Stochastic Inventory Models

Abstract

We discuss inventory systems in an independent demand setting, where demand over time is modeled as a stationary stochastic process. We begin with some basic notions and definitions on inventory management, followed by a discussion of well-known (and applied) control systems. Under periodic review and a linear cost structure, it is known that the optimal control policy has a critical level structure, hence we analyze such critical level policies in detail. After that, in an advanced section, we turn to multi-echelon or multi-stage systems. We present a complete analysis of the decomposition result proven initially by Clark and Scarf, and its analogue in distribution systems, i.e., systems with an arborescent instead of a linear structure (state-of-the-art). Computational aspects are briefly discussed after which we close with some guidelines for further reading.

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1 Stochastic Inventory Models: Definitions and Terminology (Basic)

Inventories are everywhere. As far back as ancient nomadic and agricultural civilizations, mankind has been storing products for future use. Indeed, storage of agricultural products is required to bridge the time between the (often short) harvest season and future consumption. In modern industrial organizations, inventories are indispensable to match supply and demand, for various reasons. The most common reason to store products can be attributed to the uncertainty in future demand. While organizations would like to minimize the risk of being unable to fulfill customer demand and loose both goodwill and revenue, they also like to keep their inventory costs low. Thus, it is necessary to have some level of inventory in many organizations. Upfront, inventories may also serve as a protection against quality defects or supply uncertainty, e.g., due to malfunctioning production equipment. Capacity limitations are another reason for storing products in anticipation of future needs. Note that adjusting capacity is difficult in many industries, especially those in which the cost to switch from one product to another are high. If demand is highly volatile and capacity is set to match average demand, then advance production to meet future peak demands is necessary, again resulting in inventories. This is especially true for seasonal products. The need to minimize production costs, e.g., of large batches for products that have high production set-up times or costs, leads to a build-up of inventories. These inventories are gradually consumed and upon their depletion, a new batch is produced. In all cases, inventories basically represent an alternative form of (allocated) capacity.

However, inventories also represent capital invested which therefore cannot be used other than to fulfill future demand. The Economic Production Quantity discussed in Chap. 12 appeared to be a first attempt to balance production set-up and inventory costs, in a deterministic demand setting. Such a deterministic scenario is common in a dependent demand situation in which the needs for parts and components are derived from a master production schedule at end-item level, as in an MRP driven production environment. When demand is independent, i.e., generated by external sources, it is prudent to model it as a stochastic process. That is the focus of the current chapter.

Inventory modeling, planning and control is a topic that has received significant attention in the Operations Research and Operations Management literature. The pioneering publication of Arrow et al. (1958) marked the start of an extremely fruitful period in the study of stochastic inventory models, with an emphasis on structural properties of optimal control policies. However, many models turned out to be computationally intractable, which has led to a range of heuristic approaches. The number of books on the topic is overwhelming and we will not make an attempt to review them. Instead, we only mention some important results that still provide a basis for further study. In this chapter, we limit ourselves mainly to periodic review systems with full backlogging that either consist of a single stage or involve multiple stages, mostly under centralized control.

We begin by introducing some basic definitions and terminology. Consider a planner in a warehouse who is responsible for having sufficient items in stock to fulfill anticipated future demand. Typically, the planner will have an idea on what demand can be expected in some time period ahead. He or she can order items from an external supplier if the available stock is about to be depleted, thus building inventory to meet future demand. Before we describe the multiple inventory policies a planner may use, we begin with some important definitions, cf. Silver et al. (2017).

A stock keeping unit (SKU) refers to one particular item or product, characterized by attributes such as size, color, and function.

The on-hand inventory (OH) of an SKU refers to the quantity of that SKU physically present, i.e., available on the shelf and to be used to satisfy any customer demand.

The net stock $\left( {NS} \right)$ of an SKU is defined as the on-hand inventory minus the number of backorders. Note that under reasonable circumstances, there is either on-hand inventory or there are backorders, denoted by BO. Thus, $NS = OH{-}BO$.

The inventory position (IP) of an SKU equals the on-hand inventory, plus the number of items ordered that have not yet physically arrived, minus the number of backordered items.

The (supply) lead time of an order for a particular SKU is the time that elapses between the time when the order was placed and the time when the order arrives.

In inventory theory, we discern two basic situations on how to handle a customer’s order when the desired item is temporarily not available in stock. In the case of backordering, it is assumed that the customer is willing to wait until the item becomes available again. Upon arrival of an order (usually a batch of items), backorders are fulfilled on a first-come, first-served (FCFS) basis. The alternative is lost sales, in which case it is assumed that the customer seeks another vendor to satisfy his or her demand. In general, however, a sales organization may face a mix in which some customers are willing to wait and others go elsewhere, while also some products lend themselves more easily for backordering than others (e.g. when they are unique in some sense). Note that the occurrence of a lost sales is not always easily known, because customers may not inform a vendor what they actually do if their demand is unmet by that vendor. They may purchase the product from an alternate vendor, opt for demand substitution by purchasing another product that is reasonable close in functionality to the one they were seeking from the original vendor or a competitor, or just may decide not to buy the product at all. In what follows, we will restrict our discussion to inventory models with full backordering.

Generally, an inventory planner will attempt to fulfill demand even if it is higher than expected. That is the function of safety stock, which typically is meant to deal with random demand fluctuations. In the case of smooth, fully predictable demand there is no need to reserve safety stock but when demand fluctuates over time, it may play an essential role in preventing stock-outs. The latter result in either backorders or lost sales and hence in both cases, a reduced customer service level, or additional costs, or both. We will come back on how to model possible penalty costs in the case of a stock-out, and to various definitions of a customer service level next.

We now discuss when to monitor the inventory status and possibly to place a new order at an upstream supplier or manufacturing department. Before doing so, it is important to determine whether we face a stationary or a non-stationary demand. A stationary demand process is not constant but is modeled as a stochastic process with a fixed mean and variance over time, whereas the mean and variance of a non-stationary demand processes change over time (e.g., in the case of seasonal demand patterns). In this chapter, we assume that demand is stationary with known mean and variance.

Modern warehouse management systems allow for a continuous review and update of the inventory position. Thus, in principle, orders can be placed continuously. However, many organizations choose to place orders periodically, i.e., after one or more time intervals of fixed length. In the retail sector, periodic review and ordering is common practice, simply because a large number of different SKU’s can then be ordered simultaneously. The choice of the length of the time intervals must be determined, but often follows naturally from the calendar or from distribution schedules of logistics service providers, leading to the possibility of placing a replenishment order periodically, e.g., once a day or once per week.

When reviewing stock, it seems natural to order a replenishment quantity if the inventory position is so low that a further postponement of a replenishment order may lead to a stock-out in the near future. Following this argument, often a minimum inventory level $s$ is chosen such that $s$ may cover the expected demand plus a certain fraction of the demand variation during the time it takes for the new order to arrive and become available to fulfill future demand. The determination of this so-called re-order point $s$ will be discussed in more detail below for a periodic review system.

In addition to the determination of the reorder point, which determines when to order, a companion question is: how much to order? Often, ordering costs consist of two components, a fixed cost, which is independent of the order size, and variable purchasing costs that are a linear function of the order size. Because of the fixed costs, it makes sense to order at least a minimum quantity, while on the other hand large orders may lead to high inventory holding costs (cf. the discussion on the Economic Order Quantity in Chap. 12). Two systems are often distinguished: one in which a replenishment order has a fixed size of $Q$, say, and one in which the inventory position after ordering is brought to a fixed order-up-to level $S$, say. Combined with the continuous versus periodic review options, this leads to four system control categories:

Continuous review $\left( {s,Q} \right)$ systems: As soon as the inventory position drops below a specified re-order point $s$, a replenishment order of size $Q$ is placed. The order will arrive after $L$ time units. If every customer requires exactly one unit, the reorder will be placed when the inventory position equals $s$. However, when a customer demands more than one unit, the IP may drop instantaneously below $s$.

Continuous review $\left( {s, S} \right)$ systems: As soon as the inventory position drops below a specified re-order point $s$,a replenishment order is placed such that the inventory position is returned to an order-up-to level $S$. The order will arrive after $L$ time units.

Periodic review $\left( {R, s, Q} \right)$ systems: At time points $t = 0,R,2R, \ldots ,$ the inventory status is reviewed. As soon as the inventory position is equal to or smaller than $s$ at a review point $t$, an order of size $Q$ is placed, which will arrive at time $t + L$.

