/P

2#1/P

1)2kB/k#M(1/P

2)2kB2#(1/P

1)2kBN/k2]Q2

1CI

1"CI

1in policy 2

Policy 4b

CI2"(1

2)Q

1[(Q

1/P

2!kBQ

2/P

1)#(k!kB)Q

2/P

1]"(1

2)[(1/P

2#1/P

1)!(1/P

1)2kB/k]Q2

1CI

1"(1

2)[Q2

1/D!Q2

1/P

1]"(1

2)[1/D!1/P

1]Q2

1

Policy 5

CI2"CI

2in Eq. (1)!Q2

2/P

2"(1

2)[(1/P

1!1/P

2)#(1/D!1/P

1)2kB/k!(1/D!1/P

1)(kB2#kB)/k2]Q2

2CI

1"CI

1in policy 1

Policy 6

CI2"(1

2)[kB(Q2

2/P

1!Q2

2/P

2)#(k!kB)2(Q2

2/P

1!Q2

2/P

2)]

"(12)[(1/P

1!1/P

2)(1!2kB/k#(kB2#kB)/k2)]Q2

1CI

1"CI

1in policy 2

Policy 7

CI2"CI

2in Eq.(1)!Q

2(Q

2/P

2#Q

1/P

1!Q

1/P

2)

"(12)[!(1/P

2!1/P

1)#M(1/D!1/P

1)2kB#2(1/P

2!1/P

1)N/k!(1/D!1/P

1)(kB2#kB)/k2]Q2

2CI

1" CI

1in policy 1

Policy 8

CI2"(1

2)[1/P

2!1/P

1]Q2

1CI

1"(1

2)[1/D!1/P

1]Q2

1

¹otal cost and lot size functions

For Policy 1, 3, 5, and 7

TC(k, Q2)"[(F

2#c

2CI

2)#(F

1kC#c

1CI

1)](D/Q

2)"D(F

2#F

1kC)/Q

2#(1

2)D(c

2w

2#c

1w

1)Q

2(Q

2(k), ¹C(k))

"([2(F2#F

1kC)/(c

2w

2#c

1w1)]0.5, D[2(F

2#F

1kC)]0.5(c

2w

2#c

1w

1)]0.5)

For Policy 2, 4a & 4b, and 6

TC(k, Q1)"[(F

2kC#c

2CI

2)#(F

1#c

1CI

1)](D/Q

1)

"D(F2kC#F

1)/Q

1#(1

2)D(c

2w

2#c

1w1)Q

1(Q

1(k), TC(k))

"([2(F2kC#F

1)/(c

2w

2#c

1w1)]0.5, D[2(F

2kC#F

1)]0.5(c

2w2#c

1w

1)]0.5)

For Policy 8

TC(k, Q1)"[(F

2kC#c

2CI

2)#(F

1#c

1CI

1)](D/Q

1)

Note that (w2, w

1)"[ ] in the last equation for the above CI

2and CI

1for each policy, respectively.



Fig. 4. Inventory curves for policy 4(a) and (b).

Appendix A for the analysis of the optimal lot sizeratio.) Then, given the optimal integer lot size ratio,we obtain

TC*(K)"D[2(F2K#F

1)]0.5[c

1(1/D!1/P

1)

#c2(1/P

2#1/P

1)/K]0.5

"D20.5[F1c2(1/P

2#1/P

1)/K

#F2c1(1/D!1/P

1)K

#F2c2(1/P

2#1/P

1)

#F1c1(1/D!1/P

1)]0.5

and

Q*1(K)"[2(F

2K#F

1)/Mc

1(1/D!1/P

1)

#c2(1/P

2#1/P

1)/KN]0.5. (6)

3.3. Policy 3: OºB [Q2*Q

1("Q

2/k)] for

P2)P

1

In this policy, the behaviors of process andfinished goods inventory and the calculationof CI

2, CI

1, and TC are exactly the same as those

in policy 1, i.e., the Eq. (1), (2), and Eqs. (3)— (5),respectively (see Fig. 3 in comparison withFig. 1).

