# A Mathematical Safety Stock Model for DDMRP Inventory Replenishment

Authors: Chan-Ju Lee; Suk-Chul Rim
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The inventory replenishment method of the demand-driven materials requirement planning (DDMRP) draws attention from both scholars and practitioners due to its lower average inventory and, simultaneously, lower stockout rate than existing methods. However, the safety stock of DDMRP replenishment uses a subjective guideline, which is different from any existing safety stock formula. The guideline allows the user to subjectively select the parameter values within a certain range, which jeopardizes the consistency of the inventory performance. This paper proposes an alternative safety stock formula for DDMRP replenishment, which is mathematically defined so as to be consistent. Moreover, simulation indicates that the proposed formula outperforms the DDMRP guidelines and existing safety stock formulas in terms of average inventory and stockout rate.


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## Body

## 1. Introduction

In demand-driven material requirement planning (DDMRP), replenishment is a core concept. DDMRP replenishment approach is different from existing replenishment methods (e.g., Silver et al. [1], Ballou and Srivastava [2], and Brown [3]). The typical research targets in replenishment are to determine when and how much to order. When to order is typically determined as the sum of demand during lead time (DDLT) and safety stock. DDLT is defined as the average demand times average lead time. However, regarding the safety stock, DDMRP replenishment model is very different from that of existing studies.Most of the inventory replenishment models determine the safety stock level by using the safety factor associated with a given service level, based on the probability theory [1]. However, according to Brown [3], applying a safety stock theory that assumes an incorrect demand distribution shape can cause severe consequences, so it needs to identify the correct distribution shape to determine the safety factor. However, it is difficult to determine the distribution type in practices, since it may vary depending on the span of history data.As an alternative, Goldratt et al. [4] proposed a replenishment method which does not use the safety factor based on distribution shape. He presented MTA (make-to-availability), which is an inventory replenishment and operation method. Due to its practicality, many successful cases have been reported in the TOCICO Conferences [5]. Ptak and Smith [6] further exploited MTA more specifically in DDMRP. Many successful cases of DDMRP have been reported in the literature (see Section 2.7).The safety stock of DDMRP replenishment also does not use the safety factor by service level. Instead, DDMRP replenishment guideline [6] allows the users to select the parameter values within a certain range, which will lead to inconsistent results due to its subjectivity. We need an objective formula to determine the safety stock which will obtain more consistent results.In this study, we propose a new safety stock formula which we derived from the DDMRP replenishment guidelines [6]. The proposed safety stock formula will eliminate subjectivity in determining the safety stock of DDMRP. Moreover, it does not compromise the performance of DDMRP replenishment; rather, better performance is expected in certain situations.In order to examine the performance of the proposed safety stock formula, we conduct a simulation of DDMRP replenishment model in which we compare the average inventory and stockout rate of the following three safety stock formulas: DDMRP guideline, existing safety stock formula, and the proposed safety stock formula.In the following section, we examine the research results related to the safety stock of replenishment models, and subsequently, we review the Theory of Constraints (TOC) replenishment method. In the next subsection, we review the DDMRP replenishment model which is based on the TOC and derive a new safety stock formula. Simulation model and its results are given in the following sections. In the last section, conclusion and future research agenda are presented.


## 2. Materials and Methods

### 2.1. Safety Stock of Replenishment Models and DDMRP Model

We use the following notations in the model:d: average daily demandSd: standard deviation of daily demandL: average replenishment lead timeSL: standard deviation of replenishment lead timezα: safety factor with service level ασL: standard deviation of demand during lead time


### 2.2. Safety Stock Formulas in Existing Models

Brown [3] presented a method to determine the safety stock using the standard deviation of demand during lead time and safety factor as shown in (1). Ballou and Srivastava [2] showed that when d and L are random variables, the standard deviation of demand during lead time is determined as (2):(1)SS=zασL,(2)σL=L⋅sd2+d2⋅sL2.The safety factorzα is determined from the service level α which is strategically determined. Recall that, in (1) and (2), both daily demand and replenishment lead time are assumed to follow normal distribution, which we believe is unrealistic. Silver et al. [1] state that “because lead time demand is often skewed right, the normal distribution may not be appropriate.” Clark [7] also states that the deviations from normality of the demand during lead time can be characterized almost completely by the skewness. He also states that “in a typical case the optimum value determined a stockout of 2 per cent, but the calculated values would have produced a 7 per cent stockout if skewness had been ignored,” indicating that skewness of distribution is also necessary for determining the safety factor in addition to mean and variance. Das [8] states that demand for an item during the delivery lead time can often be represented by a lognormal distribution. Brown [3] also states that exponential distribution and lognormal distribution substantially differ in the expected stockout rates for large safety factor, although the two distributions may look similar. Most existing literature in inventory management considers a certain level of stockout and safety factor based on demand and/or lead time distribution [9]. Since the safety factors differ depending on the distribution shape, the accurate distribution shape must be identified in order to determine an accurate safety factor. However, it is difficult to determine the distribution type in practices, since it may vary depending on the span of history data such as seasonality, or competitors’ behavior. Goldratt and Cox, the founder of the Theory of Constraints (TOC) [10], mentioned this point as he proposed the TOC replenishment model in MTA [11].