Periodic review $\left( {R, s, S} \right)$ systems: At time points $t = 0,R,2R, \ldots ,$ the inventory status is reviewed. As soon as the inventory position is equal to or smaller than $s$ at review point $t$, an order is placed such that the inventory position is returned to an order-up-to level $S$. The order will arrive at time $t + L$.

It is important to realize that the variable to be observed is the inventory position IP, not the net stock. In particular, if more than one replenishment order is underway, it is important also to keep track of items that may arrive soon, instead of only the net stock. Also, note that if the inventory position just before ordering is $i \le s$, and an order of size $V$ is placed, upon arrival of that order, the resulting inventory position will in general be less than $i + V$, due to the demand that was satisfied from the currently available stock after the placement of the order. In particular, in the case of an $\left( {R,s,S} \right)$ system, the inventory position just after arrival of a placed order will in general be less than $S$ (even if only one replenishment order is issued at a time). See Fig. 20.1.

Fig. 20.1

A single-item periodic review $\left( {R, s, S} \right)$ inventory system

In a periodic system we generally assume that the parameter $R$ is determined in advance, e.g., as a day or week, so we need to determine the re-order point s and either the order quantity $Q$ or the order-up-to level $S$, depending on what policy is applied. Hence, in any system two parameters are to be determined. In the case of a periodic system, when we take $s = S$, this means that at each review moment, the observed inventory position $i$ is increased with a replenishment order of size $S - i$. Such systems are generally denoted as $\left( {R,S} \right)$ systems, in which only the parameter $S$ has to be determined. These systems typically occur in large retail stores that periodically place replenishment orders for a large number of different SKU’s, that are delivered simultaneously. It may happen in non-retail (e.g., manufacturing) industries as well, however, note that in the case of larger lead times $L$, the likelihood of two or more replenishment orders for a similar SKU being underway at the same time increases.

Finally, we discuss cost parameters. As previously mentioned, there is a fixed cost $K$ associated with the placement of each replenishment order (independent of its size) while in addition a variable cost $c$ is incurred for each item purchased. For each item of a particular SKU that is stocked, an inventory holding cost $h$ is incurred which generally equals a fraction $r$ of the variable purchasing cost of that item, hence $h = rc$. Although depending on the type of industry, a fraction $r = 0.2$ (20%) is not uncommon in inventory management; generally such a percentage does not only cover the interest rate, but also the costs of storage (often based on the depreciation of an overall investment in warehouse building and equipment), the operational costs of all materials handling and support activities, insurance costs, opportunity costs, and obsolescence risks. In case a stock-out occurs, typically shortly before a replenishment order arrives, one may in addition incur a penalty cost $p$ per item short per time unit. Such costs may indeed represent physical costs (e.g., costs of a crash action or, for example in the spare parts business, the costs of downtime of a client’s system while the inventory management is contractually obliged to deliver a desired part immediately upon request). Often, the penalty costs also serve to enforce a desired customer service level, e.g., defined as the fill rate, i.e., the number of items delivered from the shelf (hence without backorders). It is intuitive, and it can be proven that higher penalty costs induce a higher customer service level and vice versa, cf. Van Houtum and Zijm (2000). In fact, there are several ways to penalize a shortage, and also several ways to define customer service levels. See Silver et al. (2017) for an overview, and Van Houtum and Zijm (2000) for the relations between various penalty functions and service levels.

Now, to find the right parameters in any of the four control categories above, one may proceed as follows. In case penalty costs are specified, a planner generally attempts to minimize the sum of the ordering, inventory holding, and penalty costs, averaged over a sufficiently long horizon. When it is difficult to assess a penalty cost, a target customer service level (CSL) is often specified. In this case, the planner may attempt to minimize the sum of the ordering and inventory holding costs, averaged over a sufficiently long horizon subject to the constraint that on average the requested service level is met. There are several direct approaches for the latter problem, but alternatively one may solve a pure cost problem with some artificial penalty costs, measure the optimal policy’s customer service level and next adjust the penalty cost after which the procedure is repeated. Because the CSL is generally increasing as a function of $p$ and vice versa, a simple bisection procedure is sufficient to determine the right penalty costs. Van Houtum and Zijm (2000) have proven that an optimal policy for the pure cost formulation, which results in a specific CSL, is also optimal for the problem in which order and inventory-holding costs are minimized, subject to a service level constraint based on the same CSL.

In practice, it is not always easy to determine the right parameters numerically. Fortunately, a number of structural properties can be exploited. The most important one is that for periodic review problems with a cost structure as defined (with a penalty cost $p$ per unit short per period), the optimal policy turns out to belong to the category of $\left( {R,s,S} \right)$ policies, see Scarf (1960). Because periodic review systems are generally used in practice, we restrict ourselves in this chapter to an analysis of $\left( {R,s,S} \right)$ systems and a reduced version, i.e. $\left( {R,S} \right)$ systems.

2 Case Study: Inventory Management at IKEA

IKEA is one of the world’s largest home furnishing companies. Its name is an abbreviation of “Ingvar Kamprad Elmtaryd Agunnaryd”, referring to the founder Ingvar Kamprad who grew up on the farm Elmtaryd in the town of Agunnaryd in Southern Sweden. Kamprad started the company in 1943, at the age of 17. Today, IKEA owns and operates more than 400 retail stores in 49 countries. Good quality products at low prices is the motto of the company. The vision of IKEA is to provide well designed, functional home furnishings at prices so low that as many people as possible will be able to afford them (www.ikea.com). To realize that vision, the various supply chain functions (procurement, inventory management, sales) are carefully tuned, in that way contributing to the company’s strong competitive position. Most IKEA products are procured from external suppliers, while the SWEDWOOD group, an IKEA subsidiary with its largest factory in Southern Poland, is responsible for the manufacture of the entire set of wooden products.

One of the principles of IKEA is to commit to a catalog of products that are stocked for a year at a guaranteed price. Another distinctive feature is that most furniture is not sold pre-assembled, but designed to be self-assembled. IKEA’s well-known wooden products are characterized by a high degree of modularity. The standard elements (modules) can be combined in an almost unlimited number of variants, allowing for a high level of customization while remaining cost effective. Modularity and standardization are indeed key elements of IKEA’s strategy.

IKEA’s retail stores are megastores, with a design that defines pathways that guide the customer in a natural way along all the functions that are part of home living. These paths end in a self-service warehouse where customers themselves pick up the requested modules (unless they are too heavy for manual handling). Storage columns in the warehouses are separated in a retrieval area (downstairs) and a bulk area (upstairs) that serve to replenish the retrieval area overnight.

Customers can order and purchase products from IKEA in three different ways, at retail stores, by phone or via the internet. In the latter two cases, the products are delivered directly to the customer’s home, from a customer distribution center and often via a local hub. Some of the articles are replenished to stores from distribution centers while others are delivered directly to stores from suppliers. Direct deliveries minimize the handling costs as well as transportation cost, but on the other hand generally drive up the stock levels. Articles delivered through distribution centers are mainly divided into fast and slow movers. The fast mover distribution centers are either used for storage or as transfer centers where articles are repacked and shipped to retail stores. Lead times from suppliers to distribution centers and stores are normally a few weeks while lead times from distribution centers to retail stores are only a few days. Slow movers are articles subject to low volume are stored in only a small number of distribution centers, each one supplying an entire market with slow moving articles. The idea is to create economies of scale and therefore reduce costs by avoiding many small inventories across distribution centers.

The in-store logistics managers use an inventory replenishment management process developed by IKEA called ‘minimum/maximum settings’ where ‘minimum’ refers to the minimum amount of products available before reordering, and ‘maximum’ defines an upper limit on the number of a particular product to order at one time. Still, IKEA feels that the inventory levels kept at retail stores and distribution centers throughout the company are generally high. In addition, inventories at the distribution centers and the retail stores are controlled independently and safety stocks should be sufficiently large to cover uncertainties from the next level of demand. In other words, the various echelons are not synchronized and the system as a whole may be considered to operate suboptimally. Service level requirements are the same for both retail stores and distribution centers, while also IKEA management realizes that the service level experienced by customers is the only one that matters. The two questions that IKEA faces are: what is an optimal inventory control policy, and does a coordination of inventory policies across retail stores and distribution centers help reduce overall inventory?

