Abstract
This chapter focuses on the replenishment aspect of static stochastic inventory models, namely the modeling of the replenishment lead time generating process. Before starting, let us briefly recapitulate the contents of the previous chapters. To begin with, we described the fundamental terminology related to speaking of inventory systems and telling one system from another in Chap. 2. We then gave an overview of the relevant literature on single-level static stochastic inventory systems in Chap. 3. We will revisit some of the papers that we introduced there in this chapter with respect to the analysis of the replenishment process. Finally, we described some basic methods of stochastic analysis in Chap. 4 that we will partially revisit in the approaches described in this chapter.
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This chapter focuses on the replenishment aspect of static stochastic inventory models, namely the modeling of the replenishment lead time generating process. Before starting, let us briefly recapitulate the contents of the previous chapters. To begin with, we described the fundamental terminology related to speaking of inventory systems and telling one system from another in Chap. 2. We then gave an overview of the relevant literature on single-level static stochastic inventory systems in Chap. 3. We will revisit some of the papers that we introduced there in this chapter with respect to the analysis of the replenishment process. Finally, we described some basic methods of stochastic analysis in Chap. 4 that we will partially revisit in the approaches described in this chapter.
Whenever the replenishment lead times in an inventory system are considered to be stochastic, it is indispensable for the analysis to specify the underlying lead time generating process. Even where lead times have the same distribution, the behavior of two systems may be very different if the underlying lead time models vary. This chapter distinguishes three lead time models that are discussed in the literature, namely the cases of:
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Non-interchangeability of demand units (Sect. 5.1)
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Replenishment order crossover (Sect. 5.2)
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Sequential arrivals of replenishment orders (Sect. 5.3)
For a brief literature overview of these three cases see Hayya et al. (2008), for example.One might add a fourth case where lead time distributions are limited in order to result in the same system behavior for all three basic cases. We consider this a special case of all three general models and discuss it in Sect. 5.4.
5.1 Non-Interchangeability
An analytically attractive idea is to consider non-interchangeable unit demands, i.e., each demand unit can only be satisfied by a specific replenishment unit arriving with a specific replenishment order. The idea was introduced in a technical paper by Washburn (1973) on a stochastic lead time extension to the classical economic order quantity problem. Similar problems are regarded by Liberatore (1979), Sphicas (1982), Sphicas and Nasri (1984) and more recently He et al. (2005). Common to all four approaches is the assumption of a constant demand rate, allowing for the synchronization of replenishment orders and prospective demand units.
What makes the assumption so attractive is the property that an arbitrary lead time distribution may be considered in the inventory model without causing any stochastic dependencies. Due to the fixed assignment, each demand unit will be fulfilled on its arrival or on the arrival of the corresponding replenishment order, whichever event is the later. The analysis can thus be focused on one replenishment cycle, and lead times may be independent.
For real inventory systems with stochastic demands, however, it is unlikely that a demand unit must inevitably be satisfied by material arriving with a specific replenishment order. It is somewhat contrary to the idea of keeping stock in order to serve an uncertain demand. Nonetheless, it is worth checking wether the assumption is met in a real system, as it significantly simplifies the analysis.
5.2 Order Crossover
Two orders are said to cross over in time if they arrive in a different sequence to that in which they were issued, i.e., the order that was issued later arrives earlier. From a theoretical point of view, this phenomenon may generally be observed in inventory systems if lead times are independent random variables.
A formal definition is given by Riezebos (2006).
Definition 18 (Order crossover).
Let A and B be two replenishment orders that were issued at time o A and o B , respectively, where o A < o B holds, i.e., A was issued earlier than B. Let A and B arrive at times a A and a B : these orders then cross if a A > a B .
Definition 18 implies corollary 1.
Corollary 1.
Let A and B be two orders that observe issue and arrival times as in Definition 18. Furthermore, let l A and l B be the lead times of these orders, so that \({a}_{A} = {o}_{A} + {l}_{A}\) and \({a}_{B} = {o}_{B} + {l}_{B}\) . Then A and B cross if and only if \({o}_{A} - {o}_{B} + {l}_{A} > {l}_{B}.\)
Proof.
Corollary 1 follows from inserting equations \({a}_{A} = {o}_{A} + {l}_{A}\) and \({a}_{B} = {o}_{B} + {l}_{B}\) into inequation \({a}_{A} > {a}_{B}\).
The corollary directly implies that lead time variability is a prerequisite for order crossover. To be more specific, crossover occurs whenever \({o}_{B} - {o}_{A} < {l}_{A} - {l}_{B}\), i.e., it can only be observed in systems with a lead time variability higher than the minimum time between two orders. (Also see Sect. 5.4.)
