Asia
Pαcifìc
Managemenr Review (200 1) 6( 1 ) ，
卜19
Inventory ratio based production switching heuristic
(RPSH) for the aggregate production planning
problem
Chun Nam Cha'and Hark Hwang"
The production switch ing heuristic (PSH ) is a viable and effective solution method for the
agg regate production planning problcm in industries where prodllction is restricted 10 discrete levels. In 111051 induSlrics. the Înventory turnover ratio is onc ofthc most important measures in evalualing thc ap propriateness 0 1' invcntory assc t. This paper proposes an inventory ratio based PSH 10
integratc this induslrìal practices into the switching mcchanism of the PSH . The proposed heuristic
has a uniquc property in lh31 the decision about changing the production r3te in a pcriod is made
differe叫 ly in accordance 、vit h the ncxt period's demand forecasted . The effcctivencss of the proposed heuristic is investigated with several well-knowll problems including the paint factory problem. The total CQsts of the pro posed heuristic are comparcd with thosc of the linear decision rule
and thc linear program model as well as the PSH in quadratic and linear cost fu nction cases. The
rcsults show that the proposed rulc outpcrforms the I'SH in 11105t test5. espec ially when the demand
variability is significant
Key w叫 'ds :
Aggrcgate Production
Planni n皂 ;
Prod uction Swi tching H e uri s t 峙 ; Invcnlory Ratio
1. Introduction
In most ma nufacturing co mpani 郎， th e aggregate production planning
practical manageria l concern . The A PP is concerned with the determination of the production , inve ntory and workforce leve ls to meet the fluctuating demand most economically utiliz in g the physica l resources of a firm which
are ass umed to be fi xed during s pecified planning hori zon. As summarized in
the survey p aper of Saad [1 句 ， many researchers have proposed diverse APP
mode ls and so luti on methods ranging from soph isticated mathematical models
to simple heuristic m odels. However , most of these model s seem to have failed
in making a practica l contribution to operating practice mainly due to their
computationa l co mple x i旬 ， unrea li stic ass umption s o r impractical solutions inherent in th e mode l. Moreover there is some inco nsi stency between dec is ions
from the proposed APP m ode l and the practical management decisions
ß ased upon th e research works of Orr [1 4] and Elmaleh and Eilon [4] ,
Mellichamp a nd Love [10] proposed and evaluated a conceptually appealing
(A PP) 的 a
Corresponding author. De part l11cnt of Industrial and Systcms En g in自ring， R，借閱πh Institute of Ind回trial
Technology. Gyeongsang National U niversity， 9∞ G缸wa-dong， Chinju. Gy曲 ngnarn 660-70 1 、 Ko間a;
FAX : +82-55-762-6599; e-mail : [email protected]
Department oflndu strial Eng in ee ri n皂， Korea Advanced Institute of Scicnce and Tech n o l ogy 、
373-1 Kusong-dong YU 5o n g - gu 可 Taejon 305-701 . Korea; FAX: +82-42-869-311 0;
e-mail :[email protected]. ac.kr
Chlln Nam
Cha αnd
/-Iark /-Il1'ang
and easy-to-use APP model named production sw itching heuristic (PSH). In the
PSH , the production and associated workforce decisions are restricted to certain
finite number of levels (e.g. , low , normal and high) and the production level in
each period is determined on the basis of the amounts of inventory level and
forecasted demand. By limiting the number of admissib le production and workforce leve ls, the PSH lessens the possibility of rescheduling production and
workforce sizes 仕equently over the planning horizon. Furthermore , unlike optimization approaches such as linear decision rllle (LDR) and linear program
(LP) model , the PSH can handle any type of cost function structure inherent in
the production system under consideration
Oliffand Burch [12] repo口ed a successful implementation of three-phase
hierarchical planning system in which the aggregate plan is generated from the
PSH for a multi-product production system in Owens-Corn ing Fiberglass. Oliff
and Leong [1 3] extended the three. level PSH to a general n-Ievel production
switching rule and devised a way to explicitly handle the overtime decision for
a discrete production system. Barman and Burch [1] proposed a more simp lified
two-Ievel PSH. Through two experim ental stud 悶， Barman and Tersine [2 , 3]
investigated the performance of the PSH when the cost coefficient or forecasted
demand data undergoes some kinds of estimation errors. Nam and Logendran
[1 1] proposed two modifications of the PSH in deciding the workforce size and
control parameters of the PSH. Hwang and Cha [7] showed that the PSH tends
to make belated decision s in se lecting production leve l and proposed an improved version of the PSH named dominant production sw itch ing heuristic
