Plane Thermoelastic Waves in Infinite Half-Space Caused Operational Research in Engineering Sciences: Theory and Applications Vol. 5, Issue 1, 2022, pp. 90-106 ISSN: 2620-1607 eISSN: 2620-1747 DOI: https://doi.org/10.31181/oresta180222016p * Corresponding author. 2017476603@ufs4free.ac.za (D. Pavolo), Chikobvu@ufs.ac.za (D. Chikobvu) ESTIMATING RUBBER COVERED CONVEYOR BELTING CURE TIMES USING MULTIPLE SIMULTANEOUS OPTIMISATIONS ENSEMBLE Domingo Pavolo*, Delson Chikobvu Department of Mathematical Statistics and Actuarial Science, Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa Received: 15 July 2021 Accepted: 08 November 2021 First online: 18 February 2022 Research paper Abstract: Multiple response surface methodology (MRSM) has been the favorite method for optimizing multiple response processes though it has two weaknesses which challenge the credibility of its solutions. The first weakness is the use of experimentally generated small sample size datasets, and the second is the selection, using classical model selection criteria, of single best models for each response for use in simultaneous optimization to obtain the optimum or desired solution. Classical model selection criteria do not always agree on the best model resulting in model uncertainty. The selection of single best models for each response for simultaneous optimization loses information in rejected models. This work proposes the use of multiple simultaneous optimizations to estimate multiple solutions that are ensembled in solving a conveyor belting cure time problem. The solution is compared with one obtained by simultaneous optimization of single best models for each response. The two results were different. However, results show that it is possible to obtain a more credible solution through ensembling of solutions from multiple simultaneous optimizations. Key words: Multiresponse surface methodology, ensembling, credibility of results, solution uncertainty, small sample size problems, simultaneous optimisation 1. Introduction The mining industry is at the heart of the Southern African Development Community (SADC) region’s economic activities and development. Conveyor belts are critical for conveyance of bulk ore over distances and through various stages of processing. The regional product quality standard minimum requirements for general purpose rubber covered conveyor belts for the mining industry were amended. The component adhesion requirement was increased from 5N/mm to 7N/mm. However, key customers were insisting on a minimum of 10N/mm adhesion and 60⁰ Shore A rubber compound cover hardness. After redesigning of the specifications of rubber Pavolo & Chikobvu/Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 90-106 91 compounds, a client manufacturing company required optimum cure times (Tc), which would ensure a minimum of 12N/mm adhesion and 60⁰ Shore A hardness, to be determined for the vulcanisation of different conveyor belt thicknesses (Rt) for use in shop-floor work instructions. Figure 1. Illustrating the conveyor belting vulcanization process problem Given the illustrated process in Figure 1, it was thus intended to estimate credible cure times (Tc) for given rubber thicknesses (Rt), as shown in Table 1 below. Table 1. Showing the expected solution 𝐑𝐭 (𝐦𝐦) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π“πœ(𝐦𝐒𝐧. ) π“πŸ• π“πŸ– π“πŸ— . . . . . . . . . . π“πŸπŸŽ For manufacturers, optimum cure time (Tc) is critical for product quality and production process productivity. Good adhesion between conveyor belt components (covers, skims and reinforcement fabrics) ensures that they do not separate during heavy duty operations in the mines. The top and bottom rubber covers protect the reinforcement fabrics, therefore hardness is essential for wear resistance to the abrasive mining operational environments. The separation of belting components during heavy duty operation and excessive rubber cover wear are the two major failures of conveyor belting during mining operations. Increasing adhesion between belting components and cover hardness ensures more belting life and therefore lower mining operational costs. Beyond just providing a solution to the client company, the study sought to recommend to the conveyor belt manufacturing industry a credible and efficient tool for converting changes in product standard requirements to production process input