Riezebos distinguishes stochastic and dynamic lead time variability, both of which are reported to be observable in real-life inventory systems, either solely or in combination. Stochastic fluctuations are related to time-invariant uncertainty in the supply processes, while dynamic fluctuations are due to systematic process changes over time. For the latter, an example of ordering tobacco leaves is given by Riezebos, where individual ordering modalities and production and transport times in different countries of origin introduce high lead time volatility to the overall ordering process. As dynamic lead times – or system dynamics in general – require an inventory system with dynamic parametrization, we will exclusively focus on the (static) stochastic lead time case in the following.
From a process perspective Song and Zipkin (1996) give a clear motivation for real world systems with static stochastic lead times at which order crossover may and may not be observed. Orders may cross over whenever they are processed in parallel, and cannot cross over if they are sequentially processed. The first case is considered in this section; the latter case in general means that lead times are dependent (see Sect. 5.3).
Two further modalities can be distinguished regarding parallel processing. On the one hand, each order may be assigned to a certain parallel process; on the other hand, we have random assignment that may not be controlled by the system that issues orders.
The first case is broadly discussed in the literature in the context of multiple sourcing, which means that a company maintains more than one supplier for the same type of goods. In this context, problems of order crossover are addressed by Ramasesh et al. (1991) and Kelle and Silver (1990), for example. Both articles consider a multiple sourcing strategy for a continuous review system with fixed order sizes, where each order quantity is split among at least two suppliers. Besides the crossover phenomenon, the related problem of determining so-called effective lead times may arise in the context of multiple sourcing. See Sculli and Wu (1981), Sculli and Shum (1990), Pam et al. (1991) and Fong et al. (2000), for example. In these papers, the effective lead time is regarded as the lead time of the first, second, etc... arrival of two or more simultaneously issued orders. In Sect. 5.2.3 we consider the more general problem of determining the time that passes between the n-th issue of an order and the n-th order arrival, where the corresponding orders may not be the same if order crossover is possible.
A detailed literature overview on multiple sourcing is given by Minner (2003).
Besides the literature on multiple sourcing, some articles directly focus on the order crossover phenomenon, where the actual cause of crossovers is of secondary interest. The related research goes back to the 1950s. See Galliher et al. (1959), for instance, for an early paper on a continuous review system in which order crossover is not exactly addressed, but considered in the model. In spite of these early works, studies on systems with replenishment processes that may randomly cause order crossovers are limited. The majority of authors consider systems where the possibility that orders may cross is simply ignored or cannot occur by definition.
In recent years, however, more attention has been dedicated to the phenomenon. See for example the works of Robinson et al. (2001), Bradley and Robinson (2005) and Robinson and Bradley (2008) on periodic review systems, and those of He et al. (1998) and Hayya et al. (1995) on continuous review systems, where the latter paper also regards the multi- (here dual-) sourcing aspect. An overview of literature concerning order crossovers when lead times are static stochastic is given by Hayya et al. (2008).
In the remainder of this section we will look more closely at three characteristics of the replenishment process when orders may crossover in time, namely the number of outstanding orders, inventory shortfall and the effective lead time.
5.2.1 Outstanding Orders
Zalkind (1978) analyzes the number of outstanding orders for a periodically distributed inventory system. It is assumed that an order is placed in every review interval, i.e., demand is always sufficiently great between two order events that a replenishment order will be placed. To motivate the calculation, we will firstly assume r = 1, and derive the general formulation afterwards. An order is considered outstanding in a period if it arrives no sooner than in the next period.