(DPSH). The DPSH conta in s a production switching mechanism that expands
the feasible solution space of the decision rule by making timely decision on the
prodllction level
However, the PSH inevitably has some disappointing features: Firstly , the
solution quality of the PSH may be short to our expectation due to the absence
of the tlexibility to absorb the tluctuations in demand data [1 , 9 , 10]. In this
regard , Lambrecht et a l. [9] suggested a procedure to adapt the control parameters ofthe PSH to the change in demand data. In a ll the PSH based APP models
mentioned , the minimum and maximum target inventory leve ls, which remain
at constant levels throughout the entire planning horizon , are utili zed in their
prod
Chlln Nam Cha and Hark Hwang
This paper intends to present and evaluate another version ofthe PSH that
adopts the concept of target inventory ratio in its switching mechanism considering the industrial practices in evaluating the inventory asset. In the proposed
heuríst悶， the upper and lower Ií m íts of the target inventory level for a planning
períod are determined by taking the next períod's demand forecast into consideration. It is expected. that the resulting variable switching rule can meet highly
tluctuating demand data more economica lI y than the static rule of the PSH. In
the next place, we develop a hybrid grid search procedure to determine the
switching control parameters more efficiently
This paper is organized as fo lI ows: In section 2 , we describe our heuristic
that adopts the concept of target inventory ratio in its switching mechanism
The hybrid grid search method for the heuristic is explained in section 3. In
section 4 , the effectiveness of the proposed heuristic is investigated through
several we lI -known problems including the classical paint factory problem. The
results are compared with those of the LDR and LP model as well as the PSH
Finally , the conclusion appears in section 5
2. Inventory Ratio based Production Switching Heuristic (RPSH)
In the PSH of Mellichamp and Love [1 呵， once the production and inventory contro l parameters L ~ N 豆 H and A ~ C that minimize the relevant costs
over a planning horizon have been determined from historical data , the
production rate P, in period is determined by
,
p,
where
=
IL
if
jH
if
IN
otherwise
F, - /' _1 < L - C
F, - 1' _1 > H - A
,
F , = forecasted demand in period
1'.1 = net inventory level al the beginning of period
L = low level produclion rale
N = normal level production rate
H = high level production rate
A = minimum acceplable target inventory level
C = maximum acceptab le target inventory level
(l )
,
According to the switching mechanism of equation (1) , it is clear that the
PSH does not respond swiftly to changing situations and tends to switch the
production level belatedly , i.e ., the low (high) level of production has to be
scheduled only when the closing inventory is already expected to exceed(fa lI
sho忱。f) the maximum(minimum) acceptable target inventory leve l. To
overcome this belated response of the PSH , Hwang and Cha [7] proposed the
DPSH as follows
Chun Nam Cha and Hark Hwang
!L
P,
=~H
IN
ifF, -I ,_I <N-C
if F， 一 九 I > N-A
otherwise
(2)
They showed that the DPSH can generate better than or at least equal solution
to that ofthe PSH with a given decision rule for the workforce size
In most industries, perhaps the most impactful measure for the aggregate
performance of inventory asset is the inventory tumover ratio which is
expressed as the amount of average inventory over the cost of goods sold in a
unit period. The inventory tumover ratio represents the number oftimes that the
inventory has turned over or has been replaced during that period [15] . Tersine
[17] and Johnson and Montgomery [8] also pointed out that the inventory ratio
to sales is one of the critical indices for evaluating inventory asset in practical
industrial managemen t. For example , 100 units of inventory can have somewhat
different interpretation in terms of its adequacy of quantity and it depends
mainly on the com pan y's sales forecas t. That 時 ， the manager of a' company who
has 1,000 units of forecast in the next period may regard it as too small to meet
the forthcoming demand. On the contrary , it probably becomes a headache for
the manager who has only 150 units of forecas t. The existing heuristics based
on the PSH do not consider this situation in their decision rule
To integrate the inventory evaluation practice into the production
planning process , the inventory ratio , R, is defined as follows
R=
_c urrenl invenlory /eνe/
"的 forecast 戶'r
the next period
inventory turnoνer
(3)
In eq uation (3) , the current inventory level and the sales forecast replace
respectively the average inventory and the cost of goods sold term s of the