3.4. Policy 4: OºB [Q2

("Q1/k))Q

1] for

P2)P

1

In this policy, two cycle inventory (CI) curves arepossible for process inventory, based on(k!kB)Q

2/P

2) or *Q

2/P

1related to the last

kCth smaller batch size. Fig. 4 shows these curves,one with P

2Fig. 4a and the other with P

2Fig. 4b.

Thus, we analyze each case individually.

3.4.1. Policy 4a: ¼hen (k!kB)Q2/P

2)Q

2/P

1,

i.e., (k!kB)P1)P

2)P

1or kB)k)kB

#P2/P

1)kC

As can be seen from the curve with P2

(Fig. 4a),since (k!kB)Q

2/P

2(Fig. 4a))Q

2/P

1in this case,

in order to minimize the process inventory whileensuring the continuous production of finishedgoods Q

1, the production start time of the last kCth

smaller batch of process inventory needs to comeafter the finish time of its immediately precedingkBth batch of process inventory and its productionfinish time needs to be synchronized with the finishtime of the depletion of its immediately precedingbatch. Thus, CI

2is the combined area of the poly-

gon with its side slopes of P2

(Fig. 4a) and kB(!P

1)’s (which is essentially the same as the area of

the quadrilateral with its side slopes of P2

(Fig. 4a)and P

1and the height of kBQ

2) and the uppermost

kCth small triangle with its side slopes ofP2(Fig. 4a) and !P

1and the height of (k!kB)Q

2.

CI1

is the area of the right big triangle with its side



slopes of P1!D and !D (See Table 1 for the

summary of cycle inventory, total cost, and lot sizefunctions). In this case, Appendix A shows that theoptimal lot size ratio is non-integer, i.e.,k*"kB#P

2/P

1. Thus, TC* and Q*

1are, by substi-

tuting kB"k!P2/P

1and kC"k#1!P

2/P

1into TC(k) for policy 4a in Table 1,

TC*(k)"D[2(F2(k#1!P

2/P

1)#F

1)]0.5

][c2(1/P

2!1/P

1)#c

1(1/D!1/P

1)

#2c2(P

2/P2

1)/k]0.5

"D20.5[2c2(P

2/P2

1)MF

2(1!P

2/P

1)

#F1N/k#F

2Mc

2(1/P

2!1/P

1)

#c1(1/D!1/P

1)Nk#2F

2c2(P

2/P2

1)

#MF2(1!P

2/P

1)#F

1N

]Mc2(1/P

2!1/P

1)#c

1(1/D!1/P

1)N]0.5

and

Q*1(k)"[2MF

2(k#1!P

2/P

1)#F

1N/Mc

2(1/P

2!1/P

1)

#c1(1/D!1/P

1)#2c

2(P

2/P2

1)/kN]0.5. (7)

3.4.2. Policy 4b: ¼hen (k!kB)Q2/P

2*Q

2/P

1,

i.e., P2)(k!kB)P

1)P

1or kB#P

2/P

1)k)kC

As shown by the curve with P2

(Fig. 4b), policy4b is slightly different from policy 4a. In this case,since (k!kB)Q

2/P

2(Fig. 4b)*Q

2/P

1, the produc-

tion start time of the last kCth smaller batch of theprocess inventory is synchronized with the finishtime of its immediately preceding kBth batch of theprocess inventory, in order to minimize the processinventory while ensuring the continuous produc-tion of finished goods. Thus, the CI at stage 2 (CI

2)

is the area of the polygon with its side slopes ofP2

(Fig. 4b) and kC (!P1)’s, which is essentially

the same as the area of the quadrilateral with itsside slopes of P

2(Fig. 4b) and P

1and the height of

kQ2

(or Q1). And the CI at stage 1 (CI

1) is the same

as that in policy 4a (i.e., the area of the right bigtriangle with its side slopes of P

1!D and !D)

(See Table 1 for the summary of cycle inventory,total cost, and lot size functions). In this case, Ap-pendix A shows again that k*"kB#P

2/P

1as in

policy 4a. Thus, by substituting kB"k!P2/P

1

and kC"k#1!P2/P

1into TC(k) for policy 4b in

Table 1, we obtain the same TC*(k) and Q*1(k) in

Eq. (7).

