The previous models established optimal inventory policies for single item models.
It is simple show thut without the presence of joint order costs, a problem with
several items each facing a constant demand can be handled by solving each item's
replenishment problem separately. In reality, management, of a single warehouse
inventory system involves coordinating inventory orders to minimize cost without
exceeding the warehouse capacity. The warehouse capacity limit* the total volume
held by the warehouse at any point in time. This constraint ties together the dif-
ferent items and necessitates careful coordination (or scheduling) of the orders.
That is, it is not only important to know bow often an itern is ordered, but ex-
actly the point in time at which each order takes place, This problem iB called the
Economic Warehouse Lot Scheduling Problem (EWLSP). The scheduling part,
hereafter called the Staggering problem, is exactly the problem of time-phasing
the placement of the orders to satisfy the warehouse capacity constraint. Unfor-
tunately, this problem has no easy solution and consequently it has attracted a
considerable amount, of attention in the last three decades.
The earliest known reference to the problem appears in Churchman et al. (1957)
and subsequently in Holt (1958) and Hadley and Whitin (1963). authors
were concerned with determining lot sizes that made an overall schedule satisfy
the capacity constraint, and not with the possibility of phasing the orders to avoid
holding the maximum volume of each item at t,he same time. Thus, they only con-
sidered what are called Independent Solutions, wherein every item is replenished
without any regard for coordination with other items,
Several authors considered another clas of policies called Rotation Cycle policies
wherein all items share the same order interval. Homer (1966) showed how to
optimally time-phase (stagger) the orders to satisfy the warehouse constraint for a
given common order interval. Page and Paul (1976), Zoller (1977) and Hall (1988)
independently rediscovered Homer's result. At the end of his paper devoted to
Rotation Cycle policies, Zoller indicates the possibility of partitioning the items
into disjoint subsets, or clusters, if the assumption of a Rotation Policy "proves
to be too restrictive." This is precisely Page and Paul's partitioning heuristic.
In their heuristic, all the items in a cluster share a common order interval. The
orders arc then optimally staggered within each cluster, but no attempt is made to
time-phase the orders of different clusters. Goyal (1978) argued that such a time-
phasing across the different clusters may lead to further reduction in warehouse
space requirements. Hartley and Thomas (1982) and Thomas and Hartley (1983)
considered the two-item case in detail.
Recently a number of studies have been concerned with the strutegic version
of the EWLSP in which the warehouse capacity is not, a constraint but rather a
decision variable. These include Hodgson and Howe (1982), Park and Yun (1985),
Hall (1988), Rosenblatt, and Rothblum (1990) and Anily (1991). In this model,
t,he inventory carrying cost; consists of two parts; one part is proportional to the
6. Economic Int Size Models with Constant Demands
92
average inventory while the second part is proportional to the peak inventory. A component of the latter cost, discussed in Silver and Peterson (1985), is the cost of leasing the storage space. This cost is typically proportional to the size of the
warehouse, and not to the invent,ories actually stored in it.
Define a policy to be a Stationary Order Size policy if all replenishments of an item are of the sarne size. Likewise, a Stationary Order Intervals policy has all orders for an item equally spaced in time. It is easily verified that an optimal Stationary Order Size (respectively, Stationary Order Interval) policy is also a Stationary Order Interval (respectively: a Stationary Order Size) policy if every order of an item is received precisely when the inventory of that item drops to zero; that is, it also satisfies the Inventory Ordering property. Thus, it is natural to consider policies that have all three properties: Stationary Order Size, Stationary
Order Interval and Zero Inventory Ordering. We call such policicxs Stationary Oyrler• Sizes and Intervals policies, in short, SOSI policies. Two "extreme" cases of SOSI policies are the Independent Solutions and the Rotation Cycle policies defined above. All the authors cited above considered SOSI policie; exclusively. Zoller claims that SOSI policies are the only rational and most authors agree that SOSI policies are much easier to implement in practice. In his Ph.D. thesis, however, Hariga (1988) investigated both time-variant and stationary order sizes.
He was motivated to study time-variant order sizes by their successful application in resolving the feasibility issue in the Economic Lot Scheduling Problem (ELSP)
(see Dobson (1987)).
The paper by Anily departs from earlier work on the EWLSP in its focus on worst-case performance of heuristics, In her paper, Anily restricts herself to the class of SOSI policies for the strategic model. She proves lower bounds on the minimum required warehouse size and on the total cost for this class of policies. She presents a partitioning heuristic of which the best, Independent Solution and the best Rotation Cycle policies are special cases. This partitioning heuristic is similar to the one proposed by Page and Paul for the tactical model, although the precise methods for finding the partition are different. Anily proves that the ratio of the cost of the best Independent Solution to her lower bound is at most She also provides a data-dependent bound for the best Rotation Cycle, derived from Jones and Inman's (1989) work on the Economic Int Size Problem. As a result, her partitioning heuristic is at least as good as either special case, and thus has a
worst case bound of relative to SOSI policies.
In this section we determine easily computable lower bounds on the cost of the EWLSP as well as some simple heuristics for the problem. These bounds are used to determine the worst-case performance of these heuristics on different versions of the problem. First, in Section 6.22, we introduce notation, state assumptions and formally define the strategic and tactical versions of the EWLSP. In Section 6.2.3, we establish the worst-case results. The discussion in this section is based on the work of Gallego et al. (1996).