On the estimation of the fill rate for the continuous (s, S) inventory system for the lost sales context.

In the continuous review reorder point, base-stock (s, S) policy, the replenishment order is launched when the inventory position reaches the reorder point, s. It is commonly assumed that the inventory position is exactly equal to the reorder point at the moment the order is launched, when actually it could be lower at that moment. This implies neglecting the possible undershoots at the reorder point, which has a direct impact on the calculation of the expected shortages per replenishment cycle. This article presents a method for an exact calculation of the fill rate (fraction of demand that is immediately satisfied from shelf) which takes explicit account of the existence of undershoots and is applicable to any discrete demand distribution function in a context of lost sales. This method is based on the determination of the stock probability vector at the moment the replenishment order is launched. Furthermore, neglecting the undershoots is shown to lead to an overestimation of the fill rate, particularly when we move farther away from the unitary demand assumption. From a practical point of view, this behaviour involves underestimating the base-stock level, S, when a target fill rate is set for its determination. The method proposed in this paper overcomes these shortcomings.

Introduction

Inventory management systems are designed according to a measure of system performance, based either on costs or on the level of customer service. Since the costs associated with the system are difficult to estimate accurately [1,2] the service model is the most commonly used in practice. In this management context, one of the most widely used service measures is the fill rate (β further on) which is traditionally known as the fraction of total demand that is delivered from available stock without shortages [3]. Recently [4] identify several fill rate expressions, including the traditional one. This interesting metric not only reveals stockout situations, but also gives information on the size of the unmet demand [5,6] as it is calculated as the ratio between the satisfied demand and the total demand during a replenishment cycle. This paper focuses on the exact estimation of the fill rate when the system is continuously reviewed for the lost sales case that implies that unfilled demand is lost.

Continuous review inventory policies are very common in practice. The main characteristic of this type of policy is that the status of the inventory is known at all times, so that any change in the stock level is reported immediately. Basically, once the inventory position (i.e. on-hand stock + on-order stock—backorders) drops to the reorder point, s, or lower, a new replenishment order is launched and received L periods later. Such a replenishment order may be constant and equal to Q, as is the case of a system (s, Q), or variable until the base-stock level, S, is reached, and thereafter managed by means of a system (s, S). This research is dedicated to the (s, S) policy given that this policy is normally used in practice to manage strategic items [7]. This policy entails an additional difficulty: to exactly reach the reorder point, the demand must be necessarily unitary. If this is not the case the inventory position may not be exactly at the reorder point, but a certain amount below it, called deficit or undershoot, which is difficult to quantify. Therefore common derivations of a (s, S) policy are based upon the assumption that the undershoots at the reorder point are negligible [6]. However, neglecting undershoots may lead to higher inventory costs or a lower service level than desired [8,9]. This is particularly true when managing intermittent, lumpy or slow-moving items for which the unitary demand assumption is not appropriate [10] although undershoots can also occur with non-intermittent demand as long as the unit sized demand assumption is not met.

Authors who take explicit account of the undershoots at the reorder point aim to estimate the undershoot distribution function in such a way that the expected value of the undershoots is incorporated into the service measures. In the literature we find several approaches to the calculation of the undershoots based on asymptotic results or the renewal theory (see, for example [5,11-14], among others) or computing probability functions and the expected value of the undershoots [9,15,16]. Our approach, on contrary, instead of deriving an approach of the distribution function of the undershoots, considers that the inventory position when the reorder point is reached may take any feasible value (i.e. from 0 to s). This entails not neglecting the undershoots at the reorder point and implies the difficulty of accurately calculating expected values of the on-hand stock at order delivery and when the reorder point is reached (exactly or exceeded). This approach makes our research innovative.

Regarding the fill rate estimations in service models, there are only a few authors that explicitly consider the presence of undershoots. The studies by [5,11,17-19] suggest methods to compute the fill rate, however they are only for the backordering context. Under a lost sales context only [15], present an approximate method for the fill rate in a continuous review policy (s, S) that includes the expected undershoot value that is obtained from an approximate distribution function.

Summarizing, on the one hand, the (s, S) policy is widely used in practice but assumes that undershoots at the reorder point are negligible. This fact may introduce deviations on service measures such as the fill rate. On the other hand, there is no optimal approach to estimate the fill rate for the lost sales context, which has been hardly discussed in the literature [20]. Despite the fact that lost sales case is frequent in many sectors most of the studies are based on the total backordering assumption and more research is needed assuming the lost sales case [21]. The objective of this paper is to bridge that gap by proposing an exact method to estimate the fill rate. To address this objective, we will analyse the different issues to be taken into account for the calculation of the fill rate in the (s, S) policy and illustrate the bias that neglecting undershoots introduces in the model.