Let L be a discrete random variable with minimum l min = 0 and a finite maximum \({l}_{\mathit{max}}\), and let the lead times be a series of independent random variables having the distribution of L. Furthermore, let \({e}_{l} \in \{ 0,1\}\) indicate wether an order placed l periods ago has arrived (0) or is outstanding (1), and \(P\{pt = ({e}_{0},{e}_{1},\ldots,{e}_{{l}_{\mathit{max}}-1})\}\) be the probability that the pattern of outstanding orders is given by \(pt = ({e}_{0},{e}_{1},\ldots,{e}_{{l}_{\mathit{max}}-1})\) during the current period. We will consider the state at the end of each period, so that an order with lead time l counts as arrived and not outstanding after l periods. Therefore \({e}_{{l}_{\mathit{max}}} = 0\) always holds. Using the properties that e l is binary and lead times are independent, the corresponding probability can be calculated according to (5.1):
As e l = 0 for \(l \geq {l}_{\mathit{max}}\), only patterns of length \({l}_{\mathit{max}} - 1\) need to be considered. In terms of e l , the number of outstanding orders is given by (5.2):
Let PT be the set of all patterns \(pt = ({e}_{0},{e}_{1},\ldots,{e}_{{l}_{\mathit{max}}-1})\) that satisfy (5.2). The probability that there are k orders outstanding is then given by (5.3):
Obviously, there are \(\left(\mathop{n}_k\right) = \frac{n!} {k!\cdot (n-k)!}\) n-element patterns \(({e}_{0},{e}_{1},\ldots,{e}_{n-1})\) that satisfy (5.2) and need to be considered in (5.3). Thus, the calculation of the probability distribution of the number of orders outstanding would have exponential computational complexity \(O({2}^{n}),n = {l}_{\mathit{max}} - 1\) if r = 1, as \(\sum\nolimits_{k=0}^{n}\left(\mathop{n}_k\right)= {2}^{n}\). This circumstance can easily be obtained from the well-studied binomial theorem (5.4), which is included here for the reader’s convenience. (Set \(x = y = 1\).) It is also obvious though, as all 2n possible patterns have to be considered to calculate the probabilities of all possible values for k:
Zalkind indicates an interesting possibility to reduce computational complexity. He introduces a set of binary discrete distributions K l that are defined according to (5.5) and (5.6):
Using this set, the probability distribution of the orders outstanding equals the convolution of \({K}_{0},{K}_{1},\ldots.\,,{K}_{{l}_{\mathit{max}}-1}\), where the computational complexity is in O((l max )2) (see Sect. 4.2.2).
This idea is also applicable to a generalized r ≥ 0, as will be shown in the following. With an arbitrary positive review interval, we have to consider that there may be periods in which no orders are issued, and thus exclude all e l that do not comply with the ordering pattern. In other words: for r > 1, only those order ages are possible for a certain subperiod t that reach back to an order period. While r = 1 means that order cycles and periods are the same, it is now necessary to focus on order cycles.
Let us first address the question of the maximum orders that may be outstanding, i.e., the maximum order cycles that have to be considered with respect to a certain maximum lead time. Considering period \(t \in \{ 1,2,\ldots,r\}\) within an order cycle r, we know that any outstanding order must be at least t periods old. Thus, an order can only be outstanding if it has been issued no later than \({l}_{\mathit{max}} - t\) periods ago. Adjusting this observation to the order cycle pattern, we derive (5.7), where \({c}_{\mathit{max}}(r,t)\) is the maximum number of cycles (including the present one) from which an outstanding order might originate:
K l also needs adjustment. We will therefore define \({K}_{c}(t,r)\) as a function indicating that an order issued c cycles ago will be outstanding in the t-th period of an order cycle of length r (5.8 and 5.9), where c = 1 is the present cycle:
By convolution of the corresponding cycle-dependent \({K}_{c}(t,r)\), we derive the distribution of outstanding orders in period t according to (5.10) if the order cycle is r:
Thus, the overall probability that the inventory system has k orders outstanding is given by (5.11). K is the mixture of r distributions K(t, r), each having influence \(\frac{1} {r}\):
Numerical Example. Let L be discretely uniformly distributed with values {1,2,3,4} and r = 1. Then we have the following relevant distributions K l :
By convolution, we obtain the following probabilities that k orders are outstanding:
For r = 2, the calculations are as follows:
5.2.2 Inventory Shortfall
In the previous section, the distribution of outstanding orders is developed for a periodically distributed inventory system. This figure, however, is of limited help for the proper configuration of an inventory system, as one is more interested in the amount of stock missing. This amount may be either part of orders that have already been issued, or may be preconsidered for the next order due date. We will use the term inventory shortfall (SF) for the combined amount, as used by Robinson et al. (2001), for example.
An exact formulation of the inventory shortfall in periodically distributed inventory systems is also developed by Zalkind (1978) using his findings on the number of orders outstanding, as described above.
As indicated, inventory shortfall can be divided into (1) the order amount that has already been issued but which has not yet arrived, and (2) the demand that has occurred since the last order was issued. The first part is described by the mixture of the distributions of demand in k ⋅r periods (5.12). The second part is the demand in (t − 1) periods (5.13), where we assume that the demand occurring in an order period is always considered with the corresponding replenishment order.
The overall distribution of the inventory shortfall in a specific period t is then obtained by proper convolution:
With r periods forming one order cycle, distribution of the inventory shortfall for the whole system (5.15) is the mixture of r distributions according to (5.14):
One may not want to use a combination of various different mixed distributions for computational convenience. Regarding (5.15) a little closer, we observe that the underlying mixed distributions may share equal elements. Thus, we can directly formulate (5.15) as mixture of convolved demand distributions (5.16), where SFP is the distribution of the number of demand periods constituting the inventory shortfall:
Numerical Example. Let L be discretely uniformly distributed with values {1,2,3,4}, D normally distributed with \((\mu = 100,\sigma = 30)\) and r = 1. Using the results of the previous section’s example (we obtain the following shortfall distribution)
Robinson et al. (2001) compare inventory shortfall with lead time demand (LTD), which is commonly used in literature and practice to adjust inventory systems. In contrast to SF, LTD is the demand that occurs from the moment an order is placed until the moment when that particular order arrives. Thus, for our first example (r = 1), LTD is calculated as follows. While mean values are equal, we notice a significant disparity of standard deviations.