original definition for inventory turnover ratio. Suppo唱e the manager wants to
have a period's ending inventory kept between certain proportions of the next
period's sales forecas t. Then the switching mechanism of the PSH can be
modified as follows
In period 1, produce at the low level , L of production if the estimated
c1 0sing inventory with the normal production level (1 , = N - F, + 1 卜 1 ) divided
by the forthcoming period' s forecasted demand (F，刊 ) is e次pectedωbe above
the maximum allowable target inventory ratio , Ru. If
Î , = N - F , + 1' - 1 divided by F'+I goes below the minimum allowable target
mventory ratJ o ， 丸， produce at the high level of production , H. And when it is
between the minimum and the maximum target inventory ratios , set the
production rate at the normal level
Using the above switching mechanism , the RPSH is suggested as follows
Chun Nam Cha and Hark Hwang
ra- -tj
LH
〈
il--
一
-
u
P、
p川
Rnr H
N
、
Wh ere
if F , - 1 ， 一 1 < N - R" . F'+I
if F , -1'- 1> N - R . F ,+I
,
(4)
otherwise
,
R = minimum acceptable target inventory ratio
R" = max imum acceptable target inventory ratio (R， 三 R，，)
FT+ I = average forecasts of FT, F T- 1 and F H (T; planning horizon)
The unique characteristic of the. RPSH is that the amount of minimum
(maximum) target inventory level varies period by period for a given R (R ,,)
whereas those of the PSH and DPSH remain constant throughout the planning
horizon. Consequently , the sw itching rule of each period varies in accordance
with the forthcom ing sales forecas t. The varying target inventory may be more
compatible with the dynamic nature of demand variation in real world
environmen t. Due to this distinguishing characteristic , the basic role of the
inventory i.e., absorbing the fluctuations in demand could be improved. The
RPSH can be modified to utilize more than two future periods ' demand forecast
in determining the target inventory level for a period
For W" the workforce size in period 人 Barman and Burch ' s [1] decision
rule gi ven by equation (5) is adopted in the RPSH. ln this ru 峙 ， C. represents the
productivity conversion factor (units/man-month) of a manufacturing factory
And Z, which is determined from a grid search method , is another decision
variable corresponding to the adju stment of workforce size when the production
level becomes low or high. ln this paper, the value of Z is determined optimally
when the production and inventory control parameters are known
,
N
-=-+
C4
p, - N
一ι一一=- Z .
C4
O::;Z::; \
(5)
3. Determination of the contro l parameters
To utilize the RPSH as a solution method for the APP , the production , inventory and workforce control parameters of the heuristic that minimize the
total cost (TC) re levant to the APP should be found firs t. In this secti 凹， we
describe a hybrid grid search procedure (HGSP) which determines the control
parameters ofthe RPSH efficiently. The HGSP shown in figure 1 is an adaptive
grid search method to find a minimum cost point within predetermined range of
each control paramete r. The minimal cost ranges for the production rates (L , N ,
的 and the target inventory ratios (丸， R,,) are determined first with larger values
for search increments . Success ively , a detailed search phase with smaller increments is invoked to determ ine the best (L , N， 的 and 阱， R ,,) within the
Chun Nam Cha and Hark Hwang
set
TC= ∞
within
search
range
甘甜
℃
nH-nH 戶iu
u
vsI
nHpuuqu
n
ob
JUMP
set next
OPTIMIZATION
(R"R ,,)
1) quadratic
to be tested
2) linear
no
Figure 1 The hybrid grid search procedure for the RPSH
ranges specitied in the tirst phase. In the HGSP , the JUMP procedure explained
in section 3.1 is utilized to reduce the search spaces for R and R". And the deci.
sion on workforce control parameter Z is made optimally without resorting to a
grid search method as described in section 3 .2
,
3.1. JUMP procedure for
阱，
R,J determination
The production rate in each period is determined by equation (4) provided
that the tixed grid points (L , N， 的 and 阱， R,,) are given in the HGSP of Figure
1. Consequently , the following sets of periods can be identitied uniquely
I(L) = the set ofperiods whose production levels are low = {II P, = L}
I(的 = the set of periods whose production levels are normal = {t IP,= N}
I(的 = the set ofperiods whose production levels are high = {t I 代=的
Detine R 1, R2 , R3 and R4 for the bounds of target inventory ratio parameters , in
which the decision on production rate in each period does not alter， 的 follows
Chun Nam Cha and Hark Hwang
I
Rl = j
. N - F , + /'_1
max {~E
t ε t(H) 、
Ft+l
'-}, if t(H) ot
|一∞
R2 =
ø
otherwise
. N-F, +1 ,_,
I min {一一ι」二斗，
if t(N) ot ø
\, el(N j'
F' +1
1+ ∞
， otherwise
. N-F， +I ， 。
I max{一一一斗」二斗 ， if t(N) 手白
的 =j 峙I(N)
+1
F,
|一∞
， otherwise
.
