CONSECUTIVE OPTIMIZORS FOR A PARTITIONING PROBLEM
WITH APPLICATIONS TO OPTIMAL
INVENTORY GROUPINGS FOR JOINT REPLENISHMENT*
A. K. Chakravarty
J. B. Orlin
and
U. G. Rothblum
Sloan W.P. No. 1383-82
November 1982
*Support received from the National Science Foundation Grant ENG-78-25182
and Grant ECS-8205022.
A. K. Chakravarty, Washington State University, Department of Management,
Pullman, Washington 99163
J. B. Orlin, Massachusetts Institute of Technology, Sloan School of
Management, Cambridge, Massachusetts 02139.
U. G. Rothblum, Yale University, School of Organization and Management,
New Haven, Connecticut 06520
CONSECUTIVE OPTIMIZORS FOR A PARTITIONING PROBLEM
WITH APPLICATIONS TO OPTIMAL INVENTORY GROUPINGS FOR JOINT REPLENISHMENT
by
A. K. Chakravarty
J. B. Orlin
and
U. G. Rothblum
Abstract
We consider several subclasses of the problem of grouping
(indexed 1, ..., n)
g(S1, ..., S ).
into
m
items
subsets so as to minimize the function
In general, these problems are very difficult to solve to
optimality, even for the case
conditions on
n
g(-)
m = 2 .
Here we provide several sufficient
which guarantee that there is an optimum partition in which
each subset consists of consecutive integers (or else the partition
satisfies a more general condition called "semi-consecutiveness").
S1
...
,
S
Moreover, by
restricting attention to "consecutive" (or "semi-consecutive") partitions we can
solve the partition problem efficiently for small values of
g
m .
If in addition,
is symmetric then the partition problem reduces to a shortest path problem,
solvable in
O(n2m)
steps.
We apply the above results to the problem of grouping inventory items into
subgroups with a common order cycle per subgroup so as to minimize the resulting
economic order quantity costs.
Under relatively minor assumptions on the cost
structure and on the set of feasible policies, we may reindex the items a priori
so as to guarantee the optimality of a "consecutive" partition.
-
1.
1 -
Let
al, ..., a
some integer
0
and
r
Introduction
bl, ...,
n, bl,
bn
br
...,
be real numbers ordered so that for
are negative,
br,
..., b
r+l'
n
are non-
negative and
a1 / b1
where for
a
> 0
.
bi = 0
If
whose coordinates are
let
N = {,
bS =
sets, say
ai / b i
/ br+l
,
to be
+*
a i / bi
As usual, we let
and
.
ar+
a
•a
or
--
/b
(1)
according as
is defined arbitrarily so
and
b
denote the vectors
b i , respectively.
For each subset
S
of
N
let
aS =
iS
bi .iS
. We consider the problem of partitioning the set
S,
...
gm(S1s
hm (-)
function
ai
..., n
S
real-valued function
where
and
ai = bi =
that inequality (1) holds.
Let
/ b
we consider
ai < O .
or
a
N
and
into
m
, including possibly some empty sets, so as to minimize a
g (S1
'' Sm
..., S ),
hm(as1' b
1
where
g ()
' a
1
is a real valued function of
h (.)
m
ai
2m
is assumed to have the form
' bs'' )
m
(2)
im
variables.
We allow that the
be nonsymmetric, in which case the cost of a partition depends
on the order in which the sets are selected.
A subset
S c N
integers, e.g.,
secutive.
is called consecutive if its elements are consecutive
{4,5,6}.
A partition
consecutive for each
P = {S1, ..., S}
i .
if there is a permutation
S (1) u ... u S
In particular, the empty set is considered to be con-
A partition
S3, S3
=
P
{S1,
..., S }
of the set of integers
are consecutive for
{S1 ,S 2 ,S 3 } = {{3,7,8}-, {1,2,91
subsets
is called consecutive if
{4,5,61}
u S1 , S3 u S 1 u S2
is
is called semi-consecutive
1, ..., m
j - 1, ..., m .
Si
such that the sets
For example, the partition
is semi-consecutive because the three
are all consecutive.
II
2-
The purpose of this paper is to provide sufficient conditions that the
optimization problem defined above has an optimal consecutive or semi-consecutive
partition.
