This model was developed to investigate the recruitment process in the replication cycle of BMV, a positive strand RNA virus. This replication cycle is highly conserved across positive strand RNA viruses, such as Severe Acute Respiratory Syndrome (SARS) and Hepatitis C, and the BMV system has been used to gain insights into interactions of the virus with host factors [12, 13, 14]. Briefly, the mathematical model describes the interaction between Protein 1a (x(t)) and RNA3 in the unstabilized (y(t)) and stabilized (z(t)) forms; for an in-depth description see [10]. The levels of Protein 1a (x(t)) and total RNA3 (y(t) + z(t)) are measured at time points designed by the experimenter. Param- eters describing Protein 1a (r x , **d** x , A) were estimated prior to estimating parameters for

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working in collaborative efforts with ecologists, biologists, and quantitative life scientists, renewed in- terest in **design** of “best” **experiments** to elucidate mechanisms has been seen [AB]. Thus, a major question that experimentalists and inverse problem investigators alike often face is how to best collect the data to enable one to efficiently and accurately estimate model parameters. This is the well-known and widely studied **optimal** **design** problem. A rather through review is given in [BHT2014]. Briefly, traditional **optimal** **design** methods (**D**-**optimal**, E-**optimal**, c-**optimal**) [AD, BW, Fed, Fed2] use infor- mation from the model to find the sampling distribution or mesh for the observation times (and/or locations in spatially distributed problems) that minimizes a **design** criterion, quite often a function of the Fisher Information Matrix (FIM). Experimental data taken on this **optimal** mesh are then expected to result in accurate parameter estimates. We outline a framework based on the FIM for a system of ordinary differential equations (ODEs) to determine when an experimenter should take samples and what variables to measure when collecting information on a physical or biological process modeled by a dynamical system.

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Model estimating and fitting are two important topics in statistical sciences. The present paper was written considering the crucial role of statistical discussions in other sciences such as industries, medicine and so on. As mentioned earlier, model fitting is one of the important discussions in statistics; thus, in most cases, fitting and obtaining a relationship between dependent variable(s) and independent variable(s) are intended to be achieved. When the effect of an independent variable on a dependent variable is investigated, two points should be taken into consideration: a) finding independent variables which affect dependent variables, and b) finding values of dependent variables which have a significant role in model fitting. Therefore, the basic motivation for designing **optimal** **experiments** is to find an ideal **design** that could result in appropriate inferences about model parameters by conducting an experiment using this **design**.

Response surface methodology (RSM) consists of a set of statistical methods that can be used to develop, improve or optimized products (Cihon, 2003). RSM typically is used in situations where several factors influence one or more performance characteristics or responses. RSM used to optimize one or more responses or to meet a given set of specification (e. g a minimum strength specification or an allowable range of values). There are three general steps to that which comprise: experimental **design**, modeling, and optimization (Gihan, 2013). In order to obtain a higher and homogeneous mixture of crucible composition, it is important that the **optimal** additives to be added to clay and graphite are investigated. **Design** of experiment is useful in the analysis and optimization of the mixture effects of several process variables influencing responses with a minimum number of experimental runs while varying the variables concurrently (Oladosu et al, 2016). In the mixture **experiments**, the measured response is assumed to depend only on the relative ratio of the components in the mixture and not on the amount of the mixture, compared to Taguchi and factorial methods (Anderson and Whitcomb, 2014). The focus of this study is therefore to investigate the influence of optimizing the mixture of clay, graphite and selective additives such as MgO and SiC on the production of crucibles with the view of developing an effective and suitable mixing ratio for clay bonded crucible production.

The in vivo pharmacokinetic study was performed on male New Zealand white rabbits (weighing around 1.8- 2.0 kg) obtained from the Animal Vaccine Institute, Gandhinagar, India. The rabbits were housed in cages in an air conditioned room (25±2°, 30-65 % RH), 1 animal per cage, with free access to pelleted food (Pranav Agro Foods Pvt. Ltd., Vadodara, India) and water. The preclinical study protocol was approved by the Institutional Animal Ethics Committee, Pharmacy Department, The M. S. University of Baroda, Vadodara, India. All the experimental procedures were carried out as per the guidelines of the Committee for the Purpose of Control and Supervision of **Experiments** on Animals (CPCSEA), Govt. of India.

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aliasing artifacts. Furthermore, even the fastest known strategies for evaluating popular dis- crepancy measures require O(N 2 **d**) operations making evaluation, let alone optimization, for discrepancy difficult even for moderate dimensions. Finally, for most discrepancy mea- sures, the **optimal** achievable values are not known. This makes it difficult to determine whether a poorly performing sample **design** (e.g., in terms of generalization (or test error) in a regression application and reconstruction error in an image reconstruction application) is due to the insufficiency of the chosen discrepancy measure or due to ineffective optimization. Another class of metrics to describe sample designs are based on geometric distances. These can be used directly by, for example, optimizing the maximin or minimax distance of a sample **design** (Schlomer et al., 2011) or indirectly by enforcing empty disk conditions. The latter is the basis for the so-called Poisson disk samples (Lagae and Dutr, 2008), which aim to generate random points such that no two samples can be closer than a given minimal distance r min , i.e. enforcing an empty disc of radius r min around each sample. Typically,

