Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 706962 |

Seong Hoon Jeong, Pauline Kongsuwan, Nguyen Khoa Viet Truong, Sangmun Shin, "Optimal Tolerance Design and Optimization for a Pharmaceutical Quality Characteristic", Mathematical Problems in Engineering, vol. 2013, Article ID 706962, 17 pages, 2013.

Optimal Tolerance Design and Optimization for a Pharmaceutical Quality Characteristic

Academic Editor: Ming Li
Received07 Nov 2012
Revised18 Jan 2013
Accepted19 Jan 2013
Published08 May 2013


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.

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, 1016].

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 Figure 1.

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, 1619]. 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 with the correlation between destructive variables and surrogate variables t. y and are treated as random variables whose means depend upon the fixed values of . Based on the dual response approach [9], the following models are recommended for destructive and surrogate variables: where and are normal random variables with mean zero and variances and , respectively; and , , , , , and 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 , and that of the surrogate variable , the estimated variance function of the performance variable and that of the surrogate variable , and the estimated correlation coefficient function between the performance variable and the surrogate variable are given as follows: where is a design factor (or a control factor); , , and are estimated regression coefficients of ; , , and are estimated regression coefficients of ; , , and are estimated regression coefficients of ; , , and are estimated regression coefficients of ; and , , and are estimated regression coefficients of. 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. and are the sample mean and the sample variance of destructive quality characteristic y; and are the sample mean and the sample variance of surrogate variable t; and is the sample correlation coefficient associated with the destructive quality characteristic and the surrogate variable t.

Run No.Control factors ( )Quality characteristics ( )Surrogate variables ( )


2.2. Variability Reduction of Destructive Characteristics

From the experimental data, the functions of process mean and variance are separately estimated by using the dual response approach. This model implies that the process mean is adjusted to the target 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 is formulated as follows: where is a process target of destructive qualitycharacteristic y, is a convex set in, and 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, 3446].

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. Let denoting the joint distribution of major quality characteristic y and surrogate variable t. Then, assumed to be a bivariate normal distribution with means and , variances and , and correlation coefficient . 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 , , , , and are already known [47]. Note that where is the marginal density function of T and is the conditional density function of , given . It can be shown that is a normal density function with mean and variance , and is a normal density function with mean and variance .

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: where 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 , the expected quality loss within a tolerance limit can be formulated as follows: From the definition of , after replacing μ with and σ with , (5) becomes

The rejection cost includes unit rework cost and unit scrap cost , which incur when a product performance falls above the upper specification limit and below the lower specification limit, respectively. The expected rework cost and the expected scrap cost are given as follows: 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 , , and as follows: For simplicity, the manufacturing cost is assumed to have a linear relationship with the tolerance range as described below: where 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: Substituting Tolerance Range = in (10), the expected manufacturing cost becomes

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 By replacing (6), (7), and (11) into (12), the expected total cost is reformulated as: To simplify (13), the marginal density function of t is converted from normal distribution function f(t) to standard normal distribution (z) by using Thus, (13) could be rewritten as where , , and . Based on the dual response approach used in the RD stage, terms and are removed from (15). Also, after applying the following properties of normal probability density functions: the expected total cost becomes

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

To investigate the optimal solution of tolerance design, the first derivative with respect to is as follows: Equating to zero and substituting with , (18) becomes Since it is nearly impossible to obtain a closed-form solution from this complex equation, Lambert W function is introduced to effectively obtain such a solution for as discussed in Appendix B. According to the property of Lambert W function ([20, 50]) such that where , , , and are not functions of , the optimal tolerance can be derived from the following relationship: According to (20), (19) after formatting becomes , ,, and in (21) can be replaced with 1, , −1/2, and , respectively. Then, the closed-form solution of is formulated and provided as follows: where ,   and ; 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 y and 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 is obtained, the optimal LSL and USL can be derived from and , 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 of are investigated. The second derivative of with respect to is as follows: The value of 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 could be written as In order to avoid imaginary values, the following condition needs to be satisfied: If both conditions in (25) and (26) are satisfied, the closed-form solution of defined in (23) represents the global minimum of . The overall framework of TD optimization and its associated key elements are provided as follows.

Given Parametersf(t): the normal probability density function of surrogate variable t, : the target of the process mean of destructive quality characteristic y, the optimal process mean of destructive quality characteristic y, derived from the RD stage,: the optimal variance of destructive quality characteristic y, derived from the RD stage,: the optimal process mean of surrogate variable t, derived from the RD stage,: the optimal variance of surrogate variable t, derived from the RD stage,  : the optimal correlation coefficient between destructive quality characteristic y and surrogate variable t, derived from the RD stage.

System Parameters

Find the optimal tolerance, which leads to the derivation of USL and LSL as follows:


Closed-Form Solution

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 (, MPa), humidity (, %), and compression speed (, 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 Figure 1, 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(), the sample variance of y(), the sample mean of t(), the sample variance of t(), and the correlation coefficient of y and t() 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.