, N-F,+/,_,
I1 min{一一~
I
~ I三!-)， if t(L)
R4 = \ lel (l.) ‘
布什
1+ ∞
手 G
， otherwise
From the RPSH of equation (4) , any R7 for the minimum target inventory ratio should satisfy the inequality F, - /' -1 > N - R; F, +I so that all the
periods tE t(的 could be planned at the high production leve l. Similarily, R;
for the maximum inventory ratio should satisfy the inequality
Ft 一九一 1 < N - R ~ Ft+ 1 for the periods tEt(L) to be planned at P,=L. At the
same time , the inequality N - R,; F， ' I 歪吭一 l卜1 至 N - R; F, +I should hold for all
R; and R,: in periods t ε t(川. By combining these inequalities , we can
conclude that any pair of (衍 ， R,; ) satisfying R 1< 阿三 R2 and R3 主 R，;< R4
will generate the same production schedule as that of current (L , N， 的 and (R{ ,
R,,). Therefore, the (前，時) in this region needs no further investigation during
the HGSP iterations while the production levels evaluated are (L , N， 的
3. 2. Optimal workforce size delermination
In the APP prob lem , the total cost tS composed of several cost terrns such
as regular payroll , workforce hiring/firing cost, overtime charge and inventory
cost as shown in equation (6). In most research works on the APP , the quadratic
cost function of LDR and the linear cost function of LP model have attracted
researcher's attention. Relevant cost components of both quadratic and linear
cost functions are summarized in table 1. In table 1, the linear cost function is an
approximation to the quadratic cost function of Holt et al. ' s LDR (1 ，的
T
TC= 主 (CRt + CH t + CO, + CI ,)
(6)
Chun Nam Cha and Hark Hwang
Table 1 Cost components of the aggregate production planning problem
C051∞ mpone nt
Quadratic ∞ st
Regular payroll c051 (CR,)
Hiring Or layofT C051 (CH,)
Overtime ωSI
(CO,)
Inventory C051
(CI ,)
Li near
function
cost function
C1W,
C1w,
C,(W,-W,.,)'
C，I W，- W，叫|
m"{("l(/', -C4 W ,)2
+C5 的 - C: 6 W , . O} m出 {CJ (P， /C.
C,(/,-C,)'
-w, 10
C ,I/ ,- C"I
In the HGSP of figure 1, the values of P, and 1, in each period are
determined uniquely from equation (4) when the production and inventory
con甘01 parameters are fixed at some values. Thus , depending on the cost
function adopted , the TC of equation (6) can be reduced to a quadratic or linear
function of workforce size , W,. Moreover, after substituting equation (5) for 眠，
equation (6) results in a simple quadratic or linear function of Z. ln the
subsequent discussion , the production levels and the inventory control
parameters are assumed to be fi xed at (L , N， 的 and (丸， R,,), respectively
(1) quadratic cost function
The sum of the cost components associated with the workforce level in
period 1, CR, + CH, + CO" is represented by equation (7)
間 (z)= C2( pr -=-互.=l YZ2 +丘主立2 z +旦旦
\L.4
+ maxl
L. 4
L.
4
C3 (乃 - N )中 +(N -P(~2C3(月 N)+ 主 fz
L.