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of an RSM problem. Recently, many researchers have considered this issue. Lucas and Ju (1992) provided a simulation study to investigate the use of split-plot designs in industrial **experiments**. Their results confirmed that steepest ascent designs produce increased precision for the subplot factors while sacrificing precision for the whole plot elements. Box (1996) explained that completely randomized **experiments** are often impractical in industry; in contrast, he indicated that split-plot **experiments** are often extremely efficient and easier to run. To illustrate the difference in the analysis, Lucas and Hazel (1997) ran the known paper- helicopter experiment as a CRD and as a split plot. Letsinger, Myers and Lentner (1996) introduced randomization designs (BRD), which are designs that have two randomisations, making them similar to split-plot designs. Vining, Kowalski, and Montgomery (2004) proposed central composite designs that were specifically designed for split- plot **experiments**. They also provided a general proof showing that for certain **design** conditions, the ordinary least squares coefficient estimates are equivalent to the generalized least squares coefficient estimates for second-order models. Kowalski, Vining, Montgomery and Borror (2004) modified the proposed central composite designs of Vining, Kowalski and Montgomery (2004) to model both the process mean and variances within a split- plot structure. Moreover, Bisgaard and Steinberg (1997) focussed on a **design** algorithm to obtain **D**-**optimal** split-plot designs. They showed that the **design** matrices for the **D**-**optimal** split-plot designs and **D**-**optimal** CRDs are typically different; moreover, they clarified that split-plot **experiments** are often more efficient than CRDs Huang, Chen and Voelkel (1998) and Bingham and Sitter (1999a) discussed minimum-aberration (MA) designs for steepest ascent **experiments**; in these works, both the whole-plot and subplot factors had two levels. The researchers provided methods for determining the MA designs and provided tables for various combinations of whole-plot and subplot factors. Some **design** issues with two-level fractional– factorial split-plot **experiments**, including where to split and where to fractionate, were presented in Bingham and Sitter (2001).

2 Experimental **design** methods are useful for making testing more economical. Results with these approaches can be achieved with less investment of time or resources. One type of experimental **design** is **optimal** experimental **design**. **Optimal** experimental **design** is a method which maximizes the amount of information gained from the experiment when the number of trials is constrained. The settings in the resulting **design** are chosen to maximize the probability of improving the model of interest. Unlike traditional factorial designs, **optimal** experimental designs require prior knowledge of the system being studied for generating the **design**. An **optimal** approach can work well when prior system knowledge is available, but results suffer if little is known about the system. **Optimal** **design** is further complicated with the addition of multiple responses.

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processes. The purpose of this paper has been to illustrate the importance of screening **experiments** and full factorial **experiments** using two catapult **experiments**. Screening **experiments** using P–B **design** were used to identify the most important control factors. P–B designs were used to separate out the ‘vital few’ main effects from the ‘trivial many’. Having identified the key factors from the screening experiment, a full factorial experiment was used to study both the main effects and the interaction effects. Apart from the use of P–B designs for screening **experiments**, one can also look at the use of powerful saturated fractional factorial designs. Instead of using a 12-run P–B **design**, it is also worthwhile looking at the use of 2 ( 7 − 4 ) fractional factorial **design** for studying seven factors in eight trials. This is a resolution III **design**, where main effects are confounded with two-way interaction effects. **Design** resolution is a summary characteristic of confounding patterns [12]. In order to study the main effects clear of two-way interactions, one could utilize the use of fold-over designs or reflected designs advocated by Box [9].

[2] Kundan Kumar, Somnath Chattopadhyaya, Hari Singh, **Optimal** material removal and effect of process parameters of cylindrical grinding machine by Taguchi method, International Journal of Advanced Engineering Research and Studies E-ISSN2249–8974, Vol. II/ Issue I/Oct.-Dec., 2012/41-45

Figure 1B shows a 3-**D** model of the effect of independent variables on PDI. The lowest PDI was obtained at the low- est amounts of oil and xanthan gum, and also at the highest amounts of surfactant and water. PDI is a measurement of inconsistency in particle size in the nanoemulsion system, and PDI value is used as an indication to measure the uniformity of particle dispersion in the nanoemulsion system. Therefore, lower PDI values are favorable. In this case, the system formed was an o/w nanoemulsion, in which the particles being measured were the oil droplets. When a small amount of xanthan gum was used, the nanoemulsion formed became thinner. The homogenization process became easier, and led to more uniformed production of particle sizes, confirmed by a low PDI result.

