# Optimal Tolerance Design and Optimization for a Pharmaceutical Quality Characteristic

Authors: Seong Hoon Jeong; Pauline Kongsuwan; Nguyen Khoa Viet Truong; Sangmun Shin
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A number of pharmaceutical quality characteristics are destructive or too costly to inspect. However, most quality improvement tools developed in the pharmaceutical research community typically assume that quality characteristics are nondestructive. This paper proposes a new design system for quality improvement by incorporating the concept of surrogate variables with the concepts of robust design (RD) and tolerance design (TD). The proposed robust-tolerance design paradigm determines the optimal factor setting and specification limits simultaneously, thereby improving quality of pharmaceutical products. In addition, the proposed methodology can provide the optimal tolerance as a mathematical closed-form solution. Finally, a numerical example and its associated sensitivity analysis for a pharmaceutical case are conducted for verification purposes. Based on the numerical example results, the proposed approach could provide robust factor settings with significant tradeoffs between quality and cost.


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## Body

## 1. Introduction

Effectively considering product quality in the early stage of product/process design could reduce production cost, which is one of the key issues in quality engineering. It has been more than twenty years that enterprises integrate product quality into their business strategies. In off-line quality improvement inspired by Taguchi [1, 2], there are three major phases as follows: (i)
system design: a design phase for an activity of proceeding and functional prototyping from an identified set of requirements,(ii)
robust design (RD): a design phase for determining the optimal factor setting which minimizes product/process variability and bias,(iii)
tolerance design (TD): a design phase for determining optimal specification limits in order to improve product quality by reducing the variability of quality characteristics.RD directly reflects a design for an optimal factor setting [3] with the aims of minimizing process bias and variability of a quality characteristic. Because of its practicability in reducing inherent uncertainty associated with design factors and system performance, the applications of RD techniques have resulted in significant improvements of product quality, manufacturability, and reliability at low cost. Even though the ad hoc robust design methods suggested by Taguchi remain controversial due to various mathematical flaws, there is little disagreement among researchers and practitioners about his basic philosophy. The controversy surrounding Taguchi’s assumptions, experimental design, and statistical analysis has been well addressed by Box [4], Box et al. [5], León et al. [6], Nair [7], and Tsui [8]. Consequently, researchers have closely examined alternative approaches using well-established statistical and optimization tools. As an alternative for modeling process relationships, Vining and Myers [9] introduced a dual response approach based on response surface methodology (RSM) by separately estimating the response functions of process mean and variance, thereby achieving the primary goal of robust design by minimizing the process variance while adjusting the process mean at the target. Many extensions of the dual response approach are reported in the literature, such as in [3, 10–16].As advanced types of automated inspection equipment become an integral part of modern manufacturing systems, the implementation of a tolerance on every product item has increased in recent years and become an attractive means of manufacturing high quality products. Automated equipment can perform rigorous inspection procedures and provide a consistent performance. When the nature of a performance variable is destructive, a surrogate variable, which is correlated to the performance variable, can be used effectively for the TD optimization under 100% inspections in many industrial applications. K. Tang and J. Tang [17] provided an excellent discussion of the overall concept for determining the optimal product specification limits under various screening inspection environments including the Deming’s all-or-none rules, Taguchi’s tolerance design, economic models for correlated variables, burn-in, and group testing.Pharmaceutical applications of RD and TD during granulation process, tablet pressing process, and coating process require a careful implementation because the primary characteristics of interest are often destructive. The primary purpose of this paper is to propose a new design modeling and optimization framework for the optimal factor setting and specification limits by incorporating the concept of surrogate variables with RD and TD. An overview of the proposed robust-tolerance design approach is presented in Figure1.Figure 1
An overview of the proposed robust-tolerance design approach.


## 2. Development of a Robust Design Method for Destructive Characteristics

### 2.1. Modification of Response Surface Methodology

Researchers have sought to combine Taguchi’s robust design principles with conventional response surface methodology (RSM) to model the response directly as a function of design variables. a second-order polynomial model is adequate to accommodate the curvature of the process mean and variance functions, as evidenced by [3, 9, 11, 13, 14, 16–19]. When the nature of a quality characteristic is destructive, a surrogate variable which is correlated to the quality characteristic is applied in many industrial applications. The response surface model associates control factors x with the correlation between destructive variables y and surrogate variables t. y and t are treated as random variables whose means depend upon the fixed values of x. Based on the dual response approach [9], the following models are recommended for destructive and surrogate variables:
(1)y(x)=α0+a1x+a2x2+εy,t(x)=b0+b1x+b2x2+εt,
where εy and εt are normal random variables with mean zero and variances σyε2 and σtε2, respectively; and α0, a1, a2, b0, b1, and b2 are model parameters. Based on the experimental results, the relationships between destructive quality characteristics and surrogate variables can be found as functional forms. Expanding the dual response approach, the estimated process mean functions of the performance variable μ^y, and that of the surrogate variable μ^t, the estimated variance function of the performance variable σ^y2 and that of the surrogate variable σ^t2, and the estimated correlation coefficient function between the performance variable and the surrogate variable ρ^yt are given as follows:
(2)μ^y(x)=αy0+αy1x+αy2x2,μ^t(x)=αt0+αt1x+αt2x2,σ^y2(x)=βy0+βy1x+βy1x2,σ^t2(x)=βt0+βt1x+βt1x2,ρ^yt(x)=γ0+γ1x+γ2x2,
where x is a design factor (or a control factor); αy0, αy1, and αy2 are estimated regression coefficients of μ^y; αt0, αt1, and αt2 are estimated regression coefficients of μ^t; βy0, βy1, and βy2 are estimated regression coefficients of σ^y2; βt0, βt1, and βt2 are estimated regression coefficients of σ^t2; and γ0, γ1, and γ2 are estimated regression coefficients ofρ^yt. To determine the regression coefficients, ordinary methods, such as least square method, could be used. An experimental format for destructive quality characteristics and their associated surrogate variables is shown in Table 1. y- and  sy2 are the sample mean and the sample variance of destructive quality characteristic y; t- and st2 are the sample mean and the sample variance of surrogate variable t; and ρyt is the sample correlation coefficient associated with the destructive quality characteristic y and the surrogate variable t.Table 1
Experimental format.
Run No.
Control factors (x)
Quality characteristics (y)
Surrogate variables (t)
x
1
x
2
⋯
x
p
y
1
y
2
⋯
y
q
μ
y
s
y
2
t
1
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2
⋯
t
q
μ
t
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2
ρ
y
t
1
x
11
x
21
⋯
x
p
1
y
11
y
21
⋯
y
q
1
μ
y
1
s
y
1
2
t
11
t
21
⋯
t
q
1
μ
t
1
s
t
1
2
ρ
y
t
1
2
x
12
x
22
⋯
x
p
2
y
12
y
22
⋯
y
q
2
μ
y
2
s
y
2
2
t
12
t
22
⋯
t
q
2
μ
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2
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2
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2
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x
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n


### 2.2. Variability Reduction of Destructive Characteristics

From the experimental data, the functions of process meanμ^(x) and variance σ^(x) are separately estimated by using the dual response approach. This model implies that the process mean is adjusted to the target τy first and then the variability is minimized. Because the primarily role of the RD approach in this paper is to reduce the variability of destructive quality characteristics, these destructive quality characteristics are considered in the RD stage, and their associated surrogate variables are considered in the TD stage for cost reduction. The RD optimization model associated with destructive quality characteristics y is formulated as follows: (3)Minimizeσ^y2(x)Subjecttoμy(x)=τyx∈Ω,
where τy is a process target of destructive qualitycharacteristic y,  Ωis{x∈S:gi(x)≤0,i=1,2,…,n},S is a convex set inRn, and gi(x)  is the ith constraint associated with the destructive quality characteristic.


## 2.1. Modification of Response Surface Methodology

Researchers have sought to combine Taguchi’s robust design principles with conventional response surface methodology (RSM) to model the response directly as a function of design variables. a second-order polynomial model is adequate to accommodate the curvature of the process mean and variance functions, as evidenced by [3, 9, 11, 13, 14, 16–19]. When the nature of a quality characteristic is destructive, a surrogate variable which is correlated to the quality characteristic is applied in many industrial applications. The response surface model associates control factors x with the correlation between destructive variables y and surrogate variables t. y and t are treated as random variables whose means depend upon the fixed values of x. Based on the dual response approach [9], the following models are recommended for destructive and surrogate variables:
(1)y(x)=α0+a1x+a2x2+εy,t(x)=b0+b1x+b2x2+εt,
where εy and εt are normal random variables with mean zero and variances σyε2 and σtε2, respectively; and α0, a1, a2, b0, b1, and b2 are model parameters. Based on the experimental results, the relationships between destructive quality characteristics and surrogate variables can be found as functional forms. Expanding the dual response approach, the estimated process mean functions of the performance variable μ^y, and that of the surrogate variable μ^t, the estimated variance function of the performance variable σ^y2 and that of the surrogate variable σ^t2, and the estimated correlation coefficient function between the performance variable and the surrogate variable ρ^yt are given as follows:
(2)μ^y(x)=αy0+αy1x+αy2x2,μ^t(x)=αt0+αt1x+αt2x2,σ^y2(x)=βy0+βy1x+βy1x2,σ^t2(x)=βt0+βt1x+βt1x2,ρ^yt(x)=γ0+γ1x+γ2x2,
where x is a design factor (or a control factor); αy0, αy1, and αy2 are estimated regression coefficients of μ^y; αt0, αt1, and αt2 are estimated regression coefficients of μ^t; βy0, βy1, and βy2 are estimated regression coefficients of σ^y2; βt0, βt1, and βt2 are estimated regression coefficients of σ^t2; and γ0, γ1, and γ2 are estimated regression coefficients ofρ^yt. To determine the regression coefficients, ordinary methods, such as least square method, could be used. An experimental format for destructive quality characteristics and their associated surrogate variables is shown in Table 1. y- and  sy2 are the sample mean and the sample variance of destructive quality characteristic y; t- and st2 are the sample mean and the sample variance of surrogate variable t; and ρyt is the sample correlation coefficient associated with the destructive quality characteristic y and the surrogate variable t.Table 1
Experimental format.
Run No.
Control factors (x)
Quality characteristics (y)
Surrogate variables (t)
x
1
x
2
⋯
x
p
y
1
y
2
⋯
y
q
μ
y
s
y
2
t
1
t
2
⋯
t
q
μ
t
s
t
2
ρ
y
t
1
x
11
x
21
⋯
x
p
1
y
11
y
21
⋯
y
q
1
μ
y
1
s
y
1
2
t
11
t
21
⋯
t
q
1
μ
t
1
s
t
1
2
ρ
y
t
1
2
x
12
x
22
⋯
x
p
2
y
12
y
22
⋯
y
q
2
μ
y
2
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y
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## 2.2. Variability Reduction of Destructive Characteristics

From the experimental data, the functions of process meanμ^(x) and variance σ^(x) are separately estimated by using the dual response approach. This model implies that the process mean is adjusted to the target τy first and then the variability is minimized. Because the primarily role of the RD approach in this paper is to reduce the variability of destructive quality characteristics, these destructive quality characteristics are considered in the RD stage, and their associated surrogate variables are considered in the TD stage for cost reduction. The RD optimization model associated with destructive quality characteristics y is formulated as follows: (3)Minimizeσ^y2(x)Subjecttoμy(x)=τyx∈Ω,
where τy is a process target of destructive qualitycharacteristic y,  Ωis{x∈S:gi(x)≤0,i=1,2,…,n},S is a convex set inRn, and gi(x)  is the ith constraint associated with the destructive quality characteristic.