Specifically, we show that if
h
m
is concave in its
then there is an optimal semi-consecutive partition.
addition,
b
0
or
Also, we show that if
a
2m
variables
Moreover, if in
0 , then there is an optimal consecutive partition.
b. = 1
1
for each
i
and , in addition,
h
m
is concave
in the "a variables" for each fixed value of the "b variables" then a consecutive
optimal partition exists.
S c {l, ..., n}
a set
a given partition
Of course, when
bS
P =
equals
S1
...,
ISI,
S }
b
P
... b S )
(bS
we have that for
S .
In this case, for
is called the shape of
m
m
of sets in a partition
P
we call
P
We observe that the number of ordered m-Dartitions that are con-
O(m!n m
secutive is
i
(e.g., Hwang [19811).
When we wish to specify the number
an m-partition.
for each
the cardinality of
1
the partition
1
-l)
and the number of semi-consecutive partitions is
Thus for a fixed value of
m
O(m!n2m-2)
we can determine an optimal consecutive or semi-
consecutive m-partition in polynomial time via complete enumeration.
Although
complete enumeration is quite impractical even for moderate sizes of
m , the
above fact contrasts sharply with the NP-completeness of the optimal partition
problem for any fixed
m
partition problem for
m
is
m
.)
changes in
2 , as the knapsack problem is a special case of the
=
2 .
(Of course, the total number of ordered m-partitions
We note that the sensitivity of optimal consecutive partitions to
m
can be studied by using the results of Denardo, Huberman and
Rothblum [1982].
We formally state our main results concerning the existence of consecutive
and semi-consecutive optimal partitions in Section 2, where we also survey a
number of places where special cases of our results are used.
In Section 3, we
show how our results apply to problems that involve optimal groupings for joint
- 3-
replenishment of inventories.
where
g (-)
Next, in Section 4, we discuss the special case
is separable and symmetric, in which case optimal consecutive
partitions can be determined in
2
(mn
)
steps and optimal semi-consecutive
partitions can be determined in
4
O(mn
)
steps.
of
NP-completeness which apply to our results.
are given in Section 6.
In Section 5, we discuss issues
Finally, proofs of our results
111
- 4 -
2.
The Main Results
We next state our main results concerning optimal partitions that are
either consecutive or semi-consecutive.
Theorem 1:
b
0
or
a
Theorem 2:
Suppose
0 .
h
The proofp are deferred to Section 6.
is concave in its
m
2m
variables and in addition
Then there exists a consecutive optimal partition.
Suppose that
h
m
is concave in its
2m
variables.
Then there
exists a semi-consecutive optimal partition.
Theorem 3:
Suppose
b.
=
1
for
i = 1, ... , n
and
h (-)
the "a variables" for each fixed value of the "b variables."
is concave in
Then there exists
a consecutive optimal partition.
Although the partition problems considered in the above Theorems are quite
restricted, there have been several realms where special cases of these results
have been used.
We next give a brief survey of such applications.
One classical partitioning problem occurs in Hypothesis Testing in Statistics.
Outcomes of a random phenomenon are partitioned into two sets, one of which corresponds to the region where the null hypothesis is accepted and the other to the
region where it is rejected.
In this case the "a variables" correspond to proba-
bility of type 1 error, the "b variables" correspond to probability of type 2 error
and the corresponding objective function is linear.
applies and an optimal consecutive partition exists.
Neyman-Pearson Lemma.
DeGroot
[1975, p. 374].
It follows that Theorem 1
This fact is the well-known
For more details on the applicaton to statical testing see
-5-
We next give a number of examples of studies where special cases of the
results of Theorem 3 were obtained.
where
[1982].
h
In particular, the special case of Theorem 3
is separable and symmetric is discussed in Chakravarty, Orlin and Rothblum
They apply the corresponding result to a problem of inventory grouping
with joint replenishment (see Section 4 for more complex such problems).
Also,
Hwang [1981] and Hwang, Sun and Yao [1982] consider restricted classes of functions
for which a corresponding optimization problem have consecutive optimizers.
Their
conditions entail monotonicity and additivity requirements, and some of their
results are special cases of Theorem 3.
The above papers provide applications
of their results to problems in storage and problems in group testing.
