Role of extreme vertex design approach on the mechanical and morphological behaviour of residual soil composite

This research work reports the usability of binary additive materials known as tile waste dust (TWD) and calcined kaolin (CK) in ameliorating the mechanical response of weak soil. The extreme vertex design (EVD) was adopted for the mixture experimental design and modelling of the mechanical properties of the soil-TWD-CK blend. In the course of this study, a total of fifteen (15) design mixture ingredients’ ratios for water, TWD, CK and soil were formulated. The key mechanical parameters considered in the study showed a considerable rate of improvement to the peak of 42%, 755 kN/m2 and 59% for California bearing ratio, unconfined compressive strength and resistance to loss in strength respectively. The development of EVD-model was achieved with the aid of the experimental derived results and fractions of component combinations through fits statistical evaluation, analysis of variance, diagnostic test, influence statistics and numerical optimization using desirability function to analyze the datasets. In a step further, the non-destructive test explored to assess the microstructural arrangement of the studied soil-additive materials displayed a substantial disparity compared to the corresponding original soil material and this is an indicator of soil improvement. From the geotechnical engineering perspective, this study elucidates the usability of waste residues as environmental friendly and sustainable materials in the field of soil re-engineering.

In the course of constructing civil engineering products such as highway infrastructures, buildings, embankments etc. necessitates the use of soil materials. Soil as a construction material forms a sensitive layer in road structures and this makes it valuable for both construction and other human activities. It may interest one to know that there are cases where the in situ soil contains a substantial quantity of montmorillonite clay mineral and it is termed as black cotton soil. Such soil possesses expansive tendencies, both montmorillonite and illite have high dominance in the structure of this soil and consequently accountable for swell-shrink behaviour associated with the soil. These clay minerals are pivotal menace to foundations of uncountable civil engineering products such as road embankments, roadways etc. However, during the wet period this soil material grasps water whereas during the dry period it drops it moisture. This seasonal alterations results to stability issues which will in return stress the foundation of civil engineering products like buildings and road pavements. Although a number of soil investigations have been documented on the numerous complications of using black cotton soil during construction activities 1,2 , it is almost impossible for civil engineers not to encounter these soil materials in the cause of construction, most especially in the Northern part of Nigeria. In the course of constructing infrastructural projects, the stable soils are utilised as infrastructural construction material whereas the unstable ones are replaced with soil materials of good geotechnical standing or improved via stabilization protocols by the incorporation of additives. The occurrence of these unstable soil materials has been a major setback for the construction industry most especially civil engineers, thus making the execution of infrastructural projects on unfavourable soil conditions practically unavoidable. The setbacks posed by the soil when encountered during infrastructural development are well reported in the literature 2,3 . To mitigate these setbacks, preventive options including soil replacement, chemical modification and special foundation systems are employed 4 . The mitigation option often applied in soft soil is amendment with conventional or nonconventional ameliorants. The act www.nature.com/scientificreports/ of amending soft soil via stabilization using conventional ameliorants is becoming an old-fashioned practice coupled with some environmental concerns. The production process of these ameliorant materials coupled with the discharge of greenhouse gases is one of the numerous threats posed by them. It is important to note that the amendment of soil with non-conventional ameliorants which are sustainable and economical, has proved to be very successful due to their ability to perform anticipated outcomes in the geotechnical parameters of soft clay soil compared to conventional ameliorants. Thus, the emerging advances in construction materials have proved that diverse ameliorants rich in alumina and silica such as sawdust ash-quarry dust 5 , cement kiln dust-rice husk ash 6 , cement kiln dust-metakaolin 7,8 , cement kiln dust 9 , periwinkle shell ash 10 , oyster shell ash 11,12 , quarry dust 13 , marble dust-rice husk ash 14 , coconut husk ash 15 , corncob ash 16 , locust bean waste ash 17,18 , groundnut