4. Unit–unit batching (UUB) models

In this section, we formulate lot sizing modelsbased on the unit batching (UB, i.e., the continuousunit transfer of a lot or batch during its production)from stage 2 to 1 and from stage 1 to customers,different from the combined operation and unitbatching (OUB) policies in the previous section.Similarly as before, for each UUB policy, we ana-lyze the optimal lot size ratio between the twostages and derive the optimal lot sizes and totalcost.

4.1. Policy 5: ººB [Q2*Q

1("Q

2/k)] for

P2*P

1

As seen from Fig. 5, due to the unit batching, theproduction of the first batch Q

1of finished goods at

stage 1 occurs almost simultaneously with the pro-duction of one large batch Q

2of process inventory

at stage 2. During the run time period Q1/P

1of the

first batch Q1

of finished goods, the process inven-tory is built at a rate of P

2!P

1, since P

2*P

1and

it is simultaneously depleted at a rate of P1

for the



first batch production of finished goods, as shownby the leftmost triangle with its side slopes ofP2!P

1and !P

1. The rest of the process inven-

tory is built at a rate of P2

and depleted at a rate ofP1during only the run time period of the rest of the

finished goods batches. Therefore, the cycle inven-tory at stage 2 (CI

2) is the combined area of the

leftmost triangle with its side slopes of P2!P

1and

!P1and the polygon with its side slopes of P

2and

upper kB (!P1)’s, which is the area of the polygon

with its side slopes of P2(OB) and kC (!P

1)’s in

policy 1 minus the area of the parallelogram withits side slopes of P

2(OB) and P

2, the base of Q

2/P

2and the height of Q

2. And the CI at stage 1 (CI

1) is

exactly the same as that in policy 1 (i.e., the area ofthe kC triangles each with its side slopes of P

1!D

and !D) (See Table 1 for the summary of cycleinventory, total cost, and lot size functions). Then,given the optimal integer lot size ratio fromAppendix A, we obtain

TC*(K)"D[2(F2#KF

1)]0.5[c

2(1/D!1/P

2)

#(c1!c

2)(1/D!1/P

1)/K]0.5

"D20.5[F2(c

1!c

2)(1/D!1/P

1)/K

#F1c2(1/D!1/P

2)K

#F1(c

1!c

2)(1/D!1/P

1)

#F2c2(1/D!1/P

2)]0.5

and

Q*2(K)"[2(F

2#KF

1)/Mc

2(1/D!1/P

2)

#(c1!c

2)(1/D!1/P

1)/KN]0.5. (8)

4.2. Policy 6: ººB [Q2

("Q1/k))Q

1] for P

2*P

1

As seen from Fig. 6, due to the unit batching inthis policy, the production start time of each smallbatch Q

2of process inventory is synchronized with

the finish time of the depletion of its immediatelypreceding batch, in order to minimize the processinventory while ensuring the continuous produc-tion of finished goods Q

1. Also note that during

each run time period of process inventory batches(i.e., Q

2/P

2for the first kB batches each with the size

of Q2and (k!kB)Q

2/P

2for the last kCth batch with

(k!kB)Q2), the process inventory is built at a rate

of P2!P

1, since P

2*P

1and it is simultaneously

depleted at a rate of P1

for the production offinished goods. Therefore, CI

2is the area of the kC

small triangles each with its side slopes of P2!P

1and !P

1, which is slightly different from that in

policy 2 depicted by the kC bigger triangles (sur-rounded by thinner lines) each with its side slopesof P

2(OB) and !P

1(see also Fig. 2 for compari-

son). And CI1

is the area of the right big trianglewith its side slopes of P

1!D and !D as in policy

2 (See Table 1 for the summary of cycle inventory,total cost, and lot size functions). Then, given theoptimal integer lot size ratio from Appendix A, weobtain

TC*(K)"D[2(KF2#F

1)]0.5[c

1(1/D!1/P

1)

#c2(1/P

1!1/P

2)/K]0.5

"D20.5[F1c2(1/P

1!1/P

2)/K

#F2c1(1/D!1/P

1)K

#F2c2(1/P

1!1/P

2)

#F1c1(1/D!1/P

1)]0.5

and

Q*1(K)"[2(KF

2#F

1)/Mc

1(1/D!1/P

1)

#c2(1/P

1!1/P

2)/KN]0.5. (9)