Finally,
Barnes and Hoffman [1982] considered a special instance of the partitioning problem
studied in Theorem 3.
The motivation for their work arose in connection with the
partitioning of eigenvalues of a matrix so as to derive estimates on a graph
theoretic partitioning problem.
They used the theory of submodular set functions
to establish the existence of consecutive optimal partitions.
Their results were
developed independently of those in Chakravarty, Orlin and Rothblum [1982], and
their proof is an interesting application of the theory of submodular set functions.
In Section 5, we show that the Barnes-Hoffman partitioning problem is
NP-complete.
III
- 6 -
3.
An Application to Inventory Grouping
Consider an economic order quantity model involving
per unit time and a fixed cost
to partition the
n
items into
Ki
m
items, where the
Di , a unit inventory holding cost
i-th item has (deterministic) demand rate
hi
n
for placing an order.
The problem is
subgroups and choose order cycles for the
groups out of a given set of allowable (joint) order cycles, so as to minimize
the net average cost per unit time.
Motivation and more detailed explanation of
This model
special cases of this model are given in Chakravarty [1982a, 1982b].
is a generalization of the joint replenishment models considered by Goyal [1974],
who study the problem in which a unit of order
Silver [1975], and Nocturne [1973]
X is determined for the group of
time
X .
item is an integer multiple of
n
items, and the order cycles for each
They give heuristic solutions to this problem.
A related model was studied by Chakravarty, Orlin and Rothblum [1982].
a. = 21h D i
Let
holds.
i
and
bi = K
where the items are labeled so that (1)
As in the ordinary EOQ model (e.g., Wagner [1969, pp. 18-19]), if item
has order cycle
then its order quantity is
T
S
per unit time of a group
c(S,T) =
I
2
of items having the same order cycle
1
tD.h
ieS
-1
K.t-
+
..., t
=
Ta
l
+ T
b
T
.
is:
(3)
iS
So, if the items are partitioned into groups
t,
Di , and the average net cost
S1, ..., S
having order cycles
, respectively, the total average cost per unit time is
m
C (S1
.Sm,
...
tj)
=
(4)
(S
j=1
The m-vector of order cycles is assumed to be taken out of a set
m-vectors of order cycles.
T
of allowable
This formulation allows one to impose joint restrictions
on the order cycles, e.g., the requirement that all order cycles are integer
-7
multiples of the smallest one, or individual restrictions, e.g., that each order
cycle be an integer in the set
{1,7,30}.
It follows that for a fixed partition
of the items, the minimum average cost per unit time is:
S1 , ..., S
gm(S,
..., Sm )
c (Sl,
inf
S,
t,
..., t)
m-
tja
= inf
teT
Evidently, as
c (S
m l'
..
S , t,
m
1
(5)
b
j
..., t)
m
'(
)
, where
subsets so as to minimize
g
is linear in
(for fixed
and
+ t
j
> 0 , the infimum defining
g
is
The problem is to find a partition of the items into
finite for each partition.
aS
S
j=l
bS
gm
is given above.
t ), we conclude that
Now, as
c(S,t)
c (S1, ..., S ,
bS
and therefore g(S, ..., S )
aS ,
bs1, ..., a
1
1
m
m
as the infimum of linear functions, is concave in aS , b,
a
bS
i.e.,
1
1
m
m
the assumptions of Theorem 1 are satisfied. Hence, our results apply and there
tl, ...
tm )
is linear in
exists an optimal partition consisting of consecutive sets.
We next consider a modified version of the model described above where the
terms
and
Ta S
and
T
lbS
f( -lbs) , where
in (3) are replaced, respectively, by expressions e (aS)
S
T
e (.)
and
f ()
are real valued functions.
This
formulation allows one to introduce discounting on the holding cost as well as
the setup cost, where the value of the discount depends on the monetary volume.
Under the above modification (5) has to be replaced by:
m
gm(S1, ..., S)
= inf
Z etj(tja S ) + fS
eT j=l
i
aj
3
It is easily seen that if the function
tions of Theorem 1 hold and if
Ki
eT
and
f
is independent of
are concave then the assumptions of Theorem 3 hold.
