Optimal Tolerance Design and Optimization for a Pharmaceutical Quality Characteristic

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.

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 Figure 1.

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: 𝑦(𝑥)=𝛼₀+𝑎₁𝑥+𝑎₂𝑥²+𝜀_𝑦,𝑡(𝑥)=𝑏₀+𝑏₁𝑥+𝑏₂𝑥²+𝜀_𝑡, (1) where 𝜀_𝑦 and 𝜀_𝑡 are normal random variables with mean zero and variances 𝜎²_𝑦𝜀 and 𝜎²_𝑡𝜀, 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 𝜎²_𝑦 and that of the surrogate variable 𝜎²_𝑡, and the estimated correlation coefficient function between the performance variable and the surrogate variable 𝜌_𝑦𝑡 are given as follows: 𝜇_𝑦(𝑥)=𝛼_𝑦0+𝛼_𝑦1𝑥+𝛼_𝑦2𝑥²,𝜇_𝑡(𝑥)=𝛼_𝑡0+𝛼_𝑡1𝑥+𝛼_𝑡2𝑥²,𝜎²_𝑦(𝑥)=𝛽_𝑦0+𝛽_𝑦1𝑥+𝛽_𝑦1𝑥²,𝜎²_𝑡(𝑥)=𝛽_𝑡0+𝛽_𝑡1𝑥+𝛽_𝑡1𝑥²,𝜌_𝑦𝑡(𝑥)=𝛾₀+𝛾₁𝑥+𝛾₂𝑥², (2) where 𝑥 is a design factor (or a control factor); 𝛼_𝑦0, 𝛼_𝑦1, and 𝛼_𝑦2 are estimated regression coefficients of 𝜇_𝑦; 𝛼_𝑡0, 𝛼_𝑡1, and 𝛼_𝑡2 are estimated regression coefficients of 𝜇_𝑡; 𝛽_𝑦0, 𝛽_𝑦1, and 𝛽_𝑦2 are estimated regression coefficients of 𝜎²_𝑦; 𝛽_𝑡0, 𝛽_𝑡1, and 𝛽_𝑡2 are estimated regression coefficients of 𝜎²_𝑡; 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 𝑠²_𝑦 are the sample mean and the sample variance of destructive quality characteristic y; 𝑡 and 𝑠²_𝑡 are the sample mean and the sample variance of surrogate variable t; and 𝜌_𝑦𝑡 is the sample correlation coefficient associated with the destructive quality characteristic 𝑦 and the surrogate variable t.

Table 1 

Experimental format.

2.2. Variability Reduction of Destructive Characteristics

From the experimental data, the functions of process mean 𝜇(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 𝜏_𝑦 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: Minimize𝜎²_𝑦(𝐱)Subjectto𝜇_𝑦(𝐱)=𝜏_𝑦𝑥∈Ω, (3) where 𝜏_𝑦 is a process target of destructive qualitycharacteristic y, Ωis{𝑥∈𝑆∶𝑔_𝑖(𝑥)≤0,𝑖=1,2,…,𝑛},𝑆 is a convex set in𝑅^𝑛, and 𝑔_𝑖(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. Let ℎ(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 𝜇_𝑦 and 𝜇_𝑡, variances 𝜎²_𝑦 and 𝜎²_𝑡, and correlation coefficient 𝜌_𝑦𝑡(=𝜎_𝑦𝑡/𝜎_𝑦𝜎_𝑡). 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 𝜇_𝑦, 𝜇_𝑡, 𝜎²_𝑦, 𝜎²_𝑡, and 𝜌_𝑦𝑡 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 𝜇_𝑡 and variance 𝜎²_𝑡, and 𝑔(𝑦∣𝑡) is a normal density function with mean 𝜇=𝜇_𝑦+𝜌_𝑦𝑡(𝜎_𝑦/𝜎_𝑡)(𝑡−𝜇_𝑡) and variance 𝜎²=𝜎²_𝑦(1−𝜌²_𝑦𝑡).