The remainder of this paper is organized as follows. Section 2 describes the inventory system, basic notation, assumptions and the description of the problem. Section 3 is dedicated to explain the mathematical derivation of the fill rate when undershoots are neglected and the exact method to compute the fill rate which is the main contribution of this paper. Section 4 presents the experimental design, discussion and practical implications of using a desired fill rate to determine the base-stock level. Finally, the main conclusions of this paper are considered in Section 5.

Problem formulation and description

System description, notation and assumptions

We consider a single echelon, single item inventory system where demand is stochastic, stationary and i.i.d., and modelled by any discrete distribution function. Note that assuming discrete demand is closer to real world companies where continuous demand is rarely frequent, especially for intermittent items, such as spare parts [22].

Additionally, we assume that:

The replenishment order is received after a constant and known lead time, L, and is added to the inventory at the end of the period in which it is received;

Demand is fulfilled with the on-hand stock at shelf.

Unfulfilled demand is lost.

Only one outstanding order is launched within any given period, which implies that s5], especially for the lost sales case [15,20].

Fig 1 shows the evolution of stock levels in a (s, S) inventory policy when the system is not out of stock (a) and when it is out of stock (b) for the lost sales case. Notation in it and in the rest of the paper is as follows:

Fig 1

Evolution of the stock in a (s, S) inventory policy and lost sales when the system is not out of stock (a) and when it is out of stock (b).

s = reorder point, ROP (units),

S = base-stock level (units),

L = lead time for the replenishment order (time),

τ = number of periods from the beginning of the cycle to the ROP (time),

z = on-hand stock at the beginning of the cycle and at order delivery (units),

z = on-hand stock when the reorder point is reached and the replenishment order is launched (units),

d = demand at instant t (units),

D = accumulated demand from the beginning of the cycle to the ROP (units),

D = accumulated demand during the lead time (units),

f(·) = probability mass function of demand at t,

F(·) = cumulative distribution function of demand during t periods,

h(·) = probability mass function of D,

H(·) = cumulative distribution function corresponding to g(·),

X = maximum {X, 0} for any expression X.

Problem description

By definition, the (s, S) policy implicitly states that both the replenishment cycle, i.e. time elapsed between two consecutive order deliveries, and the order quantity are variable. These two characteristics mean that mathematical approaches to characterize the policy are truly difficult to implement in practice. For example [23], points out that “the values of the control parameters are set in a rather arbitrary fashion” and the most common approach consists of assuming that all demand transactions are unit sized or, with the same consequence, that undershoots at ROP are not possible. It supposes that the inventory position always exactly reaches the ROP and therefore the order quantity is constant and equal to S-s, being equivalent to the policy (s, Q) with Q = S-s [24]. However, in practical environments it is uncommon to find items whose demand is unitary, which makes it necessary to consider the presence of undershoots when determining optimal parameters of the inventory system.

Another implication relates to the calculation of the total demand during the replenishment cycle. One of the main features of the continuous review policies is that there is no knowing when the reorder point will be reached. As a result, the replenishment cycles are variable, which, in a stochastic demand context, hinders the calculation of the demand distribution from the order delivery (beginning of cycle) until the reorder point is reached. Nevertheless, if the undershoots are neglected, the demand consumed from the beginning of the cycle until the reorder point is reached can be easily calculated as the difference between the on-hand stock at the beginning of the cycle and the ROP.

However, what is the cost of neglecting undershoots? The answer is related to the expected value of stockouts (also known as expected shortage) during the replenishment cycle. The assumption that the replenishment order is launched at the precise moment the reorder point is reached implies that the on-hand stock at launch is greater than it actually is, since z

On the fill rate estimation in (s, S) systems and lost sales

The fill rate is the fraction of demand that is immediately fulfilled from on-hand stock (at shelf). The most common approach to estimate this service measure consists of computing the complement of the ratio between the expected shortage per replenishment cycle (ESPRC) and the total expected demand per replenishment cycle (EDPRC), i.e.: where EDPRC can be split into the accumulated demand from the beginning of the cycle until the ROP is reached, D, and the accumulated demand from the moment an order is placed until its delivery, D.

Fill rate estimation neglecting the undershoot at ROP

This approach is based on neglecting the undershoot at ROP which leads to the assumption that the system always exactly reaches the reorder point. This assumption directly implies that demand is always unit sized and leads to the consideration of two important issues: (1) we always order the same amount. Note that the replenishment order is placed when the system reaches the ROP and therefore, equal to S-s. Indeed, the system becomes a (Q, s) system with Q = S-s; and (2) the system will only be out of stock during the lead time. Fig 2 shows the evolution of stock levels when undershoots are neglected. Both features (1) and (2) can be clearly appreciated.

