# Optimizing Safety Stock Levels in Modular Production Systems Using Component Commonality and Group Technology Philosophy: A Study Based on Simulation

Authors: Kenneth Edgar Hernandez-Ruiz; Elias Olivares-Benitez; Jose Luis Martinez-Flores; Santiago Omar Caballero-Morales
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

Modular production and component commonality are two widely used strategies in the manufacturing industry to meet customers growing needs for customized products. Using these strategies, companies can enhance their performance to achieve optimal safety stock levels. Despite the importance of safety stocks in business competition, little attention has been paid to the way to reduce them without affecting the customer service levels. This paper develops a mathematical model to reduce safety stock levels in organizations that employ modular production. To construct the model, we take advantage of the benefits of aggregate inventories, standardization of components, component commonality, and Group Technology philosophy in regard to stock levels. The model is tested through the simulation of three years of operation of two modular product systems. For each system, we calculated and compared the safety stock levels for two cases: (1) under the only presence of component commonality and (2) under the presence of both component commonality and Group Technology philosophy. The results show a reduction in safety stock levels when we linked the component commonality with the Group Technology philosophy. The paper presents a discussion of the implications of each case, features of the model, and suggestions for future research.


---

## Body

## 1. Introduction

Over the past few decades, manufacturing has faced a steady growth in market competition caused mainly by three factors: (1) globalization [1], (2) technological advances, and (3) high demand for customized products [2–4]. To meet this demand and being competitive, the companies have applied as main strategy the offering of a wider variety of products [1, 2, 5–9]. However, having a wider variety of products increases the complexity of the organization because it increases the quantity of parts or components that the company must make or buy.To meet efficiently the high demand of customized products, some researchers [1, 9, 10] suggest two methods: (1) increasing production capacity and (2) applying modular production systems. Of these strategies, the modular production ormodularity is the most popular [11, 12]. The basic idea of the modularity is to design modules or components standardized with standard interfaces in such a way that they can be combined in a high number of end products to meet customer requirements [5, 13, 14]. A truly modular production system makes it possible to manufacture custom products and to react quickly to customer requirements with virtually the same features found in the mass production [15–18]. One of the main benefits of modularity is that it leads to a reduction of the stock levels.By usingMake to Stock(MTS) andAssemble to Stock(ATS) production approaches, a manufacturer can maintain cycle stock levels according to a forecast demand. However, in an environment with uncertainly, forecasts are not reliable. For this reason, the manufacturers use an investment in safety stocks, which allow them to respond effectively to forecast errors and to overcome or mitigate the risk of stock-outs. The strategy most widely used to minimize the safety stock levels in modular production systems iscomponent commonality, which replaces two or more parts of an end product by a common part [14, 19–23].The studies for minimizing the safety stock levels in modular production systems began with Collier [24], who argued that, by increasing the degree of component commonality in a modular production system, the costs of operation and the stocks levels can be reduced because larger production batches are created and standardization improved. Thus, Baker [19] proved that the safety stock levels decrease as much as the degree of commonality increases. Baker et al. [25] examined the effect of component commonality on optimal safety stock levels in two products with a two-level inventory model, showing that replacing two unique parts with a common part results in a smaller requirements for the common part. Gerchak and Henig [20] replicated Baker et al. [25] and showed that inAssemble to Order(ATO) systems the stocks of product-specific components always increase when other components are combined with common parts. Gerchak et al. [21] extended the results of Baker et al. [25] in understanding the impact of component commonality on safety stock levels under service level constraints. Hillier [26] analyzed the effects of component commonality in the total cost of a modular production system using a multiperiod model when a common component is significantly more expensive than the unique parts. Chew et al. [27] quantified the impact of component commonality in a two-echelon assembled-to-stock system consisting of several common components and end products. The results showed that when each type of component is shared by at least two end products, the safety stock levels are reduced requiring to be held at a sufficiently high service level. Catena et al. [28] proposed four models with commonality for calculating safety stock levels for subassemblies and manufacturing components under ATO andMake to Order(MTO) systems. Persona et al. [29] extended the study of Catena et al. [28] and applied the models in two different industries to prove the reduction in the safety stock levels. Like these works, there are many other studies focused on the effect of the component commonality on safety stock levels in modular production systems such as Gerchak and Henig [30]; Eynan and Rosenblatt [31]; Fisher et al. [32]; Cheung [23].The main contribution of this work is the proposal of a model that includes Group Technology philosophy (GT) for the computation of safety stock levels in modular production systems. This inclusion extends the previous approaches that only considered component commonality. The mathematical model is tested through the simulation of three years of operation of two modular product systems. For each system, we calculated and compared the safety stock levels for two scenarios: (1) under the only presence of component commonality and (2) under the presence of both component commonality and GT. These scenarios are used to compare the performance of the mathematical model proposed with the traditional model. The results show a reduction in the safety stock levels without affecting the customer service levels (CSL).The remainder of the paper is organized as follows. Section2 shows the development of the proposed model. In Section 3 the simulation results are presented. Section 4 discusses the main implications of the model. Finally, Section 5 presents the conclusions and suggestions for future research.


## 2. Mathematical Model

According to Mikkola [6], the structure of a modular product is divided into four levels of complexity: system, subsystem, modules, and components. However, this structure is determined mainly by organizations considering their production approaches and the intended application of the products. This study divides the modular product into three levels of complexity: system, modules, and components.A modular products system assumes the existence of(i)
mend products with indexes l
=
1,2
,
…
,
m;
(ii)
n modules with indexes j
=
1,2
,
…
,
n;
(iii)
c components with indexes i
=
1,2
,
…
,
c.The number of componentsi required to build one module j is denoted as a
i
j; and the number of modules j required to build one end product l is denoted as b
j
l. Figure 1 shows this configuration and the matrices of composition of modules and components.Figure 1
General structure of a modular products system and its matrices of composition.


### 2.1. Preliminary Analysis

The structure of a modular products system supports managers in determining stock levels, because it allows them to know exactly the number of modules and components they need in an end product. In reality, all manufacturers operate with an investment in inventories, even those organizations that employ the Just-in-Time strategy. These inventories consist of raw material, work in process, products from a reverse logistic system, subassemblies, and end products. All of them are necessary for very specific reasons that, taken together, facilitate efficient performance in organizations. In the ideal case, where a manufacturing system operates in a purely deterministic environment, the forecast demand is not necessary. However, in actual practice, this does not happen; there is always uncertainty in the environment and sudden changes in the customers behavior causing inaccurate forecasts, regardless of the forecasting method used.It is well known that organizations turn to the preparation of forecasts to determine the future demand for their products. With these forecasts, they adjust their resources and production capacities in order to meet these demands. Because forecast is only approximations, the managers must always take into account a forecast value and a forecast error as metrics to determine the stocks levels. Therefore, to meet the real demand for a modular product over a period of timet, managers consider a deterministic amount and a random amount of inventory. As a consequence of the random component, safety stocks become necessary in an organization. Safety stocks are inventory that remain to meet the demand that exceeds the forecast in a period of time t
. These stocks avoid stock-out costs and lost sales. For an organization, it is important to maintain optimal safety stock levels, because they also generate costs to keep them in the warehouse. It is harmful to have excess inventory when new competing products are introduced because the available inventory becomes obsolete.In an environment of modular production, where the demand of a product in a period of timet is bigger than forecast demand, the forecast error can be cover by the safety stock as follows:(1)
D
t
=
ω
μ
+
S
S
,where D
t is demand of the periodt;
ω is unit of time (days, weeks, and months) of the periodt;
μ is the average of the forecasted demand;
SS is safety stock.Assuming that the demandD
t is normally distributed with mean μ and a variance var[D
t], by definition of the normal standard distribution and its inverse, the following expression is obtained: (2)
D
t
=
ω
μ
+
k
σ
L
.Therefore, the safety stock is defined by the next expression:(3)
S
S
=
k
σ
L
, where S
S is safety stock;
k is the safety factor that determines the probability that the real demand of a product is less than or equal to the forecasted demand plus the safety stock;
L is lead-time;
σ is standard deviation of the demand.Indeed, the safety stock SS represents the stock-out in the stock cycle during the period of time between the processes of placing and receiving an order; that is,(4)
R
O
P
=
L
μ
+
S
S
,where ROP is the reorder point and μ is the mean of the demand through the planning horizon.Expression (3) determines the safety stock for an end product, which assumes independence in its demand and does not consider the presence of component commonality. This expression only considers as key metrics the standard deviation and a safety factor k to ensure a certain CSL.However, in practice, organizations adopt different strategies to reduce the safety stock levels to maintain low costs; to start to reduce them, an organization needs to create an aggregate inventory; this consists in centralizing inventories in a single warehouse, instead of having different storage points. The aggregate inventories generate savings in inventory holding costs; the process of building part families by GT is facilitated and allows the efficient use of the component commonality.An organization that offers a wide variety of modular products as customized products must maintain an aggregation in inventories. Now, if this organization seeks to reduce the safety stock levels, in addition to this aggregation, it also must consider the application of the component commonality. Then, in this case, Expression (3) can be generalized considering the distribution of the aggregate inventory, which has a normal distribution with an aggregate demand d
′, a variance var
⁡
d
′, and a standard deviation σ
′, as follows in expressions (5). One has(5)
d
′
=
∑
l
=
1
m
d
l
,
var
⁡
d
′
=
∑
l
=
1
m
σ
l
2
+
2
∑
l
<
z
c
o
v
l
z
=
∑
l
=
1
m
σ
l
2
+
2
∑
l
<
z
ρ
l
z
σ
l
σ
z
,
σ
′
=
var
⁡
d
′
=
∑
l
=
1
m
σ
l
2
+
2
∑
l
<
z
ρ
l
z
σ
l
σ
z
,where ρ
l
z is the Pearson correlation coefficient between the final products l and z
, while the sum of the standard deviations represents those deviations where component commonality exists. From expressions (3) and (5), the safety stock for each final product l into an aggregate inventory of modular products with component commonality can be determined based on the next expression:(6)
S
S
l
=
k
σ
α
L
l
=
k
∑
l
=
1
m
σ
l
2
+
2
∑
l
<
z
ρ
l
z
σ
l
σ
z
L
l
.When the demands are independent (ρ
l
z
=
0) between all end products, expression (6) is simplified as follows:(7)
S
S
l
=
k
∑
l
=
1
m
σ
l
2
L
l
.Expressions (6) and (7) determine the safety stock levels of end products with dependent and independent demands, respectively, considering the use of the aggregate inventories and the component commonality.This preliminary analysis develops expressions (3)–(7) that already have been developed in previous literature. These expressions are presented in this paper because these are the basis to construct the proposed model.


### 2.2. Proposed Model

Under MTS and ATS production approaches, it is necessary to determine the safety stock levels for each component that exists in a modular products system, and expressions (6) and (7) are the basis for calculation.To determine the safety stock of each component of a modular products system, we must consider the degree of commonality present in the system. According to Collier [24], the structure of the system determines the degree of commonality; this is defined as the total number of modules and components in the system divided by the total number of distinct modules and components. Figure 2 shows an example of how to calculate the degree of commonality of two modular products.Figure 2
Degree of commonality in two simple modular products.Considering the existence of modules and common components, expression (6) is adjusted to determine the safety stock levels of each component in the system; therefore, in each term of the expression, we consider the participation of components in modules and modules in end products. The expression developed is as follows:(8)
S
S
i
=
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
+
2
∑
j
=
1
n
∑
l
=
1
m
-
1
∑
z
=
2
m
a
i
j
2
b
j
l
b
j
z
ρ
l
z
σ
l
σ
z
L
i
.When there is independence in demands between the end products, expression (8) is simplified as follows:(9)
S
S
i
=
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
L
i
.Expressions (8) and (9) have been developed in this paper for calculating safety stocks of the components of a modular products system with component commonality and with three levels of complexity. However, the proposed model also suggests the presence of a factor of substitution from the GT philosophy to reduce further the safety stock levels.


