Repairing constrained experimental regions and designs for mixture or nonmixture variables when some design points are unacceptable

Abstract A constrained experimental region (CER) involving mixture variables or nonmixture variables is generally specified by lower and/or upper bounds on the variables and possibly by lower and/or upper bounds on linear combinations of the variables. Additionally, the proportions of mixture variables typically are constrained to sum to unity. In some cases, an experimental design generated to explore the CER may contain unacceptable points (e.g., infeasible points or points with undesirable response values). This may not be discovered until after the experiment has been completed. In such cases, it can be desirable to repair the experimental design and the CER so that they contain only acceptable points. Two approaches for performing such repairs are discussed. The first approach repairs the CER to exclude unacceptable subregions, then uses existing optimal design methodology to select design points to repair the design. The new second approach (i) replaces each unacceptable design point with an acceptable design point and (ii) adds one new constraint associated with each replacement design point to repair the CER. Both approaches are illustrated with a three-component mixture experiment example that permits visualization. A 15-component mixture experiment example is also used to illustrate the new second approach, which is reasonable for experiments with many variables. Finally, an example involving three nonmixture variables is provided in supplementary material. Applying the approaches for repairing experimental designs and CERs to problems with both mixture and nonmixture variables is briefly discussed.


Introduction
A constrained experiment involves varying the settings of mixture or nonmixture variables in various combinations over a constrained experimental region (CER), and then observing the values of one or more response variables for each combination. For simplicity in this article, mixture or nonmixture experimental variables are denoted by x i , i ¼ 1, 2, … , k. If the k variables are mixture variables, the x i represent proportions of the variables in a mixture, such that 0 x i 1 i ¼ 1; 2; :::; k ð Þand X k i¼1 x i ¼ where, for the hth MVC, the A hi are coefficients of a linear combination of variables, and C h and D h are lower and upper bounds, respectively. In general, experimental regions with constraints of the forms in Eqs.
[1] if applicable, are polyhedral regions. See Cornell (2002) for other discussions of constraints, constrained regions, experimental designs, and data analyses for experiments involving mixture variables. In practice, subject-matter experts specify the SVCs and MVCs based on their knowledge and goals for a particular constrained experiment. Often SVCs and MVCs are specified with the goal of excluding points that are infeasible or would yield undesirable values of one or more response variables. We subsequently refer to such points as "unacceptable." In some cases, subject-matter experts may not know enough about a particular constrained experiment problem to specify SVCs and MVCs so as to exclude all unacceptable points in the CER. In other cases, SVCs and MVCs may be specified to boldly explore a larger CER, so as to (i) identify points with significantly improved values of response variables or (ii) develop models to identify the boundaries between acceptable and unacceptable points. In such cases, the CER has a higher chance of containing unacceptable points.
In this article, we address the situation where (i) SVCs and MVCs were developed to specify a CER, (ii) an experimental design was generated and implemented to explore the CER, and, (iii) after the experimental work was completed, some design points were found to be unacceptable. We discuss and illustrate two approaches for repairing a constrained experiment design and CER to include points that are acceptable. We consider the two situations where the experimental variables are all mixture variables or are all nonmixture variables. The first of the two repair approaches can directly be applied to problems involving both mixture and nonmixture variables. However, there are several possible variations of the second repair approach that we have not fully developed for experiments involving both mixture and nonmixture variables. So this article focuses on repairing experimental designs and CERs involving mixture variables or nonmixture variables and not both.
Section 2 discusses the first approach, which begins by repairing the SVCs and MVCs that specify the CER, and then repairing the experimental design. We refer to this as the "region-design repair" (RDR) approach. However, for some constrained experimental problems, there may not be sufficient basis to first repair the SVCs and MVCs. For such problems, Section 3 presents a second approach in which the experimental design is repaired first and then corresponding modifications are made to the SVCs and/or MVCs to repair the CER. We refer to this as the "design-region repair" (DRR) approach. Section 4 discusses assessing the quality of a repaired experimental design for the repaired CER. Section 5 introduces two mixture experiment examples included in this article, mentions a nonmixture variable example included in supplementary material, and notes the scope of problems addressed. Section 6 discusses a three-component mixture experiment example and illustrates applying the RDR and DRR approaches for repairing the constrained mixture experiment design and CER. Section 7 discusses a 15-component mixture experiment example that was the motivation for this work and illustrates applying the DRR approach for repairing the constrained mixture experiment design and CER. Section 8 summarizes the RDR and DRR approaches, discusses their pros and cons, and makes recommendations for choosing between the two approaches for repairing constrained experimental designs and CERs.