4.3. Policy 7: ººB [Q2*Q

1("Q

2/k)] for

P2)P

1

As shown in Fig. 7, in order to minimize theprocess inventory while ensuring the production ofthe first batch Q

1of finished goods, the production

of process inventory needs to start in advance sothat the finish time of Q

1can be synchronized at

both stages, as shown by the leftmost triangle withits side slopes of P

2and P

2!P

1. Thus, the process

inventory produced for the production of the firstbatch of finished goods is built at a rate of P

2until

the start of the production of that finished goodsbatch and depleted at a rate of P

2!P

1since

P2)P

1. And the process inventory produced for

the second to the kBth batch of finished goods isbuilt at a rate of P

2during the pure depletion

periods of finished goods at stage 1 and depleted ata rate of P

2!P

1during the run time periods of

finished goods, as shown by the kB!1 middlequadrilaterals each with its side slopes of P

2,

P2!P

1and !P

1. Finally, the process inventory

to be used for the production of the last kCthsmaller batch of finished goods is built before thestart of the production of that finished goods batchand thus depleted at the rate of P

1, as indicated by

the uppermost quadrilateral with its side slopes ofP2

and !P1. Therefore, CI

2is the combined area

of one leftmost triangle with its side slopes ofP2

and P2!P

1, the kB!1 middle quadrilaterals

each with its side slopes of P2, P

2!P

1and !P

1,

and one uppermost quadrilateral with its side

slopes of P2and !P

1. And CI

1is the area of the kC

small triangles each with its side slopes of P1!D

and !D as in policy 3 (See Table 1 for the sum-mary of cycle inventory, total cost, and lot sizefunctions). Then, given the optimal integer lot sizeratio from Appendix A, we obtain

TC*(K)"D[2(F2#KF

1)]0.5[c

2(1/D!1/P

2)

#M(c1!c

2)(1/D!1/P

1)

#2c2(1/P

2!1/P

1)N/K]0.5

"D20.5[F2M(c

1!c

2)(1/D!1/P

1)

#2c2(1/P

2!1/P

1)N/K

#F1c2(1/D!1/P

1)K

#F1M(c

1!c

2)(1/D!1/P

1)

#2c2(1/P

2!1/P

1)N

#F2c2(1/D!1/P

1)]0.5

and

Q*2(K)"[2(F

2#KF

1)/Mc

2(1/D!1/P

2)

#M(c1!c

2)(1/D!1/P

1)

#2c2(1/P

2!1/P

1)N/KN]0.5. (10)

4.4. Policy 8: ººB [Q2

("Q1/k))Q

1] for

P2)P

1

Different from operation—unit batching (OUB)policy 4 in Fig. 4, this UUB policy shown in Fig. 8yields the same cycle inventory (CI

2and CI

1)

for both (k!kB)Q2/P

2(Fig. 4a))Q

2/P

1and

(k!kB)Q2/P

2(Fig. 4b)*Q

2/P

1cases (also see the

difference between OUB policy 4 with P2

(Fig. 4aor b) and UUB policy 8 with P

2(Fig. 8a or b)). That

is, when (k!kB)Q2/P

2(Fig. 8a))Q

2/P

1, i.e., kB)

k)kB#P2/P

1)kC, the finish time of the last

kCth batch of process inventory needs to be syn-chronized with the finish time of one large batchQ

1of finished goods. This requires the start time of

the first batch of process inventory to come beforethe start time of the finished goods production,since P

2)P

1. Thus, CI

2is the combined area of

the kB quadrilaterals each with its side slopes ofP2

(Fig. 8a), P2

(Fig. 8a) !P1

and !P1

and oneuppermost small triangle with its side slopes of


Fig. 8. Inventory curves for policy 8(a) and (b).

