Impact on inventory costs with consolidation of distribution centers.

Received March 1999 and accepted February 2000

The consolidation of Distribution Centers (DCs) is a new trend in global logistics management, with a reduction in inventory costs often being cited as one of the main benefits. This paper uses an analytical modeling approach to study the

impact on facility investment and inventory costs when several DCs are consolidated into a central DC. In particular, our model suggests that consolidation leads to lower total facility investment and inventory costs if the demands are identically and independently distributed, or when they follow independent but possibly nonidentical Poisson processes. This agrees with the conclusion of the classical EOQ and newsvendor models. However, we show by an example that, for general stochastic demand processes, the total facility investment and inventory costs of a consolidated system can be infinitely worse off than that of a decentralized system. This arises mainly because the order replenishment fixed cost yields a cost component proportional to the square root of the mean value of the demand, while the demand uncertainty yields a cost component proportional to the standard deviation of the demand. Whether consolidation is cost effective or not depends on the trade-off between these two components, as indicated by an extensive numerical study. We also propose an algorithm that solves for a distribution system with the total facility investment and inventory costs within [square root]2 of the optimal.

1. Introduction and overview

As quoted by Rheem (1997), "Consolidation and centralization are proving to be an excellent move for companies that know how to manage logistics effectively". There are already many case reports on companies streamlining their distribution networks by consolidating and centralizing their distribution operations. For instance, Staples Inc., one of the biggest office products retailers in the US, has recently developed a strategy to centralize and upgrade its US distribution network (Gourley, 1997). Also, Disney Stores Inc. uses a dedicated Central DC (CDC) in Memphis to supply more than 100 000 different type of products to 360 stores across the US, Canada and Puerto Rico (Jedd, 1996). Some companies go even further to centralize their worldwide distribution network, such as Benetton, the Italian sportswear company. Benetton uses one CDC in Ponzano, Italy to serve over 6000 stores in 83 countries around the world (Dapiran, 1992). Poirier and Reiter (1996) also report that one major international computer parts company started off with five DCs in England, Maine, California, Japan and Hong Kong, but has since restructured its global logistics process and established Hong Kong as the single global DC.

The following reasons have often been cited for adopting the consolidation strategy:

* Reduced facility investment costs. A large CDC is more cost efficient to build and operate compared to having many smaller regional centers.

* Increased service quality. Centralized inventory ensures better quality control and visibility of stocks within the system. Also at a modern CDC, more value-added services can be provided at lower cost.

* Lower total inventory costs. By pooling demands together, the required amount of inventory to serve them is reduced, and benefits from increased economies of scale in purchasing and transportation operations can be achieved.

This paper uses an analytical modeling approach to quantify the effects of consolidation. In particular, we study the impact on the facility investment and inventory costs when several Regional Distribution Centers (RDCs) are consolidated into one CDC. We focus on the common intuition that consolidation reduces total inventory costs due to the risk pooling effect. The model in this paper is also partially motivated by a common question faced by logistics managers in South-East Asia, who often have to rationalize the set-up of their distribution system structure, especially after some acquisition or merger exercises. The common question posed is: "Is there a need to maintain more than one DC in this region to support regional demand?" The problem is complicated by the fact that the existing DCs may not have identical inventory-related cost structures, due to the choice of different technologies and equipment types. On the other hand, the problem is simplified by the fact that customers from different location s in the region affected by the consolidation exercise are indifferent about where their orders are shipped from, as replenishment lead-times from different potential sites of DCs are more or less identical (since they are all based in South-East Asia, with customers based mainly in North Asia).

To formalize our model, consider a firm with one production plant producing a single product for a collection of aggregated demand points with random demands. Without any loss of generality, we assume that the firm has selected a set of possible locations to set up RDCs to serve these demands. The main problem the firm faces is to choose the RDC locations so that the total facility investment and inventory costs will be minimized. We assume that the customers served by different DCs are indifferent about where their orders are shipped from, and also that the outbound transportation costs will not change drastically depending on where the orders are shipped from. The main reason for doing so is that we want to isolate the effect of DC ordering cost structure and replenishment lead-times on the cost effectiveness of the consolidation strategy. When the DCs to be consolidated are nearby (as in the Singapore context), these assumptions are valid. Our model also applies to the problem of vendor/supplier selection . The consolidation issue studied here can be translated to the issue of consolidating supplier bases (i.e., whether to serve the customers by allocating their demand to one or more suppliers). Since a different supplier gives rise to a different lead-time (depending on the location of suppliers) and ordering cost structure (e.g., EDI-linked or not etc.), our model essentially addresses the question as to whether it is beneficial to divide the demand into one or more groups, to be served by different suppliers. A surprising conclusion from this study is that it is sometimes cost-effective to separate the material flow into different groups, to exploit the cost economies offered by different suppliers.

In most previous work on inventory control, the demand-supply system is assumed given and the focus is on the determination of the optimal inventory control policy. On the other hand, in the literature on facility location problems, the inventory control aspect is seldom considered. In our problem, however, the facility location decisions and the demand assignments must take into account the impact on inventory costs at the DC. We call this the location-inventory problem.

Our first conclusion is that when the demands are identically and independently distributed (i.i.d.), consolidation is an optimal strategy for facility and inventory costs minimization. As a corollary, this implies that consolidation is optimal if the demands follow independent but possibly nonidentical Poisson processes. This agrees with the conclusion of the classical EOQ and newsvendor models. However, when the demands are some general stochastic processes, the consolidation strategy may not always lead to a reduction in the total facility investment and inventory costs. In this case, we will see that the ordering fixed cost gives rise to a cost component proportional to the square root of the mean value of demand and the demand uncertainty yields a cost component proportional to the standard deviation of the demand. Whether consolidation is cost effective or not depends on the trade-off of these two components, as indicated by an extensive numerical study we conducted in the work.

The key to the solution of the problem lies in the evaluation of the optimal cost for any given distribution system structure. For that, we will obtain tight bounds on the optimal cost, which are extensions of the bounds obtained by Gallego (1998). While Gallego takes the ordering cost as a constant, we will instead allow it to be a piece-wise linear concave increasing function of the ordering size. The bounds can be explicitly evaluated for demands of normal distributions. As noted in Tijms and Groenevelt (1984), "for most items in inventory control the demand patterns are relatively smooth so that one can in general safely use two-moment approximations based on normal demand densities". Under the normal demand assumption, and with the help of the bounds, we first construct a worst case example in which the consolidation strategy is infinitely worse off than the optimal distribution strategy (which uses more than one DC), and then propose an algorithm that solves for a distribution strategy with the total i nventory costs within [square root]2 of the optimal. The algorithm is based on the work presented in Chakravarty et al. (1985) and runs in O([n.sup.2]m) time.

The rest of the paper is organized as follows. The literature review in Section 2 shows that our work falls in the intersection of the literature on facility location and the literature on stochastic inventory control problems. The general formulation of the model is presented in Section 3. In Section 4, we consider the i.i.d. demand case, and in Section 5, the general demand case. The numerical experiment is presented in Section 6. We conclude the paper in Section 7.

2. Literature review

Typically the literature on inventory control assumes a given demand distribution, and seeks the optimal inventory control policies at the DC (see Graves et al. (1993) for a thorough review of this field, for the single-echelon and multi-echelon inventory systems). There are several recent studies that integrate inventory decisions with other supply chain activities. For instance, Federgruen and Zipkin (1984a), Anily and Federgruen (1991), and Viswanathan and Mathur (1997) integrate transportation routing with inventory decisions. These works study the single-day inventory-routing problem and capitalize on many results from the vehicle-routing literature. For the integration of production with inventory, where multiple components are required to produce one part, the interested reader is referred to Lee and Billington (1986), Cohen and Lee (1988), and Ettl et al. (1996) for more details. These models pay close attention to capturing the interdependence of base-stock levels at different stores and how this int erdependence affects the overall system performance.

