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A coordinated multicommodity (s, S) inventory system
Mathl. Comput. Modelling Vol. 18, No. 11, pp. 69-73, Printed in Great Britain.
1993
All rights reserved
Copyright@
A Coordinated Multicommodity Inventory System
08957177/93 $6.00 + 0.00 1993 Pergamon Press Ltd
(s, S)
S. KALPAKAM Department of Mathematics Indian Institute of Technology, Madras-600 036, India G. ARIVARIGNAN School of Statistics Mzulurai Kamaraj University, Madurai-625 021, India (Received May 1993; accepted June 1993) Abstract-This paper analyzes a multi-item inventory model with unit renewal demands and immediate delivery of orders under a joint replenishment policy. Apart from the various operational characteristics, expression for the long run total expected cost rate is also derived.
1. INTRODUCTION One of the factors that contribute to the complexity of the present day inventory system is the multitude of items stocked and this necessitates the study of multicommodity systems. In dealing with such systems, initially models were proposed with independently established order points. But in situations where several products compete for limited storage space, or share the same transport facility, or are produced on (procured from) the same equipment (supplier), the above strategy overlooks potential savings associated with joint ordering and, hence, will not be optimal. Thus, the coordinated, or what is known as joint replenishment, reduces ordering and setup costs and allows the user to take advantage of quantity discounts. The problem of controlling the inventory of a multi-item system under joint replenishment has been receiving considerable attention for the past two decades. In continuous review systems, a well-known policy adopted in the literature is the (S, c, s) policy introduced by Ballintfy [l] and developed by Silver [2], where the three parameters Si, ci, and si are specified for each item i, with si 5 ci < Si, under unit sized Poisson demand and constant lead times. Subsequently, several articles have appeared dealing with the above policy and a more recent article of interest is due to Federgruen, Groenevelt and Tijms [3], which deals with the general csse of compound Poisson demands and non-zero lead times. A review of inventory models under joint replenishment is provided by Goyal and Satir [4]. In (S, c, s) policy, when the inventory position of any item i reaches a level less than or equal to si an order is placed to raise its level to Si and at the same time any other item j (f i) with available inventory at or below its can-order level cj is included in the replenishment to raise its level to Sj. Although the (S, c, s) policy is advantageous in many ways, it requires the calculation of three optimal variables for each item and, hence, would be desirable to have a coordinated control policy with a lesser number of decision variables. One such is the (s, S) policy introduced in this paper, with a single reorder level s defined in terms of the total number of items in the stock. This policy is particularly suitable for
The authors thank S. K. Srinivasan for his helpful suggestions.
70
S. KALPAKAM
AND G. ARIVARIGNAN
similar items which are not substitutable (such as various brands of the same equipment), will be profitable in situations where space constraint is an important factor.
2. PROBLEM
and
FORMULATION
Consider an N-commodity inventory system with fixed maximum total capacity of S units. It is assumed that the demands are of unit size and the time between successive demand points are independent and identically distributed with a common probability density function (pdf) f(.). Also, at any demand epoch, the demand will be for the jth commodity with probability pj > 0 (j = 1,2,. . . , N) and cjpj
= 1. It may be noted that the model with Poisson demand for the
items with different rates is a special case of the above demand process. Let Ij (t) denote the net inventory level (on hand minus backorders) for item j (j = 1,2, . . . , N). Then the joint ordering policy is as follows: when the total net inventory level I(t) = C 1j (t) of all commodities drops to a prefixed reorder level s (< S), an order will be placed for (4
- Ij(t))
units of the jth commodity (j = 1,2,. . . , N), where c Sj = S. The supply of items is assumed to be instantaneous. Hence, the net inventory at any Sci+i) for some i < N shortages will not occur in commodities (l), (2), . . . , (i) and may occur in commodities (i + l), . . . , (N). We use the following notation: f(“)(t) is the n-fold convolution of f(t) with itself; F,(t) is the cumulative distribution function of f(“)(t); p is (pi, ~2,. . . ,pN); I(t) is (Ii(t), 12(t), . . . ,IN(t)); and g*(o) is the Laplace transform of any arbitrary function g(t), Re o > 0.