#### 2.2.1. Factor of Substitution

The GT is a manufacturing philosophy that brings together the components with similar physical features (shape, size, and manufacturing processes) in part families to take advantages in the optimization of many processes such as lead times, setup times, processing times, labor operations, and reworks. From GT philosophy, we define afactor of substitution of a component i as f
s
i, which considers a fraction of substitute components r
i within the part family to which the component i belongs, and by their similarities, the component i can be replaced by them in case of stock-out. An example of substitutes components are two similar screws with different drives in the heads. It assume that this substitution does not affect or change the shape, function, costs, and physical and mechanical properties of the end product. The fraction r
i is defined as follows: (10)
r
i
=
n
i
∑
i
=
1
c
N
i
-
1
/
C
,wherer
i is fraction of substitution of the component i;
n
i is amount of substitute components of the component i;
N istotal components within the part family of the component i
.By using a basic mathematical model, the factor of substitutionf
s
i decreases at a rate r
i proportionally to the number of components n
i existing within the part family. If f
s
i is the factor of substitution of a component i, then f
s
i varies at a rate Δ
f
s
i = r
i
f
s
i as much as the number of components n
i increases or decreases; in other words, for every unit that increases the value n
i in the system, f
s
i takes a proportional value to this change. Hence(11)
Δ
f
s
i
=
-
r
i
f
s
i
Δ
n
i
.In differential equation form,(12)
d
f
s
i
d
n
i
=
-
r
i
f
s
i
=
-
n
i
∑
i
=
1
c
N
i
-
1
/
C
f
s
i
.For any value ofr
i, the solution of the differential equation is presented in next expression:(13)
f
s
i
=
e
-
n
i
2
/
2
∑
i
=
1
c
N
i
-
1
/
C
.Figure3 shows some patterns of the factor of substitution under different sizes of part families. In this figure we can observe that, in a scenario with absence of substitute components (
n
i
=
0
), the factor of substitution takes a maximum value (
f
s
i
=
1
), and while increasing the value of n
i
, the value of f
s
i approximates to zero.Figure 3
Functions for the factor of substitution.


#### 2.2.2. Integration of the Proposed Model

Expressions (8) and (9) determine the safety stock levels for the components of a modular products system with commonality and under an aggregate inventory. However, these levels can be reduced including the factor of substitution in expression (8) as follows:(14)
S
S
i
=
e
-
n
i
2
/
2
∑
i
=
1
c
N
i
-
1
/
C
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
+
2
∑
j
=
1
n
∑
l
=
1
m
-
1
∑
z
=
2
m
a
i
j
2
b
j
l
b
j
z
ρ
l
z
σ
l
σ
z
L
i
.When demands are independent, expression (14) is simplified as follows:(15)
S
S
i
=
e
-
n
i
2
/
2
∑
i
=
1
c
N
i
-
1
/
C
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
L
i
.Expressions (14) and (15) are the result of joining the component commonality with the GT philosophy for reducing the safety stock levels. We can see that when f
s
i takes its maximum value, expression (14) becomes expression (8); while increasing the value by n
i, the safety stock S
S
i is reduced.The basic principle of the proposed model consists in reducing safety stock levels in modular production systems through an exponential smoothing. This reduction is in function of the degree of commonality and of the amount of componentsN
i and n
i in the part families. The reduction is given due to the possibility of sharing cyclical and safety stocks between substitute components. Thus, this strategy allows eliminating the negative effect in case of stock-outs.To evaluate the proposed model, Section3 presents two applications based on simulation regarding the operation of two different modular products systems.


## 2.1. Preliminary Analysis

The structure of a modular products system supports managers in determining stock levels, because it allows them to know exactly the number of modules and components they need in an end product. In reality, all manufacturers operate with an investment in inventories, even those organizations that employ the Just-in-Time strategy. These inventories consist of raw material, work in process, products from a reverse logistic system, subassemblies, and end products. All of them are necessary for very specific reasons that, taken together, facilitate efficient performance in organizations. In the ideal case, where a manufacturing system operates in a purely deterministic environment, the forecast demand is not necessary. However, in actual practice, this does not happen; there is always uncertainty in the environment and sudden changes in the customers behavior causing inaccurate forecasts, regardless of the forecasting method used.It is well known that organizations turn to the preparation of forecasts to determine the future demand for their products. With these forecasts, they adjust their resources and production capacities in order to meet these demands. Because forecast is only approximations, the managers must always take into account a forecast value and a forecast error as metrics to determine the stocks levels. Therefore, to meet the real demand for a modular product over a period of timet, managers consider a deterministic amount and a random amount of inventory. As a consequence of the random component, safety stocks become necessary in an organization. Safety stocks are inventory that remain to meet the demand that exceeds the forecast in a period of time t
. These stocks avoid stock-out costs and lost sales. For an organization, it is important to maintain optimal safety stock levels, because they also generate costs to keep them in the warehouse. It is harmful to have excess inventory when new competing products are introduced because the available inventory becomes obsolete.In an environment of modular production, where the demand of a product in a period of timet is bigger than forecast demand, the forecast error can be cover by the safety stock as follows:(1)
D
t
=
ω
μ
+
S
S
,where D
t is demand of the periodt;
ω is unit of time (days, weeks, and months) of the periodt;
μ is the average of the forecasted demand;
SS is safety stock.Assuming that the demandD
t is normally distributed with mean μ and a variance var[D
t], by definition of the normal standard distribution and its inverse, the following expression is obtained: (2)
D
t
=
ω
μ
+
k
σ
L
.Therefore, the safety stock is defined by the next expression:(3)
S
S
=
k
σ
L
, where S
S is safety stock;
k is the safety factor that determines the probability that the real demand of a product is less than or equal to the forecasted demand plus the safety stock;
L is lead-time;
σ is standard deviation of the demand.Indeed, the safety stock SS represents the stock-out in the stock cycle during the period of time between the processes of placing and receiving an order; that is,(4)
R
O
P
=
L
μ
+
S
S
,where ROP is the reorder point and μ is the mean of the demand through the planning horizon.Expression (3) determines the safety stock for an end product, which assumes independence in its demand and does not consider the presence of component commonality. This expression only considers as key metrics the standard deviation and a safety factor k to ensure a certain CSL.However, in practice, organizations adopt different strategies to reduce the safety stock levels to maintain low costs; to start to reduce them, an organization needs to create an aggregate inventory; this consists in centralizing inventories in a single warehouse, instead of having different storage points. The aggregate inventories generate savings in inventory holding costs; the process of building part families by GT is facilitated and allows the efficient use of the component commonality.An organization that offers a wide variety of modular products as customized products must maintain an aggregation in inventories. Now, if this organization seeks to reduce the safety stock levels, in addition to this aggregation, it also must consider the application of the component commonality. Then, in this case, Expression (3) can be generalized considering the distribution of the aggregate inventory, which has a normal distribution with an aggregate demand d
′, a variance var
⁡
d
′, and a standard deviation σ
′, as follows in expressions (5). One has(5)
d
′
=
∑
l
=
1
m
d
l
,
var
⁡
d
′
=
∑
l
=
1
m
σ
l
2
+
2
∑
l
<
z
c
o
v
l
z
=
∑
l
=
1
m
σ
l
2
+
2
∑
l
<
z
ρ
l
z
σ
l
σ
z
,
σ
′
=
var
⁡
d
′
=
∑
l
=
1
m
σ
l
2
+
2
∑
l
<
z
ρ
l
z
σ
l
σ
z
,where ρ
l
z is the Pearson correlation coefficient between the final products l and z
, while the sum of the standard deviations represents those deviations where component commonality exists. From expressions (3) and (5), the safety stock for each final product l into an aggregate inventory of modular products with component commonality can be determined based on the next expression:(6)
S
S
l
=
k
σ
α
L
l
=
k
∑
l
=
1
m
σ
l
2
+
2
∑
l
<
z
ρ
l
z
σ
l
σ
z
L
l
.When the demands are independent (ρ
l
z
=
0) between all end products, expression (6) is simplified as follows:(7)
S
S
l
=
k
∑
l
=
1
m
σ
l
2
L
l
.Expressions (6) and (7) determine the safety stock levels of end products with dependent and independent demands, respectively, considering the use of the aggregate inventories and the component commonality.This preliminary analysis develops expressions (3)–(7) that already have been developed in previous literature. These expressions are presented in this paper because these are the basis to construct the proposed model.


## 2.2. Proposed Model

Under MTS and ATS production approaches, it is necessary to determine the safety stock levels for each component that exists in a modular products system, and expressions (6) and (7) are the basis for calculation.To determine the safety stock of each component of a modular products system, we must consider the degree of commonality present in the system. According to Collier [24], the structure of the system determines the degree of commonality; this is defined as the total number of modules and components in the system divided by the total number of distinct modules and components. Figure 2 shows an example of how to calculate the degree of commonality of two modular products.Figure 2
Degree of commonality in two simple modular products.Considering the existence of modules and common components, expression (6) is adjusted to determine the safety stock levels of each component in the system; therefore, in each term of the expression, we consider the participation of components in modules and modules in end products. The expression developed is as follows:(8)
S
S
i
=
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
+
2
∑
j
=
1
n
∑
l
=
1
m
-
1
∑
z
=
2
m
a
i
j
2
b
j
l
b
j
z
ρ
l
z
σ
l
σ
z
L
i
.When there is independence in demands between the end products, expression (8) is simplified as follows:(9)
S
S
i
=
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
L
i
.Expressions (8) and (9) have been developed in this paper for calculating safety stocks of the components of a modular products system with component commonality and with three levels of complexity. However, the proposed model also suggests the presence of a factor of substitution from the GT philosophy to reduce further the safety stock levels.


### 2.2.1. Factor of Substitution

The GT is a manufacturing philosophy that brings together the components with similar physical features (shape, size, and manufacturing processes) in part families to take advantages in the optimization of many processes such as lead times, setup times, processing times, labor operations, and reworks. From GT philosophy, we define afactor of substitution of a component i as f
s
i, which considers a fraction of substitute components r
i within the part family to which the component i belongs, and by their similarities, the component i can be replaced by them in case of stock-out. An example of substitutes components are two similar screws with different drives in the heads. It assume that this substitution does not affect or change the shape, function, costs, and physical and mechanical properties of the end product. The fraction r
i is defined as follows: (10)
r
i
=
n
i
∑
i
=
1
c
N
i
-
1
/
C
,wherer
i is fraction of substitution of the component i;
n
i is amount of substitute components of the component i;
N istotal components within the part family of the component i
.By using a basic mathematical model, the factor of substitutionf
s
i decreases at a rate r
i proportionally to the number of components n
i existing within the part family. If f
s
i is the factor of substitution of a component i, then f
s
i varies at a rate Δ
f
s
i = r
i
f
s
i as much as the number of components n
i increases or decreases; in other words, for every unit that increases the value n
i in the system, f
s
i takes a proportional value to this change. Hence(11)
Δ
f
s
i
=
-
r
i
f
s
i
Δ
n
i
.In differential equation form,(12)
d
f
s
i
d
n
i
=
-
r
i
f
s
i
=
-
n
i
∑
i
=
1
c
N
i
-
1
/
C
f
s
i
.For any value ofr
i, the solution of the differential equation is presented in next expression:(13)
f
s
i
=
e
-
n
i
2
/
2
∑
i
=
1
c
N
i
-
1
/
C
.Figure3 shows some patterns of the factor of substitution under different sizes of part families. In this figure we can observe that, in a scenario with absence of substitute components (
n
i
=
0
), the factor of substitution takes a maximum value (
f
s
i
=
1
), and while increasing the value of n
i
, the value of f
s
i approximates to zero.Figure 3
Functions for the factor of substitution.