In the literature on facility location problems, a standard formulation includes fixed facility setup costs and transportation costs (Geoffrion and Power, 1995; Revelle and Laporte, 1996), and usually the demands are static and deterministic. The focus is mainly on the transportation costs of the system, while the inventory costs at the DC are often ignored or simplified to be independent of the system structure. An exception is in Barahona and Jensen (1996) who studied a version of the distribution network design problem for computer spare parts. Their model finds the location of warehouses and the customer assignment, minimizing the total cost of building the warehouses and maintaining the inventories at the various locations. Their focus, however, is mainly algorithmic, and very restrictive assumptions are imposed on the inventory costs to make the model tractable.

The work in this paper can be viewed as a facility location model that captures the impact of the inventory related costs with more sophistication, and with stochastic demands at the customer locations. Also, instead of being purely algorithmic, we derive structural insights into the cost effectiveness of consolidation. Our problem is complicated by the fact that closed form expressions for the optimal inventory costs, conditioned on the demand assignment, are not available for most demand distributions. Our analysis in this paper is only made possible by several recent advances in the area of stochastic inventory theory, due mainly to the seminal works of Zheng (1992) and Gallego (1998). These works derived structural insights and bounds for the optimal inventory cost functions which we will utilize later in this paper.

Warehouse consolidation has also been studied in the context of risk-pooling of inventories. Eppen (1979) provides the seminal piece on the potential benefits of risk-pooling in this setting. He demonstrated that the expected cost of the decentralized system grows linearly with the number of DC (m), whereas the expected cost of the centralized system grows with [square root]m. Thus, we have the so-called "risk-pooling incentive" for centralizing the inventories. Eppen assumed that the ordering, holding and penalty-cost structures are identical across all possible DC locations. When the cost structures are not identical, he indicated that it might not be cost effective to centralize demand. Our counter example in Section 5 validates this conjecture. Several other researchers have also looked at the risk-pooling issue from different perspectives. We refer the readers to Eppen and Schrage (1981), Federgruen and Zipkin (1984b) and Schwarz (1989) for a detailed discussion. Most of the research so far, however, as sume a given network structure and seeks only to understand the impact of risk-pooling in the inventory context.

3. Model formulation

In the location-inventory problem, there are a total of n demand points, indexed by j = 1,2,..., n. We assume that the demand for the product at demand point j can be modeled by a continuous stochastic process {[d.sub.j](t): t [greater than or equal] 0}, with mean rate [[micro].sub.j], per unit time and variance [[[sigma].sup.2]sub.j]. The processes {[d.sub.j](t): t [greater than or equal] 0, j = 1,2,...,n} are independent. We will only consider distribution strategies that serve the demands at each demand point exclusively from a single DC (i.e., splitting of demand is not allowed).

The firm has selected in possible DC locations, indexed by i = 1,2,...,m. If a DC is set up at location i, a fixed setup cost [f.sub.i] is incurred, and the lead-time to supply from the plant to this location is [L.sub.i]. For the purpose of this model, we assume that [f.sub.i] can be amortized over the time period of consideration i.e., [f.sub.i] is a per unit time "charge" (to pay back a loan over a very long horizon). On this basis, [f.sub.i] can also be treated in the model as the direct variable cost involved in running a DC and this variable cost is DC-size dependent. The main results in this paper remain valid for the case when [f.sub.i] is a concave function of the amount of demand assigned to DC i. For ease of exposition, we will assume that [f.sub.i] is a constant. We further assume that the plant has unlimited production capacity. The inventory costs at DC i include an ordering cost [k.sub.i](Q) per order, Q being the ordering size, and proportional inventory holding (resp. penalty) costs accumula ting at a constant rate [h.sub.i] (resp. [p.sub.i]) per unit stock (resp. backorder) per unit time. The ordering cost [k.sub.i](Q) is modeled as a piece-wise linear concave increasing function of the ordering size Q to account for the economies of scale in purchasing and transportation operations.

We now define

[y.sub.i] = {1 if a DC is to be set-up at location i,

{0 otherwise,

and

[x.sub.ij] = {1 if demand point j is assigned to the DC setup at location i,

{0 otherwise.

Let Y = {[y.sub.i] : i = 1,...,m} and X = {[x.sub.ij] : I = 1,...,m, j = 1,...,n}. Thus, the aggregate demand process assigned to DC I is [[[sigma].sup.n].sub.j=1] [x.sub.ij][d.sub.j](t). Next, we denote the total demand during lead-time [L.sub.i] by [D.sub.i](X), which depends on the demand assignement X. Under mild assumptions on [d.sub.j](t), j = 1,...,n, the optimal long run average inventory cost can be shown to be given by (Zheng, 1992; Gallego, 1998)

[C.sub.i]([r.sub.i](X), [Q.sub.i](X),X) = [min.sub.r,Q] [k.sub.i](Q)[[[sigma].sup.n].sub.j=1] [x.sub.ij][[micro].sub.j] + [[[integral].sup.r+Q].sub.r] [G.sub.i](y, X) dy/Q, (1)

where [r.sub.i](X) is the optimal reoder point, [Q.sub.i](X) the optimal ordering size, and

[G.sub.i](y, X) = E([h.sub.i][(y - [D.sub.i](X)).sup.+] + [p.sub.i][([D.sub.i](X) - y).sup.+]). (2)

The term [k.sub.i](Q) [[[sigma].sub.n].sub.j=1] [x.sub.ij][[micro].sub.j]/Q reflects the average ordering cost when the average demand is [[[sigma].sup.n].sub.j=1][x.sub.ij][[micro].sub.j] (which depends on the demand assignment decision). The term

([[[integral].sup.r+Q].sub.r] [G.sub.i](y, X)dy)/Q,

reflects the average holding and stockout cost of the system, as the inventory level of the system can be shown to be uniformly distributed between r and r + Q (Zheng, 1992).

The location-inventory problem studied in this paper can thus be formulated as follows:

Problem (P) min [[sigma].sub.i][f.sub.i][y.sub.i] + [[sigma].sub.i][C.sub.i]([r.sub.i](X), [Q.sub.i](X),X),

subject to

[[sigma].sub.i][x.sub.ij] = 1 for j = 1,...,n,

[x.sub.ij] [less than or equal to] [y.sub.i] for j = 1,...n,

[x.sub.ij] [greater than or equal to] 0, [y.sub.i] [epsilon] {0, 1} for all i,j.

4. Indentical and independent demand processes

In this section, we consider the special case of i.i.d. demand processes. To simplify the notation, we assume [[micro].sub.j] = 1, for j = 1,...,n. Since demand processes are identically distributed, the lead-time demand [D.sub.i](X) at DC I depends only on the number of demand points assigned to it. If the latter is [n.sub.i], we use [D.sub.i]([n.sub.i]) to denote the lead-time demand and also write

[G.sub.i](y, X) [equivalent] [G.sub.i](y, [n.sub.i]) = E([h.sub.i][(y - [D.sub.i]([n.sub.i])).sup.+] + [p.sub.i][(D.sub.i]([n.sub.i]) - y).sup.+]).