3 Periodic Review Stochastic Inventory Systems with Backlogging (Advanced)

In this section, we analyze periodic review stochastic inventory systems in which we seek the policy that minimizes the average inventory holding and ordering costs, subject to a service level constraint. A fixed order cost $K$ is incurred whenever a replenishment order is placed, while in addition there is a variable cost $c$ for each item purchased. In addition, we incur an inventory holding cost $h = rc$ per item per period. For convenience, we assume that stock is monitored at the end of each period, while ordering takes place at the beginning of a period. Orders also arrive (and its SKU’s are ready to be used) at the beginning of a period. Demand arises throughout any period and is modeled by a random variable that denotes the total demand in such a period. Because we assume a stationary demand process, we can denote the random periodic demand variable as $u_{R}$, with mean $\mu_{R}$ and standard deviation $\sigma_{R}$. We may choose $u_{R}$ to be either a discrete or a continuous random variable. Although discrete variables may be more natural, they also give rise to cumbersome notations and computational problems, therefore we have chosen here to model demand as a continuous random variable, with pdf $F_{R}$ and density function $f_{R}$.

A fundamental equation in many stochastic inventory models is the so-called newsvendor equation (see e.g. Silver et al. 2017) which will also form the start of our discussion. The name refers to the problem of a newsvendor who wonders how many newspapers to purchase at the beginning of a day. He or she earns an additional revenue on each newspaper sold but has to deal with unsold newspapers at the end of the day, which will incur disposition or salvage costs. Some reflection shows that this problem is similar to the determination of the inventory holding and penalty costs at the end of a period. At the beginning of the period, the total net inventory of an SKU is brought up to some quantity, say $y$, and demand is satisfied as much as possible during the period. Recall that in an inventory system with backordering, the net inventory may be negative (which is not realistic for the newsvendor problem). In particular, let

$$ H\left( y \right) = h\int_{0}^{y} {\left( {y - u} \right)f_{R} \left( u \right)du + p} \int_{y}^{\infty } {\left( {u - y} \right)f_{R} \left( u \right)du,\quad \quad y \ge 0,} $$

$$ H\left( y \right) = p\int_{0}^{\infty } {\left( {u - y} \right)f_{R} \left( u \right)du,\quad \quad y < 0,} $$

where, as before, $f_{R}$ denotes the density function of demand in a period of length $R$.

Now, assume that at the beginning of the period the net stock equals $x$ and that we wish to bring the stock up to a level $y$, which takes effect immediately (hence, with zero lead time). If a fixed cost $K$ is incurred when placing an order (next to the variable costs for each product), the total expected costs over the period are equal to $K + c\left( {y - x} \right) + H\left( y \right)$, if $y > x$, and $H\left( x \right)$, if $y = x$ (that is, if we order nothing). Note that $H\left( y \right)$ is a strictly convex function of $y$ and hence the same holds for $cy + H\left( y \right)$.

The following derivation closely follows Ross (1970). Let $cy + H\left( y \right)$ be minimized in $y = S$ and defined $s$ as the smallest value for which

$$ cs + H\left( s \right) = K + cS + H\left( S \right) $$

Now, we distinguish the following three cases:

$x > S$: then $cy + H\left( y \right) > cx + H\left( x \right)$ for all $y > x$, hence $K + c\left( {y - x} \right) + H\left( y \right) > H\left( x \right)$ for all $y > x$. Therefore, it is optimal not to place an order.

$s \le x \le S$: then $K + cy + H\left( y \right) \ge cx + H\left( x \right)$ for all $y > x$, hence $K + c\left( {y - x} \right) + H\left( y \right) \ge H\left( x \right)$. Once again, it is optimal not to place an order.

$x < s$: then $\mathop {\hbox{min} }\nolimits_{y \ge x} \left\{ {K + cy + H\left( y \right)} \right\} = K + cS + H\left( S \right) < cx + H\left( x \right)$, hence

$$ \mathop {\hbox{min} }\limits_{y \ge x} \left\{ {K + c\left( {y - x} \right) + H\left( y \right)} \right\} = K + c\left( {S - x} \right) + H\left( S \right) < H\left( x \right) $$

Therefore, in this case it is optimal to bring the inventory up to level $S$.

Now, for an $n$-period inventory problem, the total costs $C_{n} \left( x \right)$ in a discounted cost framework satisfies the following dynamic programming recursion:

$$ C_{n} \left( x \right) = \mathop {\hbox{min} }\limits_{y \ge x} \left\{ {K\partial \left( {y,x} \right) + c\left( {y - x} \right) + H\left( y \right) + \alpha \int_{0}^{\infty } {C_{n - 1} \left( {y - u} \right)f_{R} \left( u \right)du} } \right\} $$

(20.1)

where $\partial \left( {y,x} \right) = 1$ if $y > x$ and $\partial \left( {y,x} \right) = 0$ if $y = x$, and $\alpha < 1$ is a periodic discount factor.

Scarf (1960) initially showed, using a Dynamic Programming formulation, that for a stationary demand process, the optimal control policy under a discounted cost framework in each period $n$ is of the type $\left( {s_{n} ,S_{n} } \right)$. Using these results, Iglehart (1963) demonstrated that the sequence $\left( {s_{n} ,S_{n} } \right)$ converges to two fixed values $\left( {s,S} \right)$ for $n \to \infty$. Thus, in an infinite horizon problem, the same $\left( {s,S} \right)$-policy appears to be optimal in each period among all possible control policies. The same result holds when switching to an average cost criterion by considering the limiting behavior of $C_{n} \left( x \right)/n$ for $n \to \infty$ and letting $\alpha \to 1$.

The result also remains valid when assuming positive lead times $L$, where for convenience we assume that $L$ is a multiple of $R$ (note that this is not really a restriction because $R$ can be set to a small value, e.g., equal to one day). Basically, one needs to keep track of not only the current net stock position $x$ but also the goods that will arrive in the next $L - 1$ future periods $x_{1} , x_{2} , \ldots ,x_{L - 1}$ (based on decisions taken in the past), thereby leading to an $L$-dimensional state space. Note that $x + \sum\nolimits_{j = 1}^{L - 1} {x_{j} }$ equals the inventory position. Any decision to order an amount of goods $z$ at the beginning of period $t$ will take effect only when the order actually arrives at the beginning of period $t + L$. Using a reduction argument, Scarf (1960) showed that the dynamic programming recursion for the corresponding multi-variable costs $C_{n} \left( {x,x_{1} , x_{2} , \ldots ,x_{L - 1} } \right)$ can be reduced to a recursion similar to (20.1) for a related one-dimensional function $\hat{C}_{n} \left( {x + \sum\nolimits_{j = 1}^{L - 1} {x_{j} } } \right)$, in which $H\left( y \right)$ is replaced by a more complicated $L$-fold integral function. Using this reduction, it is then easily shown that the optimal policy in the stationary infinite horizon problem again is of the $\left( {s,S} \right)$-type, where the variable under consideration is the inventory position $\hat{x} = x + \sum\nolimits_{j = 1}^{L - 1} {x_{j} }$, i.e., if $\hat{x} \le s$ we order $S - \hat{x}$, while if $\hat{x} > s$, we order nothing.

The structural result on the optimality of an $\left( {s,S} \right)$-strategy for all possible control policies is extremely important but unfortunately, the calculation of the parameters $s$ and $S$ is a bit complicated, although various authors have developed algorithms that allow a fast computation (see e.g., Zheng and Federgruen 1991, who study the equivalent discrete state space problem). Below, we follow another approach which views the problem from a joint cost and service based perspective, instead of considering only a pure cost approach. Although we make slightly different choices when selecting the parameters $s$ and $S,$ our analysis resembles the one developed by Tijms and Groenevelt (1984). See also Silver et al. (2017).

Instead of minimizing the sum of the ordering, inventory holding, and penalty costs, we now consider an infinite horizon stationary demand problem in which we wish to minimize the average expected ordering and inventory holding costs, subject to a service level constraint. (Note that the average purchasing costs are constant and hence can be ignored in any optimization procedure). More precisely, we wish to consider policies which guarantee a long term average fill rate $\beta$, where the fill rate is defined as the fraction of demand that can be satisfied immediately from the available stock. In other words, the fraction of demand to be satisfied from physical stock should be larger than or equal to $\beta$ in the long run, while, as before, all unsatisfied demand is backlogged. Again, by equivalence it can be shown that the optimal policy for this problem is of the $\left( {s,S} \right)$-type. Now, we turn to the computation of the control parameters.

It may be helpful to review the role of the fixed ordering cost $K$ in relation to the inventory holding cost $h$. Typically, one may expect $S - s$ to increase as a function of the ratio $K/h$, as also argued in the discussion on the Economic Order Quantity in Chap. 12. Note however that, because we observe the inventory position only at discrete points in time, the order size will not be fixed but equal $S - s + z$ where $z$ is the so-called undershoot, i.e., the difference between the reorder point s and the actually observed inventory position at the time of re-order. The choice of the re-order point $s$ should reflect the target customer service level or fill rate $\beta$ and should take into consideration the expected undershoot.