Fig 2

Evolution of the stock in a (s, S) inventory policy when undershoots are neglected.

When considering the effect of neglecting undershoots on the fill rate estimation, we see that this greatly reduces the mathematical complexity of it, since knowing the probability distribution of stock levels at the precise moment when the ROP is reached is probably the most complicated issue of its calculation. If the stock exactly reaches the ROP and the system is only out of stock if D > s, then the expected shortage per replenishment cycle is straightforwardly: To compute the expected total demand per replenishment cycle (EDPRC) we add up the accumulated demand from the beginning of the cycle until the ROP is reached, D, and the accumulated demand over the lead time, D. The latter is easily computed when the demand distribution is known. However, to compute D, we need to know the on-hand stock balance at the beginning of the cycle, i.e. at order delivery, which, in a lost sales context, is obtained thus z0 = S–s + E[s—D]+. Therefore, taking into account that D has to guarantee that the ROP is exactly reached, then D = z0 –s = S– 2s + E[s—D]+.

Consequently, the expression to estimate the fill rate when undershoots are neglected, named as “classic” further on, for the lost sales case that applies to any discrete distribution is:

General fill rate estimation considering the undershoot at ROP

To estimate the fill rate without any assumptions regarding undershoots at ROP (or demand size) means we have to address three important issues. Firstly, replenishment orders are variable and depend on the stock level when the ROP is reached, i.e. Q = S—z. In a stochastic context, to compute the expected value of z we need to estimate its probability vector using transition matrices over the cycle. Secondly, the estimation of the total demand and concretely D is not straightforward and also depends on the expected values of stock levels and on the unknown number of periods until the ROP is reached, τ. Thirdly, stockouts may occur not only during the lead time, but also beforehand, in such a way that it occurs at the very moment the ROP is reached. In this section we address how these three issues affect the estimation of the fill rate and how to solve them to finally obtain an exact expression to compute it when undershoots are considered for the lost sales case and discrete i.i.d. demands.

The importance of knowing the stock levels over the replenishment cycle to compute the fill rate is based on the fact that there is no way of knowing the satisfied demand or, therefore, the expected backorders, without knowing the probability distribution of the on-hand stock levels when the ROP is reached. The stock level of policy (s, S) satisfies the Markov property, i.e. the future state only depends on the current situation. As a result, on-hand stock levels can be represented by a Markov process and the probability transition matrix of the on-hand stock between two consecutive replenishment cycles needs to be determined so that: where is an arbitrary vector and the principal left eigenvector of , i.e. the probability vector of the on-hand stock levels at order delivery.

In order to calculate it is necessary to know the probability transition matrices from the beginning of the cycle until the ROP is reached, , and from the moment the ROP is reached and the order is launched until it is received L units of time later, , in such a way that . Once is calculated, we can obtain resulting from the convergence of the expression (4). The probability vector of the on-hand stock at ROP is calculated using the following expression: . How to obtain and is detailed in S1 Appendix.

The second important issue is related to the calculation of the expected demand during the cycle, EDPRC, since it is mathematically complicated to know the accumulated demand until the inventory position exceeds the ROP, D. As mentioned above, the main problem resides in the fact that the number of units of time up to this moment, τ, is unknown. To address this problem, we correct the derivation presented by [10] for the (s, Q) policy and adjust it to cover the special features of the (s, S) policy. S2 Appendix is dedicated to the mathematical derivation of the demand distribution from the beginning of the cycle until the ROP: h(·). Once D is estimated, EDPRC is the sum of D and D.

Thirdly, to compute the expected shortage per replenishment cycle, ESPRC, it is necessary to consider that stockouts may appear at the same instant the ROP is reached. Thus, two stockout situations are identified: (1) the on-hand stock is zero and the net stock is negative at the very moment the ROP is reached (Fig 3A); and (2) the on-hand stock is depleted once the ROP is reached (Fig 3B). In the case of (1): While in the case of (2):

Fig 3

Evolution of a (s, S) inventory policy that illustrates the ESPRC(1) and ESPRC(2) cases.

Finally, the exact fill rate in a context of discrete demand and lost sales considering undershoots is obtained through the following expression:

Experimental design, discussion and practical implications

Experimental design

This section illustrates the performance of β and β against the simulated fill rate, β, which is computed as the complement of the ratio between the expected shortage and the total demand per replenishment cycle when considering 20,000 consecutive periods. This simulation uses the data from Table 1 for the cases whenever β > 0.50 with Pure Poisson demands which fulfil the smooth and intermittent categories according to [25] categorization framework. Thirty replications were applied to each case. The averages of the values obtained are the ones to be used as the final β.