2. RDR approach: First repair the constrained experimental region, then repair the experimental design The RDR approach consists of two steps. In the first step, a subject-matter expert specifies modified SVCs and/or MVCs that exclude the unacceptable design points while retaining the acceptable design points. For this approach to be applicable, the subject-matter expert must have sufficient basis (e.g., knowledge, insights from the completed experimental design, results of additional scoping tests) to specify the modified SVCs and MVCs so that the repaired CER will exclude unacceptable points. Here, a "scoping test" means experimentally investigating a combination of the experimental variables to assess its acceptability.
In the second step, optimal experimental design methods (Atkinson, Donev, and Tobias 2007) are used to augment the subset of acceptable experiment design points with new design points from the repaired CER. Using optimal experimental design methods to augment designs has been discussed previously in the literature (e.g., Section 19 of Atkinson, Donev, and Tobias 2007;Mitchell 1974;Section 9.3 of Myers, Montgomery, and Anderson-Cook 2016). However, literature discussions of using these methods to repair experimental designs are very limited (e.g., Lewis, Mathieu, and Phan-Tan-Luu 2005; NIST/ SEMATECH 2016) and do not address repairing the CER. In this article, we are not concerned with the specific optimality criterion (e.g., D-optimality, I-optimality) or optimality algorithm (e.g., point exchange, coordinate exchange) used. This second step of the RDR approach is general and may be applied with any optimality criterion and optimality algorithm appropriate for a given constrained experiment problem. See Atkinson, Donev, and Tobias (2007) and Myers, Montgomery, and Anderson-Cook (2016) for additional discussion of optimality criteria and algorithms for optimal experimental design.
The RDR approach may be used for constrained experimental design problems involving mixture variables, nonmixture variables, or both (i.e., mixtureprocess variable and mixture-amount experimental problems). However, as mentioned earlier, the focus of this article is on problems involving only mixture variables or only nonmixture variables.
3. DRR approach: Replace unacceptable design points with acceptable design points, and then repair the constrained experimental region The DRR approach also consists of two steps. Section 3.1 discusses the first step, in which each unacceptable design point is replaced by a new acceptable design point. Section 3.2 discusses the second step, in which a new MVC is generated corresponding to each replacement design point. Section 3.3 discusses possible modifications of the DRR approach if the repairs to the CER make it desirable to use a replacement baseline point or center point. Section 3.4 discusses how the DRR CER is obtained.

DRR approach, Step 1
Two options for the first step of the DRR approach (replacing each unacceptable design point with an acceptable design point) are discussed. Option 1 relies on a subject-matter expert utilizing knowledge, experience, and possibly scoping tests to identify an acceptable replacement design point for each unacceptable design point. However, Option 1 may require a considerable number of scoping tests when the number of variables is large and/or there are more than a small number of unacceptable design points. So a second option that obtains each replacement design point with fewer scoping tests, regardless of the number of experimental variables, is desirable. Option 2 of the first step of the DRR approach sufficiently shrinks each unacceptable design point (denoted x U ) toward the center point or baseline point of the experimental design (denoted x C ) to obtain an acceptable design point. Often a center point of a CER specified by SVCs and/or MVCs is obtained by averaging the extreme vertices of the region (e.g., see Piepel 1983aPiepel , 1988. However, there may be experiment-specific reasons why a baseline point different from such a center point of a CER is included in an experimental design. For example, it may be desirable to have the lower and upper limits for one or more variables be asymmetric relative to their values in a baseline point. Under Option 2, the formula for a replacement design point (denoted x R ) on the line joining x U and x C , with shrinkage factor 0 s 1, is given by Note that s ¼ 0 means no shrinkage (so that x R ¼x U ), while s ¼ 1 means complete shrinkage (so that x R ¼x C ). Also note that Eq.