We now define

[r.sub.i](Q, [n.sub.i]) = arg [min.sub.r] [[[integral].sup.r+Q].sub.r] [G.sub.i](y, [n.sub.i])dy for Q [greater than] 0.

Then [r.sub.i](Q, [n.sub.i]) corresponds to the optimal reorder point in the stochastic inventory problem when the order quantity is fixed at Q. According to Zheng (1992), we have

[G.sub.i]([r.sub.i](Q, [n.sub.i]),[n.sub.i]) = [G.sub.i]([r.sub.i](Q, [n.sub.i]) + Q, [n.sub.i]),

[H.sub.i](Q, [n.sub.i]) [equivalent] {[min.sub.y] [G.sub.i](y, [n.sub.i]), if Q = 0, is convex in Q,

{G([r.sub.i](Q, [n.sub.i]), [n.sub.i]), otherwise,

and

[S.sub.i](Q, [n.sub.i]) [equivalent] [[[integral].sup.[r.sub.i](Q, [n.sub.i])+Q].sub.[r.sub.i](Q, [n.sub.i])] [G.sub.i](y, [n.sub.i])dy = [[[integral].sup.Q].sub.0] [H.sub.i](y, [n.sub.i])dy

is also convex in Q. Hence,

[S.sub.i]([alpha]Q, [n.sub.i]) [less than or equal to] [alpha][S.sub.i](Q, [n.sub.i]), for [alpha] [less than or equal to] 1. (3)

As the proofs to the above results are quite intricats, we refer the readers to the original paper by Zheng (1992) for their derivations. Our first result shows that consolidation is optimal in this case:

Theorem 1. The consolidation strategy is optimal for the location-inventory problem when the demand processes are i.i.d..

Proof. Let (X = {[x.sub.ij]}, Y = {[y.sub.i]}) be an optimal solution to Problem (P), then [n.sub.i] = [[sigma].sub.j] [[micro].sub.j][x.sub.ij] = [[sigma].sub.j][x.sub.ij] and the optimal value of Problem (P) is given by

[[[sigma].sup.m].sub.i=1]([chi]([n.sub.i] [greater than] 0)[f.sub.i] + [min.sub.r,Q]([k.sub.i](Q)[n.sub.i] + [[[integral].sup.r+Q].sub.r] [G.sub.i](y, [n.sub.i])dy)/Q). (4)

Now if all n demand points are consolidated to DC i, the total cost would be

[f.sub.i] + [min.sub.r,Q]([k.sub.i](Q)n + [[[integral].sup.r+Q].sub.r] [G.sub.i](y, n)dy)/Q.

We consider the weighted average of the above values with weight for i as [n.sub.i]/n:

A [equivalent] [[[sigma].sup.m].sub.i=1] [n.sub.i]/n([f.sub.i] + [min.sub.r,Q] ([k.sub.i](Q)n + [[[integral].sup.r+Q].sub.r] [G.sub.i](y, n)dy)/Q),

[less than or equal to] [[[sigma].sup.m].sub.i=1]([chi]([n.sub.i] [greater than] 0)[f.sub.i]

+ [min.sub.r,Q] ([k.sub.i](Q)[n.sub.i] + [[[integral].sup.r+Q].sub.r] [n.sub.i]/n [G.sub.i](y, n)dy)/Q).

Note that if all n demand points are consilidated to DC i, the total demand [D.sub.i](n) = [d.sub.1] + [d.sub.2] + ... + [d.sub.n] is the sum of n i.i.d. random variables. Then [n.sub.i][D.sub.i](n) has [n.sub.i] copies of [d.sub.1] + [d.sub.2] + ... + [d.sub.n], which can be rearranged and written as a sum of n groups of [n.sub.i] i.i.d. random variables. For example, the first group is [d.sub.1] + [d.sub.2] + ... + [d.sub.n], the Second group can be [d.sub.[n.sub.i]+1] + [d.sub.[n.sub.i]+2] + ... + [d.sub.[2n.sub.i]], etc. As each group the same distribution as D([n.sub.i])., we denote them by [[([[D.sup.k].sub.i]([n.sub.i])).sup.n].sub.k=1]. Therefore,

[n.sub.i]/n[G.sub.i](y,n) = [n.sub.i]/n E[([h.sub.i](y - [D.sub.i](n)).sup.+] + [p.sub.i][([D.sub.i](n) - y).sup.+]),

=1/n E([h.sub.i][([n.sub.i]y - [n.sub.i][D.sub.i](n)).sup.+] + [p.sub.i][([n.sub.i][D.sub.i](n) - [n.sub.i]y).sup.+]),

=1/n E([h.sub.i] ([[[sigma].sup.n].sub.k=1][([n.sub.i]/n y - [[D.sup.k].sub.i]([n.sub.i]))).sup.+]

+ [p.sub.i] [([[[sigma].sup.n].sub.k=1]([[D.sup.k].sub.i]([n.sub.i]) - [n.sub.i]/n y)).sup.+]),

[less than or equal to] 1/n [[[sigma].sup.n].sub.k=1] E([h.sub.i][([n.sub.i]/n y - [[D.sup.k].sub.i]([n.sub.i])).sup.+]

+ [p.sub.i] [([[D.sup.k].sub.i]([n.sub.i]) - [n.sub.i]/n y).sup.+]),

= E [([h.sub.i] ([n.sub.i]/n y - [D.sub.i]([n.sub.i])).sup.+] + [p.sub.i][([D.sub.i]([n.sub.i]) - [n.sub.i]/n y ).sup.+]),

=[G.sub.i] ([n.sub.i]/n y, [n.sub.i]).

The inequality follows from the fact that [(x + y).sup.+] [less than or equal to] [(x).sup.+] + [(y).sup.+]. Thus,

A [less than or equal to] [[[sigma].sup.m].sub.i=1] ([chi]([n.sub.i] [greater than] 0)[f.sub.i]

+ [min.sub.r,Q] ([k.sub.i](Q)[n.sub.i] + [[[integral].sup.r+Q].sub.r] [G.sub.i]([n.sub.i]/n y, [n.sub.i])dy)/Q),

= [[[sigma].sup.m].sub.i=1] ([chi]([n.sub.i] [greater than] 0)[f.sub.i]

+ [min.sub.r,Q] ([k.sub.i](Q)[n.sub.i] + n/[n.sub.i] [[[integral].sup.[n.sub.i](r+Q)/n].sub.[n.sub.i]r/n] [G.sub.i](y,[n.sub.i])dy)/Q),

= [[[sigma].sup.m].sub.i=1] ([chi]([n.sub.i] [greater than]0)[f.sub.i]

+ [min.sub.r,Q] ([k.sub.i]=(Q)[n.sub.i] + n/[n.sub.i] [[[integral].sup.r+[n.sub.i]Q/n].sub.r] [G.sub.i](y,[n.sub.i])dy)/Q),

[less than or equal to] [[[sigma].sup.m].sub.i=1] ([chi]([n.sub.i] [greater than] 0)[f.sub.i]

+ [min.sub.r,Q] ([k.sub.i](Q)[n.sub.i] + [[[integral].sup.r+Q].sub.r] [G.sub.i](y,[n.sub.i])dy)/Q).

The last inequality follows from (3). As stated in (4), the right-hand side is the optimal value corresponding to solution (X = {[x.sub.ij]}, Y = {[y.sub.i]}). Thus, we conclude that consolidating to one of the m DCs will give rise to the optimal cost.

Remark. The above result is a little surprising, since in general [S.sub.i](Q, [n.sub.i])is not concave in [n.sub.i].