By analogy it is easily shown that the demand process in subsequent periods is a renewal process (cf. Ross 1970), i.e., a sequence of independent identically distributed random variables $u_{n}$ with common distribution function $F_{R}$. For such a renewal process it is well known that in the long run, if one picks an arbitrary point in time, the expected time $t$ until the next renewal (often called the residual life time) has a density function equal to $\frac{1}{{\mu_{R} }}\left( {1 - F_{R} \left( t \right)} \right)$. Some reflection shows that in our inventory model the undershoot z exactly behaves like the residual lifetime in the analogous renewal process. Therefore, if $S - s$ is sufficiently large (hence if $K/h$ is large), the density function $f_{z}$ of the undershoot z approximately satisfies

$$ f_{z} \left( z \right) = \frac{1}{{\mu_{R} }}\left( {1 - F_{R} \left( z \right)} \right) $$

From the above, it is easy to show using partial integration, that the mean $\mu_{z}$ of z equals

$$ \mu_{z} = \mathop \int \limits_{0}^{\infty } zf_{z} \left( z \right)dz = \frac{1}{{\mu_{R} }}\mathop \int \limits_{0}^{\infty } z\left( {1 - F_{R} \left( z \right)} \right)dz = \frac{1}{{2\mu_{R} }}\mathop \int \limits_{0}^{\infty } z^{2} f_{R} \left( z \right)dz = \frac{{\sigma_{R}^{2} + \mu_{R}^{2} }}{{2\mu_{R} }} $$

where $\sigma_{R}^{2}$ denotes the variance of the periodic demand. At the end of this section we discuss what to do if $K/h$ is not ‘sufficiently’ large, i.e., if $K$ is small or even zero.

For now, if we decide to order, the average order size will be equal to $S - s + \frac{{\sigma_{R}^{2} + \mu_{R}^{2} }}{{2\mu_{R} }}$. Therefore, it seems reasonable, following the EOQ derivation, to make

$$ S - s + \frac{{\sigma_{R}^{2} + \mu_{R}^{2} }}{{2\mu_{R} }} = \sqrt {\frac{2KD}{h}} $$

where $D$ is the expected annual demand, i.e. $D = (N_{demand} /R) \mu_{R}$, $N_{demand}$ is the number of “demand days per year” (not necessarily equal to 365) and $R$ is expressed in days. However, this does not mean that an order always equals $\sqrt {\frac{2KD}{h}}$, but merely that, once we know how to calculate $s$, the order-up-to level $S$ follows immediately.

As mentioned previously, once we order, say at time $t$, the net stock position is influenced only $L$ periods later, i.e., at the beginning of period $t + L$. Hence, the remaining stock $s - z$ at time $t$ should be sufficient to cover demand in the next $L$ periods. In other words, at the end of period $t + L - 1$ (just before the order arrives), the net stock is equal to $s - z - u_{L}$, where $u_{L}$ denotes the demand during the lead time $L$. Note that we either observe some positive net stock (equal to the on-hand inventory $OH$) or a negative net stock (i.e., a backorder position $BO$). Let $x = z +$ $u_{L}$, with density function $f_{x}$, then we may compute the means $\mu_{OH}$ and $\mu_{BO}$ at the end of period $t + L - 1$, again by using partial integration, as follows:

$$ \begin{aligned} \mu_{OH} & = \int\limits_{0}^{s} {\left( {s - x} \right)f_{x} \left( x \right)dx = } \int\limits_{0}^{s} {\left( {s - x} \right)} \left[ {\int\limits_{0}^{x} {f_{z} \left( {x - u} \right)f_{L} \left( u \right)du} } \right]dx \\ & = \frac{1}{{\mu_{R} }}\int\limits_{0}^{s} {\left( {s - x} \right)} \left[ {\int\limits_{0}^{x} {\left( {1 - F_{R} \left( {x - u} \right)} \right)f_{L} \left( u \right)du} } \right]dx \\ & = \frac{1}{{\mu_{R} }}\int\limits_{0}^{s} {\left( {s - x} \right)\left( {F_{L} \left( x \right) - F_{R + L} \left( x \right)} \right)dx} \\ & = \frac{1}{{2\mu_{R} }}\left[ {\int\limits_{0}^{s} {\left( {s - x} \right)^{2} f_{L} \left( x \right)dx - } \int\limits_{0}^{s} {\left( {s - x} \right)^{2} f_{R + L} \left( x \right)dx} } \right] \\ \end{aligned} $$

and

$$ \begin{aligned} \mu_{BO} & = \int\limits_{s}^{\infty } {\left( {x - s} \right)f_{x} \left( x \right)dx = \int\limits_{s}^{\infty } {\left( {x - s} \right)} } \left[ {\int\limits_{0}^{x} {f_{z} \left( {x - u} \right)f_{L} \left( u \right)du} } \right]dx \\ & = \frac{1}{{\mu_{R} }}\int\limits_{s}^{\infty } {\left( {x - s} \right)} \left[ {\int\limits_{0}^{x} {\left( {1 - F_{R} \left( {x - u} \right)} \right)f_{L} \left( u \right)du} } \right]dx \\ & = \frac{1}{{\mu_{R} }}\int\limits_{s}^{\infty } {\left( {x - s} \right)\left( {F_{L} \left( x \right) - F_{R + L} \left( x \right)} \right)dx} \\ & = \frac{1}{{\mu_{R} }}\int\limits_{s}^{\infty } {\left( {x - s} \right)\left[ {\left( {1 - F_{R + L} \left( x \right)} \right) - \left( {1 - F_{L} \left( x \right)} \right)} \right]dx} \\ & = \frac{1}{{2\mu_{R} }}\left[ {\int\limits_{s}^{\infty } {\left( {x - s} \right)^{2} f_{R + L} \left( x \right)dx - } \int\limits_{s}^{\infty } {\left( {x - s} \right)^{2} f_{L} \left( x \right)dx} } \right] \\ \end{aligned} $$

The safety stock $SS$ is generally defined as the net stock just before an order arrives, i.e.,

$$ SS = s - \mu_{z} - \mu_{L} = s - \frac{{\sigma_{R}^{2} + \mu_{R}^{2} }}{{2\mu_{R} }} - \mu_{L} $$

The reader may easily verify that $SS = \mu_{OH} - \mu_{BO}$.

The re-order point $s$ is now determined from the fill rate condition, i.e.

$$ \mu_{BO} = \left( {1 - \beta } \right) \sqrt {\frac{2KD}{h}} , $$

or

$$ \frac{1}{{2\mu_{R} }}\left[ {\int\limits_{s}^{\infty } {\left( {x - s} \right)^{2} f_{R + L} \left( x \right)dx - } \int\limits_{s}^{\infty } {\left( {x - s} \right)^{2} f_{L} \left( x \right)dx} } \right] = \left( {1 - \beta } \right)\sqrt {\frac{2KD}{h}} $$

(20.2)

and subsequently the order-up-to level $S$ from

$$ S = s - \frac{{\sigma_{R}^{2} + \mu_{R}^{2} }}{{2\mu_{R} }} + \sqrt {\frac{2KD}{h}} $$

Note that for an arbitrary demand distribution function, the left part of Eq. (20.2) may not be easy to determine. For a normal distribution function tables exist, see for example, Silver et al. (2017). However, we follow another approach. Because both $u_{R + L}$ and $u_{L}$ are the addition of a number of independent, identically distributed (iid) random variables with known mean and variance, their mean and variance are easily determined. Next, we may fit a mixture of Erlang distributions on the mean and variance or each variable (cf. De Kok 1989; or Tijms 1994) which enables an easy calculation of the left-hand expression of (16.2). It is well known that the class of mixtures of Erlang distributions is dense in the space of all pdf’s, i.e., any pdf can be approximated arbitrarily close by a mixture of Erlang distributions (Schassberger 1973).

Finally, we consider the case that the ratio $K/h$ is small or equal to $0$. From the initial discussion of $\left( {R,s,S} \right)$-policies, it immediately follows that $s = S,$ if $K = 0$. Following the same logic as before, we then find that the optimal policy in a periodic review infinite horizon inventory model is of the type $\left( {R,S} \right)$, i.e., at the beginning of each period we bring the inventory position up to level $S$. This means that the stock used to fulfill the previous period demand is replenished. Indeed, $\left( {R,S} \right)$-systems are often used when a large number of different SKU’s is ordered periodically (e.g., every week) from the same supplier or by using one logistics service provider. If a periodic delivery schedule for a large number of different SKU’s has been predetermined, then the main ordering costs are fixed throughout the year and hence we may ignore them in the optimization procedure.