3
(6)
are concave then the assumi
and the functions
e
In either case we con-
clude that there exists an optimal grouping of the products where each set is
I
-8-
consecutive.
We emphasize that concavity of the functions
e
and
f
is
reasonable as it represents higher discounting as the volume increases.
A special case of the general model allows the decision maker to choose the
order cycles of the groups independently out of a given set
of the enumeration of the group), i.e.,
In this case
gm
T = U x U x ... x U
(which is independent
for some set
U c R
is symmetric and separable and the results of (the forthcoming)
Section 4 apply.
In particular, the problem of finding an optimal grouping can
be reduced to a shortest path problem.
Finally, we consider the case in which the cost of ordering in period
P
is
concave in the total quantity ordered in that period, where we might have a simultaneous reorder from different groups.
If we let
P
be the least common multiple
of the order cycles, then the total cost of ordering is:
P-1
I
p=O
where
tjlp
e(
I
tja
)
(7)
tjIP
means that
t.
is an integral divisor of
p .
As before, the infimum
of concave functions is concave, and thus there is a consecutive optimal partition.
-9-
4.
The Symmetric Separable Case
We next consider the case where
some function
h : R
2
is symmetric and separable, i.e., for
g
R
m
g (S1,
S)
...
=
h(as
j=1
J
) .b
j
In this case the order of the subsets in a consecutive partition does not matter.
In particular, without loss of generality, one can assume that in an (optimal)
consecutive partition the indices in
precede the indices in
Si
Sj
for
i < j
Chakravarty, Orlin and Rothblum [1982] showed that the problem of determining
an optimal consecutive partition reduces to the problem of finding the shortest
path between two nodes in a graph where the number of edges in the path is at
most
m .
0(mn2 )
This problem can be solved in
steps using a standard dynamic
programming recursion.
In the symmetric and separable case, one can determine an optimal semiconsecutive partition as follows.
O < i
of
Then
n .
j
Sij
into
1
ij
f
Let
t
- ih'(S
fi-
=
fij
subsets.
iJ)
for
Let
Sij = {i+l, ..., j}
for all
for each value of
with
be the optimal value of a semi-consecutive partitioning
Finally, let
0 < i
h'(S) - h(as,bS)
for each
he
S c N
j < n.
min
h'(Sik U S ) + ft (S)
Itis
that
easy
tosee
the above recursions can be computed in
It is easy to see that t
i,j
a
recursions
bove
can be
computed in
0(n4)
4
(8)
steps
0(n
) steps
t , and thus the total number of computations is
(mn4
)
.
- 10 -
The NP-Completeness of an Optimal Partition Problem
5.
Barnes and Hoffman [19821 consider the following partition problem.
of
n
m 1 di = N,
ii
for
al, ... ,
integers
partition
i = 1, ..., m
N
into
non-negative integers
m
and
a
m
subsets
S,
...
Sm
Given a set
dl, ..., dm,
where
ISi
= di
i=
such that
and so as to maximize
2
m
I d.
j=l 3
1
(9)
(
ai )
itS.
Barnes and Hoffman demonstrated that there is always an optimal partition consisting of consecutive subsets.
in which
m
Moreover, since they were interested in problems
was quite small (e.g.,
m = 3) , implicit enumeration of all feasible
consecutive partitions was quite practical.
The Barnes-Hoffman partition problem satisfies the conditions of Theorem 3
(except for the fact that they maximize a convex function rather than minimize a
concave function).
Thus the optimality of consecutive partitions is a special case
of Theorem 3.
Although the Barnes-Hoffman problem is polynomially solvable for fixed
m
the corresponding recognition problem is NP-complete if
jointly with
n .
m ,
is allowed to vary
We next prove the NP-completeness via a transformation from
3-partition.
3-PARTITION
INPUT:
non-negative integers
QUESTION:
such that
bl, ..., b3k
Is it possible to partition
such that
{1, ..., 3k}
b
into
k
k
2
for
subsets
i = 1,
_..,
3k
S1 , ..., Sk
-
11 -
3k
bi = (l/k) I bi
itS.
i=l
for
j = 1, ...,k
?
(10)
We consider the version of 3-Partition in which the subsets are not specified
to havw three elements each.
Garey and Johnson [1973] showed that the above
variant of the 3-Partition problem is NP-complete.