Referring to a technical paper by Zalkind (1976), Robinson et al. (2001) give proof that E{SF} = E{LTD} and Var{SF} ≤ Var{LTD} hold in periodically distributed inventory systems when r = 1.
5.2.3 Effective Lead Time
Hayya et al. (2008) introduce the concept of effective lead times (ELT) to the crossover context in order to describe the replenishment order arrival time series when order crossover is possible.
Definition 19 (Effective lead time).
Let o i be the issue time of the i-th issued replenishment order in an inventory system and let a i be the arrival time of the i-th arriving order, where o i and a i may correspond to two different orders. Then \(el{t}_{i} = {o}_{i} - {a}_{i}\) is the i-th effective lead time observed in the system. ELT is the random distribution of the elt i .
Note that ELT = L holds if no crossovers may occur in the considered inventory system.
Hayya et al. derive expressions to calculate the effective lead time when the time horizon is limited to two and three periods, mainly to illustrate the computational complexity. The idea is to enumerate all possible sequences of arrivals for orders that were issued in k consecutive periods and calculate the probabilities that each sequence will occur. Applying their approach to a time horizon of t periods, the ELT is the mixture of t! distributions, assuming that each order may observe a lead time of t or more periods.
In the following we will use this idea to derive expressions for a steady-state analysis. Instead of restricting the time horizon, we will restrict the maximum number of orders that may be involved in a crossover. For clarity, we will distinguish purchase orders and deliveries in the remainder of this section. We will refer to an order as a purchase order at the time it is issued, whereas we will speak of a delivery if we consider its arrival. For example, if the first and second order of a sequence cross over, we say that the first purchase order is the second delivery and the second purchase order is the first delivery.
Furthermore, let us introduce the following notation to describe the arrival sequence of purchase orders. Let:
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k denote the number of consecutive purchase orders that are examined for a certain purpose
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m be the maximum number of orders in an arrival sequence that any of the orders may cross
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\(a{s}_{s}^{(k,m)}\) be the arrival sequence s of orders that are identified by their positions in the order issue sequence, in which each order may cross over with a maximum of m other orders
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\(a{s}_{s,i}^{(k,m)}\) be the issue position of the i-th order in the arrival sequence \(a{s}_{s}^{(k,m)}\), and let
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\(A{S}^{(k,m)}\) be the set of all possible arrival sequences of k orders, where each order may crossover with a maximum of m other orders
Example. The arrival sequence \(a{s}_{1}^{(3,1)} = (n - 1,n + 1,n)\) indicates that the n-th and (n + 1)-th purchase order are involved in a crossover so that each order crosses the other one and the (n + 1)-th purchase order arrives before the n-th purchase order.
With this notation, we are ready to develop an approach to compute the effective lead times. We will start with the case of m = 1, i.e., a maximum of two orders may be involved in a crossover, and then proceed to the general case.
One Order Case. In the event that each order may only cross one other order at most, we observe that only those orders may cross over that have been consecutively issued, i.e. the n-th purchase order can only cross over with either its direct predecessor or its direct successor. From the supplier’s perspective, the n-th delivery may only be the (n − 1)-th, n-th or (n + 1)-th purchase order.