[4] applies regardless of whether there are mixture variables or nonmixture variables. It is easy to verify that, for all s, x R satisfies the constraints in Eqs.
[1], [2], and [3] because regions specified by the SCCs and MCCs are convex. Option 2 may also require using scoping tests to determine the shrinkage factor needed to obtain an acceptable replacement design point for each unacceptable original design point. The shrinkage factor s in Eq.
[4] need not be the same for different unacceptable design points. Using scoping tests to determine the shrinkage factor for a given unacceptable design point is illustrated for a three-component mixture experiment example in Section 6.
We propose Options 1 and 2 not as optimum options for the first step of the DRR approach, but rather as two reasonable options for replacing unacceptable design points with acceptable design points. Compared with Option 1, Option 2 will tend to (i) require a smaller total number of scoping tests and (ii) result in a smaller CER. On the other hand, Option 1 allows for more use of the subject-matter expert's knowledge in developing the replacement design points.

DRR approach, Step 2
In the second step of the DRR approach, a new MVC corresponding to each replacement design point is created, which repairs the CER. That is, each new MVC excludes the corresponding unacceptable design point and attempts to exclude nearby points in the original CER that are unacceptable. The ultimate result is a modified set of SVCs and MVCs that specifies a repaired CER that excludes the unacceptable design points and includes the original acceptable design points as well as the new repaired design points.
We propose the new MVC corresponding to each replacement design point to be based on the equation of the plane going through the replacement design point (x R ) that is orthogonal to the vector from x C to x R . Because any vector x in the desired plane containing x R is orthogonal to the vector determined by x C and x R , the dot product of these vectors equals zero.
Hence, the vector formula for the desired plane is given by regardless of whether the variables are mixture or nonmixture variables. Expressing Eq.
[5] as an algebraic equation yields To construct an MVC from Eq.
[6], it must be determined whether the "¼" sign should be replaced by the "!" or " " inequality sign. Because x C always lies inside the repaired CER, evaluating Eq.
[6] at x C (i.e., indicates which inequality is appropriate. If evaluating Eq.
[6] at x C yields a positive value, the MVC is given by This inequality may also be written as with the constant on the right side being a lower bound for the expression in terms of the experimental variables on the left side. If evaluating Eq.
[6] at x C yields a negative value, the MVC is given by This inequality may also be written as with the constant on the right side being an upper bound for the expression in terms of the experimental variables on the left side.

Discussion of the center point or baseline point
Option 2 of the first step of the DRR approach and the second step of the DRR approach utilize the center point or a baseline point (x C ) of the original experimental design. The methods in Sections 3.1 and 3.2 assume that this point is still a reasonable center point or baseline point for the repaired CER. If not, then the x C from the original design can be changed to a new x C for the repaired CER and used in the Sections 3.1 and 3.2 calculations. If x C in the original constrained experimental design was a baseline point chosen by a subject-matter expert, then the expert would choose the new x C for the repaired CER. However, suppose that the original x C was a center point that no longer adequately represents the center of the repaired CER. Then the steps of the calculation in Sections 3.1 and 3.2 could be iterated a sufficient number of times to obtain a representative center point. The center point for a given iteration could be calculated as the average of the vertices of the CER for that iteration.

Repaired constrained experimental region
The original SVCs and any MVCs, plus the new MVCs, specify the repaired CER. However, this statement must be qualified in two ways. First, it is possible (though unlikely) that a new MVC is actually a SVC. Second, the new MVCs may make one or more of the original SVCs and/or MVCs unnecessary. Piepel (1983bPiepel ( , 1988 discusses methods for assessing whether constraints on mixture and/or nonmixture variables are unnecessary. If there are unnecessary constraints, they should be deleted from the current set of SVCs and MVCs to yield the final set of SVCs and MVCs specifying the repaired CER.