P2

(Fig. 8a) and P2

(Fig. 8a) !P1. Similarly,

when (k!kB)Q2/P

2(Fig. 8b)*Q

2/P

1, i.e.,

kB#P2/P

1)k)kC, the finish time of the last

kCth batch of process inventory also needs to besynchronized with the finish time of one large fin-ished goods batch and the start time of the firstbatch of process inventory needs to come well be-fore the start time of the finished goods production.Further, since (k!kB)Q

2/P

2(Fig. 8b)*Q

2/P

1, the

start time of the last kCth batch of process inven-tory does not have to be the same as the finish timeof its immediately preceding batch in order to min-imize the process inventory while ensuring the con-tinuous production of finished goods. Thus, CI

2is

the combined area of the polygon with its sideslopes of P

2(Fig. 8b), P

2(Fig. 8b) !P

1and kB

(!P1)’s and one uppermost triangle with its side

slopes of P2(Fig. 8b) and P

2(Fig. 8b) !P

1. There-

fore, in both cases of UUB policy 8, CI2is essential-

ly the area of the left big triangle with its side slopesof P

2(Fig. 8a or b) and P

1(i.e., the rightmost

thinner slant line), the base of Q2/P

2, and the height

of kQ2. And CI

1is the same as that in OUB policy

4(a,b) (i.e., the area of the right big triangle with itsside slopes of P

1!D and !D) (See Table 1 for the

summary of cycle inventory, total cost, and lot sizefunctions).

Thus, it is obvious that TC (k, Q1) is minimized

when k*"K"kC"1. Intuitively, since CI2

issame for any k, there is no need for k('1) setups.Therefore, when P

2)P

1, policy 8 with (Q

2(Q

1)

is always inferior to policy 8 with (Q2"Q

1) which

is a subset of policy 7 with (Q2*Q

1).

5. Optimal solution approach

In this section, we present a simple, easy-to-implement, optimal solution approach to the two-stage operation—unit batching (OUB) andunit—unit batching (UUB) models developed. Theoptimal solution procedure is based on simplecalculus and total cost comparisons. LetTC

p(k

p)"TC(k) for policy p where k

pdenotes a re-

laxed non-integer lot size ratio from the optimalinteger K for pO4 and the optimal non-integerk for p"4. Then, TC* for each policy can begenerally stated as

TCp(k

p)"D20.5[A

p/k

p#B

pkp#C

p]0.5 for pO8,

(11)

where Ap, B

p, and C

p'0 and constants in

Eqs. (5)—(10) for p"1,2, 7, respectively.Note that TC

i(k

i)"TC

j(k

j) for i, j"pO8 if

kp"1, that TC

1(k

1)"TC

3(k

3), that TC

2(k

2)"

TC4(k

4) if P

2"P

1, and that p"8 is not included

here, since TC7(k

7))TC

8(k

8). (See Table 2 for the

summary of Ap, B

p, and C

p.)

To find the curvature of TCp(k

p) in Eq. (11), let

Gp(k

p)"A

p/k

p#B

pkp#C

p. (12)

Now, we analyze the curvature of Gp(k

p) in

Eq. (12), since the basic curvature between TCpand

Gp

is essentially the same. From the first- andsecond-order conditions, we obtain

G@p"!A

p/k2

p#B

p* ()) 0

if kp*()) k0

p"(A

p/B

p)0.5

and

GAp"2A

p/k3

p'0.

Thus, Gp

is convex with its minimum at

k0p"(A

p/B

p)0.5 for pO8. (13)

Now, given the analysis above with the total costcomparison procedure described in Appendix B,we derive the following general optimal solutionprocedure:


Table 2Summary of total cost components (A

p, B

p, C

p, a

p, b

p)

Note that ApB

p"a

pbp

and Cp"a

p#b

pin Eq. (11) or (5)—(10) for policy p"1,2, 7, respectively.

A1"F

2(c

1!c

2)(1/D!1/P

1), B

1"F

1c2(1/D#1/P

2)

C1"F

1(c

1!c

2)(1/D!1/P

1)#F

2c2(1/D#1/P

2)"a

1#b

1

A2"F

1c2(1/P

2#1/P

1), B

2"F

2c1(1/D!1/P

1)

C2"F

2c2(1/P

2#1/P

1)#F

1c1(1/D!1/P

1)"a

2#b

2

A3"A

1, B

3"B

1, C

3"C

1, a

3"a

1, b

3"b

1

A4"2c

2(P

2/P2

1)MF

2(1!P

2/P

1)#F

1N, B

4"F

2Mc

2(1/P

2!1/P

1)#c

1(1/D!1/P

1)N

C4"2F

2c2(P

2/P2

1)#MF

2(1!P

2/P

1)#F

1NMc

2(1/P

2!1/P

1)#c

1(1/D!1/P

1)N"a

4#b

4

A5"F

2(c

1!c

2)(1/D!1/P

1), B

5"F

1c2(1/D!1/P

2)