Table 1

Set of data.

Lead Time L = 2, 3, 4

Base-stock S = 5, 6, 7, 8, 9

Reorder point s = 2, 3, 4

Demand Variability (Poisson distributed) λ = 0.1, 0.5, 0.75, 1, 1.25, 1.5

Results and discussion

Fig 4A and 4B show the comparison between β and β and between β and β respectively (S1 Table). As can be seen, β tends to overestimate β. This behaviour was to be expected if we analyse the impact of neglecting the undershoots. For example, if the inventory position and the on-hand stock are one unit above the ROP at a given moment t, and an order for 2 units is received at t+1, a replenishment order is launched when the inventory position and the on-hand stock are equal to exactly s-1 units. Therefore, the available on-hand stock remaining on the shelves to meet demand during the lead time is one unit less than that estimated when the undershoots are neglected. Thus, β is higher than the actual value and therefore the stockouts are larger than expected.

Fig 4

β and β vs. β.

This example goes to show why neglecting the undershoots introduces a significant bias into the classic fill rate calculation, a bias which will grow as the demand increases (further removed from the unit). Table 2 details the average and standard deviation of the errors made by β as compared to β with regard to the value of λ. What can be seen in Table 2 is that when the demand rate increases, meaning it is more difficult to assume the unitary demand hypothesis, the average and deviation of the errors made by β increase. However, this example illustrates that undershoots appears also for non-intermittent demand, which justifies that undershoots should be taken into account explicitly in the calculation of the fill rate.

Table 2

Average and standard deviation of errors between β and β.

	Error (%)
λ	Average	Standard Deviation
0.1	0.026	0.031
0.5	1.406	0.673
0.75	2.478	0.709
1	3.132	0.822
1.25	3.311	1.251
1.5	4.764	7.750

Practical implications

Service measures are usually used to determine the control parameters of the inventory policy. In practice, for one specific SKU the base-stock is determined to fulfil a target fill rate taking into account its criticality. At this point, if the method used to calculate the fill rate is one which systematically overestimates its real value, as occurs in the case of β, the base-stock is lower than that required to reach the target fill rate, and therefore the system will be less protected against stockouts than managers might expect. Table 3 presents an illustrative example that helps to demonstrate the idea. In this case, if a target fill rate of 0.9 is set, according to the classic estimation of the fill rate, a base-stock of 7 units is deemed enough to reach the target fill rate, when actually it is 10 units that are needed to guarantee it. Another interesting detail that is shown in Table 3 is related to the fact that small differences in the fill rate can lead to important differences in the value of S. For the same example, a difference between both fill rates equal to only 0.01 (β = 0.90 and β = 0.91) leads to a difference of 3 units in the base-stock level. Therefore, despite the difference between the values of the fill rate seeming to be insignificant in some cases, it may become extremely important in the context of the base-stocks values and managers have to be forewarned about this fact.

Table 3

Base-stock level computed by β and β with Poisson demand with λ = 1, s = 2, L = 2.

S	β Classic	β Exact
5	0.85	0.79
6	0.88	0.83
7	0.90	0.86
8	0.92	0.88
9	0.93	0.89
10	0.94	0.91
11	0.94	0.92
12	0.95	0.92
13	0.95	0.93
14	0.96	0.93
15	0.96	0.94
16	0.96	0.94
17	0.97	0.95

One of the traditional problems in inventory management consists of how to compute accurately the optimal value of s and S simultaneously. This problem may be solved using a cost approach trying to find the optimal parameters that minimize the total costs of the system. However, the difficulty of this approach is the computation of the penalty cost, since it includes not only the direct cost of losing demand but also an indirect cost of losing customers goodwill and image of the company. For this reason, the most common approach in practical environments is to find the optimal parameters of the inventory system that guarantee the achievement of a target service level. In addition, this approach is more attractive to practitioners as it is easier for them to interpret the fill rate they strive to offer their customers [26]. However, to implement the service approach it is first required to have appropriate expression to compute accurately the service level because small errors in the calculation of the service level might have an important impact on the determination of optimal parameters [27]. In this sense, Table 4 presents the optimal value of s and S obtained with the classic and the proposed method when demand follows a Poisson distribution with λ = 1 and L = 2. Note that the cells shaded in grey do not meet the hypothesis of only one outstanding order launched. As can be seen, there are different combination of s and S that achieve the same target fill rate. For example, a target fill rate of 0.95 can be achieved when s = 2 and S = 13 or when s = 3 and S = 8 according to the classic method. However, the real value of the fill rate with these parameters is 92.9% and 92.8%, respectively. Then, if these combinations are used as optimal parameters, the system might incur unexpected stockout situations, which lead to higher stockout costs of the system. In fact, the optimal parameters to achieve a 0.95 target fill rate are s = 3 and S = 11.