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: QualityLoss𝑦=𝜃𝑦−𝜏², (4) 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: 𝐸QualityLoss𝑦=^USL_LSL^∞_−∞𝜃𝑦−𝜏²ℎ=𝑦,𝑡𝑑𝑦𝑑𝑡_USLLSL𝜃𝜎²+𝜇−𝜏²𝑓(𝑡)𝑑𝑡. (5) From the definition of 𝑔(𝑦∣𝑡), after replacing μ with 𝜇_𝑦+𝜌_𝑦𝑡(𝜎_𝑦/𝜎_𝑡)(𝑡−𝜇_𝑡) and σ with 𝜎²_𝑦(1−𝜌²_𝑦𝑡), (5) becomes 𝐸QualityLoss𝑦=𝜃^USL_LSL𝜎²_𝑦1−𝜌²_𝑦𝑡+𝜇_𝑦+𝜌²_𝑦𝑡𝜎_𝑦𝑡−𝜇_𝑡𝜎_𝑡−𝜏²𝑓(𝑡)𝑑𝑡. (6)

The rejection cost includes unit rework cost 𝐶_rework and unit scrap cost 𝐶_scrab, 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: 𝐸[ReworkCost]=𝐶_rework(𝑃[USL≤𝑇])=𝐶_rework^∞_USL𝑓(𝑡)𝐸𝑑𝑡,ScrapCost=𝐶_scrap(𝑃[𝑇≤LSL])=𝐶_scrap^LSL_−∞𝑓(𝑡)𝑑𝑡. (7) 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, 𝜇_𝑡, and 𝜎_𝑡 as follows: =𝜇ToleranceRange=USL−LSL_𝑡+𝛿𝜎_𝑡−𝜇_𝑡−𝛿𝜎_𝑡=2𝛿𝜎_𝑡. (8) For simplicity, the manufacturing cost is assumed to have a linear relationship with the tolerance range as described below: MfgCost=𝑎₀+𝑎₁ToleranceRange+𝜀_mfg, (9) where 𝜀_mfg 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: 𝐸MfgCost=𝑎₀+𝑎₁ToleranceRange. (10) Substituting Tolerance Range = 2𝛿𝜎_𝑡 in (10), the expected manufacturing cost becomes 𝐸MfgCost=𝑎₀+2𝑎₁𝛿𝜎_𝑡. (11)

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 𝐸[]𝑦[].TotalCost=𝐸QualityLoss+𝐸ReworkCost+𝐸ScrapCost+𝐸MfgCost (12) By replacing (6), (7), and (11) into (12), the expected total cost is reformulated as: 𝐸[TotalCost]=𝜃^USL_LSL𝜎²_𝑦1−𝜌²_𝑦𝑡+𝜇_𝑦+𝜌²_𝑦𝑡𝜎_𝑦𝑡−𝜇_𝑡𝜎_𝑡−𝜏²𝑓(𝑡)𝑑𝑡+𝐶_rework^∞_USL𝑓(𝑡)𝑑𝑡+𝐶_scrap^LSL_−∞𝑓(𝑡)𝑑𝑡+𝑎₀+2𝑎₁𝛿𝜎_𝑡. (13) 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 𝑧=𝑡−𝜇_𝑡𝜎_𝑡. (14) Thus, (13) could be rewritten as 𝐸[TotalCost]=𝜃^𝛿_−𝛿𝜎²_𝑦1−𝜌²_𝑦𝑡+𝜇_𝑦+𝜌_𝑦𝑡𝜎_𝑦𝑧−𝜏²𝜙(𝑧)𝑑𝑧+𝐶_rework^∞_𝛿𝜙(𝑧)𝑑𝑧+𝐶_scrap^−𝛿_−∞𝜙(𝑧)𝑑𝑧+𝑎₀+2𝑎₁𝜎_𝑡𝛿, (15) where USL=𝜇_𝑦+𝛿𝜎_𝑦, LSL=𝜇_𝑦−𝛿𝜎_𝑦, and 𝛿>0. Based on the dual response approach used in the RD stage, terms 𝜇_𝑦 and 𝜏 are removed from (15). Also, after applying the following properties of normal probability density functions: ^𝑟_−∞𝜙(𝑧)𝑑𝑧=Φ(𝑟),^𝑟_−∞𝑧𝜙(𝑧)𝑑𝑧=−𝜙_𝑟−∞(𝑟),𝑧²𝜙(𝑧)𝑑𝑧=Φ(𝑟)−𝑟𝜙(𝑟), (16) the expected total cost becomes 𝐸[TotalCost]=2𝑎₁𝜎_𝑡𝛿−2𝜌²_𝑦𝑡𝜎²_𝑦(𝛿)+𝜃𝛿𝜙2𝜎²_𝑦1−𝜌²_𝑦𝑡𝜃+2𝜌²_𝑦𝑡𝜎²_𝑦𝜃−𝐶_rework−𝐶_scrapΦ(𝛿)+𝐶_rework+𝐶_scrap−𝜃𝜎²_𝑦+𝑎₀. (17)