We thus have three purchase order candidates that may be the n-th delivery, and \(A{S}^{(3,1)} =\{ (n - 1,n,n + 1),(n,n - 1,n + 1),(n - 1,n + 1,n)\}\) is the complete set of possible arrival sequences that involve the n-th purchase order. The possible cases that lead to a certain elt of the n-th delivery can be directly derived from these sequences. To do this, we require the middle delivery to exactly meet the designated elt, while the preceding and successive deliveries’ lead times need to ensure that the assumed arrival sequence is met, i.e., the preceding delivery may not arrive later and the successive delivery may not arrive earlier than the middle delivery. To save subindexes, we will strictly connect the lead time L to purchase orders and the effective lead time ELT to deliveries. Thus, the n-th delivery observes an effective lead time of elt under the following condition:
Note that the cases in (5.18) are not free of conjunctions, e.g., the special case of \({L}_{n-1} = elt + r \wedge {L}_{n} = elt \wedge {L}_{n+1} = elt - r\) is included in all three of them. Expanding (5.18) to comparisons with the ‘ < ’, ‘ > ’ and ‘ = ’ comparative operators only leads to a conjunction-free representation (5.19), which can be reduced to (5.20) if we again allow for the ‘ ≤ ’ and ‘ ≥ ’ comparative operators:
We can derive the effective lead time distribution from the conjunction-free set of cases (5.20) if a maximum of two orders may be involved in a crossover (5.21):
Numerical Example. Let L be discretely uniformly distributed with values \(\{1,2,3,4\}\) and r = 2. Then the corresponding ELT is calculated as follows:
General Case. As a first approach to generalize the above findings, let us think of how many purchase orders may possibly arrive as the n-th delivery if we allow each order to cross m other orders at most. Let us first consider an arrival sequence in which all deliveries arrive in the same order as they were issued, except the n-th delivery, which has a relative delay. Let this delayed n-th delivery be issued p order cycles before the n-th purchase order. Then the n-th delivery is obviously overtaken by p − 1 other orders, and \(m = p - 1\) is the maximum number of orders that are crossed by another one. Following on this situation, we observe that we may only decrease m by somehow moving the n-th delivery to an earlier position in the sequence. Obviously, the number of orders that overtake the n-th delivery can neither be changed by swapping two earlier deliveries, nor by swapping two later deliveries. Even swapping an earlier with a later delivery will not change the situation because it either holds for both deliveries that they have been issued after the n-th delivery and thus one of them still overtakes it, or, if the earlier delivery was also issued before the n-th delivery, this particular order is overtaken by \(p - 1 + 1 = p\) orders at least. The latter case would thus induce m ≥ p because the particular order is overtaken by all orders that have overtaken the n-th delivery, and furthermore also the n-th delivery itself. The analogous argumentation holds for the case that the n-th delivery was issued p periods after the n-th purchase order. Thus, we have a maximum of \({k}^{{_\ast}} = 2 \cdot m + 1\) purchase orders that may finally arrive in one particular period if no more than m order may be crossed by another one.
Regarding the one order case, we have already noticed that we may not freely permutate these k ∗ orders to retrieve the possible arrival sequences. For example, \(A{S}^{(3,1)}\) may not contain \(a{s}_{x} = (n + 1,n,n - 1)\) as \(a{s}_{x,1} = n + 1\) as well as \(a{s}_{x,3} = n - 1\) would indicate an order that crosses with two other orders, which would be one more than is allowed. Therefore, \(A{S}^{({k}^{{_\ast}},m) }\), \(m \geq 1\) will always be a set of restricted permutations to describe all possible arrival sequences. Thus, we call an arrival sequence \(a{s}_{s}^{({k}^{{_\ast}},m) }\) allowed if
holds for the corresponding elements.
According to the condition, the i-th delivery must arrive earlier than any delivery j that has been issued more than m periods later. I.e, no issue position \(a{s}_{s,j}^{({k}^{{_\ast}},m) }\) may be positioned left of \(a{s}_{s,i}^{({k}^{{_\ast}},m) }\) in the arrival sequence that exceeds the issue position of i by m or more cycles. Let us consider an example here. Let m = 3. Then we need to regard k ∗ = 7 consecutively issued orders to cover all possible arrival sequences relative to the middle purchase order. Let us regard an index space \(i \in \{-3,-2,\ldots,3\}\) to emphasize that the sequence regarded is arranged around the middle purchase order indexed with 0. The sequence \(a{s}_{1}^{(7,3)} = (-3,-1,-2,3,2,1,0)\) is allowed as \(a{s}_{i} \leq a{s}_{j} + 3\) holds for every index pair i, j fulfilling i < j, whereas the sequence \(a{s}_{2}^{(7,3)} = (-2,-3, {\bf {3}},2,-{\bf{1}},0,1)\) is not allowed, as the pair \(a{s}_{2,3}^{(7,3)} = 3\), \(a{s}_{2,5}^{(7,3)} = -1\) dissatisfies the condition.
In the literature, several systematic methods are proposed to solve the problem of enumerating restricted permutations. (See Vatter 2008 for an illustrative overview.) For our application, we found the method of generating trees appropriate, both in terms of computational efficiency and confirmability of the approach for constructing the sequences. This method was introduced by Chung et al. (1978); further applications are given by West (1995), West (1996) and Merlini and Verri (2000), for example.
Given the problem of generating all sequences of k consecutive numbers that satisfy a certain set of rules, the basic idea is to iteratively derive the allowed sequences for x ≤ k numbers from the allowed insertions into the allowed sequences for x − 1 numbers. For the present problem, we start with x = 1 and obviously obtain only one trivial sequence (0 − m) that is always allowed. In a following step x, we derive further sequences by inserting x behind the element in position x − 1, \(x - 2,\ldots \) as long as the next insertion would result in a forbidden sequence. We can stop here, as all further insertions will obviously be forbidden as well. Figure 5.1 displays the generating tree of allowed sequences for k = 3 and m = 1.
