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The purpose of this paper is to make a theoretical comprehensive efficiency evaluation of a nonlinear analysis method based on the Woodbury formula from the efficiency of the solution of linear equations in each incremental step and the selected iterative algorithms.

First, this study employs the time complexity theory to quantitatively compare the efficiency of the Woodbury formula and the LDLT factorization method which is a commonly used method to solve linear equations. Moreover, the performance of iterative algorithms also significantly effects the efficiency of the analysis. Thus, the three-point method with a convergence order of eight is employed to solve the equilibrium equations of the nonlinear analysis method based on the Woodbury formula, aiming to improve the iterative performance of the Newton–Raphson (N–R) method.

First, the result shows that the asymptotic time complexity of the Woodbury formula is much lower than that of the LDLT factorization method when the number of inelastic degrees of freedom (IDOFs) is much less than that of DOFs, indicating that the Woodbury formula is more efficient for local nonlinear problems. Moreover, the time complexity comparison of the N–R method and the three-point method indicates that the three-point method is more efficient than the N–R method for local nonlinear problems with large-scale structures or a larger ratio of IDOFs number to the DOFs number.

This study theoretically evaluates the efficiency of nonlinear analysis method based on the Woodbury formula, and quantitatively shows the application condition of the comparative methods. The comparison result provides a theoretical basis for the selection of algorithms for different nonlinear problems.

Clarify the effects of potato flour (PF) addition on dough properties and quality characteristics of pie bread, thus providing a theoretical basis for potato-wheat-yogurt pie bread processing.

The seven pie bread formulations were prepared with addition of different amount of PF (0, 10, 20, 30, 40, 50 and 60 per cent) by replacing wheat flour (WF). The influence of replacement of PF with WF on the dough development, gas behavior, microstructural, gluten-starch interaction inside dough matrix and the nutritional and textural properties of pie bread was analyzed.

Compared to dough with WF, dough height was decreased significantly but total gas volume showed no significant difference with the addition of PF (10-60 per cent), and the dough height was decreased gradually with the increasing addition of PF (p = 0.0012). Scanning electron microscopy and confocal laser scanning microscopy showed that dough with 10-30 per cent PF revealed smooth surface and continuous network structure was also existed between starch and gluten, this was similar to the structure of wheat dough, while the surface was rough and the connection was not continuous when the PF addition was further increased from 40 to 60 per cent. Nuclear magnetic resonance showed dough with 10-60 per cent PF presented larger content of immobilized water compared to the wheat dough, and its content was also increased with the increasing addition of PF (p = 0.0008). Pie bread with 10-60 per cent PF exhibited lower fat, carbohydrate content and energy value compared to the wheat pie bread, and the fat, carbohydrate, energy was further decreased with the increasing addition of PF (p = 0.0012, 0.0002, 0.0007, respectively), while the dietary fiber content increased (p = 0.0011). In addition, specific volume and height/diameter ratio showed no significant difference in pie bread with 0-40 per cent PF, which were 2.1-2.4 cm³/g and 0.16-0.19, respectively (p = 0.004), whereas they were reduced significantly when the PF was increased to 60 per cent (p = 0.001, p = 0.002). In addition, the lightness (L) of pie bread decreased gradually with the increasing amount of PF, and the browning index was increased. Pie bread with 0-40 per cent PF showed similar hardness, springness and chewiness value, which were also increased when the PF was increased to 60 per cent (p = 0.001). The above results indicated that PF could improve the nutritional quality of pie bread, and the appropriate addition amount should be no more than 40 per cent.

The mechanisms of this study should be clarified for further study.

PF was an excellent food ingredient and could be used for pie bread making at 30 per cent replacement level to overcome use of WF.

The results of this study can provide a new kind of potato staple food with high nutrition and low energy for market and consumers.

PF was first used to produce yogurt pie bread to improve the nutritional properties of traditional WF yogurt pie bread.

This study aims to examine the halal brand image, halal logo and halal awareness of non-Muslim customers on attitude and behavioral intention.

This study uses a quantitative survey approach to 400 respondents consisting of 400 non-Muslim millennials in Indonesia, Malaysia and Singapore who have to buy halal cosmetics two years later. The sampling technique is purposive sampling. The analysis technique used in this study is structural equation modeling.

This study showed that halal logo, halal awareness and halal brand image have an effect on customer attitude. Then, the halal logo, halal awareness and halal brand image have an influence on behavioral intention. The existence of this positive signal has provided a great opportunity for businesses to make profits by meeting the demand for the halal market. Consumption of halal cosmetics produced by manufacturers attracts many non-Muslim consumers because of the safety, comfort and cleanliness of product ingredients, which of them must be given by the halal product, especially for cosmetics.

This study broadens the understanding of the attitudes and behavioral intentions of non-Muslim consumers toward halal cosmetic products. Furthermore, this research has provided a deeper interpretation of non-Muslim understanding of halal logos, halal brand images and halal awareness which have been minimal in research studies.

This study contributes to the literature related to the behavioral intentions of millennial non-Muslim consumers for halal cosmetics. Therefore, respondents in the study were specific, that is, non-Muslims who are millennial generation in the Indonesia, Malaysia and Singapore context.