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 𝛿 is as follows: 𝜕𝐸[TotalCost]𝜕𝛿=2𝑎₁𝜎_𝑡(𝛿)×+𝜙2𝜎²_𝑦1−𝜌²_𝑦𝑡𝜃−𝐶_rework−𝐶_scrap+2𝜌²_𝑦𝑡𝜎²_𝑦𝜃𝛿²𝜙(𝛿). (18) Equating 𝜕𝐸[TotalCost]/𝜕𝛿 to zero and substituting 𝜙(𝛿) with 𝑒^{(−1/2)𝛿2}/√2𝜋, (18) becomes 2𝑎₁𝜎_𝑡+2𝜎²_𝑦1−𝜌²_𝑦𝑡𝜃−𝐶_rework−𝐶_scrap+2𝜌²_𝑦𝑡𝜎²_𝑦𝜃𝛿²×𝑒^−𝛿2/2√2𝜋=0. (19) 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 𝛿 as discussed in Appendix B. According to the property of Lambert W function ([20, 50]) such that 𝜉₁𝛿²+𝜉₂𝑒^𝜉3𝛿2=𝜉₄, (20) where 𝜉₁, 𝜉₂, 𝜉₃, and 𝜉₄ are not functions of 𝛿, the optimal tolerance 𝛿^∗ can be derived from the following relationship: 𝛿=±Lambert𝑊𝜉₃𝜉₄𝑒^𝜉2𝜉3/𝜉₁𝜉₃−𝜉₂. (21) According to (20), (19) after formatting becomes 𝛿²+2𝜎²_𝑦1−𝜌²_𝑦𝑡𝜃−𝐶_rework−𝐶_scrap2𝜌²_𝑦𝑡𝜎²_𝑦𝜃𝑒^−𝛿2/2=−𝑎₁𝜎_𝑡√2𝜋𝜌²_𝑦𝑡𝜎²_𝑦𝜃. (22)𝜉₁, 𝜉₂,𝜉₃, and 𝜉₄ in (21) can be replaced with 1, [2𝜎²_𝑦(1−𝜌²_𝑦𝑡)𝜃−𝐶_rework−𝐶_scrap]/2𝜌²_𝑦𝑡𝜎²_𝑦𝜃, −1/2, and −𝑎₁𝜎_𝑡√2𝜋/𝜌²_𝑦𝑡𝜎²_𝑦𝜃, respectively. Then, the closed-form solution of 𝛿^∗ is formulated and provided as follows: 𝛿^∗𝑎=±−2Lambert𝑊₁𝜎₁√2𝜋𝑒^{((𝐶rework+𝐶scrap−2𝜎2𝑦(1−𝜌2𝑦𝑡)𝜃)/(4𝜌2𝑦𝑡𝜎2𝑦𝜃))}2𝜌²_𝑦𝑡𝜎²_𝑦𝜃+𝐶_rework+𝐶_scrap−2𝜎²_𝑦1−𝜌²_𝑦𝑡𝜃2𝜌²_𝑦𝑡𝜎²_𝑦𝜃^1/2, (23) where 𝜎²_𝑦,𝜎²_𝑡,𝜌_𝑦𝑡,𝜃,   𝐶_rework and 𝐶_scrap; 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 𝛿^∗ is obtained, the optimal LSL and USL can be derived from 𝜇^∗_𝑡−𝛿^∗𝜎^∗_𝑡 and 𝜇^∗_𝑡+𝛿^∗𝜎^∗_𝑡, 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 𝐸[TotalCost] are investigated. The second derivative of 𝐸[TotalCost] with respect to 𝛿 is as follows: 𝜕²𝐸[TotalCost]𝜕𝛿²(𝛿)×𝐶=𝛿𝜙_rework+𝐶_scrap−2𝜎²_𝑦𝜃×1−3𝜌²_𝑦𝑡+𝜌²_𝑦𝑡𝛿². (24) The value of 𝐸[TotalCost]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 0≤𝛿≤𝐶_rework+𝐶_scrap−2𝜎²_𝑦𝜃2𝜌²_𝑦𝑡𝜎²_𝑦𝜃+3. (25) In order to avoid imaginary values, the following condition needs to be satisfied: 𝐶_rework+𝐶_scrap−2𝜎²_𝑦𝜃2𝜌²_𝑦𝑡𝜎²_𝑦𝜃≥−3. (26) 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, 𝜎²_𝑦: 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, 𝜎²_𝑡: the optimal variance of surrogate variable t, derived from the RD stage,  𝜌_𝑦𝑡: the optimal correlation coefficient between destructive quality characteristic y and surrogate variable t, derived from the RD stage.