3. PRELIMINARY
RESULTS
Let 0 = To < Tl < Ts < . . . denote the successive epochs of reorders. As an order is placed for every Q demands, the sequence of interarrival times between successive reorders, Y, = T, - T,_l (n = 1,2,. . . ) are independent and identically distributed with pdf f(&)(t). Hence, {Tn} forms a renewal process. Let Dj(o, b) denote the number of demands for the jth commodity
(j = 1,2,. . . , N) in any
interval (a,b). Let D(a, b) = c Dj(o, b) and let D(a, b) = (&(a, b), . . . , DN(~, b)). Then it can be seen that P[D(a, b) = D 1 D(a, b) = cd],where D = (dl,dz, . . . , dN) and d = cdj, has the multinomial distribution with parameters d, pl, 172,. . . ,pN and is given by
M(D’p’d)
(3.1)
=
denote the ordering quantity for the jth commodity at Tn. Then Oj(Tn) V j. Since D(Tn_l,Tn) = Q for all n, from (3.1) it can be seen that O(T,) is multinomial with parameters Q and p. Hence (Ol(Tn), . . . ,ON(%.))
Let Oj(Tn) Dj(T,-l,T,),
E(O(W)
=QP,
= =
(3.2)
Vn.
4. ANALYSIS Consider the vector inventory level process {I(t), t 2 0) whose state space B is given by IfB= {(Xl, z2, . . ..~N)~Sj-Q+l<5j~Sj}.Vj=11,2,...,Nands+1~~Xj~S. J
InventorySystem
71
Since the resupplies are executed instantaneously, at times of reorders (which are also demand points) we have I(TY) = S. Hence, the stochastic process {I(t), t >_ 0) is a regenerative process with the renewal process {Tn} imbedded in it. Starting with a reorder (replenishment) initially, define (P(x,t) = PiI(t) = xl, where x = ($1, 52,. . . , 2~) E B. Let z = C xj. Then c$(x, t) satisfies the renewal equation j t f@‘(u) 4(x, t - u) du, 4(x, t) = K(x, t) + (4.1) s0 where K(x,t)
= P[I(t) = x, Tl > t] and is given by = P[D(O,t) = S - x, D(O,t) = S - XT]
K(x,t)
= P(D(0, t) = S - x] P[D(O, t) = S - x ( D(0, t) = S - z] = [FS--S(t) - ~s-z+l(t)l
M(S
- x;
P, S - z).
(4.2)
Taking Laplace transform on both sides of equation (4.1), we obtain, using (4.2), 4*(x
7 a)
[f*(41S-z
=
11-
f*(a)1
ws - xi PYS -
x)
(4.3)
41 - f*wl
The Laplace transform of the marginal inventory level distribution $~(z~,QI) of the ith commodity is given by K(%“)
= c 4*(x,o) “XEB,
where ix = (21, x2,. . . ,zi__l, 5i+1,. . . ) XN), B=
{
ix]Sj-Q+l
j#i,
and s+l-zi
1
for all i = 1,2, . . . , N, and the sum is taken on N - 1 variables, viz., xi, x2,. . . , xi-l, xi+l, . . . , XN. The calculation of the above quantities is not simple as the sum is taken over N - 1 variables and, hence, becomes more difficult as N increases. The computation could be made easier if the multiple sum is replaced by a single one. This is achieved by simplifying the component C [f*(cr)lS-” M(S - x; P, S - z) of the above sum. More specifically, for fixed xi, it is shown ‘XEBi that c [f*(a)]S-” ‘XEBi
M(S - x; p, s - Z)
=
[f*(~)pp-”
Q-1g+zi ( si- p+‘)
[f*(a)
(1
_
pi)]’
(4*4)
r=O
(for sake of continuity the details of the long complex derivation of the above is deferred to the appendix). Hence, d;lxica)
=
si - Xi I1- f*Ca)IV*(ff)PilSi--zi Q-1-Sa+2i r QIP - f*Q(Q)I r=O
c (
+
r
[f*(cr)
>
(1
_
p,)jy
t
,
xi = Si - Q + 1,. . .) Si.
(4.5)
REMARK. The above analysis can be performed on similar lines even if we start with an arbitrary demand point other than a reorder epoch, and this would only involve notational complexity.