C5"F

1(c

1!c

2)(1/D!1/P

1)#F

2c2(1/D!1/P

2)"a

5#b

5

A6"F

1c2(1/P

1!1/P

2), B

6"F

2c1(1/D!1/P

1)

C6"F

2c2(1/P

1!1/P

2)#F

1c1(1/D!1/P

1)"a

6#b

6

A7"F

2M(c

1!c

2)(1/D!1/P

1)#2c

2(1/P

2!1/P

1)N, B

7"F

1c2(1/D!1/P

1)

C7"F

1M(c

1!c

2)(1/D!1/P

1)#2c

2(1/P

2!1/P

1)N#F

2c2(1/D!1/P

1)"a

7#b

7

Table 3Illustrative examples of inventory batching model comparisons

Convention: Data (D, F1, F

2, P

1, P

2, c

1, c

2), Solution (optimal policy p; k

p, k

p`1; k*, Q*

2, Q*

1, TC*)

Szendrovits’ OOB versus our OUB and UUB in Szendrovits’ [1] Examples 1—3:

Example 1: Data (100, 24, 300, 200, 500, 3, 2)

Solution!OOB (1; 4.30, n/a; 4, 155.51 units, 38.88 units, $509.29)Solution!OUB (1; 1.614, 0.273; 2, 162.062 units, 81.031 units, $429.465)Solution!UUB (5; 1.976, 0.179; 2, 193.963 units, 96.982 units, $358.831)

Example 2: Data (100, 300, 12, 200, 500, 3, 2)

Solution!OOB (2; n/a, 2.83; 3, 38.77 units, 116.32 units, $577.72)Solution!OUB (2; 0.091, 4.830; 5, 40.224 units, 201.121 units, $357.994)Solution!UUB (6; 0.112, 3.162; 3, 66.273 units, 198.820 units, $337.994)

Example 3: Data (100, 600, 4, 500, 200, 3, 2)

Solution!OOB (4; n/a, 3.387; 3.4, 49.82 units, 169.38 units, $727.35)Solution!OUB (4; 0.042, 4.008; 4.008, 50.000 units, 200.400 units, $617.199)Solution!UUB (7, 8; 0.091, 1.000; 1, 200.666 units, 200.666 units, $601.997)

Karimi’s ººB (KUUB) versus our unrestricted UUB in Karimi’s [2] example II:Example II: Data (30, 300, 100, 50, 600, 8, 5)

Solution!KUUB (n/a; n/a, n/a; n/a, 64 units, 64 units, $377.89)Solution!UUB (6; 0.290, 1.606; 2, 40.489 units, 80.978 units, $370.473)

Note that the solution figures are subject to rounding errors.

Step 0. For OUB (UUB) policies, let p"M1 (5) ifP2*P

1, 3 (7) otherwiseN.

Step 1. Calculate k0j"(A

j/B

j)0.5 for j"p, p#1.

Step 2. If p"7, k*"maxM1, int(k0p)N. Else

k*"maxM1, Mint(k0p) if p0

p)p0

p`1, Mk0

p`1for

p#1"4, int(k0p`1

) for p#1O4N otherwiseNN,where int(k0

p)"k0

prounded to the nearest integer,

p0p"a0.5

p#b0.5

p, and a

p#b

p"C

pin Eqs. (11) and


(13) or in Eqs. (5)— (10) for p"1,2, 7, respectively.(See Table 2 for the summary of a

pand b

p.)

Step 3. Calculate Q*1(k*), Q*

2(k*), and TC*(k*)

from a proper equation among Eqs. (5)— (10) forp"1,2, 7, respectively.

6. Numerical examples

Using the examples in Szendrovits [1] andKarimi [2], we compare the total inventory cost(TC) (1) between the Szendrovits’ operation—opera-tion batching (OOB) and our operation—unitbatching (OUB) and unit—unit batching (UUB)models developed and (2) between the Karimi’scyclic unit—unit batching (KUUB) and our unre-stricted unit—unit batching (UUB) model. The re-sults summarized in Table 3 definitely show thesignificant TC difference from our optimal policies,indicating a possible serious problem of blind ap-plications of lot sizing and inventory batchingmodels.