Theorem 4.
is NP-complete.
The recognition version of the Barnes-Hoffman partition problem
- 12 -
6.
Proofs
Proof of Theorem 1.
We consider only the case where
from analogous arguments.
m = 2 .
be ignored.
some set
0
as the case where
The case
m = 1
is trivial.
We first remark that the case where
0
follows
and that for such
ai = bi = 0
=
S, aN\
S
aN\SaN
can write the optimal partitioning problem with
and
m = 2
m
of
We next consider the
Noting that each 2-partition may be written as
S c N
a
Our proof follows by induction on the number
subsets in the partitions.
case
b
for some
{S, N \ S}
bN\S
i
can
for
bN
b sS
we
as the following
optimization problem:
min
ScN
g2(S, N\S) = min
ScN
= min
ScN
Since the set of ordered pairs
which will be denoted by
h2(as, bs, aN\S' bN\S)
aN - aS' bN - b
h2 (aS , b,
{(as, b)
(11)
: S c N}
S)
is finite, its convex hull,
C , is a convex polyhedron.
It now follows from
(11)
that
amin
g 2 (S, N\S) = min
ScN
ScN
>
h 2 (as, b,
min
(x,y)2cC
h2 (x, y, a-
h (x, y, aN - x, bN - y)
Since the function
aN - as, b
(12)
-b
x, bN - y)
is concave in the variables
the latter minimization problem attains a minimum at an extreme point of
therefore suffices to show that for each extreme point
(x , y )
exists a set
and
S
c N
such that
a consecutive partition.
(x , y )
(as,
bs*)
of
C
{S , N \ S
(x, y) ,
C .
there
is
It
- 13 -
Let
(x , y )
be an extreme point of
C .
well known that there exists a linear function
minimum over
C
f(x, y) = ax +
at
y
(x , y ) .
where
as the minimizer of
observe that as
ax
The set
S
+
f
a
and
over
C
Since
B
f
on
C
is a polyhedron, it is
that attains its unique
is linear it has a representation
The uniqueness of
y
=
ax + By
b i < 0}
(as, b)
=
min
ScN
f
aa+bS=min
ScN
C
that
(x , y )
-
a = 0
then
{i :b. >
N \ S
}
if
B
O0 , and
B < 0 .
are consecutive.
S
Let
P
=
b
0
*
<
S
}-t
m
3 .
S}
be an optimal m-partition of
max Sj .
ij
We see this as follows.
Observe that
__
and
For
1' 2i} > i
min {M1,
then
/ )
Finally,
S
and
N
S
=
S
and
m = 2
of
N , let
d(S) = max S - min S
that minimizes
We claim that for each
d(S)
j = 1, ..., m ,
Suppose that not all subsets are
S2 , min S 1 < i < max S1 .
3
B > 0
For each subset
for some
=
if
This completes our proof of Theorem 1 when
Then we may determine two sets of
i
.(13)
N \ S
a > 0
In either case we trivially have that both
consecutive.
M.
s/sll
is consecutive.
If
and
then S * {
: ai / bi > and N \ S
are consecutive.
with repsect to all such (optimal) partitions.
Sj
S
0
U{ : i E S} , min S = min {i : i
{S1, ..., S }
b i)
as the minimizer of
assures that both
- {i : b i <
We next consider the case where
max S = max
(aa.+
iES
*
(x , y )
We consider a number of cases separately.
S = {i : a i / b i < - /
and if
In either case (1) assures that both
if
Next
(aS*, bs*)
We next show that the fact that
are consecutive.
0 .
is clearly optimal for the last optimization
Hence, we conclude from the uniqueness of
over
B
(x , y )
: S c N} ,
*
problem.
a
assures that we do not have
is the convex hull of
= {i : aa i +
f
C
are real numbers.
