3.2. Response Surface Model Fitting and Verification
The second-order model is often used for response surface function fitting, and its form is as shown in Equation (1) [
27].
In Equation (1), y represents the response value, and b0, bi, bii, and bij represent constant coefficients, linear coefficients, quadratic term coefficients, and interaction coefficients, respectively. Here, xi and xj represent independent variables.
The slump fitting function is shown in Equation (2):
The 28-d UCS fitting equation is shown in Equation (3):
The cost fitting function is shown in Equation (4):
Variance analysis was performed on the above response surface functions, and the results are shown in
Table 5.
Table 5 shows that the
p values of the regression models of the slump, 28-d UCS, and cost are all <0.01, indicating that these three mathematical models are statistically significant. In addition, the critical value of F, F-tab = F
0.05 (14, 15) = 2.42, and the calculated F values of the statistical models of the slump, 28-d UCS, and cost are 27.97, 10.08, and 32.69, respectively, which are all greater than 2.42, indicating that the statistical models of the slump, 28-d UCS, and cost are significant at a significance level of
= 0.05.
The R-squared values of the fitting equations of the slump, 28-d UCS, and cost are 0.9631, 0.9039, and 0.9683, respectively, which indicates that the three statistical models can explain the changes in response values of 96.31%, 90.39%, and 96.83%, respectively, indicating that the experimental error is not obvious.
To further visually show the correlation between the model and the experimental values, a comparison chart between the predicted values of the slump, 28-d UCS, and cost and their actual values is plotted, as shown in
Figure 3.
Figure 3 shows the scatter diagrams of the actual measurement results of 30 sets of experiments in the CCD design as the x coordinate and the response surface prediction results as the y coordinate. If all the points in the figure fall near the straight line y = x, then the correlation degree of the response surface model is high. Therefore,
Figure 3 can compare the relationship between the predicted values and the experimental values.
3.3. Effect of Response Surface Parameters on the Slump of the CPB Mix
The analysis of variance of the slump response surface model is shown in
Table 6.
The
p values of the factor C solid content and factor D air-entraining agent dosage in
Table 6 are both less than 0.01, indicating that the main effect of solid content and air-entraining agent on slump is very significant. The influence of factor C and factor D on the CPB mixture slump is discussed below.
As the solid content increases, the slump decreases. Comparing the results of test number 18 and test number 2 in
Table 4, under the same test conditions, the solid content decreased from 70% to 66%, and the slump value increased from 90 mm to 220 mm with a growth rate of 144.44%. The results show that the increase in solid content reduces the slump value significantly. On the one hand, because of the increase in solid content, the distance between the aggregate particles of the CPB mixture is reduced. In the slump test, the sliding between aggregate particles is more likely to generate friction, and the fluidity deteriorates, thereby reducing the slump value [
15,
28]. On the other hand, an increase in solid content leads to an increase in the consistency of the CPB slurry, a decrease in fluidity, and a decrease in the slump value [
29].
The influence of the air-entraining agent dosage on the slump value of the CPB mixture is shown in
Table 4. The comparison of the results of test number 8 and test number 13 in
Table 4 shows that under the same test conditions, the air-entraining agent dosage increased from 0 to 0.4%, and the slump value increased from 25 mm to 230 mm, with a growth rate of 820.00%. It can be seen that the increase in air-entraining agent dosage significantly increases the slump value. The reasons are as follows: First, BA is a loose, porous, sharp-edged aggregate with an extremely rough surface. The high friction between the BA particles reduces the fluidity of the CPB mixture, leading to a decrease in the slump value. In addition, BA, as a light aggregate, has a light weight, and it is difficult for it to collapse under only a high friction force with low self-gravitation. The air-entraining agent mixes the air bubbles between the aggregates. Under the support of the air bubbles, the friction between the BA particles is greatly reduced. The air bubbles act as ball bearings in the CPB slurry, which greatly reduces the surface friction between aggregate particles, and increased slurry fluidity leads to a larger slump value [
30].