Assessing repaired experimental designs
With both the RDR and DRR approaches, it is expected that the design properties of the repaired experimental design will not be as desirable as the properties of an experimental design that would have been developed if the repaired CER was available in the first place. To assess the magnitude of this consequence of repairing an experimental design, various design properties can be calculated and compared for the repaired design and a new design containing the same number of design points within the repaired CER. Examples of design properties that can be compared include fraction of design space (FDS) plots (Zahran, Anderson-Cook, and Myers 2003) and efficiencies related to the D-optimality, I-optimality, Aoptimality, and G-optimality criteria. For simplicity in this article, we focus on optimal designs generated using the D-optimality criterion and the I-optimality criterion Minimize In Eqs.
[9] and [10], X represents the design matrix, f(X) denotes the model matrix that expands the design matrix according to the assumed model form represented by f(Á), x denotes a point in v, f(x) denotes the point expanded according to the assumed model form, and a superscript T denotes vector or matrix transpose. An I-optimal design per Eq.
[10] minimizes the average prediction variance of the assumed model f(Á) over the CER denoted by v (Goos, Jones, and Syafitri 2016).
In this article, we assess the relative D-and I-efficiencies of a repaired design compared with a new design on a repaired CER, given respectively by [11] and where X Rep is a repaired design for the repaired region (denoted RR), X New is a new design for the RR, x is any point in the RR, and p is the number of parameters in the assumed model form used to generate the designs. These relative efficiencies do not depend on the numbers of points in the repaired and new designs because those numbers of points are the same for problems addressed in this article. The D-Eff Rel and I-Eff Rel in Eqs.
[11] and [12] are defined so that values greater than 1.0 correspond to the repaired design being less efficient (worse) than the new design. As a rule of thumb, we consider repaired designs with D-Eff Rel and I-Eff Rel less than 1.20 to be acceptable for practical use.

Examples to illustrate the RDR and DRR approaches
In Sections 6 and 7, two mixture experiment examples with 3 and 15 mixture variables (i.e., components) are used to illustrate the RDR and DRR approaches for repairing a constrained experiment design and CER. In online supplementary material, an example involving three nonmixture variables is used to illustrate the RDR and DRR approaches.
To focus on the specifics of the approaches proposed, we assumed (i) the number of replacement design points is equal to the number of unacceptable design points and (ii) there is no block effect between the original set of design points and the set of replacement design points. Repairing designs with more new points than unacceptable points is straightforward with the RDR approach. Situations where there may be a block effect between the original and replacement design points will need to be addressed in the future.
6. Three-component mixture experiment example Snee (1985) discussed an aerosol propellant example in which there were three mixture components. The proportion of a pressurizing agent was constrained as 0.2 x 2 0.8, to avoid pressures that were too low (lower bound) or too high (upper bound). However, there were solubility problems for three subregions of the CER, which led to modified SVCs and MVCs and a revised experimental design (see Figure 8 of Snee 1985).
In this section, a modification of the Snee (1985) example is used. We retained the mixture CER of Snee, specified by the SVCs of 0 x 1 0:8; 0:2 x 2 0:8; and 0 x 3 0:8: These SVCs specify a two-dimensional polyhedral region with four vertices and four edges, as shown in Figure 1. We chose a different original experimental design than Snee (1985), to illustrate different kinds of design points and CER repairs. The nine original design points for this modified example are marked in column D of Table 1 and are shown in Figure 1. This constrained mixture experiment design contains the four vertices, four edge midpoints, and center point (average of the four vertices) of the CER. For this illustrative example, all four of the vertices are designated as being unacceptable mixtures (with triangles as plotting symbols in Figure 1).

Illustration of the RDR approach
Recall from Section 2 that the first step of the RDR approach involves a subject-matter expert specifying a modified set of SVCs and (if applicable) MVCs that excludes the unacceptable design points while retaining the acceptable design points from the original experimental design. For this example, suppose the subject-matter expert specifies the following modified SVCs and two new MVCs: 0 x 1 0:65 0:2 x 2 0:8 0 x 3 0:7 7x 1 À3x 2 þ17x 3 ! 0 17x 1 À3x 2 þ7x 3 ! 0 : [14] Note that the upper bound of x 1 was reduced from 0.8 to 0.65, while the upper bound of x 3 was reduced from 0.8 to 0.7. These two modifications to the SVCs remove portions of the original CER near two of the unacceptable design points (#1 and #4 in Table 1).