### 2.3. TOC Replenishment Model

Goldratt claimed that the stock should be strategically stored in a position within a timeframe of which customers are willing to wait for their orders, and the make-to-availability (MTA) methodology is used for products that must be continuously guaranteed. MTA ensures the complete availability of a certain finished product in a certain warehouse, occupying a selected market [12]. The TOC replenishment method is used in the MTA methodology. Goldratt et al. [4] described this concept in his book “Isn’t it obvious,” and Schragenheim et al. [11] described specific application methods for the concept in the book “Supply Chain Management at Warp Speed.”The TOC replenishment model is similar to the periodic review model, while the safety stock formula is different. Rather than using a specific formula, it is recommended to determine the safety stock as 50% of the average demand during lead time and replenishment time, as (3). Note that the safety stock formula of TOC in (3) does not use safety factor, and it does not need to identify the probability distributions:(3)SSTOC=0.5dT+L.Goldratt reduced the subjective factor here. Safety stock starts by setting 50% of the demand during the replenishment lead time regardless of the shape of distribution. And then, according to the trend of demand, the maximum level of stock is adjusted by the given rule instead of the safe stock adjustment. It differs from the existing replenishment in that it focuses on the maximum stock level instead of focusing on accurate safety stock calculations.In MTA operation, TOC replenishment not only yields good inventory performance, but also focuses on ensuring availability by providing guidance for planning, execution, and control. The effectiveness of MTA method with TOC replenishment has been verified through a series of cases presentations in the TOCICO Conference [5]. One recent example is the milk run replenishment presented by Bonatsos [13] at the TOCICO Conference 2013, in which the average monthly stock decreased, while inventory turnover increased. Han and Kwon [14] compared the forecast-driven approach and TOC replenishment, to conclude that TOC replenishment is superior in terms of stockout, while both methods obtain similar average inventory level.TOC provides an opportunity to convert inventory management from forecast-driven to demand-driven. Christopher stated that most organizations base inventory management on the forecast-driven approach rather than demand-driven approach [15]. He also states that the demand-driven approach is market-sensitive, which means that the supply chain can identify and respond to real demand changes, so it is more effective in maintaining lower inventory than the forecast-driven approach while minimizing stockout. All TOC methods including MTA are based on this demand-driven concept. The demand-driven supply network (DDSN) was defined in AMR research [16]. In addition, Chan et al. [17] studied the demand-driven approach from the perspective of collaborative supply chains. Ptak and Smith [6] reinterpreted the TOC as the demand-driven concept and structured it into the DDMRP.


### 2.4. DDMRP Components

Ptak and Smith [6] introduced DDMRP as the Third Edition of Orlicky’s Material Requirements Planning. Together with Goldratt, Schragenheim and Ptak [18] authored the book “Necessary but Not Sufficient.” DDMRP [6] complements the weakness of materials requirement planning (MRP) by integrating the replenishment method with the visibility of the demand-driven method into a single system. Figure 1 shows the components of DDMRP.Figure 1
DDMRP components [6].The first stage in Figure1, strategic inventory positioning, is the stage in which inventory position is determined. In DDMRP, inventory positioning is very important. On the Bill of Material (BOM), the strategically determined stock item is referred to as the decoupling point. At this decoupling point, stock is replenished. As customers require shorter lead time, it directly affects the stock level. Since variability increases for longer lead time, the stock of appropriate items is strategically placed, based on the BOM, thereby decoupling the linked variability. Since the end item is produced using parts and components in stock, the lead time offered to customers can be shortened.In the second stage, buffer profiles and levels, the buffer levels are determined and the maximum level of stock (M) is defined. In the third stage, like the TOC replenishment method, buffer levels are adjusted when item characteristics change. In the fourth stage, the replenishment amount is calculated. The fifth stage, visible and collaborative execution, is the control step similar to the TOC replenishment method.


### 2.5. DDMRP Replenishment and Its Guideline

The DDMRP replenishment model is similar to the min-max model. In the DDMRP replenishment model, a maximum level of stock (M) is expressed as the sum of three components: yellow zone (YZ), red zone (RZ), and green zone (GZ). YZ is defined as average daily demand multiplied by average lead time, as in (4), so it corresponds to the demand during lead time in the existing methods. RZ represents the safety stock in the existing replenishment models. Note that, in the reorder point model (or min-max model), an order is placed when the inventory position falls below the reorder point, which is the sum of DDLT and safety stock. Likewise, in DDMRP replenishment, an order is placed when the inventory position falls below the top of yellow, which is the sum of DDLT (i.e., YZ) and RZ, as in (10). So RZ clearly corresponds to the safety stock in the existing replenishment models.RZ is expressed as the sum of red zone base (RB) and red zone safety (RS) as in (5). RB is defined as YZ multiplied by the lead time factor in Table 1, as in (6). RS is defined as RB multiplied by the variability factor in Table 2, as in (7). GZ represents the order quantity and defined as YZ multiplied by the lead time factor in Table 1, as in (8). Top of green (TOG) is the sum of YZ, RZ, and GZ, as in (9), and represents the maximum stock level (M) in the existing replenishment models. Top of yellow (TOY) is the sum of YZ and RZ, as in (10), and represents the reorder point in the existing models:(4)YZ=d⋅L,(5)RZ=RB+RS,(6)RB=YZ⋅FL,(7)RS=RB⋅FV,(8)GZ=d⋅L⋅FL,(9)TOG=GZ+YZ+RZ,(10)TOY=YZ+RZ.Table 1
Guideline for lead time factorFL [19].
Lead time
F
L (%)
Purchased part (days)
Long
20–40
26+
Medium
41–60
11–25
Short
61–100
1–10Table 2
Guideline for variability factorFV [19].
Variation
F
V (%)
High
61–100
Medium
41–60
Low
20–40The lead time factor (FL) and variability factor (FV) in (6) and (7) can be determined by the user using the guidelines [19] shown in Tables 1 and 2, respectively. Note that RS in (7) does not explicitly consider the lead time variability. The guideline [6] is presented for three relative variability levels (high, medium, or low); three relative length of lead time (long, medium, or short); three item types (make, purchased, or distributed); and two types of order quantities (minimum order quantity or not), total of 54 cases. Among those, for items to buy, the guideline [6, 19] presented the lead time factor as in Table 1.