Enumerating the allowed sequences for combinations of k and m, we observe rapid combinatorial growth. Table 5.1 displays the result using a straightforward Java implementation, without explicit memory management, which limits the generating method to a few hundred thousand permutations. Note that we give the number of allowed sequences for all (computable) pairs of 1 ≤ k ≤ 20, 1 ≤ m ≤ 7, where in terms of the ELT we are only interested in the highlighted pairs for which \(k = 2 \cdot m + 1(= {k}^{{_\ast}})\) holds.
However, the very sequence is not exactly what we are after. Remember, we are interested in the probability that an order will arrive at a certain position in the arrival sequence. This position is already determined by the set of orders that have arrived before, regardless of the sequence in which they arrived. Calculating the probabilities, we will focus on the middle position only and ask wether an order will arrive either on the exact position, before or after it.
Example. In the event that the range of the independent lead time allows each order to crossover with a maximum of two other orders (m = 2), we have to extend the time window observed to five orders (k ∗ = 5). The number of allowed sequences is now 25. Equation (5.23) displays all possible arrival sequences for orders issued at order period n relative to a middle period (n = 0):
Regarding the first four sequences of the first line, we note that they are equal in terms of orders arriving before and after the middle element. Thus, those four sequences are covered by one case \((\{-1,0\},-2,\{1,2\})\). Similarly, other sequences can be reduced to one case of the described form, leading to (5.24). We will refer to these cases as partly defined sequences.
Using the enumeration scheme described above, we can derive the relevant partly defined sequences from the set of allowed sequences. This approach, however, appears to be inefficient both in terms of computational effort and memory requirements. We therefore propose Algorithm ?? to directly enumerate the partly defined sequences for any given (odd) number k ∗. The basic idea of the algorithm is as follows. Given \({k}^{{_\ast}} = 2 \cdot m + 1\) numbers, we must consider each number as the middle element. Choosing one of these numbers determines the set of numbers that may be assigned to the left of it. Regarding (5.24), only \(\{-2,-1,0\}\) may be assigned to the left of 1, for example. Thus, the algorithm comes up with two nested loops: one outer loop that defines the middle element (me), and one inner loop that chooses one possible left element (ce). To prevent duplicate partly defined sequences, it is ensured in the following that ce is the largest of all left side numbers. Therefore, only some numbers apply for ce, namely those between se and le. (See the structured exposition of the algorithm for details.)
Some numbers are definitely required on the left side of me for certain pairs of me and ce. Regarding (5.24) again, if me = 0 and ce = 1, then − 2 must not be placed on the right of me. Admittedly, the example of (5.24) is too small to observe the full phenomenon, as − 2 must always either be on the left, or be me. Thus imagine \(m = 3,{k}^{{_\ast}} = 7\), then \(ce = 0,me = 1\) would require all numbers smaller than \(\max \{ce,me\} - m = -2\) on the left. A base sequence BS is constructed if these required elements are added.
If all slots on the left side have already been taken in a base sequence, the corresponding pair of ce and le only results in one partly defined sequence, which can easily be completed by assigning all numbers to the right that have not been assigned so far.
If there are x > 0 slots remaining, these may freely be filled with all possible x-subsets of all numbers that may (but need not) stand left of ce and me. Again, the sequences are completed by adding all remaining unassigned numbers to the right.
Experiments suggest that the total number of different partly defined sequences is \((m + 2) \cdot {2}^{m-1}\), but providing proof for this assumption remains an open task.
Let PSQ be the set of all partly defined sequences for a given m that we are now able to enumerate, using Algorithm ??. Furthermore, let psq be one member of PSQ, and psq i the value of element i in a specific psq, i.e., the position in the order issue sequence. We can then describe the probabilities of occurrence of certain effective lead times subject to a specific psq according to (5.25) and (5.26):
Numerical Example. The partly defined sequence
resolves to
We are now already able to state (5.27) as upper bound for the efficient lead time probabilities:
Equation (5.27) does not describe the exact probabilities, as the cases we derive from PSQ are not generally free of conjunctions, e.g., the special case of \(P\{W = elt + i \cdot r\}\ \forall i\) is included in every psq. We have already observed this phenomenon regarding the special case of m = 1. To derive a form that is free of conjunctions, we can use the technique of expansion and reduction as introduced above. Each non-simple comparative operator (i.e., ‘ ≤ ’ and ‘ ≥ ’) has to be subdivided into two cases of simple comparative operators, leading to 2φ expanded cases, where φ is the number of factors that include a non-simple comparative operator. Expression (5.27) directly implies that we derive 22 ⋅m + 1 expanded cases from each partly defined sequence.