In addition, it can be seen from the analysis of variance that the
p value of
AB and
BD is less than 0.05, indicating that the interaction between
AB and
BD is more obvious than the other interactions. The
p value of
CD is less than 0.0001, indicating that the interaction of CD is extremely significant. The three-dimensional surface of the response surface of the interaction of
AB,
BD, and
CD is shown in
Figure 4. In
Figure 4, the three-dimensional response surface is clearly curved, indicating that there is a significant interaction between the factors. In
Figure 4a, as A increases and B decreases, the curvature of the slump response surface (i.e., the slump growth rate) increases. The response surface of the slump in
Figure 4b shows that there is a significant interaction between B and D, wherein as B decreases, D increases and the slump growth rate decreases. As shown in
Figure 4c, as C decreases, D increases and the value of the slump increases, but the curvature of the surface decreases, indicating that the slump growth rate decreases.
3.4. Effect of Response Surface Parameters on the 28-d UCS of CPB
The analysis of variance of the response surface model of the 28-d UCS is shown in
Table 7. The
p values of the four main effects A, B, C, and D in
Table 7 are all <0.01, indicating that the four main effects have a significant influence on the 28-d UCS main effect. The following section analyzes and discusses the effects of A, B, C, and D on the 28-d UCS.
The test data of test number 5 and test number 20 are compared in
Table 4. Under the same test conditions, the aggregate-binder ratio changed from 2.75 to 3.75, the 28-d UCS decreased from 4.16 MPa to 2.03 MPa, and the decrease rate was 51.20%. The results show that the aggregate-binder ratio increases and that the 28-d UCS decreases. Interestingly, this result is consistent with the findings of Lee [
31] et al. but is in contrast to the findings of Park [
32] et al. The reason for this discrepancy is that the elemental content in the raw materials is different. In the study of Park et al. [
32], the Ca content of the BA used was 52.7%, while the Ca content of the BA used in this study was only 5.3%. The BA activity is extremely low, and the active Si and Al are difficult to release under the stimulation of NaOH. However, the slag is excited by NaOH to produce a polymer that endows the slag particles with a cohesive force, wrapping the BA, filling the internal pores, and making the whole structure bind together [
33]. Therefore, in this study, it is considered that the hydration products produced by alkali-activated slag are the main source of strength of hardened CPB materials. As the aggregate-binder ratio increases, the amount of binder in the CPB material decreases, and the hydration products decrease, resulting in a decrease in the compactness of the CPB material and a decrease in the CPB strength.
Comparing the 28-d UCS of test number 14 and test number 10 in
Table 4, it can be seen that under the same test conditions, the alkali dosage is reduced from 6% to 2%, and the 28-d UCS is reduced from 3.54 MPa to 1.97 MPa, with a decrease rate of 44.35%. The reasons for this result may be that NaOH is an activator of alkali-activated slag binder. The increase in alkali dosage causes more activated alumina and silica leaching and generates more hydration products through polycondensation. The hydration products connect the slag particles and BA particles, which fill the pores, resulting in an increase in the 28-d UCS [
15,
34].
Comparing the 28-d UCS of test number 18 and test number 2 in
Table 4, under the same test conditions, when the solid content is reduced from 70% to 66%, the 28-d UCS is reduced from 3.96 MPa to 2.05 MPa, and the increase rate is 48.23%. It shows that the solid content is reduced, and the 28-d UCS is significantly reduced. The reason is that the increase in solid content is equivalent to the decrease in water content. On the one hand, the initial porosity of the CPB was reduced, and the alkali-activated slag binder matrix was more tightly bonded to the BA particles. On the other hand, the reduction of water content indirectly increases the alkali concentration, and the alkaline environment promotes the hydration process of slag.
The 28-d UCS values of test number 8 and test number 13 are compared in
Table 4. Under the same test conditions, the air-entraining agent dosage was increased from 0% to 0.4%, and the 28-d UCS decreased from 6.1 MPa to 2.24 MPa, with a reduction rate of 63.28%. The results showed that the increase in air-entraining agent dosage reduced the 28-d UCS value. The reason for this result may be that the incorporation of air-entraining agents introduces holes in the matrix of the original average gel material. Microcracks are formed around the holes. Under the load, stress concentration occurs, which causes the structure to easily break and reduces the strength [
30].