### 2.6. Order Spike and Replenishment Quantity in DDMRP

To address the replenishment quantity (RQ) calculation in DDMRP, we first need to address the order spike. The order spike is a qualifying quantity of known cumulative demand within a qualifying time window that threatens the integrity of the buffer [19]. That means that the qualifying level, namely, the order spike threshold (OST), and the qualifying time window, namely, the order spike horizon (OSH) must be defined [6]. If the total orders received for an item on a specific date are greater than the threshold, then the amount is regarded as the order spike, as shown on the fourth day in Figure 2. Note that the cumulated order on day 9 is not regarded as an order spike since it is currently out of the order spike horizon (5 days). OSH is typically set to be the replenishment lead time. Ptak and Smith [6] recommend OST value to be determined as 50% of RZ.Figure 2
An example of order spike on day 4.With order spikes, replenishment is requested in advance for the daily lumpy demand with confirmed order, allowing earlier response to the lumpy demand. This results in a substantial reduction of stockout. Clark [7] explains this as follows: “The distribution of daily demand was not only skewed but even bimodal. The demand during lead time is characterized by three parameters, mean, standard deviation, and the coefficient of skewness.” Safety stock that does not reflect skewness cannot respond to lumpy demand; however, the order spike concept can effectively respond to lumpy demand without considering skewness.In DDMRP replenishment, order is placed when the available stock is below TOY. However, there is a difference in the method of determining the replenishment quantity. Generally, the replenishment quantity is the maximum stock minus available stock. Recall that Brown [20] defines available stock (AS) as on-hand (H) plus schedule receipt (R) minus allocated quantity (A), as in (11). However, in DDMRP, the replenishment order quantity (RQ) is defined as top of green (TOG) minus available stock (AS) plus order spike quantity (k), as in (12):(11)AS=H+R−A,(12)RQ=TOG−AS+k.


### 2.7. DDMRP-Related Research

Success stories from companies that have applied the DDMRP method have been continuously announced. According to the Demand Driven Institute [21], improved performance of numerous enterprises, such as Royal Engineered Composites, Figeac Aero, Allergan, Stemcell Technologies, Michelin, MIC, Oregon Freeze Dry, and PZ Cussons, is evidence of the high practicality of this method.While there have been many DDMRP-related studies, most of them compare with MRP. Lee and Jang [22] demonstrated DDMRP’s superiority through a system dynamic simulation with DDMRP and MRP. Ihme [23] applied DDMRP to a printer ink manufacturer, demonstrating improved performance, in which the stock level and stockout are reduced by approximately 45% and 95%, respectively. Miclo et al. [24] used a discrete-event simulation approach to conduct an objective and quantitative comparison between DDMRP and MRP. Shofa and Widyarto [25] presented a paper studying MRP and DDMRP to effectively control production in the automobile industry. Similar to these examples, most of the other studies were comparisons of MRP and DDMRP. Other than these examples, Rim et al. [26] developed an algorithm to optimize strategic inventory position in BOM. Our study is unique in that we present a mathematical approach to determine the red zone level for the DDMRP replenishment, which has not been reported in the literature.


### 2.8. Excessive Deviation of the Safety Stock in DDMRP

In DDMRP guideline for the safety stock (RZ), the lead time factor and variability factor are selected arbitrarily within a certain range given in Tables1 and 2. The critical weakness of DDMRP replenishment method is that excessively large deviation can be obtained by subjective choice of the factors, so that the performance will be naturally inconsistent. For example, consider the average demand of 1,000 units and the average lead time of 5 days. The guideline from Table 1 will offer the user to choose the lead time factor between 0.61 and 1.0, and you may choose the variability factor between 0.2 and 1.0 if no information is available about the demand variability. Recall that, from (5) to (7), RZ=RB+RS=dLFL1+FV. So the maximum safety stock will be determined as 1,00051.001+1.00=10,000, while minimum as 1,0000.611+0.20=3,660. Safety stock can be determined as three times larger than the minimum, which is excessively inconsistent. For medium and long lead time cases, the gap between minimum and maximum is about 2.4 time or 3.3 times greater than safety stock level, respectively, as shown in Table 3.Table 3
Safety stock level obtained from the various parameter values.
F
V
Low
Medium
High
Min
Max
Min
Max
Min
Max
Lead time
F
L
0.20
0.40
0.41
0.60
0.61
1.00
Long
35
Min
0.20
8,400
9,870
11,270
Max
0.40
19,600
22,400
28,000
Medium
20
Min
0.41
9,840
11,562
13,202
Max
0.60
16,800
19,200
24,000
Short
5
Min
0.61
3,660
4,301
4,911
Max
1.00
7,000
8,000
10,000
∗Empty cells indicate irrelevant combinations.Such an excessive inconsistency of the existing guideline for safety stock level in [6] is the core motivation of this study. In the following section, we will present an alternative approach to determine the safety stock level of DDMRP replenishment, which will result in more consistent and better performance in terms of average inventory and shortage.


## 2.1. Safety Stock of Replenishment Models and DDMRP Model

We use the following notations in the model:d: average daily demandSd: standard deviation of daily demandL: average replenishment lead timeSL: standard deviation of replenishment lead timezα: safety factor with service level ασL: standard deviation of demand during lead time