parameters. Quality and productivity are critical manufacturing industry competitive factors and the speed of successfully implementing change is critical in any industry as it gives first mover advantages. This work is of interest, therefore, to operations researchers, industrial engineers and business management strategists in the conveyor belting manufacturing industry. Estimating Rubber Covered Conveyor Belting Cure Times Using Multiple Simultaneous Optimizations Ensemble 92 The authors could not find anywhere in literature were cure times per given conveyor belt thickness were estimated for the vulcanisation process of general purpose rubber covered conveyor belting. Literature only gives methodologies for estimating the cure times of different rubber compounds. A rubber covered conveyor belt is constructed from a rubber cover compound, a rubber skim compound (which provides the bonding strength between components) and reinforcement fabric. These components individually contribute to the overall vulcanization time due to different heat conductivities. In this work, the sufficiency of the contemporary multiple response surface methodology (MRSM) framework in estimating a credible solution to the problem was critiqued and two major weaknesses identified. Firstly, it is statistically difficult to extract credible process information from small sample size MRSM datasets. Secondly, the selection of single best models for each response for simultaneous optimization is prone to (1) loss of information in the rejected response models and (2) model uncertainty as model selection criteria do not always agree on the best model. This work proposes the use of multiple simultaneous optimizations to estimate multiple solutions that are then ensembled, to account for the two weaknesses in the MRSM framework, in solving the conveyor belting cure time problem. Results suggest that the proposed ensemble system can provide a credible solution to the problem. 2. Literature Review 2.1. Rubber technology perspective A number of techniques have been proposed in rubber technology literature to estimate the cure time of rubber products such as nuclear magnetic resonance spectroscopy, differential scanning calorimetry, dynamic mechanical analysis, adaptive neuro-fuzzy inference systems, equivalent cure concept, and artificial neural networks and finite element analysis (Gatos and Karger-Kocsis, 2004; KaraagΓ’c et al., 2011; Gough, 2017). The accepted basic tool of cure time estimation is the rheograph (Appendix A) which shows how the shear strength of a sample of rubber changes with time during vulcanisation. The rheograph does not consider the case where there are different layers of constituent rubber compounds and other materials such as conveyor reinforcement fabric (nylon and/or polyester). The conveyor belting case requires a multiple factor and multiple response simultaneous optimisation solution methodology, hence the shift to multiple response surface methodology (MRSM). 2.2 Multiple response surface methodology (MRSM) Figure 2. Showing the contemporary MRSM framework Pavolo & Chikobvu/Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 90-106 93 MRSM is an important tool for optimising manufacturing processes in industry. It is a collection of mathematical and statistical techniques that are useful for the modelling and analysis of problems in which multiple responses are influenced by several variables and the objective of the analysis is to optimize the responses by determining the best settings of the input variables (Myers et al., 2016; Hejazi et al., 2017; Khuri, 2017). In Figure 2, the MRSM dataset generation stage, stage (1), involves designing and running screening and MRSM experiments (Myers et al., 2016). The stage (3) are the solution methodologies for estimating the operating conditions that optimise all the responses or at least keep them in desired ranges. MRSM experimental designs are constructed to eliminate or minimise correlations between chosen variables which allows independent estimation of variable effects and their potential interactions (Myers et al., 2016; Khuri, 2017; MΓ€kelΓ€, 2017). Examples include central composite designs (CCD), Box-Behnken, Orthogonal Arrays, Placket- Burman, and computer-generated optimal designs (Myers et al., 2016; Khuri, 2017; Alhorn et al., 2019). The strength of MRSM is in efficient experimental designs (Khuri, 2017). However, statistically, it is difficult to extract credible population information from small sample size datasets (Rawlings et al., 1998; Yuan and Yang, 2005; Xu and Goodacre, 2018; Jenkins and Quintana-Ascencio, 2020). This is the first weakness that requires to be accounted for to obtain credible solutions. Optimisation in MRSM is multi-objective in nature, and is performed after regression modelling and model selection of single β€œbest” models for each response (Myers et al., 2016; Khuri, 2017). MRSM solution methodologies rely heavily on classical model selection criteria for choosing the best model for each response for simultaneous optimisation. This the second weakness of the contemporary MRSM framework. Problems associated with the contemporary MRSM contextual framework are presented in Figure 3 below. Figure 3. Problems related to the current MRSM contextual framework Estimating Rubber Covered Conveyor Belting Cure Times Using Multiple Simultaneous Optimizations Ensemble 94 In this paper, the authors proposed and utilised a novel solution methodology that accounted for the two weaknesses to obtain a credible solution. 3. Solution Methodology The MRSM dataset generation for the rubber covered conveyor belting problem is explained in detail in Pavolo and Chikobvu (2020). The dataset was adopted as is and is shown in Table 2. Table 2. The two-factor CCD experiment MRSM dataset Run T (min.) Rt (mm) Ave. Hardness (0shore A) Ave. Adhesion(N/mm) 1 16 7.2 60 10.60 2 30 7.2 63 13.34 3 16 22.8 53 6.20 4 30 22.8 61 12.10 5 23 15 58 11.80 6 23 15 58 12.10 7 13 15 44 6.5 8 33 15 63 13.30 9 23 4 63 13.30 10 23 26 56 3.50 11 23 15 58 12.20 12 23 15 57 12.30 13 23 15 58 12.10 Ensemble-based systems have been recommended for small sample size situations in literature (Kittler, 1998; Burnham and Anderson, 2002; Polikar, 2006; Yang et al., 2016; Ahangi et al., 2019). An ensemble system was considered the best option for accounting for the weaknesses of the contemporary MRSM framework and delivering a credible solution. The solution methodology is summarised in Figure 4. Figure 4. Showing the solution methodology flow diagram Pavolo & Chikobvu/Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 90-106 95 At response surface analysis, the one hardness model with response surface conformity was adopted as is from Pavolo and Chikobvu (2020). However, in this work, all the adhesion response models were assumed to be response surface conforming. The estimated cure time solution was compared with one from a methodology structured after the contemporary MRSM contextual framework. Figure 5 shows the strategies used in the solution methodology to deal with each problem listed in Figure 3. Figure 5. Showing the strategies employed to deal with problems Figure 6 summarises the problems of the contemporary solution methodologies and presents the advantages of the ensembling methodology from literature. Figure 6. The advantages of the solution methodology vs. problems of contemporary MRSM Estimating Rubber Covered Conveyor Belting Cure Times Using Multiple Simultaneous Optimizations Ensemble 96 The formulae used for computation of theoretical accuracy are shown below. Validation was computed against the minimum targeted response values as the sample size was too small to be split into a fitting set and a validation set. MSPEval(min.): The validation minimum mean squared prediction error (MSPEval(min.)) of a response model measures the minimum squared deviation of the model predictions from the targeted. MSPEval(min.) is given below for a sample size n. MSPEval(min.)= βˆ‘ (π’π’Š=𝟏 π’€π’Šβˆ’π’€π‘») 𝟐 𝒏 , (1) where π’€π’Š is the ith estimated response, 𝒀𝑻 is the a response value. MSPEsimul: The mean squared prediction error at simultaneous optimisation (MSPEsimul) of a response model in an adhesion – hardness model pair indicates the mean squared deviation of the model predictions from the targeted and is given below for a sample size n as: MSPEsimul= βˆ‘ (π’π’Š=𝟏 π’€π’Šβˆ’π’€π‘») 𝟐 𝒏 , (2) where π’€π’Š is the ith estimated response value at simultaneous optimisation. The MSPEsimul bias-variance decomposition estimates were integrated by arithmetic averaging to estimate the bias-variance-covariance decomposition of the MSPEsimul of the ensembled results (Geman et al., 1992; Ueda and Nakano, 1996). MSE(𝑓) = bias(𝑓)2 + var(𝑓) (3) And the expected ensemble