KM18:11-F
S. KALPAKAM AND G. AFUVARIGNAN
72
5. STEADY
STATE RESULTS
Let4(x) = Jim, 4(x, t), if it exists. Then --*
qqx) = liliocYc#J*(x,~) = M(S - x;p,s Applying
L’hospital
mw-”
P - f*(a)1
1 - [f*(4lQ
*
rule to the above, we have
4(x>= The limiting
x) liFo
-
value &(zi)
M(S-x;p,S-2)
Q
of the marginal
qbi(Xi) = ;
’
inventory
,c
distribution
qs -
x; p, s
-
of the ith commodity
is given
x).
‘XEBi
Substituting
Q = 0 in equation
f$i(xj)
=
(4.5), we obtain 9-1-si+m
c (
$-
Sj
_
r
7-0
The mean inventory level Aj and the expected in the steady state are, respectively, given by
xi + r
number
>
(1 -Pi>'.
of backorders
Bj for the jth commodity,
Bj = -CX~~(Z). Aj = CX~~(X)> s
r>O
The expected
ordering
quantity
for the jth commodity,
in the limiting
case is given by
E(uj) = QPj. Since {T,} forms a renewal case is given by
process with pdf f(Q)(t), T=L,
where m is the mean time between
(from (3.2)) th e mean reorder
rate y in the stationary
Qm
two successive
demands.
A typical optimization problem incorporating the above factors is to find the values that minimise the long run total expected cost rate C(s, S) given by
C(s,S) = ;
s and S
+x(ajAj+bjBj),
j
where K is the set up cost per reorder, cj, the unit purchase price of item j, oj and bj, the holding and backorder costs per item per unit time, respectively, for the jth commodity.
APPENDIX DERIVATION Let l&(k) =
ix(
Cz,=k; Tk#.i
OF THE RESULT
xj=Sj-Q+l,...,Sj-l,Sj,
that li =
s-s< u k---s+1 -q
(4.4) j#i
l&(k).
. Thenitcanbeseen
InventorySystem Hence we have, for fixed
c f*-‘(a!) =
7:1
xi,
M(S - x; p, s
-
x)
c f*‘-zk4(s _xl,y*T,“s, _xN) fJ Pi?-) n=l, n#i
“XEB,
f k=:E i i z(k)
*‘s-=‘((r)(s
=
’
=
sic (
2
k=s+l-xi
si -
‘XEIl!i(k)
1lj ($I:.:;! 1rI
z
(Si - Xi)!
(S - xi - k)
’
p(s”-“i)
_ x)!p(s’-““)
n
f*S-=t-k (cl) x,
N
pp-Gd
(Sn- 0
n=l,n_#i
(Since S - x = S - xi - nTtxn = S - xi - k, for all ‘X E Bi(k))
X
(p_,,l!
S-&-k
V
(1 - pi)s-s,-”
> j,,
“XEBi (k)
s-s,
s- y,” r; - “)[f*(o!>(l
c
= [f*(a)pi]Si-“”
Ic=s+1-s,
71
- pi)]s-s’-”
z
(
Sn-Gl x
c
(S-$-k)!
{
fi n=l,n#i
‘XEB. (rc)
3_xn), n
>
(Since C (S, - x,) = S - Si - C x, = S - Si - k for all ix E l&(k)). n#i n#i
c
(s
‘XEB, (k)
_
si _ k)!
fi
As
‘pn’5-“3s”n 71
n=l,n#i
the sum of all probabilities of a multinomial distribution with parameters S - Si - k, p/(1 n=1,2 )..., i-1,i+1,..* , N, its value is equal to one. Hence, we have
is
c
[f*(a)]S-”
‘Ye&(k)
M(S-x;
p, S-x)
= [f*(a)pi]s’-“’
Q-1-Si+zi c r=o
si - $ + ‘)
--pi),
[f*(a) (1 -pi)]T.