shell ash [19][20][21] , lime-rice husk ash 22 , bambara nut shell ash 23 , yam peel ash 24 , metakaolin 25 , mine tailings 26 , palm bunch ash 27 , cement kiln dust-periwinkle shell ash 28,29 , cow bone ash-waste glass powder 30 , sawdust ash-lime 31 , rice husk ash-sisal fibre 32 , coffee husk ash 33 , construction waste 34 , sawdust 35 , nanosilica 36 , electric arc furnace dust 37 , iron ore tailing 38 and so on have been utilised in the amendment studies of deficient soils and concrete structures. Conclusively, as a result of the positive outcomes recorded by the incorporation of solid wastes in soil amelioration studies, a comprehensive review was carried out on the trends in expansive soil stabilization using calcium-based stabilizer materials 39 . From their all-inclusive review study, it was documented that calcium-based stabilizer materials (CSMs) display pozzolanic tendencies thereby influencing the engineering, geotechnical and microstructural properties of clayey soil materials. However, it is worth noting that the main motivation behind the use of these ameliorants/additives is the decline in greenhouse gas production and the cost of purchasing conventional binding agents (Cement and lime). Remarkable leaps and bounds have been accomplished in the field of civil engineering construction materials using single non-conventional ameliorants. Attah et al., 25 studied modelling and predicting CBR values of lateritic soil treated with metakaolin (non-conventional ameliorant) and concluded that there was an enhancement in the strength values of the soil when 20% metakaolin was added by air-dried weight of the lateritic soil. Interestingly, a different approach has been explored by soil experts, which involves the use of multi non-conventional additives thereby in some cases incorporating non-conventional ameliorants combined with a little fraction of the conventional binders for soil improvement. The essence of this recent approach is to ensure that there will be adequate pozzolana in the soil matrix reacting with silica and alumina which will activate high strength in the studied soil. Similarly, Ekpo et al., 28 also combined periwinkle shell ash and cement kiln dust to strengthen selected lateritic soils. In a related development, Etim et al., 40 improved the properties of lateritic soil by using the combination of lime and periwinkle shell ash and reported the microstructural and geotechnical behaviour of both the untreated and optimally treated soils. The enhancement in the soil properties was linked with the pozzolanic interplay as a result of the incorporation of lime-periwinkle shell ash blend into the soil.
It is worth mentioning, the recent experimental investigation carried out by soil experts has proved the continuous efforts towards the amelioration of deficient soil using single or multi-additives. In the course of ameliorating this deficient soil, considerable improvement in the properties of the soil has been documented. However, notwithstanding the improvement recorded by most soil experts, only a few have deployed optimization tools to arrive at an appropriate target in the properties of the soil. Thus, the latest innovations in Civil engineering practice have prompted specialists in materials and construction engineering to deploy optimization techniques with the sole aim of building precise and reliable models for elucidating engineering problems 9,41,42 . This development occasioned the possibility of formulating equations to become accustomed to the technical hitches connected with modelling the behaviour of soil materials 43 . Remarkably, numerous materials and construction engineering scientists have looked into the achievability of utilizing diverse optimization tools and the conducted experimental outcomes revealed optimistic outcomes such as: extreme vertex design (EVD) 44 , response surface methodology (RSM) 34,45 , artificial neural network (ANN) 46 , gene expression programming (GEP) 47,48 , Taguchi optimization method 49,50 , Scheffe's optimization approach 7 , artificial intelligence models 51 and so on. In view of this, the EVD approach is a mixture blending system which utilises a smaller fraction within the simplex and has been widely used in Civil engineering. Also, EVD is an offline optimization measure whose objective is to optimise most engineering parameters and processes. It also aids in the design of experiment process and achieving the best possible mixture combinations during optimization protocols. Moreover, to the best of the authors' knowledge, the major advantage of EVD compared to other optimization strategies is the flexibility in terms of the imposition of other constraints factor levels by specifying both the upper and lower bounds on the components via the designation of linear constraints for blends 52 . This has prompted numerous research experts to explore the utilization of EVD in Civil engineering materials 9,49,52 .