7. Conclusions

In this study, we examined the previously unex-plored two-stage operation—unit batching andunit—unit batching models with different lot sizingand inventory batching policies, which are fre-quently found in many real production and inven-tory systems. Through the analysis of lot size ratio,we have shown analytically that the optimal lotsizing policy between the two stages can be eithernon-increasing or non-decreasing and that theoptimal lot size ratio can be either integer ornon-integer. We developed a simple, optimalsolution approach to the models studied, whichcan also be used effectively as system myopicheuristic approach to multi-stage serial produc-tion and inventory systems as in Schwarz andSchrage [7].

Finally, considering the practical significance, itis much desired that future research is directedtoward the analysis of other two-stage lot sizingand inventory policies such as transfer batchingand extensions to more general multi-stage systems

and capacitated production and inventory systemswith setup time and transportation or materialhandling time and/or cost explicitly considered.

Acknowledgements

The author wishes to thank anonymous refereesfor valuable comments and suggestions.

Appendix A

A.1. Optimality of the lot size ratio

For policy pO4, TC(k) in Eq. (4) and Table 1can be generally written as

TC(k)"D20.5F0.5[a!b/k#c/k2]0.5, (A.1)

where F"MF2#F

1kC if p"1, 3, 5, 7, F

2kC#F

1otherwiseN and Ma, b, cN for p"

1,3: Mc2(1/P

2#1/P

1)#c

1(1/D!1/P

1),

(c1!c

2)(1/D!1/P

1)2kB,

(c1!c

2)(1/D!1/P

1)(kB2#kB)N,

2: Mc2(1/P

2#1/P

1)#c

1(1/D!1/P

1),

c2(1/P

2#1/P

1)2kB,

c2(1/P

2#1/P

1)(kB2#kB)N,

5: Mc2(1/P

1!1/P

2)#c

1(1/D!1/P

1),

(c1!c

2)(1/D!1/P

1)2kB,

(c1!c

2)(1/D!1/P

1)(kB2#kB)N,

6: Mc2(1/P

1!1/P

2)#c

1(1/D!1/P

1),

c2(1/P

1!1/P

2)2kB,

c2(1/P

1!1/P

2)(kB2#kB)N,

7: Mc1(1/D!1/P

1)!c

2(1/P

2!1/P

1),

(c1!c

2)(1/D!1/P

1)2kB!2c

2(1/P

2!1/P

1),

(c1!c

2)(1/D!1/P

1)(kB2#kB)N.

Let H(k)"a!b/k#c/k2. Observe a,b, c'0for pO8. Then, given the interval of k"[kB, kC],

H@(k)"LH(k)/Lk

"b/k2!2c/k3

"(bk!2c)/k3(0


since bk!2c(0, and

HA(k)"LH@(k)/Lk

"!2b/k3#6c/k4

"!2(bk!3c)/k4'0

since bk!3c(bk!2c(0.Thus, H(k) is monotone decreasing convex with

its minimum at k"kC"integer and hencek*"K"kC"kB.

For p"4, when kB)k)kB#P2/P

1)kC,

TC(k) has the same structure as that in (A.1) where

Ma, b, cN"Mc2(1/P

2#1/P

1)#c

1(1/D!1/P

1),

c2(1/P

2#1/P

1)2kB, c

2M(1/P

2)2kB2#(1/P

1)2kBN

and a, b, c'0. Thus, H(k) is monotone decreasingconvex with its minimum at k*"kB#P

2/P

1"

non-integer. When kB#P2/P

1)k)kC, TC(k)"

D[2(F2kC#F

1)]0.5[a!b/k]0.5 in Table 1. Let

H(k)"a!b/k, where a"c2(1/P

2#1/P

1)#

c1(1/D!1/P

1)'0 and b"c

2(1/P

1)2kB'0. Then,

H@(k)"b/k2'0 and HA(k)"!2b/k3(0. Thus,H(k) is monotone increasing concave with its min-imum at k*"kB#P

2/P

1"non-integer. There-

fore, for policy 4, k*"kB#P2/P

1"non-integer

and kC"kB#1.