MSE is, E{MSE(𝑓𝑒𝑛𝑠. )} = Bias2 + ( 1 π‘˜ ) Γ—Variance + (1- 1 π‘˜ ) Γ—Covariance (4) Where k is the number of base models in the ensemble. Prediction Accuracy Compromise: Define Prediction Accuracy Compromise (PAC) as the difference between MSPEsimul and the MSPEval(min.) of a response model. PAC gives a picture of how models compromise their accuracy in the process of simultaneous optimisation. Then % PAC will be the percentage change in MSPEval(min.) to achieve simultaneous optimisation. % PAC = 100% x (MSPEsimul - MSPEval(min.)) / MSPEval(min.) (5) Relative accuracy: The Relative Accuracy is computed for each base model in the ensemble relative to the ensemble result and is given by: Relative Accuracy = π‘΅π’–π’Žπ’ƒπ’†π’“ 𝒐𝒇 𝑩𝒂𝒔𝒆 𝑴𝒐𝒅𝒆𝒍 𝒐𝒓 π‘¬π’π’”π’†π’Žπ’ƒπ’π’† π‘ͺ𝒐𝒓𝒓𝒆𝒄𝒕 π‘·π’“π’†π’…π’Šπ’„π’•π’Šπ’π’π’” 𝑻𝒐𝒕𝒂𝒍 π‘΅π’–π’Žπ’ƒπ’†π’“ 𝒐𝒇 𝑰𝒏𝒔𝒕𝒂𝒏𝒄𝒆𝒔 (6) Pavolo & Chikobvu/Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 90-106 97 4. Results 4.1 All possible regression modelling results Tables 3 and 4 below show the all possible ordinary least squares (OLS) regression models for both responses after removal of response models that did not conform to the recommendations of the screening experiment at dataset generation. Table 3. The twenty-five OLS adhesion all possible regression response models MODEL π›ƒπŸŽ π›ƒπŸ π›ƒπŸ π›ƒπŸπŸ π›ƒπŸπŸ π›ƒπŸπŸ Tc.Rt 12.2600 -0.0039 Tc, Rt 7.9500 0.3244 -0.3127 Tc, Tc .Rt 3.2600 0.5100 -0.0124 Tc, Rt2 6.1800 0.3244 -0.0111 R, Tc.Rt 15.4100 -0.7910 0.0208 Rt, Tc 2 11.6700 -0.3127 0.0067 Tc.Rt, Tc 2 8.9600 -0.0119 0.0105 Tc.Rt, Rt2 10.4970 0.0203 -0.0258 Tc 2, Rt2 9.9100 0.0066 -0.0111 Tc, Rt, Tc.Rt 12.9400 0.1070 -0.6460 0.0145 Tc, Rt, Tc 2 2.4100 0.8350 -0.3127 -0.0111 Tc, Rt, Rt 2 3.6100 0.3244 0.3800 -0.0231 Tc, Tc.Rt, Tc 2 -2.2800 1.0200 -0.0124 -0.0111 Tc, Tc.Rt, Rt 2 9.1400 0.0910 0.0156 -0.0224 Tc, Tc 2, Rt 2 -0.2500 0.9190 -0.0129 -0.0112 Rt, Tc.Rt, Tc 2 15.2400 -0.7710 0.0199 0.0003 Rt, Tc.Rt, Rt 2 11.0800 -0.0980 0.0208 -0.0231 Rt, Tc 2, Rt 2 7.5200 0.3580 0.0066 -0.0224 Tc.Rt, Tc 2, Rt 2 10.3900 0.0189 0.0005 -0.0249 Tc, Rt, Tc.Rt, Tc 2 7.4000 0.6180 -0.6460 0.0145 -0.0111 Tc, Rt, Tc.Rt, Rt 2 8.6100 0.1070 0.0470 0.0145 -0.0231 Tc, Rt, Tc 2, Rt 2 -4.2500 1.0210 0.4300 -0.0151 -0.0248 Tc, Tc.Rt, Tc 2, Rt 2 1.9500 0.7590 0.0168 -0.0149 -0.0234 Rt, Tc.Rt, Tc 2, Rt 2 11.2100 -0.1130 0.0215 -0.0003 -0.0232 Tc, Rt, Tc.Rt, Tc 2, Rt 2 0.7400 0.8040 0.0970 0.0145 -0.0151 -0.0248 Table 4. The twenty-five OLS hardness all possible regression response models MODEL π›ƒπŸŽ π›ƒπŸ π›ƒπŸ π›ƒπŸπŸ π›ƒπŸπŸ π›ƒπŸπŸ Tc.Rt 56.1800 0.0040 Tc, Rt 48.4600 0.5130 -0.1800 Tc, Tc .Rt 45.7500 0.6040 0.0061 Tc, Rt2 46.5300 0.5130 -0.0030 R, Tc.Rt 60.2500 -0.9610 0.0339 Rt, Tc 2 54.8400 -0.1800 -0.0097 Tc.Rt, Tc 2 52.8900 0.0045 -0.0111 Tc.Rt, Rt2 55.0800 0.0209 -0.0181 Tc 2, Rt2 52.9000 0.0097 -0.0030 Tc, Rt, Tc.Rt 57.5000 0.0320 -0.7180 0.0321 Tc, Rt, Tc 2 18.0000 3.3200 -0.1800 -0.0610 Tc, Rt, Rt 2 57.5100 0.5130 -1.6290 0.0483 Tc, Tc.Rt, Tc 2 15.3000 3.4100 -0.0061 -0.0610 Tc, Tc.Rt, Rt 2 41.9300 0.8760 -0.0242 0.0146 Tc, Tc 2, Rt 2 15.9000 3.3400 -0.06160 -0.0034 Rt, Tc.Rt, Tc 2 65.0600 -1.4960 0.0572 -0.00860 Estimating Rubber Covered Conveyor Belting Cure Times Using Multiple Simultaneous Optimizations Ensemble 98 Rt, Tc.Rt, Rt 2 69.3100 -2.4090 0.0339 0.0483 Rt, Tc 2, Rt 2 64.0100 -1.6610 0.0098 0.0494 Tc.Rt, Tc 2, Rt 2 52.6600 -0.0094 0.0127 0.0039 Tc, Rt, Tc.Rt, Tc 2 29.1000 2.8400 -0.9180 0.0321 -0.0610 Tc, Rt, Tc.Rt, Rt 2 68.6000 0.0320 -2.3660 0.0321 0.0483 Tc, Rt, Tc 2, Rt 2 13.4000 3.5300 -0.0196 -0.0592 0.0108 Tc, Tc.Rt, Tc 2, Rt 2 29.4200 3.0020 -1.4500 -0.0541 0.0423 Rt, Tc.Rt, Tc 2, Rt 2 73.31000 -2.8470 0.0540 -0.0074 0.0048 Tc, Rt, Tc.Rt, Tc 2, Rt 2 40.5000 2.5210 -2.1870 0.0321 -0.0541 0.0423 Hardness response model [Tc, Rt, Tc.Rt, Tc2] was the only hardness model with a conforming response surface. 