(
REFERENCES J.L. Ballintfy, On a basic class of inventory problems, Management Science 10, 287-297 (1964). E.A. Silver, A control system for coordinated inventory replenishment, International Journal of Pmduction Research 12, 647-671 (1974). A. Federgruen, H. Groenevelt and H.C. Tijms, Coordinated replenishment in a multi-item inventory system with compound Poisson demands, Management Science 30, 344-357 (1984). SK. Goyal and T. Satir, Joint replenishment inventory control: Deterministic and stochastic models, European Journal of Operations Research 38, 2-13 (1989).
1993
All rights reserved
Copyright@
A Coordinated Multicommodity Inventory System
08957177/93 $6.00 + 0.00 1993 Pergamon Press Ltd
(s, S)
S. KALPAKAM Department of Mathematics Indian Institute of Technology, Madras-600 036, India G. ARIVARIGNAN School of Statistics Mzulurai Kamaraj University, Madurai-625 021, India (Received May 1993; accepted June 1993) Abstract-This paper analyzes a multi-item inventory model with unit renewal demands and immediate delivery of orders under a joint replenishment policy. Apart from the various operational characteristics, expression for the long run total expected cost rate is also derived.
1. INTRODUCTION One of the factors that contribute to the complexity of the present day inventory system is the multitude of items stocked and this necessitates the study of multicommodity systems. In dealing with such systems, initially models were proposed with independently established order points. But in situations where several products compete for limited storage space, or share the same transport facility, or are produced on (procured from) the same equipment (supplier), the above strategy overlooks potential savings associated with joint ordering and, hence, will not be optimal. Thus, the coordinated, or what is known as joint replenishment, reduces ordering and setup costs and allows the user to take advantage of quantity discounts. The problem of controlling the inventory of a multi-item system under joint replenishment has been receiving considerable attention for the past two decades. In continuous review systems, a well-known policy adopted in the literature is the (S, c, s) policy introduced by Ballintfy [l] and developed by Silver [2], where the three parameters Si, ci, and si are specified for each item i, with si 5 ci < Si, under unit sized Poisson demand and constant lead times. Subsequently, several articles have appeared dealing with the above policy and a more recent article of interest is due to Federgruen, Groenevelt and Tijms [3], which deals with the general csse of compound Poisson demands and non-zero lead times. A review of inventory models under joint replenishment is provided by Goyal and Satir [4]. In (S, c, s) policy, when the inventory position of any item i reaches a level less than or equal to si an order is placed to raise its level to Si and at the same time any other item j (f i) with available inventory at or below its can-order level cj is included in the replenishment to raise its level to Sj. Although the (S, c, s) policy is advantageous in many ways, it requires the calculation of three optimal variables for each item and, hence, would be desirable to have a coordinated control policy with a lesser number of decision variables. One such is the (s, S) policy introduced in this paper, with a single reorder level s defined in terms of the total number of items in the stock. This policy is particularly suitable for
The authors thank S. K. Srinivasan for his helpful suggestions.
70
S. KALPAKAM
AND G. ARIVARIGNAN
similar items which are not substitutable (such as various brands of the same equipment), will be profitable in situations where space constraint is an important factor.
2. PROBLEM
and
FORMULATION
Consider an N-commodity inventory system with fixed maximum total capacity of S units. It is assumed that the demands are of unit size and the time between successive demand points are independent and identically distributed with a common probability density function (pdf) f(.). Also, at any demand epoch, the demand will be for the jth commodity with probability pj > 0 (j = 1,2,. . . , N) and cjpj
= 1. It may be noted that the model with Poisson demand for the
items with different rates is a special case of the above demand process. Let Ij (t) denote the net inventory level (on hand minus backorders) for item j (j = 1,2, . . . , N). Then the joint ordering policy is as follows: when the total net inventory level I(t) = C 1j (t) of all commodities drops to a prefixed reorder level s (< S), an order will be placed for (4
- Ij(t))
units of the jth commodity (j = 1,2,. . . , N), where c Sj = S. The supply of items is assumed to be instantaneous. Hence, the net inventory at any
3. PRELIMINARY
RESULTS
Let 0 = To < Tl < Ts < . . . denote the successive epochs of reorders. As an order is placed for every Q demands, the sequence of interarrival times between successive reorders, Y, = T, - T,_l (n = 1,2,. . . ) are independent and identically distributed with pdf f(&)(t). Hence, {Tn} forms a renewal process. Let Dj(o, b) denote the number of demands for the jth commodity
(j = 1,2,. . . , N) in any
interval (a,b). Let D(a, b) = c Dj(o, b) and let D(a, b) = (&(a, b), . . . , DN(~, b)). Then it can be seen that P[D(a, b) = D 1 D(a, b) = cd],where D = (dl,dz, . . . , dN) and d = cdj, has the multinomial distribution with parameters d, pl, 172,. . . ,pN and is given by
M(D’p’d)
(3.1)
=
denote the ordering quantity for the jth commodity at Tn. Then Oj(Tn) V j. Since D(Tn_l,Tn) = Q for all n, from (3.1) it can be seen that O(T,) is multinomial with parameters Q and p. Hence (Ol(Tn), . . . ,ON(%.))