System Parameters𝐸[][],𝐸𝑦TotalCost=𝐸QualityLoss+𝐸MfgCost+𝐸ReworkCost+𝐸ScrapCostQualityLoss=𝜃^USL_LSL𝜎²_𝑦1−𝜌²_𝑦𝑡+𝜇_𝑦+𝜌²_𝑦𝑡𝜎_𝑦𝑡−𝜇_𝑡𝜎_𝑡−𝜏²𝑓(𝑡)𝐸𝑑𝑡,MfgCost=𝑎₀+2𝑎₁𝛿𝜎_𝑡,𝐸[ReworkCost]=𝐶_rework^∞_USL𝑓(𝑡)𝐸𝑑𝑡,ScrapCost=𝐶_scrap^LSL_−∞𝑓(𝑡)𝑑𝑡. (27)

Find 𝛿^∗∶ the optimal tolerance, which leads to the derivation of USL and LSL as follows: USL=𝜇_𝑡+𝛿^∗𝜎_𝑡,LSL=𝜇_𝑡−𝛿^∗𝜎_𝑡. (28)

Minimize𝐸[TotalCost]=2𝑎₁𝜎_𝑡𝛿−2𝜌²_𝑦𝑡𝜎²_𝑦(𝛿)+𝜃𝛿𝜙2𝜎²_𝑦1−𝜌²_𝑦𝑡𝜃+2𝜌²_𝑦𝑡𝜎²_𝑦𝜃−𝐶_rework−𝐶_scrapΦ(𝛿)+𝐶_rework+𝐶_scrap−𝜃𝜎²_𝑦+𝑎₀. (29)

Closed-Form Solution𝛿^∗×𝑎=±−2Lambert𝑊₁𝜎₁√2𝜋𝑒^{((𝐶rework+𝐶scrap−2𝜎2𝑦(1−𝜌2𝑦𝑡)𝜃)/4𝜌2𝑦𝑡𝜎2𝑦𝜃)}2𝜌²_𝑦𝑡𝜎²_𝑦𝜃+𝐶_rework+𝐶_scrap−2𝜎²_𝑦1−𝜌²_𝑦𝑡𝜃2𝜌²_𝑦𝑡𝜎²_𝑦𝜃^1/2 (30)

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 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(𝜇_𝑦), the sample variance of y(𝑠²_𝑦), the sample mean of t(𝜇_𝑡), the sample variance of t(𝑠²_𝑡), and the correlation coefficient of y and t(𝜌_𝑦𝑡) 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.

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: 𝜇_𝑦=8.09+0.631𝑥₁+0.757𝑥₂+0.724𝑥₃−0.726𝑥₁𝑥₂+2.13𝑥₁𝑥₃+0.398𝑥₂𝑥₃−0.344𝑥²₁+0.551𝑥²₂−0.973𝑥²₃;𝜎²_𝑦=0.494+0.0219𝑥₁+0.0344𝑥₂+0.0237𝑥₃−0.0328𝑥₁𝑥₂+0.0755𝑥₁𝑥₃+0.0157𝑥₂𝑥₃−0.0111𝑥²₁+0.0246𝑥²₂−0.0367𝑥²₃;𝜇_𝑡=1.57+0.0641𝑥₁+0.0706𝑥₂+0.0752𝑥₃−0.0549𝑥₁𝑥₂+0.210𝑥₁𝑥₃+0.0476𝑥₂𝑥₃−0.0290𝑥²₁+0.0573𝑥²₂−0.0845𝑥²₃;𝜎²_𝑡=0.00731+0.000027𝑥₁−0.000343𝑥₂+0.000073𝑥₃−0.000125𝑥₁𝑥₂−0.00112𝑥₁𝑥₃−0.000125𝑥₂𝑥₃−0.000307𝑥²₁−0.000484𝑥²₂+0.000046𝑥²₃,𝜌_𝑦𝑡=0.858−0.0130𝑥₁+0.0061𝑥₂+0.0045𝑥₃+0.0139𝑥₁𝑥₂+0.0751𝑥₁𝑥₃+0.0069𝑥₂𝑥₃+0.0180𝑥²₁+0.0325𝑥²₂−0.0156𝑥²₃. (31)