Moving onward, recent investigation have shown the level of potency displayed by residues such CK or TWP when utilised as single additive or combined with other additives in geo-engineering protocols. Howbeit, to the best of the author's knowledge there exist infinitesimal or no known studies documented in open literature that evaluates the role of blending CK-TWP in ameliorating the geotechnical performance of an expansive soil using the principles EVD. Hence, this research becomes necessary and crucial. In this manuscript, one of the author's hotspots is the use of scanning electron microscopy (SEM) to elucidate mutual interference between the studied soil and additives/ameliorants thereby detecting the amelioration potentials of the additives/ameliorants. The main focus of employing SEM is to also visualise soil-ameliorants interplay at the microscale efficiently and as well provide both qualitative and quantitative reports. The motivation behind the use of SEM in this study is not unconnected with the scanty use of qualitative tests in optimization studies and as well close the gap of the existing studies in soil re-engineering. Finally, it is presumed that blending CK-TWP would be a cheaper and sustainable option compared to the conventional stabilization agents such as lime, bitumen and cement. The benefit of the present investigation will entail in two folds: enhancing the soil performance and as well re-utilising / disposing of waste. Moreover, it is believed that the upshot of this study will create an enabling strategy for waste management.

Materials.
A potentially fine-grained soil also known as black cotton soil engaged in this current optimization exercise was acquired from its deposit in the municipality of Deba, Gombe State, Nigeria (which lies within latitude 10° 12ʹ 42.73ʹʹ N, longitude 11° 23ʹ 13.56ʹʹ E) as seen in the GIS plot in Fig. 1. The soil understudy is known to have a distinctive mineralogical and plasticity response. After excavating the soil material at an average depth of about 1.0 m, it was stacked and transported to the laboratory for the experimentation exercise. Before the experimental tests, it was sundried, pulverised and sieved with the relevant British Standard (BS) sieves. The two additive materials include: calcined kaolin (CK) and tile waste dust (TWD). The raw kaolin was gotten from a locality along the Enugu-Port Harcourt Highway. It was subjected to a controlled thermal activation/ calcination and the ash residue gotten after the calcination protocols were allowed to cool and sieved via a 75 μm sieve to obtain a finely powdered substance for experimentation. The calcined kaolin had a specific gravity of 2.58. The tile wastes were gotten from a construction site in the Uyo metropolis, having a specific gravity of 2.75 as shown in Fig. 2. The dominant clay minerals obtainable in the unaltered soil were detected via means of the X-ray diffraction strategy whereas the elemental oxide compositions of the additive materials used in the study (soil, calcined kaolin and tile waste dust) were assessed utilizing the X-ray fluorescent strategy as shown in Fig. 3.  However, in the course of this study it can be seen that studied soil in it natural form has poor geotechnical performance and as such it calls for soil improvement. Secondly, the utilisation of TWD and CK as additive materials was connected to some reasons such as: [i] Incorporation of both TWD and CK will ascertain that there will be enough pozzolana in the soil matrix to react with both silica and alumina components of the soil in the presence of moisture.
[ii] Both TWD and CK are waste materials, reutilising them will curb environmental nuisance.
California bearing ratio test (CBR). The procedure established by British Standard was engaged to find out the CBR values of virgin soil samples as well as the soil-additives admixed with the mix ratios developed by EVD strategy. The guiding principles of the British Standard Light compaction strategy were utilized in executing these experimentation exercises on the soil materials. This test involves the use of 2360 cm 3 CBR mould and compacting the soil-additives uniformly with a 2.5 kg rammer in three layers. Thereafter, the compacted soil samples were subjected to a curing duration of 6 days. On completion of the 6 days of curing, the CBR samples were submerged in water for 48 h before testing (Nigerian General Specification) 53 .
Unconfined compressive strength test (UCS). Expansive soil material was subjected to amelioration protocols with the mixtures of CK and TDW via the EVD method. Based on the outlines ascribed in BS 1377 part 4 54 , the soil mixtures were meticulously mixed and utilised for the UCS experimentation. The soil samples were compacted using British Standard Light (BSL) compaction energy and were allowed to cure for 7 days. Before the curing, the specimens were exposed to loadings axially in the UCS machine and for each experimentation, two samples were taken and the average values were documented.
Durability. Durability assessment could as well be referred to as the resistance to loss in strength when exposing soil material to an unfriendly environment such as water. Considering tropical regions like ours, the durability assessment was conducted according to the steps outlined in Ola 55 , which involves the ratio of UCS of the specimen wax-cured for 7 days and de-waxed top and bottom before being soaked for another 7 days to the UCS of the specimen cured for 14 days.