After the reduction, the resulting cases may again be combined to expressions that allow for the non-simple comparative operators. This may be done by repeatedly combining cases that differ in only one comparison operator. To understand the algorithm, let us introduce the following notation. A case of k ∗ comparisons is represented by k ∗ numbers x, \(x \in \{-2,-1,0,1,2\}\), where we encode the comparison operators according to Table 5.2.
The basic idea of the approach displayed in Algorithm ?? is to repeatedly combine two cases that meet two conditions. Firstly, both cases are equal except for one element at position i and secondly, x i 1 and x i 2, i.e., the elements at position i of both sequences, must either be both smaller than or equal to zero, or they must both be greater than or equal to zero. When these conditions are met, we can replace the two sequences by one, with entry − 2 if both x i 1 and x i 2 were smaller or equal to 0 and 2 otherwise.
Table 5.3 summarizes the cardinalities of PSQ, the expanded cases (EXP), the reduced cases (RED) and finally the combined cases (COM). Where we have an indication, we denote the theoretical closed expression in the second line. Clearly, the expansion step limits the approach, especially in terms of memory.
Using the algorithm, we derive the following expressions for m = 2, 3 and 4. We give the expressions for m = 2 in encoded form (5.28) and in plain form (5.29) to further illustrate the encoding described above. The expressions for m = 3 (5.30) and m = 4 (5.31) are given in encoded form only.
Numerical Example. Let L be a discrete uniform distribution with possible values l ∈ { 1, 2, 3, 4, 5}, furthermore let ELT(L, r) describe the effective lead time of a replenishment process, where orders are issued every r-th period and replenishment lead times have the distribution of L. Table 5.4 displays the resulting effective lead time probabilities and first two moments for different parameters r.
Regarding these results, we observe a stable mean value and a declining standard deviation when lowering r. This leads us to the following two conjectures.
Conjecture 1.
E(L) = E(ELT(L, r)) holds for all r > 0.
Conjecture 2.
Var(L) ≤ Var(ELT(L, r)) holds for all r > 0.
Incomplete Proof. We may prove the two conjectures for the special case of m = 1, which means that two consecutive orders placed at period n and n + 1 observe either \(el{t}_{n} = {l}_{n},el{t}_{n+1} = {l}_{n+1}\) (no-crossover) or \(el{t}_{n} = {l}_{n+1} + r,el{t}_{n+1} = {l}_{n} - r\) (crossover).
Conjecture 1 holds because of
i.e., the mean effective lead time of the two orders equals their underlying mean lead times, whether they cross over or not.
Considering the variance (Conjecture 2), we have to examine whether
holds in the event that those two orders cross over. This can be shown by the following conversions, assuming r > 0:
The last line proves the conjecture for m = 1, because we regard the crossover case here, meaning that l n ≥ l n + 1 holds.
We leave the question open to future research as to wether the two conjectures hold for general assumptions and close with a last unproven conjecture, based on both empirical test results and the author’s intuition.
Conjecture 3.
Var(ELT(L, r 1)) ≤ Var(ELT(L, r 2)) holds for all r 1, r 2 > 0, r 1 ≤ r 2.
5.3 Sequential Arrivals
Based on an idea due to Kaplan (1970), Zipkin (1986b) introduces a general set of conditions to define a replenishment lead time process that rules out order crossover. Let {U(t) : t ∈ ℝ} be a real-valued, stationary, ergodic stochastic process that satisfies the following conditions. (See Zipkin 1986b, p. 770) U(t) may be interpreted as the age of the oldest order at time t.
-
1.
U(t) ≥ 0 and E[U(t)] < ∞
-
2.
t − U(t) is nondecreasing
-
3.
Sample paths of {U(t)} are continuous to the right
-
4.
{U(t)} is independent of the placement and size of orders and the demand process
U(t) may be interpreted as the age of the oldest order arriving at time t. According to (1), U(t) returns meaningful (non-negative) lead times with a finite expected value, i.e., the process is stationary. To understand that condition (2) rules out order crossover, consider two orders arriving in t 1 and t 2, t 1 ≤ t 2. These two orders would cross over, if and only if \(U({t}_{2}) > U({t}_{1}) + ({t}_{2} - {t}_{1})\). Reformulating the inequation to \({t}_{1} - U({t}_{1}) > {t}_{2} - U({t}_{2})\), we clearly see that this is contradictory to (2). Condition (3) is a rather technical condition that ensures continuity of U(t). Typically, we observe jump discontinuities in sample paths of U(t) for those values t at which an (oldest) order arrives. Nonetheless, U(t) is continuous to the right if an order is immediately removed from the stack as soon as it arrives, so that the age of the oldest order at the very point of arrival is then determined by the order that was previously the second oldest. Finally, (4) is self-explanatory.