## 3. Development of an Economic Tolerance Design Method for Destructive Characteristics

Functional performance and cost are two factors primarily affecting the design of tolerances [20]. Thus, the determination of optimal tolerance involves a tradeoff between the level of quality (based on functional performance) and costs (associated with the tolerance). In manufacturing processes, the following tolerances could be specified on quality characteristics:(i)
a tight tolerance which causes slow processing rates and high manufacturing costs due to additional manufacturing operations, additional care on part of operators, and need of expensive measuring and processing equipments,(ii)
a loose tolerance which causes low manufacturing costs while decreasing product/process quality.In order to facilitate the economic tradeoff, researchers typically express quality in monetary terms using quality loss functions. As a result, proper implementation of the TD optimization under 100% inspections might lead to a substantial cost savings in scrap and rework costs and provide consistently high product quality throughout the manufacturing process. Ultimately, the selection of specification limits for a product is particularly important, since it directly affects the process defective rate, rejection cost, and loss to the customer due to the variability of product performance.TD involves the problem of determining optimal specification limits from the viewpoint of cost reduction and functional performance. For example, Chase et al. [21], Kim and Cho [22], Speckhart [23], and Spotts [24] considered the reduction of manufacturing cost in tolerance allocation problems whereas Fathi [25], Kim and Cho [26], and Phillips and Cho [27] studied the issue of TD from the viewpoint of functional performance and expressing it in monetary terms by using Taguchi quality loss concept. In an integrated study considering the effect of both cost reduction and functional performance, Tang [28] developed an economic model for selecting the most profitable tolerance in situations where inspection cost is a linear function. When two viewpoints, such as cost and functional performance, are integrated, a tradeoff is often necessary. Along this line, Jeang [29] considered the simultaneous optimization of manufacturing cost, rejection cost, and quality loss by using a process capability index to establish a relationship between tolerance and standard deviation. When historical data of losses are available, a regression analysis could be applied, as exemplified by Phillips and Cho [27], who studied the minimization of the quality loss and the rejection cost. They developed optimization models using the first-order and second-order empirical loss functions. Wang and Liang [30] studied a tolerance optimization problem in the context of machining processes, such as milling, turning, drilling, reaming, boring, and grinding. Recently, Shin and Cho [31] studied two separate process parameters, such as process mean and variability, in the biobjective framework. Peng et al. [32] and Prabhaharan et al. [33] considered optimal process tolerances and stack-up conditions for mechanical assemblies as a combinatorial optimization problem. Destructive characteristics are often required in many situations, for instance, glass industry. Thus, an implementation of surrogate variables might be more practical. For surrogate variable-based TD optimization procedures, readers could refer to [17, 34–46].


### 3.1. Tolerance Design and Surrogate Variables

If the nature of major quality characteristics involves destructive testing, it is impossible to inspect all major quality characteristics themselves. In this case, it is more economical to use surrogate variables highly correlated with the major quality characteristics and relatively inexpensive to measure as screening variables. Leth(y,t) denoting the joint distribution of major quality characteristic y and surrogate variable t. Then, h(y,t) assumed to be a bivariate normal distribution with means μy and μt, variances σy2 and σt2, and correlation coefficient ρyt(=σyt/σyσt). In the RD stage, the optimal setting of control factors is determined by setting all the major characteristics to their target values. Therefore, the values of these parameters μy, μt, σy2, σt2, and ρyt are already known [47]. Note that h(y,t)=g(y∣t)f(t) where f(t) is the marginal density function of T and g(y∣t) is the conditional density function of y, given T=t. It can be shown that f(t) is a normal density function with mean μt and variance σt2, and g(y∣t) is a normal density function with mean μ=μy+ρyt(σy/σt)(t-μt)  and variance σ2=σy2(1-ρyt2).TD determines the tolerance limit of surrogate variables by minimizing cost incurred by both customer and manufacturer sides [47]. In the side of customers, it has been assumed that if the performance of a product falls within a tolerance limit, the product is acceptable to customers and its quality loss is zero. However, it is more reasonable to assume that quality loss occurs when a product performance deviates from its target although it is in the tolerance limit. Within the tolerance limit, the expected quality loss is evaluated by integrating a quality loss function and a probability density function over a region of interest. In this paper, the following quadratic quality loss function is applied:
(4)QualityLoss(y)=θ(y-τ)2,
where θ is a positive loss coefficient. Let LSL and USL be lower and upper bounds of a tolerance limit for surrogate variable T, respectively. By integrating the loss function defined in (4) and the product performance h(y,t), the expected quality loss within a tolerance limit can be formulated as follows:
(5)E[QualityLoss(y)]=∫LSLUSL∫-∞∞θ(y-τ)2h(y,t)dydt=∫LSLUSLθ(σ2+(μ-τ)2)f(t)dt.
From the definition of g(y∣t), after replacing μ with μy+ρyt(σy/σt)(t-μt) and σ with σy2(1-ρyt2), (5) becomes
(6)E[QualityLoss(y)]=θ∫LSLUSL([μy+ρyt2σy(t-μt)σt-τ]2σy2(1-ρyt2)+[μy+ρyt2σy(t-μt)σt-τ]2)f(t)dt.The rejection cost includes unit rework costCrework and unit scrap cost Cscrab, which incur when a product performance falls above the upper specification limit and below the lower specification limit, respectively. The expected rework cost E[ReworkCost] and the expected scrap cost E[ScrapCost] are given as follows:
(7)E[ReworkCost]=Crework(P[USL≤T])=Crework∫USL∞f(t)dt,E[ScrapCost]=Cscrap(P[T≤LSL])=Cscrap∫-∞LSLf(t)dt.
To achieve tight tolerance, additional manufacturing operations and trainings, which probably decrease manufacturing processing rates and increase manufacturing cost, are usually required. The manufacturing cost contributes to a significant portion of the product unit cost, and its exclusion from the TD optimization model might result in a suboptimal tolerance. A tolerance range is defined in terms of δ, μt, and σt as follows:
(8)ToleranceRange=USL-LSL=(μt+δσt)-(μt-δσt)=2δσt.
For simplicity, the manufacturing cost is assumed to have a linear relationship with the tolerance range as described below:
(9)MfgCost=a0+a1ToleranceRange+εmfg,
where εmfg represents the least-squares regression error. This linear manufacturing cost-tolerance model is also implemented in [48, 49]. The expected manufacturing cost could then be written as:
(10)E[MfgCost]=a0+a1ToleranceRange.
Substituting Tolerance Range = 2δσt in (10), the expected manufacturing cost becomes
(11)E[MfgCost]=a0+2a1δσt.


### 3.2. The Proposed Optimization Model for Tolerance Design

The objective of economic TD is to find the tolerance limit which minimizes the expected total cost including expected quality loss, expected rework cost, expected scrap cost, and expected manufacturing cost. Thus, the expected total cost is mathematically expressed as(12)E[TotalCost]=E[QualityLoss(y)]+E[ReworkCost]+E[ScrapCost]+E[MfgCost].
By replacing (6), (7), and (11) into (12), the expected total cost is reformulated as:
(13)E[TotalCost]=θ∫LSLUSL(+[μy+ρyt2σy(t-μt)σt-τ]2)σy2(1-ρyt2)+[μy+ρyt2σy(t-μt)σt-τ]2)f(t)dt+Crework∫USL∞f(t)dt+Cscrap∫-∞LSLf(t)dt+a0+2a1δσt.
To simplify (13), the marginal density function of t is converted from normal distribution function f(t) to standard normal distribution ϕ(z) by using
(14)z=(t-μt)σt.
Thus, (13) could be rewritten as
(15)E[TotalCost]=θ∫-δδ(+(μy+ρytσyz-τ)2)σy2(1-ρyt2)+(μy+ρytσyz-τ)2)ϕ(z)dz+Crework∫δ∞ϕ(z)dz+Cscrap∫-∞-δϕ(z)dz+a0+2a1σtδ,
where USL=μy+δσy, LSL=μy-δσy, and δ>0. Based on the dual response approach used in the RD stage, terms μy and τ are removed from (15). Also, after applying the following properties of normal probability density functions:
(16)∫-∞rϕ(z)dz=Φ(r),∫-∞rzϕ(z)dz=-ϕ(r),∫-∞rz2ϕ(z)dz=Φ(r)-rϕ(r),
the expected total cost becomes
(17)E[TotalCost]=2a1σtδ-2ρyt2σy2θδϕ(δ)+{2σy2(1-ρyt2)θ+2ρyt2σy2θ-Crework-Cscrap}Φ(δ)+Crework+Cscrap-θσy2+a0.