The new repaired CER specified by the constraints in Eq.
[14] is shown in Figure 2. The two new MVCs, listed in Eq.
[14] and seen in Figure 2, remove portions of the original CER near the other two unacceptable design points (#2 and #3 in Table 1). Appendix A discusses how the two MVCs in Eq. [14] were developed.
In the second step of the RDR approach, the five original design points that were acceptable (#5-9 in Table 1) were augmented with four replacement design points chosen using the D-optimality criterion. For this example, a point-exchange algorithm was employed with the vertices and edge centroids of the repaired CER used as candidate points. A coordinate-exchange algorithm without candidate points could also have been used, with a D-or I-optimality criterion. For the D-optimality criterion, it was assumed that the six-term Scheff e quadratic mixture model E y ð Þ ¼ b 1 x 1 þb 2 x 2 þb 3 x 3 þb 12 x 1 x 2 þb 13 x 1 x 3 þb 23 x 2 x 3 [15] would adequately approximate the true unknown response surface(s) over the repaired CER. The four replacement design points chosen using the D-optimality criterion are marked in column E of Table 1. The nine-point repaired experimental design is shown in Figure 2. The four new points in the repaired mixture design are seen to be vertices of the repaired CER. However, the repaired design "replaced" point #1 with points #10 and #11 and "replaced" point #4 with points #12 and #13. Neither point #2 or #3 was "replaced" in the repaired design. We mention this as interesting, not because it is necessarily bad.
To assess the RDR design per Section 4, a new nine-point D-optimal design for the RDR CER was generated in Design-Expert (Stat-Ease 2017) using a point-exchange algorithm with the vertices, edge center points, and overall center point used as candidate points. The model in Eq.
[15] was assumed as an adequate representation of the response surface of interest. The points in the resulting new design are marked in column F of Table 1. The relative efficiencies (from Eqs. [11] and [12]) of the repaired design relative to the new design generated using the D-optimal criterion are D-Eff Rel ¼ 1.030 and I-Eff Rel ¼ 0.924. These values indicate that the RDR design has only a slightly worse D-efficiency and a better I-efficiency than the new D-optimal design for the repaired CER.

Illustration of the DRR approach
We now illustrate the DRR approach using Option 2 (see Section 3) for the modified three-component aerosol propellant example. The first step of the DRR approach is to generate one replacement design point for each unacceptable design point. Using Option 2, the replacement design points were obtained by shrinking the original design points that were unacceptable (#1 through #4 in Table 1) toward the original design center point (#9 in Table 1). We judged the original center point as sufficiently representative to use as the center point for the repaired CER.
In practice, it may be necessary to perform a small number of scoping tests to determine the amount of shrinkage appropriate for each unacceptable design point. Figure 3 shows a line connecting one  unacceptable design point (#1 in Table 1) to the original design center point (#9 in Table 1). This line represents the different points possible from shrinking this unacceptable design point toward the center point. The first scoping test (numbered 1. in Figure 3) shrunk the unacceptable design point with x 1 ¼ 0.8 to a point with x 1 ¼ 0.7. For the purpose of this conceptual illustration, the values of x 2 and x 3 for this and subsequent shrunken versions of point #1 are not important. Suppose that the first scoping test determined that the corresponding mixture is still unacceptable (represented by a smaller triangle in Figure 3). Then a second scoping test (numbered 2. in Figure 3) shrunk the unacceptable design point to a point with x 1 ¼ 0.6, which was determined to be acceptable (represented by a smaller square in Figure  3). A third scoping test investigated the point with x 1 ¼ 0.65 that is halfway between the points in the two previous scoping tests, which was determined to also be acceptable (represented by a larger square in Figure 3). It was decided to use this third scoping-test point with x 1 ¼ 0.65 as the replacement point for the corresponding unacceptable design point.