Morphological interaction. Both the unaltered and multiple-additive blended soil were morphologically and analytically visualized via scanning electron microscopy (SEM). The Phenom world electron microscopy testing equipment was engaged for conducting this test. The SEM analysis provides a series of micrographs which unravels the bonding, surface texture and geometry of the specimen 56 . Lately, the deployment of qualitative analysis such as SEM in optimization studies has been the hotspot for experts in the field of materials and construction engineering 57 www.nature.com/scientificreports/ Design of simplex and factor space. Moreover, as a result of multiple constraints imposition on the four mixture components, the resulting design factor space takes the form of a hyper-polyhedron simplex. Through the evaluation of the constraints of the components, the feasible region where the experimental region lies within the constrained portion of the simplex is derived. Also, through design matrix computation, the degrees-of-freedom (df) evaluation is achieved using a model of special-cubic as presented in Table 2. The essence is to guarantee the validity of the lack-of-fit test where fewer degrees of freedom would result in the inability of the test to identify lack-of-fit. The simplex factor space and contour plot for the 4-component mixtures were obtained with the help of design expert software and show the positioning of the experimental points within three space types of planes, vertices and edges of the simplex as shown in Figs. 4, 5. Information matrix measures which provide the leverage and build the type of the derived experimental points for the three space types were further calculated and presented in Table 3. The results obtained implied an average leverage value of 0.9333 with one lack of fit build type for experimental run 5 at edge space type. Furthermore, the criterion optimality algorithm and I-optimal designs also known as IV (integrated variance) which offer the lowest average prediction of the variance across the experimental regions were carried out to evaluate the design experimental space to obtain a Condition Coefficient Matrix of 15,203.166; minimum, maximum and average mean variance of 0.23, 0.625 and 2.262 respectively; G-efficiency of 41.3% which is associated with the inverse of maximum variance; the scaled optimal criterion of 1803.097 and I-optimality value of 0.61945. The derived results implied adequate multicollinearity which would ensure acceptable model goodness of fit behaviour during model development.
Design of experimental mix proportions. Details obtained from the computed information matrix of the mixture design are utilized to formulate the components mixture ratio and the corresponding number of experimental runs. The actualization of these mixed ingredients fractions and the number of runs required were possible through the evaluation of the imposed components constraints and also from the criterion optimal algorithm computation outcome to ensure essential selection of the experimental points within the feasible factor space. Fifteen runs of experiments for the experimental exercise were obtained from the computation procedure as shown in Table 4; presenting the actual and pseudo components fractional values for the factor levels and providing the fundamental template for the evaluation of the 4-components mixture to stabilize the expansive soil using TWD-CK blends.

Results and discussion
As detailed in Table 5, is the elementary distinctive response of the representative expansive soil in its virgin state, while the particle size analysis curve is presented in Table 6. The particle size distribution exercise of the soil depicts a minute fraction of gravel (0.475%), a sand fraction of 27% and a large fraction of fine (72.028%). Sadly, the high fine content in the soil contributes to the expansive and plastic behaviour of the soil. Also, the strength indices of the studied soil were found not to meet the minimum benchmark. The oxide composition of constituent materials in this experimental investigation is shown in Table 7 which indicates good pozzolanic behaviour of the additives to improve the expansive soil's mechanical characteristics for foundation purposes.
Presented in Fig. 6 is the Differential scanning calorimetry (DSC) of the raw kaolin. The endothermic peak reactions took place between 552 and 953.9 °C this is connected with dihydroxylation of minerals 63 . This is a pointer towards the formation phase of calcined kaolin (CK). Interestingly, this DSC outcome is comparable  Table 8 presents the soaked CBR values for soil-calcined kaolin mixtures with tile waste dust. It shows that by adding the additives into the soil the CBR values of the treated soil mixtures yield prominent enhancement in terms of strength. The value of the CBR increased from an initial value of 9.41% to an optimal value of 61.4% as shown in a contour plot in Fig. 7. By comparing the CBR outcomes of the natural soil with the optimally treated soil sample, the rate of increment is about 552%. This remarking influence of an increase in CBR values results from some reasons. Primarily, the combination of calcium ions and water facilitates chemical interplay within the soil mixtures which thereby activates the establishment of new products such as CSH and CAH which has the affinity of influencing bonding properties. Another promising reason may be attributed to the high dominance of cementitious content in the soil matrix which would need hydration to ensure robust pozzolanic interplay as a result of the flocculation and agglomeration of clay particles that occurred. This chemical reaction forms a soil matrix that could efficiently enhance the non-bonded soil particles speedily and as well promote higher strength. A similar phenomenon was also documented by 9,13,59,60 .