In reality, demand distributions are usually non-identical. However, when the demand are Poisson processes of the same mean rate, Theorem 1 would apply. If the rates are different, we can split the processed into i.i.d. sums of a 'basic unit' is found, a continuous approximation scheme can then be applied. Thus, it can be seen that Theorem 1 would apply if the demands are independent Poisson processed. Notice that if demands are generated from many independent individual points, the Poisson process is a good approximation to the demands. Thus, Theorem 1 has important ramifications in the design of the optimal distribution strategy.

5. General independent demand processes

For general demand processes, the optimal inventory cost function is very complex. We first seek to bound the optimal inventory cost with more manageable functions. With the help of these bounds, we present an example to show that consolidation can be infinitely worse off than the optimal decentralized system. We will also propose an algorithm that solves for a distribution strategy with the total inventory costs within [square root]2 of the optimal.

5.1. Bounds for optimal inventory cost

Consider a single location, single product, continuous review stochastic inventory system in which all stockouts are backordered. The demand is stochastic with mean rate [micro], and the lead-time demand is D. Replenishments need a positive lead-time L for delivery and incur a fixed cost of k(Q), which is a piece-wise linear concave increasing function of the ordering size Q reflecting the economies of scale in the purchasing and transportation costs. The other costs include proportional inventory holding (penalty) costs accumulating at a constant h(p) per unit stock (backorder) per unit time.

Define

G(y) = E[h[(y - D).sup.+] + p[(D - y).sup.+]],

and denote [y.sub.0] = [min.sub.y] G(y). Let [C.sup.d] be the solution value of the minimum inventory cost when the random demand is approximated by its mean value. Assuming k(Q) = K to be a constant independent of Q, Gallego (1998) (improving a result of Zheng) obtains lower and upper bounds for

[C.sup.*] [equivalent] [min.sub.r,Q] C(r,Q) [equivalent] [min.sub.r,Q] ((k(Q)[micro] + [[[integral].sup.r+Q].sub.r] G(y)dy)/Q).

We use the following theorem from Zheng (1992) and Gallego (1998) [C.sup.d] + G([y.sub.0]) [greater than or equal to] [C.sup.*] [greater then or equal to] [square root] [([C.sup.d]).sup.2] + G[([y.sub.0]).sup.2]. (5)

G([y.sub.0]) is the buffer cost. The bounds are tight when the demand is deterministic. We now extend Gallego's bounds to the case where k(Q) is a piece-wise linear concave increasing function.

Theorem 2. If k(Q) is a piece-wise linear concave increasing function, then

[min.sub.Q] ([micro]k(Q)/Q + hp/h + p Q/2) + G([y.sub.0]) [greater than or equal to] [C.sup.*]

[greater than or equal to] [min.sub.Q] ([micro]k(Q)/Q + hp/h + p Q/2 + G[([y.sub.0]).sup.2]/2Q h + p/hp), (6)

which implies that both the lower and upper bounds are within [square root]2 of the optimal.

The proof we used below is based on a piece-wise linear convex approximation to the optimal inventory cost function. The bound in Gallego (1998) is derived using a different argument.

Proof. Let [a.sub.1] = E[D\D [less than or equal to] [y.sub.0]] and [a.sub.2] = E[D\D [greater than] [y.sub.0]], and define a two-point distribution [D.sub.e] as follows:

P([D.sub.e] = [a.sub.1]) = P(D [less than or equal to] [y.sub.0]) = p/h + p,

P([D.sub.e] = [a.sub.2]) = P (D [greater than] [y.sub.0]) = h/h + p.

[D.sub.e] is related to D through

E([D.sub.e]) = [a.sub.1] p/h + p + [a.sub.2] h/h + p,

= E[D\D [less than or equal to] [y.sub.0]]P(D [less than or equal to] [y.sub.0]) + E[D\D [greater than] [y.sub.0]]P(D [greater than] [y.sub.0]),

= E(D) = [micro]L.

By Jensen's inequality, we have

G(y) = E[h[(y - D).sup.+] + p[(D - y).sup.+]],

= E[h[(y - D).sup.+] + p[(D - y).sup.+]\D [less then or equal to][y.sub.0]]P(D [less then or equal to] [y.sub.0])

+ E[h[(y - D).sup.+] + p[(D - y).sup.+]\D [greater than] [y.sub.0]]P(D [greater than] [y.sub.0]),

[greater than or equal to] [h[(y - E[D\D [less than or equal to] [y.sup.0]]).sup.+]

+ p[(E[D\D [less than or equal to] [y.sub.0]] - y).sup.+]]P(D [less than or equal to] [y.sub.0])

+ [h[(y - E[D\D [greater than] [y.sub.0]]).sup.+]

+ p[(E[D\D [greater than] [y.sub.0]] - y).sup.+]P(D [greater than] [y.sub.0]),

= [h[(y - [a.sub.1]).sup.+] + p[([a.sub.1] - y).sup.+]]P(D [less than or equal to] [y.sub.0])

+ [h[(y - [a.sub.2]).sup.+] + p[(a.sub.2] - y).sup.+]]P(D [greater than] [y.sub.0]),

= E[h[(y - [D.sub.e]).sup.+] + p[([D.sub.e] - y).sup.+]].

Let

[G.sub.e](y) [equivalent] E[h[(y - [D.sub.e]).sup.+] + p[([D.sub.e] - y).sup.+]],

= [h[(y - [a.sub.1]).sup.+] + p[([a.sub.1] - y).sup.+]]P(D [less than or equal to] [y.sub.0])

+ [h[(y - [a.sub.2]).sup.+] + p[([a.sub.2] - y).sup.+]]P(D [greater than] [y.sub.0]).

It can be easily verified that:

[G.sub.e](y) = {p[[micro]L - y], y [less than or equal to] [a.sub.1],

{hp/h + p ([a.sub.2] - [a.sub.1]), [a.sub.1] [less than] y [less than] [a.sub.2],

{h[y - [micro]L], y [greater than] [a.sub.2].

On the other hand,

G([y.sub.0]) = E[h[([y.sub.0] - D).sup.+] + p[(D - [y.sub.0]).sup.+]],

= h([y.sub.0] - [a.sub.1]) P/h + P + p([a.sub.2] - [y.sub.0]) h/h + p,

= hp/h + p ([a.sub.2] - [a.sub.1]).

Hence we have

[G.sub.e](y) = {[G.sub.d](y), for y [less than or equal to] [a.sub.1] and y [greater than or equal to] [a.sub.2],

{G([y.sub.0]), for [a.sub.1] [less than] y [less than] [a.sub.2].

Figure 1 depicts the relationship of the three functions, G(.), [G.sub.e](.), and [G.sub.d](.). Let

[[C.sup.*].sub.e] [equivalent] [min.sub.r,Q] [c.sub.e] (Q, r) [equivalent] [min.sub.r,Q] (([micro]k(Q) + [[[integral].sup.r+Q].sub.r] [G.sub.e](y)dy)/Q), (7)

then [[C.sup.*].sub.e] [less than or equal to] [C.sup.*]. To evaluate [[C.sup.*].sub.e], we follow the procedure taken by Zheng (1992) to first fix Q and find the optimal r(Q) and then find the optimal [[Q.sup.*].sub.e]. From Fig. 1, it can be seen that for Q [less than or equal to] [a.sub.2] - [a.sub.1], the optimal r(Q) = [a.sub.1], and when Q = [a.sub.2] - [a.sub.1], we get the minimal value

([micro]k(Q)+ [[[integral].sup.[a.sub.1]+Q].sub.[a.sub.1]] [G.sub.e](y)dy) / ([a.sub.2] - [a.sub.1]) = [micro]k(Q) hp/G([y.sub.0]) h + p + G([y.sub.0]), (8)

where we have used the fact that

[a.sub.2] - [a.sub.1] = G([y.sub.0]) h + p/hp.