## 2.2. Safety Stock Formulas in Existing Models

Brown [3] presented a method to determine the safety stock using the standard deviation of demand during lead time and safety factor as shown in (1). Ballou and Srivastava [2] showed that when d and L are random variables, the standard deviation of demand during lead time is determined as (2):(1)SS=zασL,(2)σL=L⋅sd2+d2⋅sL2.The safety factorzα is determined from the service level α which is strategically determined. Recall that, in (1) and (2), both daily demand and replenishment lead time are assumed to follow normal distribution, which we believe is unrealistic. Silver et al. [1] state that “because lead time demand is often skewed right, the normal distribution may not be appropriate.” Clark [7] also states that the deviations from normality of the demand during lead time can be characterized almost completely by the skewness. He also states that “in a typical case the optimum value determined a stockout of 2 per cent, but the calculated values would have produced a 7 per cent stockout if skewness had been ignored,” indicating that skewness of distribution is also necessary for determining the safety factor in addition to mean and variance. Das [8] states that demand for an item during the delivery lead time can often be represented by a lognormal distribution. Brown [3] also states that exponential distribution and lognormal distribution substantially differ in the expected stockout rates for large safety factor, although the two distributions may look similar. Most existing literature in inventory management considers a certain level of stockout and safety factor based on demand and/or lead time distribution [9]. Since the safety factors differ depending on the distribution shape, the accurate distribution shape must be identified in order to determine an accurate safety factor. However, it is difficult to determine the distribution type in practices, since it may vary depending on the span of history data such as seasonality, or competitors’ behavior. Goldratt and Cox, the founder of the Theory of Constraints (TOC) [10], mentioned this point as he proposed the TOC replenishment model in MTA [11].


## 2.3. TOC Replenishment Model

Goldratt claimed that the stock should be strategically stored in a position within a timeframe of which customers are willing to wait for their orders, and the make-to-availability (MTA) methodology is used for products that must be continuously guaranteed. MTA ensures the complete availability of a certain finished product in a certain warehouse, occupying a selected market [12]. The TOC replenishment method is used in the MTA methodology. Goldratt et al. [4] described this concept in his book “Isn’t it obvious,” and Schragenheim et al. [11] described specific application methods for the concept in the book “Supply Chain Management at Warp Speed.”The TOC replenishment model is similar to the periodic review model, while the safety stock formula is different. Rather than using a specific formula, it is recommended to determine the safety stock as 50% of the average demand during lead time and replenishment time, as (3). Note that the safety stock formula of TOC in (3) does not use safety factor, and it does not need to identify the probability distributions:(3)SSTOC=0.5dT+L.Goldratt reduced the subjective factor here. Safety stock starts by setting 50% of the demand during the replenishment lead time regardless of the shape of distribution. And then, according to the trend of demand, the maximum level of stock is adjusted by the given rule instead of the safe stock adjustment. It differs from the existing replenishment in that it focuses on the maximum stock level instead of focusing on accurate safety stock calculations.In MTA operation, TOC replenishment not only yields good inventory performance, but also focuses on ensuring availability by providing guidance for planning, execution, and control. The effectiveness of MTA method with TOC replenishment has been verified through a series of cases presentations in the TOCICO Conference [5]. One recent example is the milk run replenishment presented by Bonatsos [13] at the TOCICO Conference 2013, in which the average monthly stock decreased, while inventory turnover increased. Han and Kwon [14] compared the forecast-driven approach and TOC replenishment, to conclude that TOC replenishment is superior in terms of stockout, while both methods obtain similar average inventory level.TOC provides an opportunity to convert inventory management from forecast-driven to demand-driven. Christopher stated that most organizations base inventory management on the forecast-driven approach rather than demand-driven approach [15]. He also states that the demand-driven approach is market-sensitive, which means that the supply chain can identify and respond to real demand changes, so it is more effective in maintaining lower inventory than the forecast-driven approach while minimizing stockout. All TOC methods including MTA are based on this demand-driven concept. The demand-driven supply network (DDSN) was defined in AMR research [16]. In addition, Chan et al. [17] studied the demand-driven approach from the perspective of collaborative supply chains. Ptak and Smith [6] reinterpreted the TOC as the demand-driven concept and structured it into the DDMRP.


## 2.4. DDMRP Components

Ptak and Smith [6] introduced DDMRP as the Third Edition of Orlicky’s Material Requirements Planning. Together with Goldratt, Schragenheim and Ptak [18] authored the book “Necessary but Not Sufficient.” DDMRP [6] complements the weakness of materials requirement planning (MRP) by integrating the replenishment method with the visibility of the demand-driven method into a single system. Figure 1 shows the components of DDMRP.Figure 1
DDMRP components [6].The first stage in Figure1, strategic inventory positioning, is the stage in which inventory position is determined. In DDMRP, inventory positioning is very important. On the Bill of Material (BOM), the strategically determined stock item is referred to as the decoupling point. At this decoupling point, stock is replenished. As customers require shorter lead time, it directly affects the stock level. Since variability increases for longer lead time, the stock of appropriate items is strategically placed, based on the BOM, thereby decoupling the linked variability. Since the end item is produced using parts and components in stock, the lead time offered to customers can be shortened.In the second stage, buffer profiles and levels, the buffer levels are determined and the maximum level of stock (M) is defined. In the third stage, like the TOC replenishment method, buffer levels are adjusted when item characteristics change. In the fourth stage, the replenishment amount is calculated. The fifth stage, visible and collaborative execution, is the control step similar to the TOC replenishment method.