### 2.2.2. Integration of the Proposed Model

Expressions (8) and (9) determine the safety stock levels for the components of a modular products system with commonality and under an aggregate inventory. However, these levels can be reduced including the factor of substitution in expression (8) as follows:(14)
S
S
i
=
e
-
n
i
2
/
2
∑
i
=
1
c
N
i
-
1
/
C
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
+
2
∑
j
=
1
n
∑
l
=
1
m
-
1
∑
z
=
2
m
a
i
j
2
b
j
l
b
j
z
ρ
l
z
σ
l
σ
z
L
i
.When demands are independent, expression (14) is simplified as follows:(15)
S
S
i
=
e
-
n
i
2
/
2
∑
i
=
1
c
N
i
-
1
/
C
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
L
i
.Expressions (14) and (15) are the result of joining the component commonality with the GT philosophy for reducing the safety stock levels. We can see that when f
s
i takes its maximum value, expression (14) becomes expression (8); while increasing the value by n
i, the safety stock S
S
i is reduced.The basic principle of the proposed model consists in reducing safety stock levels in modular production systems through an exponential smoothing. This reduction is in function of the degree of commonality and of the amount of componentsN
i and n
i in the part families. The reduction is given due to the possibility of sharing cyclical and safety stocks between substitute components. Thus, this strategy allows eliminating the negative effect in case of stock-outs.To evaluate the proposed model, Section3 presents two applications based on simulation regarding the operation of two different modular products systems.


## 2.2.1. Factor of Substitution

The GT is a manufacturing philosophy that brings together the components with similar physical features (shape, size, and manufacturing processes) in part families to take advantages in the optimization of many processes such as lead times, setup times, processing times, labor operations, and reworks. From GT philosophy, we define afactor of substitution of a component i as f
s
i, which considers a fraction of substitute components r
i within the part family to which the component i belongs, and by their similarities, the component i can be replaced by them in case of stock-out. An example of substitutes components are two similar screws with different drives in the heads. It assume that this substitution does not affect or change the shape, function, costs, and physical and mechanical properties of the end product. The fraction r
i is defined as follows: (10)
r
i
=
n
i
∑
i
=
1
c
N
i
-
1
/
C
,wherer
i is fraction of substitution of the component i;
n
i is amount of substitute components of the component i;
N istotal components within the part family of the component i
.By using a basic mathematical model, the factor of substitutionf
s
i decreases at a rate r
i proportionally to the number of components n
i existing within the part family. If f
s
i is the factor of substitution of a component i, then f
s
i varies at a rate Δ
f
s
i = r
i
f
s
i as much as the number of components n
i increases or decreases; in other words, for every unit that increases the value n
i in the system, f
s
i takes a proportional value to this change. Hence(11)
Δ
f
s
i
=
-
r
i
f
s
i
Δ
n
i
.In differential equation form,(12)
d
f
s
i
d
n
i
=
-
r
i
f
s
i
=
-
n
i
∑
i
=
1
c
N
i
-
1
/
C
f
s
i
.For any value ofr
i, the solution of the differential equation is presented in next expression:(13)
f
s
i
=
e
-
n
i
2
/
2
∑
i
=
1
c
N
i
-
1
/
C
.Figure3 shows some patterns of the factor of substitution under different sizes of part families. In this figure we can observe that, in a scenario with absence of substitute components (
n
i
=
0
), the factor of substitution takes a maximum value (
f
s
i
=
1
), and while increasing the value of n
i
, the value of f
s
i approximates to zero.Figure 3
Functions for the factor of substitution.


## 2.2.2. Integration of the Proposed Model

Expressions (8) and (9) determine the safety stock levels for the components of a modular products system with commonality and under an aggregate inventory. However, these levels can be reduced including the factor of substitution in expression (8) as follows:(14)
S
S
i
=
e
-
n
i
2
/
2
∑
i
=
1
c
N
i
-
1
/
C
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
+
2
∑
j
=
1
n
∑
l
=
1
m
-
1
∑
z
=
2
m
a
i
j
2
b
j
l
b
j
z
ρ
l
z
σ
l
σ
z
L
i
.When demands are independent, expression (14) is simplified as follows:(15)
S
S
i
=
e
-
n
i
2
/
2
∑
i
=
1
c
N
i
-
1
/
C
k
∑
j
=
1
n
∑
l
=
1
m
a
i
j
b
j
l
σ
l
2
L
i
.Expressions (14) and (15) are the result of joining the component commonality with the GT philosophy for reducing the safety stock levels. We can see that when f
s
i takes its maximum value, expression (14) becomes expression (8); while increasing the value by n
i, the safety stock S
S
i is reduced.The basic principle of the proposed model consists in reducing safety stock levels in modular production systems through an exponential smoothing. This reduction is in function of the degree of commonality and of the amount of componentsN
i and n
i in the part families. The reduction is given due to the possibility of sharing cyclical and safety stocks between substitute components. Thus, this strategy allows eliminating the negative effect in case of stock-outs.To evaluate the proposed model, Section3 presents two applications based on simulation regarding the operation of two different modular products systems.


## 3.Application and Results

The proposed model is evaluated through two applications based on the simulation of three years of operation of two modular products systems, where each system has its own structure and its own degree of component commonality. The analysis of each application makes the comparison of performance between safety stocks calculated for two scenarios: (1) under the only presence of component commonality and (2) with the presence of component commonality and the factor of substitution from GT philosophy.The simulations have been developed using C++ language, where after generating demands for end products, the safety stocks, reorder points, initial and final inventories, components used, and stock-outs for each component are calculated. The results for each scenario in both applications have been derived from thirty simulation runs. Figure4 shows the general simulation process.Figure 4
Simulation process.To make the applications simple, we suppose the following conditions held:(1)
A continuous review (Q, R) policy is used to determine the safety stock levels.
(2)
There is one period of lead-time for first application.
(3)
There are two periods of lead-time for second application.
(4)
In each period simulated, demand for end productsl follows a normal distribution, with mean μ
l and standard deviation σ
l.
(5)
For each case, it assumes aggregate inventories for production.
(6)
The CSL is 90%(
k
=
1.28
).
(7)
There is a buffer of enough substitute components to meet the demand in case of stock-outs.
(8)
The substitution of components does not affect or change the shape, function, costs, and physical and mechanical properties of the end product.
(9)
Each component analyzed belongs to a different part family.


### 3.1. Application One

This application simulates and analyzes two modular products (EP1 and EP2) of an enterprise that offers a high variety of different modular products to its customers. To manufacture all its products, the enterprise applies GT philosophy to arrange its components in part families. The two products analyzed are assembled from 3 modules and 4 components as shown in Figure5. The structure of these two products has a degree of commonality of 1.7.Figure 5
A system of two modular products and its matrices of composition.We suppose that the demands of these two end products are independent and normally distributed with meansμ
E
P
1
=
700 and μ
E
P
2
=
1100, standard deviations σ
E
P
1
=
80 and σ
E
P
2
=
120. Table 1 shows the initial inventories and order quantities used for the two scenarios analyzed.Table 1
Initial inventories and quantity orders.
c
i
1
2
3
4
I. Initial
10000
9500
10500
9000
Q
i
7000
8000
7000
4000Case 1 (safety stocks with commonality).
Based on the structure of Figure4 and considering that the system handles aggregation in inventories with only component commonality, the safety stock for each component is determined by applying expression (9). Table 2 shows the calculated values for safety stocks and reorder points.Table 2
Safety stock levels and reorder points.
c
i
1
2
3
4
S
S
i
185
212
185
154
R
O
P
i
1985
2712
1985
1254The results of the thirty simulation runs for this case are shown in Table3. This table shows that five stock-outs occurred in five simulation runs, generating a stock-out average equal to 1. The average inventories for each component also are showed.Table 3
Results of the simulation: Case1.
Stock-outs
Average inventories
Amount of stock-outs
Runs with stock-outs
Average of stock-outs
c1
c2
c3
c4
5
5
1
5563
6847
5804
3701Case 2 (safety stocks with commonality and factor of substitution).
In this case, we suppose that each componenti of the two modular products analyzed belongs to a part family, the experiment assumes that each part family has its own total number of components N
i with n
i substitute components for the component i
, and each substitute component has a large buffer to meet any demand. Table 4 shows the details with respect to the composition of the part families of each component i of the system analyzed.Table 4
Part families composition.
c
i
1
2
3
4
N
i
25
30
45
60
n
i
3
6
4
6For an organization, obtaining the data in Table4 is not a difficult task; they can be obtained through the construction of part families and identifying the substitute components for each member of the families. Once this information is structured, it can be used to calculate the safety stock levels for each component.Given the information in Table4, safety stock for each component of the system is determined using the expression (15), which considers the aggregation in inventories, existence of component commonality, independent demands, and the factor of substitution. Table 5 shows the calculated values for safety stocks and reorder points.Table 5
Safety stock levels and reorder points.
c
i
1
2
3
4
S
S
i
166
135
152
98
R
O
P
i
1966
2635
1952
1198The results of the thirty simulation runs for this case are shown in Table6. This table shows that eight stock-outs occurred in six simulation runs, generating a stock-out average equal to 1.3. The average inventories for each component also are showed.Table 6
Results of the simulation: Case2.
Stock-outs
Average inventories
Amount of stock-outs
Runs with stock-outs
Average of stock-outs
c1
c2
c3
c4
8
6
1.3
5542
6795
5785
3648