Table 4

Base-stock level and ROP computed by β and β with Poisson demand with λ = 1 and L = 2.

	Classic
s / S	5	6	7	8	9	10	11	12	13	14	15
1	0.779	0.815	0.841	0.860	0.876	0.888	0.898	0.906	0.914	0.920	0.925
2	0.847	0.881	0.902	0.917	0.928	0.937	0.943	0.949	0.953	0.957	0.96
3			0.948	0.958	0.965	0.97	0.973	0.976	0.979	0.981	0.982
4					0.985	0.988	0.989	0.991	0.992	0.993	0.993
5							0.996	0.997	0.997	0.998	0.998
6									0.999	0.999	0.999
7											1
	Exact
s / S	5	6	7	8	9	10	11	12	13	14	15
1	0.733	0.772	0.801	0.823	0.841	0.855	0.867	0.878	0.886	0.894	0.901
2	0.794	0.835	0.861	0.881	0.895	0.906	0.915	0.923	0.929	0.934	0.939
3			0.912	0.928	0.939	0.946	0.952	0.957	0.961	0.964	0.967
4					0.968	0.973	0.977	0.979	0.981	0.983	0.985
5							0.99	0.991	0.992	0.993	0.994
6									0.997	0.997	0.998
7											0.999

Conclusions

This paper proposes, for the first time, an exact method for the calculation of the fill rate when the inventory is managed by the continuous review reorder point, base-stock (s, S) policy in a context of lost sales and discrete demand. As opposed to the classic approach (β), the exact method (β) takes explicit account of the presence of undershoots; consequently, it is not assumed that the replenishment order is launched when the inventory position exactly reaches the reorder point, but rather when it is equal to or lower than it.

Section 5 analyses the performance of β and β as opposed to a simulated fill rate. Results show that β introduces a bias into the computation of the metric since it overestimates the fill rate. This performance was to be expected if we consider that neglecting the undershoots implies assuming that we have more on-hand stock available to meet future demand during the lead time than we actually do. Deviations presented by β are more important when using it to compute the base-stock of the system, since small differences in the fill rate become higher when dealing with stock levels. The result of this article has a very important practical implication in regards to the design of (s, S) systems. In practice, the base-stock of the policy is determined to guarantee a target fill rate. If the method used to calculate the fill rate is one which systematically overestimates its real value, as occurs in the case of β, the established base-stock is lower than that required to reach the target service level and therefore the system will be less protected against possible stockouts than the inventory managers might expect.

As the Introduction section points out, neglecting the undershoots at the reorder point has important implications in inventory management. After analysing the results of this paper, it is clear that: (i) the impact of considering a replenishment order as constant and equal to s-S may not be ignored; (ii) it is necessary to take into account the variability of the replenishment cycle for accurate computation of the fill rate and the parameters of the (s, S) system; (iii) the mathematical complexity of the computation of the probability of every stock level when the reorder point is reached is acceptable for the benefit of an accurate fill rate estimation. The method presented in this paper eliminates the problems presented by the classical method, β, and thus entails a robust non-skewed method to exactly calculate the fill rate in the discrete demand and lost sales context, and therefore contributes to the operational research as the only exact method available in this context.

Limitations of this study mainly focus on the hypotheses regarding the i.i.d. demand condition. While this is a common assumption when dealing with theoretical inventory models, extending these methods such as the one addressed in this article to demand contexts that do not satisfy the i.i.d. condition may be beneficial for practical settings. Finally, the exact method derived in this paper, in addition to determining the customer service performance of the inventory management system, can be used as a constraint in deriving optimal stock policy values. This application of the exact method proposed in this article constitutes a challenging further research of this work.

β β and β for the data from Table 1.

(XLSX)

Click here for additional data file.

Calculation of the transition matrixes and .

(DOCX)

Click here for additional data file.

Derivation of the demand distribution from the beginning of the cycle until ROP.

(DOCX)

Click here for additional data file.

19 Oct 2021

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

Baogui Xin, Ph.D.

Academic Editor

PLOS ONE

Journal Requirements:

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Partly

Reviewer #2: Partly

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: I Don't Know

Reviewer #2: N/A

**********

3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: No

**********

4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: Referee report

In this draft, the authors have reported their efforts in proposing an exact method to estimate the fill rate in a base-stock (s,S) policy. Overall, this paper is good and it needs substantial changes. Following major points are recommended to revise the present article:

1- Literature survey should be improved by reviewing recently related studies. The references are too old.