### 3.3. Determination of the Closed-Form Solution for Tolerance Optimization

To investigate the optimal solution of tolerance design, the first derivative with respect toδ is as follows:
(18)∂E[TotalCost]∂δ=2a1σt+ϕ(δ)×[2σy2(1-ρyt2)θ-Crework-Cscrap]+2ρyt2σy2θδ2ϕ(δ).
Equating ∂E[TotalCost]/∂δ to zero and substituting ϕ(δ) with e(-1/2)δ2/2π, (18) becomes
(19)2a1σt+(2σy2(1-ρyt2)θ-Crework-Cscrap+2ρyt2σy2θδ2)×(e-δ2/22π)=0.
Since it is nearly impossible to obtain a closed-form solution from this complex equation, Lambert Wfunction is introduced to effectively obtain such a solution for δ as discussed in Appendix B. According to the property of Lambert W function ([20, 50]) such that
(20)ξ1(δ2+ξ2)eξ3δ2=ξ4,
where ξ1, ξ2, ξ3, and ξ4 are not functions of δ, the optimal tolerance δ* can be derived from the following relationship:
(21)δ=±LambertW(ξ3ξ4eξ2ξ3/ξ1)ξ3-ξ2.
According to (20), (19) after formatting becomes
(22)(δ2+2σy2(1-ρyt2)θ-Crework-Cscrap2ρyt2σy2θ)e-δ2/2=-a1σt2πρyt2σy2θ.ξ1, ξ2,ξ3, and ξ4 in (21) can be replaced with 1, [2σy2(1-ρyt2)θ-Crework-Cscrap]/2ρyt2σy2θ,  −1/2, and -a1σt2π/ρyt2σy2θ, respectively. Then, the closed-form solution of δ* is formulated and provided as follows:
(23)δ*=±(+(Crework+Cscrap-2σy2(1-ρyt2)θ2ρyt2σy2θ))1l2-2LambertW(a1σ12πe((Crework+Cscrap-2σy2(1-ρyt2)θ)/(4ρyt2σy2θ))2ρyt2σy2θ)+(Crework+Cscrap-2σy2(1-ρyt2)θ2ρyt2σy2θ)(+(Crework+Cscrap-2σy2(1-ρyt2)θ2ρyt2σy2θ))1l2-2LambertW)1/2,
where σy2,σt2,ρyt,θ,   Crework and Cscrap; are the optimal variance of the destructive quality characteristic y, the optimal variance of the surrogate variable t, the correlation coefficient associated with the destructive quality characteristic yand the surrogate variable t, the positive loss coefficient, unit rework cost, and unit scrap cost, respectively. Although the closed-form solution shown in (23) consists of the Lambert W function, its computation is remarkably simple. After δ* is obtained, the optimal LSL and USL can be derived from μt*-δ*σt* and μt*+δ*σt*, respectively.


### 3.4. Determination of the Conditions for Convexity

To verify the validity of the optimal solution, the second derivative is computed and the conditions for obtaining the minimum value ofE[Total Cost] are investigated. The second derivative of E[Total Cost] with respect to δ is as follows:
(24)∂2E[TotalCost]∂δ2=δϕ(δ)×[Crework+Cscrap-2σy2θ×(1-3ρyt2+ρyt2δ2)].
The value of E[Total Cost]at the stationary points will be minimum if the second derivative is greater than or equal to zero. Therefore, the interval for a minimum value of δ could be written as
(25)0≤δ≤Crework+Cscrap-2σy2θ2ρyt2σy2θ+3.
In order to avoid imaginary values, the following condition needs to be satisfied:
(26)Crework+Cscrap-2σy2θ2ρyt2σy2θ≥-3.
If both conditions in (25) and (26) are satisfied, the closed-form solution of δ* defined in (23) represents the global minimum of E[TotalCost]. The overall framework of TD optimization and its associated key elements are provided as follows.Given Parameters
f(t): the normal probability density function of surrogate variable t,
τ: the target of the process mean of destructive quality characteristic y,
μ
y


: the optimal process mean of destructive quality characteristic y, derived from the RD stage,
σ
y
2: the optimal variance of destructive quality characteristic y, derived from the RD stage,
μ
t: the optimal process mean of surrogate variable t, derived from the RD stage,
σ
t
2: the optimal variance of surrogate variable t, derived from the RD stage,
ρ
y
t: the optimal correlation coefficient between destructive quality characteristic y and surrogate variable t, derived from the RD stage.System Parameters
(27)
E
[
Total
Cost
]
=
E
[
Quality
Loss
]
+
E
[
Mfg
Cost
]
+
E
[
Rework
Cost
]
+
E
[
Scrap
Cost
]
,
E
[
Quality
Loss
(
y
)
]
=
θ
∫
LSL
USL
(
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
+
ρ
y
t
2
σ
y
(
t
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μ
t
)
σ
t
-
τ
{
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
f
μ
y
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
σ
y
2
(
1
-
ρ
y
t
2
)
+
{
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
f
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
f
μ
y
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
f
(
t
)
d
t
,
E
[
Mfg
Cost
]
=
a
0
+
2
a
1
δ
σ
t
,
E
[
Rework
Cost
]
=
C
rework
∫
USL
∞
f
(
t
)
d
t
,
E
[
Scrap
Cost
]
=
C
scrap
∫
-
∞
LSL
f
(
t
)
d
t
.Find
δ
*
: the optimal tolerance, which leads to the derivation of USL and LSL as follows:
(28)USL=μt+δ*σt,LSL=μt-δ*σt.Minimize
(29)
E
[
Total
Cost
]
=
2
a
1
σ
t
δ
-
2
ρ
y
t
2
σ
y
2
θ
δ
ϕ
(
δ
)
+
{
2
σ
y
2
(
1
-
ρ
y
t
2
)
θ
+
2
ρ
y
t
2
σ
y
2
θ
-
C
rework
-
C
scrap
}
Φ
(
δ
)
+
C
rework
+
C
scrap
-
θ
σ
y
2
+
a
0
.Closed-Form Solution
(30)
δ
*
=
±
(
(
C
rework
+
C
scrap
-
2
σ
y
2
(
1
-
ρ
y
t
2
)
θ
2
ρ
y
t
2
σ
y
2
θ
)
)
-
2
Lambert
W
×
(
a
1
σ
1
2
π
e
(
(
C
rework
+
C
scrap
-
2
σ
y
2
(
1
-
ρ
y
t
2
)
θ
)
/
4
ρ
y
t
2
σ
y
2
θ
)
2
ρ
y
t
2
σ
y
2
θ
)
+
(
C
rework
+
C
scrap
-
2
σ
y
2
(
1
-
ρ
y
t
2
)
θ
2
ρ
y
t
2
σ
y
2
θ
)
)
1
/
2


## 3.1. Tolerance Design and Surrogate Variables

If the nature of major quality characteristics involves destructive testing, it is impossible to inspect all major quality characteristics themselves. In this case, it is more economical to use surrogate variables highly correlated with the major quality characteristics and relatively inexpensive to measure as screening variables. Leth(y,t) denoting the joint distribution of major quality characteristic y and surrogate variable t. Then, h(y,t) assumed to be a bivariate normal distribution with means μy and μt, variances σy2 and σt2, and correlation coefficient ρyt(=σyt/σyσt). In the RD stage, the optimal setting of control factors is determined by setting all the major characteristics to their target values. Therefore, the values of these parameters μy, μt, σy2, σt2, and ρyt are already known [47]. Note that h(y,t)=g(y∣t)f(t) where f(t) is the marginal density function of T and g(y∣t) is the conditional density function of y, given T=t. It can be shown that f(t) is a normal density function with mean μt and variance σt2, and g(y∣t) is a normal density function with mean μ=μy+ρyt(σy/σt)(t-μt)  and variance σ2=σy2(1-ρyt2).TD determines the tolerance limit of surrogate variables by minimizing cost incurred by both customer and manufacturer sides [47]. In the side of customers, it has been assumed that if the performance of a product falls within a tolerance limit, the product is acceptable to customers and its quality loss is zero. However, it is more reasonable to assume that quality loss occurs when a product performance deviates from its target although it is in the tolerance limit. Within the tolerance limit, the expected quality loss is evaluated by integrating a quality loss function and a probability density function over a region of interest. In this paper, the following quadratic quality loss function is applied:
(4)QualityLoss(y)=θ(y-τ)2,
where θ is a positive loss coefficient. Let LSL and USL be lower and upper bounds of a tolerance limit for surrogate variable T, respectively. By integrating the loss function defined in (4) and the product performance h(y,t), the expected quality loss within a tolerance limit can be formulated as follows:
(5)E[QualityLoss(y)]=∫LSLUSL∫-∞∞θ(y-τ)2h(y,t)dydt=∫LSLUSLθ(σ2+(μ-τ)2)f(t)dt.
From the definition of g(y∣t), after replacing μ with μy+ρyt(σy/σt)(t-μt) and σ with σy2(1-ρyt2), (5) becomes
(6)E[QualityLoss(y)]=θ∫LSLUSL([μy+ρyt2σy(t-μt)σt-τ]2σy2(1-ρyt2)+[μy+ρyt2σy(t-μt)σt-τ]2)f(t)dt.The rejection cost includes unit rework costCrework and unit scrap cost Cscrab, which incur when a product performance falls above the upper specification limit and below the lower specification limit, respectively. The expected rework cost E[ReworkCost] and the expected scrap cost E[ScrapCost] are given as follows:
(7)E[ReworkCost]=Crework(P[USL≤T])=Crework∫USL∞f(t)dt,E[ScrapCost]=Cscrap(P[T≤LSL])=Cscrap∫-∞LSLf(t)dt.
To achieve tight tolerance, additional manufacturing operations and trainings, which probably decrease manufacturing processing rates and increase manufacturing cost, are usually required. The manufacturing cost contributes to a significant portion of the product unit cost, and its exclusion from the TD optimization model might result in a suboptimal tolerance. A tolerance range is defined in terms of δ, μt, and σt as follows:
(8)ToleranceRange=USL-LSL=(μt+δσt)-(μt-δσt)=2δσt.
For simplicity, the manufacturing cost is assumed to have a linear relationship with the tolerance range as described below:
(9)MfgCost=a0+a1ToleranceRange+εmfg,
where εmfg represents the least-squares regression error. This linear manufacturing cost-tolerance model is also implemented in [48, 49]. The expected manufacturing cost could then be written as:
(10)E[MfgCost]=a0+a1ToleranceRange.
Substituting Tolerance Range = 2δσt in (10), the expected manufacturing cost becomes
(11)E[MfgCost]=a0+2a1δσt.