Section B.1 of Appendix B illustrates applying Eq.
[4] to calculate the mixture coordinates for the replacement point with x 1 ¼ 0.65. The resulting replacement design point for point #1 is listed as point #18 in Table 1. A similar process was used to calculate the mixture coordinates of the replacement design points for unacceptable points #2, #3, and #4. A few scoping tests were performed to determine the appropriate amount of shrinkage for each point, with values of s ¼ 0.129, 0.129, and 2/11 obtained. These values were then used to calculate the coordinates of the replacement design points for points #2,#3,and #4 using Eq. [4], as illustrated in Section B.1. Those replacement design points are listed as points #19, #20, and #21, respectively, as marked in column G of Table 1.
The second step of the DRR approach is to calculate the coefficients of the MVC expression corresponding to each replacement design point. Each such MVC expression is the equation of a plane orthogonal to the line from the design center point to the corresponding replacement design point. As discussed in Section 3, the general formula of the plane corresponding to an MCC is given by Eq. [6]. Ultimately, the MVC is given by Eq.
In summary, the repaired nine-point experimental design using Option 2 of the DRR approach consists of Points #5-9 and #18-21 in Table 1. Figure 4 shows the repaired experimental design points (plotted as squares) and the repaired CER specified by the only SVC that remains necessary (0.2 x 2 ) and the MVCs in Eqs. [16] and [17]. Technically the x 2 0.8 SVC is made unnecessary by the last two MVCs in Eq.
[17] because those constraints allow only one point with x 2 ¼ 0.8 in the repaired CER. Finally, the light gray lines in Figure 4 that join each repaired design point to the center point are seen to be orthogonal to the four edges of the repaired CER corresponding to the new MVCs in Eqs. [16] and [17].
To assess the DRR design per Section 4, a new nine-point D-optimal design for the DRR CER was . Constrained region and nine-point mixture experiment design repaired using the DRR approach and Option 2 for the modified aerosol propellant example. The acceptable original design points are shown as circles, the unacceptable original design points are shown as triangles, and the replacement design points are shown as squares. The light gray lines connecting the center point to each repaired design point are seen to be orthogonal to the edges of the constrained experimental region resulting from the new multiple variable constraints. generated using the same methods described at the end of Section 6.1. The points in the resulting new Doptimal design are marked in column H of Table 1. The relative efficiencies (from Eqs. [11] and [12]) of the DRR design relative to the corresponding new design (both generated using the D-optimality criterion) are D-Eff Rel ¼ 1.128 and I-Eff Rel ¼ 1.077. These values are < 1.20 per Section 4, suggesting the corresponding repaired design has efficiencies not too much worse than the new design generated for the repaired CER.
7. Fifteen-component mixture experiment example Piepel et al. (2016) discussed the development of a layered mixture experiment design (Cooley et al. 2003;Piepel, Anderson, and Redgate 1993;Piepel, Cooley, and Jones 2005) to investigate the effects of 15 simulated low-activity waste (LAW) glass components on various glass properties. The layered experimental design contained 35 points, including 18 outer-layer points, 13 inner-layer points, a baseline point investigated three times, and a point from a different lab's study. For this example, we consider only the SVCs and MVCs for the outer layer, which are listed in Table 2. The baseline point is also listed in Table 2.
The 18 outer-layer design points (selected from the set of outer-layer vertices, as discussed by Piepel et al. 2016) are listed in Table 3.
The goal of the study was to investigate glass compositions with higher proportions of LAW components, which increased the chances that some outerlayer experimental design points may be unacceptable. After the glasses were melted and cooled, six of the 18 outer-layer experimental design points (#2, #10, #11, #15, #16, and #18 in Table 3) were found to be unacceptable because of component solubility problems (components not dissolving into the glass melt) or crystallinity problems (crystals formed upon fast cooling of the glass).
This example illustrates using Option 1 of the DRR approach, as discussed in Section 3. Glass scientists performed a series of scoping tests that modified the composition of each unacceptable glass until an acceptable replacement glass composition was obtained. The compositions of the six replacement glasses are listed as points #20-25 in Table 3.