If $R$ is predetermined in an $\left( {R,S} \right)$ system (which is usually the case) then we are left with a one parameter problem, i.e., the determination of $S$. Let us take a service perspective again, i.e., we wish to minimize inventory holding costs, subject to a fill rate constraint. Note that an order placed at time $t$ will arrive at time $t + L$, and any order issued later will not arrive earlier than at time $t + L + R$. That means, the inventory position after ordering at time $t$ should be sufficient to cover the cumulative demand in the time interval $\left[ {t, t + L + R} \right)$. Therefore, the expected on-hand inventory at $t + L + R$ (i.e., just before arrival of the next order) equals

$$ \mu_{OH} = \int\limits_{0}^{S} {\left( {S - x} \right)f_{L + R} \left( x \right)dx} $$

Similarly, the expected backorder position at time $t + L + R$, before the arrival of the next order, is

$$ \mu_{BO} = \int\limits_{S}^{\infty } {\left( {x - S} \right)f_{L + R} \left( x \right)dx} $$

Therefore, noting that on average we order an amount $\mu_{R}$ at every review instance, to satisfy a target fill rate $\beta$, $S$ should be chosen such that

$$ \int\limits_{S}^{\infty } {\left( {x - S} \right)f_{L + R} \left( x \right)dx = \left( {1 - \beta } \right)\mu_{R} } $$

In the special case of $\left( {R,S} \right)$ inventory systems, we can find an explicit relation between cost and service models. Assume, as before, that inventories are charged at a rate $h = rc$ per item per period, while a shortage is penalized at a rate $p$ per item per period (all costs are charged at the end of a period, just before a new order arrives). Note that, if the net stock at the beginning of interval $t + L$ ($t$ being a review time instant) equals $y$ (just after arrival of an order placed at time $t$) then the expected costs just before the next review moment $t + L + R$ are equal to $H\left( y \right)$ (see the Newsboy equation discussed in the beginning of Sect. 20.3). Hence, if at time $t$ we issue an order such as to return the inventory position back to $S$, then the expected costs as a result of that action at the end of period $t + L + R$ are

$$ \begin{aligned} C\left( S \right) & = \int\limits_{0}^{\infty } {H\left( {S - u} \right)f_{L} \left( u \right)du} \\ & = h\int\limits_{0}^{S} {\left( {S - u} \right)f_{L + R} \left( u \right)du + p} \int\limits_{S}^{\infty } {\left( {u - S} \right)f_{L + R} \left( u \right)du} \\ \end{aligned} $$

where the last equality follows from taking convolutions of $f_{R}$ and $f_{L}$. Putting the derivative of $C\left( S \right)$ equal to zero yields the following.

$$ hF_{L + R} \left( S \right) - p\left( {1 - F_{L + R} \left( S \right)} \right) = 0 $$

or

$$ F_{L + R} \left( S \right) = \frac{p}{h + p} $$

Thus, knowing the distribution of demand in a time-interval of length $R + L$, we can explicitly derive the optimal order-up-to level $S$ as a function of $p$, and also the corresponding fill rate $\beta$, via

$$ \beta = 1 - \frac{1}{{\mu_{R} }}\int\limits_{S}^{\infty } {\left( {x - S} \right)f_{L + R} \left( x \right)dx} $$

It is intuitively clear, and it can be proven rigorously, that the fill rate determined in this way is a monotone increasing function of $p$, and vice versa. Hence, by applying a simple bisection procedure, it is easy to determine the value of $p$ (and $S$) that corresponds to a target fill rate.

In the next section, we will discuss multi-stage or multi-echelon inventory systems, in which at each stage $n$ an $\left( {R,S_{n} } \right)$-policy is applied, again under a pure cost framework. The determination of the fill rate at the most downstream stage (i.e., the stage facing independent market demand) appears to be a bit more tricky but builds on the cost analysis of a single-stage $\left( {R,S} \right)$ system.

4 Multi-stage, Periodic Review Inventory Systems (State-of-the-Art)

In this section, we discuss the so-called multi-stage or multi-echelon systems. In the first part, we concentrate on strictly linear systems, consisting of $N$ stages where each stage receives goods from its predecessor and delivers products to its successor. The most upstream stage orders materials from an external supplier and the most downstream stage faces customer demand, see Fig. 20.2.

Fig. 20.2

A linear multi-stage inventory system

One may think of a chain consisting of a production facility, followed by a central warehouse, followed by a retail shop. Inventory holding costs are incurred at each stage where the final stage may incur additional penalty costs. For example, due to an out-of-stock situation, customer demand cannot be satisfied immediately and must be backlogged. After analyzing serial systems, we turn to 2-stage distribution systems, in which a central depot supplies several local warehouses.

4.1 Linear Multi-stage Inventory Systems

Consider first a two-stage linear inventory system, consisting of two stages, denoted by $I_{2}$ and $I_{1}$ (following the flow of goods). $I_{2}$ can order material from an external supplier (with unlimited stock), which is received after a lead time $L_{2}$, while $I_{1}$ orders material from $I_{2}$, which is received after a lead time $L_{1}$, assuming all materials are available at $I_{2}$ when ordered. If $I_{2}$ is short of materials, it delivers as much as possible while the remaining goods are backordered. $I_{2}$ charges an inventory holding cost per item per period equal to $h_{2}$, for items stored at the installation and still in transit to $I_{1}$. Items available at $I_{1}$ are charged at an inventory holding cost of $h_{2} + h_{1}$ per item per period while a penalty $p$ per item per period is charged in case of a shortage at stage 1. The additional holding costs $h_{1}$ in stage 1 may reflect the value added for items in transit between $I_{2}$ and $I_{1}$, e.g., when the transition from $I_{2}$ to $I_{1}$ represents a production phase, whereas the upstream transition to $I_{2}$ is the materials supply from an external source. Note that a backlog at $I_{2}$ is not charged (at least not directly).

Now define the net echelon stock of $I_{1}$ simply as its net stock, and the net echelon stock of $I_{2}$ as the stock at stage 2 plus all items in transit from $I_{2}$ to $I_{1}$ as well as the net stock of $I_{1}$. Hence, the net echelon stock of $I_{2}$ includes the net (echelon) stock of $I_{1}$. Note that the net echelon stock of $I_{2}$ can be negative although there may still be items in transit to $I_{1}$. The echelon inventory position of each stage is defined as that stage’s net stock plus all products ordered but not yet received. Note that the inventory position of $I_{1}$ does not simply include all items on their way from $I_{2}$, but possibly also items still on their way to $I_{2}$ (i.e., when $I_{2}$ faces a backlog), in which case $I_{1}$ may experience a lead time larger than $L_{1}$ for these delayed items.

The concept of echelon stock has been introduced by Clark (1958), after which Clark and Scarf (1960) were the first to show that in a multi-stage system with no fixed ordering costs, an optimal inventory control policy is of the type $\left( {S_{1} ,S_{2} } \right)$, meaning that at the beginning of each period each stage $I_{n}$ brings its echelon inventory position up to $S_{n}$ $n = 1,2$, in an infinite horizon discounted cost framework. The average cost analysis presented below is due to Langenhoff and Zijm (1990) while the computational analysis based upon this framework stems from Van Houtum and Zijm (1991).

Define $v_{1}$ and $v_{2}$ as the net echelon stock of $I_{1}$ and $I_{2}$, respectively, hence $v_{1} \le v_{2} .$ Consider the following cases:

• a. $v_{1} \ge 0$. Because $v_{2} \ge v_{1} ,$ the stock at $I_{2}$ plus in transit between $I_{2}$ and $I_{1}$ equals $v_{2} - v_{1}$. Then the total costs summed over the two stages are equal to

$$ (h_{1} + h_{2} )v_{1} + h_{2} \left( {v_{2} - v_{1} } \right) = (h_{1} + h_{2} )v_{1} - h_{2} v_{1} + h_{2} v_{2} $$

• b. $v_{1} < 0$. Then $I_{1}$ faces a backlog while the physical stock at $I_{2}$ plus in transit between $I_{2}$ and $I_{1}$ equals $v_{2} - v_{1}$ (note that $v_{2}$ may be both positive or negative). The total costs summed of the two stages are therefore equal to

$$ h_{2} \left( {v_{2} - v_{1} } \right) + p( - v_{1} ) = p( - v_{1} ) - h_{2} v_{1} + h_{2} v_{2} $$