Let Oj(Tn) Dj(T,-l,T,),
E(O(W)
=QP,
= =
(3.2)
Vn.
4. ANALYSIS Consider the vector inventory level process {I(t), t 2 0) whose state space B is given by IfB= {(Xl, z2, . . ..~N)~Sj-Q+l<5j~Sj}.Vj=11,2,...,Nands+1~~Xj~S. J
InventorySystem
71
Since the resupplies are executed instantaneously, at times of reorders (which are also demand points) we have I(TY) = S. Hence, the stochastic process {I(t), t >_ 0) is a regenerative process with the renewal process {Tn} imbedded in it. Starting with a reorder (replenishment) initially, define (P(x,t) = PiI(t) = xl, where x = ($1, 52,. . . , 2~) E B. Let z = C xj. Then c$(x, t) satisfies the renewal equation j t f@‘(u) 4(x, t - u) du, 4(x, t) = K(x, t) + (4.1) s0 where K(x,t)
= P[I(t) = x, Tl > t] and is given by = P[D(O,t) = S - x, D(O,t) = S - XT]
K(x,t)
= P(D(0, t) = S - x] P[D(O, t) = S - x ( D(0, t) = S - z] = [FS--S(t) - ~s-z+l(t)l
M(S
- x;
P, S - z).
(4.2)
Taking Laplace transform on both sides of equation (4.1), we obtain, using (4.2), 4*(x
7 a)
[f*(41S-z
=
11-
f*(a)1
ws - xi PYS -
x)
(4.3)
41 - f*wl
The Laplace transform of the marginal inventory level distribution $~(z~,QI) of the ith commodity is given by K(%“)
= c 4*(x,o) “XEB,
where ix = (21, x2,. . . ,zi__l, 5i+1,. . . ) XN), B=
{
ix]Sj-Q+l
j#i,
and s+l-zi
1
for all i = 1,2, . . . , N, and the sum is taken on N - 1 variables, viz., xi, x2,. . . , xi-l, xi+l, . . . , XN. The calculation of the above quantities is not simple as the sum is taken over N - 1 variables and, hence, becomes more difficult as N increases. The computation could be made easier if the multiple sum is replaced by a single one. This is achieved by simplifying the component C [f*(cr)lS-” M(S - x; P, S - z) of the above sum. More specifically, for fixed xi, it is shown ‘XEBi that c [f*(a)]S-” ‘XEBi
M(S - x; p, s - Z)
=
[f*(~)pp-”
Q-1g+zi ( si- p+‘)
[f*(a)
(1
_
pi)]’
(4*4)
r=O
(for sake of continuity the details of the long complex derivation of the above is deferred to the appendix). Hence, d;lxica)
=
si - Xi I1- f*Ca)IV*(ff)PilSi--zi Q-1-Sa+2i r QIP - f*Q(Q)I r=O
c (
+
r
[f*(cr)
>
(1
_
p,)jy
t
,
xi = Si - Q + 1,. . .) Si.
(4.5)
REMARK. The above analysis can be performed on similar lines even if we start with an arbitrary demand point other than a reorder epoch, and this would only involve notational complexity.