Impact of an optimal blend of ameliorants on California bearing ratio (CBR).
Impact of an optimal blend of ameliorants on unconfined compressive strength. The outcomes of unconfined compressive strength (UCS) carried out on the soil samples in the course of this experimental work are presented in Table 8. By comparing the UCS upshots of the unaltered soil to that of additive-altered soil, an amazing increase was witnessed with increasing calcined kaolin (CK) inclusion level irrespective of tile waste powder (TWP) content. The inclusion of these binary additives instigated an increase in UCS from 102 to 755 KN/m 2 as shown in Table 3 and this translate to approximately seven times increment as shown in a contour plot in Fig. 7. The causes for this UCS improvement mechanism due to additive inclusion can be accredited to three reasons and they include: [1] Filling the voids within expansive soil grains by the binary additives during compaction which thereby imparts higher density to the soil, [2] alteration of the soil physical structure due to the higher specific gravity of the additives (i.e. CK and TWP having 2.58 and 2.5, respectively) in comparison with that of the BCS (2.40), and [3] formation of pozzolanic cementitious compounds between CK, TWP and BCS.
In the presence of moisture, the soil reacts with the additives thereby releasing the calcium ions for replacement and exchange in the diffuse double layer of clay particles. The result aligns with that obtained by 9,62 . www.nature.com/scientificreports/

Impact of an optimal blend of ameliorants on resistance to loss in strength (RLS).
In the course of experimenting durability of the soil specimens, the obtained upshot is shown in Table 3. The durability value of 33% (which implies a loss in strength of 67%) was noticed to be the least possible value compared to 59% (which implies a loss in strength of 41%) being the utmost value. The least value of 33% and peak value of 59% were achievable as a result of mix combination ratios of 3.24:6.48:0.81:5.67 and 1.62:3.65:5.26:5.67 for water, tile waste dust, calcined kaolin and soil respectively. Consequently, the initial RLS value of 33% could be linked with the high dominance of swelling minerals in the parent soil whereas the peak value of 59% might not be unrelated with the presence of additives. It is important to note that the additives considered in this study are rich in the oxides of alumina and silica which are responsible for pozzolanic activities within the soil matrix and onward increment in mechanical strength. The peak RLS value in this study is low compared to the 80% requirement specified in 55 . Howbeit, considering the soaking duration of 7 days against the 4 days as earlier specified by Ola 55 , the peak RLS of 59% was recorded at mixture ratios of 1.62:3.65:5.26:5.67 for water, tile waste dust, calcined www.nature.com/scientificreports/ kaolin and soil respectively, looks promising. This experimental outcome is not far from the upshots of previous investigators who deployed other solid waste materials 7,65 .

Impact of an optimal blend of ameliorants on micro-morphological test.
Scanning electron microscope (SEM) observation. Lately, researchers with specialty in the field of materials and construction engineering have deployed a scanning electron microscope (SEM) to unravel the microstructural response of both soil and concrete materials 7,60,61,66 . The main motivation for the authors to subject both the unstructured and additive structured soil samples to the SEM testing was to establish the morphological amendment of the tested soil based on the optimization exercise. Figure 8 (a and b) shows the SEM micrograph of the raw soil sample at 300X and 500X magnifications. The unstructured soil material unveils a lumpy clay particle with a blackish appearance because of the absence of hydration products. As shown in the micrographs of the additive structured soil (Fig. 9), it results in a fairly whitish surface appearance with staggered bonding. On a second note, the inconsistencies noticed between the unstructured and additives-structured soil could be related to the incorporation of additives. The presence of these additives activated the dissolution of silicates thereby reacting with calcium and as well enhance the pozzolanic protocols within the soil matrix and the production of cemented materials. It is believed that the cemented composite bonds the aggregates and improves the strength response of the additive-structured soil [67][68][69][70] .