+
(7)
4J
C3(月一 N)2 + 叫令。|
[n equation (7) , the regular payroll and hiringlfiring costs over the planning
horizon can be rewritten as equation (8) where WO denotes the workforce size at
tho I,eginning of 1 = [
Ch !l n Nam Cha and Hark
Hl咱ng
α(Z)= C2 {(于Y+ 玄(守守平哨二斗→L守)
+十(卡C ，玄旦宇乏
f主叫叫叫叫
2丹扣(何P， -N卅〈恰寺吉一吾討叫)}Z
(8)
+C， 早已N(是一等) +叫
Next consider the overtime cost term of equation (7). Following
substitutions are introduced for the coefficients of the function
α = C J (P , - N)' 主 O ; b=(N-P，)
( 2 C (P , J
甘len，
N)+ 去} ; c
=
C J (P , - N)' + C 5
P， 一去 N
例
the overtime cost in period t, CO正Z)， can be rewritten as equation (10)
CO ,(Z)= m叫 C O;(Z) = aZ ' +bZ+c ， O}
(10)
Unlike the regular payroll and hiring/firing cost terms , the CO ,(Z) of
equation (10) can be expressed by a piecewise quadratic function according to
the interval of Z which is divided by the coefficients a , b , and c or equivalently
by the production rate in period t
Firstly, in the periods of P, = N, CO正Z) becomes a constant shown below
(l l )
CO,(Z)=m
, Secondl'y, consider the periods of low level productio肌 Let
Z( and
Z i (Z( s; zU be two solutions ofthe quadratic equation CO;(Z)=O w 仙 P，
= L. If these solutions are real numbers , the range of Z for COl Z) > 0 can be
identified as shown in table 2. If the equation has no solution , CO ,(Z) > 0 for all
values ofO 主 Z 主 l
Finally , for the periods of high level production , let Z ,h (豆 1) denote the
= 0 with P, = H. Table J
smaller solution of 伽 quad叫ic equation C
shows the scope of Z for CO正Z)> O when zt exists. If there ex 的ts no real
solution , COρ) > 0 for all values of 0 豆 Z 豆 l
0; (Z )
Chun Nam Cha and Hark Hwang
Table 2 The range of Z for CO,(Z) > 0 when P, = L
z;
[1 ， ∞ ]
Z(
主丘Æ
0 至 Z <ZI
(0 , 1)
Table 3 The range of Z for
COρ)
> 0 when P , = H
1 <z~ < ∞
。三 Z 主 l
From the preceding analyses , it is cJ ear that the CO,(Z), and hence TCW,(Z) of
equation (7) , can be rewritten separately by a quadratic function according to
the interval of Z forrned by 抖 ， Z! and zt. For an example case of
0< Z: <
< Z < 1, tigure 2 shows the different functions for COρ) when the
production rate is low , normal or high level , respectively . With the aid of tables
2 and 3 , one can construct a quadratic total cost function (6) by adding
functions CR( Z) and L1~lcq(Z) in the interva l. Then a g lobal optimum value of
Z can be identitied easily among the local so lutions obtained separately in each
interval of Z
t
z;
CO ,(Z)
?, = L case
p = ,r 1
ltt
(Cs 一 C，jC4 )N
一一一._ . _ . -
、覓
ZI
一一τ
?, =N
-，..-
司、 ___ _1
Figure 2 Shape of CO,(Z) when production rate is low, norrnal or high
10
z
CllIln NIα m
Cha and Hark Hwang
(2) linear cost function
In case of the linear cost function , the cost components associated with the
decision on workforce size over the planning horizon is expressed as follows
,
T
‘
TC W(Z) = I 玉- KC1(fl- N)+ C2 1乃一月 - d Z )
1=1 '--4
+C1N + C3 max(( 乃 - N) (l- Z) ,O))
( 12)
1n this case , the overtime cost is charged only in the periods of high level
production. Let TCW,(Z) and TCW2( Z) denote the cost of changing workforce
size in the first period and the other surplus cost terms in TCW( Z), respective ly
And let n(的 be the number of periods in the set I(的 defined in section 3.1
TCW
|且
1
N
_
N
(Z ) = C 2 1 .:.l一~Z
+一 - Wol=laZ
C4
- 1'
1 C4
1
,
1
T
TC W2(Z)= j 云一 1 C 1 :L(們 一 N)
1'--41
+ bl
(13)
- n(H)(H - N )C3
1=1
+C2 ￡|月一凡 dlZ
+ NTC 1 +n(H)(M-N )C 3?