Note that due to (2), lead times modeled in accordance with U(t) are not independent in general, but form a continuous-time, continuous-state Markov process. See Ehrhardt (1984) or Nahmias (1979) for further insights into formulating lead time processes.
The recursion defined in (5.32) and (5.33) gives an example of a sampling process satisfying (1)–(4), where S(L n ) is a sampling function of an arbitrarily distributed L n , and A(n) returns the issue date of the n-th order. \(A(n) - A(n - 1)\) is thus the time between the issue of the n − 1-st and n-th order. For the case of an (r, S) policy, obviously \(A(n) - A(n - 1) = r\) holds for every pair of two consecutive orders:
Let \(A(n) - A(n - 1) = r\ \forall \ n\) and \(({L}_{n}^{{_\ast}}\vert {L}_{n-1}^{{_\ast}} = {l}_{n-1})\) be the lead time generating process defined by the recursion above. Then the probability distribution of the latter is given by (5.34):
Note that (5.34) forms a Markov chain with state transition probabilities p ab given by \(P\{{L}_{n}^{{_\ast}} = a\vert {L}_{n-1}^{{_\ast}} = b\}\). On analyzing Markov chains in general, see Meyn and Tweedie (2009), for example.
Let us assume in the following that the L n are discrete distributions with a common finite set of integer states with probabilities p l . Then (5.34) forms a discrete-time, discrete-state Markov chain, where steady state probabilities can easily be calculated.
Numerical Example. Let L 1, L 2, …, L n be identically discretely uniformly distributed with states {1, 2, 3, 4} and r = 1. Then the matrix of state transitions resolves to (5.35):
To determine the steady state probabilities, we derive a system of linear equations (5.36), where we can drop one of the first four lines:
The example lead time process has p ∗ = { 0. 09375, 0. 28125, 0. 375, 0. 25} steady state probabilities.
Alternatively to the recursion defined in (5.32) and (5.33), it may be reasonable to make use of a truncated distribution, where the probability distribution of \(({L}_{n}^{{_\ast}}\vert {L}_{n-1}^{{_\ast}} = {l}_{n-1})\) is then given by (5.36). See also Sect. 4.4.
Here, we assume that the probability mass of the forbidden values l n for a given l n − 1 is transferred proportionally to the remaining allowed values. In terms of a probability experiment, this would be equivalent to repeatedly drawing from L n and discarding forbidden values until we obtain an allowed value. We obtain the following state transition matrix for the above example:
This lead time process has p ∗ = { 0. 0625, 0. 1875, 0. 375, 0. 375} steady state probabilities.
Remark. When lead times are dependent the proceeds for calculating the number of outstanding orders and the amount of stock outstanding as introduced in Sects. 5.2.1 and 5.2.2 are not applicable. We leave it open to future research to create appropriate methods to solve these problems.
5.4 Limited Distributions
While many authors make the assumption that lead times are independent and orders never cross, this is generally self-contradictory. (See e.g. Chen and Zheng 1992.) However, one may specify a system where both assumptions are met, an idea that is due to Hadley and Whitin (1963). Let us have a closer look at corollary 1, p. 58, to understand what we necessarily have to assume.
The corollary implies that in the case of two orders A and B, o A < o B do not cross if and only if \({o}_{B} - {o}_{A} \leq {l}_{A} - {l}_{B}\), i.e., the difference between the two order lead times must be smaller than the difference between the order issue dates. Let R be a random variable denoting the time between issuing two consecutive orders and let R n be the random time between two orders A and B that represent the i-th and \((n + i + 1)\)-th order in the issue sequence, so that \({R}_{n} = {O}_{B} - {O}_{A}\), where the issuing times of A and B are random. Then A and B do not cross if and only if either R n ≤ 0 or R n ≥ L A − L B .
Excluding the rather theoretical case of a system with \(R \equiv 0\), R n ≤ 0 is never fulfilled. R n ≥ L A − L B is generally fulfilled if Min{R n } ≥ Max{L A − L B }. With R ≥ 0, we have Min{R n } = Min{R}, while \(\mathit{Max}\{{L}_{A} - {L}_{B}\} = {L}_{\mathit{span}}\), \({L}_{\mathit{span}} = {l}_{\mathit{max}} - {l}_{\mathit{min}}\). In other words, in the event that the replenishment lead time span is shorter than the minimum time between two consecutive order placements, order crossover can be ruled out even with independent lead times.
For an application using limited distributions in inventory management, see Sphicas and Nasri (1984), for example.
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Wensing, T. (2011). Replenishment Processes. In: Periodic Review Inventory Systems. Lecture Notes in Economics and Mathematical Systems, vol 651. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20479-1_5
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