Development, formulation and validation of the extreme vertex design (EVD) model
For data processing of the derived experimental responses which is essential for the development of the EVD model in the 4-components mixture optimization, quadratic (square-root, λ = 0.5) power transformation was adopted. The experimental responses obtained for CBR, UCS and RLS indicated 11-42%, 410-755 kN/m 2 and 33-59% with max-to-min ratios of 3.81818, 1.84146 and 1.78788 respectively. Statistical analysis was carried out through fits summary, diagnostic computation and influences, numerical and graphical optimization to ascertain the optimal 4-components mixture ratio to improve the engineering behaviour of problematic soil. After the development of the model coefficients, Post analysis, point prediction, confirmation and model simulation   Fit-summary statistics. This statistical assessment is first executed on the system datasets to derive their viability and adequacy which would enable the actualization of the requirements for the EVD modelling measures using a mixture coding of L_Pseudo. The fit-summary computation provides important information which would aid the choice of starting point suitable for the development of the EVD model. The essential collection of statistical indicators such as lack-of-fit, coefficient-of-determination (R-sqd), sum-of-squares (SS) and predicted-sum-of-squares (PRESS) is calculated as presented in Tables 9, 10 [1] ↓ exo Figure 6. Differential scanning calorimetry (DSC) result of raw kaolin used in the paper.            www.nature.com/scientificreports/

Diagnostics plots.
To achieve the corroboration of the regression model assumptions and to elucidate if the experimental observations utilized for the EVD-model development were significantly unjustifiable and at the same time pose a noticeable influence on the analysis results. Studentized residuals were adopted to accomplish this statistical analysis which is used to detect outliers which significantly differ from regular observations as a result of a verifiable error in the mixture design. Moreover, varying normal distributions observed are mapped to a unit standard normal-distribution using the studentized residuals for the CBR, UCS and RLS responses respectively. Statistical-diagnostic tests carried out include predicted vs. residual, normal probability test, predicted vs. actual, run plot vs. residuals, and box-cox power transformation.
Graph of normal-probability. The graphical results evaluating the normal probability help to confirm that the derived errors also known as residuals follow or assume normal distribution positioned nearest to the regression line. The plot illustrates the relationship between the externally studentized residuals on the x-axis against the normal probability (%) on the y-axis for the CBR, UCS and RLS response variables as shown in Fig. 10. Details such as definite configurations like curvilinear shapes are examined carefully to ascertain the possibility of deploying suitable model transformation of the response parameters. The results derived from the plots indicate the majority of the points from the horizontal axis are within the range of -1 to + 1 studentized residuals and also within 10-90% normal probability for the three response cases under study.
Graph of predicted versus studentized residuals. The constant variance assumption of the residuals is validated for the three response parameters through the residuals vs. predicted statistical diagnostic test. The plot presents the model-predicted values on the x-axis against the externally studentized residuals on the y-axis of the graph as shown in Fig. 11. The presented graphical results indicated clustering of the derived model-predicted values at the zero studentized point for the majority of the observations. The plot for CBR response was observed to be positioned at the limits of ± 4 within the boundary points of ± 6.254, the responses for UCS and RLS were thus observed to show a similar trend as they are situated at ± 2 limits within the boundary points of ± 3.8273.
Experimental run versus studentized residuals Plot. The examination for plundering factors which pose a significant impact on the response variables during the statistical calculations is achieved using this diagnostic computation graph. The plot presents the experimental run's effects on the target responses on the x-axis against the externally studentized residuals on the y-axis as shown in Fig. 12. From the graphical results similar to the residual vs. predicted diagnostic plots, the observed design points were situated at the boundary points of ± 4 for the CBR response and ± 2 for both the UCS and RLS response which signifies a time-dependent lurking variable in the background of the presented scattered plot.