For Q [greater than] [a.sub.2] - [a.sub.1], according to Lemma 2 in Zheng (1992), the optimal r(Q) is to be solved from the following condition:

[G.sub.e](r) = [G.sub.e](r + Q). (9)

Again from Fig. 1, we can see that to make Equation (9) hold, r(Q) [less than or equal to] [a.sub.1]. Since for y [less than or equal to] [a.sub.1], [G.sub.e](y) = [G.sub.d](y) then Equation (9) is the same as

[G.sub.d](r) = [G.sub.d](r+Q),

whose solution is given in Zheng (1992) as [r.sub.d](Q) = [micro]L - hQ/(h+p). Futhermore,

[micro]k(Q) + [[[integral].sup.[r.sub.d]+Q].sub.[r.sub.d]] [G.sub.e](y)dy/Q

= ([micro]k(Q) + [[[integral].sup.[r.sub.d]+Q].sub.[r.sub.d]] [G.sub.d](y)dy + 1/2 G([y.sub.0])([a.sub.2] - [a.sub.1])) / Q,

= ([micro][k(Q) + 1/2[micro] G([y.sub.0])([a.sub.2] - [a.sub.1])] + [[[integral].sup.[r.sub.d]+Q].sub.[r.sub.d]] [G.sub.d](y)dy) / Q.

The optimal [Q.sup.*] is clearly greater than the EOQ solution

[square root]2[micro][k([0.sup.+]) + 1/2[micro]G([y.sub.0])([a.sub.2] - [a.sub.1])] h + p/hp

= [square root]2[micro]k([0.sup.+]) h + p/hp + [([a.sub.2] - [a.sub.1]).sup.2],

which is greater than [a.sub.2] - [a.sub.1]. The corresponding cost value is smaller than

2[micro][k([0.sup.+]) + (1/2[micro])G([y.sub.0])([a.sub.2] - [a.sub.1])]/[square root]2[micro][k([0.sup.+]) + (1/2[micro])G([y.sub.0])([a.sub.2] - [a.sub.1])](h + p)/(hp)

= [square root]2[micro][k([0.sup.+]) + 1/2[micro]G([y.sub.0])([a.sub.2] - [a.sub.1])] hp/h + p

= [square root]2[micro][k([0.sup.+]) + 1/2[micro]G[([y.sub.0]).sup.2] h + p/hp] hp/h + p

= [square root]2[micro]([0.sup.+]) hp/ h + p + G[([y.sub.0]).sup.2]

[less than or equal to] [square root][([micro]k([0.sup.+])(hp/(h + p))/G([y.sub.0]) + G([y.sub.0])).sup.2].

The above is thus smaller than the minimal value in (8) obtained for the case Q [less than or equal to] [a.sub.2] - [a.sub.1]. Therefore

[[C.sup.*].sub.e] = [min.sub.Q] [micro]k(Q)/Q + hp/h + p Q/2 + G([y.sub.0])([a.sub.2] - [a.sub.1])/2Q

= [min.sub.Q]([micro]k(Q)/Q + hp/h + p Q/2 + G[([y.sub.0]).sup.2]/2Q h + p/hp),

[greater than or equal to] [square root]2[micro]k([0.sup.+]) hp/h + p + G[([y.sub.0]).sup.2].

To prove the upper bound, we note that the function [G.sub.f](y) = [G.sub.d](y - [y.sub.0] + [micro]L) + G([y.sub.0]) strictly dominates G(y), and hence the optimal cost is bounded above by

[min.sub.r,Q] ( [micro]k(Q) + [[[integral].sup.r+Q].sub.r] [G.sub.f](y)dy)/Q

= G([y.sub.0]) + [min.sub.Q]([micro]K(Q)/Q + hp/h + p Q/2).

Let Q' denote the batch size that attains the minimum in the upper bound. Then k(Q') = f + cQ' for some f and c. The lower bound is at least

c[micro] + [square root]2[micro]f hp/h + p + G[([y.sub.0]).sup.2],

with equality only when the optimal batch size is attained at a running point.

On the other hand, the upper bound is given by

[min.sub.Q]([micro]k(Q)/Q + hp/h + p Q/2) + G([y.sub.0])

[less than or equal to] c[micro] + G([y.sub.0]) + [min.sub.Q]([micro]f/Q + hp/h + p Q/2)

(by concavity of K(Q)),

= c[micro] + G([y.sub.0]) + [square root]2[micro]f hp/h + p,

[less than or equal to] c[micro] + [square root]2(2[micro]f hp/h + p + G[([y.sub.0]).sup.2]).

In general, G([y.sub.0]) is not easy to characterize in closed form. If the demand D follows normal distribution, however, it can be shown to be proportional to the standard deviation. Theorem 2 can thus be simplified as follows.

Corollary 1. If the demand per unit time is i.i.d. and normally distributed with mean [micro] and standard deviation [sigma], then there exists a positive number

k = [square root]L [min.sub.z] E [h [(z - D - [micro]L/[sigma][square root]L).sup.+] + p [(D - [micro]L/[sigma][square root]L - z).sup.+]],

such that G ([y.sub.0]) = k[sigma] and

[min.sub.Q] ([micro]k(Q)/Q + hp/h + p Q/2) + k[sigma]

[greater than or equal to] [C.sup.*] [greater than or equal to] [min.sub.Q] ([micro]k(Q)/Q + hp/h + p Q/2 + [(k[sigma]).sup.2]/2Q h + p/hp).

The proof of this corollary follows from standard arguments in inventory theory. For the sake of completeness, we include the easy proof.

Proof. Under the above assumption, the lead-time demand D [sim] N([micro]L, [[sigma].sup.2]L). Then the optimal newsvendor quantity [y.sub.0] is given by [micro]L + z[sigma][square root]L for some positive number z which depends on h and p, and correspondingly,

G([y.sub.0]) = [min.sub.y] E[h[(y - D).sup.+] +p[(D - y).sup.+]],

= [min.sub.z] E[h[([micro]L + z[sigma][square root]L - D).sup.+] + p[(D - [micro]L - z[sigma] [square root]L).sup.+],

= [sigma][square root]L [min.sub.z] E [h [(z - D - [micro]L/[sigma][square root]L).sup.+] + p [(D - [micro]L/[sigma] [square root]L - z).sup.+]],

= k[sigma],

where in the last equation

k = [square root]L [min.sub.z] E [h [(z - D - [micro]L/[sigma] [square root]L).sup.+] + p [(D - [micro]L/[sigma] [square root]L - z).sup.+]],

depends on h, p, and L. Note that

D - [micro]L/[sigma] [square root]L [sim] N (0, 1),

and

[z.sup.*] = [min.sub.z] E [h [(z - D - [micro]L/[sigma] [square root]L).sup.+] + p [(D - [micro]L/[sigma] [square root]L - z).sup.+]],

which can be obtained by solving a classical newsvendor model.