From the analysis of variance of the 28-d UCS response surface model in
Table 7, the
p value of each interaction term is greater than 0.05, showing that the influence of each interaction term on the 28-d UCS is not obvious, so they are not analyzed here. However, the addition of a quadratic term and the interaction term increases the correlation coefficient of the fitting equation, so this study still uses the second-order function with the interaction term as the 28-d UCS response surface regression model.
3.5. Effect of Response Surface Parameters on the Cost of CPB
The analysis of variance of the cost response surface model is shown in
Table 8. Among the
p values, the
p values of factors A, B, and D are all less than 0.01, which indicates that the effects of A, B, and D on the cost of the CPB are extremely significant. The following is an analysis of the impact of A, B, and D on cost.
The cost values of test number 5 and test number 20 are compared in
Table 4. Under the same test conditions, the aggregate-binder ratio changes from 2.75 to 3.75, the cost decreases from 8.36 USD to 5.99 USD, and the reduction rate is 28.35%. This result shows that the larger the aggregate-binder ratio is, the lower the cost. The reason is that when the alkali dosage, solid content, and air-entraining agent dosage are the same, the aggregate-binder ratio increases, the amount of slag per unit volume increases, and the cost increases.
The costs of test number 14 and test number 10 are compared in
Table 4. Under the same test conditions, the alkali dosage is reduced from 6% to 2%, the cost is reduced from 7.84 USD to 5.61 USD, and the reduction rate is 28.44%. This shows that the alkali dosage is reduced, and the cost is significantly reduced. The reason may be that in the alkali-activated slag CPB material, under the same aggregate-binder ratio, solid content, and air-entraining agent dosage, the adjustment of the alkali dosage affects the apparent density of the material and is close to zero. However, the more NaOH the CPB mix dissolves, the higher its cost.
Comparing the cost values of test number 13 and test number 8 in
Table 4, it can be seen that under the same test conditions, the air-entraining agent dosage is reduced from 0.4% to 0%, and the cost is reduced from 6.81 USD to 6.29 USD with a reduction rate of 7.64%. This shows that the air-entraining agent dosage is reduced, and the cost is reduced. Even if the dosage of the air-entraining agent is increased, the apparent density of the CPB mixture is reduced, that is, the quality of the raw materials used per unit volume is reduced. However, the unit price of the air-entraining agent is much higher than the unit prices of the other raw materials (see
Table 3 for the unit prices of the raw materials). Therefore, the unit price of the air-entraining agent dominates the impact of the cost of materials. Therefore, the dosage of the air-entraining agent increases, and the cost of CPB materials increases.
Table 8 shows the analysis of variance of the response surface model of cost; the
p value of each interaction term is greater than 0.05, showing that the influence of each interaction item on the cost is not obvious, so they are not analyzed here. The addition to the interaction term increases the correlation coefficient of the fitting equation, this study still uses the second-order function with the interaction term as the response surface regression model of cost.
3.6. Multi-Objective Optimization
This paper refers to the desirability function method in [
35] to deal with the multi-objective optimization problem to obtain the optimal mix ratio of CPB.
The general idea of multi-objective optimization is as follows: First, establish the single desirability function di of every response according to the type (“maximum,” “minimum,” and “target”). Then, according to the experimental results of the CCD design, the lower limit, upper limit, and response surface equations of each response, namely, Lowi, Highi, and Yi, are brought into the desirability function of a single response. The overall desirability function, D, is the geometric mean of all single desirability functions. Nonlinear programming is performed on D. When D obtains the maximum value, the parameter value of each single satisfaction function is the optimal mixture ratio.