4.2. Simultaneous optimisation results Table 5 shows the simultaneous optimisation of the adhesion-hardness model pair [Tc.Rt, Rt2] - [Tc, Rt, Tc.Rt, Tc2] using an Excel spreadsheet tool. The rest of the adhesion response models were similarly optimised with the same hardness model. Table 5. Showing simultaneous optimisation on an Excel spreadsheet Tc Rt Adhesion Hardness (min.) (mm) [T*Rt, Rt2] e2 [T, Rt, T* Rt, T2 ] e2 21 7 12 12.2169 0.0470 60 60.1317 0.0173 22 8 12 12.4186 0.1752 60 60.3616 0.1308 22 9 12 12.4266 0.1820 60 60.1498 0.0224 23 10 12 12.5860 0.3434 60 60.3540 0.1253 23 11 12 12.5111 0.2612 60 60.1743 0.0304 24 12 12 12.6282 0.3946 60 60.3528 0.1245 24 13 12 12.4704 0.2213 60 60.2052 0.0421 24 14 12 12.2610 0.0681 60 60.0576 0.0033 25 15 12 12.3045 0.0927 60 60.2425 0.0588 25 16 12 12.0122 0.0001 60 60.1270 0.0161 26 17 12 12.0134 0.0002 60 60.2862 0.0819 27 18 12 12.0036 0.0000 60 60.3876 0.1502 29 19 12 12.3685 0.1358 60 60.4041 0.1633 30 20 12 12.3570 0.1274 60 60.3000 0.0900 Ave. 12.3270 MPSE:0.1464 Ave.: 60.2525 MPSE: 0.0755 Bias: 0.3270 Var.: 0.0394 Bias: 0.2525 Var.: 0.0117 Table 6 shows cure time estimates which each adhesion response model gave at simultaneous optimisation. Table 6. Showing the cure time estimate results for each adhesion response model Rt (mm) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Tc.Rt 21 22 22 23 24 25 26 26 27 28 29 30 Tc, Tc.Rt 21 22 22 23 24 25 26 27 28 29 30 Tc, Rt 2 21 22 22 23 23 24 24 25 26 27 28 30 Rt ,Tc.Rt, 21 22 22 23 24 25 26 27 28 28 29 29 30 30 Tc, Rt 2 21 22 22 23 24 25 26 27 28 29 30 30 Tc.Rt, Tc 2 22 23 23 24 25 26 26 27 28 29 30 Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 25 26 27 29 30 Tc 2, Rt 2 21 22 22 23 24 25 26 27 28 29 30 30 Tc, Rt, Tc.Rt 21 22 22 23 24 25 26 27 27 28 29 30 30 31 Tc ,Rt, Tc 2 21 22 22 23 24 25 26 27 28 29 30 Tc, Rt, Rt 2 22 22 22 23 23 24 24 24 25 26 27 28 30 31 Tc, Tc.Rt, Tc 2, 21 22 22 23 24 25 26 27 28 29 30 Tc, Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Pavolo & Chikobvu/Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 90-106 99 Tc, Tc 2, Rt 2 21 22 22 23 23 23 24 24 25 26 28 30 Rt ,Tc.Rt, Tc 2 21 22 22 23 24 25 26 27 28 28 29 29 30 30 Rt ,Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 26 28 29 30 Rt, Tc 2, Rt 2 22 22 22 22 23 23 24 25 26 27 28 29 30 31 Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Tc, Rt, Tc.Rt, Tc 2 21 22 22 23 23 24 25 26 27 28 29 30 Tc, Rt, Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Tc, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 25 26 27 29 30 Tc, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 23 24 24 25 25 26 27 30 Rt, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Tc, Rt, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 25 25 27 28 31 Table 7 shows the remaining thirteen adhesion response models with their cure time estimates after dropping those results that did not give estimates for the full rubber thickness range. Table 7. Showing the adhesion response models with simultaneous optimisation cure time estimates for the full rubber thickness range Rt(mm) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 MODEL Rt ,Tc.Rt, 21 22 22 23 24 25 26 27 28 28 29 29 30 30 Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 25 26 27 29 30 Tc, Rt, Tc.Rt 21 22 22 23 24 25 26 27 27 28 29 30 30 31 Tc, Rt, Rt 2 22 22 22 23 23 24 24 24 25 26 27 28 30 31 Tc, Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Rt ,Tc.Rt, Tc 2 21 22 22 23 24 25 26 27 28 28 29 29 30 30 Rt ,Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 26 28 29 30 Rt, Tc 2, Rt 2 22 22 22 22 23 23 24 25 26 27 28 29 30 31 Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Tc, Rt, Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Tc, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 25 26 27 29 30 Rt, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Tc, Rt, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 25 25 27 28 31 A frequency analysis of the occurrence of the different cure time results is given in Table 8. There were only seven possible cure time estimate solutions in Table 7. The solution with the highest occurrence had a frequency of five. Table 8. Showing frequency of occurrence of cure time estimates results Rt 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Frequency 1 21 22 22 23 24 25 26 27 27 28 29 30 30 31 1 2 22 22 22 23 23 24 24 24 25 26 27 28 30 31 1 3 22 22 22 22 23 23 24 25 26 27 28 29 30 31 1 4 21 22 22 23 23 24 24 24 25 25 25 27 28 31 1 5 21 22 22 23 24 25 26 27 28 28 29 29 30 30 2 6 21 22 22 23 23 24 24 24 25 25 26 27 29 30 2 7 21 22 22 23 23 24 24 24 25 26 27 28 29 30 5 Estimating Rubber Covered Conveyor Belting Cure Times Using Multiple Simultaneous Optimizations Ensemble 100 4.3 Integration results Table 9 shows the results of integrating the cure time estimates of Table 6 using arithmetic averaging (A. Ave.) and majority vote (M. Vote). Table 9. Showing