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Complex technology not only provides potential economic benefits but also increases the difficulty of application. Whether and how upstream technological complexity affects downstream manufacturers' innovation through vertical separation structure is worth discussing, but it has not been effectively discussed.

Through theoretical analysis and empirical testing, this article discusses the cost effect and market competition effect caused by upstream technological complexity on downstream manufacturers and further elucidates the impact of upstream technological complexity on downstream manufacturers' innovation.

Research has found that the impact of upstream technological complexity on the downstream manufacturers' innovation depends on the cost effect and market competition effect. The cost effect caused by the complexity of upstream technology inhibits the innovation of downstream manufacturers. In contrast, the market competition effect promotes the innovation of downstream manufacturers. There are differences in the cost effect and market competition effect of upstream technological complexity on different types of downstream manufacturers, so there is also significant heterogeneity in the impact of upstream technological complexity on innovation of different types of downstream manufacturers.

The conclusions of this article improve the understanding of the relationship between upstream technological complexity and downstream innovation and provide helpful implications for industrial chain innovation.

The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d≥12. Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases.

Let p_[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p_[1,r;t](n) is the generalisation of the two-colour partition function p_[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.

Using q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.

In the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.

The partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.

In order to make full use of the generalized greyness of interval grey number, this paper analyzes the properties and its applications of generalized greyness.

Firstly, the static properties of generalized greyness in bounded background domain, infinite background domain and infinitesimal background domain are analyzed. Then, this paper gives the dynamic properties of generalized greyness in bounded background domain, infinite background domain and infinitesimal background domain and explains the dialectical principle contained in it. Finally, the generalized greyness is used to judge the effectiveness of interval grey number transformation.

The results show that the generalized greyness of interval grey number has relativity, normativity, unity, eternity and conservation. The static and dynamic properties of generalized greyness are the same in the infinite and infinitesimal background domain, while they are different in the bounded background domain. The generalized greyness can be used as an index to judge whether the grey number transformation is greyness or information preservation.

The research shows that the generalized greyness can be used as an index to judge the validity of the grey number transformation and also can be applied in grey evaluation, grey decision-making and grey prediction and so on.

The paper succeeds in realizing the mathematical principle of “white is black”, the “greyness clock-slow effect” of the value domain of interval grey number and the generalized greyness conservation principle, which provides a theoretical basis for the rational use of generalized greyness of interval grey number.

The purpose of this paper is to develop the linguistic q-rung orthopair fuzzy set (LqROFS) information VIKOR method based on the bi-direction Choquet integral (BDCI), taking into account the correlation between information. The method can enrich the existing studies related to LqROFS information and better solve the problem of MAGDM problem.

Since applying Choquet integral (CI) depict information interaction is a common operation in MAGDM. However, the traditional CI has some limitations. The unidirectional alignment may affect the MAGDM results. Therefore, a LqROFS-VIKOR method based on BDCI is proposed, where BDCI is used to aggregate the decision matrix. Furthermore, it is not reasonable to apply exact numbers to express the similarity between two qualitative data. Then a new method of defining similarity using linguistics is proposed. The similarity is used to calculate attribute weights.

The validity and potential application of MAGMD method with linguistic q-rung orthopair fuzzy information based on BDCI are demonstrated in a numerical examples study.

According to the study of available literature, the current research on LqROFS is incomplete. The existing studies of both similarity and aggregate operators have certain shortcomings. The definition of similarity proposed in this paper is more in line with reality. And compared with the existing methods, the BDCI-based aggregate operator can describe the interaction between information more reasonably. Based on this VIKOR method based on BDCI under the LqROFS environment can better select the alternative.

The purpose of this paper is to maximize the average profit of the supply chain by calculating the order quantity, the number of shipments during the production time of the vendor, the number of shipments during the supply cycle of the vendor and the time when the retailer’s inventory level reaches to zero.

A production and inventory model for degrading commodities with stochastic demand and two-level partial trade credit in a supply chain is presented. The model’s applicability and the processes' feasibility for solving are verified by GAMS software with BARON.

The impact of the model’s parameters on the vendor and retailer’s average profit was found through sensitivity analysis. The effect of the model’s parameters on the supply chain’s average profit was also found. Moreover, the reasons for this effect were given.

First, decision-makers may use this model to increase the supply chain's average profit. Second, the proposed model takes a general form. Third, the policymakers can also adjust the model’s parameters according to their preferences to get the desired results.

First, this paper develops an inventory and production model for perishable goods. Second, it is believed that the demand is random because the demand is affected by many factors, which make the study more realistic. Third, this paper studies production and inventory problems from the supply chain perspective. Finally, the interest for partial trade credit is calculated. The interest caused by stochastic shortages is also considered and calculated.

In this paper, the authors study the nonlinear matrix equation Xp=Q±A(X-1+B)-1AT, that occurs in many applications such as in filtering, network systems, optimal control and control theory.

The authors present some theoretical results for the existence of the solution of this nonlinear matrix equation. Then the authors propose two iterative schemes without inversion to find the solution to the nonlinear matrix equation based on Newton's method and fixed-point iteration. Also the authors show that the proposed iterative schemes converge to the solution of the nonlinear matrix equation, under situations.

The efficiency indices of the proposed schemes are presented, and since the initial guesses of the proposed iterative schemes have a high cost, the authors reduce their cost by changing them. Therefore, compared to the previous scheme, the proposed schemes have superior efficiency indices.