## 2.5. DDMRP Replenishment and Its Guideline

The DDMRP replenishment model is similar to the min-max model. In the DDMRP replenishment model, a maximum level of stock (M) is expressed as the sum of three components: yellow zone (YZ), red zone (RZ), and green zone (GZ). YZ is defined as average daily demand multiplied by average lead time, as in (4), so it corresponds to the demand during lead time in the existing methods. RZ represents the safety stock in the existing replenishment models. Note that, in the reorder point model (or min-max model), an order is placed when the inventory position falls below the reorder point, which is the sum of DDLT and safety stock. Likewise, in DDMRP replenishment, an order is placed when the inventory position falls below the top of yellow, which is the sum of DDLT (i.e., YZ) and RZ, as in (10). So RZ clearly corresponds to the safety stock in the existing replenishment models.RZ is expressed as the sum of red zone base (RB) and red zone safety (RS) as in (5). RB is defined as YZ multiplied by the lead time factor in Table 1, as in (6). RS is defined as RB multiplied by the variability factor in Table 2, as in (7). GZ represents the order quantity and defined as YZ multiplied by the lead time factor in Table 1, as in (8). Top of green (TOG) is the sum of YZ, RZ, and GZ, as in (9), and represents the maximum stock level (M) in the existing replenishment models. Top of yellow (TOY) is the sum of YZ and RZ, as in (10), and represents the reorder point in the existing models:(4)YZ=d⋅L,(5)RZ=RB+RS,(6)RB=YZ⋅FL,(7)RS=RB⋅FV,(8)GZ=d⋅L⋅FL,(9)TOG=GZ+YZ+RZ,(10)TOY=YZ+RZ.Table 1
Guideline for lead time factorFL [19].
Lead time
F
L (%)
Purchased part (days)
Long
20–40
26+
Medium
41–60
11–25
Short
61–100
1–10Table 2
Guideline for variability factorFV [19].
Variation
F
V (%)
High
61–100
Medium
41–60
Low
20–40The lead time factor (FL) and variability factor (FV) in (6) and (7) can be determined by the user using the guidelines [19] shown in Tables 1 and 2, respectively. Note that RS in (7) does not explicitly consider the lead time variability. The guideline [6] is presented for three relative variability levels (high, medium, or low); three relative length of lead time (long, medium, or short); three item types (make, purchased, or distributed); and two types of order quantities (minimum order quantity or not), total of 54 cases. Among those, for items to buy, the guideline [6, 19] presented the lead time factor as in Table 1.


## 2.6. Order Spike and Replenishment Quantity in DDMRP

To address the replenishment quantity (RQ) calculation in DDMRP, we first need to address the order spike. The order spike is a qualifying quantity of known cumulative demand within a qualifying time window that threatens the integrity of the buffer [19]. That means that the qualifying level, namely, the order spike threshold (OST), and the qualifying time window, namely, the order spike horizon (OSH) must be defined [6]. If the total orders received for an item on a specific date are greater than the threshold, then the amount is regarded as the order spike, as shown on the fourth day in Figure 2. Note that the cumulated order on day 9 is not regarded as an order spike since it is currently out of the order spike horizon (5 days). OSH is typically set to be the replenishment lead time. Ptak and Smith [6] recommend OST value to be determined as 50% of RZ.Figure 2
An example of order spike on day 4.With order spikes, replenishment is requested in advance for the daily lumpy demand with confirmed order, allowing earlier response to the lumpy demand. This results in a substantial reduction of stockout. Clark [7] explains this as follows: “The distribution of daily demand was not only skewed but even bimodal. The demand during lead time is characterized by three parameters, mean, standard deviation, and the coefficient of skewness.” Safety stock that does not reflect skewness cannot respond to lumpy demand; however, the order spike concept can effectively respond to lumpy demand without considering skewness.In DDMRP replenishment, order is placed when the available stock is below TOY. However, there is a difference in the method of determining the replenishment quantity. Generally, the replenishment quantity is the maximum stock minus available stock. Recall that Brown [20] defines available stock (AS) as on-hand (H) plus schedule receipt (R) minus allocated quantity (A), as in (11). However, in DDMRP, the replenishment order quantity (RQ) is defined as top of green (TOG) minus available stock (AS) plus order spike quantity (k), as in (12):(11)AS=H+R−A,(12)RQ=TOG−AS+k.


## 2.7. DDMRP-Related Research

Success stories from companies that have applied the DDMRP method have been continuously announced. According to the Demand Driven Institute [21], improved performance of numerous enterprises, such as Royal Engineered Composites, Figeac Aero, Allergan, Stemcell Technologies, Michelin, MIC, Oregon Freeze Dry, and PZ Cussons, is evidence of the high practicality of this method.While there have been many DDMRP-related studies, most of them compare with MRP. Lee and Jang [22] demonstrated DDMRP’s superiority through a system dynamic simulation with DDMRP and MRP. Ihme [23] applied DDMRP to a printer ink manufacturer, demonstrating improved performance, in which the stock level and stockout are reduced by approximately 45% and 95%, respectively. Miclo et al. [24] used a discrete-event simulation approach to conduct an objective and quantitative comparison between DDMRP and MRP. Shofa and Widyarto [25] presented a paper studying MRP and DDMRP to effectively control production in the automobile industry. Similar to these examples, most of the other studies were comparisons of MRP and DDMRP. Other than these examples, Rim et al. [26] developed an algorithm to optimize strategic inventory position in BOM. Our study is unique in that we present a mathematical approach to determine the red zone level for the DDMRP replenishment, which has not been reported in the literature.


## 2.8. Excessive Deviation of the Safety Stock in DDMRP

In DDMRP guideline for the safety stock (RZ), the lead time factor and variability factor are selected arbitrarily within a certain range given in Tables1 and 2. The critical weakness of DDMRP replenishment method is that excessively large deviation can be obtained by subjective choice of the factors, so that the performance will be naturally inconsistent. For example, consider the average demand of 1,000 units and the average lead time of 5 days. The guideline from Table 1 will offer the user to choose the lead time factor between 0.61 and 1.0, and you may choose the variability factor between 0.2 and 1.0 if no information is available about the demand variability. Recall that, from (5) to (7), RZ=RB+RS=dLFL1+FV. So the maximum safety stock will be determined as 1,00051.001+1.00=10,000, while minimum as 1,0000.611+0.20=3,660. Safety stock can be determined as three times larger than the minimum, which is excessively inconsistent. For medium and long lead time cases, the gap between minimum and maximum is about 2.4 time or 3.3 times greater than safety stock level, respectively, as shown in Table 3.Table 3
Safety stock level obtained from the various parameter values.
F
V
Low
Medium
High
Min
Max
Min
Max
Min
Max
Lead time
F
L
0.20
0.40
0.41
0.60
0.61
1.00
Long
35
Min
0.20
8,400
9,870
11,270
Max
0.40
19,600
22,400
28,000
Medium
20
Min
0.41
9,840
11,562
13,202
Max
0.60
16,800
19,200
24,000
Short
5
Min
0.61
3,660
4,301
4,911
Max
1.00
7,000
8,000
10,000
∗Empty cells indicate irrelevant combinations.Such an excessive inconsistency of the existing guideline for safety stock level in [6] is the core motivation of this study. In the following section, we will present an alternative approach to determine the safety stock level of DDMRP replenishment, which will result in more consistent and better performance in terms of average inventory and shortage.