1= 2
(1 4)
= cZ + d
From equation (11) , the 2 of equation (13) minimizes TCW ,(Z) if a
* 0, that is ,
P , *N
z﹒
主
a
C ， W。 一 N
P
,
-
(1 5)
N
The optimal va lue Z' of Z can be determined simply from the decision
rule summarized in tab le 4. 1n figure 3, graphical presentations are depicted for
two example cases and we can conclude that Z' =Z" in case of O<c<lal. Similarly ,
when the condition of c<-Ial ho lds , the optimal value of Z becomes Z' = 1
Table 3 Decision rule for optimal 2 determ ination in linear cost model
\ >4 \
c~O
c<O
c訓。|Z=|O C〈|。|
Z s;O
。手。
z=O
z =z
。< 2 < 1
.t~ 1
a= 0 (P ,=N)
2=0
2= 1
Z'=O
11
學?
c~-Ial
z.=1
z.= 1
Z =1
Chun Nam Cha and Hark Hwang
TCW
TCW2
、
、
、、
/~/
c-.-
∞
TCW1( Z)
、、一 . -'
r‘τ
、
、
、
、
、
、
、、
、
。
(a)
z =z.
0 < c < IαI
Z
case
TCW
Z
(b) c
f/
←
一
。
、'，一
、
、
、
、
、
、
、
、
、
、、
、 、
、
- 、、、、 Z﹒ = 1
fr
w
n
Z
TCW2( Z)
< 一 |αI case
Figure 3 Examples for Z' determination in case of a>O, -a<b<O, 0 <2' < 1
4. Computational experiences
In this section , we evaluate the effectiveness of the RPSH through the
paint factory problem of Mellichamp and Love [10] . Using both quadratic and
linear cost functions , the RPSH solutions are compared with the LDR (for quadratic cost function) , LP (for linear cost function) , PSH and DPSH solutions under the assumption of perfect forecas t. To find the production rate (L , N， 的 and
target inventory ratio (丸， R,,), the search ranges were set to [min 吭， max F,] and
[0.5 , 1.5] in the HGSP , respectively. For L , N and H , the increments were set to
20 and 5 for the initial and succeeding detailed search procedures , respectively
12
Chlln Nam Cha and liark Ii、附ng
,
For R and R的 the increments of 0.05 and 0.01 were used in stages. AlI the
procedures were written in programming language C and executed on an IBM
compatible personal computer
4. 1. Quadratic cost mode/
The cost coefficients of the paint factory shown below were used for the
quadratic cost function of table 1. The initial inventory level (10) and workforce
size (Wo) were given by 263 and 訓 ， respectively
C 1=340 , C2=64.3 , C)=0.2 , C4=5 .67, Cs=51 .2, C6 =28 1, C 7=0.0825 , Cg=320
For the demand forecast of Mellichamp and Love [10] , table 4 gives the
cost details of the solutions resulting from the LDR, PSH and RPSH. The total
cost ofthe RPSH is lower than the LDR solution by 0.65%. And compared with
the PSH solution , the RPSH reduces the total cost by 0.85%. As was explained
by Hwang and Cha [7] , the slightly poor performance compared with the RPSH
of the LDR which is supposed to give the minimum cost may come from not
considering the effect of ending inventory , 111 The lower part of table 4
contains the results ofa simple analysis which considers the effect ofthe ending
inventory. From this analysis , we can see that the LDR provides the lowest cost
so lution and the RPSH gives the cost which is 1.87% higher than that of the
LDR on the basis of the net total cost
Table 5 shows the production and workforce levels over the entire
planning periods under each rule. The production level and target inventory
control parameters of the RPSH are determined as L = 350 , N = 375 , H = 465 ,
R = 0.52 , R" = 0.80 and the workforce size control parameter as Z = 0.51. Note
that , in periods 8 and 10 , the RPSH triggers different production rates even
,
,
Table 4 Cost analysis for the paint factory problem with quadratic cost function
Cost component
Regular payroll
Hiring/layoff
Overtime
lnventory
LDR+
$287 ,053
$3 , 127
$3 ,867
$1 ,261
PSH+
$273 ,439
$9 ,549
$9,922
$3 ,011
RPSH
$272,300
$8 , 150
$10,337
$2 ,609
Total cost
(1)
($l2O9O5O,3O0%8)
($l2O9O52,9l2%l)
主(9空9JL5盟%主)
Total Production
(2)
4,696.45
263
370.45
107 .4 5
$6,7 56
$288.552
(100.00%)