5.2. Worst case example

Consider a scenario where we have two possible DC locations and two demand points. At the DC, the ordering costs are [k.sub.i](Q) = [K.sub.i], i = 1, 2. At the demand points, the demands are normally distributed with mean [[micro].sub.j] and standard deviation [[sigma].sub.j], j = 1, 2. For the DC, we define [A.sub.i] = [2K.sub.i][h.sub.i][p.sub.i]/([h.sub.i] + [p.sub.i]) and [B.sub.i] = [[k.sup.2].sub.i], i = 1, 2, where [k.sub.i] is the positive number as defined in Corollary 1. To isolate the effect due to ordering cost and demand uncertainty, we may ignore the effect on the parameters due to the holding (h) and penalty (p ) cost, by proper normalization. Thus [A.sub.i] and [B.sub.i] roughly indicate the magnitude of the ordering cost and the replenishment lead-time at DC i respectively.

By (6), if we consolidate the demands at DC 1, the total inventory cost is at least ([square root][A.sub.1]([[micro].sub.1] + [[micro].sub.2]) + [square root][B.sub.1] ([[[sigma].sup.2].sub.1] + [[[sigma].sup.2].sub.2]))/[square root]2; if we consolidate at DC 2, the total inventory cost is at least ([square root][A.sub.2]([[micro].sub.1] + [[micro].sub.2]) + [square root][B.sub.2]([[[sigma].sup.2].sub.1] + [[[sigma].sup.2].sub.2]))/[square root]2. However, if demand 1 is served by DC 1 and demand 2 by DC 2 respectively, the total cost is only at most [square root][A.sub.1][[micro].sub.1] + [square root][B.sub.1][[[sigma].sup.2].sub.1] + [square root][A.sub.2][[micro].sub.2] + [square root][B.sub.2][[[sigma].sup.2].sub.2]. Now we choose ([[micro].sub.1] = 1/[epsilon], [[sigma].sub.1] = 1) and ([[micro].sub.2] = 1/[[epsilon].sup.4], [[sigma].sub.2] = [epsilon]) for some arbitrarily small positive constant [epsilon]. Also we can make [A.sub.1] = [[epsilon].sup.2] and [A.sub.2] = [[epsilon].sup.6] by suitable c hoices of [K.sub.1] and [K.sub.2]. We can simultaneously make [B.sub.1] = [epsilon] and [B.sub.2] = 1/[epsilon] with certain choices of [L.sub.1] and [L.sub.2]. With these choices of the parameters, it can be verified that the cost for the decentralized structure is at most [theta]([square root][epsilon]), while both of the consolidated solutions have cost of at least [theta](1/[square root][epsilon]). Hence, the consolidated solutions are infinitely worse off than the optimal decentralized structure.

In the above example, the ordering cost of DC 2 (O([[epsilon].sup.6])) is much smaller than that of DC 1 (O([[epsilon].sup.2])), although the replenishment lead-time (O(1/[epsilon])) is much larger than that of DC 1 (O([epsilon])). Furthermore, a large demand ([[micro].sub.2] = O(1/[[epsilon].sup.4])) with low variability ([[sigma].sub.2] = [epsilon]) is assigned to DC 2, whereas a small volume with relative large variability ([[micro].sub.1] = 1/[epsilon], [[sigma].sub.1] = 1) is assigned to DC 1. Note that such demand assignment is ideal for the specific cost structures of the respective DC. If we consolidate the volume and serve all the customers using only one of the two DCs, the above calculation shows that the final inventory cost will be excessive. This is the fundamental reason why consolidation strategy may not work for all demand scenarios.

5.3. The [square root]2-approximation algorithm

Suppose we replace the optimal inventory cost function in (P) by its upper bound. Consider the following problem:

(P) min [[sigma].sub.i], [f.sub.i][y.sub.i]

+ [[sigma].sub.i]([min.sub.Q] (([[sigma].sub.j] [[micro].sub.j][x.sub.ij])[k.sub.i](Q)/Q + hp/h + p Q/2) + [G.sub.i](X)),

subject to

[x.sub.ij] [less than or equal to] [y.sub.i], for all i, j,

[[sigma].sub.i] [x.sub.ij] = 1 for all j,

[x.sub.ij], [y.sub.i] [epsilon] {0, 1} for all i.

[G.sub.i](X) denotes the optimal buffer cost at DC i under the demand assignment X. Let P(XY) denote the objective value of the solution (X,Y) in problem (P). Problems (P) and (P) are related by the following theorem:

Theorem 3. Suppose ([X.sup.*] = {[x.sub.ij]}, [Y.sup.*] = {[y.sub.i]}) is an optimal solution to Problem (P), then ([X.sup.*], [Y.sup.*]) is a feasible solution to Problem (P). Let P([X.sup.*],[Y.sup.*]) be the value of this solution and [P.sup.*] the optimal value, we have P([X.sup.*],[Y.sup.*]) [less than or equal to] [square root]2[P.sup.*].

Proof. By Theorem 2, P(X,Y) [less than or equal to] [square root]2P(X,Y) for any (X,Y). Thus, [P.sup.*] [less than or equal to] [square root]2[P.sup.*]. Hence, P([X.sup.*],[Y.sup.*]) [less than or equal to] P([X.sup.*],[Y.sup.*]) = [P.sup.*] [less than or equal to] [square root]2[P.sup.*].

When the demands are normally distributed, Corollary 1 gives an explicit expression for [G.sub.i](X). Then, Problem (P) is equivalent to the following concave partitioning problem:

Find a partition ([S.sub.1], [S.sub.2], ..., [S.sub.i]) of the set [1, 2, ..., n] (i.e., [[union].sub.i][S.sub.i] = [1, 2, ..., n], [S.sub.i] [intersection] [S.sub.j] = 0 for I [neq] j)to minimize the function

[[[sigma].sup.m].sub.i=1][g.sub.i]([[sigma].sub.j[epsilon][S.sub.i]][ [micro].sub.j], [[sigma].sub.j[epsilon][S.sub.j]][[[sigma].sup.2].sub.j]),

where

[g.sub.i](a, b) = [f.sub.i][delta](a) + [min.sub.Q](a[k.sub.i](Q)/Q + [h.sub.i][P.sub.i]/[h.sub.i] + [P.sub.i] Q/2) + [k.sub.i][square root]b,

and

[delta](a) = { 1 if a [greater than] 0,

{ 0 otherwise.

Such problems have been extensively studied in the operations research literature. In particular, Chakravarty et al. (1985) showed that whenever [g.sub.i](a, b) is concave in both a and b for all i, and if

[[micro].sub.1]/[[[sigma].sup.2].sub.1] [less than or equal to] [[micro].sub.2]/[[[sigma].sup.2].sub.2] [less than or equal to] ... [less than or equal to] [[micro].sub.n]/[[[sigma].sup.2].sub.n], (10)

then there is an optimal partition with a pleasing property. The sets [S.sub.i] in the optimal partition is consecutive, i.e., if k [less than] l are both in [S.sub.i], then j [epsilon] S if k [less than] j [less than] l.

In general, such structural results are not strong enough to guarantee polynomial time construction of the optimal solution, since the cost function of any consecutive subsets depends on which location they are assigned to (i.e., dependent on i). In the special case when the cost function [g.sub.i](a, b) is independent of the index i, Chakravarty et al. (1985) showed that the optimal consecutive optimizer can be constructed by a simple shortest path computation. The problem has also been extended in several ways by many others, notably Anily and Federgruen (1991), and Gal and Klots (1995).