## 3.2. The Proposed Optimization Model for Tolerance Design

The objective of economic TD is to find the tolerance limit which minimizes the expected total cost including expected quality loss, expected rework cost, expected scrap cost, and expected manufacturing cost. Thus, the expected total cost is mathematically expressed as(12)E[TotalCost]=E[QualityLoss(y)]+E[ReworkCost]+E[ScrapCost]+E[MfgCost].
By replacing (6), (7), and (11) into (12), the expected total cost is reformulated as:
(13)E[TotalCost]=θ∫LSLUSL(+[μy+ρyt2σy(t-μt)σt-τ]2)σy2(1-ρyt2)+[μy+ρyt2σy(t-μt)σt-τ]2)f(t)dt+Crework∫USL∞f(t)dt+Cscrap∫-∞LSLf(t)dt+a0+2a1δσt.
To simplify (13), the marginal density function of t is converted from normal distribution function f(t) to standard normal distribution ϕ(z) by using
(14)z=(t-μt)σt.
Thus, (13) could be rewritten as
(15)E[TotalCost]=θ∫-δδ(+(μy+ρytσyz-τ)2)σy2(1-ρyt2)+(μy+ρytσyz-τ)2)ϕ(z)dz+Crework∫δ∞ϕ(z)dz+Cscrap∫-∞-δϕ(z)dz+a0+2a1σtδ,
where USL=μy+δσy, LSL=μy-δσy, and δ>0. Based on the dual response approach used in the RD stage, terms μy and τ are removed from (15). Also, after applying the following properties of normal probability density functions:
(16)∫-∞rϕ(z)dz=Φ(r),∫-∞rzϕ(z)dz=-ϕ(r),∫-∞rz2ϕ(z)dz=Φ(r)-rϕ(r),
the expected total cost becomes
(17)E[TotalCost]=2a1σtδ-2ρyt2σy2θδϕ(δ)+{2σy2(1-ρyt2)θ+2ρyt2σy2θ-Crework-Cscrap}Φ(δ)+Crework+Cscrap-θσy2+a0.


## 3.3. Determination of the Closed-Form Solution for Tolerance Optimization

To investigate the optimal solution of tolerance design, the first derivative with respect toδ is as follows:
(18)∂E[TotalCost]∂δ=2a1σt+ϕ(δ)×[2σy2(1-ρyt2)θ-Crework-Cscrap]+2ρyt2σy2θδ2ϕ(δ).
Equating ∂E[TotalCost]/∂δ to zero and substituting ϕ(δ) with e(-1/2)δ2/2π, (18) becomes
(19)2a1σt+(2σy2(1-ρyt2)θ-Crework-Cscrap+2ρyt2σy2θδ2)×(e-δ2/22π)=0.
Since it is nearly impossible to obtain a closed-form solution from this complex equation, Lambert Wfunction is introduced to effectively obtain such a solution for δ as discussed in Appendix B. According to the property of Lambert W function ([20, 50]) such that
(20)ξ1(δ2+ξ2)eξ3δ2=ξ4,
where ξ1, ξ2, ξ3, and ξ4 are not functions of δ, the optimal tolerance δ* can be derived from the following relationship:
(21)δ=±LambertW(ξ3ξ4eξ2ξ3/ξ1)ξ3-ξ2.
According to (20), (19) after formatting becomes
(22)(δ2+2σy2(1-ρyt2)θ-Crework-Cscrap2ρyt2σy2θ)e-δ2/2=-a1σt2πρyt2σy2θ.ξ1, ξ2,ξ3, and ξ4 in (21) can be replaced with 1, [2σy2(1-ρyt2)θ-Crework-Cscrap]/2ρyt2σy2θ,  −1/2, and -a1σt2π/ρyt2σy2θ, respectively. Then, the closed-form solution of δ* is formulated and provided as follows:
(23)δ*=±(+(Crework+Cscrap-2σy2(1-ρyt2)θ2ρyt2σy2θ))1l2-2LambertW(a1σ12πe((Crework+Cscrap-2σy2(1-ρyt2)θ)/(4ρyt2σy2θ))2ρyt2σy2θ)+(Crework+Cscrap-2σy2(1-ρyt2)θ2ρyt2σy2θ)(+(Crework+Cscrap-2σy2(1-ρyt2)θ2ρyt2σy2θ))1l2-2LambertW)1/2,
where σy2,σt2,ρyt,θ,   Crework and Cscrap; are the optimal variance of the destructive quality characteristic y, the optimal variance of the surrogate variable t, the correlation coefficient associated with the destructive quality characteristic yand the surrogate variable t, the positive loss coefficient, unit rework cost, and unit scrap cost, respectively. Although the closed-form solution shown in (23) consists of the Lambert W function, its computation is remarkably simple. After δ* is obtained, the optimal LSL and USL can be derived from μt*-δ*σt* and μt*+δ*σt*, respectively.


## 3.4. Determination of the Conditions for Convexity

To verify the validity of the optimal solution, the second derivative is computed and the conditions for obtaining the minimum value ofE[Total Cost] are investigated. The second derivative of E[Total Cost] with respect to δ is as follows:
(24)∂2E[TotalCost]∂δ2=δϕ(δ)×[Crework+Cscrap-2σy2θ×(1-3ρyt2+ρyt2δ2)].
The value of E[Total Cost]at the stationary points will be minimum if the second derivative is greater than or equal to zero. Therefore, the interval for a minimum value of δ could be written as
(25)0≤δ≤Crework+Cscrap-2σy2θ2ρyt2σy2θ+3.
In order to avoid imaginary values, the following condition needs to be satisfied:
(26)Crework+Cscrap-2σy2θ2ρyt2σy2θ≥-3.
If both conditions in (25) and (26) are satisfied, the closed-form solution of δ* defined in (23) represents the global minimum of E[TotalCost]. The overall framework of TD optimization and its associated key elements are provided as follows.Given Parameters
f(t): the normal probability density function of surrogate variable t,
τ: the target of the process mean of destructive quality characteristic y,
μ
y


: the optimal process mean of destructive quality characteristic y, derived from the RD stage,
σ
y
2: the optimal variance of destructive quality characteristic y, derived from the RD stage,
μ
t: the optimal process mean of surrogate variable t, derived from the RD stage,
σ
t
2: the optimal variance of surrogate variable t, derived from the RD stage,
ρ
y
t: the optimal correlation coefficient between destructive quality characteristic y and surrogate variable t, derived from the RD stage.System Parameters
(27)
E
[
Total
Cost
]
=
E
[
Quality
Loss
]
+
E
[
Mfg
Cost
]
+
E
[
Rework
Cost
]
+
E
[
Scrap
Cost
]
,
E
[
Quality
Loss
(
y
)
]
=
θ
∫
LSL
USL
(
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
f
μ
y
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
σ
y
2
(
1
-
ρ
y
t
2
)
+
{
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
f
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
τ
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
f
μ
y
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
}
2
)
(
σ
y
2
(
1
-
ρ
y
t
2
)
f
(
t
)
d
t
,
E
[
Mfg
Cost
]
=
a
0
+
2
a
1
δ
σ
t
,
E
[
Rework
Cost
]
=
C
rework
∫
USL
∞
f
(
t
)
d
t
,
E
[
Scrap
Cost
]
=
C
scrap
∫
-
∞
LSL
f
(
t
)
d
t
.Find
δ
*
: the optimal tolerance, which leads to the derivation of USL and LSL as follows:
(28)USL=μt+δ*σt,LSL=μt-δ*σt.Minimize
(29)
E
[
Total
Cost
]
=
2
a
1
σ
t
δ
-
2
ρ
y
t
2
σ
y
2
θ
δ
ϕ
(
δ
)
+
{
2
σ
y
2
(
1
-
ρ
y
t
2
)
θ
+
2
ρ
y
t
2
σ
y
2
θ
-
C
rework
-
C
scrap
}
Φ
(
δ
)
+
C
rework
+
C
scrap
-
θ
σ
y
2
+
a
0
.Closed-Form Solution
(30)
δ
*
=
±
(
(
C
rework
+
C
scrap
-
2
σ
y
2
(
1
-
ρ
y
t
2
)
θ
2
ρ
y
t
2
σ
y
2
θ
)
)
-
2
Lambert
W
×
(
a
1
σ
1
2
π
e
(
(
C
rework
+
C
scrap
-
2
σ
y
2
(
1
-
ρ
y
t
2
)
θ
)
/
4
ρ
y
t
2
σ
y
2
θ
)
2
ρ
y
t
2
σ
y
2
θ
)
+
(
C
rework
+
C
scrap
-
2
σ
y
2
(
1
-
ρ
y
t
2
)
θ
2
ρ
y
t
2
σ
y
2
θ
)
)
1
/
2