the integration of the thirteen cure time estimate results 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Rel. MODEL Acc. Rt ,Tc.Rt, 21 22 22 23 24 25 26 27 28 28 29 29 30 30 36% Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 25 26 27 29 30 79% Tc, Rt, Tc.Rt 21 22 22 23 24 25 26 27 27 28 29 30 30 31 29% Tc, Rt, Rt 2 22 22 22 23 23 24 24 24 25 26 27 28 30 31 86% Tc, Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 100% Rt ,Tc.Rt, Tc 2 21 22 22 23 24 25 26 27 28 28 29 29 30 30 36% Rt ,Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 26 28 29 30 100% Rt, Tc 2, Rt 2 22 22 22 22 23 23 24 25 26 27 28 29 30 31 29% Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 100% Tc, Rt, Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 100% Tc, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 25 26 27 29 30 79% Rt, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 100% Tc, Rt, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 25 25 27 28 31 64% AVE 21 22 22 23 23 24 24 25 26 26 27 28 29 30 M. Vote 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Three observations to note: (1)The two integration methods did not agree on two cure time estimates for rubber thicknesses 14 and 15 mm; (2) Some adhesion- hardness pairs had relative accuracy less than 50%; and (3) The majority vote result was equivalent to the result with the highest frequency in Table 8. Table 10 shows the bias-variance-covariance decomposition of the MSPEsimul of the ensemble of results. Table 10. Showing the bias-variance-covariance decomposition of the MSPEsimul MSPE Bias Var. Covar MSPE Bias Var. Covar. MODEL Rt ,Tc.Rt, 0.1430 0.3849 0.0609 0.1841 0.4015 0.0229 Tc.Rt, Rt 2 0.1464 0.3270 0.0394 0.0755 0.2500 0.0117 Tc, Rt, Tc.Rt 0.1298 0.2980 0.0410 0.1726 0.3738 0.0329 Tc, Rt, Rt 2 0.1121 0.2878 0.0292 0.1048 0.2890 0.0202 Tc, Tc.Rt, Rt 2 0.1417 0.3543 0.0162 0.0957 0.2852 0.0144 Rt ,Tc.Rt, Tc 2 0.1297 0.2636 0.0602 0.1841 0.4015 0.0229 Rt ,Tc.Rt, Rt 2 0.1513 0.3699 0.0144 0.0957 0.2852 0.0144 Rt, Tc 2, Rt 2 0.5463 0.3323 0.0161 0.0957 0.2852 0.0144 Tc.Rt, Tc 2, Rt 2 0.1051 0.3006 0.0147 0.0957 0.2852 0.0144 Tc, Rt, Tc.Rt, Rt 2 0.0992 0.2774 0.0223 0.0957 0.2852 0.0144 Tc, Tc.Rt, Tc 2, Rt 2 0.4951 0.6186 0.1125 0.0755 0.2500 0.0117 Rt, Tc.Rt, Tc 2, Rt 2 0.1265 0.3323 0.0161 0.0957 0.2852 0.0144 Tc, Rt, Tc.Rt, Tc 2, Rt 2 0.1561 0.6155 0.1373 0.0651 0.2200 0.0117 AVE 0.21864 0.3586 0.0446 0.0937 0.1105 0.2998 0.0174 0.0209 There were six adhesion-hardness model pairs that have the same accuracy values on the hardness side. Generally, for a high adhesion side MSPEsimul, there was a low MSPEsimul on the hardness side. This pattern, however, did not seem to have any significant relationship with the accuracy of the cure time estimates. Pavolo & Chikobvu/Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 90-106 101 Table 11 gives the percentage accuracy compromise of the base model pairs of the ensemble at simultaneous optimisation. Table 11. Showing the PAC results at simultaneous optimisation Rt MSPEvalmin MSE PAC(H) PAC(A) Adhesion Model % % Rt, Tc.Rt 0.0242 2.3855 310 487 Tc.Rt, Rt2 0.0321 1.3382 69 357 Tc, Rt, Tc.Rt 0.0405 2.3441 285 221 Tc, Rt, Rt 2 0.0458 1.4710 145 134 Tc, Tc.Rt, Rt 2 0.0456 1.2815 114 211 Rt, Tc.Rt, Tc 2 0.0194 2.5300 114 569 Rt, Tc.Rt, Rt 2 0.0226 1.3200 114 220 Rt, Tc 2, Rt 2 0.0445 1.8069 114 1128 Tc.Rt, Tc 2, Rt 2 0.0474 1.3377 114 219 Tc, Rt, Tc.Rt, Rt 2 0.0407 1.2789 113 144 Tc, Tc.Rt, Tc 2, Rt 2 0.0348 0.9842 33 1323 Rt, Tc.Rt, Tc 2, Rt 2 0.0345 1.3689 114 267 Tc, Rt, Tc.Rt, Tc 2, Rt 2 0.0366 0.9762 69 1323 Table 11 shows that it’s very difficult, were simultaneous optimisation is concerned, to find a model pair that has all the model accuracy criteria aligning. ➒ The response model pair with best MSPEvalmin (0.0194) had the worst MSE (2.53). ➒ The adhesion response model with the best MSE (0.9762) compromised the worst (% accuracy compromise, PAC(A) = 1323%) to achieve simultaneous optimisation. ➒ The response model pair with the best PAC on the hardness side (33%), had the worst PAC on the adhesion side (1323%). ➒ There were six adhesion-hardness model response pairs with the same PAC(H) value. These six were not necessarily the same ones with similar MSPEsimul values on the hardness side. The average PAC(H) was lower compared to the PAC(A). This suggests that response models do not necessary compromise the same to achieve simultaneous optimisation. 