Identific

If there is more than a single inventory keeping facility, the interaction between these facilities may require the coordination of replenishments. The integrated models that optimize inventory decisions of several vertically connected facilities in a supply chain are known as multi-stage inventory models. Interest in multi-stage models with deterministic stationary demand was initiated by Goyal (1976) and Schwarz (1973). Papers in this area focus on the interaction between one inventory keeping

facility (namely, a supplier, a distribution center, or a warehouse) and one or more stocking facilities (namely, retailers) under deterministic stationary demand. These papers establish the foundations for the single warehouse single retailer (SWSR) and the single warehouse multi-retailer (SWMR) deterministic lot-sizing literature. Both SWSR and SWMR lot-sizing models, under deterministic stationary demand with infinite planning horizon and instantaneous replenishments, appear as subproblems in this dissertation. Hence, we refer in detail to these models and their solution approaches in our analysis.

The SWSR lot-sizing problem under deterministic stationary demand is the basis for the multi-stage inventory models since it considers two stock-keeping (inventory-keeping) locations, i.e., a single warehouse and a single retailer. The problem is to find the order quantities of the warehouse and the retailer, Q_w and Q_r, respectively, so that the total cost at the warehouse and the retailer, including the inventory ordering and inventory holding costs, is minimized. The traditional SWSR lot-sizing problem assumes that the warehouse and the retailer cooperate and determine their inven-tory replenishment strategies using a centralized approach. Using this centralized approach, it is desirable to coordinate the inventory decisions of the warehouse and the retailer. For this purpose, many researchers consider lot-for-lot and integer-ratio policies. The lot-for-lot policy refers to an integrated replenishment strategy for the warehouse and the retailer where the order quantity Qw of the warehouse is equiv-alent to the order quantity Qr of the retailer, i.e., Qw = Qr. This policy definitely coordinates the inventory replenishment decisions in the supply chain.

On the other hand, the integer ratio policy states that the warehouse’s order quantity is an integer-multiple n of the retailer’s order quantity, that is Qw = nQr. Since its origination (Goyal, 1976), this model has been referred to as the basic deterministic model in multi-stage inventory systems in many of the inventory books

(e.g. Silver et al., 1998, p.477). We introduce the following notation for this problem:

D deterministic, constant demand rate at the retailer, in units/unit time.

Kw fixed ordering (setup) cost associated with a replenishment at the warehouse.

Kr fixed ordering cost associated with a replenishment at the retailer.

h^′_w the inventory holding cost per unit per unit time at the warehouse.

h^′_r the inventory holding cost per unit per unit time at the retailer, h^′_r > h^′_w. Tw the reorder interval of the warehouse, Tw = Qw/D.

Tr the reorder interval of the retailer, Tr= Qr/D.

Under integer ratio policies, the main objective of the SWSR lot-sizing problem is to determine the integer ratio n and the retailer’s order quantity Qr while minimizing the total average annual cost in the system. The total average annual cost for the system under integer ratio policies is given as

KrD Qr

+h^′_rQr

2 +KwD nQr

+h^′_w(n− 1)Qr

2 ,

where the first two terms are average ordering and inventory holding costs at the retailer and the last two terms are the similar costs incurred by the warehouse.

Note from Figure 5 that the inventory profile at the warehouse does not follow the usual sawtooth pattern, even though the demand at the retailer is deterministic and constant. This is due to withdrawals of size Qr every Tr time units from the warehouse’s inventory. With conventional definitions of inventories, the determina-tion of average inventory levels become more complicated than the sawtooth pattern as shown with the above formulation. Therefore, many researchers prefer to use a concept known as echelon inventory, introduced by Clark and Scarf (1960). The ech-elon inventory of stage j (in a general multi-stage setting) is defined as the number of units in the system that are at, or have passed through, stage j but have as yet

FIGURE 5. Inventory Levels for the SWMR Lot-sizing Problem.