## 4. Numerical Example

To illustrate the application of the proposed approach, a problem of measuring tablet hardness in pharmaceutical manufacturing process is provided. Three factors, which are pressure (x1, MPa), humidity (x2, %), and compression speed (x3, mm/min), were in control. The hardness (y, kg), required at five [18, 51], was considered as a destructive quality characteristic because the tablet could be easily destroyed during the measurement. Process Analytical Technology (PAT) tools such as Raman, Near-Infrared (NIR), and Focused Beam Reflectance Measurement (FBRM) were used to understand and control the pharmaceutical process. In addition, Terahertz Pulsed Spectroscopy (TPS) and Terahertz Pulsed Imaging (TPI) techniques were used and terahertz refractive index (t) was introduced as a surrogate variable for the tablet hardness (y) [18, 51].From the overview of the proposed robust-tolerance approach, provided in Figure1, there are three main steps: process measurement, RD optimization, and TD optimization with Lambert W function. Central composite design (CCD) with three replications was applied to collect the experimental data of the destructive quality characteristic y and its associated surrogate variable t. Then, the sample mean of y(μy), the sample variance of y(sy2), the sample mean of t(μt), the sample variance of t(st2), and the correlation coefficient of y and t(ρyt) were calculated. Table 2 shows all the experimental data obtained from step one. The other two steps are explained in Sections 4.1 and 4.2, respectively.Table 2
Experimental data for a pharmaceutical manufacturing process.
Control factors (x)
Destructive quality characteristichardness (y)
Surrogate variableterahertz refractive index (t)
x
1
x
2
x
3
y
1
y
2
y
3
μ
y
s
y
2
t
1
t
2
t
3
μ
t
s
t
2
ρ
y
t
-
1
-
1
-
1
6.478
7.518
6.281
6.759
0.442
1.477
1.560
1.407
1.481
0.006
0.948
-
1
-
1
1
2.000
3.004
2.345
2.450
0.260
1.043
1.180
1.000
1.074
0.009
0.840
-
1
1
-
1
8.839
10.220
9.088
9.382
0.542
1.670
1.787
1.641
1.699
0.006
0.937
-
1
1
1
5.699
6.750
5.581
6.010
0.414
1.403
1.460
1.284
1.382
0.008
0.807
1
-
1
-
1
5.495
6.817
6.280
6.197
0.443
1.351
1.505
1.343
1.400
0.008
0.779
1
-
1
1
9.600
10.570
9.103
9.757
0.554
1.753
1.801
1.648
1.734
0.006
0.922
1
1
-
1
4.987
5.985
4.807
5.260
0.403
1.327
1.362
1.205
1.298
0.007
0.774
1
1
1
10.756
11.940
10.510
11.070
0.586
1.903
2.000
1.866
1.923
0.005
0.994
-
1.68
0
0
7.113
7.848
6.443
7.135
0.494
1.480
1.523
1.380
1.461
0.005
0.968
1.68
0
0
7.473
8.470
7.143
7.695
0.477
1.570
1.626
1.466
1.554
0.007
0.900
0
-
1.68
0
8.657
9.609
8.195
8.820
0.520
1.655
1.718
1.567
1.647
0.006
0.957
0
1.68
0
10.647
12.000
10.557
11.070
0.653
1.828
1.936
1.804
1.856
0.005
0.993
0
0
-
1.68
2.705
3.804
3.089
3.199
0.312
1.082
1.213
1.050
1.115
0.007
0.858
0
0
1.68
7.779
8.888
7.546
8.071
0.514
1.610
1.657
1.490
1.586
0.007
0.820
0
0
0
8.431
8.511
7.271
8.071
0.482
1.651
1.561
1.475
1.563
0.008
0.828
0
0
0
7.379
7.286
6.173
6.946
0.451
1.475
1.551
1.383
1.470
0.007
0.858
0
0
0
6.993
7.887
6.521
7.134
0.481
1.519
1.549
1.392
1.487
0.007
0.868
0
0
0
8.856
8.856
7.623
8.445
0.507
1.700
1.609
1.527
1.612
0.008
0.849
0
0
0
7.299
8.657
7.694
7.883
0.488
1.537
1.654
1.480
1.557
0.008
0.818
0
0
0
10.443
10.300
9.092
9.945
0.550
1.739
1.788
1.631
1.720
0.006
0.918


### 4.1. Stage I: Robust Design Optimization

By applying RSM to the experimental data shown in Table2, fitted functions of the mean and the variance of y, the mean and the variance of t, and the correlation coefficient of y and, t are obtained and respectively, provided as follows:
(31)μ^y=8.09+0.631x1+0.757x2+0.724x3-0.726x1x2+2.13x1x3+0.398x2x3-0.344x12+0.551x22-0.973x32;σ^y2=0.494+0.0219x1+0.0344x2+0.0237x3-0.0328x1x2+0.0755x1x3+0.0157x2x3-0.0111x12+0.0246x22-0.0367x32;μ^t=1.57+0.0641x1+0.0706x2+0.0752x3-0.0549x1x2+0.210x1x3+0.0476x2x3-0.0290x12+0.0573x22-0.0845x32;σ^t2=0.00731+0.000027x1-0.000343x2+0.000073x3-0.000125x1x2-0.00112x1x3-0.000125x2x3-0.000307x12-0.000484x22+0.000046x32,ρ^yt=0.858-0.0130x1+0.0061x2+0.0045x3+0.0139x1x2+0.0751x1x3+0.0069x2x3+0.0180x12+0.0325x22-0.0156x32.To demonstrate the statistical analysis, ANOVA for the five responses based on significant levelα at 0.05 was conducted as shown in Tables 3, 4, 5, 6, and 7. The result could be summarized as follows. For μ^y, two main effects (i.e. x2 and  x3), the interaction between x1 and x3, and the quadratic effect of x3 are significant; for σ^y2, x2, the interaction between x1 and x3, and the quadratic effect of x3 are significant; for μ^t, all the main effects (i.e. x1, x2, and x3), the interaction between x1 and x3, and the quadratic effect of x3 are significant; for σ^t2, only the interaction between x1 and x3 is significant; and for ρ^yt, the interaction between x1 and x3 and the quadratic effect of x2 are significant.Table 3
ANOVA of the mean ofy (μ^y).
Predictor
Coef
SE Coef
T
P
Constant
8.0877
0.4857
16.65
0.000
x
1
0.6315
0.3222
1.96
0.078
x
2
0.7573
0.3222
2.35
0.041
x
3
0.7236
0.3222
2.25
0.049
x
1
x
2
-
0.7259
0.4210
-
1.72
0.115
x
1
x
3
2.1314
0.4210
5.06
0.000
x
2
x
3
0.3984
0.4210
0.95
0.366
x
1
x
1
-
0.3435
0.3136
-
1.10
0.299
x
2
x
2
0.5507
0.3136
1.76
0.110
x
3
x
3
-
0.9727
0.3136
-
3.10
0.011
S
=
1.1908
R
2 = 85.5%
R
2 (adj) = 72.4%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
83.371
9.263
6.53
0.004
Residual error
10
14.18
1.418
Total
19
97.55Table 4
ANOVA of the variance ofy (σ^y2).
Predictor
Coef
SE Coef
T
P
Constant
0.4944
0.0225
21.92
0.000
x
1
0.0219
0.0149
1.46
0.174
x
2
0.0343
0.0149
2.30
0.044
x
3
0.0237
0.0149
1.58
0.144
x
1
x
2
-
0.0327
0.0195
-
1.67
0.125
x
1
x
3
0.0755
0.0195
3.86
0.003
x
2
x
3
0.0157
0.0195
0.81
0.439
x
1
x
1
-
0.0110
0.0145
-
0.76
0.465
x
2
x
2
0.0246
0.0145
1.69
0.122
x
3
x
3
-
0.0366
0.0145
-
2.52
0.030
S
=
0.0553138
R
2 = 79.6%
R
2 (adj) = 61.2%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
0.119074
0.01323
4.32
0.016
Residual error
10
0.030596
0.00306
Total
19
0.149671Table 5
ANOVA of the mean oft (μ^t).
Predictor
Coef
SE Coef
T
P
Constant
1.5693
0.0417
37.63
0.000
x
1
0.0641
0.0276
2.32
0.043
x
2
0.0706
0.0276
2.55
0.029
x
3
0.0752
0.0276
2.72
0.022
x
1
x
2
-
0.0548
0.0361
-
1.52
0.160
x
1
x
3
0.2103
0.0361
5.82
0.000
x
2
x
3
0.0476
0.0361
1.32
0.217
x
1
x
1
-
0.0289
0.0269
-
1.08
0.307
x
2
x
2
0.0572
0.0269
2.13
0.059
x
3
x
3
-
0.0844
0.0269
-
3.14
0.011
S
=
0.102261
R
2 = 88.1%
R
2 (adj) = 77.4%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
0.77375
0.08597
8.22
0.001
Residual error
10
0.10457
0.01046
Total
19
0.87833Table 6
ANOVA of the variance oft (σ^t2).
Predictor
Coef
SE Coef
T
P
Constant
0.0073
0.0003
19.34
0.000
x
1
0.0000
0.0002
0.11
0.917
x
2
-
0.0003
0.0002
-
1.37
0.201
x
3
0.0000
0.0002
0.29
0.776
x
1
x
2
-
0.0001
0.0003
-
0.38
0.711
x
1
x
3
-
0.0011
0.0003
-
3.43
0.006
x
2
x
3
-
0.0001
0.0003
-
0.38
0.711
x
1
x
1
-
0.0003
0.0002
-
1.26
0.237
x
2
x
2
-
0.0004
0.0002
-
1.98
0.076
x
3
x
3
0.0000
0.0002
0.19
0.853
S
=
0.000926452
R
2 = 65.9%
R
2 (adj) = 35.3%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
1.66169
E
-
05
1.84632
E
-
06
2.15
0.124
Residual error
10
8.58313
E
-
06
8.58313
E
-
07
Total
19
0.0000252Table 7
ANOVA of the correlation coefficient ofy and t (ρ^yt).
Predictor
Coef
SE Coef
T
P
Constant
0.8579
0.0167
51.14
0.000
x
1
-
0.0129
0.0111
-
1.17
0.270
x
2
0.0061
0.0111
0.55
0.595
x
3
0.0044
0.0111
0.40
0.696
x
1
x
2
0.0138
0.0145
0.95
0.363
x
1
x
3
0.0751
0.0145
5.17
0.000
x
2
x
3
0.0068
0.0145
0.47
0.647
x
1
x
1
0.0180
0.0108
1.66
0.127
x
2
x
2
0.0325
0.0108
3.00
0.013
x
3
x
3
-
0.0155
0.0108
-
1.44
0.181
S
=
0.0411335
R
2 = 81.5%
R
2 (adj) = 64.8%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
0.074522
0.00828
4.89
0.01
Residual error
10
0.01692
0.001692
Total
19
0.091441The variability of destructive quality characteristicy could be reduced by applying the RD model shown in (3) with τy=5. The optimal solutions (i.e. x1*= −0.3000, x2*= −0.6219, and  x3*= −1.6810) and then the optimal parameters associated with y and t (i.e. μy*= 5.0000, σy2*= 0.3792, μt*= 1.3061, σt2*= 0.0070, and ρyt*= 0.8684) were obtained and provided in Table 8. Figures 2, 3, and 4 present the graphical representations of all the fitted response functions (i.e. μ^y, σ^y2, μ^t, σ^t2, and ρ^yt) with respect to the optimal setting of each input factor (i.e. x1*, x2*, and x3*, resp.).Table 8
Optimal solutions from the RD stage.
Control factors (x)
Hardness (y)
Terahertz refractive index (t)
ρ
y
t
*
x
1
*
x
2
*
x
3
*
μ
y
*
σ
y
2
*
μ
t
*
σ
t
2
*
-
0.2999
-
0.6219
-
1.6810
5.0000
0.3792
1.3061
0.0070
0.8684Figure 2
Plot ofμ^y, σ^y2, μ^t, σ^t2, and ρ^yt with respect to x1*.Figure 3
Plot ofμ^y, σ^y2, μ^t, σ^t2, and ρ^yt with respect to x2*.Figure 4
Plot ofμ^y, σ^y2, μ^t, σ^t2, and ρ^yt with respect to x3*.