Appendix B

B.1. Total cost comparisons

Let TCp(k

p)"TC(k) for policy p where k

pde-

notes a relaxed non-integer lot size ratio from theoptimal integer K for pO4 and the optimal non-integer k for p"4. For OUB policies, if P

2*P

1,

TC comparison is made between G1

and G2

inEq. (12), i.e., TC

1in Eq. (5) and TC

2in Eq. (6), and

if P2)P

1, between G

3and G

4in Eq. (12), i.e., TC

3in Eq. (5) and TC

4in Eq. (7). For UUB policies, if

P2*P

1, comparison is made between G

5and

G6

in Eq. (12), i.e., TC5

in Eq. (8) and TC6

inEq. (9), and if P

2)P

1, no comparison is made due

to TC7)TC

8. Recall k0

p"(A

p/B

p)0.5 for pO8 in

Eq. (13). Define int(k) as k rounded to the nearest k.

B.0. Let p"M1 Eq. (5) if P2*P

1, 3 Eq. (7) other-

wise, for OUB (UUB) policiesN. If p"7, k* is

from B.1.—B.2. Else k* is determined by one ofthe four possible cases, B.1.—B.4.

B.1. If k0p

and k0p`1

)1, then k*p"k*

p`1"1 and

TC*p"TC*

p`1. Thus k*"1.

B.2. If k0p'1 and k0

p`1)1, then k*

p"k0

pand

k*p`1

"1. Since Gp!G

p`1)0, i.e., TC*

p)

TC*p`1

, k*"int(k*p)"int(k0

p).

B.3. If k0p)1 and k0

p`1'1, then k*

p"1 and

k*p`1

"k0p`1

. Since Gp!G

p`1*0, i.e.,

TC*p*TC*

p`1, k*"Mint(k0

p`1) for p#1O4,

k0p`1

for p#1"4N.B.4. If k0

pand k0

p`1'1, then k*

p"k0

pand

k*p`1

"k0p`1

.If p0

p)p0

p`1, then k*"k*

p"k0

p, since

Gp!G

p`1)0, i.e., TC*

p)TC*

p`1,

where p0p"a0.5

p#b0.5

p, a

pbp"A

pBp, and

ap#b

p"C

pin Eq. (11) or in Eqs. (5)—(10)

for p"1,2, 7, respectively. (See Table 2 forthe summary of a

pand b

p.)

Else k*"Mint(k*p`1

)"int(k0p`1

) for p#1O4,k*p`1

"k0p`1

for p#1"4N.

Proof of B.1.—B.4. The proof of B.1.—B.4. comesdirectly from the convexity of G

pand TC

pin

Eqs. (11)—(13) with the optimal lot size ratio inAppendix A. To prove the relationship betweenppand p

p`1in B.4., consider a comparison between

TCp

and TCp`1

, i.e., Gp

and Gp`1

.

Gp!G

p`1

"[Ap/(A

p/B

p)0.5#B

p/(A

p/B

p)0.5#C

p]

![Ap`1

/(Ap`1

/Bp`1

)0.5

#Bp`1

/(Ap`1

/Bp`1

)0.5#Cp`1

]

"[2(ApBp)0.5#C

p]![2(A

p`1Bp`1

)0.5#Cp`1

]

"[2(apbp)0.5!2(a

p`1bp`1

)0.5]

#[(ap#b

p)!(a

p`1#b

p`1)]

"(a0.5p

#b0.5p

)2!(a0.5p`1

#b0.5p`1

)2

"[(a0.5p

#b0.5p

)#(a0.5p`1

#b0.5p`1

)][(a0.5p

#b0.5p

)

!(a0.5p`1

#b0.5p`1

)]

"(pop#po

p`1)(po

p!po

p`1)*()) 0 if po

p

*()) pop`1

.

Therefore, the proof is complete. h


References

[1] A.Z. Szendrovits, Non-integer optimal lot size ratios intwo-stage production/inventory systems, InternationalJournal of Production Research 21 (3) (1983) 323—336.

[2] I.A. Karimi, Optimal cycle times in a two-stage serialsystem with set-up and inventory costs, IIE Transactions21 (4) (1989) 324—332.

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Optimal two-stage lot sizing and inventory batching policies

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