2- You calculate the exact value of the fill rate by considering the value of S as a parameter. Can you set S as a decision variable and calculate its exact and optimal value considering the constant fill rate?

3- You may design and solve an inventory problem based on the base-stock policy, and use the proposed exact method to obtain the optimal value of s and S. In this way, the effectiveness of the exact method in reducing inventory system costs compared to the classical method can be seen.

4- In the section of “practical implications”, you refer to Table 4, but you provide the information in Table 3 (See line 283 and 287). This is also the case with Table 2 (See line 271).

5- In the conclusion section, you may explore the future extension of this proposed method mentioning major limitations of it.

Reviewer #2: Given possible undershoots at the reorder point, the authors proposed a method to accurately estimate the fill rate for the continuous (s, S) inventory system for the lost sales context. However, the paper has the following problems.

1. The undershoots at the reorder point might be only possible for lumpy and batch production situations. Now many companies use real-time systems and automate reorder process with built-in trigger functions, the undershoot issues might not be relevant. Thus, the authors need to clearly identify the limitations of the research.

2. For the exact approach, the authors assume that a Markov Chain transition process is followed. How about a non-Markov chain process？

**********

6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

14 Nov 2021

RESPONSE TO ACADEMIC EDITOR AND REVIEWERS

The authors would firstly like to express our gratitude to the academic editor and the reviewers for all the comments they have made. All the changes made to the manuscript following their comments have contributed to an increase in the quality of our work.

In this document, we answer all the comments made by them in the same order and showing where the changes have been made in the final version of the manuscript. Please note that our answers to the different concerns are written in blue and highlighted also in blue in the final version of the manuscript (Revised Manuscript with Track Changes file).

RESPONSE TO THE ACADEMIC EDITOR

1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at

https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and

https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

Format has been updated to fulfill PLOS ONE’s style requirements.

Furthermore in the new version of the manuscript the Appendix has been converted to S1 Appendix and S2 Appendix following the recommendations about Supporting Information; funding information has been removes from the acknowledgments; and format of headings, normal text, references, Equations, Figures and Tables has been reviewed.

2. We note that the grant information you provided in the ‘Funding Information’ and ‘Financial Disclosure’ sections do not match.

When you resubmit, please ensure that you provide the correct grant numbers for the awards you received for your study in the ‘Funding Information’ section.

The information has been duly updated.

3. In your Data Availability statement, you have not specified where the minimal data set underlying the results described in your manuscript can be found. PLOS defines a study's minimal data set as the underlying data used to reach the conclusions drawn in the manuscript and any additional data required to replicate the reported study findings in their entirety. All PLOS journals require that the minimal data set be made fully available. For more information about our data policy, please see http://journals.plos.org/plosone/s/data-availability.

Upon re-submitting your revised manuscript, please upload your study’s minimal underlying data set as either Supporting Information files or to a stable, public repository and include the relevant URLs, DOIs, or accession numbers within your revised cover letter. For a list of acceptable repositories, please see http://journals.plos.org/plosone/s/data-availability#loc-recommended-repositories. Any potentially identifying patient information must be fully anonymized.

Important: If there are ethical or legal restrictions to sharing your data publicly, please explain these restrictions in detail. Please see our guidelines for more information on what we consider unacceptable restrictions to publicly sharing data: http://journals.plos.org/plosone/s/data-availability#loc-unacceptable-data-access-restrictions. Note that it is not acceptable for the authors to be the sole named individuals responsible for ensuring data access.

We will update your Data Availability statement to reflect the information you provide in your cover letter.

According to the data policy, a new file (S1 Table) has been added to the submission to support the results of our paper. Also appendixes have been converted as Support information files.

4. Please amend the manuscript submission data (via Edit Submission) to include author Eugenia Babiloni.

The confusion in the names of the authors has been duly corrected, as Eugenia Babiloni and Maria Eugenia are the same person. Her name now appears correctly. Thank you for your suggestion.

5. Please amend your authorship list in your manuscript file to include author María Eugenia

See previous comment.

6. Please ensure that you refer to Figure 4 in your text as, if accepted, production will need this reference to link the reader to the figure.

Thank you for your comment. In the new version of the manuscript, Fig. 4 is quoted in the text.

7. We note you have included a table to which you do not refer in the text of your manuscript. Please ensure that you refer to Table 1 in your text; if accepted, production will need this reference to link the reader to the Table.

Thank you for your comment. In the new version of the manuscript Table 1 is quoted in the text.