4.4 Ensemble review results Elimination of model pairs with relative accuracy less than 50% left nine adhesion- hardness pairs in the ensemble. The arithmetic average and majority vote results of the reviewed ensemble were equal throughout the whole rubber thickness range as shown in Table 12. Estimating Rubber Covered Conveyor Belting Cure Times Using Multiple Simultaneous Optimizations Ensemble 102 Table 12. Showing the result of eliminating response models with relative accuracy<50% Rt (mm) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Rel. Model Acc. Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 25 26 27 29 30 79% Tc, Rt, Rt 2 22 22 22 23 23 24 24 24 25 26 27 28 30 31 86% Tc, Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 100% Rt ,Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 100% Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 100% Tc, Rt, Tc.Rt, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 29 30 100% Tc, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 25 26 27 29 30 79% Rt, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 26 27 28 28 30 100% Tc, Rt, Tc.Rt, Tc 2, Rt 2 21 22 22 23 23 24 24 24 25 25 25 27 28 31 64% AVE 21 22 22 23 23 24 24 24 25 26 27 28 29 30 M. Vote 21 22 22 23 23 24 24 24 25 26 27 28 29 30 Table 13 shows that the accuracy results at simultaneous optimisation significantly improved, but more on the hardness side than the adhesion side. Table 13. Showing the accuracy results of the reviewed ensemble MSPE Bias Var. Covar MSPE Bias Var. Covar. MODEL Tc.Rt, Rt 2 0.1464 0.3270 0.0394 0.0755 0.2500 0.0117 Tc, Rt, Rt 2 0.1121 0.2878 0.0292 0.1048 0.2890 0.0202 Tc, Tc.Rt, Rt 2 0.1417 0.3543 0.0162 0.0957 0.2852 0.0144 Rt ,Tc.Rt, Rt 2 0.1513 0.3699 0.0144 0.0957 0.2852 0.0144 Tc.Rt, Tc 2, Rt 2 0.1051 0.3006 0.0147 0.0957 0.2852 0.0144 Tc, Rt, Tc.Rt, Rt 2 0.0992 0.2774 0.0223 0.0957 0.2852 0.0144 Tc, Tc.Rt, Tc 2, Rt 2 0.4951 0.6186 0.1125 0.0755 0.2500 0.0117 Rt, Tc.Rt, Tc 2, Rt 2 0.1265 0.3323 0.0161 0.0957 0.2852 0.0144 Tc, Rt, Tc.Rt, Tc 2, Rt 2 0.1561 0.6155 0.1373 0.0651 0.2200 0.0117 Arithmetic Ave 0.2104 0.3870 0.0447 0.0619 0.0888 0.2706 0.0148 0.0157 If the base models were the five adhesion response models with the same cure time estimates the theoretical accuracy of the ensemble would be as shown in Table 14. It appeared, for this problem, that when the cure time estimates for different adhesion- hardness pairs were the same, the theoretical accuracy on the hardness response side was the same. Pavolo & Chikobvu/Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 90-106 103 Table 14. Accuracy results of the ensemble with five base models with similar Tc estimates MSPE Bias Var. Covar MSPE Bias Var. Covar. MODEL Tc, Tc.Rt, Rt 2 0.1417 0.3543 0.0162 0.0957 0.2852 0.0144 Rt, Tc.Rt, Rt 2 0.1513 0.3699 0.0144 0.0957 0.2852 0.0144 Tc.Rt, Tc 2, Rt 2 0.1051 0.3006 0.0147 0.0957 0.2852 0.0144 Tc, Rt ,Tc.Rt, Rt 2 0.0992 0.2774 0.0223 0.0957 0.2852 0.0144 Rt, Tc.Rt, Tc 2, Rt 2 0.1265 0.3323 0.0161 0.0957 0.2852 0.0144 Arithmetic Ave 0.1248 0.3269 0.0167 0.0182 0.0957 0.2852 0.0144 0.0144 4.5 Multiple MS criteria best model selection methodology results The first result was for the selection of the best model using majority vote of fifteen different model selection criteria. The formulae used to compute the criteria values are shown in the Appendix. Table 15 shows the model selection criteria values and their votes. Adhesion response model [Tc.Rt, Rt2] is the obvious best with a vote of 10. This minimises uncertainty. Model [Tc, Tc.Rt, Tc2, Rt2] follows behind with a vote of 6. Table 15. Showing multiple MS criteria selections MODEL Tc, Rt, Tc.Rt, Tc 2, Rt 2 Tc, Tc.Rt, Tc 2, Rt 2 Rt, Tc.Rt, Rt 2 Tc.Rt, Rt 2 Tc, Rt, Rt 2 Tc, Tc.Rt, R2 (pr.) 26.5 51.5 49.9 65.4 49.4 52 Adeq. pr. 10.4 4.1 5.7 1.8 11.9 5.4 Cp-k 1.0 0.1 0 0 2.1 0 PRESS 88.1 59.3 81.3 42.1 62.2 58.6 AIC 11.7 9.8 11.6 9.8 13 17.3 BIC 15.1 12.6 13.9 11.5 15.3 19 AICc 20.2 14.8 14.3 11 15.7 18.5 APCp 2.9 2.4 2.6 2.9 2.6 4.0 SBC 1.9 1.7 4.9 4.8 6.1 11.9 HQc 1.1 1.1 4.3 4.5 5.6 11.5 KICc 87.3 64.4 51.2 38.1 52.6 45.6 HQ 0.5 0.5 4.2 4.2 5.6 11.7 KIC 20.7 17.8 18.6 15.8 20 23.3 MKIC 18.2 12 9.1 5.4 9.8 10.7 TIC 13.7 11.8 13.6 11.8 15 19.3 VOTE 2 6 1 10 1 1 The simultaneous optimisation results of both response models [Tc.Rt, Rt2] and [Tc, Tc.Rt, Tc2, Rt2] are shown in Tables 12 and 13. The two have