Warehouse’s inventory level

Retailer’s inventory level Q_w

Q_r

Time Time

Tr

Tr

2Tr

2Tr

3Tr

3Tr

Tw

Actual physical inventory level Echelon inventory of the warehouse

not been specifically committed to the external demand. In Figure 5, we provide the inventory profiles of a warehouse and the retailer when n = 3. This figure also illustrates the echelon inventory at the warehouse. As can be seen from the figure, it is simple to compute the average echelon inventory since the sawtooth pattern at the warehouse re-emerges. However, in order to estimate inventory holding costs at the warehouse and the retailer, we should define the echelon inventory holding costs as the incremental cost of moving the product from the warehouse to the retailer so that the inventory costs are not double-counted. For instance, for the SWSR problem, the

echelon holding cost for the warehouse is hs = h^′_s, and the echelon holding cost for the retailer is hr = h^′_r− hs. With the echelon inventory concept, the total average annual cost for the system under integer ratio policies is given as

KrD Qr

+ hrQr

2 + KsD nQr

+hsnQr

2 .

Since the SWSR lot-sizing problem appears as a subproblem in several integrated location-inventory models in Chapter IV, the solution methods for this problem are further discussed in that chapter.

The SWMR lot-sizing problem under deterministic stationary demand is a gen-eralization of several classical inventory models including the SWSR lot-sizing prob-lem. Arkin, Joneja, and Roundy (1989) show that the SWMR lot-sizing problem is NP-hard. An optimal inventory policy for the SWMR lot-sizing problem has not been found. However, considering an infinite horizon and instantaneous deliveries, Schwarz (1973) proves that if an optimal policy for the deterministic stationary de-mand SWMR lot-sizing problem exists, it has the following properties:

• Zero Inventory Ordering: Each facility orders when its inventory is zero.

• Last Minute Ordering: The warehouse orders only when at least one retailer orders.

• Stationarity Between Orders: At each retailer, all orders placed between two successive orders at the warehouse are of equal size.

Still, the structure of the optimal inventory policy for the deterministic SWMR prob-lem may be exceedingly complex. Even if it could be computed efficiently, its com-plexity would make it unattractive to implement in practice (Graves and Schwarz, 1977). However, in a seminal paper, Roundy (1985) shows that the best power-of-two

policy, where the replenishment intervals are chosen as power-of-two multiples of a base period, has an average cost that is within either 2% (when the base period is variable) or 6% (when the base period is fixed) of a lower bound of the minimum cost value. The SWMR inventory models have been studied extensively since this seminal work; for details, see the review paper by Muckstadt and Roundy (1993).

Simchi-Levi et al. (2004) summarizes the classical SWMR model and its solution under power-of-two policies. The notation for the SWMR problem is as follows:

I set of retailers, indexed by i∈ I.

Di deterministic and stationary demand rate at retailer i∈ I.

K0 fixed ordering (setup) cost associated with a replenishment at the warehouse.

Ki fixed ordering cost associated with a replenishment at retailer i∈ I.

h^′₀ the inventory holding cost per unit per unit time at the warehousee.

h^′_i the inventory holding cost per unit per unit time at retailer i∈ I, h^′i ≥ h^′0. h0 echelon holding cost rate at the warehouse, h0 = h^′₀.

hi echelon holding cost rate at retailer i∈ I, hⁱ = h^′_i− h⁰. Tb base planning period.

T0 the reorder interval of the warehouse.

Ti the reorder interval of the retailer i∈ I.

T the reorder interval’s vector, T = (T0, T1, . . . , Tn).

Using echelon inventory, the SWMR problem is formulated as follows:

Min Z(T) = K0

T0

+X

i∈I

1

2h₀D_i max{T0, T_i} +X

i∈I

Ki

Ti

+X

i∈I

1

2h_iD_iT_i (SWMR) subject to

Ti = 2^viTb and vi ∈ Z, for i = 0, . . . , n. (2.10)

T∈ Rⁿ⁺¹+ . (2.11)

In the objective function of the SWMR problem, the first two terms represent the average annual ordering and holding costs at the warehouse, respectively, and the last two terms represent the total average annual ordering and holding costs at the retailers, respectively. Note that, with constraint (2.10), the reorder intervals of the retailers and the warehouse are restricted to a value that is a power-of-two multiple of a base period, Tb.

In this dissertation, the SWMR problem appears as a subproblem in several integrated location-inventory models, especially in the models discussed in Chapters IV and VI.