Predicted versus experimental result graphical plots. This diagnostic statistical analysis plot essentially helps to obtain a unitary or group of values which are difficult to accurately estimate by the generated EVD-model in a straight line graph of the laboratory outcomes on the x-axis and the predicted values on the y-axis as shown in Fig. 13. The essence of this diagnostic computation is to assess the relationships between the square root transformed model predicted and actual values by how well the designed experimental data fit the regression line. The details derived from the graphical plots for the three responses indicated a good and acceptable correlation between the model estimated and laboratory values. www.nature.com/scientificreports/ Power transformation using box-cox plot. This analytical computation plot offers suitable guidelines for the selection of an adequate power transformation model to examine the significant impacts on the response parameters by the factor levels. The results derived from this diagnostic assessment help to attain the best lambda outcomes on the horizontal axis of the plot which are derived at the minimum point of the parabolic curve corresponding to the sum of squares residuals' natural logarithm on the vertical axis as shown in Fig. 14. From the presented graphs for CBR response showed current lambda of 0.5, confidence intervals (C.I.) of -0.07 and 1.41 lambda for low and high limits respectively, and best lambda of 0.6. Also, the UCS response showed current and best lambda of 0.5 and 2.06 respectively, low and high C.I. of -0.38 and 4.55 respectively while the RLS response showed current and best lambda of 0.5 and -0.02 respectively, low and high C.I. of -2.2 and 1.98 respectively. The cook's distance. The Cook's distance is adopted in regression analysis to ascertain the influence of data points or outliers in explanatory variables concerning the target response variables. This computation also helps to determine the feasible experimental planes where negative effects are achieved and also where acceptable correlation outcomes are derived. The plot presents runs of experiments on the x-axis against the calculated cook's distance result as shown in Fig. 15. From the derived results, the experimental runs were mostly positioned at the    www.nature.com/scientificreports/ imposed criteria or components constraints for the factor levels and the corresponding optimized dependent variables simultaneously. The expected goal of this optimization exercise is set to maximize the target responses while the four mixture components are set at the in-range option to ascertain the optimal proportion of the factor levels to produce maximum responses bounded by the derived upper and lower limits from the experimental details as shown in Table 38.
The computed optimization solution derived from analytical proceedings of the 4-component mixture experiment design showing seventeen optimized solutions is presented in Table 39. From the derived results, solution number 13 was preferentially selected showing an optimal desirability score of 1.0 at a combination ratio of 0.100:0.097:0.450:0.0353 for water, TWD, CK and soil respectively to produce maximized CBR, UCS and RLS response of 44.611%, 810.687kN/m 2 and 63.448% respectively.
Graphical optimization plots of ramps and bar. The analytical outputs derived from the 4-components mixture design optimization are presented graphically through ramps plot and bar charts for clarity and understanding of the obtained results. The ramp and bar chart illustrates the optimal solution for the factor levels in red colours while the target responses are in blue colouration to elucidate the obtained desirability function computation outputs as shown in Figs. [18][19]. The graphical optimization results presented indicated the explanatory and response variables' desirability score of 1.0, while the combined optimal desirability score of 1.0 was calculated which signifies adequate performance taking into consideration the imposed factors criteria.
Optimization trace plot. Trace plots are deployed to assess all mixture components' effects in the feasible experimental space considering the derived optimal responses and it's similar to perturbation plots adapted for analysis in non-mixture designs. The referenced optimized mixture blend is derived using this graphical assessment Optimization contour plot. The contour plot is a two-dimensional graphical analysis computational tool that helps the constrained experimental space visualization to observe the numerical factor points against the mixture components within the simplex through an iterative optimization solution as shown in Fig. 21. The pre- 3D surface plots. 3-dimensional surface plots are graphical illustrations of the interactions and connectivity behaviour between the factor levels to the response variables. The 3-dimensional graphs for the mixture components concerning the dependent variables and desirability function optimal solution and presenting the corresponding experimental design points' response surface is shown in Fig. 22. The plotted results indicated optimal responses of 44.611%, 810.687 kN/m 2 and 63.448% for CBR, UCS and RLS respectively.

EVD-Model post-analysis and simulation.