The principles of CPB optimization in this study are as follows: the slump reaches the target value of 200 mm, the 28-d UCS is maximized, and the cost is minimized. That is, under the condition that the CPB slurry working performance is satisfactory, the strength of the backfill is maximized, and the cost of the CPB material per cubic meter is minimized. Therefore, the type of slump is “target,” the type of 28d-UCS is “maximum,” and the type of cost is “minimum.”
First, a single response satisfaction function di based on a response surface model of the slump, 28-d UCS, and cost is established.
For the single response, the satisfaction function
di of the slump is a function for the goal as a target and should be calculated according to Equation (5) [
35].
where
di represents the satisfaction function of the i-th response surface, which is the satisfaction function of the slump.
Yi is the i-th response, which is the response surface function for the slump.
Lowi is the lower limit of the i-th response value, which is the minimum value of the slump test result in the CCD design (25mm).
Highi is the upper limit of the i-th response value, which is the maximum value of the slump test result in the CCD design (260mm).
Ti is the target value of the i-th response surface, and the target value of the slump in this study is 200 mm.
wti represents the weight factor of the i-th satisfaction function, wherein 0.1 ≤
wti ≤ 10. The weight factor
wti can change the shape of the satisfaction function. When
wti is equal to 1,
di changes from 0 to 1 in a linear form. If
wti is less than 1, the degree of emphasis on the target is low,
di is a convex function, and the rate of change from 0 to 1 gradually slows. If
wti is greater than 1, the degree of emphasis on the target is higher, the
di function is concave, and the rate of change from 0 to 1 gradually increases. In this study, the weight factor
wti = 1 is selected.
Similarly, di represents the satisfaction function of 28d-UCS. Yi is the response surface function for 28d-UCS. Lowi is the minimum value of the 28d-UCS test result in the CCD design, which is 0.44MPa. Highi is the maximum value of the slump test result in the CCD design, which is 6.10MPa; the weight factor wti is equal to 1.
The 28-d UCS single response satisfaction function is a function for the goal as a maximum and should be calculated according to Equation (6) [
35].
The cost single response satisfaction function is a function for the goal as a minimum, which should be calculated according to Equation (7) [
35].
Similarly, di represents the satisfaction function of cost. Yi is the response surface function for cost. Lowi is the minimum value of the cost in the CCD design, which is 5.20 USD. Highi is the maximum value of the cost in the CCD design, which is 8.36 USD; and the weight factor wti is equal to 1.
Finally, an overall satisfaction function (i.e., desirability function)
D is established, which is equal to the geometric mean of the desired goals
di of all responding individuals [
35], which is
where
ri represents the importance degree of each response,
, and the greater
ri the more important it is. This study refers to [
36] and considers that the slump, 28-d UCS, and cost are equally important, that is,
r1 =
r2 =
r3 = 1/3. With the above single response satisfaction functions as the constraint conditions, nonlinear programming is performed on
D to obtain a set of mixture ratios, and the one with the highest
D value is selected as the optimal mixture ratio.
Through optimization analysis, the maximum point of optimization result D is found. At this time, the aggregate-binder ratio was 3.28, the alkali dosage was 3.00%, the solid content was 67.44%, and the air-entraining agent dosage was 0.10%. The predicted value of slump is 200 mm, the predicted value of 28-d UCS is 2.94 MPa, and the predicted cost is 5.59 USD.
Table 9 compares the experimental and predicted values of the optimized mixture.
Under the same test conditions, the CPB is configured with the optimal mix ratio. The measured slump is 205 mm, the 28-d UCS is 2.93 MPa, and the cost is 5.70 USD. The absolute relative deviation (
ARD) of the predicted and experimental values is calculated according to the following formula [
37].
The ARDs for slump, 28-d UCS, and cost were 2.44%, −0.34%, and 2.00%, respectively.
In addition, the UCS of CPB changes with curing age, as shown in
Figure 5. With an increasing age of 1 d, 3 d, 7 d, 14 d, and 28 d, the UCS values were 0.11 MPa, 0.68 MPa, 2.00 MPa, 2.91 MPa, and 2.93 MPa, respectively.