## 3. Proposed Safety Stock Model in DDMRP Replenishment

In this paper, we propose a new safety stock (RZ) formula, which will mitigate the vulnerability of RZ of DDMRP replenishment, based on the following direction: first, the formula is still based on the DDMRP guidelines, but avoids inconsistency due to subjective choice of values within a range. Second, parameter values must be obtainable from the historical data. Third, both demand variability and lead time variability are considered. Fourth, safety factor based on the service level is not used. Fifth, according to the demand-driven concept, shortage must be minimized, while average inventory not be increased. Since RZ comprises the two components RB and RS, based on the above direction, we will propose the modifiedRBp and RSp, respectively, as follows.


### 3.1. ProposedRBp

Recall that, from (4) and (6), the existing RB is defined as in (13). Ptak and Smith [19] allow a subjective choice of a value FL within the range given in Table 1 to determine RB:(13)RB=dLFL.In Table1, it is implied that for shorter lead time case, the larger portion of DDLT and for the longer lead time case the smaller portion of DDLT be used as RB. Instead of such a subjective and qualitative relation, we propose an alternative quantitative expression of inverse relation that FL is expressed as 1/L. Then, the resulting RB1 is expressed as shown in the following equation:(14)RB1=dL1/L=dL.Now, in Figure3, the highest plotting curve represents f1L=LFL, where FL is computed by the linear interpolation for the three ranges given in Table 1. Although the long lead time in Table 1 is defined as over 25 days, for linear interpolation, we need to set the upper limit of the range. We consider 90 days as the longest lead time since most imported make-to-order (MTO) items often take about three months. The lowest curve represents f2L=L as in (14). It is clearly observed that the two curves differ significantly. From the simulation in our earlier study [27], we observed that DDMRP guideline yields excessive inventory for long lead time and some shortage for shorter lead time and higher variability. We also observed that f2L is too low to avoid shortage [27].Figure 3
f
1
L, f2L, and f3L.Hence, we adjustf2L to devise an alternative function f3L in such a way that f3L is f2L amplified by βfor 0<β<2 and shifted by γ for 0<γ<10, as shown in the following equation:(15)RBp=dβL+γ.The performance ofRBp in (15) will depend on the parameter values β and γ. So in order to find the pair of β and γ that yields the best performance, that is, the lowest average inventory with no shortage, for lead time of 5, 20, 35, and 60 days, CVd = 0.7 and CVL = 0.4, we perform a simulation for the range 0<β<2 and 0<γ<10, to obtain β∗ and γ∗ to be 1.02 and 1.15, respectively. The solid line in Figure 3 shows f3L. Therefore, we propose RBp as(16)RBp=d1.02L+1.15.


### 3.2. ProposedRSp

Recall that DDMRP defined the safety stock level as the sum of the two terms, one is RB which considers lead time factor and the other is RS which considers demand variability. That is, DDMRP’s safety stock does not explicitly consider lead time variability. Hence, we want an alternative expression for RS to be the one which considers both demand variability and lead time variability. This naturally leads the authors to considerσL, the existing standard deviation of the demand during lead time when both demand and lead time varies, as in (2). So, we propose RS to be equal to σL. Then (2) can be rewritten as (17) (recall that the coefficient of variation (CV) is defined as CV=σ/μ):(17)σL=L⋅sd2+d2⋅sL2=d⋅L⋅CVd2+L⋅CVL2.From (14), (17) can be rewritten as follows:(18)RS1=RB1CVd2+CVL2⋅L.UnlikeFV in (7), CVd2+CVL2⋅L is a factor that considers both demand variability and lead time variability. Using the adjusted RBp in (16), RSp is written as follows:(19)RSp=RBpCVd2+CVL2⋅L=dβL+γCVd2+CVL2⋅L.Consequently, from (16) and (19), we propose (20) as an alternative quantitative formula to the safety stock of DDMRP replenishment model:(20)RZp=RBp+RSp=d1.02L+1.151+CVd2+CVL2⋅L.Now the proposed quantitative model does not useFL and FV from the guideline, so it is free from arbitrariness and will yield a consistent result. In the next section, we will examine the average performance of the proposed safety stock formula (20) against the DDMRP guideline and existing safety stock formula using simulation.