Post-analysis and model simulation was carried out after the EVD-model development, numerical and graphical optimization to validate the derived factor values, point prediction and confirmation to achieve sample means and coefficient table. Also, a simulation exercise of the EVD-model generated to assess its applicability and performance is further achieved.
Point prediction computation. This post-analysis tool essentially provides an avenue for the factor values to be reviewed which utilizes the fit statistical computation of the developed model and the prescribed settings for the factor tools to determine interval estimates after the analysis is successfully carried out as presented in Table 40.  EVD model simulation. The simulation of the EVD-model is carried out after the derivation of the coefficient table and post-analysis process. It is also the last phase of the statistical analysis of the 4-components mixture design optimization and is executed to validate the necessary statistical assessments carried out to develop the EVD-model to simulate real-life conditions which provide adequate guidance for contractors, field operators,  www.nature.com/scientificreports/   71 .

Comparison with previous studies
The recent hotspot in the field of materials and construction engineering is the utilization of waste residue in conjunction with a suitable optimization technique in soil amelioration protocols. The use of waste residue is a result of the high cost of cement, depletion of the ozone layer due to cement production and as well as promoting  www.nature.com/scientificreports/ sustainable environment whereas the deployment of optimization technique is to aid in model building and as well reduce the process of the random number of experiments experienced in the trial and error approach. Thus, this necessitated the deployment of extreme vertex design of experiment approach in the current study. Notably, in this current experimental study, the CBR upshots of soil-calcined kaolin mixtures with tile waste dust demonstrated an appreciating trend with the addition of additive dosages. In contrast to the study of Alaneme et al. 9 who engaged EVD in the optimization of a single additive for the amelioration of expansive soil, they reported an increasing CBR with the incorporation of additive material. However, both studies used black expansive clay soils but in their study, a single additive was considered whereas in this current study binary additives were considered and this might not be unrelated to the rate of strength gained. Ikeagwuani et al. 59 established coherently that optimization of multi-additives via means of Taguchi mixture experimental design in a certain way influenced the mechanical response (differential free swell, California bearing ratio, and unconfined compressive www.nature.com/scientificreports/ strength) of expansive soil. Surprisingly, the durability response recorded in this study increased with an increase in the dose of additives to a peak value of 59%. The durability outcome for the optimal soil additive mixture fell short of the 80% requirement and it seems to be consistent with those of 66 . In summary, the extreme vertex design optimization strategy used in this experimental study endorses the explorations of 9,44 who disclosed the effectiveness and efficiency of EVD in civil engineering materials.

Conclusion
This study documents the upshots of successful soil re-engineering protocols of a weak soil ameliorated with the combination of TWD and CK based on the principles of EVD strategy. Based on AASHTO classification and USCS scheme, the soil in its original form falls under the class of A-7-6 material with group index of 14 and CH, respectively. The upshot of this optimization study indicates that using the mixture ratio of 0.10:0.23:0.32:0.35 for water, TWD, CK and soil respectively, optimally enhanced the key parameters including CBR, UCS and RLS. The documented rate of enhancement in the soil's mechanical might not be far from the pozzzolanic interplay within www.nature.com/scientificreports/ the soil structure as a result of the introduction of additive materials, which was later confirmed qualitatively via microstructural experiments such as scanning electron microscopy (SEM). From the geo-engineering point of view, the benefit of the present investigation entails in numerous folds: [i] the integration of waste materials as ameliorants promotes a sustainable environment thereby regulating the depletion of natural resources and possible occurrence of global warming, [ii] it will create a cost-effective strategy in the eradication of mammoth amount of waste materials are generated into the society thereby reducing nuisance and cleaner environment and [iii] in the course of infrastructural developments, the practice of waste materials reutilization in soil reengineering as either cement or lime surrogate materials facilitates lessening in cost of Civil engineering projects. The use of EVD strategy have shown a huge success thus, the use of other optimization techniques such as Taguchi, response surface methodology etc. would be a welcome development as it will create room for further studies and as much a more efficient optimal ratio could be achievable.  www.nature.com/scientificreports/   www.nature.com/scientificreports/ www.nature.com/scientificreports/

Data availability
All data generated or analyzed during this study are included in this published article.