#### 3.1.1. A Performance Comparison between the Two Cases

Tables3 and 6 show the results of each case; these tables show that there are a greater number of stock-outs and lower average inventories in Case  2 as a result of reducing safety stock levels. However, the experiment involves the possibility of using substitute components to meet this demand, while stock-outs in Case  1 become in shortage costs. Figure 6 shows the difference between safety stock levels calculated in each case for each component in the system. An example of one simulation run for both cases is shown in Table 7.Table 7
Application 1: demands, components used, and initial and final inventories.
Period (month)
Demands
Case1
Case2
Initial inventory
Components used
Final inventory
Initial inventory
Components used
Final inventory
FP1
FP2
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
1
724
1062
10000
9500
10500
9000
1786
2510
1786
1062
8214
6990
8714
7938
10000
9500
10500
9000
1786
2510
1786
1062
8214
6990
8714
7938
2
772
1076
8214
6990
8714
7938
1848
2620
1848
1076
6366
4370
6866
6862
8214
6990
8714
7938
1848
2620
1848
1076
6366
4370
6866
6862
3
800
1197
6366
4370
6866
6862
1997
2797
1997
1197
4369
1573
4869
5665
6366
4370
6866
6862
1997
2797
1997
1197
4369
1573
4869
5665
4
740
1062
4369
9573
4869
5665
1802
2542
1802
1062
2567
7031
3067
4603
4369
9573
4869
5665
1802
2542
1802
1062
2567
7031
3067
4603
5
728
1189
2567
7031
3067
4603
1917
2645
1917
1189
650
4386
1150
3414
2567
7031
3067
4603
1917
2645
1917
1189
650
4386
1150
3414
6
770
1142
7650
4386
8150
3414
1912
2682
1912
1142
5738
1704
6238
2272
7650
4386
8150
3414
1912
2682
1912
1142
5738
1704
6238
2272
7
718
1012
5738
9704
6238
2272
1730
2448
1730
1012
4008
7256
4508
1260
5738
9704
6238
2272
1730
2448
1730
1012
4008
7256
4508
1260
8
737
1289
4008
7256
4508
1260
2026
2763
2026
1289
1982
4493
2482
0
4008
7256
4508
1260
2026
2763
2026
1289
1982
4493
2482
0
9
794
1220
8982
4493
2482
4000
2014
2808
2014
1220
6968
1685
468
2780
1982
4493
2482
4000
2014
2808
2014
1220
0
1685
468
2780
10
683
1014
6968
9685
7468
2780
1697
2380
1697
1014
5271
7305
5771
1766
7000
9685
7468
2780
1697
2380
1697
1014
5303
7305
5771
1766
11
630
1292
5271
7305
5771
1766
1922
2552
1922
1292
3349
4753
3849
474
5303
7305
5771
1766
1922
2552
1922
1292
3381
4753
3849
474
12
673
1281
3349
4753
3849
4474
1954
2627
1954
1281
1395
2126
1895
3193
3381
4753
3849
4474
1954
2627
1954
1281
1427
2126
1895
3193
13
585
1017
8395
10126
8895
3193
1602
2187
1602
1017
6793
7939
7293
2176
8427
10126
8895
3193
1602
2187
1602
1017
6825
7939
7293
2176
14
671
1090
6793
7939
7293
2176
1761
2432
1761
1090
5032
5507
5532
1086
6825
7939
7293
2176
1761
2432
1761
1090
5064
5507
5532
1086
15
677
1186
5032
5507
5532
5086
1863
2540
1863
1186
3169
2967
3669
3900
5064
5507
5532
5086
1863
2540
1863
1186
3201
2967
3669
3900
16
691
1117
3169
2967
3669
3900
1808
2499
1808
1117
1361
468
1861
2783
3201
2967
3669
3900
1808
2499
1808
1117
1393
468
1861
2783
17
614
1093
8361
8468
8861
2783
1707
2321
1707
1093
6654
6147
7154
1690
8393
8468
8861
2783
1707
2321
1707
1093
6686
6147
7154
1690
18
543
1123
6654
6147
7154
1690
1666
2209
1666
1123
4988
3938
5488
567
6686
6147
7154
1690
1666
2209
1666
1123
5020
3938
5488
567
19
626
981
4988
3938
5488
4567
1607
2233
1607
981
3381
1705
3881
3586
5020
3938
5488
4567
1607
2233
1607
981
3413
1705
3881
3586
20
610
1288
3381
9705
3881
3586
1898
2508
1898
1288
1483
7197
1983
2298
3413
9705
3881
3586
1898
2508
1898
1288
1515
7197
1983
2298
21
691
1135
8483
7197
8983
2298
1826
2517
1826
1135
6657
4680
7157
1163
8515
7197
1983
2298
1826
2517
1826
1135
6689
4680
157
1163
22
773
1103
6657
4680
7157
5163
1876
2649
1876
1103
4781
2031
5281
4060
6689
4680
7157
5163
1876
2649
1876
1103
4813
2031
5281
4060
23
616
1230
4781
10031
5281
4060
1846
2462
1846
1230
2935
7569
3435
2830
4813
10031
5281
4060
1846
2462
1846
1230
2967
7569
3435
2830
24
631
1006
2935
7569
3435
2830
1637
2268
1637
1006
1298
5301
1798
1824
2967
7569
3435
2830
1637
2268
1637
1006
1330
5301
1798
1824
25
705
972
8298
5301
8798
1824
1677
2382
1677
972
6621
2919
7121
852
8330
5301
8798
1824
1677
2382
1677
972
6653
2919
7121
852
26
664
1207
6621
2919
7121
4852
1871
2535
1871
1207
4750
384
5250
3645
6653
2919
7121
4852
1871
2535
1871
1207
4782
384
5250
3645
27
729
1116
4750
8384
5250
3645
1845
2574
1845
1116
2905
5810
3405
2529
4782
8384
5250
3645
1845
2574
1845
1116
2937
5810
3405
2529
28
772
1181
2905
5810
3405
2529
1953
2725
1953
1181
952
3085
1452
1348
2937
5810
3405
2529
1953
2725
1953
1181
984
3085
1452
1348
29
582
1032
7952
3085
8452
1348
1614
2196
1614
1032
6338
889
6838
316
7984
3085
8452
1348
1614
2196
1614
1032
6370
889
6838
316
30
766
1105
6338
8889
6838
4316
1871
2637
1871
1105
4467
6252
4967
3211
6370
8889
6838
4316
1871
2637
1871
1105
4499
6252
4967
3211
31
622
1120
4467
6252
4967
3211
1742
2364
1742
1120
2725
3888
3225
2091
4499
6252
4967
3211
1742
2364
1742
1120
2757
3888
3225
2091
32
705
1255
2725
3888
3225
2091
1960
2665
1960
1255
765
1223
1265
836
2757
3888
3225
2091
1960
2665
1960
1255
797
1223
1265
836
33
679
1024
7765
9223
8265
4836
1703
2382
1703
1024
6062
6841
6562
3812
7797
9223
8265
4836
1703
2382
1703
1024
6094
6841
6562
3812
34
709
1059
6062
6841
6562
3812
1768
2477
1768
1059
4294
4364
4794
2753
6094
6841
6562
3812
1768
2477
1768
1059
4326
4364
4794
2753
35
728
1156
4294
4364
4794
2753
1884
2612
1884
1156
2410
1752
2910
1597
4326
4364
4794
2753
1884
2612
1884
1156
2442
1752
2910
1597
36
671
1217
2410
9752
2910
1597
1888
2559
1888
1217
522
7193
1022
380
2442
9752
2910
1597
1888
2559
1888
1217
554
7193
1022
380Figure 6
Safety stock levels for each component in both scenarios.


### 3.2. Application Two

Similar to the previous application, this application analyzes and simulates a system of modular products composed of five end products EP1, EP2,…, EP5, which are assembled from five modules and five components as shown in Figure7. The structure has a degree of commonality of 2.5. The system uses the ATO strategy to meet the needs of its customers.Figure 7
A system of five modular products and its matrices of composition.For this modular products system, we suppose that the demands of the end products are dependent. Table8 shows the initial inventories and the order quantities of each component of the system, whereas Table 9 shows the means of the demands, the standard deviations, and the correlation coefficients of the end products. The information of both Tables 8 and 9 are used for the two cases analyzed.Table 8
Initial inventories and quantity orders.
c
i
1
2
3
4
5
I. Initial
10000
10000
10000
10000
10000
Q
i
6000
12000
14000
4000
4000Table 9
Correlation coefficients, means of demands, and standard deviations.
ρ
l
z
EP1
EP2
EP3
EP4
EP5
EP1
—
0.72
0.61
0.75
0.8
EP2
0.72
—
0.55
0.66
0.91
EP3
0.61
0.55
—
0.32
0.53
EP4
0.75
0.66
0.32
—
0.44
EP5
0.8
0.91
0.53
0.44
—
μ
i
550
450
650
600
450
σ
i
80
75
95
105
75Case 1 (safety stocks with commonality).
In an environment where there are dependent demands between end products, determining safety stocks of each component of the system is performed by applying expression (8). Table 10 shows the calculated values for safety stocks as well as reorder points for each component.Table 10
Safety stocks and reorder points.
c
i
1
2
3
4
5
S
S
i
333
390
411
261
278
R
O
P
i
4033
4790
5711
2261
2378The results of the thirty simulation runs for this case are shown in Table11. This table shows that eight stock-outs occurred in eight simulation runs, generating a stock-out average equal to 1. The average inventories for each component also are showed.Table 11
Results of the simulation; Case1.
Stock-outs
Average inventories
Amount of stock-outs
Runs with stock-outs
Average of stock-outs
c1
c2
c3
c4
c5
8
8
1
5375
8452
10024
3951
3978Case 2 (safety stocks with commonality and factor of substitution).
Similar to Case  2 of the application 1, each componenti of the modular products system belongs to a part family. Each part family has its own total number of components N
i with a number n
i of substitute components by the component i, and each substitute component has a large buffer to meet any demand. Table 12 shows the details regarding the composition of the part families of each component i of the system analyzed.Table 12
Part families composition.
c
i
1
2
3
4
5
N
i
9
15
18
9
13
n
i
4
7
5
5
6For this case, the safety stocks are calculated using expression (14), because this expression considers the existence of aggregation in inventories, commonality, dependent demands, and factor of substitution. Table 13 shows the calculated values of the safety stocks and reorder points for each component of the system.Table 13
Safety stocks and reorder points.
c
i
1
2
3
4
5
S
S
i
176
55
152
97
66
R
O
P
i
3876
4455
5452
2097
2166The results of the thirty simulation runs for this case are shown in Table14. This table shows that forty-four stock-outs occurred in twenty-six simulation runs, generating a stock-out average equal to 1.7. The average inventories for each component also are showed.Table 14
Results of the simulation: Case2.
Stock-outs
Average inventories
Amount of stock-outs
Runs with stock-outs
Average of stock-outs
c1
c2
c3
c4
c5
44
26
1.7
5239
8169
9728
3821
3820