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of Patterson (1964), who excluded the simpler fi xed rotation **experiments** that study the eff ects of treatments on the crops of a single rotation. (Th ese can involve some of the problems discussed in below but are inherently much easier to handle.) To illustrate the concepts, Table 1 shows one block of an experiment by Glynne and Slope (1959), which was designed to assess the eff ects of previous cropping by bean (Vicia faba L.) or potato (Solanum tuberosum L.) on the incidence of eyespot (Oculimacula yallundae and Oculimacula acuformis) in winter wheat (Triticum aestivum L.). Th e crops in Years 1 and 2 are treatment crops that set the scene for Year 3, when test crops are grown to enable the diff erences between the sequences to be assessed. Th e two winter wheat crops in Year 2 also act as partial test crops in that they allow the eff ects from the wheat and potato crops in Year 1 to be assessed. No analysis was made of the bean or potato yields.

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2.Results in this study imply that a fewer number of gears which are **optimal** can provide a performance equivalent to the multi-speed gear system. As expected, reducing the number of gears without a loss of physiological efficiency would result in easier gear shifting.

2.3.3. Preparation of Orally Dissolving Films of ketorolac tromethamine and olopatadine-HCl Using **D**-**optimal** **design**: Fast dissolving oral films were prepared by solvent casting technique. The formulations were prepared as per Table (2). The hydrophilic polymers namely Hydroxy propyl methyl cellulose (HPMC E5) and Kollicoat IR were accurately weighed and dissolved in 10 mL distilled water and 0.1 mL glycerol was added as a plasticizer. Kollidone was added as superdisintegrant. Then OPD-HCl - Kleptose complex in ratio (1:1) and KTR- T were added to the polymeric dispersion under constant stirring with a magnetic stirrer for 15 minutes. The resultant homogeneous solution was poured into a petridish. Then the films were dried in an oven at 50ºC for 24 h. The films were then carefully removed and cut into strips of dimension 3×3 cm, and the dried films were wrapped in a butter paper, covered with an aluminum foil and stored in an air tight glass container for further investigations (31) .

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When both the number h of undirected edges and the number g of nodes in a chain component are very large, instead of using the **optimal** designs, we may use the approximate designs via sampling DAGs. We checked several standard graphs found at the Bayesian Network Repository (http://compbio.cs.huji.ac.il/Repository/). We extracted their chain components and found that most of their chain components have tree structures and their sizes are not large. For example, ALARM with 37 nodes has 4 chain components with only two nodes in each component, HailFinder with 56 nodes has only one component with 18 nodes, Carpo with 60 nodes has 9 components with at most 7 nodes in each component, Diabets with 413 nodes has 25 components with at most 3 nodes, and Mumin 2 to Mumin 4 with over 1000 nodes have at most 21 components with at most 35 nodes. Moreover, all of those largest chain components have tree structures, and thus we can easily carry out **optimal** designs as discussed in Example 2.

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The aim of this work is to develop microsphere formulation and study the impact of drug-polymer ratio, surfactant concentration and stirring speed on particle size and drug entrapment during preparation. Clotrimazole-loaded Eudragit S100 microspheres were prepared by emulsion solvent evaporation method. A three level factorial Box-Behnken **design** was used to characterize and optimize the formulation. The result of analysis of variance test indicated that the test is significant. The optimum drug:polymer ratio (X1), surfactant concentration (X2) and stirring speed (X3) was found to be 1:4, 1.5% and 1000 rpm, respectively to obtain particle size and maximum entrapment efficiency of microspheres as 35.6 µm and 83.3%, respectively. The drug release from microsphere based cream exhibited a controlled release pattern.

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Since n 1 and n 2 are required to be the integers. For practical purpose if the solution is non integer then the NLPP is solved using Branch and Bound method instead of rounding the non integer sample sizes to the nearest integral values. However in some situation for small samples the rounded off allocation may become infeasible and non **optimal** In order to get the integer value we use Branch and bound method as given below for problem (p1). Using LINGO-13.0, we obtain the integer optimum allocation as n 1 =4 and n 2 = 21 and the **optimal** value is 96.

In our numerical study, we observe that standardized maximin **D**-**optimal** designs with respect to rectangular or triangular parameter spaces have at most three support points. A third support point appears if a ”large” Θ is chosen, since these Ψ −∞ -**optimal** designs must be ”good” despite a high level

L., is an herb widely used throughout the world. It is one of the most popular plants in Morocco. For the purpose of examining the factors affecting extraction of this plant’s essential oil by hydrodistillation, a screening study by Hadamard matrix type Placket and Burman was conducted. After an appropriate choice of seven variables, sixteen **experiments** lead to a mathematical model of first degree ponse function (yield) to factors. After the experiment’s realization and data analysis, we concluded that five factors have a significant effect on the hydrodistillation process, namely: the extracting the material/water ratio and the heating temperature with 0.015 respectively. Moreover, the plant material’s drying and

In this paper, it is proposed the use of canonical forms to solve a problem non-standard of **optimal** experimental designs laid out by Atkinson et al. (1995) [1] upon calculating the optimum dose of a fly insecticide. The main difficulty arises by adding uncertainty about gender since they differ in the response and the experiment only senses applied on the whole population. The witty transformation of the problem to a canonical version reduces the parameter dependence leading to analytical expression of the **optimal** weights. From these, we are able to compute **D**-**optimal** designs for several cases. In particular, it is constructed **optimal** designs for two and three dose levels.