Forward Integration Research Articles

Benchmarking methods for computing local sensitivities in ordinary differential equation models at dynamic and steady states.

Estimating parameters of dynamic models from experimental data is a challenging, and often computationally-demanding task. It requires a large number of model simulations and objective function gradient computations, if gradient-based optimization is used. In many cases, steady-state computation is a part of model simulation, either due to steady-state data or an assumption that the system is at steady state at the initial time point. Various methods are available for steady-state and gradient computation. Yet, the most efficient pair of methods (one for steady states, one for gradients) for a particular model is often not clear. In order to facilitate the selection of methods, we explore six method pairs for computing the steady state and sensitivities at steady state using six real-world problems. The method pairs involve numerical integration or Newton's method to compute the steady-state, and-for both forward and adjoint sensitivity analysis-numerical integration or a tailored method to compute the sensitivities at steady-state. Our evaluation shows that all method pairs provide accurate steady-state and gradient values, and that the two method pairs that combine numerical integration for the steady-state with a tailored method for the sensitivities at steady-state were the most robust, and amongst the most computationally-efficient. We also observed that while Newton's method for steady-state computation yields a substantial speedup compared to numerical integration, it may lead to a large number of simulation failures. Overall, our study provides a concise overview across current methods for computing sensitivities at steady state. While our study shows that there is no universally-best method pair, it also provides guidance to modelers in choosing the right methods for a problem at hand.

The First Validation of Aerosol Optical Parameters Retrieved from the Terrestrial Ecosystem Carbon Inventory Satellite (TECIS) and Its Application

In August 2022, China successfully launched the Terrestrial Ecosystem Carbon Inventory Satellite (TECIS). The primary payload of this satellite is an onboard multi-beam lidar system, which is capable of observing aerosol optical parameters on a global scale. This pioneering study used the Fernald forward integration method to retrieve aerosol optical parameters based on the Level 2 data of the TECIS, including the aerosol depolarization ratio, aerosol backscatter coefficient, aerosol extinction coefficient, and aerosol optical depth (AOD). The validation of the TECIS-retrieved aerosol optical parameters was conducted using CALIPSO Level 1 and Level 2 data, with relative errors within 30%. A comparison of the AOD retrieved from the TECIS with the AERONET and MODIS AOD products yielded correlation coefficients greater than 0.7 and 0.6, respectively. The relative error of aerosol optical parameter profiles compared with ground-based measurements for CALIPSO was within 40%. Additionally, the correlation coefficients R2 with MODIS and AERONET AOD were approximately between 0.5 and 0.7, indicating the high accuracy of TECIS retrievals. Utilizing the TECIS retrieval results, combined with ground air quality monitoring data and HYSPLIT outcomes, a typical dust transport event was analyzed from 2 to 7 April 2023. The results indicate that dust was transported from the Taklamakan Desert in Xinjiang, China, to Henan and Anhui provinces, with a gradual decrease in the aerosol depolarization ratio and backscatter coefficient during the transport process, causing varying degrees of pollution in the downstream regions. This research verifies the accuracy of the retrieval algorithm through multi-source data comparison and demonstrates the potential application of the TECIS in the field of aerosol science for the first time. It enables the fine-scale regional monitoring of atmospheric aerosols and provides reliable data support for the three-dimensional distribution of global aerosols and related scientific applications.

Integration of manufacturing and services: Examining its effect on resource allocation and manufacturing labor productivity

The integration of manufacturing and services is a key factor driving the development of the manufacturing sector. Thus, this study uses Chinese province–industry data to determine the effect of such integration on resource allocation and manufacturing labor productivity. Additionally, heterogeneity is systematically examined, while avenues for its improvement are explored. Based on the results, dividing integration into forward and backward integration shows that both types significantly enhance manufacturing labor productivity by improving resource allocation efficiency, with forward integration demonstrating a more pronounced effect. This mechanism mainly manifests in capital- and technology-intensive industries but not in labor-intensive ones. Notably, while the integration of manufacturing and services enhances both economic and social efficiency by curbing capital over-allocation, it creates the problem of labor over-allocation. These conclusions provide value insights for promoting industrial transformation and upgrading.

Global value chains and inward FDI: An empirical investigation of European firms

AbstractThis paper empirically investigates whether and how the level of GVC integration of a given market may explain the presence of foreign‐owned firms. Using firm‐level data from 28 European Union countries during the period 2008–2014, we provide evidence that a greater country‐sector‐level GVC participation, via both backward and forward linkages, exerts a positive effect on a firm's likelihood to receive FDI. These findings appear particularly strong for new EU Member States and services industries when looking at the differences across countries and sectors. Interestingly, when exploring the role of country‐sector position along the GVC, we find that FDI gains from backward GVC integration are more prominent if the markets are associated with the final stages of the supply chain, whereas those from forward GVC integration are greater when the markets are associated with the initial stages, in line with the smile curve hypothesis.

