This model is a variant of the joint replenishment models considered by Goyal [1974], Silver [1976], and Nocturne [1973] who study the. ZBRSParrrranP-^--.------.
A Partitioning Problem with Additive Objective with an Application to Optimal Inventory Groupings for Joint Replenishment A. K. CHAKRAVARTY Washington State University, Pullman, Washington
J. B. ORLIN Massachusetts Institute of Technology, Cambridge,Massachusetts
U. G. ROTHBLUM Yale University, New Haven, Connecticut (Received December 1980; accepted January 1982) We consider a problem of optimal grouping and provide conditions under which an optimal partition of an ordered set S = {r1, -*t, r,) consists of subsets of consecutive elements. We transform the problem into the problem of finding a shortest path on a directed acyclic graph with n + 1 vertices (for which efficient algorithms exist). These results may be used to solve the problem of grouping n items in stock into subgroups with a common order cycle per group so as to minimize the resulting economic order quantity costs.
ET S = {(r,
, r,) be a set of n real numbers ordered so that rl < r2 < ... < r. (The results follow for the case where there are weak ..
inequalities as well. Strict inequalities are considered because they allow the notational convenience of letting the ri's rather than i's be the elements of S.) Associated with each subset T C S is a cost g(T). The set partition problem with an additive objective is to partition S into nonempty subsets S, · , Sm so as to minimize the sum g(S1) + · . + g(Sm), where g(.) is a function mapping the nonempty subsets of S into the reals. A number of variants of this problem, in which the number of sets in the partition is restricted and/or empty sets are allowed, are considered in the extension section. For general functions g(.) this problem is NP-hard as is proven at the end of this note. Subject classification: 331 optimal inventory groupings for joint replenishment, 625 a partitioning problem with additive objective. Operations Research Vol. 30, No. 5, September-October 1982
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0030-364X/82/3005-1018 $01.25 O 1982 Operations Research Society of America
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A subset S' C S is consecutive if the indices of its elements are consecutive integers, e.g., r4 , r5, r6}. In this note we consider a special case of the above problem where the function g(.) is concave in the subset sum for fixed cardinality of the subset, i.e., there exist concave functions f(. ), f2(), ... , f( ) such that
g(T) = fi(ZT rj) when iT = i. We transform this special case into a shortest path problem on a directed acyclic graph with n + 1 vertices. The transformation is based on the following lemma. If g(.) is concave in the subset sum for fixed cardinalityof the subset then there is an optimum set partition in which each subset is consecutive.
LEMMA.
Proof. It suffices to show that the conclusion of the lemma holds for our partition problem amended by the requirement that the partitions consist of precisely m sets, where m is an arbitrary (fixed) positive integer. Our proof follows by induction on m. We first consider the case where m = .2. For any subset Q C S let .. , rk} and rQ = r,,Q ri. Let k E {1, ... , n) be fixed. Let K = r, N= ({r-k+l, - , rn). Then minITI=k g(T) + g(S\T)
=
mrni Tl=k fk(rT) + fn-k(rs minrK_ x
rN
rT)
fk(X) + fn-k(rS
-
X).
The latter minimization problem attains a minimum at an extreme point of the interval [rK, rN] since fk(x) + fn-k(rs - x) is concave in x. Thus the optimal set partition, when the cardinality of SI is restricted to be k, is either T = K or else T = N. The conclusion of our lemma for the case m = 2 now follows directly. We next consider the general case where the number of sets is required to be some fixed integer m Ž 3. For each subset S of f(r, ... , r,}, let h(S) = max(i:ri E S) - min(i:ri E S). Let S1, ... , Sm be an optimal set partitioning such that XEiml h(Si) is minimal with respect to all such (optimal) partitionings. We claim that each Si is consecutive for i = 1, .. , m. We see this as follows. Suppose that not all subsets are consecutive. Then we may determine two subsets, say SI and S2, such that for j = 1, 2, aj = min(i:r E Si) and by = max(i:ri E Si) and al < a2 < bl. Let Si' and S2' be an optimal partitioning of Si U S2 into sets that are consecutive with respect to Si U S2 (this is possible by the argument for the case m = 2). Then h(S1') + h(S2') s max(b2, b) - al- 1, and h(S1 ) + h(S2) = b 2 - a2 + bl - a, implying that h(S 1) + h(S2 ) > h(S1') + h(S2 '). This contradicts our choice of S,
...
, Sm.