C
min
(x,y)EC
Since
P , say
j = 1, 2,
and
S1
let
and
S2 ,such
mj = min S
max {ml1, m2 } < i .
that
and
Hence,
II
- 14 -
{ml
min {M1 , M2} -max
Let
{S,
S}
be an optimal parition of
consecutive with respect to
(14)
O
m2 }
S1 u S2
S1 u S 2
where
into two subsets which are
S3, ..., S
m = 2
possible by the established result of Theorem 1 when
that
h
m
is concave in the variables corresponding to
remaining variables are fixed).
d(Si) + d(S')
remain fixed (as is
and the observation
S1
1
and
S2
2
when the
Then
< max {M1 , M2 } - min {ml, m2
(15)
- 1
and therefore, using (14), we conclude that
d(S 1) + d(S2) = M 1 - m 1 + M 2
-
(16)
2
= max {M1 , M2 } + min {M1 , M 2} - min {ml, m 2} - max {m1 , m 2}
>
d(S{) + d(S') + 1 .
This contradicts the choice of
L
S1 , ..., Sm , completing our proof.
Proof of Theorem 2.
Our proof follows by induction on the number
of subsets in the partitions.
m = 2 .
We next consider the case
is trivial.
m = 1
The case
m
The argument
used in the proof of Theorem 1 shows that it suffices to prove that for every
pair of real numbers
a
B , the set
and
S
= {i :
bi < 0}
ai +
N , or equivalently, that
perty that {S , N \ S 3 is a semi-consecutive partition of
N \ S
either
S
or
ately.
If
a > O
then
is a consecutive set.
S
=
ai / bi < -
We consider a number of cases separ-
{i = 1, ..., r : a i / bi > -B /
{i = r+l, ..., n : ai / b i < -B / a}
and if
has the pro-
a < 0
/ a} u {i = r+l, ..., n : ai / bi > -B
a
u
then
S
= {i - 1, ..., r
/ a
.
In the former case
- 15 -
if
$ > 0
and
= {i : b
S
8
then
and
= {i : bi < 0
S
..., r}
= {1l,
is not consecutive if
N \ S
{S , N \ S}
and therefore the partition
i, a i > 0 = b i)
is semi-
S
In eitner case we have that
<0 .
if
> 0)
0
X
(though in the latter case
is consecutive
and for some
a = 0
Finally, if
consecutive.
{S , N \ S }
Hence, in either case the partition
is consecutive.
N \ S
is consecutive and in the latter case (1) assures that
S
(1) assures that
r > 1
is semi-
consecutive.
Next assume that the conclusions of Theorem 2 hold whenever the number
of sets in the partitioning problem is less than
partitioning problems where the number of sets is
m
(
m .
3), and consider
We prove the existence
of a semi-consecutive optimal partition for the corresponding partitioning
problems by induction on the number of elements in the partitioned set
The cases where the number of elements in
forward.
less than
is
N
Next assume that each partitioning problem of a set consisting of
n
sets, where the assumptions of Theorem 2 are
m
elements into
We next consider such
= S n {1, ..., r
S
S c N, let
For each subset
where the number of elements in
N
partitioning problems of a set
r
S,
Si,...
..., S
m
S 1 U ... U S+
1
m
N+
all consecutive.
S+
...
,
S+
S },
N
fixed and repartition
such that the sets in the new (optimal) partition of
N
are
The above arguments establish the existence of an optimal
r = 0
for which
r = n
or
S1, ..., S , SS, ..
1m
then the partition
secutive and therefore semi-consecutive.
of this proof that
S+ = S n {r+l, ..., n},
Similarly, by applying an appropriate modification of
{Sl' ..., S I
m
We remark that if
n
fixed and, by applying Theorem 1, repartition
Theorem 1, one can hold the (new) sets
= N
is
such that the sets in the new (optimal) partition of
are all consecutive.
S1 u ... u S
and
N
Given any optimal partition, say {S1,...
is defined through (1).
one can hold
partition
are straight
1, 2 or 3
satisfied, has a semi-consecutive optimal partition.
where
N
1
____P__IIII·_l__l__1__1__1111111____1
r < n.
S.
m
are all consecutive.
S
{S , ...,
1
I
m
is con-
Henceforth, we assume through the end
II
- 16 -
We continue our proof by assuming that the partitioning problem of
m
sets has an optimal partition
> 1
IS
...,
S }
for which the sets
or
> 1
combine all the elements in
and a "b" value
have that if
bT
T
T
Let
T = S
Si
.