4,605
263
279
16
$1 ,006
$294,915
(102.21%)
4,580
263
254
-9
-$566
$293 ,962
(10 1. 87%)
t且omrIyvnevd~1t fofterorreyv n臼 (3)
IEInnnvitdeiainI1
Yalue of inventory
Net total cost
(4)
(1)-(4)
一，
Barman and Bu叫 [1]
(4)=$62.88 x(3) , where $62.88/unit=(1) of LDRI(2) of LDR
13
ChunNam
Cha 叫 ld
Hark Hwang
though the net requirements shown in the last column are equa l. This result
illustrates the adaptive sw itching capability of the RPSH reacting responsively
to the amount of forthcoming period ' s demand . The minimum and maximum
target inventory leve ls adj usted by the next peri od' s forecast are depicted in
figure 4
Barman and Burch [1) tested th e effect iveness of the PSH w ith four kinds
of demand seri es that have relatively high co巴ffic i en ts of variation (CV) as
shown in table 6 . Table 7 presents the relative performance of each rule for the
dem and data of table 6. In all cases , the RPSH produced better soluti ons than
the PSH. With the quadrat ic cost funct酬 . the effectiveness of the proposed
Table 5 The producti on and workforce plans for quadratic cost function
PSH
LDR
I
2
3
4
5
6
7
8
9
10
11
12
F,
430
447
440
316
397
375
292
458
400
350
284
400
P,
467.57
441 .3 9
4 15.4 0
380.91
3 77. 10
368.26
359 .4 3
382.24
376 .3 9
364.06
362.89
400.81
W,
78. 15
75 .27
72. 5 1
70.08
68 .2 9
67 .08
66.52
66.77
67 .3 7
68.50
70 .4 2
73 .3 1
戶，
440
440
440
365
365
365
365
365
365
365
365
365
W,
74.96
74.96
74.96
64.3 7
64 .3 7
64 .3 7
64.37
64 .3 7
64 .3 7
64 .37
64 .37
64 .3 7
RPSH
P,
465
465
375
375
375
350
375
375
375
350
350
350
W,
74 .27
74.27
66.14
66. 14
66 .14
63 .88
66. 14
66.14
66.14
63.88
63 .88
63 .88
F ,-!,.,
167
149
124
65
87
87
29
112
137
112
46
96
-、
J
A句句
3?"
∞∞∞∞∞
﹒l
。 」一一ι--'
4
6
~demand -含一 min inv
9
-+- max
10
11
inv
Figure 4 Variable target inventory levels ofthe RPSH
14
12 I
CJwn Nam Cha and Hark Hwang
he uri stic decreases as the variability of demand increases. Its total cost deviates
0 .96% from the LDR so lution when the CV in the dem and is 0 .5, which are still
lower than that of the PSH (7 .65%). Fig ure 5 shows the relative performance of
total costs from the LDR , PSH , DPSH and RPSH for the five example problems
tested. The DPSH costs are the results from Hwan g a nd Cha [7]. Except for
CV=0. 16 case (demand d ata of table 5), the RPSH outperforms the other
heuristics. For the prob le ms tested , the HGSP found the best control parameters
ofthe RPSH within 28 seconds
Tabl e 6 Demand data for various coefficients of variation
2
3
4
3
6
7
8
9
10
11
12
CV =O.3
510
528
500
280
415
242
250
480
475
252
247
410
CV=0.2
430
460
480
304
415
320
262
470
439
315
284
410
CV=O .4
549
56 1
566
251
43 1
194
190
512
496
201
221
417
CV=0.5
589
626
591
221
466
150
171
532
501
101
194
447
Table 7 Costs ofthe LDR , PSH and RPSH with demand data oftable 6
。2
。1
0.4
0.5
EUnu +
。仇
CV
PSH+
RPSH
$296 ,373
$297 ,75 1
$294 ,091
( 100%)
( 100 .4 6%)
(99.23%)
$302. 102
$310,275
$302 ,659
( 100. 18%)
(100%)
( 102.7 1%)
$311 , 162
$325 ,038
$312 , 190
(100%)
( 104 .4 6%)
(100 .3 3%)
$322 ,734
( 100%)
Barm an and
$347 ‘408
$325 ,828
( 107.65%)
Bu 叫1 [1)
( 100.96%)
15
Chun Nam Cha and Hα rk /-lll'ang
的m 闊的
104
法
102
l∞
98
c. v
%
。 16
。2
。3
。4
0.5
-e- LDR -含一 PSH --+- DPSH """*- RPSH
Figure 5 Compari son ofrelative total costs for example problems
4.2. Linear COSl Model
Bannan and Burch [1] evaluated the perfonnance of the PSH for the paint
factory problem with an approximated linear cost function . The total cost function with the followin g cost coefficients in the linear cost function of table 1
was tested
C t=340 , C2=208 .9, C J =630 , C, =5.67 , Cs=5.51 , C6=320
To prevent the inventory depl etion in the last period and the invemory