#### 3.2.1. A Performance Comparison between the Two Cases

Tables11 and 14 show the results of each case; these tables show that there are a greater number of stock-outs and lower average inventories in Case  2 as a result of reducing safety stock levels. Similar to that in application one, the experiment involves the possibility of using substitute components to meet this demand, while stock-outs in Case  1 become in shortage costs. Figure 8 shows the difference between safety stock levels calculated in each case for each component in the system. An example of one simulation run for both cases is shown in Table 15.Table 15
Application 2: demands, components used, and initial and final inventories.
Period (month)
Case1
Case2
Initial inventory
Components used
Final inventory
Initial inventory
Components used
Final inventory
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
1
10000
10000
10000
10000
10000
2033
2471
2809
973
879
7967
7529
7191
9027
9121
10000
10000
10000
10000
10000
2033
2471
2809
973
879
7967
7529
7191
9027
9121
2
7967
7529
7191
9027
9121
1419
1850
2293
940
967
6548
5679
4898
8087
8154
7967
7529
7191
9027
9121
1419
1850
2293
940
967
6548
5679
4898
8087
8154
3
6548
5679
4898
8087
8154
1746
2201
2656
1053
1063
4802
3478
2242
7034
7091
6548
5679
4898
8087
8154
1746
2201
2656
1053
1063
4802
3478
2242
7034
7091
4
4802
3478
16242
7034
7091
1964
2240
2611
961
1054
2838
1238
13631
6073
6037
4802
3478
16242
7034
7091
1964
2240
2611
961
1054
2838
1238
13631
6073
6037
5
2838
13238
13631
6073
6037
1925
2371
2826
998
1262
913
10867
10805
5075
4775
2838
13238
13631
6073
6037
1925
2371
2826
998
1262
913
10867
10805
5075
4775
6
6913
10867
10805
5075
4775
1934
2003
2387
812
977
4979
8864
8418
4263
3798
6913
10867
10805
5075
4775
1934
2003
2387
812
977
4979
8864
8418
4263
3798
7
4979
8864
8418
4263
3798
1586
1681
2113
850
712
3393
7183
6305
3413
3086
4979
8864
8418
4263
3798
1586
1681
2113
850
712
3393
7183
6305
3413
3086
8
3393
7183
6305
3413
3086
1905
2175
2682
1014
1133
1488
5008
3623
2399
1953
3393
7183
6305
3413
3086
1905
2175
2682
1014
1133
1488
5008
3623
2399
1953
9
7488
5008
3623
2399
1953
1715
2114
2586
1035
1197
5773
2894
1037
1364
756
7488
5008
3623
2399
1953
1715
2114
2586
1035
1197
5773
2894
1037
1364
756
10
5773
2894
15037
1364
4756
2267
2601
3022
1068
1158
3506
293
12015
296
3598
5773
2894
15037
1364
4756
2267
2601
3022
1068
1158
3506
293
12015
296
3598
11
3506
12293
12015
4296
3598
1603
1874
2324
857
1075
1903
10419
9691
3439
2523
3506
12293
12015
4296
3598
1603
1874
2324
857
1075
1903
10419
9691
3439
2523
12
7903
10419
9691
3439
2523
1842
2152
2830
1192
1251
6061
8267
6861
2247
1272
7903
10419
9691
3439
2523
1842
2152
2830
1192
1251
6061
8267
6861
2247
1272
13
6061
8267
6861
2247
1272
1959
2069
2522
916
968
4102
6198
4339
1331
304
6061
8267
6861
2247
1272
1959
2069
2522
916
968
4102
6198
4339
1331
304
14
4102
6198
4339
5331
4304
2029
2330
2795
1032
958
2073
3868
1544
4299
3346
4102
6198
4339
1331
4304
2029
2330
2795
1032
958
2073
3868
1544
299
3346
15
2073
3868
15544
4299
3346
2286
2408
2838
1042
855
0
1460
12706
3257
2491
2073
3868
15544
4299
3346
2286
2408
2838
1042
855
0
1460
12706
3257
2491
16
6000
13460
12706
3257
2491
1746
2087
2486
965
960
4254
11373
10220
2292
1531
6000
13460
12706
3257
2491
1746
2087
2486
965
960
4254
11373
10220
2292
1531
17
4254
11373
10220
2292
1531
1774
2141
2603
972
1123
2480
9232
7617
1320
408
4254
11373
10220
2292
1531
1774
2141
2603
972
1123
2480
9232
7617
1320
408
18
2480
9232
7617
1320
4408
1838
1929
2468
933
891
642
7303
5149
387
3517
2480
9232
7617
1320
4408
1838
1929
2468
933
891
642
7303
5149
387
3517
19
6642
7303
5149
4387
3517
2027
2359
2731
1017
881
4615
4944
2418
3370
2636
6642
7303
5149
4387
3517
2027
2359
2731
1017
881
4615
4944
2418
3370
2636
20
4615
4944
16418
3370
2636
1981
2506
2942
1101
1065
2634
2438
13476
2269
1571
4615
4944
16418
3370
2636
1981
2506
2942
1101
1065
2634
2438
13476
2269
1571
21
2634
2438
13476
2269
1571
1335
1524
1884
733
945
1299
914
11592
1536
626
2634
2438
13476
2269
1571
1335
1524
1884
733
945
1299
914
11592
1536
626
22
7299
12914
11592
1536
4626
1287
1834
2219
838
1121
6012
11080
9373
698
3505
7299
12914
11592
1536
4626
1287
1834
2219
838
1121
6012
11080
9373
698
3505
23
6012
11080
9373
4698
3505
1917
2149
2528
938
973
4095
8931
6845
3760
2532
6012
11080
9373
4698
3505
1917
2149
2528
938
973
4095
8931
6845
3760
2532
24
4095
8931
6845
3760
2532
1925
2210
2694
1003
1236
2170
6721
4151
2757
1296
4095
8931
6845
3760
2532
1925
2210
2694
1003
1236
2170
6721
4151
2757
1296
25
2170
6721
4151
2757
1296
1931
2323
2924
1214
979
239
4398
1227
1543
317
2170
6721
4151
2757
1296
1931
2323
2924
1214
979
239
4398
1227
1543
317
26
6239
4398
15227
1543
4317
1667
2046
2401
854
1015
4572
2352
12826
689
3302
6239
4398
15227
1543
4317
1667
2046
2401
854
1015
4572
2352
12826
689
3302
27
4572
14352
12826
4689
3302
2232
2543
2991
1108
1029
2340
11809
9835
3581
2273
4572
14352
12826
4689
3302
2232
2543
2991
1108
1029
2340
11809
9835
3581
2273
28
2340
11809
9835
3581
2273
2146
2447
2942
1115
1002
194
9362
6893
2466
1271
2340
11809
9835
3581
2273
2146
2447
2942
1115
1002
194
9362
6893
2466
1271
29
6194
9362
6893
2466
5271
1813
2394
2879
1110
1290
4381
6968
4014
1356
3981
6194
9362
6893
2466
1271
1813
2394
2879
1110
1290
4381
6968
4014
1356
0
30
4381
6968
4014
1356
3981
2314
2622
2892
938
1081
2067
4346
1122
418
2900
4381
6968
4014
1356
4000
2314
2622
2892
938
1081
2067
4346
1122
418
2919
31
2067
4346
15122
4418
2900
2049
2326
2745
976
847
18
2020
12377
3442
2053
2067
4346
15122
4418
2919
2049
2326
2745
976
847
18
2020
12377
3442
2072
32
6018
14020
12377
3442
2053
1898
2179
2551
932
887
4120
11841
9826
2510
1166
6018
14020
12377
3442
2072
1898
2179
2551
932
887
4120
11841
9826
2510
1185
33
4120
11841
9826
2510
5166
1890
2326
2802
1068
997
2230
9515
7024
1442
4169
4120
11841
9826
2510
5185
1890
2326
2802
1068
997
2230
9515
7024
1442
4188
34
2230
9515
7024
1442
4169
1831
2137
2583
1031
865
399
7378
4441
411
3304
2230
9515
7024
1442
4188
1831
2137
2583
1031
865
399
7378
4441
411
3323
35
6399
7378
4441
4411
3304
1932
2145
2735
1078
1060
4467
5233
1706
3333
2244
6399
7378
4441
4411
3323
1932
2145
2735
1078
1060
4467
5233
1706
3333
2263
36
4467
5233
15706
3333
2244
1435
1874
2223
786
1039
3032
3359
13483
2547
1205
4467
5233
15706
3333
2263
1435
1874
2223
786
1039
3032
3359
13483
2547
1224Figure 8
Safety stock levels for each component in both scenarios.


## 3.1. Application One

This application simulates and analyzes two modular products (EP1 and EP2) of an enterprise that offers a high variety of different modular products to its customers. To manufacture all its products, the enterprise applies GT philosophy to arrange its components in part families. The two products analyzed are assembled from 3 modules and 4 components as shown in Figure5. The structure of these two products has a degree of commonality of 1.7.Figure 5
A system of two modular products and its matrices of composition.We suppose that the demands of these two end products are independent and normally distributed with meansμ
E
P
1
=
700 and μ
E
P
2
=
1100, standard deviations σ
E
P
1
=
80 and σ
E
P
2
=
120. Table 1 shows the initial inventories and order quantities used for the two scenarios analyzed.Table 1
Initial inventories and quantity orders.
c
i
1
2
3
4
I. Initial
10000
9500
10500
9000
Q
i
7000
8000
7000
4000Case 1 (safety stocks with commonality).
Based on the structure of Figure4 and considering that the system handles aggregation in inventories with only component commonality, the safety stock for each component is determined by applying expression (9). Table 2 shows the calculated values for safety stocks and reorder points.Table 2
Safety stock levels and reorder points.
c
i
1
2
3
4
S
S
i
185
212
185
154
R
O
P
i
1985
2712
1985
1254The results of the thirty simulation runs for this case are shown in Table3. This table shows that five stock-outs occurred in five simulation runs, generating a stock-out average equal to 1. The average inventories for each component also are showed.Table 3
Results of the simulation: Case1.
Stock-outs
Average inventories
Amount of stock-outs
Runs with stock-outs
Average of stock-outs
c1
c2
c3
c4
5
5
1
5563
6847
5804
3701Case 2 (safety stocks with commonality and factor of substitution).
In this case, we suppose that each componenti of the two modular products analyzed belongs to a part family, the experiment assumes that each part family has its own total number of components N
i with n
i substitute components for the component i
, and each substitute component has a large buffer to meet any demand. Table 4 shows the details with respect to the composition of the part families of each component i of the system analyzed.Table 4
Part families composition.
c
i
1
2
3
4
N
i
25
30
45
60
n
i
3
6
4
6For an organization, obtaining the data in Table4 is not a difficult task; they can be obtained through the construction of part families and identifying the substitute components for each member of the families. Once this information is structured, it can be used to calculate the safety stock levels for each component.Given the information in Table4, safety stock for each component of the system is determined using the expression (15), which considers the aggregation in inventories, existence of component commonality, independent demands, and the factor of substitution. Table 5 shows the calculated values for safety stocks and reorder points.Table 5
Safety stock levels and reorder points.
c
i
1
2
3
4
S
S
i
166
135
152
98
R
O
P
i
1966
2635
1952
1198The results of the thirty simulation runs for this case are shown in Table6. This table shows that eight stock-outs occurred in six simulation runs, generating a stock-out average equal to 1.3. The average inventories for each component also are showed.Table 6
Results of the simulation: Case2.
Stock-outs
Average inventories
Amount of stock-outs
Runs with stock-outs
Average of stock-outs
c1
c2
c3
c4
8
6
1.3
5542
6795
5785
3648