To demonstrate the statistical analysis, ANOVA for the five responses based on significant level 𝛼 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 𝜎²_𝑦, 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 𝜎²_𝑡, 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 of 𝑦 (𝜇_𝑦).

Table 4 

ANOVA of the variance of y (𝜎²_𝑦).

Table 5 

ANOVA of the mean of t (𝜇_𝑡).

Table 6 

ANOVA of the variance of t (𝜎²_𝑡).

Table 7 

ANOVA of the correlation coefficient of y and t (𝜌_𝑦𝑡).

The variability of destructive quality characteristic y could be reduced by applying the RD model shown in (3) with 𝜏_𝑦=5. The optimal solutions (i.e. 𝑥^∗₁= −0.3000, 𝑥^∗₂= −0.6219, and 𝑥^∗₃= −1.6810) and then the optimal parameters associated with y and t (i.e. 𝜇^∗_𝑦= 5.0000, 𝜎_𝑦^2∗= 0.3792, 𝜇^∗_𝑡= 1.3061, 𝜎_𝑡^2∗= 0.0070, and 𝜌^∗_𝑦𝑡= 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, 𝜎²_𝑦, 𝜇_𝑡, 𝜎²_𝑡, and 𝜌_𝑦𝑡) with respect to the optimal setting of each input factor (i.e. 𝑥^∗₁, 𝑥^∗₂, and 𝑥^∗₃, resp.).

Table 8 

Optimal solutions from the RD stage.

Figure 2 

Plot of 𝜇𝑦, 𝜎2𝑦, 𝜇𝑡, 𝜎2𝑡, and 𝜌𝑦𝑡 with respect to 𝑥∗1.

Figure 3 

Plot of 𝜇𝑦, 𝜎2𝑦, 𝜇𝑡, 𝜎2𝑡, and 𝜌𝑦𝑡 with respect to 𝑥∗2.

Figure 4 

Plot of 𝜇𝑦, 𝜎2𝑦, 𝜇𝑡, 𝜎2𝑡, and 𝜌𝑦𝑡 with respect to 𝑥∗3.

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 𝐶_rework and 𝐶_scrap 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)2/2𝜎2𝑡)}/2𝜋𝜎²_𝑡 𝜏     5.0000 𝜇_𝑦   5.0000 𝜎²_𝑦   0.3792 𝜇_𝑡   1.3061 𝜎²_𝑡   0.0070 𝜌_𝑦𝑡   0.8684.

System Parameters 𝐸[][],𝐸𝑦×TotalCost=𝐸QualityLoss+𝐸MfgCost+𝐸ReworkCost+𝐸ScrapCostQualityLoss=250_USLLSL𝜎²_𝑦1−𝜌²_𝑦𝑡+𝜇_𝑦+𝜌²_𝑦𝑡𝜎_𝑦(𝑡−𝜇_𝑡)𝜎_𝑡−5²(𝑡)𝐸×𝑓𝑑𝑡,MfgCost=100−0.4𝛿𝜎_𝑡.𝐸[ReworkCost]=80^∞_USL𝑓(𝑡)𝐸𝑑𝑡,ScrapCost=80^LSL_−∞𝑓(𝑡)𝑑𝑡, (32)

Find 𝛿^∗∶ the optimal tolerance, which leads to the derivation of USL and LSL as follows: USL=𝜇_𝑡+𝛿^∗𝜎_𝑡,LSL=𝜇_𝑡−𝛿^∗𝜎_𝑡. (33)

Minimize𝐸[TotalCost]=−0.4𝜎_𝑡𝛿−500𝜌²_𝑦𝑡𝜎²_𝑦(𝛿)+𝛿𝜑500𝜎²_𝑦1−𝜌²_𝑦𝑡+500𝜌²_𝑦𝑡𝜎²_𝑦(𝛿)−160×Φ+260−250𝜎²_𝑦. (34)

Closed-Form Solution𝛿^∗×=±−2Lambert𝑊−0.2𝜎₁√2𝜋𝑒^{((160−500𝜎2𝑦(1−𝜌2𝑦𝑡))/1000𝜌2𝑦𝑡𝜎2𝑦)}500𝜌²_𝑦𝑡𝜎²_𝑦+160−500𝜎²_𝑦1−𝜌²_𝑦𝑡500𝜌²_𝑦𝑡𝜎²_𝑦^1/2=0.8910 (35)USL^∗ = 1.3806 and LSL^∗ = 1.2317.