Strategic Choices of General Contractors in the Context of China’s Industry Chain of Construction Industrialization

Amidst the challenges of economic downturn and construction industrialization, the profits obtained by general contractors through comparative advantage strategies are slowly decreasing, and thus, new strategic choices are required. The collaborative division of labor effect in the industry chain can improve profits and labor productivity, which is an important driving force for enterprise transformation and development. Therefore, a need arises to improve the profits of general contractors in the industrial chain system composed of prefabricated component suppliers, general contractors, and building development enterprises. Accordingly, this paper constructs a backward integration, forward integration, and bidirectional integration Stackelberg game model based on the proportion of resource investments, with general contractors as the main decision-making body. It then compares and analyzes the optimal decision-making values in different situations to study the optimal strategic decision-making problem of general contractors. Research results indicate the following. (1) All three integrated strategies can improve the profits of general contractors. When the proportion of resource investment meets certain conditions, the profits of general contractors under the bidirectional integration strategy are the highest, while the sustainable performance of the industrial chain and prefabricated buildings can be increased and the coordination of the industrial chain can be achieved. Thus, it is the best choice for a general contractor. (2) As a prefabricated component supplier needs to carry out continuous technological innovation activity to obtain a cumulative effect, the return on investment of forward integration is less than that of backward integration. (3) General contractors may consider choosing to carry out bidirectional integration strategies of forward integration followed by backward integration.

Experimental backward integration for state-dependent differential Riccati equation (SDDRE): A case study on flapping-wing flying robot

Backward integration (BI) is a classical approach for solving optimal control arising from linear quadratic regulator (LQR) design in differential form. It proposes a two-round solution, the first one starting from a final boundary condition to the initial one to generate the optimal gain, and the next one, solving the control system in a forward loop. Implementation of the BI on the nonlinear optimal control, the state-dependent differential Riccati equation (SDDRE), is a challenge since in the backward loop, the state information is missing and is not straightforward like the LQR. Hence, a control for backward motion is required to regulate the system from the terminal to the initial desired states. While there have been some valuable works on the theoretical implementation of the BI in simulations for the SDDRE, this approach has not been reported in experiments for the best knowledge of the authors. Here in this work, the SDDRE is solved using the BI and two backward and forward solution steps, and it is experimentally applied for a flapping-wing flying robot (FWFR). The execution of the control system is done onboard using Raspberry Pi 4B as the processor which has limited computational capacity. The trajectory tracking of a line in a closed limited space is proposed for the FWFR flight. The objective is to position the robot bird at the corner of the rectangular limited space with minimum error in translation. The results have been presented for 21 flights to show the repeatability and presented the best-case minimum error of 10 (cm) at the end of the trajectory, in the YZ-plane. Considering approximately 12 (m) flight path, the error is found less than 1 percent of the travel distance. The results were compared with forward integration to confirm the correctness of the computation. The experimental flight dataset, MATLAB simulation codes, and experimental Python codes are available as supplementary material for this work in the online version of the paper.

Methodological considerations in assessing countermovement jumps with handheld accentuated eccentric loading

ABSTRACT This study aimed to compare the agreement between three-dimensional motion capture and vertical ground reaction force (vGRF) in identifying the point of dumbbell (DB) release during a countermovement jump with accentuated eccentric loading (CMJAEL), and to examine the influence of the vGRF analysis method on the reliability and magnitude of CMJAEL variables. Twenty participants (10 male, 10 female) completed five maximal effort CMJAEL at 20% and 30% of body mass (CMJAEL20 and CMJAEL30, respectively) using DBs. There was large variability between methods in both loading conditions, as indicated by the wide limits of agreement (CMJAEL20 = −0.22 to 0.07 s; CMJAEL30 = −0.29 to 0.14 s). Variables were calculated from the vGRF data, and compared between four methods (forward integration (FI), backward integration (BI), FI adjusted at bottom position (BP), FI adjusted at DB release point (DR)). Greater absolute reliability was observed for variables from DR (CV% ≤ 7.28) compared to BP (CV% ≤ 13.74), although relative reliability was superior following the BP method (ICC ≥ 0.781 vs ≥ 0.606, respectively). The vGRF method shows promise in pinpointing the DB release point when only force platforms are accessible, and a combination of FI and BI analyses is advised to understand CMJAEL dynamics.