### 3.1.1. A Performance Comparison between the Two Cases

Tables3 and 6 show the results of each case; these tables show that there are a greater number of stock-outs and lower average inventories in Case  2 as a result of reducing safety stock levels. However, the experiment involves the possibility of using substitute components to meet this demand, while stock-outs in Case  1 become in shortage costs. Figure 6 shows the difference between safety stock levels calculated in each case for each component in the system. An example of one simulation run for both cases is shown in Table 7.Table 7
Application 1: demands, components used, and initial and final inventories.
Period (month)
Demands
Case1
Case2
Initial inventory
Components used
Final inventory
Initial inventory
Components used
Final inventory
FP1
FP2
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
1
724
1062
10000
9500
10500
9000
1786
2510
1786
1062
8214
6990
8714
7938
10000
9500
10500
9000
1786
2510
1786
1062
8214
6990
8714
7938
2
772
1076
8214
6990
8714
7938
1848
2620
1848
1076
6366
4370
6866
6862
8214
6990
8714
7938
1848
2620
1848
1076
6366
4370
6866
6862
3
800
1197
6366
4370
6866
6862
1997
2797
1997
1197
4369
1573
4869
5665
6366
4370
6866
6862
1997
2797
1997
1197
4369
1573
4869
5665
4
740
1062
4369
9573
4869
5665
1802
2542
1802
1062
2567
7031
3067
4603
4369
9573
4869
5665
1802
2542
1802
1062
2567
7031
3067
4603
5
728
1189
2567
7031
3067
4603
1917
2645
1917
1189
650
4386
1150
3414
2567
7031
3067
4603
1917
2645
1917
1189
650
4386
1150
3414
6
770
1142
7650
4386
8150
3414
1912
2682
1912
1142
5738
1704
6238
2272
7650
4386
8150
3414
1912
2682
1912
1142
5738
1704
6238
2272
7
718
1012
5738
9704
6238
2272
1730
2448
1730
1012
4008
7256
4508
1260
5738
9704
6238
2272
1730
2448
1730
1012
4008
7256
4508
1260
8
737
1289
4008
7256
4508
1260
2026
2763
2026
1289
1982
4493
2482
0
4008
7256
4508
1260
2026
2763
2026
1289
1982
4493
2482
0
9
794
1220
8982
4493
2482
4000
2014
2808
2014
1220
6968
1685
468
2780
1982
4493
2482
4000
2014
2808
2014
1220
0
1685
468
2780
10
683
1014
6968
9685
7468
2780
1697
2380
1697
1014
5271
7305
5771
1766
7000
9685
7468
2780
1697
2380
1697
1014
5303
7305
5771
1766
11
630
1292
5271
7305
5771
1766
1922
2552
1922
1292
3349
4753
3849
474
5303
7305
5771
1766
1922
2552
1922
1292
3381
4753
3849
474
12
673
1281
3349
4753
3849
4474
1954
2627
1954
1281
1395
2126
1895
3193
3381
4753
3849
4474
1954
2627
1954
1281
1427
2126
1895
3193
13
585
1017
8395
10126
8895
3193
1602
2187
1602
1017
6793
7939
7293
2176
8427
10126
8895
3193
1602
2187
1602
1017
6825
7939
7293
2176
14
671
1090
6793
7939
7293
2176
1761
2432
1761
1090
5032
5507
5532
1086
6825
7939
7293
2176
1761
2432
1761
1090
5064
5507
5532
1086
15
677
1186
5032
5507
5532
5086
1863
2540
1863
1186
3169
2967
3669
3900
5064
5507
5532
5086
1863
2540
1863
1186
3201
2967
3669
3900
16
691
1117
3169
2967
3669
3900
1808
2499
1808
1117
1361
468
1861
2783
3201
2967
3669
3900
1808
2499
1808
1117
1393
468
1861
2783
17
614
1093
8361
8468
8861
2783
1707
2321
1707
1093
6654
6147
7154
1690
8393
8468
8861
2783
1707
2321
1707
1093
6686
6147
7154
1690
18
543
1123
6654
6147
7154
1690
1666
2209
1666
1123
4988
3938
5488
567
6686
6147
7154
1690
1666
2209
1666
1123
5020
3938
5488
567
19
626
981
4988
3938
5488
4567
1607
2233
1607
981
3381
1705
3881
3586
5020
3938
5488
4567
1607
2233
1607
981
3413
1705
3881
3586
20
610
1288
3381
9705
3881
3586
1898
2508
1898
1288
1483
7197
1983
2298
3413
9705
3881
3586
1898
2508
1898
1288
1515
7197
1983
2298
21
691
1135
8483
7197
8983
2298
1826
2517
1826
1135
6657
4680
7157
1163
8515
7197
1983
2298
1826
2517
1826
1135
6689
4680
157
1163
22
773
1103
6657
4680
7157
5163
1876
2649
1876
1103
4781
2031
5281
4060
6689
4680
7157
5163
1876
2649
1876
1103
4813
2031
5281
4060
23
616
1230
4781
10031
5281
4060
1846
2462
1846
1230
2935
7569
3435
2830
4813
10031
5281
4060
1846
2462
1846
1230
2967
7569
3435
2830
24
631
1006
2935
7569
3435
2830
1637
2268
1637
1006
1298
5301
1798
1824
2967
7569
3435
2830
1637
2268
1637
1006
1330
5301
1798
1824
25
705
972
8298
5301
8798
1824
1677
2382
1677
972
6621
2919
7121
852
8330
5301
8798
1824
1677
2382
1677
972
6653
2919
7121
852
26
664
1207
6621
2919
7121
4852
1871
2535
1871
1207
4750
384
5250
3645
6653
2919
7121
4852
1871
2535
1871
1207
4782
384
5250
3645
27
729
1116
4750
8384
5250
3645
1845
2574
1845
1116
2905
5810
3405
2529
4782
8384
5250
3645
1845
2574
1845
1116
2937
5810
3405
2529
28
772
1181
2905
5810
3405
2529
1953
2725
1953
1181
952
3085
1452
1348
2937
5810
3405
2529
1953
2725
1953
1181
984
3085
1452
1348
29
582
1032
7952
3085
8452
1348
1614
2196
1614
1032
6338
889
6838
316
7984
3085
8452
1348
1614
2196
1614
1032
6370
889
6838
316
30
766
1105
6338
8889
6838
4316
1871
2637
1871
1105
4467
6252
4967
3211
6370
8889
6838
4316
1871
2637
1871
1105
4499
6252
4967
3211
31
622
1120
4467
6252
4967
3211
1742
2364
1742
1120
2725
3888
3225
2091
4499
6252
4967
3211
1742
2364
1742
1120
2757
3888
3225
2091
32
705
1255
2725
3888
3225
2091
1960
2665
1960
1255
765
1223
1265
836
2757
3888
3225
2091
1960
2665
1960
1255
797
1223
1265
836
33
679
1024
7765
9223
8265
4836
1703
2382
1703
1024
6062
6841
6562
3812
7797
9223
8265
4836
1703
2382
1703
1024
6094
6841
6562
3812
34
709
1059
6062
6841
6562
3812
1768
2477
1768
1059
4294
4364
4794
2753
6094
6841
6562
3812
1768
2477
1768
1059
4326
4364
4794
2753
35
728
1156
4294
4364
4794
2753
1884
2612
1884
1156
2410
1752
2910
1597
4326
4364
4794
2753
1884
2612
1884
1156
2442
1752
2910
1597
36
671
1217
2410
9752
2910
1597
1888
2559
1888
1217
522
7193
1022
380
2442
9752
2910
1597
1888
2559
1888
1217
554
7193
1022
380Figure 6
Safety stock levels for each component in both scenarios.


## 3.1.1. A Performance Comparison between the Two Cases

Tables3 and 6 show the results of each case; these tables show that there are a greater number of stock-outs and lower average inventories in Case  2 as a result of reducing safety stock levels. However, the experiment involves the possibility of using substitute components to meet this demand, while stock-outs in Case  1 become in shortage costs. Figure 6 shows the difference between safety stock levels calculated in each case for each component in the system. An example of one simulation run for both cases is shown in Table 7.Table 7
Application 1: demands, components used, and initial and final inventories.
Period (month)
Demands
Case1
Case2
Initial inventory
Components used
Final inventory
Initial inventory
Components used
Final inventory
FP1
FP2
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
1
724
1062
10000
9500
10500
9000
1786
2510
1786
1062
8214
6990
8714
7938
10000
9500
10500
9000
1786
2510
1786
1062
8214
6990
8714
7938
2
772
1076
8214
6990
8714
7938
1848
2620
1848
1076
6366
4370
6866
6862
8214
6990
8714
7938
1848
2620
1848
1076
6366
4370
6866
6862
3
800
1197
6366
4370
6866
6862
1997
2797
1997
1197
4369
1573
4869
5665
6366
4370
6866
6862
1997
2797
1997
1197
4369
1573
4869
5665
4
740
1062
4369
9573
4869
5665
1802
2542
1802
1062
2567
7031
3067
4603
4369
9573
4869
5665
1802
2542
1802
1062
2567
7031
3067
4603
5
728
1189
2567
7031
3067
4603
1917
2645
1917
1189
650
4386
1150
3414
2567
7031
3067
4603
1917
2645
1917
1189
650
4386
1150
3414
6
770
1142
7650
4386
8150
3414
1912
2682
1912
1142
5738
1704
6238
2272
7650
4386
8150
3414
1912
2682
1912
1142
5738
1704
6238
2272
7
718
1012
5738
9704
6238
2272
1730
2448
1730
1012
4008
7256
4508
1260
5738
9704
6238
2272
1730
2448
1730
1012
4008
7256
4508
1260
8
737
1289
4008
7256
4508
1260
2026
2763
2026
1289
1982
4493
2482
0
4008
7256
4508
1260
2026
2763
2026
1289
1982
4493
2482
0
9
794
1220
8982
4493
2482
4000
2014
2808
2014
1220
6968
1685
468
2780
1982
4493
2482
4000
2014
2808
2014
1220
0
1685
468
2780
10
683
1014
6968
9685
7468
2780
1697
2380
1697
1014
5271
7305
5771
1766
7000
9685
7468
2780
1697
2380
1697
1014
5303
7305
5771
1766
11
630
1292
5271
7305
5771
1766
1922
2552
1922
1292
3349
4753
3849
474
5303
7305
5771
1766
1922
2552
1922
1292
3381
4753
3849
474
12
673
1281
3349
4753
3849
4474
1954
2627
1954
1281
1395
2126
1895
3193
3381
4753
3849
4474
1954
2627
1954
1281
1427
2126
1895
3193
13
585
1017
8395
10126
8895
3193
1602
2187
1602
1017
6793
7939
7293
2176
8427
10126
8895
3193
1602
2187
1602
1017
6825
7939
7293
2176
14
671
1090
6793
7939
7293
2176
1761
2432
1761
1090
5032
5507
5532
1086
6825
7939
7293
2176
1761
2432
1761
1090
5064
5507
5532
1086
15
677
1186
5032
5507
5532
5086
1863
2540
1863
1186
3169
2967
3669
3900
5064
5507
5532
5086
1863
2540
1863
1186
3201
2967
3669
3900
16
691
1117
3169
2967
3669
3900
1808
2499
1808
1117
1361
468
1861
2783
3201
2967
3669
3900
1808
2499
1808
1117
1393
468
1861
2783
17
614
1093
8361
8468
8861
2783
1707
2321
1707
1093
6654
6147
7154
1690
8393
8468
8861
2783
1707
2321
1707
1093
6686
6147
7154
1690
18
543
1123
6654
6147
7154
1690
1666
2209
1666
1123
4988
3938
5488
567
6686
6147
7154
1690
1666
2209
1666
1123
5020
3938
5488
567
19
626
981
4988
3938
5488
4567
1607
2233
1607
981
3381
1705
3881
3586
5020
3938
5488
4567
1607
2233
1607
981
3413
1705
3881
3586
20
610
1288
3381
9705
3881
3586
1898
2508
1898
1288
1483
7197
1983
2298
3413
9705
3881
3586
1898
2508
1898
1288
1515
7197
1983
2298
21
691
1135
8483
7197
8983
2298
1826
2517
1826
1135
6657
4680
7157
1163
8515
7197
1983
2298
1826
2517
1826
1135
6689
4680
157
1163
22
773
1103
6657
4680
7157
5163
1876
2649
1876
1103
4781
2031
5281
4060
6689
4680
7157
5163
1876
2649
1876
1103
4813
2031
5281
4060
23
616
1230
4781
10031
5281
4060
1846
2462
1846
1230
2935
7569
3435
2830
4813
10031
5281
4060
1846
2462
1846
1230
2967
7569
3435
2830
24
631
1006
2935
7569
3435
2830
1637
2268
1637
1006
1298
5301
1798
1824
2967
7569
3435
2830
1637
2268
1637
1006
1330
5301
1798
1824
25
705
972
8298
5301
8798
1824
1677
2382
1677
972
6621
2919
7121
852
8330
5301
8798
1824
1677
2382
1677
972
6653
2919
7121
852
26
664
1207
6621
2919
7121
4852
1871
2535
1871
1207
4750
384
5250
3645
6653
2919
7121
4852
1871
2535
1871
1207
4782
384
5250
3645
27
729
1116
4750
8384
5250
3645
1845
2574
1845
1116
2905
5810
3405
2529
4782
8384
5250
3645
1845
2574
1845
1116
2937
5810
3405
2529
28
772
1181
2905
5810
3405
2529
1953
2725
1953
1181
952
3085
1452
1348
2937
5810
3405
2529
1953
2725
1953
1181
984
3085
1452
1348
29
582
1032
7952
3085
8452
1348
1614
2196
1614
1032
6338
889
6838
316
7984
3085
8452
1348
1614
2196
1614
1032
6370
889
6838
316
30
766
1105
6338
8889
6838
4316
1871
2637
1871
1105
4467
6252
4967
3211
6370
8889
6838
4316
1871
2637
1871
1105
4499
6252
4967
3211
31
622
1120
4467
6252
4967
3211
1742
2364
1742
1120
2725
3888
3225
2091
4499
6252
4967
3211
1742
2364
1742
1120
2757
3888
3225
2091
32
705
1255
2725
3888
3225
2091
1960
2665
1960
1255
765
1223
1265
836
2757
3888
3225
2091
1960
2665
1960
1255
797
1223
1265
836
33
679
1024
7765
9223
8265
4836
1703
2382
1703
1024
6062
6841
6562
3812
7797
9223
8265
4836
1703
2382
1703
1024
6094
6841
6562
3812
34
709
1059
6062
6841
6562
3812
1768
2477
1768
1059
4294
4364
4794
2753
6094
6841
6562
3812
1768
2477
1768
1059
4326
4364
4794
2753
35
728
1156
4294
4364
4794
2753
1884
2612
1884
1156
2410
1752
2910
1597
4326
4364
4794
2753
1884
2612
1884
1156
2442
1752
2910
1597
36
671
1217
2410
9752
2910
1597
1888
2559
1888
1217
522
7193
1022
380
2442
9752
2910
1597
1888
2559
1888
1217
554
7193
1022
380Figure 6
Safety stock levels for each component in both scenarios.