From the above analysis, we conclude that it is natural to attribute costs $h_{2} v_{2}$ to the echelon stock $v_{2}$, independent of its sign, while we attribute either $(h_{1} + h_{2} )v_{1} - h_{2} v_{1}$ (if $v_{1} \ge 0$) or $p( - v_{1} ) - h_{2} v_{1}$ (if $v_{1} < 0$) to $x_{1}$. Note that the second term in the costs attributed to $v_{1}$ is again independent of its sign. If the net echelon stock of $I_{1}$ and $I_{2}$ is increased to $y_{1}$ and $y_{2}$ at the beginning of a period respectively, then the expected costs at the end of the period are equal to $H_{1} \left( {y_{1} } \right) + H_{2} \left( {y_{2} } \right)$, where

$$ \begin{aligned} H_{1} \left( {y_{1} } \right) & = \left( {h_{1} + h_{2} } \right)\int\limits_{0}^{{y_{1} }} {\left( {y_{1} - u} \right)f_{R} \left( u \right)du + p} \int\limits_{{y_{1} }}^{\infty } {\left( {u - y_{1} } \right)f_{R} \left( u \right)du} \\ & \quad - h_{2} \int\limits_{0}^{\infty } {\left( {y_{1} - u} \right)f_{R} \left( u \right)du} \\ & = h_{1} \int\limits_{0}^{\infty } {\left( {y_{1} - u} \right)f_{R} \left( u \right)du} \\ & \quad + \left( {p + h_{1} + h_{2} } \right)\int\limits_{{y_{1} }}^{\infty } {\left( {u - y_{1} } \right)f_{R} \left( u \right)du} \\ \end{aligned} $$

$$ H_{2} \left( {y_{2} } \right) = h_{2} \int\limits_{0}^{\infty } {\left( {y_{2} - u} \right)f_{R} \left( u \right)du} $$

Finally, suppose that at some review time $t,$ an order is placed by $I_{2}$ so as to increase its echelon inventory position to $Y_{2}$ and, at time $t + L_{2}$, an order is placed by $I_{2}$ so as to increase its echelon inventory position to $Y_{1}$. Note that if there is a shortage of items in stage $I_{2}$) at time $t + L_{2}$, $I_{2}$ may not be able to ship the requested amount to $I_{1} ,$ in which case it faces a backlog. Define, similar as in the preceding section,

$$ C_{n} \left( {Y_{n} } \right) = \int\limits_{0}^{\infty } {H_{n} \left( {Y_{n} - u} \right)f_{{L_{n} }} \left( u \right)du,\quad \quad n = 1,2} $$

It is now easily verified that the expected costs at the end of period $t + L_{2} + L_{1}$ are equal to

$$ C_{2} \left( {Y_{1} ,Y_{2} } \right) = C_{1} \left( {Y_{1} } \right) + C_{2} \left( {Y_{2} } \right) + \int\limits_{{Y_{2} - Y_{1} }}^{\infty } {\left( {C_{1} \left( {Y_{2} - u} \right) - C_{1} \left( {Y_{1} } \right)} \right)f_{{L_{2} }} \left( u \right)du} $$

where the last term reflects the situation that at time $t + L_{2}$ installation $I_{2}$ has a net echelon stock of $Y_{2} - u_{{L_{2} }} < Y_{1}$ (where $u_{{L_{2} }}$ denotes the demand in the interval $\left[ {t,t + L_{2} } \right)$), hence $I_{2}$ is unable to raise the inventory position of $I_{1}$ to $Y_{1}$. Therefore $I_{1}$ will only have a net stock position of $Y_{2} - u_{{L_{2} }}$ just after arrival of the (partial) order from $I_{2}$ at the beginning of period $t + L_{2} + L_{1}$. That is, the last term reflects some indirect penalty cost for $I_{2}$.

Langenhoff and Zijm (1990) show that the parameters $\left( {S_{1} ,S_{2} } \right)$ that optimize the overall cost function can be derived sequentially by first minimizing $C_{1} \left( {Y_{1} } \right)$, yielding $S_{1}$, and subsequently $C_{2} \left( {S_{1} ,Y_{2} } \right)$, yielding $S_{2}$. This decomposition result was initially proved by Clark and Scarf (1960) in a discounted cost framework and significantly reduces the computation of the optimal base stock policies in a two-stage system.

The results above can be extended to an $N$-stage linear system (with inventory holding costs $h_{n}$ and lead times $L_{n} ,n = 1,2, \ldots ,N$, and penalty costs $p$ for the most downstream stage) as follows. Define

$$ H_{1} \left(y_{1} \right) = h_{1} \int\limits_{0}^{\infty} \left(y_{1} - u \right)f_{R}(u)du + \left( p + \sum\limits_{n = 1}^{N} h_{n}\right) \int\limits_{y_{1}}^{\infty} \left(u - y_{1}\right) f_{R}(u)du $$

$$ H_{n} \left( {y_{n} } \right) = h_{n} \int\limits_{0}^{\infty } {\left( {y_{n} - u} \right)f_{R} \left( u \right)du,\quad \quad n = 2, \ldots N} $$

Next, define

$$ C_{n} \left( {Y_{n} } \right) = \int\limits_{0}^{\infty } {H_{n} \left( {Y_{n} - u} \right)f_{{L_{n} }} \left( u \right)du,\quad \quad n = 1,2, \ldots , N} $$

and recursively

$$ \begin{aligned} C_{n} & \left( {Y_{1} ,Y_{2} , \ldots ,Y_{n} } \right) = C_{n} \left( {Y_{n} } \right) + C_{n - 1} \left( {Y_{1} ,Y_{2} , \ldots ,Y_{n - 1} } \right) \\ & + \int\limits_{{Y_{n} - Y_{n - 1} }}^{\infty } {\left( {C_{n - 1} \left( {Y_{1} ,Y_{2} , \ldots ,Y_{n - 2} ,Y_{n} - u)} \right) - C_{n - 1} \left( {Y_{1} ,Y_{2} , \ldots ,Y_{n - 1} } \right)} \right)f_{{L_{n} }} \left( u \right)du} \\ \end{aligned} $$

The optimal control policy for the $N$-stage serial inventory system is a base-stock policy characterized by parameters $S_{1} ,S_{2} , \ldots ,S_{N}$. These parameters can be determined sequentially by minimizing $C_{1} \left( {Y_{1} } \right)$, $C_{2} \left( {S_{1} ,Y_{2} } \right)$, and so forth … up to $C_{N} \left( {S_{1} ,S_{2} , \ldots ,S_{N - 1} ,Y_{N} } \right)$.

The importance of the concept of net echelon stock and echelon inventory position (Clark 1958) can hardly be overestimated. The fact that the echelon stock covers all inventories for each stage, downstream up to delivery to the market, implies that stages using echelon stock based control policies always base their decision on the forecasted or actual market demand, and not on the demand of the next downstream stage. It is this phenomenon that prevents the amplification of stock variation (as a reaction on demand variation) in a multi-stage system, initially discussed by Forrester (1961) and later analyzed as the so-called Bullwhip effect by e.g., Lee et al. (1997). Naturally, under a centralized control policy, prevention of the Bullwhip effect is easier than in a multi-stage system involving multiple companies or stakeholders. In the latter case, contracts between these companies that include penalties in case of delivery failures should serve to prevent temporary shortages, see e.g., Cachon and Zipkin (1999) or Zijm and Timmer (2008).

4.2 Two-Stage Distribution Systems

Next, we turn to two-stage distribution models in which one central depot regularly ships products to a number of local warehouses, while stock in this central depot is replenished by an external supplier who is always able to deliver. As before, both the depot and each local warehouse places an order each review time such that its echelon inventory position is brought back to a target level (i.e., stages apply an order-up-to level policy). Any demand experienced at any local warehouse as well as at the depot that cannot be fulfilled immediately is backlogged.

What makes the control problem in a distribution system different from that of a serial system is the fact that an allocation decision must be made in case of a (temporary) shortage of depot stock. That is, if the sum of the demands of the local warehouses exceeds the available stock in the central depot at any review and ordering instant, we must decide how much to ship to each local warehouse while backlogging the remaining demand.