### 4.2. Stage II: Tolerance Design Optimization

After the variability was reduced in the RD stage and the optimal setting for the hardness (y) and the Terahertz Refractive Index (t) were obtained, the proposed economic TD framework was conducted with respect to t in order to find the optimal tolerance δ*, which could be easily found by the closed-form solution shown in (23) and then lead to the derivation of the optimal lower and upper specification limits (i.e., LSL* and USL*), respectively. When the inspection was implemented, the expected quality loss (E[Quality Loss (y)]) occurred as shown in (6) where the positive loss coefficient θ  was 250. The expected manufacturing cost (E[Mfg Cost]) presented in (11) was described as a polynomial model where a0 and a1 were 100 and −0.2, respectively. Also, when the product performance fell above the upper limit or below the lower limit of the production specification, the expected rework cost (E[Rework Cost]) and the expected scrap cost (E[Scrap Cost]) were calculated as, respectively, defined in (7) where both Crework and Cscrap were set to an arbitrary value such as 80. All the parameters, objectives, and optimal solutions involved in this TD stage are provided as follows.Given Conditions
f(t)  e-((t-μt)2/2σt2)/2πσt2
τ      5.0000
μ
y   5.0000
σ
y
2   0.3792
μ
t   1.3061
σ
t
2   0.0070
ρ
y
t   0.8684.System Parameters
(32)
E
[
Total
Cost
]
=
E
[
Quality
Loss
]
+
E
[
Mfg
Cost
]
+
E
[
Rework
Cost
]
+
E
[
Scrap
Cost
]
,
E
[
Quality
Loss
(
y
)
]
=
250
×
∫
LSL
USL
(
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
2
)
σ
y
2
(
1
-
ρ
y
t
2
)
+
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
2
(
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
2
)
σ
y
2
(
1
-
ρ
y
t
2
)
(
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
2
)
σ
y
2
(
1
-
ρ
y
t
2
)
)
×
f
(
t
)
d
t
,
E
[
Mfg
Cost
]
=
100
-
0.4
δ
σ
t
.
E
[
Rework
Cost
]
=
80
∫
USL
∞
f
(
t
)
d
t
,
E
[
Scrap
Cost
]
=
80
∫
-
∞
LSL
f
(
t
)
d
t
,Find
δ
*
: the optimal tolerance, which leads to the derivation of USL and LSL as follows:
(33)USL=μt+δ*σt,LSL=μt-δ*σt.Minimize
(34)
E
[
Total
Cost
]
=
-
0.4
σ
t
δ
-
500
ρ
y
t
2
σ
y
2
δ
φ
(
δ
)
+
{
500
σ
y
2
(
1
-
ρ
y
t
2
)
+
500
ρ
y
t
2
σ
y
2
-
160
}
×
Φ
(
δ
)
+
260
-
250
σ
y
2
.Closed-Form Solution
(35)
δ
*
=
±
(
+
(
160
-
500
σ
y
2
(
1
-
ρ
y
t
2
)
500
ρ
y
t
2
σ
y
2
)
)
-
2
Lambert
W
×
(
-
0.2
σ
1
2
π
e
(
(
160
-
500
σ
y
2
(
1
-
ρ
y
t
2
)
)
/
1000
ρ
y
t
2
σ
y
2
)
500
ρ
y
t
2
σ
y
2
)
+
(
160
-
500
σ
y
2
(
1
-
ρ
y
t
2
)
500
ρ
y
t
2
σ
y
2
)
)
1
/
2
=
0.8910
USL
* = 1.3806 and LSL* = 1.2317.The optimal toleranceδ* is 0.8910 and then USL* and LSL* are 1.3806 and 1.2317, respectively. Figure 5 illustrates the relationship between δ and all relevant costs which are expected total cost, expected manufacturing cost, cost incurred by the expected quality loss, and expected rejection cost including expected rework cost and expected scrap cost.Figure 5
Plots ofE[Total Cost], E[Mfg Cost], E[Quality Loss], and E[Rework Cost] + E[Scrap Cost] with respect to δ.In order to verify the optimality conditions ofδ*, the plots of the first and the second derivatives of the expected total cost function with respect to δ are provided in Figures 6 and 7, respectively. The former shows that, at δ*, the value of ∂E[TotalCost]/∂δ is zero; while the latter shows that, at δ*, the expected total cost is minimum because the value of ∂2E[TotalCost]/∂δ2 is greater than zero. Also, the value of δ* is between 0 and 1.6712, which is relevant to the interval defined in (25).Figure 6
Plot of∂E[TotalCost]/∂δ with respect to δ.Figure 7
Plot of∂2E[TotalCost]/∂δ2 with respect to δ.Three sensitivity analyses were conducted in order to determine the impact ofθ, Crework, and Cscrap on E[ReworkCost] around δ* A full data set of each sensitivity analysis is provided in Tables 9, 10, and 11, respectively. First is the effect of θ while Crework and Cscrap are held at 80. Figure 8 identifies that while the value of θ changes from 100 to 400 with an increment of 25, the values of E[ReworkCost], E[ScrapCost], E[MfgCost], and E[TotalCost] increase; and while the value of δ decreases, E[QualityLoss] has a concave pattern.Table 9
Effects ofθ on minimum costs (Crework=80 and Cscrap=80).
θ
δ
*
E [Rework Cost]
E [Scrap Cost]
E [Quality Loss]
E [Mfg Cost]
E [Total Cost]
100
1.5737
4.6227
4.6227
23.1282
99.9474
132.3210
125
1.3838
6.6567
6.6567
24.3612
99.9537
137.6283
150
1.2414
8.5787
8.5787
25.0187
99.9585
142.1345
175
1.1288
10.3601
10.3601
25.3359
99.9623
146.0184
200
1.0363
12.0022
12.0022
25.4404
99.9653
149.4102
225
0.9583
13.5167
13.5167
25.4044
99.9680
152.4057
250
0.8910
14.9177
14.9177
25.2706
99.9702
155.0762
275
0.8319
16.2194
16.2194
25.0646
99.9722
157.4755
300
0.7792
17.4346
17.4346
24.8024
99.9739
159.6456
325
0.7317
18.5747
18.5747
24.4940
99.9755
161.6189
350
0.6883
19.6496
19.6496
24.1454
99.9770
163.4216
375
0.6484
20.6679
20.6679
23.7604
99.9783
165.0745
400
0.6114
21.6372
21.6372
23.3408
99.9796
166.5948Table 10
Effects ofCrework on minimum costs (θ=250 and Cscrap=80).
C
rework
δ
*
E [Rework Cost]
E [Scrap Cost]
E [Quality Loss]
E [Mfg Cost]
E [Total Cost]
40
0.7169
9.4683
18.9365
18.2914
99.9760
146.6723
48
0.7550
10.8066
18.0109
19.7268
99.9748
148.5190
56
0.7912
12.0078
17.1541
21.1413
99.9735
150.2767
64
0.8258
13.0859
16.3573
22.5362
99.9724
151.9518
72
0.8590
14.0524
15.6138
23.9125
99.9713
153.5500
80
0.8910
14.9177
14.9177
25.2706
99.9702
155.0762
88
0.9219
15.6907
14.2643
26.6108
99.9692
156.5349
96
0.9517
16.3794
13.6495
27.9333
99.9682
157.9303
104
0.9807
16.9908
13.0699
29.2381
99.9672
159.2660
112
1.0088
17.5314
12.5224
30.5253
99.9663
160.5454
120
1.0362
18.0068
12.0045
31.7949
99.9653
161.7715
128
1.0629
18.4222
11.5139
33.0467
99.9645
162.9472
136
1.0889
18.7823
11.0484
34.2808
99.9636
164.0751
144
1.1143
19.0915
10.6064
35.4970
99.9627
165.1576
152
1.1391
19.3537
10.1861
36.6954
99.9619
166.1971
160
1.1634
19.5724
9.7862
37.8758
99.9611
167.1955Table 11
Effects ofCscrap on minimum costs (θ=250 and Crework=80).
C
scrap
δ
*
E [Rework Cost]
E [Scrap Cost]
E [Quality Loss]
E [Mfg Cost]
E [Total Cost]
40
0.7169
18.9365
9.4683
18.2914
99.9760
146.6723
48
0.7550
18.0109
10.8066
19.7268
99.9748
148.5190
56
0.7912
17.1541
12.0078
21.1413
99.9735
150.2767
64
0.8258
16.3573
13.0859
22.5362
99.9724
151.9518
72
0.8590
15.6138
14.0524
23.9125
99.9713
153.5500
80
0.8910
14.9177
14.9177
25.2706
99.9702
155.0762
88
0.9219
14.2643
15.6907
26.6108
99.9692
156.5349
96
0.9517
13.6495
16.3794
27.9333
99.9682
157.9303
104
0.9807
13.0699
16.9908
29.2381
99.9672
159.2660
112
1.0088
12.5224
17.5314
30.5253
99.9663
160.5454
120
1.0362
12.0045
18.0068
31.7949
99.9653
161.7715
128
1.0629
11.5139
18.4222
33.0467
99.9645
162.9472
136
1.0889
11.0484
18.7823
34.2808
99.9636
164.0751
144
1.1143
10.6064
19.0915
35.4970
99.9627
165.1576
152
1.1391
10.1861
19.3537
36.6954
99.9619
166.1971
160
1.1634
9.7862
19.5724
37.8758
99.9611
167.1955Figure 8
The effect ofθ on minimum costs (Crework= 80 and Cscrap= 80).Second is the effect ofCrework while θ and Cscrap are held at 250 and 80, respectively. Figure 9 demonstrates that while the value of Crework changes from 40 to 160 with an increment of 8, the values of δ*, E[ReworkCost], E[QualityLoss], and E[TotalCost] increase; and while the values of E[ScrapCost] and E[MfgCost] decrease, the rejection cost (which is a summation of E[ReworkCost] and E[ScrapCost]) has a concave pattern.Figure 9
The effect ofCrework on minimum costs (θ = 250 and Cscrap = 80).Last is the effect ofCscrap while θ and Crework are held at 250 and 80, respectively. Figure 10 represents that while the value of Cscrap changes from 40 to 160 with an increment of 8, the values of, E[ScrapCost], E[QualityLoss], and E[TotalCost] increase. Similar to the second sensitivity analysis, while the values of E[ReworkCost] and E[MfgCost] decrease, the rejection cost has a concave pattern.Figure 10
The effect ofCscrap  on minimum costs (θ = 250 and Crework = 80).