The six new MVCs to repair the 15-component CER corresponding to the six replacement design points were calculated using Eq.  Table 4. For simplicity, all six of the MVCs in Table 4 are written in the form P 15 i¼1 A i x i þ A 0 ! 0, so that each expression evaluates to a nonnegative value if a point is on or inside that constraint.  The 35-point DRR layered design could be assessed, as discussed in Section 4, by generating a new 35-point layered design for the DRR CER using the same methods used to generate the original design (Piepel et al. 2016). However, because of the amount of work involved to do so and the relatively small changes made to obtain the replacement design points, such an assessment was judged not to be necessary.

Summary and discussion
This article presented the RDR and DRR approaches for repairing constrained experimental designs and regions involving mixture or nonmixture variables when some of the original design points are found to be unacceptable after the experimental work is completed.
The RDR approach (presented in Section 2) involves first repairing the CER, then repairing the constrained experimental design. The RDR approach relies on subject-matter experts having sufficient basis (e.g., prior knowledge, results of the original design, new scoping tests) to specify modified SVCs and/or MVCs to repair the CER. Then the original design points that were acceptable are augmented by replacement design points chosen from the repaired CER using an optimal experimental design approach. Any combination of optimal design criterion (e.g., D-optimal, I-optimal) and optimal design algorithm (e.g., point exchange, coordinate exchange) appropriate for a given problem may be used.
The DRR approach (presented in Section 3) takes the opposite approach, first replacing the unacceptable design points to repair the experimental design, then developing new MVCs to repair the CER. Options 1 and 2 for replacing unacceptable design points were presented and compared. The second step of the DRR approach involves generating a new MVC   corresponding to each of the replacement design points. The plane comprising a new MVC is orthogonal to the line determined by the replacement design point and the center or baseline point of the design. We now discuss some possible limitations of the RDR and DRR approaches. First, a repaired experimental design containing the same number of points as the original experimental design may not provide adequate design properties for the repaired CER. In such cases, the RDR approach easily provides for including more points in the repaired experimental design. The DRR approach provides only one-for-one replacements of unacceptable design points and, hence, does not provide for repairing a design with more design points than were unacceptable. Second, in this article, it was assumed there is no block effect between the set of acceptable original design points and the set of repaired design points. Existing and possibly new research is needed to address situations where there may be a block effect, especially if only one design point needs to be repaired. Third, the repaired CER obtained with the DRR approach may be smaller and not contain some acceptable combinations of variable values that would be included in the repaired CER with the RDR approach. However, many scoping tests may be required to specify revised SVCs and MVCs that include such combinations with the RDR approach, especially for more than a few experimental variables.
Ultimately, the choice between the RDR and DRR approaches depends on (i) the number of experimental variables, (ii) the number of unacceptable design points, (iii) the knowledge of subject-matter experts, (iv) how many scoping tests may be needed, and (v) the budget for repairing a constrained experimental design and region. The RDR approach is generally preferable because it follows the usual experimental design approach of specifying the (repaired) CER and then selecting the (repaired) design points. Also, the RDR approach can be implemented using commercially available optimal design software with optimal design augmentation capabilities. However, the RDR approach requires that a subject-matter expert be able to specify a modified set of SVCs and MVCs that eliminate unacceptable points from the CER. If the subject-matter expert does not have sufficient knowledge or it would require too many scoping tests to provide a basis for the RDR approach, then the DRR approach using Option 1 or 2 is a reasonable alternative. The computations required by the DRR approach are straightforward and can easily be programed in any programing language, software package scripting language, or Excel.

About the authors
Dr. Piepel is a Laboratory Fellow Statistician in the Applied Statistics and Computational Modeling Group. He is a Fellow of the American Society for Quality and the American Statistical Association. His email address is greg.piepel@pnnl.gov.
Mr. Cooley is a Senior Research Statistician in the Applied Statistics and Computational Modeling Group. His email address is scott.cooley@pnnl.gov.