Below, we formally define the two-stage distribution model, cf. Fig. 20.3. The upstream stage (the central depot) is indexed by $N + 1$, while the downstream stages (the local warehouses) are numbered $1,2, \ldots , N$. All stages order at the same time, i.e., at the beginning of a period of length $R$. Lead times are denoted by $L_{N + 1}$ (from external supplier to central depot) and $L_{1} , L_{2} , \ldots ,L_{N}$ (from central depot to the local warehouses), inventory holding costs are $h_{N + 1}$ at the central depot and $h_{N + 1} + h_{n}$ for all stock in transfer to or available at local warehouse $n, n = 1, \ldots ,N$. Each local warehouse $n$ incurs a penalty cost $p_{n}$ per unit per period in the event of a shortage. All costs are incurred based on the net stock at the end of a period. The periodic demand experienced by local warehouse $n$ is denoted by its distribution function $F_{R}^{\left( n \right)}$ (with density function $f_{R}^{\left( n \right)}$) and hence the cumulative periodic customer demand is determined by the convolution $F_{R} = F_{R}^{\left( 1 \right)} *F_{R}^{\left( 2 \right)} * \cdots *F_{R}^{\left( N \right)}$. The cost functions for the local warehouses are derived similarly to those for the final stage in a serial system. Let $y_{n}$ denote the net stock of local warehouse $n \left( {n = 1, \ldots ,N} \right)$ and let $y_{N + 1}$ denote the net echelon stock of the central depot, all at the beginning of a period of length $R$, just after arrival of possible orders. Because costs are incurred at the end of a period, we can define, as before,

Fig. 20.3

A two-stage distribution inventory model

$$ \begin{aligned} H_{n} \left( {y_{n} } \right) & = h_{n} \int\limits_{0}^{\infty } {\left( {y_{n} - u} \right)f_{R}^{\left( n \right)} \left( u \right)du + } \\ & \quad \quad \left( {p_{n} + h_{n} + h_{N + 1} } \right)\int\limits_{{x_{n} }}^{\infty } {\left( {u - y_{n} } \right)f_{R}^{\left( n \right)} \left( u \right)du\quad \quad n = 1,2, \ldots ,N} \\ \end{aligned} $$

$$ H_{N + 1} \left( {y_{N + 1} } \right) = h_{N + 1} \int\limits_{0}^{\infty } {\left( {y_{N + 1} - u} \right)f_{R} \left( u \right)du} $$

(recall that the demand density $f_{R}$ is the convolution of the individual local warehouse demand densities $f_{R}^{\left( n \right)}$, and hence represents the cumulative downstream periodic demand that equals the decrease of the net inventory position of the central depot in a period of length $R$). Now, suppose that at some review time $t$, the central depot manager orders an amount of goods from the external supplier and thereby brings the echelon inventory position of the central depot up to level $Y_{N + 1}$. The order arrives at time $t + L_{N + 1}$, after which the net echelon stock of the central depot equals $Y_{N + 1} - u_{{L_{N + 1} }}$ where $u_{{L_{N + 1} }}$ is the cumulative customer demand experienced between $t$ and $t + L_{N + 1} .$ Recall that the net echelon stock of a stage includes all inventories downstream as well. Because costs are incurred at the end of the next period of length $R$ we define, as before

$$ C_{N + 1} \left( {Y_{N + 1} } \right) = \int\limits_{0}^{\infty } {H_{N + 1} \left( {y_{N + 1} - u} \right)f_{{L_{N + 1} }} \left( u \right)du} $$

(where $f_{{L_{N + 1} }}$ is the convolution of the individual local warehouse demand densities in a period of length $L_{N + 1}$). Next, let the managers of the local warehouses each place an order to raise their inventory position to $Y_{n} ,n = 1, \ldots ,N,$ at time $t + L_{N + 1}$. Now, if

$$ Y_{N + 1} - u_{{L_{N + 1} }} \ge \mathop \sum \limits_{n = 1}^{N} Y_{n} $$

then all local warehouse orders placed at time $t + L_{N + 1}$ can be fulfilled by the central depot manager, and the remaining stock is held at the central depot for at least one more period. If, on the other hand

$$ Y_{N + 1} - u_{{L_{N + 1} }} < \mathop \sum \limits_{n = 1}^{N} Y_{n} $$

then a decision has to be made on how much products to send to each local warehouse, i.e., how much stock to allocate to each local warehouse. In other words: what is a good allocation policy in case the central depot temporarily falls short? How to select order-up-to-levels $z_{n} = z_{n} \left( {Y_{N + 1} - u_{{L_{N + 1} }} } \right)$ such that

$$ \mathop \sum \limits_{n = 1}^{N} z_{n} \left( {Y_{N + 1} - u_{{L_{N + 1} }} } \right) = Y_{N + 1} - u_{{L_{N + 1} }} $$

To answer that question, we need to determine the cost effects of any partial order fulfillment $z_{n}$ of local warehouse $n$. Note that, if at time $t + L_{N + 1}$ the manager of local warehouse $n$ increases its inventory position to $z_{n}$, the net stock of that warehouse at time $t + L_{N + 1} + L_{n}$ equals $z_{n} - u_{{L_{n} }}^{\left( n \right)}$, where $u_{{L_{n} }}^{\left( n \right)}$ denotes the demand experienced by warehouse $n$ in a time period of length $L_{n}$. Therefore, the costs experienced by local warehouse n at time $t + L_{N + 1} + L_{n} + R$ equal

$$ C_{n} \left( {z_{n} } \right) = \int\limits_{0}^{\infty } {H_{n} \left( {z_{n} - u} \right)f_{{L_{n} }}^{\left( n \right)} \left( u \right)du,\quad \quad n = 1,2, \ldots ,N,} $$

Hence, it seems reasonable to apply a simple myopic allocation policy (MYAL), by solving the following non-linear optimization problem

$$ \mathop {\hbox{min} }\limits_{{z_{1} , \ldots ,z_{N} }} \mathop \sum \limits_{n = 1}^{N} C_{n} \left( {z_{n} } \right) $$

subject to

$$ \mathop \sum \limits_{n = 1}^{N} z_{n} = v_{N + 1} $$

$$ z_{n} \ge w_{n} , n = 1,2, \ldots ,N $$

where $v_{N + 1}$ denotes the available net echelon stock of the central depot and $w_{n}$ is the inventory position of local warehouse $n$ just before ordering.

Because the functions $C_{n} \left( {y_{n} } \right)$ are convex, this non-linear optimization problem is easily solved, see e.g., Langenhoff and Zijm (1990). The problem however is that the optimal solution and therefore also the joint optimal costs are functions of not only $v_{N + 1}$ but also of $w_{n} , n = 1,2, \ldots ,N$, i.e., of the specific inventory positions just before ordering at the local warehouses and not just of their sum $\mathop \sum \nolimits_{n = 1}^{N} w_{n}$. Therefore, a decomposition result similar to the one for serial systems is not straightforward and indeed Clark and Scarf (1960) already observed that the optimal policies in a two-stage distribution network are not necessarily base stock policies. However, by making a simple additional assumption, a decomposition result similar to the one for serial systems is still within reach.

The second constraint in the above formulated allocation problem states that we cannot lower the net inventory position of any local warehouse. Without that constraint the following scenario could occur. In the case of a serious temporary shortage of stock at the central depot, a cost-optimal solution is only possible if the net inventory position at some local warehouses is reduced to help other warehouses to bring their stock to an acceptable level, by applying lateral transshipments. If we do not allow lateral transshipments and still want to skip the final inequalities in the constraint set, we must assume that a cost-optimal solution can be reached without lowering any local warehouse’s net inventory position. This is called the balance assumption (cf. Eppen and Schrage 1981), i.e., we assume that a very odd distribution of inventories over the various local warehouses never occurs and, although the central depot stock may not be sufficient to raise each local warehouse’s inventory position to its target base stock level, the cost-optimal solution allows each local warehouse to at least not reduce its inventory position (and for most of them, to increase it). Under the balance assumption we may apply a Relaxed Myopic Allocation (REMYAL) policy which only depends on $v_{N + 1}$, as a solution of

$$ \mathop {\hbox{min} }\limits_{{z_{1} , \ldots ,z_{N} }} \mathop \sum \limits_{n = 1}^{N} C_{n} \left( {z_{n} } \right) $$

subject to

$$ \mathop \sum \limits_{n = 1}^{N} z_{n} = v_{N + 1} $$

The assumption implies that the solution still satisfies $z_{n} \left( {v_{N + 1} } \right) \ge w_{n} , n = 1,2, \ldots ,N$, i.e., that negative allocation quantities never occur. Fortunately, it turns out that the balance assumption is not a very serious restriction, cf. Van Donselaar and Wijngaard (1987), Eppen and Schrage (1981), Federgruen and Zipkin (1984), not because imbalance never occurs but because its impact on the optimal costs are almost negligible, see also Doğru et al. (2009).