8. Please include a copy of Table 4 which you refer to in your text on page 13.

Thank you for your comment. There was a typographical error in the numbering of the tables, which has been conveniently corrected. In the new version of the manuscript, Table 1, Table 2 and Table 3 are referenced properly in the document. In the new version of the manuscript, Table 4 is referred to the new example included under the “Practical implications” section.

RESPONSE TO REVIEWER 1

1- Literature survey should be improved by reviewing recently related studies. The references are too old.

The literature has been carefully reviewed. Older references have been deleted or replaced by more recent ones where possible.

In summary, the references that have been removed are (numbering of the old version of the manuscript):

7. Silver EA. A modified formula for calculating customer service under continuous inventory review. AIIE Trans. 1970;2: 241–245.

10. Karlin S, Scarf H. Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, Stanford, CA; 1958.

17. Vincent P. Exact fill rates for items with erratic demand patterns. INFOR Inf Syst Oper Res. 1985;23: 171–181.

24. Hadley G, Whitin T. Analysis of Inventory Systems. Englewood Cliffs, NJ: Prentice-Hall; 1963.

References that have been added in the new version of the manuscript are:

4. Luo P, Bai L, Zhang J, Gill R. A computational study on fill rate expressions for single-stage periodic review under normal demand and constant lead time. Oper Res Lett. 2014;42: 414–417. doi:10.1016/j.orl.2014.07.004

9. Gutierrez M, Rivera FA. Undershoot and order quantity probability distributions in periodic review, reorder point, order-up-to-level inventory systems with continuous demand. Appl Math Model. 2021;91: 791–814.

26. Gutgutia A, Jha JK. A closed-form solution for the distribution free continuous review integrated inventory model. Oper Res. 2018;18: 159–186. doi:https://doi.org/10.1007/s12351-016-0258-5.

27. Babiloni E, Guijarro E. Fill rate: from its definition to its calculation for the continuous (s, Q) inventory system with discrete demands and lost sales. Cent Eur J Operarion Res. 2020;28: 35–43. doi:https://doi.org/10.1007/s10100-018-0546-7.

2- You calculate the exact value of the fill rate by considering the value of S as a parameter. Can you set S as a decision variable and calculate its exact and optimal value considering the constant fill rate?

The new version of the manuscript includes an explanation under “Practical implication” section (see Table 4) to illustrate that, given a target fill rate, we can find different combinations of optimal s and S. As this example shows, if the classic fill rate is used to determine the optimal values of the inventory policy (s and S), the systems is not actually meeting the target fill rate and, as a consequence, the systems is less protected against stockout situations than expected. (See the Response to Reviewers file for detais).

3- You may design and solve an inventory problem based on the base-stock policy, and use the proposed exact method to obtain the optimal value of s and S. In this way, the effectiveness of the exact method in reducing inventory system costs compared to the classical method can be seen.

Thank you for this comment. Following the Reviewer’s suggestion, we have included a new example in section “Practical implications” to illustrate how the proposed method can be used to find the optimal values of s and S when a target fill rate is given. As can be observed in the new Table 4, the use of the classic method may lead to unexpected stockout situations, and consequently to higher stock-out costs of the system. Please see comment 2 of this revision for explanation of the new version of the practical implications section.

In any case, we agree with the reviewer that it would be a very interesting contribution to derive a method for the optimal derivation of policy parameters, for which the objective of this paper is a major contribution. Following his recommendations, the application of the exact method proposed in this paper for the determination of control parameters of the inventory policy has been included as a further research as follows:

“[…].Finally, the exact method derived in this paper, in addition to determining the customer service performance of the inventory management system, can be used as a constraint in deriving optimal stock policy values. This application of the exact method proposed in this article constitutes a challenging further research of this work.”

4- In the section of “practical implications”, you refer to Table 4, but you provide the information in Table 3 (See line 283 and 287). This is also the case with Table 2 (See line 271).

Thank you for your comment. There was a typographical error in the numbering of the tables, which has been conveniently corrected. In the new version of the manuscript, Table 1,Table 2 and Table 3 are referenced properly in the document. In the new version of the manuscript, Table 4 is referred to the new example included under the “Practical implications” section.

5- In the conclusion section, you may explore the future extension of this proposed method mentioning major limitations of it.

Thank you for the suggestion. In the latest version of the manuscript, a paragraph on the limitations of this study has been introduced as follows:

“Limitations of this study mainly focus on the hypotheses regarding the i.i.d. demand condition. While this is a common assumption when dealing with theoretical inventory models, extending these methods such as the one addressed in this article to demand contexts that do not satisfy the i.i.d. condition may be beneficial for practical settings. […]”

RESPONSE TO REVIEWER 2

1. The undershoots at the reorder point might be only possible for lumpy and batch production situations. Now many companies use real-time systems and automate reorder process with built-in trigger functions, the undershoot issues might not be relevant. Thus, the authors need to clearly identify the limitations of the research.