KM18:11-F
S. KALPAKAM AND G. AFUVARIGNAN
72
5. STEADY
STATE RESULTS
Let4(x) = Jim, 4(x, t), if it exists. Then --*
qqx) = liliocYc#J*(x,~) = M(S - x;p,s Applying
L’hospital
mw-”
P - f*(a)1
1 - [f*(4lQ
*
rule to the above, we have
4(x>= The limiting
x) liFo
-
value &(zi)
M(S-x;p,S-2)
Q
of the marginal
qbi(Xi) = ;
’
inventory
,c
distribution
qs -
x; p, s
-
of the ith commodity
is given
x).
‘XEBi
Substituting
Q = 0 in equation
f$i(xj)
=
(4.5), we obtain 9-1-si+m
c (
$-
Sj
_
r
7-0
The mean inventory level Aj and the expected in the steady state are, respectively, given by
xi + r
number
>
(1 -Pi>'.
of backorders
Bj for the jth commodity,
Bj = -CX~~(Z). Aj = CX~~(X)> s
r>O
The expected
ordering
quantity
for the jth commodity,
in the limiting
case is given by
E(uj) = QPj. Since {T,} forms a renewal case is given by
process with pdf f(Q)(t), T=L,
where m is the mean time between
(from (3.2)) th e mean reorder
rate y in the stationary
Qm
two successive
demands.
A typical optimization problem incorporating the above factors is to find the values that minimise the long run total expected cost rate C(s, S) given by
C(s,S) = ;
s and S
+x(ajAj+bjBj),
j
where K is the set up cost per reorder, cj, the unit purchase price of item j, oj and bj, the holding and backorder costs per item per unit time, respectively, for the jth commodity.
APPENDIX DERIVATION Let l&(k) =
ix(
Cz,=k; Tk#.i
OF THE RESULT
xj=Sj-Q+l,...,Sj-l,Sj,
that li =
s-s< u k---s+1 -q
(4.4) j#i
l&(k).
. Thenitcanbeseen
InventorySystem Hence we have, for fixed
c f*-‘(a!) =
7:1
xi,
M(S - x; p, s
-
x)
c f*‘-zk4(s _xl,y*T,“s, _xN) fJ Pi?-) n=l, n#i
“XEB,
f k=:E i i z(k)
*‘s-=‘((r)(s
=
’
=
sic (
2
k=s+l-xi
si -
‘XEIl!i(k)
1lj ($I:.:;! 1rI
z
(Si - Xi)!
(S - xi - k)
’
p(s”-“i)
_ x)!p(s’-““)
n
f*S-=t-k (cl) x,
N
pp-Gd
(Sn- 0
n=l,n_#i
(Since S - x = S - xi - nTtxn = S - xi - k, for all ‘X E Bi(k))
X
(p_,,l!
S-&-k
V
(1 - pi)s-s,-”
> j,,
“XEBi (k)
s-s,
s- y,” r; - “)[f*(o!>(l
c
= [f*(a)pi]Si-“”
Ic=s+1-s,
71
- pi)]s-s’-”
z
(
Sn-Gl x
c
(S-$-k)!
{
fi n=l,n#i
‘XEB. (rc)
3_xn), n
>
(Since C (S, - x,) = S - Si - C x, = S - Si - k for all ix E l&(k)). n#i n#i
c
(s
‘XEB, (k)
_
si _ k)!
fi
As
‘pn’5-“3s”n 71
n=l,n#i
the sum of all probabilities of a multinomial distribution with parameters S - Si - k, p/(1 n=1,2 )..., i-1,i+1,..* , N, its value is equal to one. Hence, we have
is
c
[f*(a)]S-”
‘Ye&(k)
M(S-x;
p, S-x)
= [f*(a)pi]s’-“’
Q-1-Si+zi c r=o
si - $ + ‘)
--pi),
[f*(a) (1 -pi)]T.
(
REFERENCES J.L. Ballintfy, On a basic class of inventory problems, Management Science 10, 287-297 (1964). E.A. Silver, A control system for coordinated inventory replenishment, International Journal of Pmduction Research 12, 647-671 (1974). A. Federgruen, H. Groenevelt and H.C. Tijms, Coordinated replenishment in a multi-item inventory system with compound Poisson demands, Management Science 30, 344-357 (1984). SK. Goyal and T. Satir, Joint replenishment inventory control: Deterministic and stochastic models, European Journal of Operations Research 38, 2-13 (1989).