Although the cost function in our instance does depend on the location (i.e., index i), the following proposition shows that the dependence can be removed:

Proposition 1. Let h(a, b) = [min.sub.i=1,...,m] [g.sub.i](a, b). Then

[min.sub.partition([S.sub.l],...,[S.sub.m])] [[[sigma].sup.m].sub.i=1][g.sub.i]([[sigma].sub.j[epsilon][S.sub.i]][ [micro].sub.j], [[sigma].sub.j[epsilon][S.sub.j][[[sigma].sup.2].sub.j])

= [min.sub.partition([S.sub.l],...,[S.sub.m])] [[[sigma].sup.m].sub.i=1] h ([[sigma].sub.j[epsilon][S.sub.i]][[micro].sub.j], [[sigma].sub.j[epsilon][S.sub.i]][[[sigma].sup.2].sub.j]).

Furthermore, there is a consecutive optimal partition solution to the right-hand side.

Hence, the optimal consecutive partition can be obtained by finding the shortest path in the graph as shown in Fig. 2.

In this graph, the cost of the edge (i,j) is given by

C(i,j) = h ([[[sigma].sup.j].sub.k=i+1] [[micro].sub.k], [[[sigma].sup.j].sub.k=i+1] [[[sigma].sup.2].sub.k])

= [min.sub.l=l, ...,m] [g.sub.l] ([[[sigma].sup.j].sub.k=i+1] [[micro].sub.k], [[[sigma].sup.j].sub.k=i+1] [[[sigma].sup.2].sub.k]), which can be computed in O([n.sup.2]m) time. Since the graph is acyclic, the standard dynamic programming recursion procedure can find the shortest path in O([n.sup.2]) steps. In summary, the concave partitioning problem, and hence Problem (P), can be solved in O([n.sup.2]m) time.

Proof. Let ([[S.sup.*].sub.1],..., [[S.sup.*].sub.m]) be an optimal consecutive partition to the left-hand side. Suppose for some i,

[g.sub.i] ([[sigma].sub.j[epsilon][[S.sup.*].sub.i]] [[micro].sub.j], [[sigma].sub.j[epsilon][[S.sup.*].sub.i]] [[[sigma].sup.2].sub.j]) [greater than] h ([[sigma].sub.j[epsilon][[S.sup.*].sub.i]] [[micro].sub.j], [[sigma].sub.j[epsilon][[S.sup.*].sub.i]] [[[sigma].sup.2].sub.j]) = [g.sub.k] ([[sigma].sub.j[epsilon][[S.sup.*].sub.i]] [[micro].sub.j], [[sigma].sub.j[epsilon][[S.sup.*].sub.i]] [[[sigma].sup.2].sub.j]),

for some location k. Hence

[g.sub.k] ([[sigma].sub.j[epsilon][[S.sup.*].sub.i][union][[S.sup.*].sub.k]] [[micro].sub.j], [[sigma].sub.j[epsilon][[S.sup.*].sub.i][union][[S.sup.*].sub.k]] [[[sigma].sup.2].sub.j]),

= [f.sub.k][delta] ([[sigma].sub.j[epsilon][[S.sup.*].sub.i][union][[S.sup.*].sub.k]] [[micro].sub.j]) + [min.sub.Q] (([[sigma].sub.j[epsilon][[S.sup.*].sub.i][union][[S.sup.*].sub.k]][[ micro].sub.j]]) [K.sub.k](Q)/Q + [h.sub.kpk]/[h.sub.k]+[p.sub.k] Q/2)

+ [square root][[k.sup.2].sub.k] [[sigma].sub.j[epsilon][[S.sup.*].sub.i][union][[S.sup.*].sub.k]] [[[sigma].sup.2].sub.j],

[less than or equal to][f.sub.k][delta]([[sigma].sub.j[epsilon][[S.sup.*].sub.i]][[micro] .sub.j]) + [f.sub.k][sigma]([[sigma].sub.j[epsilon][[S.sup.*].sub.k]][[micro].su b.j])

+ [square root][[k.sup.2].sub.k] [[sigma].sub.j[epsilon][[S.sup.*].sub.i]] [[[sigma].sup.2].sub.j] + [square root][[k.sup.2].sub.k] [[sigma].sub.j[epsilon][[S.sup.*].sub.k]] [[[sigma].sup.2].sub.j]

+ [square root] ([2K.sub.k][h.sub.k][p.sub.k]/[h.sub.k]+[p.sub.k] [[sigma].sub.j[epsilon][[S.sup.*].sub.i]][[micro].sub.j]) + ([[k.sup.2].sub.k] [[sigma].sub.j[epsilon][[S.sup.*].sub.i]][[[sigma].sup.2].sub.j])

+ [square root] ([2K.sub.k][h.sub.k][p.sub.k]/[h.sub.k]+[p.sub.k] [[sigma].sub.j[epsilon][[S.sup.*].sub.k]][[micro].sub.j]) + ([[k.sup.2].sub.k] [[sigma].sub.j[epsilon][[S.sup.*].sub.k]][[[sigma].sup.2].sub.j]),

[less than][g.sub.i]([[sigma].sub.j[epsilon][[S.sup.*].sub.i]][[micro].sub. j], [[sigma].sub.j[epsilon][[S.sup.*].sub.i]][[[sigma].sup.2].sub.j]) + [g.sub.k] ([[sigma].sub.j[epsilon][[S.sup.*].sub.k]][[micro].sub.j], [[sigma].sub.j[epsilon][[S.sup.*].sub.k]][[[sigma].sup.2].sub.j]).

Thus the partition ([T.sub.1],...,[T.sub.m]) (where [T.sub.j] = [[S.sup.*].sub.j] if j [neq] i, k; [T.sub.i], = 0; and [T.sub.k] = [[S.sup.*].sub.i] [union] [[S.sup.*].sub.k]) will be a strictly better partition, resulting in a contradiction.

6. Numerical experiment

We consider the effectiveness of the consolidation strategy on a system with two regional DCs. As stated in the introduction, we are concerned here only with the facility investment and inventory costs when evaluating the consolidation strategy. The worst case example above shows that consolidation can be "infinitely bad". In our numerical experiment, we want to study the likelihood of such situation and to understand the trade-off between the facility investment and inventory costs. To this end, instead of simulating the stochastic behavior of the system (which gives only insights to distribution-specific instances), we have opted to compare the upper bounds obtained from the normal approximation to determine how likely it is for consolidation to be more than [square root]2 times off optimum. The experiment reveals that the likelihood of consolidation being "infinitely bad" depends on the difference between the ratios [[micro].sub.1]/[[[sigma].sup.2].sub.1] and [[micro].sub.2]/[[[sigma].sup.2].sub.2].

In the experiment, we assume [k.sub.i](Q) = [K.sub.i], (constant) for i = 1,2 to simplify calculations. Then, we have

[min.sub.Q] ([[sigma].sub.j][[micro].sub.j][x.sub.ij])[k.sup.i](Q)/Q + [h.sub.i][p.sub.i]/[h.sub.i] + [p.sub.i] Q/2 = [A.sub.i] [square root][[sigma].sub.j][[micro].sub.j][x.sub.ij] for some [A.sub.i],

and by Corollary 1,

[G.sub.i](x) = [B.sub.i] [square root] [[sigma].sub.j] [[[sigma].sup.2].sub.j][x.sub.ij] for some [B.sub.i].

Experiment:

Given a set of parameters ([A.sub.1], [B.sub.1], [A.sub.2], [B.sub.2]), derived from a system with two DCs, how likely is it that consolidating the two DCs into one DC will be infinitely worse-oft?