Finally, the accuracy and effectiveness of the proposed schemes in comparison to an existing scheme are demonstrated by various numerical examples. Moreover, as an application, by using the proposed schemes, the authors can get the optimal controller state feedback of $x(t+1) = A x(t) + C v(t)$.

This paper analytically models and numerically investigates two operating levers, namely optimization of product price and optimization of product quality in the context of a manufacturer that sells the products directly in the marketplace. The study attempts to identify how optimizing product quality and product price can fulfill a manufacturer's economic aims such as maximization of the manufacturer's profit and market demand. Anchored to the extant literature, the demand is modeled as a parametric joint multiplicative function of price and quality. Further, price is modeled as a function of product quality.

First, the authors evolve the analytical expression for the manufacturer's profit. Thereafter, following the mathematical principles of unconstrained optimization, the authors arrive at the conditions for optimal product quality and product price. The authors further perform numerical experiments to understand the behavior of economic dimensions such as profit and demand with respect to sensitivities associated with cost, quality and price.

The authors find that under product quality optimization, the optimal product quality is a unique solution in that a highest possible theoretical manufacturer's profit is obtained. However, in the case of product price optimization, the optimal product price is non-unique and is a function of product quality. The authors further find that in the context of functional quality-level expectations, product quality optimization as an operating lever gives a better dividend. However, in the case of higher product quality expectations, product price optimization performs better than product quality optimization. Further, several novel findings are also obtained from numerical experimentations.

The findings of the authors' study have implications for types of industries characterized by relatively low as well as relatively high competitive intensity. Further, as opposed to several extant studies that have often carried out joint optimization of quality and price, the authors' study is one of the first to study the impact of product price and product quality on the manufacturer's economic objective in a disparate and focused manner, thus capturing individual effects.

The purpose of this paper is to develop a multi-attribute group decision-making (MAGDM) method under the q-rung orthopair trapezoidal fuzzy environment, which calculates the interaction between the criteria depending on the proposed q-rung orthopair trapezoidal fuzzy aggregation Choquet integral (q-ROTrFACI) and employ TODIM (an acronym in Portuguese of Interactive and Multi-criteria Decision Making) to consider the risk psychology of decision-makers, to determine the optimal ranking of alternatives.

In MAGDM, q-rung orthopair trapezoidal fuzzy numbers (q-ROTrFNs) are efficient to indicate the quantitative vagueness of decision-makers. The q-ROTrFACI operator is defined and some properties are proved. Then, a novel similarity measure is developed by fusing the area and coordinates of the q-rung orthopair trapezoidal fuzzy function. Based on the above, a Choquet integral-based TODIM (CI-TODIM) method to consider the risk psychology of decision-makers is proposed and two cases are provided to prove superiority of the method.

The paper investigates q-ROTrFACI operator to productively solve problems with interdependent criteria. Then, an approach is proposed to determine the center point of q--ROTrFNs and a q-rung orthopair trapezoidal fuzzy similarity is constructed. Furthermore, CI-TODIM method is devised based on the proposed q-ROTrFACI operator and similarity in q-rung orthopair trapezoidal fuzzy context. The illustration example of business models' solutions and hypertension health management are given to demonstrate the effectiveness and superiority of proposed method.

The paper develops a novel CI-TODIM method that effectively solves the MAGDM problems under the premise of fully considering the priority of criteria and the risk preference of decision-makers, which provides guiding advantages for practical decision-making and enriches the application of decision-making theory.

The purpose of this paper is to investigate the optimal production and financing strategies for the closed-loop supply chain (CLSC) composed of a capital-constrained original equipment manufacturer (OEM) and a risk-averse authorized remanufacturer (RM).

The authors formulate four models with different scenarios, namely, the OEM has sufficient capital; the OEM has limited capital without financing; the OEM adopts debt financing strategy; and the OEM adopts equity financing strategy. The equilibrium solutions of each scenario are obtained by backward induction method, the influences of risk aversion coefficient on the equilibrium solutions are examined and the OEM's optimal financing strategy is found by comparison analysis.

When the OEM's initial capital is limited and the equity dividend ratio is less than a certain threshold, the equity financing strategy is more advantageous for the OEM. However, if the OEM's initial capital is extremely scarce and the dividend proportion is large, the OEM prefers the debt financing strategy. When considering financing, consumer surplus always decreases as the risk aversion factor increases; the debt financing strategy is more environmentally friendly compared with the equity financing strategy. Only the debt financing strategy can make both members in the CLSC achieve a win-win situation in a certain region when the dividend ratio is sufficiently large.

It will be more fascinating if the model extends to such a case that the production operation situation in the CLSC composed of multiple OEMs in multiple periods. Furthermore, the remanufacturer's risk-averse information is asymmetry may be more realistic in our daily life.

There are three main differences from the existing research. One is that the remanufacturer's risk aversion originates from the uncertain remanufacturing cost instead of the uncertain market demand. Another is that the boundary conditions of the OEM prefer to adopt debt financing is obtained through the envelope theorem with Lagrange multiplier method. Last but not the least, this paper provides a good theoretical reference and practical guidance for the OEM to make the rational financing strategy selection in face of different degree of capital scarcity in the CLSC system. The value of the three aspects provides a theoretical basis for the optimal operation decisions of capital-constrained manufacturer considering the remanufacturer's risk aversion in the CLSC operation system.