>1
One can
into a single element and attach an "a" value
to this combined element.
ai / bbi
a
..., Sm
We consider only the case where
as the alternative case follows from similar arguments.
aT
S,
into
~~mm
1'
with either
{S1,
N
for each
i
E
T
Evidently, as
then
ac
T
N
aT / b T < B
we
It
follows that the corresponding partitioning problem where the elements in
T
are
combined into a single element satisfies (1) and the other assumptions of Theorem
2 .
As the number of elements in the partitioned set of this modified problem is
less than
n , we conclude from the induction assumption that a semi-consecutive
optimal partition exists.
It immediately follows that the original problem too
has a semi-consecutive optimal solution.
It remains to consider the case where any optimal partition
{S1 , ..., S }
m
for which
Si, ...
each of the sets
{S1 , ..., S }
and
S
are all consecutive has the property that
S
..
Si, ...
...
S,
S
,
has at most a single element.
Let
be such an optimal partition (whose existence follows from our
earlier arguments).
r
S , S
Let
S
and S
be the sets in
r+l , respectively (recall that
1
r < n ).
{S1 , ..., S }
which contain
We establish existence
of a semi-consecutive optimal partition by considering two cases.
First assume that either
S
or
S
q
P
is consecutive.
We consider only the
former case as the latter case follows from identical arguments.
By using the
induction assumption concerning partitioning problems of a set into
one can fix
S
and repartition
P
u {St : t = 1, ..., m, t
t
resulting partition of this set, say
{S
: t = 1, ..., m, t
p
m - 1
sets,
such that the
p}
will be semi-
consecutive and will not reduce the value of the objective function.
It easily
- 17 -
follows that, with
partition of
Sp = Sp
1,{l'
..., S}
is a semi-consecutive optimal
N
S
We next consider the case where neither
ISpI
1
P
that as
and
ISp
p
1
1
IS+
P
,
r+l < j
and for some
1
n ,
and
p
argument shows that for some
{St : t = 1, ..., m, t
of Theorem 2 for the case
into sets
Sp
and
S
Now, if either of the sets
element, one can apply
m = 2 ,
where
{i, r, r+l, j
respect to
q}
= {i
|s
p
S+
p
A similar
q .
fixed and apply the (established) conclusions
to repartition
{Sp,
}
S
P
u S
= {i, r, r+l, j}
q
is a semi-consecutive partition with
and where the value of the objective is not reduced.
S
P
,S
q
,
p
,
consists of more than a single
q
a combination of the arguments used earlier to establish
> 1 , one can repartition
number of nonempty sets among
partition is consecutive.
at least two elements.
r+l
We can next hold
.
the existence of a semi-consecutive optimal partition.
example
p
In particular,
i < r , S
1
p, t
P
= {j} .
S
Also observe
is not consecutive, we have that
S
As
is consecutive.
S+ = {r+l} .
q
and
S = {r}
P
we have that
JS+J
q
S
q
nor
p
As
into
= k
k-l
sets where
k
is the
where each of the sets in the new
..., S }
m
{S1 ,
IN-
N-
Specifically, if for
we have that one of these sets contains
By combining the elements in that set and applying the
induction assumption concerning partitioning problems of sets consisting of less
than
n
elements into
optimal partition.
m
sets, we conclude the existence of a semi-consecutive
We continue by assuming that the sets
each
S,
contains at most a single element, in which case each contains precisely one
element.
It follows from the semi-consecutiveness of the partition {S , S }
p
q
with respect to {i, r, r+l, j}
that either
In either case, the set containing
r
S
P
= {r, r+l}. or
(which also contains
S
q
= {r, rl} .
r+l ) is consecutive
and our earlier arguments assure the existence of a semi-consecutive optimal
O
partition.
7"·*"81s-----·-------san-araar-·ll
____1
- 18 -
Proof of Theorem 3:
Chakravarty, Orlin and Rothblum established a special case of Theorem 3
g' (
where
)
Their arguments apply to the general
is symmetric and separable.
D
case considered in Theorem 3 and are omitted.
Proof of Theorem 4:
Let
bl, ..., b3k
be the input for a given 3-Partition problem.
We will
show below how to transform the 3-Partition problem into the Barnes-Hoffman partition problem.
The transformation is based on the following elementary lemma.