The optimal tolerance 𝛿^∗ 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 of E[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 of 𝛿^∗, 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 of 𝜃, 𝐶_rework, and 𝐶_scrap 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 𝐶_rework and 𝐶_scrap 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 of 𝜃 on minimum costs (𝐶_rework=80 and 𝐶_scrap=80).

Table 10 

Effects of 𝐶_rework on minimum costs (𝜃=250 and 𝐶_scrap=80).

Table 11 

Effects of 𝐶_scrap on minimum costs (𝜃=250 and 𝐶_rework=80).

Figure 8 

The effect of 𝜃 on minimum costs (𝐶rework= 80 and 𝐶scrap= 80).

Second is the effect of 𝐶_rework while 𝜃 and 𝐶_scrap are held at 250 and 80, respectively. Figure 9 demonstrates that while the value of 𝐶_rework 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 of 𝐶rework on minimum costs (𝜃 = 250 and 𝐶scrap = 80).

Last is the effect of 𝐶_scrap while 𝜃 and 𝐶_rework are held at 250 and 80, respectively. Figure 10 represents that while the value of 𝐶_scrap 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 of 𝐶scrap 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 Lambert W 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.

Appendices

A. Sensitivity Analysis

This section provides a result of the sensitivity analyses conducted for the numerical example in Section 4. Table 9 shows effects of 𝜃 while holding 𝐶_rework and 𝐶_scrap at 80 and while changing values of 𝜃 from 100 to 400 with an increment of 25. Table 10 shows effects of 𝐶_rework while holding 𝜃 and 𝐶_scrap at 250 and 80, respectively and while changing values of 𝐶_rework from 40 to 160 with an increment of 8. Table 11 shows effects of 𝐶_scrap while holding 𝜃 and 𝐶_rework at 250 and 80, respectively and while changing values of 𝐶_scrap from 40 to 160 with an increment of 8.

B. The Lambert W Function

Since the Lambert W function is defined as𝑓(Ψ)=Ψ𝑒^Ψ, Lambert W (Ψ) allows solving such functional equation as 𝑔(Ψ)𝑒^𝑔(Ψ)=Ψ and 𝑔(Ψ)=𝑒^{Lambert𝑊(ln(Ψ))}, and Ψ𝑒^Ψ=𝜔 and Ψ=Lambert𝑊(𝜔) ([20, 53]). As shown in Section 3.3, this functional equation is encountered in the tolerance optimization models involving normally distributed random variables and use of the Lambert W function facilitates obtaining a closed-form solution for the optimal specification limits. Based on Lambert W principle [52], Shin et al. [20] consider the equation as follows: Δ₄=Δ₁𝑥+Δ₂𝑒^Δ3𝑥. (B.1) By dividing Δ₁, multiplying Δ₃, and denoting Ψ=Δ₃(𝑥+Δ₂), (B.1) becomes Δ₃Δ₄𝑒^Δ2Δ3Δ₁=Ψ𝑒^Ψ1. (B.2) Replacing Δ₃Δ₄𝑒^Δ2Δ3/Δ₁ by 𝜔, (B.2) then becomes Ψ=Lambert𝑊(𝜔). (B.3) Denoting 𝜔=Δ₃Δ₄𝑒^Δ2Δ3/Δ₁, (B.3) can then be ΔΨ=Lambert𝑊³Δ₄𝑒^Δ2Δ3Δ₁. (B.4) Substituting Ψ to Δ₃(𝑥+Δ₂), the solution in terms of x can then be obtained as follows: 𝑥=±Lambert𝑊Δ₃Δ₄𝑒^Δ2Δ3/Δ₁Δ₃−Δ₂. (B.5)

Acknowledgment

This work was supported by the Dong-A University research fund.