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Technical Notes EXTENSIONS
The proof of the lemma shows that the set partition problem amended by the requirement that there are precisely m sets in the partition, where m is an arbitrary (fixed) positive integer, has an optimal solution consisting of consecutive sets. It is easily seen that the conclusion of the lemma also holds when the number of the sets in the partition is required to be in any given (fixed) subset of f1, , n} (e.g., the number of sets has to be at most (or at least) m where m is an arbitrary (fixed) positive integer). Also, the conclusion of the lemma remains unchanged when empty sets are allowed in the partition. A REDUCTION TO A SHORTEST PATH PROBLEM Let G = (V, E) be an acyclic directed graph with vertex set V= {1,. , n + 1). For each i < j, let dij = g(fri, ... , rj-}) be the length of edge (i, j). It is easily seen that a partition of (ri, ... , r,} whose sets are consecutive corresponds to a path in G from vertex 1 to vertex n + 1, and the value of the (additive) objective of the partition equals the length of the corresponding path. Thus, the lemma implies that an optimal partition of (ri, ... , rn} may be determined by finding a minimum length path from vertex 1 to vertex n + 1. Of course, restrictions on the number of sets in the partition correspond to restrictions on the number of edges in the corresponding paths. In particular, when no such restrictions are imposed, the shortest path can be computed in 0(n 2 ) steps (e.g., Dijkstra [1959]). When the number of sets is restricted to be m (or at most m) the shortest path can be computed in O(mn2 ) steps (e.g., Bellman [1958]). In fact, the latter method computes the shortest path for each of the m problems where the number of arcs is restricted to be j, where j = 1, *.., m. Consequently the optimal partition in which empty sets are allowed with or without restrictions can be solved in 0(n3 ) sfeps. AN APPLICATION TO INVENTORY GROUPING Consider an economic order quantity model involving n items, where the ith item has demand rate Di and a unit inventory cost h per time unit. The fixed cost of simultaneously ordering a group of k # 0 items is F(k) > 0 for some real-valued function F(.), independent of the items in the group. The problem is to partition the n items into nonempty subgroups, consisting of one or more items apiece, and choose order cycles for these subgroups so as to minimize the net average cost per unit time, where the cost for each group is assumed independent of the order cycles of the other groups. For a more detailed explanation of and motivation for the model, see Chakravarty [1979]. This model is a variant of the joint replenishment models considered by Goyal [1974], Silver [1976], and Nocturne [1973] who study the
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t
Chakravarty et al.
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problem in which an order time r is Set for the group of n items, and the order cycle for each item is an integer 'multiple of T.They give heuristic solutions to this problem. ... r,. For Let ri = Dih where the items are labeled so that r notational convenience we will identify a subset T of (1, *.., n} with ({rl E T). As in the ordinary EOQ model (e.g., Wagner [1969], pp. 1819) if item i has order cycle t, then its economic order quantity is tD1 and the average net cost per unit time of a group of items T having the same order cycle t is: c(T, t) = t-(F(f TI) + (t/2)(eT Dihi) = t- 1F(ITI) + (t/2)rT. So, for a fixed subset T, the minimum average cost per unit time is:
j
g(T) =mint>o c(T, t) = (2F(j TI)rT)1/ 2 (obtained by setting the order cycle at (2F(j T )rT1 )1/ 2). When ITJ = i, we have that g(T) = fi(rT) where fi(x) = (2F(i)x) 1/2 is a concave function of x. The problem is to find a partition of the items into nonempty subsets SI, ... , Sm, so as to minimize Z=l g(Si) where g(.) is given above. Our above analysis shows that there is an optimal partition consisting of consecutive subsets and the problem can be reduced to a shortest path problem, which can be solved in 0(n 2) steps. Of course, the extensions of our general model can be used when there are restrictions on the number of groups or when there are costs involved for empty orders. COMPUTATIONAL COMPLEXITY As the following example shows, without the concavity assumption, the problem is NP-hard. (Our example uses a convex function g(. ).) Consider the case in which S = {rl, ...
, r3 ),
o g(T) = {-lr
+0
if rT= n-rs
otherwise.
In this case there is a partition with objective value 0 if and only if there is a partition of S into n subsets each with sum n-1rs. This is the 3partition problem proved NP-hard in the strong sense in Garey and Johnson [1978]. So, the partition problem with arbitrary function g(.) with or without restriction on the number of sets in the partition (as long as these restrictions allow n sets in the partition) is NP-hard. ACKNOWLEDGMENT We thank the anonymous referees for their helpful suggestions.
J
REFERENCES BELLMAN, R. E. 1958. CHAKRAVARTY, A. K.
On a Routing Problem. Quart. Appl. Math. 16, 87-90. 1979. Optimum Grouping of Inventory Items, USM Tech-
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Technical Notes
nical Report No. 2/79, University of Science of Malaysia, Penang, Malaysia. DIJKSTRA, E. 1959. A Note on Two Problems in Connection with Graphs. Numerische Mathematik 1, 269-271. GAREY, M. R., AND D. S. JOHNSON. 1978. Strong NP-Completeness Results: Motivation, Examples, and Implications, J. Assoc. Comput. Mach. 25, 499-508. GOYAL, S. K. 1974. Determination of Optimum Packaging Frequency of Items Jointly Replenished. Mgmt. Sci. 21, 436-443. NOCTURNE, D. J. 1973. Economic Order Frequency for Several Items Jointly Replenished. Mgmt. Sci. 19, 1093-1096. SILVER, E. A. 1976. A Simple Method of Determining Order Quantities in Joint Replenishment under Deterministic Demands. Mgmt. Sci. 22, 1351-1361. WAGNER, H. M. 1969. Principlesof Operations Research. Prentice Hall.
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