Thank you for this comment. We have realized that the previous version of the manuscript did not clearly state the importance of the presence of undershoots in the (s, S) inventory policy since they appear when demand is not unit sized. Therefore, the problem of neglecting undershoots affects not only lumpy or batch production demand, but also non-intermittent demand. In fact, the illustrative examples used in this paper consider a demand rate of 1 unit and, even in that case, the classic method underestimates the real value of the fill rate due to the fact of neglecting undershoots.

In order to avoid possible misunderstandings, we have better clarified the importance of undershoots in the “Introduction” section:

“[…] This policy entails an additional difficulty: to exactly reach the reorder point, the demand must be necessarily unitary. If this is not the case the inventory position may not be exactly at the reorder point, but a certain amount below it, called deficit or undershoot, which is difficult to quantify. Therefore common derivations of a (s, S) policy are based upon the assumption that the undershoots at the reorder point are negligible [6]. However, neglecting undershoots may lead to higher inventory costs or a lower service level than desired [8,9]. This is particularly true when managing intermittent, lumpy or slow-moving items for which the unitary demand assumption is not appropriate [10] although undershoots can also occur with non-intermittent demand as long as the unit sized demand assumption is not met.”

Also in “Problem description” section, at the end of the first paragraph:

“[…] However, in practical environments it is uncommon to find items whose demand is unitary, which makes it necessary to consider the presence of undershoots when determining optimal parameters of the inventory system.”

And at the end of “Results and discussion” section, when we comment the results of Table 2:

“[…] What can be seen in Table 2 is that when the demand rate increases, meaning it is more difficult to assume the unitary demand hypothesis, the average and deviation of the errors made by �  Classic increase. However, this example illustrates that undershoots appears also for non-intermittent demand, which justifies that undershoots should be taken into account explicitly in the calculation of the fill rate.”

Furthermore, we have better explained the novelty of the proposed method, which is based on calculating the probability vector of the on-hand stock at ROP, instead of approximating the undershoots. In this way, our approach is valid both when undershoots appear and when they do not. In the new version of the manuscript, the third paragraph of “Introduction” section reads:

“[…] Our approach, on contrary, instead of deriving an approach of the distribution function of the undershoots, considers that the inventory position when the reorder point is reached may take any feasible value (i.e. from 0 to s). This entails not neglecting the undershoots at the reorder point and implies the difficulty of accurately calculating expected values of the on-hand stock at order delivery and when the reorder point is reached (exactly or exceeded). This approach makes our research innovative.”

Lastly, regarding the limitations and further research of this work, we have introduced a new paragraph at the end of the “Conclusions” section as follows:

“Limitations of this study mainly focus on the hypotheses regarding the i.i.d. demand condition. While this is a common assumption when dealing with theoretical inventory models, extending these methods such as the one addressed in this article to demand contexts that do not satisfy the i.i.d. condition may be beneficial for practical settings. Finally, the exact method derived in this paper, in addition to determining the customer service performance of the inventory management system, can be used as a constraint in deriving optimal stock policy values. This application of the exact method proposed in this article constitutes a challenging further research of this work.”

2. For the exact approach, the authors assume that a Markov Chain transition process is followed. How about a non-Markov chain process？

Thank you for pointing out that in the previous wording it looked like it might be a non-Markov process. We have partially rewritten the paragraph prior to equation (4) as follows:

“The stock level of policy (s, S) satisfies the Markov property, i.e. the future state only depends on the current situation. As a result, on-hand stock levels can be represented by a Markov process and the probability transition matrix of the on-hand stock between two consecutive replenishment cycles needs to be determined so that:”

Submitted filename: Response to Reviewers.docx

Click here for additional data file.

25 Jan 2022

On the estimation of the fill rate for the continuous ( s, S ) inventory system for the lost sales context

PONE-D-21-14224R1

Dear Dr. Babiloni,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at http://www.editorialmanager.com/pone/, click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org.

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

Kind regards,

Baogui Xin, Ph.D.

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #2: All comments have been addressed

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #2: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #2: N/A

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #2: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #2: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #2: The authors have addressed my comments in the previous round. I have no further comments in this round.

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #2: No

31 Jan 2022

PONE-D-21-14224R1

On the estimation of the fill rate for the continuous (s, S) inventory system for the lost sales context

Dear Dr. Babiloni:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

If we can help with anything else, please email us at plosone@plos.org.

Thank you for submitting your work to PLOS ONE and supporting open access.

Kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Professor Baogui Xin

Academic Editor

PLOS ONE