To eliminate the trivial cases, we will always generate [A.sub.i] and [B.sub.i] such that [A.sub.1][B.sub.1]/[A.sub.2] = [B.sub.2]. Otherwise, if [A.sub.1] [greater than or equal to] [A.sub.2], and [B.sub.1] [greater than or equal to] [B.sub.2] (or vice-versa), then consolidation is clearly optimal. We generate two types of problem instances: in Type 1, [A.sub.1],[B.sub.1],[A.sub.2] are uniformly picked in [0, 1]; and in Type 2, [A.sub.1],[B.sub.1] are uniformly picked in [0,1] and [A.sub.2] = [[A.sup.3].sub.1]. In the latter, [A.sub.2] [much less than] [A.sub.1] and hence [B.sub.1] [much less than] [B.sub.2]

We next discuss how we generate the parameters for [[micro].sub.i], [[[sigma].sup.2].sub.i]. Suppose the optimal assignment partitions the demand points into two groups. Let [micro](i) and a [[sigma].sup.2](i) denote the total demand rate and variance assigned to DC i, i = 1, 2. We choose

[micro](1)/[[sigma].sup.2](1) = a [less than] [micro](2)/[[sigma].sup.2](2) = b,

and for both types of problems, we fix a = 10, and take b from the set {l00, 1000, 10000, 100 000, 1 000 000}. Note that since the optimal partition has the consecutive property, then by (10), a [greater than or equal to] [min.sub.i]([[micro].sub.i],/[[[sigma].sup.2].sub.i]) and b [less than or equal to] max([[micro].sub.i]/[[[sigma].sup.2].sub.i]). Hence the ratio of a and b is a lower bound of the difference in the ratios [{[[micro].sub.i]/[[[sigma].sup.2].sub.i]}.sub.=1,2]. Now we let

[micro](1) + [micro](2)/[[sigma].sup.2](1) + [[sigma].sup.2](2) = c,

which represents the situation when the demands are consolidated, and fix [[sigma].sup.2](2) = 1. In this way, [[sigma].sup.2](1) = (b - c)/(c - a), and [micro](1),[micro](2) can also be expressed as a function of a, b, c.

We first generate the 100 values of ([A.sub.1],[B.sub.1],[A.sub.2] [B.sub.2]) (for Type 1 and Type 2 problems separately), then for each ([A.sub.1],[B.sub.1],[A.sub.2],[B.sub.2]), with (a,b) ranging over the set {(10, 100), (10,1000),... ,(10,1 000 000)}, we generate 1000 c's uniformly from (a, b,). We check next how many instances of (a, b, c) give rise to cases where consolidation is not optimal for the given set of parameters ([A.sub.1],[B.sub.1],[A.sub.2],[B.sub.2]).

For the experiments, we want to observe the number of times (S) when the cost parameters ([A.sub.1],[B.sub.1],[A.sub.2],[B.sub.2]) give rise to problem instances for which consolidation may affect the inventory cost adversely (indicated by the rule that upper bound function obtained from consolidation in either DC is larger than the total cost without consolidation). We obtain insights to this by searching over various combinations of (a, b, c). For such cost parameters, we also tabulate the number of problem instances (P) out of 1000 choices of (a, b, c) for which consolidation may affect the inventory cost adversely.

For Type 1 problems, the value of S is as listed in Table 1. The result shows that the consolidation strategy is pretty robust in these situations. For instance, given 100 generations of the parameters ([A.sub.1],[B.sub.1],[A.sub.2],[B.sub.2]), when (a, b) = (10,100), we generated 1000 different values of c between a and b but could not find any instances where consolidation may be inferior. Furthermore, for the Type 1 problems, at most 14 out of 100 generations of ([A.sub.1],[B.sub.1],[A.sub.2],[B.sub.2]) give rise to problem instances for which consolidation may have an adverse effect. This happens in the case when (a,b) = (10,10 000). When such situations arise, the distribution of the number P is shown in Fig. 3. The y-axis indicates the number of c's (out of 1000) where consolidation is inferior. The x-axis indicates the number of the experimental trials (out of 100). The graph indicates that when at times where the parameters ([A.sub.1],[B.sub.1],[A.sub.2],[B.sub.2]) are not favorable for consolidation, then almost any choice of c will result in a case where consolidation may be more than [square root]2 times off optimum.

The negative effect of consolidation is more apparent when the differences of [A.sub.1],[B.sub.1],[A.sub.2],[B.sub.2] are large. This is seen in Type 2 problems, for which the S values are listed in Table 2.

The result shows that consolidation strategy may not be very effective for Type 2 problems, when b/a[greater than]10, as significantly more of the 100 generations of ([A.sub.1],[B.sub.1],[A.sub.2],[B.sub.2]) give rise to problem instances for which consolidation may not be optimal. The distribution of the number P is shown in Fig. 4. We observe that in this case when the ratio b/a is around 10, no negative example has been generated. This illustrates that the consolidation strategy could be useful for moderate values of b/a.

In both problem types generated, whenever (a, b) = (10,100), we are not able to generate any example where consolidation may be inferior. Thus the numerical experiments seem to support the hypothesis:

Given any set of parameters ([A.sub.1],[B.sub.1],[A.sub.2],[B.sub.2]), derived from a system with two DCs, it is likely that consolidation will be a good strategy, whenever the ratios of the mean to variance for each demand location are similar (not more than 10 times apart).

7. Conclusion

The location-inventory problem in this paper is motivated by the new trend of consolidating and centralizing distribution operations in supply chain management. We analyzed the impact on the total facility investment and inventory costs of the consolidation strategy. The key insight gained through this study is that unlike in the classical EOQ and newsvendor models, consolidation does not always lead to reduction in the total facility investment and inventory costs, and the differences in the ratios [[micro].sub.i]/[[[sigma].sup.2].sub.i] of the demand processes play an important role on the effectiveness of the consolidation strategy. When the differences are small (less than 10), the chances of the consolidation strategy being effective are very good. In particular, for Poisson demand processes which have the same such ratios of value one, the consolidation strategy is actually optimal. It is also optimal for i.i.d. demands.

The paper can be extended in many directions. We can consider multi-echelon location-inventory problems. Lim et al. (1998) have obtained some results in this direction. Also, the problem with multiple products would be both challenging and interesting.

Acknowledgements

The authors wish to thank the Associate Editor and two referees for their suggestions that have made the paper more focused and clearer. This research was partially supported by a grant from the Singapore-MIT Alliance Fellowship 1999-2001 and NUS Research Project 3960012.

Biographies

Chung Piaw Tea is an Assistant Professor in the Department of Decision Sciences. Well grounded in methodology. Dr Tea has published in OR and SODA on network optimization problems in the broad areas of inventory and logistics management.

Ou Jihong is a Senior Lecturer in the Department of Decision Sciences. Dr Ou has a track record in publishing in internationally refereed journals. He is currently Associate Editor of the Asia Pacific Journal of Operational Research. His current research portfolio includes supply chain modelling and stochastic optimisation.

Mark Goh is an Associate Professor and is currently serving as the Logistics Management Coordinator in the department. He has published widely in internationally refereed journals. His current research activities include supply chain management and strategy. He was also the Vice President of the Singapore OR society, and sits on the Editorial Board of Supply Chain Management.

Gontributed by the Inventory Department

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                The values of S used in the Type I problems
(a,b) (10,100) (10,1000) (10,10 000) (10,100 000) (10,1 000 000)
S        0        13         14           8             12
                       The S values associated with
                            the Type 2 problem
(a,b) (10,100) (10,1000) (10,10 000) (10,100 000) (10,1 000 000)
S        0        20         38           41            36