Let
Lemma.
al, ..., a
n
and
i=l ai .
be the data for the
m
Then the maximum cost partition has
Barnes-Hoffman partition problem.
value at most
dl, ..., d
Moreover, it is possible to achieve this
bound if and only if it is possible to determine a feasible partition
S1, ...
S
such that
a. = a
1
m
same subset
ai) 2
"(
IS
and equality holds if and only if
a
iS.
j
for all
t
in the
By the Cauchy-Schwarz inequality
iS.
j
.
i,t
S'
Proof of Lemma.
a. = a
for every two indices
t
i,t
S
j
.
12
m
I I-1
.
j=l
Thus:
n
( I
isS.
i)C jsj2I
2
i=l
and equality holds if and only if
a.2
ai = a t
for every two indices
i,t
in the same subset.
To complete the proof of the theorem, let b = k -1
subset sum for each subset of the 3-partition).
and let
di = bii
for
i = 1, ... ,
i~~..
ai+jS
i+j b = j
for
1 < i
3k .
b
and
Let
Finally, let
O0
j
m-l .
3k b )
i=l bi
i=l
(b
bi , let
is the
m
3k
-
Thus there are exactly
b
indices
19
i
-
for which
ai = j
By the previous lemma, we can achieve a value of
n.l a
determine a feasible partition
in the same subset.
{PO, ..., Pk 1 }
0
k-i
bj
a 3-partition of
"a"
bl, ..., b3 k
value is
partition
...,
b1 , ..., b 3 k .
t ,
such that
m
then let
where
{S1, ..., S }
{b,
...
1'
at = i
such that
,
S
S
for
bk
3kt
S
*D
_
= at
t
for all
Then let
such that
Pi
Then
...
{P,
i,t
consists of
1
Pk
is
{P0 , ... , Pk-1 }
consist of
elements whose corresponding
b. E P .
0
t
b.
is a 3-partition
We have thus seen that there is a feasible
such that the
"a"
value for each
Si
is constant if
Thus the Barnes-Hoffman partition
Since the recognition variant is clearly in the class
the latter problem is NP-complete.
--
a
Conversely, if
and only if there is a 3-partition for the b's .
problem is NP-hard.
j = 0, ..., m-1
if only if we can
Suppose we can achieve this latter result.
be a partition of
those elements
of
S1,
for any
NP ,
I
- 20 -
References
Barnes, E.R., and A.J. Hoffman (1982), "Partitioning, spectra, and linear
programming". To appear in the Proceedings of the Silver Jubilee Conference
on Combinatorics, University of Waterloo, Ontario, Canada.
Chakravarty, A.K. (1982a), "Multi-item inventory aggregation into groups",
Journal of Operational Research Scciety (U.K.) 32, 19-26.
Chakravarty, A.K. (1982b), "Consecutiveness rule for inventory grouping with
integer multiple constrained group-review periods", unpublished manuscript.
Chakravarty, A.K., J.B. Orlin and U.G. Rothblum (1982), "A partitioning problem
with additive objective with an application to optimal inventory groupings
for joint replenishment", Operations. Research 30, pp. 1018-1022.
DeGroot, M.H. (1975), Probability and Statistics, Addison-Wesley.
Denardo, E.V., G. Huberman and U.G. Rothblum (1982), "Optimal locations on a line
are interleaved", Operations Research 30, pp. 745-759.
Garey, M.R. and D.S. Johnson (1978), "Strong NP-completeness results:
examples, and implication," S, Assoc. Comput. Mach. 25, 499-508.
motivation,
Goyal, S.K. (1974), "Determination of Optimum packing frequence of items jointly
replenished", Management Science 21, pp. 436-443.
Hwang, F.K. (1981), "Optimal partitions," Journal of ODtimization Theory and
Applications 34, pp. 1-10.
Hwang, F.K., J. Sun and Y. Yao (1982), "Optimal set partitioning," an unpublished
manuscript.
Nocturne, D. (1973), "Economic order frequency for several items jointly replenished", Management Science 19, 1093.
Silver, E.A. (1976), "A simple method of determining order quantities in joint
replenishments under deterministic demands", Management Science 22, 1351.
Wagner, H.M. (1969), Principles of Operations Research, Prentice-Hall.