## 4.1. Stage I: Robust Design Optimization

By applying RSM to the experimental data shown in Table2, fitted functions of the mean and the variance of y, the mean and the variance of t, and the correlation coefficient of y and, t are obtained and respectively, provided as follows:
(31)μ^y=8.09+0.631x1+0.757x2+0.724x3-0.726x1x2+2.13x1x3+0.398x2x3-0.344x12+0.551x22-0.973x32;σ^y2=0.494+0.0219x1+0.0344x2+0.0237x3-0.0328x1x2+0.0755x1x3+0.0157x2x3-0.0111x12+0.0246x22-0.0367x32;μ^t=1.57+0.0641x1+0.0706x2+0.0752x3-0.0549x1x2+0.210x1x3+0.0476x2x3-0.0290x12+0.0573x22-0.0845x32;σ^t2=0.00731+0.000027x1-0.000343x2+0.000073x3-0.000125x1x2-0.00112x1x3-0.000125x2x3-0.000307x12-0.000484x22+0.000046x32,ρ^yt=0.858-0.0130x1+0.0061x2+0.0045x3+0.0139x1x2+0.0751x1x3+0.0069x2x3+0.0180x12+0.0325x22-0.0156x32.To demonstrate the statistical analysis, ANOVA for the five responses based on significant levelα at 0.05 was conducted as shown in Tables 3, 4, 5, 6, and 7. The result could be summarized as follows. For μ^y, two main effects (i.e. x2 and  x3), the interaction between x1 and x3, and the quadratic effect of x3 are significant; for σ^y2, x2, the interaction between x1 and x3, and the quadratic effect of x3 are significant; for μ^t, all the main effects (i.e. x1, x2, and x3), the interaction between x1 and x3, and the quadratic effect of x3 are significant; for σ^t2, only the interaction between x1 and x3 is significant; and for ρ^yt, the interaction between x1 and x3 and the quadratic effect of x2 are significant.Table 3
ANOVA of the mean ofy (μ^y).
Predictor
Coef
SE Coef
T
P
Constant
8.0877
0.4857
16.65
0.000
x
1
0.6315
0.3222
1.96
0.078
x
2
0.7573
0.3222
2.35
0.041
x
3
0.7236
0.3222
2.25
0.049
x
1
x
2
-
0.7259
0.4210
-
1.72
0.115
x
1
x
3
2.1314
0.4210
5.06
0.000
x
2
x
3
0.3984
0.4210
0.95
0.366
x
1
x
1
-
0.3435
0.3136
-
1.10
0.299
x
2
x
2
0.5507
0.3136
1.76
0.110
x
3
x
3
-
0.9727
0.3136
-
3.10
0.011
S
=
1.1908
R
2 = 85.5%
R
2 (adj) = 72.4%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
83.371
9.263
6.53
0.004
Residual error
10
14.18
1.418
Total
19
97.55Table 4
ANOVA of the variance ofy (σ^y2).
Predictor
Coef
SE Coef
T
P
Constant
0.4944
0.0225
21.92
0.000
x
1
0.0219
0.0149
1.46
0.174
x
2
0.0343
0.0149
2.30
0.044
x
3
0.0237
0.0149
1.58
0.144
x
1
x
2
-
0.0327
0.0195
-
1.67
0.125
x
1
x
3
0.0755
0.0195
3.86
0.003
x
2
x
3
0.0157
0.0195
0.81
0.439
x
1
x
1
-
0.0110
0.0145
-
0.76
0.465
x
2
x
2
0.0246
0.0145
1.69
0.122
x
3
x
3
-
0.0366
0.0145
-
2.52
0.030
S
=
0.0553138
R
2 = 79.6%
R
2 (adj) = 61.2%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
0.119074
0.01323
4.32
0.016
Residual error
10
0.030596
0.00306
Total
19
0.149671Table 5
ANOVA of the mean oft (μ^t).
Predictor
Coef
SE Coef
T
P
Constant
1.5693
0.0417
37.63
0.000
x
1
0.0641
0.0276
2.32
0.043
x
2
0.0706
0.0276
2.55
0.029
x
3
0.0752
0.0276
2.72
0.022
x
1
x
2
-
0.0548
0.0361
-
1.52
0.160
x
1
x
3
0.2103
0.0361
5.82
0.000
x
2
x
3
0.0476
0.0361
1.32
0.217
x
1
x
1
-
0.0289
0.0269
-
1.08
0.307
x
2
x
2
0.0572
0.0269
2.13
0.059
x
3
x
3
-
0.0844
0.0269
-
3.14
0.011
S
=
0.102261
R
2 = 88.1%
R
2 (adj) = 77.4%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
0.77375
0.08597
8.22
0.001
Residual error
10
0.10457
0.01046
Total
19
0.87833Table 6
ANOVA of the variance oft (σ^t2).
Predictor
Coef
SE Coef
T
P
Constant
0.0073
0.0003
19.34
0.000
x
1
0.0000
0.0002
0.11
0.917
x
2
-
0.0003
0.0002
-
1.37
0.201
x
3
0.0000
0.0002
0.29
0.776
x
1
x
2
-
0.0001
0.0003
-
0.38
0.711
x
1
x
3
-
0.0011
0.0003
-
3.43
0.006
x
2
x
3
-
0.0001
0.0003
-
0.38
0.711
x
1
x
1
-
0.0003
0.0002
-
1.26
0.237
x
2
x
2
-
0.0004
0.0002
-
1.98
0.076
x
3
x
3
0.0000
0.0002
0.19
0.853
S
=
0.000926452
R
2 = 65.9%
R
2 (adj) = 35.3%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
1.66169
E
-
05
1.84632
E
-
06
2.15
0.124
Residual error
10
8.58313
E
-
06
8.58313
E
-
07
Total
19
0.0000252Table 7
ANOVA of the correlation coefficient ofy and t (ρ^yt).
Predictor
Coef
SE Coef
T
P
Constant
0.8579
0.0167
51.14
0.000
x
1
-
0.0129
0.0111
-
1.17
0.270
x
2
0.0061
0.0111
0.55
0.595
x
3
0.0044
0.0111
0.40
0.696
x
1
x
2
0.0138
0.0145
0.95
0.363
x
1
x
3
0.0751
0.0145
5.17
0.000
x
2
x
3
0.0068
0.0145
0.47
0.647
x
1
x
1
0.0180
0.0108
1.66
0.127
x
2
x
2
0.0325
0.0108
3.00
0.013
x
3
x
3
-
0.0155
0.0108
-
1.44
0.181
S
=
0.0411335
R
2 = 81.5%
R
2 (adj) = 64.8%
Analysis of variance
Source
DF
SS
MS
F
P
Regression
9
0.074522
0.00828
4.89
0.01
Residual error
10
0.01692
0.001692
Total
19
0.091441The variability of destructive quality characteristicy could be reduced by applying the RD model shown in (3) with τy=5. The optimal solutions (i.e. x1*= −0.3000, x2*= −0.6219, and  x3*= −1.6810) and then the optimal parameters associated with y and t (i.e. μy*= 5.0000, σy2*= 0.3792, μt*= 1.3061, σt2*= 0.0070, and ρyt*= 0.8684) were obtained and provided in Table 8. Figures 2, 3, and 4 present the graphical representations of all the fitted response functions (i.e. μ^y, σ^y2, μ^t, σ^t2, and ρ^yt) with respect to the optimal setting of each input factor (i.e. x1*, x2*, and x3*, resp.).Table 8
Optimal solutions from the RD stage.
Control factors (x)
Hardness (y)
Terahertz refractive index (t)
ρ
y
t
*
x
1
*
x
2
*
x
3
*
μ
y
*
σ
y
2
*
μ
t
*
σ
t
2
*
-
0.2999
-
0.6219
-
1.6810
5.0000
0.3792
1.3061
0.0070
0.8684Figure 2
Plot ofμ^y, σ^y2, μ^t, σ^t2, and ρ^yt with respect to x1*.Figure 3
Plot ofμ^y, σ^y2, μ^t, σ^t2, and ρ^yt with respect to x2*.Figure 4
Plot ofμ^y, σ^y2, μ^t, σ^t2, and ρ^yt with respect to x3*.