## 3.1. ProposedRBp

Recall that, from (4) and (6), the existing RB is defined as in (13). Ptak and Smith [19] allow a subjective choice of a value FL within the range given in Table 1 to determine RB:(13)RB=dLFL.In Table1, it is implied that for shorter lead time case, the larger portion of DDLT and for the longer lead time case the smaller portion of DDLT be used as RB. Instead of such a subjective and qualitative relation, we propose an alternative quantitative expression of inverse relation that FL is expressed as 1/L. Then, the resulting RB1 is expressed as shown in the following equation:(14)RB1=dL1/L=dL.Now, in Figure3, the highest plotting curve represents f1L=LFL, where FL is computed by the linear interpolation for the three ranges given in Table 1. Although the long lead time in Table 1 is defined as over 25 days, for linear interpolation, we need to set the upper limit of the range. We consider 90 days as the longest lead time since most imported make-to-order (MTO) items often take about three months. The lowest curve represents f2L=L as in (14). It is clearly observed that the two curves differ significantly. From the simulation in our earlier study [27], we observed that DDMRP guideline yields excessive inventory for long lead time and some shortage for shorter lead time and higher variability. We also observed that f2L is too low to avoid shortage [27].Figure 3
f
1
L, f2L, and f3L.Hence, we adjustf2L to devise an alternative function f3L in such a way that f3L is f2L amplified by βfor 0<β<2 and shifted by γ for 0<γ<10, as shown in the following equation:(15)RBp=dβL+γ.The performance ofRBp in (15) will depend on the parameter values β and γ. So in order to find the pair of β and γ that yields the best performance, that is, the lowest average inventory with no shortage, for lead time of 5, 20, 35, and 60 days, CVd = 0.7 and CVL = 0.4, we perform a simulation for the range 0<β<2 and 0<γ<10, to obtain β∗ and γ∗ to be 1.02 and 1.15, respectively. The solid line in Figure 3 shows f3L. Therefore, we propose RBp as(16)RBp=d1.02L+1.15.


## 3.2. ProposedRSp

Recall that DDMRP defined the safety stock level as the sum of the two terms, one is RB which considers lead time factor and the other is RS which considers demand variability. That is, DDMRP’s safety stock does not explicitly consider lead time variability. Hence, we want an alternative expression for RS to be the one which considers both demand variability and lead time variability. This naturally leads the authors to considerσL, the existing standard deviation of the demand during lead time when both demand and lead time varies, as in (2). So, we propose RS to be equal to σL. Then (2) can be rewritten as (17) (recall that the coefficient of variation (CV) is defined as CV=σ/μ):(17)σL=L⋅sd2+d2⋅sL2=d⋅L⋅CVd2+L⋅CVL2.From (14), (17) can be rewritten as follows:(18)RS1=RB1CVd2+CVL2⋅L.UnlikeFV in (7), CVd2+CVL2⋅L is a factor that considers both demand variability and lead time variability. Using the adjusted RBp in (16), RSp is written as follows:(19)RSp=RBpCVd2+CVL2⋅L=dβL+γCVd2+CVL2⋅L.Consequently, from (16) and (19), we propose (20) as an alternative quantitative formula to the safety stock of DDMRP replenishment model:(20)RZp=RBp+RSp=d1.02L+1.151+CVd2+CVL2⋅L.Now the proposed quantitative model does not useFL and FV from the guideline, so it is free from arbitrariness and will yield a consistent result. In the next section, we will examine the average performance of the proposed safety stock formula (20) against the DDMRP guideline and existing safety stock formula using simulation.


## 4. Simulation

In order to evaluate the average performance of the proposed safety stock level given in (20), we conduct a simulation using Excel. We compare three safety stock models: the existing DDMRP safety stock (E-SS) shown in (5), traditional safety stock (T-SS) shown in (1), and the proposed safety stock (P-SS) shown in (20). We assume that backorder is accepted. Note that the three models differ only in determining its safety stock level, and replenishment methods are the same, that is, place a replenishment order when the available stock level falls below the reorder point (which is TOY for E-SS and P-SS, and dL+zασL for T-SS), and order quantity includes order spike as in (12). For each model, we repeat simulation for 10 times, with each run length of 365 days.


### 4.1. Parameters

We consider the following cases: (1) three average lead times (5, 20, and 35 days) which correspond to short, medium, and long lead time, respectively; (2) six coefficient of variation values of lead time (0.00, 0.02, 0.04, 0.06, 0.08, and 010); and (3) three coefficient of variation values of demand (0.3, 0.5, and 0.7), which correspond to low, medium, and high demand variability, respectively. A total of 54 cases are simulated. For E-SS, using linear interpolation from Table1, the lead time factors (FL) are obtained to be 82.7%, 47.8%, and 37.2% for 5, 20, and 35 days of lead time, respectively. For T-SS, safety factor zα is selected as 5.0 to ensure “close to zero” shortage.We use average daily demand of 1,000 units. Ptak suggested that minimum order quantity (MOQ) be used if it is smaller than the order quantity in (12). In this simulation, we use MOQ of 2,000 units as the order quantity.Regarding the distribution of the daily demand and replenishment lead time, in order to simulate the proposed model using parameters that are close to the actual data in reality, we select lognormal distribution as the appropriate realistic distribution based on the following statements: Silver et al. [1] state that because lead time demand is often skewed right, the normal distribution may not be an appropriate; Das [8] states that demand for an item during the delivery lead time can often be represented by a lognormal distribution; and Tadikamalla [28] states that the percentiles and cumulative distribution function values of the lognormal distribution can easily be calculated from the corresponding standard normal values.To generate random daily demand and lead time data following lognormal distribution, the Excel LOGINV function is used as in (21) and (22), respectively:(21)Daily demand=ROUNDLOGINVRAND,D1,D2,0,where D1=LNd−D22/2,D2=LNSd2/d2+1.(22)Lead time=ROUNDLOGINVRAND,L1,L2,0,where L1=LNL−L22/2,L2=LNSL2/L2+1.