## 3.2. Application Two

Similar to the previous application, this application analyzes and simulates a system of modular products composed of five end products EP1, EP2,…, EP5, which are assembled from five modules and five components as shown in Figure7. The structure has a degree of commonality of 2.5. The system uses the ATO strategy to meet the needs of its customers.Figure 7
A system of five modular products and its matrices of composition.For this modular products system, we suppose that the demands of the end products are dependent. Table8 shows the initial inventories and the order quantities of each component of the system, whereas Table 9 shows the means of the demands, the standard deviations, and the correlation coefficients of the end products. The information of both Tables 8 and 9 are used for the two cases analyzed.Table 8
Initial inventories and quantity orders.
c
i
1
2
3
4
5
I. Initial
10000
10000
10000
10000
10000
Q
i
6000
12000
14000
4000
4000Table 9
Correlation coefficients, means of demands, and standard deviations.
ρ
l
z
EP1
EP2
EP3
EP4
EP5
EP1
—
0.72
0.61
0.75
0.8
EP2
0.72
—
0.55
0.66
0.91
EP3
0.61
0.55
—
0.32
0.53
EP4
0.75
0.66
0.32
—
0.44
EP5
0.8
0.91
0.53
0.44
—
μ
i
550
450
650
600
450
σ
i
80
75
95
105
75Case 1 (safety stocks with commonality).
In an environment where there are dependent demands between end products, determining safety stocks of each component of the system is performed by applying expression (8). Table 10 shows the calculated values for safety stocks as well as reorder points for each component.Table 10
Safety stocks and reorder points.
c
i
1
2
3
4
5
S
S
i
333
390
411
261
278
R
O
P
i
4033
4790
5711
2261
2378The results of the thirty simulation runs for this case are shown in Table11. This table shows that eight stock-outs occurred in eight simulation runs, generating a stock-out average equal to 1. The average inventories for each component also are showed.Table 11
Results of the simulation; Case1.
Stock-outs
Average inventories
Amount of stock-outs
Runs with stock-outs
Average of stock-outs
c1
c2
c3
c4
c5
8
8
1
5375
8452
10024
3951
3978Case 2 (safety stocks with commonality and factor of substitution).
Similar to Case  2 of the application 1, each componenti of the modular products system belongs to a part family. Each part family has its own total number of components N
i with a number n
i of substitute components by the component i, and each substitute component has a large buffer to meet any demand. Table 12 shows the details regarding the composition of the part families of each component i of the system analyzed.Table 12
Part families composition.
c
i
1
2
3
4
5
N
i
9
15
18
9
13
n
i
4
7
5
5
6For this case, the safety stocks are calculated using expression (14), because this expression considers the existence of aggregation in inventories, commonality, dependent demands, and factor of substitution. Table 13 shows the calculated values of the safety stocks and reorder points for each component of the system.Table 13
Safety stocks and reorder points.
c
i
1
2
3
4
5
S
S
i
176
55
152
97
66
R
O
P
i
3876
4455
5452
2097
2166The results of the thirty simulation runs for this case are shown in Table14. This table shows that forty-four stock-outs occurred in twenty-six simulation runs, generating a stock-out average equal to 1.7. The average inventories for each component also are showed.Table 14
Results of the simulation: Case2.
Stock-outs
Average inventories
Amount of stock-outs
Runs with stock-outs
Average of stock-outs
c1
c2
c3
c4
c5
44
26
1.7
5239
8169
9728
3821
3820


### 3.2.1. A Performance Comparison between the Two Cases

Tables11 and 14 show the results of each case; these tables show that there are a greater number of stock-outs and lower average inventories in Case  2 as a result of reducing safety stock levels. Similar to that in application one, the experiment involves the possibility of using substitute components to meet this demand, while stock-outs in Case  1 become in shortage costs. Figure 8 shows the difference between safety stock levels calculated in each case for each component in the system. An example of one simulation run for both cases is shown in Table 15.Table 15
Application 2: demands, components used, and initial and final inventories.
Period (month)
Case1
Case2
Initial inventory
Components used
Final inventory
Initial inventory
Components used
Final inventory
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
1
10000
10000
10000
10000
10000
2033
2471
2809
973
879
7967
7529
7191
9027
9121
10000
10000
10000
10000
10000
2033
2471
2809
973
879
7967
7529
7191
9027
9121
2
7967
7529
7191
9027
9121
1419
1850
2293
940
967
6548
5679
4898
8087
8154
7967
7529
7191
9027
9121
1419
1850
2293
940
967
6548
5679
4898
8087
8154
3
6548
5679
4898
8087
8154
1746
2201
2656
1053
1063
4802
3478
2242
7034
7091
6548
5679
4898
8087
8154
1746
2201
2656
1053
1063
4802
3478
2242
7034
7091
4
4802
3478
16242
7034
7091
1964
2240
2611
961
1054
2838
1238
13631
6073
6037
4802
3478
16242
7034
7091
1964
2240
2611
961
1054
2838
1238
13631
6073
6037
5
2838
13238
13631
6073
6037
1925
2371
2826
998
1262
913
10867
10805
5075
4775
2838
13238
13631
6073
6037
1925
2371
2826
998
1262
913
10867
10805
5075
4775
6
6913
10867
10805
5075
4775
1934
2003
2387
812
977
4979
8864
8418
4263
3798
6913
10867
10805
5075
4775
1934
2003
2387
812
977
4979
8864
8418
4263
3798
7
4979
8864
8418
4263
3798
1586
1681
2113
850
712
3393
7183
6305
3413
3086
4979
8864
8418
4263
3798
1586
1681
2113
850
712
3393
7183
6305
3413
3086
8
3393
7183
6305
3413
3086
1905
2175
2682
1014
1133
1488
5008
3623
2399
1953
3393
7183
6305
3413
3086
1905
2175
2682
1014
1133
1488
5008
3623
2399
1953
9
7488
5008
3623
2399
1953
1715
2114
2586
1035
1197
5773
2894
1037
1364
756
7488
5008
3623
2399
1953
1715
2114
2586
1035
1197
5773
2894
1037
1364
756
10
5773
2894
15037
1364
4756
2267
2601
3022
1068
1158
3506
293
12015
296
3598
5773
2894
15037
1364
4756
2267
2601
3022
1068
1158
3506
293
12015
296
3598
11
3506
12293
12015
4296
3598
1603
1874
2324
857
1075
1903
10419
9691
3439
2523
3506
12293
12015
4296
3598
1603
1874
2324
857
1075
1903
10419
9691
3439
2523
12
7903
10419
9691
3439
2523
1842
2152
2830
1192
1251
6061
8267
6861
2247
1272
7903
10419
9691
3439
2523
1842
2152
2830
1192
1251
6061
8267
6861
2247
1272
13
6061
8267
6861
2247
1272
1959
2069
2522
916
968
4102
6198
4339
1331
304
6061
8267
6861
2247
1272
1959
2069
2522
916
968
4102
6198
4339
1331
304
14
4102
6198
4339
5331
4304
2029
2330
2795
1032
958
2073
3868
1544
4299
3346
4102
6198
4339
1331
4304
2029
2330
2795
1032
958
2073
3868
1544
299
3346
15
2073
3868
15544
4299
3346
2286
2408
2838
1042
855
0
1460
12706
3257
2491
2073
3868
15544
4299
3346
2286
2408
2838
1042
855
0
1460
12706
3257
2491
16
6000
13460
12706
3257
2491
1746
2087
2486
965
960
4254
11373
10220
2292
1531
6000
13460
12706
3257
2491
1746
2087
2486
965
960
4254
11373
10220
2292
1531
17
4254
11373
10220
2292
1531
1774
2141
2603
972
1123
2480
9232
7617
1320
408
4254
11373
10220
2292
1531
1774
2141
2603
972
1123
2480
9232
7617
1320
408
18
2480
9232
7617
1320
4408
1838
1929
2468
933
891
642
7303
5149
387
3517
2480
9232
7617
1320
4408
1838
1929
2468
933
891
642
7303
5149
387
3517
19
6642
7303
5149
4387
3517
2027
2359
2731
1017
881
4615
4944
2418
3370
2636
6642
7303
5149
4387
3517
2027
2359
2731
1017
881
4615
4944
2418
3370
2636
20
4615
4944
16418
3370
2636
1981
2506
2942
1101
1065
2634
2438
13476
2269
1571
4615
4944
16418
3370
2636
1981
2506
2942
1101
1065
2634
2438
13476
2269
1571
21
2634
2438
13476
2269
1571
1335
1524
1884
733
945
1299
914
11592
1536
626
2634
2438
13476
2269
1571
1335
1524
1884
733
945
1299
914
11592
1536
626
22
7299
12914
11592
1536
4626
1287
1834
2219
838
1121
6012
11080
9373
698
3505
7299
12914
11592
1536
4626
1287
1834
2219
838
1121
6012
11080
9373
698
3505
23
6012
11080
9373
4698
3505
1917
2149
2528
938
973
4095
8931
6845
3760
2532
6012
11080
9373
4698
3505
1917
2149
2528
938
973
4095
8931
6845
3760
2532
24
4095
8931
6845
3760
2532
1925
2210
2694
1003
1236
2170
6721
4151
2757
1296
4095
8931
6845
3760
2532
1925
2210
2694
1003
1236
2170
6721
4151
2757
1296
25
2170
6721
4151
2757
1296
1931
2323
2924
1214
979
239
4398
1227
1543
317
2170
6721
4151
2757
1296
1931
2323
2924
1214
979
239
4398
1227
1543
317
26
6239
4398
15227
1543
4317
1667
2046
2401
854
1015
4572
2352
12826
689
3302
6239
4398
15227
1543
4317
1667
2046
2401
854
1015
4572
2352
12826
689
3302
27
4572
14352
12826
4689
3302
2232
2543
2991
1108
1029
2340
11809
9835
3581
2273
4572
14352
12826
4689
3302
2232
2543
2991
1108
1029
2340
11809
9835
3581
2273
28
2340
11809
9835
3581
2273
2146
2447
2942
1115
1002
194
9362
6893
2466
1271
2340
11809
9835
3581
2273
2146
2447
2942
1115
1002
194
9362
6893
2466
1271
29
6194
9362
6893
2466
5271
1813
2394
2879
1110
1290
4381
6968
4014
1356
3981
6194
9362
6893
2466
1271
1813
2394
2879
1110
1290
4381
6968
4014
1356
0
30
4381
6968
4014
1356
3981
2314
2622
2892
938
1081
2067
4346
1122
418
2900
4381
6968
4014
1356
4000
2314
2622
2892
938
1081
2067
4346
1122
418
2919
31
2067
4346
15122
4418
2900
2049
2326
2745
976
847
18
2020
12377
3442
2053
2067
4346
15122
4418
2919
2049
2326
2745
976
847
18
2020
12377
3442
2072
32
6018
14020
12377
3442
2053
1898
2179
2551
932
887
4120
11841
9826
2510
1166
6018
14020
12377
3442
2072
1898
2179
2551
932
887
4120
11841
9826
2510
1185
33
4120
11841
9826
2510
5166
1890
2326
2802
1068
997
2230
9515
7024
1442
4169
4120
11841
9826
2510
5185
1890
2326
2802
1068
997
2230
9515
7024
1442
4188
34
2230
9515
7024
1442
4169
1831
2137
2583
1031
865
399
7378
4441
411
3304
2230
9515
7024
1442
4188
1831
2137
2583
1031
865
399
7378
4441
411
3323
35
6399
7378
4441
4411
3304
1932
2145
2735
1078
1060
4467
5233
1706
3333
2244
6399
7378
4441
4411
3323
1932
2145
2735
1078
1060
4467
5233
1706
3333
2263
36
4467
5233
15706
3333
2244
1435
1874
2223
786
1039
3032
3359
13483
2547
1205
4467
5233
15706
3333
2263
1435
1874
2223
786
1039
3032
3359
13483
2547
1224Figure 8
Safety stock levels for each component in both scenarios.