shortage in each period , additional con strain的， I t 2? /o=263 and ~ 1，主0， were
added to the LP model of Hanssmann and Hess [5]
The optimal LP solution , the PSH solution of Barman and Burch [1] and
the RPSH solutions are presented in tables 8 and 9 for the original demand data
ofCV=0. 16. From table 8 , it can be seen that the total cost from the RPSH 時，
unlike the test with the quadratic cost function , slightly higher than that of the
PSH . For this example problem , the control parameters that generate the same
production plan as that ofthe PSH could not be found
Further comparisons were made for the demand data of table 6 and the re司
sults are listed in table 10. Regardless of the variability in demand data , all the
heuristics generated near optimal solutions with the linear cost structure. Even
though the PSH is quite effective , the proposed heuristic dominates the PSH in
all the cases with CV higher than 0.2
16
Chun
N，α m
Cha and Hark H lI'ang
Table 8 Cost a na lysis for th e paint factory problem with linear cost function
PSH+
LP
$275 , 176
$4 ,239
Cost component
Regu1ar payroll
Hiring/1ayoff
Overtime
1nventory
Total cost
$275 ,238
$3 ,657
。
。
$1. 987
$28 1. 402
( 100.00%)
Bannan an d Burch [ 1]
$2 ,909
$28 1,804
( 100. 14%)
RPSH
$275 ,2 38
$3 ,473
。
$3 ,90 1
$282 ,612
(100 .4 3%)
Table 9 The production and wo rkforce plans for lin ear cost function
2
3
4
5
6
7
8
9
10
11
12
RPSH+
PSH
LP
f
F,
P,
W,
P,
W,
P,
W,
F ,-!,.!
430
447
440
3 16
397
375
292
458
400
350
284
400
459.28
459 .28
41 8 .4 4
375.00
375.00
375.00
375. 00
37500
344.25
344.25
344.25
344.25
8 1.00
450
450
450
360
360
360
360
360
360
360
79.37
79.37
79 .37
6 3.4 9
63 .4 9
63 .4 9
63.49
63 .4 9
63 .4 9
63 .4 9
63 .4 9
63 .4 9
435
435
435
365
365
365
365
365
365
365
365
365
76.72
76.72
76.72
64.37
64 .3 7
64.37
64 .3 7
64 .3 7
64 .3 7
167
179
184
65
97
107
34
127
162
147
66
101
~
8 1.00
73.80
66 .14
66. 14
66. 14
66. 14
66. 14
6日 7 1
60.7 1
60 .7 1
360
60.7 1
360
RF O.57 , R,,=O.65 and Z'=1
64 .3 7
64 .3 7
6 4.3 7
Table 10 Costs of the LP , PSH and RPSH with demand data oftable 6
cv
。2
0.3
。4
0.5
LP
$28 1. 86 1
(100%)
$285 ,549
( 100%)
PSH'
RPSH
$282 ,2 18
(100. 13%)
$286 ,4 86
(100.33%)
$282 ,2 18
(100. 13%)
$285 ,783
(1 00.08%)
$288 ,9 14
$290 , 170
$289,373
( 100%)
( 100 .4 3%)
(100.16%)
$293 ,055
$294 ,685
$293 ,255
( 100%)
( 100.07%)
(叩0 . 56%。
solutions from HYPER L1 NDO
+ Bannan and Burch [1]
17
Chun Nam Cha and Hark Hwang
5. Conclusions
For the aggregate production planning problem , this paper proposes an
inventory ratio based production switching heuristic (RPSH) on the basis of the
PSH. The key advantages of the PSH are its s imple structure and easy-to-use
sw itching mechanism governing the change of the production leve l. To reflect
some industrial practices on the eva lu ation of inv entory asset , the concept of
target inventory ratio is integrated into the switching mechanism of the RPSH
and the hybrid grid search procedure is dev ised to find the sw itching contro l
parameters of the heuristic efficiently
Through the performance tests with several well-known aggregate production planning problems, it is confirmed that the proposed heuristic can generate better so lution s than those of the PSH especially when the variability of
demand data is significan t. Since the proposed heuristic has a managerially appealing and easy-to-use structure than the PSH , the results of this study can
contribute to narrow the gap between practice and theory in the area of aggregate production planning. For a future research , it is wo吋h studying the effect of
tak in g multipl e periods forecasts into account in calculating the inventory ratio
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