## 4.1. Parameters

We consider the following cases: (1) three average lead times (5, 20, and 35 days) which correspond to short, medium, and long lead time, respectively; (2) six coefficient of variation values of lead time (0.00, 0.02, 0.04, 0.06, 0.08, and 010); and (3) three coefficient of variation values of demand (0.3, 0.5, and 0.7), which correspond to low, medium, and high demand variability, respectively. A total of 54 cases are simulated. For E-SS, using linear interpolation from Table1, the lead time factors (FL) are obtained to be 82.7%, 47.8%, and 37.2% for 5, 20, and 35 days of lead time, respectively. For T-SS, safety factor zα is selected as 5.0 to ensure “close to zero” shortage.We use average daily demand of 1,000 units. Ptak suggested that minimum order quantity (MOQ) be used if it is smaller than the order quantity in (12). In this simulation, we use MOQ of 2,000 units as the order quantity.Regarding the distribution of the daily demand and replenishment lead time, in order to simulate the proposed model using parameters that are close to the actual data in reality, we select lognormal distribution as the appropriate realistic distribution based on the following statements: Silver et al. [1] state that because lead time demand is often skewed right, the normal distribution may not be an appropriate; Das [8] states that demand for an item during the delivery lead time can often be represented by a lognormal distribution; and Tadikamalla [28] states that the percentiles and cumulative distribution function values of the lognormal distribution can easily be calculated from the corresponding standard normal values.To generate random daily demand and lead time data following lognormal distribution, the Excel LOGINV function is used as in (21) and (22), respectively:(21)Daily demand=ROUNDLOGINVRAND,D1,D2,0,where D1=LNd−D22/2,D2=LNSd2/d2+1.(22)Lead time=ROUNDLOGINVRAND,L1,L2,0,where L1=LNL−L22/2,L2=LNSL2/L2+1.


## 5. Results and Discussion

The performance of the inventory management method is typically measured by the average inventory level and shortage rate. Since DDMRP replenishment model is known to yield “close to zero” shortage, we set safety factorzα as 5.0 for T-SS to ensure “close to zero” shortage and to compare the average inventory level among the three models, as shown in the following figures for L = 5, 20, and 35, respectively.Note that the average inventory of E-SS remains unchanged as CVL varies, since DDMRP model does not consider lead time variability, although it considers the relative magnitude of lead time as long, medium, or short, as in Table1. It is observed in Figure 4 that the average inventory of the proposed model P-SS is lower than E-SS in all cases and lower than T-SS when CVd = 0.5 or 0.7.Figure 4
Average inventory forL = 5.For medium lead time of 20 days, Figure5 shows that the average inventory level of P-SS is significantly lower than E-SS in all cases and lower than T-SS in all cases except when CVd = 0.3 and CVL is very small. Note that the average inventory of T-SS substantially increases as CVd increases, while that of P-SS increases by a small magnitude.Figure 5
Average inventory forL = 20.For long lead time of 35 days, Figure6 shows that the average inventory level of P-SS is far lower than E-SS in all cases and lower than T-SS in all cases except when CVd = 0.3 and CVL is very small. Recall that zero shortage is ensured in all cases.Figure 6
Average inventory forL = 35.Consequently, simulation results indicate that the safety stock model that we proposed in (20) significantly outperforms the existing DDMRP replenishment method for all cases and significantly also outperforms the traditional safety stock model for almost all cases, under the same “zero shortage” situation. Hence, we believe that the proposed safety stock formula will provide more consistent results and, therefore, can be effectively used in practices.The above simulation result is for 0 ≤ CVL ≤ 0.1, which we believe sufficiently covers practical uncertainty in today’s business. However, for academic curiosity, we have conducted further simulation for CVL = 0.2, 0.3, and 0.4, as it may bring more insights to the validity of the proposed formula. For CVL = 0.4, shortage occurs 5 times in E-SS, while only one time in P-SS. It is observed that the average inventory of P-SS is higher than that of E-SS only when lead time = 5, and CVL = 0.3 and 0.4, CVd = 0.3 and 0.5; or CVL = 0.4, CVd = 0.7. That is, it turns out that the proposed model outperforms the existing DDMRP replenishment model in most of the cases for higher demand variability as well.


## 6. Conclusion

In this paper, we propose an alternative model to determine the safety stock level for the existing DDMRP inventory replenishment model. The motivation of this study is twofold. Firstly, the existing guideline to determine the safety stock level in DDMRP lacks consistency as it allows the users to select values subjectively from a range. Secondly, the existing DDMRP replenishment yields excessively high inventory level in the practical parameter value range, although it yields lower inventory than the traditional safety stock formula for some cases.Hence, we propose a method to mathematically determine the safety stock level for DDMRP, which will yield lower inventory level than the existing DDMRP guideline, while still maintaining the shortage rate close to zero. The proposed model basically follows the frame of the existing DDMRP inventory replenishment model, in which the safety stock comprises red zone base (RB) and red zone safety (RS). Instead of using subjective guideline given in Tables1 and 2, we propose an alternative mathematical model for RB and RS.The proposed RB reflects the observation that the existing guideline yields excessive inventory for longer lead time and some shortage for shorter lead time. Within the valid range of parameter values that will yield better performance than the existing guideline, we identified the best modifying parametersβ and γ for the formula using simulation for one year for ten times. The proposed RS is devised in that not only demand variability but also lead time variability is mathematically considered.We have compared using simulation the average inventory level of the proposed safety stock formula against the existing DDMRP guideline and traditional safety stock formula over the range of practical parameter values. Skewness of demand and lead time distribution has been considered by using lognormal distribution since we believe it is more realistic.Under “no shortage” situation, the average inventory of the proposed formula outperforms the existing guideline for all cases and outperforms traditional safety stock formula for all cases except when the demand variability is small. Hence, we conclude that the proposed safety stock formula provides more consistent results and, therefore, can be effectively used in practices.Recall that the existing inventory replenishment models require a certain demand and/or lead time distribution in order to determine the optimal safety factor (i.e., service level) to minimize the sum of inventory and shortage cost. However, the proposed model does not use the safety factor for a specific demand and/or lead time distributions. Therefore, our proposed model can be regarded as more practical as it is simpler than the existing models and yet outperforms the existing models in terms of average inventory level.For further study, we suggest that alternative distributions for demand and/or lead time such as Gamma and Weibull distribution be used for simulation. We also suggest that simulation expands the range of the parameter values including the coefficient of variation of demand beyond 0.7 up to 1.0.



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*Source: 6496309-2019-09-19.xml*