## 4.2. Stage II: Tolerance Design Optimization

After the variability was reduced in the RD stage and the optimal setting for the hardness (y) and the Terahertz Refractive Index (t) were obtained, the proposed economic TD framework was conducted with respect to t in order to find the optimal tolerance δ*, which could be easily found by the closed-form solution shown in (23) and then lead to the derivation of the optimal lower and upper specification limits (i.e., LSL* and USL*), respectively. When the inspection was implemented, the expected quality loss (E[Quality Loss (y)]) occurred as shown in (6) where the positive loss coefficient θ  was 250. The expected manufacturing cost (E[Mfg Cost]) presented in (11) was described as a polynomial model where a0 and a1 were 100 and −0.2, respectively. Also, when the product performance fell above the upper limit or below the lower limit of the production specification, the expected rework cost (E[Rework Cost]) and the expected scrap cost (E[Scrap Cost]) were calculated as, respectively, defined in (7) where both Crework and Cscrap were set to an arbitrary value such as 80. All the parameters, objectives, and optimal solutions involved in this TD stage are provided as follows.Given Conditions
f(t)  e-((t-μt)2/2σt2)/2πσt2
τ      5.0000
μ
y   5.0000
σ
y
2   0.3792
μ
t   1.3061
σ
t
2   0.0070
ρ
y
t   0.8684.System Parameters
(32)
E
[
Total
Cost
]
=
E
[
Quality
Loss
]
+
E
[
Mfg
Cost
]
+
E
[
Rework
Cost
]
+
E
[
Scrap
Cost
]
,
E
[
Quality
Loss
(
y
)
]
=
250
×
∫
LSL
USL
(
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
2
)
σ
y
2
(
1
-
ρ
y
t
2
)
+
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
2
(
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
2
)
σ
y
2
(
1
-
ρ
y
t
2
)
(
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
{
μ
y
+
ρ
y
t
2
σ
y
(
t
-
μ
t
)
σ
t
-
5
}
2
)
σ
y
2
(
1
-
ρ
y
t
2
)
)
×
f
(
t
)
d
t
,
E
[
Mfg
Cost
]
=
100
-
0.4
δ
σ
t
.
E
[
Rework
Cost
]
=
80
∫
USL
∞
f
(
t
)
d
t
,
E
[
Scrap
Cost
]
=
80
∫
-
∞
LSL
f
(
t
)
d
t
,Find
δ
*
: the optimal tolerance, which leads to the derivation of USL and LSL as follows:
(33)USL=μt+δ*σt,LSL=μt-δ*σt.Minimize
(34)
E
[
Total
Cost
]
=
-
0.4
σ
t
δ
-
500
ρ
y
t
2
σ
y
2
δ
φ
(
δ
)
+
{
500
σ
y
2
(
1
-
ρ
y
t
2
)
+
500
ρ
y
t
2
σ
y
2
-
160
}
×
Φ
(
δ
)
+
260
-
250
σ
y
2
.Closed-Form Solution
(35)
δ
*
=
±
(
+
(
160
-
500
σ
y
2
(
1
-
ρ
y
t
2
)
500
ρ
y
t
2
σ
y
2
)
)
-
2
Lambert
W
×
(
-
0.2
σ
1
2
π
e
(
(
160
-
500
σ
y
2
(
1
-
ρ
y
t
2
)
)
/
1000
ρ
y
t
2
σ
y
2
)
500
ρ
y
t
2
σ
y
2
)
+
(
160
-
500
σ
y
2
(
1
-
ρ
y
t
2
)
500
ρ
y
t
2
σ
y
2
)
)
1
/
2
=
0.8910
USL
* = 1.3806 and LSL* = 1.2317.The optimal toleranceδ* is 0.8910 and then USL* and LSL* are 1.3806 and 1.2317, respectively. Figure 5 illustrates the relationship between δ and all relevant costs which are expected total cost, expected manufacturing cost, cost incurred by the expected quality loss, and expected rejection cost including expected rework cost and expected scrap cost.Figure 5
Plots ofE[Total Cost], E[Mfg Cost], E[Quality Loss], and E[Rework Cost] + E[Scrap Cost] with respect to δ.In order to verify the optimality conditions ofδ*, the plots of the first and the second derivatives of the expected total cost function with respect to δ are provided in Figures 6 and 7, respectively. The former shows that, at δ*, the value of ∂E[TotalCost]/∂δ is zero; while the latter shows that, at δ*, the expected total cost is minimum because the value of ∂2E[TotalCost]/∂δ2 is greater than zero. Also, the value of δ* is between 0 and 1.6712, which is relevant to the interval defined in (25).Figure 6
Plot of∂E[TotalCost]/∂δ with respect to δ.Figure 7
Plot of∂2E[TotalCost]/∂δ2 with respect to δ.Three sensitivity analyses were conducted in order to determine the impact ofθ, Crework, and Cscrap on E[ReworkCost] around δ* A full data set of each sensitivity analysis is provided in Tables 9, 10, and 11, respectively. First is the effect of θ while Crework and Cscrap are held at 80. Figure 8 identifies that while the value of θ changes from 100 to 400 with an increment of 25, the values of E[ReworkCost], E[ScrapCost], E[MfgCost], and E[TotalCost] increase; and while the value of δ decreases, E[QualityLoss] has a concave pattern.Table 9
Effects ofθ on minimum costs (Crework=80 and Cscrap=80).
θ
δ
*
E [Rework Cost]
E [Scrap Cost]
E [Quality Loss]
E [Mfg Cost]
E [Total Cost]
100
1.5737
4.6227
4.6227
23.1282
99.9474
132.3210
125
1.3838
6.6567
6.6567
24.3612
99.9537
137.6283
150
1.2414
8.5787
8.5787
25.0187
99.9585
142.1345
175
1.1288
10.3601
10.3601
25.3359
99.9623
146.0184
200
1.0363
12.0022
12.0022
25.4404
99.9653
149.4102
225
0.9583
13.5167
13.5167
25.4044
99.9680
152.4057
250
0.8910
14.9177
14.9177
25.2706
99.9702
155.0762
275
0.8319
16.2194
16.2194
25.0646
99.9722
157.4755
300
0.7792
17.4346
17.4346
24.8024
99.9739
159.6456
325
0.7317
18.5747
18.5747
24.4940
99.9755
161.6189
350
0.6883
19.6496
19.6496
24.1454
99.9770
163.4216
375
0.6484
20.6679
20.6679
23.7604
99.9783
165.0745
400
0.6114
21.6372
21.6372
23.3408
99.9796
166.5948Table 10
Effects ofCrework on minimum costs (θ=250 and Cscrap=80).
C
rework
δ
*
E [Rework Cost]
E [Scrap Cost]
E [Quality Loss]
E [Mfg Cost]
E [Total Cost]
40
0.7169
9.4683
18.9365
18.2914
99.9760
146.6723
48
0.7550
10.8066
18.0109
19.7268
99.9748
148.5190
56
0.7912
12.0078
17.1541
21.1413
99.9735
150.2767
64
0.8258
13.0859
16.3573
22.5362
99.9724
151.9518
72
0.8590
14.0524
15.6138
23.9125
99.9713
153.5500
80
0.8910
14.9177
14.9177
25.2706
99.9702
155.0762
88
0.9219
15.6907
14.2643
26.6108
99.9692
156.5349
96
0.9517
16.3794
13.6495
27.9333
99.9682
157.9303
104
0.9807
16.9908
13.0699
29.2381
99.9672
159.2660
112
1.0088
17.5314
12.5224
30.5253
99.9663
160.5454
120
1.0362
18.0068
12.0045
31.7949
99.9653
161.7715
128
1.0629
18.4222
11.5139
33.0467
99.9645
162.9472
136
1.0889
18.7823
11.0484
34.2808
99.9636
164.0751
144
1.1143
19.0915
10.6064
35.4970
99.9627
165.1576
152
1.1391
19.3537
10.1861
36.6954
99.9619
166.1971
160
1.1634
19.5724
9.7862
37.8758
99.9611
167.1955Table 11
Effects ofCscrap on minimum costs (θ=250 and Crework=80).
C
scrap
δ
*
E [Rework Cost]
E [Scrap Cost]
E [Quality Loss]
E [Mfg Cost]
E [Total Cost]
40
0.7169
18.9365
9.4683
18.2914
99.9760
146.6723
48
0.7550
18.0109
10.8066
19.7268
99.9748
148.5190
56
0.7912
17.1541
12.0078
21.1413
99.9735
150.2767
64
0.8258
16.3573
13.0859
22.5362
99.9724
151.9518
72
0.8590
15.6138
14.0524
23.9125
99.9713
153.5500
80
0.8910
14.9177
14.9177
25.2706
99.9702
155.0762
88
0.9219
14.2643
15.6907
26.6108
99.9692
156.5349
96
0.9517
13.6495
16.3794
27.9333
99.9682
157.9303
104
0.9807
13.0699
16.9908
29.2381
99.9672
159.2660
112
1.0088
12.5224
17.5314
30.5253
99.9663
160.5454
120
1.0362
12.0045
18.0068
31.7949
99.9653
161.7715
128
1.0629
11.5139
18.4222
33.0467
99.9645
162.9472
136
1.0889
11.0484
18.7823
34.2808
99.9636
164.0751
144
1.1143
10.6064
19.0915
35.4970
99.9627
165.1576
152
1.1391
10.1861
19.3537
36.6954
99.9619
166.1971
160
1.1634
9.7862
19.5724
37.8758
99.9611
167.1955Figure 8
The effect ofθ on minimum costs (Crework= 80 and Cscrap= 80).Second is the effect ofCrework while θ and Cscrap are held at 250 and 80, respectively. Figure 9 demonstrates that while the value of Crework changes from 40 to 160 with an increment of 8, the values of δ*, E[ReworkCost], E[QualityLoss], and E[TotalCost] increase; and while the values of E[ScrapCost] and E[MfgCost] decrease, the rejection cost (which is a summation of E[ReworkCost] and E[ScrapCost]) has a concave pattern.Figure 9
The effect ofCrework on minimum costs (θ = 250 and Cscrap = 80).Last is the effect ofCscrap while θ and Crework are held at 250 and 80, respectively. Figure 10 represents that while the value of Cscrap changes from 40 to 160 with an increment of 8, the values of, E[ScrapCost], E[QualityLoss], and E[TotalCost] increase. Similar to the second sensitivity analysis, while the values of E[ReworkCost] and E[MfgCost] decrease, the rejection cost has a concave pattern.Figure 10
The effect ofCscrap  on minimum costs (θ = 250 and Crework = 80).


## 5. Conclusions and Future Research

This paper proposed the optimization schemes for determining an optimal factor setting and tolerances for a destructive pharmaceutical characteristic. In reality, there are a number of destructive quality characteristics in pharmaceutical processes. Conceptualizing destructive pharmaceutical characteristics for the purpose of optimizing such pharmaceutical parameters, to our best knowledge, might be the first attempt in this field. To find the optimal factor setting with minimal variability, RD is an effective method. It is more effective when applied in the early stage of process design. Then, to remove possible variability, TD based on the concept of Taguchi’s offline quality management is applied. With the result from RD stage, optimal tolerance and specification limits are obtained.The proposed approach integrated the RD and TD concepts with the use of RSM and surrogate variables within the framework of destructive characteristics in order to achieve the target requirement with minimal variability and handle the destructive quality characteristic under 100% inspection. The experimental design format to find the relationship between a destructive quality characteristic and its associated surrogate variable was also proposed. Moreover, mathematical procedures for finding optimal tolerance and its closed-form solution by applying the LambertW function principle were provided. The numerical example showed that the proposed approach not only delivered a robust factor setting but also considered tradeoff between the level of quality and cost.Based on the results of this paper, a significant extension to support multiple quality characteristics can be a potential future research topic. In addition, current interesting issues, related to fractals associated with fractal dimension and Hurst parameter [52–58] in the field of pharmaceutical science and technology, can be incorporated into the proposed robust-tolerance design optimization method. By observing the fractal phenomena, these possible further approaches may achieve the realization of process understanding (PU) and process analysis technology (PAT) in pharmaceutical processes.



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*Source: 706962-2013-05-08.xml*