## 3.2.1. A Performance Comparison between the Two Cases

Tables11 and 14 show the results of each case; these tables show that there are a greater number of stock-outs and lower average inventories in Case  2 as a result of reducing safety stock levels. Similar to that in application one, the experiment involves the possibility of using substitute components to meet this demand, while stock-outs in Case  1 become in shortage costs. Figure 8 shows the difference between safety stock levels calculated in each case for each component in the system. An example of one simulation run for both cases is shown in Table 15.Table 15
Application 2: demands, components used, and initial and final inventories.
Period (month)
Case1
Case2
Initial inventory
Components used
Final inventory
Initial inventory
Components used
Final inventory
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
c1
c2
c3
c4
c5
1
10000
10000
10000
10000
10000
2033
2471
2809
973
879
7967
7529
7191
9027
9121
10000
10000
10000
10000
10000
2033
2471
2809
973
879
7967
7529
7191
9027
9121
2
7967
7529
7191
9027
9121
1419
1850
2293
940
967
6548
5679
4898
8087
8154
7967
7529
7191
9027
9121
1419
1850
2293
940
967
6548
5679
4898
8087
8154
3
6548
5679
4898
8087
8154
1746
2201
2656
1053
1063
4802
3478
2242
7034
7091
6548
5679
4898
8087
8154
1746
2201
2656
1053
1063
4802
3478
2242
7034
7091
4
4802
3478
16242
7034
7091
1964
2240
2611
961
1054
2838
1238
13631
6073
6037
4802
3478
16242
7034
7091
1964
2240
2611
961
1054
2838
1238
13631
6073
6037
5
2838
13238
13631
6073
6037
1925
2371
2826
998
1262
913
10867
10805
5075
4775
2838
13238
13631
6073
6037
1925
2371
2826
998
1262
913
10867
10805
5075
4775
6
6913
10867
10805
5075
4775
1934
2003
2387
812
977
4979
8864
8418
4263
3798
6913
10867
10805
5075
4775
1934
2003
2387
812
977
4979
8864
8418
4263
3798
7
4979
8864
8418
4263
3798
1586
1681
2113
850
712
3393
7183
6305
3413
3086
4979
8864
8418
4263
3798
1586
1681
2113
850
712
3393
7183
6305
3413
3086
8
3393
7183
6305
3413
3086
1905
2175
2682
1014
1133
1488
5008
3623
2399
1953
3393
7183
6305
3413
3086
1905
2175
2682
1014
1133
1488
5008
3623
2399
1953
9
7488
5008
3623
2399
1953
1715
2114
2586
1035
1197
5773
2894
1037
1364
756
7488
5008
3623
2399
1953
1715
2114
2586
1035
1197
5773
2894
1037
1364
756
10
5773
2894
15037
1364
4756
2267
2601
3022
1068
1158
3506
293
12015
296
3598
5773
2894
15037
1364
4756
2267
2601
3022
1068
1158
3506
293
12015
296
3598
11
3506
12293
12015
4296
3598
1603
1874
2324
857
1075
1903
10419
9691
3439
2523
3506
12293
12015
4296
3598
1603
1874
2324
857
1075
1903
10419
9691
3439
2523
12
7903
10419
9691
3439
2523
1842
2152
2830
1192
1251
6061
8267
6861
2247
1272
7903
10419
9691
3439
2523
1842
2152
2830
1192
1251
6061
8267
6861
2247
1272
13
6061
8267
6861
2247
1272
1959
2069
2522
916
968
4102
6198
4339
1331
304
6061
8267
6861
2247
1272
1959
2069
2522
916
968
4102
6198
4339
1331
304
14
4102
6198
4339
5331
4304
2029
2330
2795
1032
958
2073
3868
1544
4299
3346
4102
6198
4339
1331
4304
2029
2330
2795
1032
958
2073
3868
1544
299
3346
15
2073
3868
15544
4299
3346
2286
2408
2838
1042
855
0
1460
12706
3257
2491
2073
3868
15544
4299
3346
2286
2408
2838
1042
855
0
1460
12706
3257
2491
16
6000
13460
12706
3257
2491
1746
2087
2486
965
960
4254
11373
10220
2292
1531
6000
13460
12706
3257
2491
1746
2087
2486
965
960
4254
11373
10220
2292
1531
17
4254
11373
10220
2292
1531
1774
2141
2603
972
1123
2480
9232
7617
1320
408
4254
11373
10220
2292
1531
1774
2141
2603
972
1123
2480
9232
7617
1320
408
18
2480
9232
7617
1320
4408
1838
1929
2468
933
891
642
7303
5149
387
3517
2480
9232
7617
1320
4408
1838
1929
2468
933
891
642
7303
5149
387
3517
19
6642
7303
5149
4387
3517
2027
2359
2731
1017
881
4615
4944
2418
3370
2636
6642
7303
5149
4387
3517
2027
2359
2731
1017
881
4615
4944
2418
3370
2636
20
4615
4944
16418
3370
2636
1981
2506
2942
1101
1065
2634
2438
13476
2269
1571
4615
4944
16418
3370
2636
1981
2506
2942
1101
1065
2634
2438
13476
2269
1571
21
2634
2438
13476
2269
1571
1335
1524
1884
733
945
1299
914
11592
1536
626
2634
2438
13476
2269
1571
1335
1524
1884
733
945
1299
914
11592
1536
626
22
7299
12914
11592
1536
4626
1287
1834
2219
838
1121
6012
11080
9373
698
3505
7299
12914
11592
1536
4626
1287
1834
2219
838
1121
6012
11080
9373
698
3505
23
6012
11080
9373
4698
3505
1917
2149
2528
938
973
4095
8931
6845
3760
2532
6012
11080
9373
4698
3505
1917
2149
2528
938
973
4095
8931
6845
3760
2532
24
4095
8931
6845
3760
2532
1925
2210
2694
1003
1236
2170
6721
4151
2757
1296
4095
8931
6845
3760
2532
1925
2210
2694
1003
1236
2170
6721
4151
2757
1296
25
2170
6721
4151
2757
1296
1931
2323
2924
1214
979
239
4398
1227
1543
317
2170
6721
4151
2757
1296
1931
2323
2924
1214
979
239
4398
1227
1543
317
26
6239
4398
15227
1543
4317
1667
2046
2401
854
1015
4572
2352
12826
689
3302
6239
4398
15227
1543
4317
1667
2046
2401
854
1015
4572
2352
12826
689
3302
27
4572
14352
12826
4689
3302
2232
2543
2991
1108
1029
2340
11809
9835
3581
2273
4572
14352
12826
4689
3302
2232
2543
2991
1108
1029
2340
11809
9835
3581
2273
28
2340
11809
9835
3581
2273
2146
2447
2942
1115
1002
194
9362
6893
2466
1271
2340
11809
9835
3581
2273
2146
2447
2942
1115
1002
194
9362
6893
2466
1271
29
6194
9362
6893
2466
5271
1813
2394
2879
1110
1290
4381
6968
4014
1356
3981
6194
9362
6893
2466
1271
1813
2394
2879
1110
1290
4381
6968
4014
1356
0
30
4381
6968
4014
1356
3981
2314
2622
2892
938
1081
2067
4346
1122
418
2900
4381
6968
4014
1356
4000
2314
2622
2892
938
1081
2067
4346
1122
418
2919
31
2067
4346
15122
4418
2900
2049
2326
2745
976
847
18
2020
12377
3442
2053
2067
4346
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12377
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1224Figure 8
Safety stock levels for each component in both scenarios.


## 4. Discussion

Currently, uncertainty in demand and variability in consumer behavior hinder decision-making concerning the determination of the optimal safety stock levels. As a result, an organization requires a sufficient level of these inventories to ensure a high CSL. Greater safety stock levels represent higher costs. Of course, safety stocks are not the only factor that can negatively affect the CSL; there are other factors that also affect it directly, such as product quality, after-sales services, and production capacity. Integrating all these factors to ensure a high CSL is an important area for research.The proposed model in this paper solves the problem of reducing safety stock levels without affecting CSL in organizations that employ modular production as a strategy to offer a wide variety of products. The model suggests that the safety stock level for a component belonging to a modular products system can be reduced through exploiting the benefits of the aggregation of inventories, standardization of components, the degree of commonality, and the GT philosophy. The model assumes that the negative effect of increasing the amount of stock-outs can be eliminated when using substitute components to meet demands.The model facilitates basic economies of substitution, such as costs reduction and inventory reduction as well as reduction in system complexity. For best performance, the model requires a high degree of integration between all areas of the organization, because that forces organizations to adopt the best strategies to achieve a proper standardization and to define an efficient methodology for the construction of part families.


## 5. Conclusions and Suggestions for Future Research

In this paper, we explored and quantified the positive effect of linking component commonality and GT philosophy on safety stock levels for organizations that employ modular production. Specifically, we achieved two objectives: (1) we showed an efficient way of reducing safety stock levels, and (2) we expanded knowledge regarding the positive relationship between component commonality and the GT philosophy.The main contribution of this paper is the development of an efficient model for reducing safety stocks levels in modular production systems by linking component commonality with GT philosophy, thus creating a factor of substitution for components with great physical similarities. The factor of substitution constitutes an extension to the basic theory of quantitative models used for determining safety stock levels in organizations employing modular production. The model is a good option for companies whose holding costs are bigger than their shortage costs.It is important to mention that, traditionally, researchers have developed mathematical models to try to determine the optimal safety stock levels for different strategies of production. They assumed that the demand that exists within the elapsed time period between when an order is made and when it is received follows a normal distribution. However, some studies have shown that the normal distribution is not the best representation of the demand behavior during a waiting time, but for ease of operation, a model that assumes a normal distribution represents the basis for other applications with higher requirements.The proposed model is compared with the traditional method through a simulation study for two modular products systems with different degrees of commonality, where, for each component in both systems, we calculated the safety stocks, reorder points, initial and final inventories, and the usability of components. The result shows a reduction in safety stock levels allowing a high CSL.For future research, the model can be tested using components belonging to the same part family and these can be substituted between them; also the model can be adjusted to different demand distributions to evaluate the performance. It is possible to supplement it with a study assessing the impact on total costs under different scenarios and different operational strategies. Finally, the model can be extended to products of more than three levels of complexity.



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*Source: 9510201-2016-08-18.xml*