Control factors ( )Destructive quality characteristic
hardness ( )
Surrogate variable
terahertz refractive index ( )

00 2.7053.8043.0893.1990.3121.0821.2131.0501.1150.0070.858

4.1. Stage I: Robust Design Optimization

By applying RSM to the experimental data shown in Table 2, 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:

To demonstrate the statistical analysis, ANOVA for the five responses based on significant level at 0.05 was conducted as shown in Tables 3, 4, 5, 6, and 7. The result could be summarized as follows. For , two main effects (i.e. and ), the interaction between and , and the quadratic effect of are significant; for , , the interaction between and , and the quadratic effect of are significant; for , all the main effects (i.e. , , and ), the interaction between and , and the quadratic effect of are significant; for , only the interaction between and is significant; and for , the interaction between and and the quadratic effect of are significant.

PredictorCoefSE Coef

Constant8.0877 0.4857 16.65 0.000
0.6315 0.3222 1.96 0.078
0.7573 0.3222 2.35 0.041
0.7236 0.3222 2.25 0.049
0.4210 0.115
2.1314 0.4210 5.06 0.000
0.3984 0.4210 0.95 0.366
0.3136 0.299
0.5507 0.3136 1.76 0.110
0.3136 0.011

= 85.5% (adj) = 72.4%

Analysis of variance


Residual error1014.181.418


PredictorCoefSE Coef

Constant0.4944 0.0225 21.92 0.000
0.02190.0149 1.46 0.174
0.03430.0149 2.30 0.044
0.02370.0149 1.58 0.144
0.0195 0.125
0.07550.0195 3.86 0.003
0.01570.0195 0.81 0.439
0.0145 0.465
0.02460.0145 1.69 0.122
0.0145 0.030

= 79.6% (adj) = 61.2%

Analysis of variance


Regression90.1190740.013234.32 0.016
Residual error100.0305960.00306


PredictorCoefSE Coef

Constant1.5693 0.0417 37.63 0.000
0.0641 0.0276 2.32 0.043
0.0706 0.0276 2.55 0.029
0.0752 0.0276 2.72 0.022
0.0361 0.160
0.2103 0.0361 5.82 0.000
0.0476 0.0361 1.32 0.217
0.0269 0.307
0.0572 0.0269 2.13 0.059
0.0269 0.011

= 88.1% (adj) = 77.4%

Analysis of variance


Residual error100.104570.01046


PredictorCoefSE Coef

Constant0.0073 0.0003 19.34 0.000
0.0000 0.0002 0.11 0.917
0.0002 0.201
0.0000 0.0002 0.29 0.776
0.0003 0.711
0.0003 0.006
0.0003 0.711
0.0002 0.237
0.0002 0.076
0.0000 0.0002 0.19 0.853

= 65.9% (adj) = 35.3%

Analysis of variance


Regression9 2.150.124
Residual error10


PredictorCoefSE Coef

Constant0.8579 0.0167 51.14 0.000
0.0111 0.270
0.0061 0.0111 0.55 0.595
0.0044 0.0111 0.40 0.696
0.0138 0.0145 0.95 0.363
0.0751 0.0145 5.17 0.000
0.0068 0.0145 0.47 0.647
0.0180 0.0108 1.66 0.127
0.0325 0.0108 3.00 0.013
0.0108 0.181

= 81.5% (adj) = 64.8%

Analysis of variance


Residual error100.016920.001692


The variability of destructive quality characteristic y could be reduced by applying the RD model shown in (3) with . The optimal solutions (i.e. = −0.3000, = −0.6219, and = −1.6810) and then the optimal parameters associated with y and t (i.e. = 5.0000, = 0.3792, = 1.3061, = 0.0070, and = 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. , , , , and ) with respect to the optimal setting of each input factor (i.e. , , and , resp.).

Control factors ( )Hardness ( )Terahertz refractive index ( )


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 , 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., and ), respectively. When the inspection was implemented, the expected quality loss (E[Quality Loss (y)]) occurred as shown in (6) where the positive loss coefficient was 250. The expected manufacturing cost (E[Mfg Cost]) presented in (11) was described as a polynomial model where and 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 and 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 Conditionsf(t)       5.0000   5.0000   0.3792   1.3061   0.0070   0.8684.

System Parameters

Find the optimal tolerance, which leads to the derivation of USL and LSL as follows:


Closed-Form Solution = 1.3806 and = 1.2317.

The optimal tolerance is 0.8910 and then and are 1.3806 and 1.2317, respectively. Figure 5 illustrates the relationship between 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.

In order to verify the optimality conditions of , the plots of the first and the second derivatives of the expected total cost function with respect to are provided in Figures 6 and 7, respectively. The former shows that, at , the value of is zero; while the latter shows that, at , the expected total cost is minimum because the value of is greater than zero. Also, the value of is between 0 and 1.6712, which is relevant to the interval defined in (25).

Three sensitivity analyses were conducted in order to determine the impact of , , and on