Revolutionizing Fast Food Marketing Strategy using Kotler's Approach

This study investigates marketing strategies and service quality's impact on consumer decisions to purchase fast food at M2M, employing Kotler's marketing mix strategy. Using a qualitative approach, the research explores elements of the marketing mix—product, pricing, promotion, and distribution—influencing marketing success. Interviews with M2M managers and on-site observations provide insights. Findings highlight segmentation, targeting, and positioning processes, emphasizing mass marketing via social media. Key aspects include market penetration through affordable pricing and forward integration by collaborating with street food vendors. Implications suggest enhancing product innovation and understanding competitors' offerings. This research contributes to understanding effective marketing strategies in the fast-food industry, informing future business practices. Highlight: Impact of marketing mix on consumer behavior. Influence of service quality on purchase decisions. Application of Kotler's strategy in fast food industry. Keyword: Marketing strategy, Service quality, Consumer decisions, Fast food industry, Kotler's marketing mix.

MISTER-T: An open-source software package for quantum optimal control of multi-electron systems on arbitrary geometries

We present an open-source software package, MISTER-T (Manipulating an Interacting System of Total Electrons in Real-Time), for the quantum optimal control of interacting electrons within a time-dependent Kohn-Sham formalism. In contrast to other implementations restricted to simple models on rectangular domains, our method enables quantum optimal control calculations for multi-electron systems (in the effective mass formulation) on nonuniform meshes with arbitrary two-dimensional cross-sectional geometries. Our approach is enabled by forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency. We provide several examples of the versatility and efficiency of the MISTER-T code in handling complex geometries and quantum control mechanisms. The capabilities of the MISTER-T code provide insight into the implications of varying propagation times and local control mechanisms to understand a variety of strategies for manipulating electron dynamics in these complex systems. Program summaryProgram Title: MISTER-TCPC Library link to program files:https://doi.org/10.17632/psymy4ddnw.1Licensing provisions: GNU General Public License 3Programming language: MATLABSupplementary material: animated movies of total electron densities under the influence of optimal control fields for (1) an asymmetric double-well potential for long propagation times, (2) an asymmetric double-well potential for short propagation times, and (3) a triple-well potential with a position-dependent effective mass.Nature of problem: The MISTER-T code solves quantum optimal control problems for interacting electrons within a time-dependent Kohn-Sham formalism. It can handle two-dimensional systems with arbitrary cross-sectional geometries within the effective mass formulation. The user-friendly code uses forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency.Solution method: iterative solution of the quantum optimal control equations using finite element methods, effective mass formulation, pseudoskeleton decomposition, sparse matrix linear algebra, and nonuniform fast Fourier transforms.

Convergence analysis of a spectral numerical method for a peridynamic formulation of Richards’ equation

We study the implementation of a Chebyshev spectral method with forward Euler integrator proposed in Berardi et al.(2023) to investigate a peridynamic nonlocal formulation of Richards’ equation. We prove the convergence of the fully-discretization of the model showing the existence and uniqueness of a solution to the weak formulation of the method by using the compactness properties of the approximated solution and exploiting the stability of the numerical scheme. We further support our results through numerical simulations, using initial conditions with different order of smoothness, showing reliability and robustness of the theoretical findings presented in the paper.

SCFI: Efficient forward fine-grained control flow integrity based on coarse-grained ISA extensions

Code reuse attack (CRA) is a severe threat to computer systems. To mitigate this threat, control-flow integrity (CFI) was proposed to restrict control-flow transfers. Processor vendors developed Instruction Set Architecture (ISA) extensions to enforce forward CFI on processors, such as Indirect Branch Tracking (IBT) developed by Intel and Branch Target Identification (BTI) developed by ARM. However, these extensions are coarse-grained and unable to provide enough security protection.We combine existing coarse-grained ISA extension with novel software modifications and propose a new fine-grained forward CFI implementation called Slot-based CFI (SCFI). SCFI consists of two software modifications: Slot Policy and ID Policy. Slot policy utilises lower bits of code address as labels to implement fine-grained CFI. ID Policy provides multi-label support to solve the destination equivalence problem. SCFI strengthens security by improving coarse-grained ISA extension to fine-grained CFI (improving Average Indirect Targets Allowed (AIA) from 752 to 6.24). SCFI costs negligible performance overhead (on average 0.78% in SPEC2006). It also has good performance in complex programs (1.12% in 471.omnetpp and 1.79% in 483.xalancbmk). In addition, SCFI only needs lightweight software instrument without hardware modification and supports complex control-flow graphs.

A new lattice-based password authenticated key exchange scheme with anonymity and reusable key.