the same cure time estimate and theoretical accuracy results on the hardness side. The cure time estimate result, however, was different from the multiple simultaneous optimisations ensemble one. 5. Conclusion The dilemma of choosing from two different solutions, see Table 16, both of which standing on strong positions makes the problem at hand challenging. However, an objective analysis and critic of each position helps in separating the two. Estimating Rubber Covered Conveyor Belting Cure Times Using Multiple Simultaneous Optimizations Ensemble 104 Table 16. Multiple MS criteria solution (S6) vs. Ensemble solution (S7) Rt 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Vote S6 21 22 22 23 23 24 24 24 25 25 26 27 29 30 2 S7 21 22 22 23 23 24 24 24 25 26 27 28 29 30 5 The multiple simultaneous optimisations ensemble cure time estimate solution (S7) shows credibility in that (i) it is the most frequent solution from the adhesion- hardness model pairs, (ii) there is agreement between the two integration methodologies used, and (iii) by design, it fairly accounts for all the listed problems of the contemporary MRSM contextual framework. It accounts for dataset uncertainty, loss of information, model over-/underfitting, and model parameter bias by utilising multiple models and minimising discarded models. It minimises model uncertainty and small sample size inefficiency by totally avoiding the use of classical model selection criteria. On the other hand, seven of the ten model selection criteria that voted for the best single adhesion response model [Tc.Rt, Rt2] are information criteria and three are prediction model selection criteria. This implies that the response model has the best parsimonious fit to the MRSM dataset, of all the 25 OLS adhesion response models, and has good prediction capability. However, the cure time estimate solution (S6) is not considered the best in credibility because (i) model selection criteria have a small sample size inefficiency problem, (ii) they do not deal with the problem of model parameter bias, (iii) dataset uncertainty and (iv) since the methodology is structured as the contemporary MRSM framework, it loses information in discarded models by the selection and use of one model per response in simultaneous optimisation. It should be emphasised that where the model with the best parsimonious fit to the dataset is required, response model [Tc.Rt, Rt2] is the model. The arguments above clearly separate the most credible cure time solution (S7) from the model with the best parsimonious fit to the MRSM dataset (S6]. The multiple simultaneous optimisations ensemble, therefore, is both logically and empirically a better way of obtaining credible results compared to the current MRSM contextual framework which must first select a best model for each response before simultaneous optimisation. The multiple simultaneous optimisations ensemble is thus recommended to the rubber covered conveyor belting manufacturing industry for use in reviewing cure times when adhesion and cover hardness minimum quality standard requirements change. The use of targeted values in validation is worth mentioning here, as well, since the size of the MRSM dataset is small and it would be senseless to split it. The other option would have been to use cross validation which would have taken back the solution methodology to the weakness of the contemporary MRSM framework. In itself, the practice is worth considering where targeted quality values have to be effectively converted to production process parameters. Noting the fact that the multiple simultaneous optimisations ensemble worked well on a two factors and two responses problem, it then makes it imperative to investigate its generalisability to other more complex MRSM problems. As the number of factors and responses increase, the number of models to deal with quickly multiplies. This will definitely require software and intelligent algorithms to manage complexity and still achieve credible results. Pavolo & Chikobvu/Oper. Res. Eng. Sci. Theor. Appl. 5(1) (2022) 90-106 105 Acknowledgements: We would like to thank the editor and anonymous referees for their valuable contributions that helped us to improve the quality and presentation of this paper. References Ahangi, A., Langroudi, A. F., Yazdanpanah, F., & Mirroshandel, S. A. (2019). 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