The optimal REMYAL policy is easily found by applying a simple Lagrange multiplier technique, based on the convexity of the functions $C_{n} \left( {z_{n} } \right)$, from the following set of equations (in which $\lambda$ is the Lagrange multiplier)

$$ \frac{\partial }{{\partial y_{n} }}C_{n} \left( {z_{n} } \right) = h_{n} - \left( { p_{n} + h_{n} + h_{N + 1} } \right)\left( {1 - F_{{L_{n} + R}}^{\left( n \right)} \left( {z_{n} } \right)} \right) = \lambda , $$

$$ n = 1,2, \ldots ,N $$

$$ \mathop \sum \limits_{n = 1}^{N} z_{n} = v_{N + 1} $$

We now have a set of $N + 1$ equations for the $N + 1$ unknown variables $z_{1} , \ldots ,z_{N} , \lambda ,$ which is easily solved numerically. If all downstream cost parameters are identical, i.e., if $h_{n} = h$ and $p_{n} = p$ for $n = 1,2, \ldots ,N$, then we even find a closed-form expression, i.e.

$$ F_{{L_{n} + R}}^{\left( n \right)} \left( {z_{n} } \right) = \frac{{p + h_{N + 1} + \lambda }}{{p + h_{N + 1} + h}} \quad \quad n = 1,2, \ldots ,N $$

which is known as the equal fractile rule, cf. Eppen and Schrage (1981) who in addition assumed equal lead times. Moreover, if the demand distribution functions $F_{{L_{n} + R}}^{\left( n \right)}$ satisfy a normalization property, i.e., if there exists a distribution function $\varphi$ such that

$$ F_{{L_{n} + R}}^{\left( n \right)} \left( {z_{n} } \right) = \varphi \left( {\frac{{z_{n} - \mu_{{L_{n} + R}}^{\left( n \right)} }}{{\sigma_{{L_{n} + R}}^{\left( n \right)} }}} \right),\quad \quad n = 1,2, \ldots N, $$

where $\mu_{{L_{n} + R}}^{\left( n \right)}$ and $\sigma_{{L_{n} + R}}^{\left( n \right)}$ denote the mean and variance of demand experienced by local warehouse n in a period of length $L_{n} + R$, then it is easy to show that

$$ z_{n} = z_{n} \left( {v_{N + 1} } \right) = \mu_{{L_{n} + R}}^{\left( n \right)} + \sigma_{{L_{n} + R}}^{\left( n \right)} \frac{{v_{N + 1} - \hat{\mu }}}{{\hat{\sigma }}} $$

(20.3)

with $\hat{\mu } = \mathop \sum \nolimits_{n = 1}^{N} \mu_{{L_{n} + R}}^{\left( n \right)}$ and $\hat{\sigma } = \sqrt {\mathop \sum \nolimits_{n = 1}^{N} \left( {\sigma_{{L_{n} + R}}^{\left( n \right)} } \right)^{2} }$. In other words: the allocation quantity $z_{n} \left( {v_{N + 1} } \right)$ is a linear function of $v_{N + 1}$. The normalization property holds for a number of demand distribution functions, including the normal distribution, and is trivially satisfied if all local warehouses behave identically, i.e., face similar demand distributions and have equal cost and lead time parameters.

Under the balance assumption, a decomposition result similar to the one for serial systems can also be proven for distribution systems. Define the following cost function

$$ \begin{aligned} & C_{N + 1} \left( {Y_{1} ,Y_{2} , \ldots ,Y_{N} ,Y_{N + 1} } \right) = C_{N + 1} \left( {Y_{N + 1} } \right) + \mathop \sum \limits_{n = 1}^{N} C_{n} \left( {Y_{n} } \right) \\ & \quad + \int\limits_{{Y_{N + 1} - \mathop \sum \nolimits_{n = 1}^{N} Y_{n} }}^{\infty } {\mathop \sum \limits_{n = 1}^{N} \{ C_{n} \left( {z_{n} (Y_{N + 1} - u} \right)) - C_{n} \left( {Y_{n} } \right)\} f_{{L_{N + 1} }} \left( u \right)du} \\ \end{aligned} $$

where the final term denotes the induced penalty costs which are incurred if the central depot cannot immediately satisfy the demand of the $N$ downstream local warehouses, i.e. $Y_{N + 1} - u_{{L_{N + 1} }} < \mathop \sum \nolimits_{n = 1}^{N} Y_{n}$ and therefore has to allocate the available stock according to the allocation rule $z_{n} \left( {Y_{N + 1} - u_{{L_{N + 1} }} } \right)$. It can be shown that the optimal control policy for such a distribution system is a base stock policy, which is characterized by order-up-to levels $S_{1} ,S_{2} , \ldots ,S_{N} ,S_{N + 1} .$ For $n = 1, \ldots ,N$, the optimal parameters $S_{n}$ are obtained from minimizing the cost functions $C_{n} \left( {Y_{n} } \right)$. Subsequently, $S_{N + 1}$ is obtained from minimizing the cost function $C_{N + 1} \left( {S_{1} ,S_{2} , \ldots ,S_{N} ,Y_{N + 1} } \right)$. Hence, as before, the multi-dimensional optimization problem can be reduced to a series of one-dimensional (convex) optimization problems.

Note that the calculation of the base stock level for the central depot requires the calculation of the allocation function $z_{n} \left( {v_{N + 1} } \right)$ for $n = 1,2, \ldots N$, and in principle for all possible values of $v_{N + 1}$. This severely complicates numerical solutions. If $z_{n} \left( {v_{N + 1} } \right)$ is linear, as for instance in (20.3), it appears to be possible to express $C_{N + 1} \left( {Y_{1} ,Y_{2} , \ldots ,Y_{N} ,Y_{N + 1} } \right)$ as a type of newsvendor function which facilitates the calculation considerably. In the final section, we briefly discuss numerical solution procedures. For a complete analysis, the reader is referred to Van Houtum and Zijm (1991, 1997) and to Van Houtum et al. (1996).

5 Computational Procedures, Based on Incomplete Convolutions (State-of-the-Art)

When evaluating the cost functions of multi-stage serial or distribution systems, we often encounter the situation that the demand distribution functions are generally unknown and at best characterized by their mean and variance, or alternatively by their mean and coefficient of variation. Even if the distribution functions are known exactly (which is rarely the case in any practical setting), we need numerical approximations for the integral forms in the cost function. To that end, Van Houtum and Zijm (1991) propose to approximate the distribution functions by mixtures of Erlang distributions, or to fit mixtures of Erlang distributions on the mean and coefficient of variation. This is a well-known procedure, based on a result of Schassberger (1973) who proved that any arbitrary distribution function can be approximated arbitrarily close by a mixture of Erlang distributions. Accurate fitting procedures are discussed by Tijms (1994). Such mixtures appear to be fairly tractable, even when arising in sometimes complex convolutions as is the case in multi-period, multi-echelon inventory systems. A detailed description of the numerical analysis of multi-echelon systems by means of approximations based on mixtures of Erlang distributions is beyond the scope of this chapter. The reader is referred to Van Houtum and Zijm (1991) for details, including all proofs of results mentioned above.

6 Further Reading

As mentioned in Sect. 20.1, there are numerous papers and books dealing with mathematical inventory theory. A rather complete overview is presented in Silver et al. (2017), which is an update of former versions by partially the same authors. A deeper analytical treatment of some topics can be found in Axsäter (2000), while Zipkin (2000) presents an excellent overview of the foundations of inventory theory. A more managerial oriented book is Fogarty et al. (1991). The case study of IKEA is based on information from a Master’s Thesis project carried out at Lund University, see Rasmusson and Sunesson (2009). In addition, there exists significant theory on spare parts inventory management, see e.g., Sherbrooke (2004), Muckstadt (2005) and Van Houtum and Kranenburg (2015), and Chap. 22 on Maintenance Service Logistics in this volume.

Multi-stage or multi-echelon inventory systems under centralized control were initially studied by Clark and Scarf (1960, 1962) and, in quite a different framework, by Forrester (1961). However, significant progress was made in the 1980s, see e.g. Schwartz (1981), Federgruen and Zipkin (1984) and a large number of followers, see for example, Tayur et al. (1999). The analysis of multi-echelon inventory systems presented in this chapter is based on Langenhoff and Zijm (1990), Van Houtum and Zijm (1991) and follow-up papers. Diks and De Kok (1998, 1999) take a direct service-oriented (instead of a pure cost) approach) and develop several accurate approximation algorithms, see also Van der Heijden (1997), Van der Heijden et al. (1997). Capacitated multi-stage production-inventory systems have also been studied in a queueing framework, see e.g., Buzacott and Shanthikumar (1993). Contract management in multi-stage systems under decentralized control (multiple independent players) have been studied by Cachon and Zipkin (1999), Zijm and Timmer (2008), who apply a framework closely related to the one presented in this chapter, and others.