In this article, we propose a novel bilateral generalization inhomogenous short integer solution (BiGISIS)-based password-authenticated key exchange (PAKE) scheme for post-quantum era security. The hardness assumption of the constructed PAKE is based on newly proposed hard lattice problem, BiGISIS. The main aim of this article is to provide a solution for the post-quantum secure PAKE scheme, which is one of the open problems in the literature. The proposed PAKE is the first BiGISIS-based PAKE that satisfies anonymity and reusable key features. The bilateral-pasteurization (BiP) approach is used to obtain the reusable key, and anonymity is achieved thanks to the additional identity components and hash functions. The reusable key structure reduces the time in the key generation, and anonymity prevents illegal user login attempts. The security analysis is done by following the real-or-random (RoR) model assumptions. As a result of security examinations, perfect forward secrecy (PFS) and integrity are satisfied, and the resistance against eavesdropping, manipulation-based attack (MBA), hash function simulation, impersonation, signal leakage attack (SLA), man-in-the-middle (MitM), known-key security (KKS), and offline password dictionary attack (PDA) is captured. According to the comparison analysis, the proposed PAKE is the first SLA-resistant lattice-based PAKE with reusable key and anonymity properties.

Putting communities and citizens at the heart of what we do - The development of a continuum of integration within a locality in the Northwest of the UK to enhance health well-being and reduce inequalities.

The Northwest of England has seen health inequality and poor health outcome be a persistent feature for the population. The area has gained economic advancement in recent decades, but these have not been equally shared, and many population groups suffer disproportionately (1). Rochdale is a multi-ethnic locality within Greater Manchester. Around a quarter of local people have at least one or more long-term conditions and a third of residents live within 10% of the most deprived neighbourhoods nationally(2). 
 The Health and Care Act (2022) aims to transform the commissioning and delivery of health and social care across the UK with a far stronger emphasis on integration and neighbourhood working as well as statutory responsibilities for the newly formed Integrated Care systems (3). For Rochdale, as part of Greater Manchester, this journey started pre-2016 with devolution and the movement of the 6-billion-pound budget to the Greater Manchester Devolved Authority. Devolution provided the opportunity to implement an ambitious plan to transform the health and well-being across communities through a focus on system-wide integration, harnessing community assets, and transforming the relationship between the system, and communities.
 The Rochdale system took a radial approach to locality development, this started with thinking differently about the formation of the locality board that strategically plans for the location. The Rochdale approach that started with the concept of “in and out of scope” was not progressive or relevant when attempting to work with wider partners and the community. Equally, the changes in statutory responsibility potentially lead to greater complications in finances, accountability, and decision-making, these could all act as barriers to the integration of the service and the development of new models of care. Rochdale’s starting point was that “everything is in scope” and focused on a continuum for integration. Which enabled system partners to move to different levels of integration in a flexible way depending on the needs of the communities (from common interests to full integration with shared workforce and infrastructure). In this approach, the system is working towards ensuring clear accountability and responsibility for all parts of the continuum – for example, understanding quality, performance, and financial accountability at each level, while enabling a broad spectrum of engagement. The adoption of the continuum aims to be inclusive of a wider spectrum of partners magnifying the voice of held-back communities at a strategic planning level and enabling organisations to have clear expectations in driving forward sustainable change and integration. The learning for an international audience from the past 6 years of systems working and the development of the continuum approach is that this approach has the potential to cut through some of the barriers to integration, especially for organisations that may not form part of the central integration agenda enabling more innovative solutions to tackling inequalities. 
 
 
 Greater Manchester Combined Authority [GMCA] (2015) Taking Charge of Our Health and Social Care in Greater Manchester, Retrieved from:
 
 http://greatermanchester- a.gov.uk/download/downloads/id/125/taking_charge_of_our_health_and_social_car e_in_greater_manchester.pdf 
 
 Rochdale Borough council (2020) Rochdale Borough Locality Plan. https://www.rochdale.gov.uk/downloads/download/462/rochdale-borough-locality-plan
 Health and Care Act (2022)
 
 

Chaotic Capture of a Retrograde Moon by Venus and the Reversal of Its Spin

Planets are surrounded by fractal surfaces (traditionally called Hill spheres), separating the inner zones of long-term stable orbital motion of their satellites from the outer space where the gravitational pull from the Sun takes over. Through this surface, external minor bodies in trajectories loosely co-orbital to a planet can be stochastically captured by the planet without any assistance from external perturbative forces, and can become moons chaotically orbiting the planet for extended periods of time. Using state-of-the-art orbital integrators, we simulate such capture events for Venus, resulting in long-term attachment phases by reversing the forward integration of a moon initially attached to the planet and escaping it after an extended period of time. Chaotic capture of a retrograde moon from a prograde heliocentric orbit appears to be more probable because the Hill sphere is almost four times larger in area for a retrograde orbit than for a prograde orbit. Simulated capture trajectories include cases with attachment phases up to 860,000 years for prograde moons and up to 370,000 years for retrograde moons. Although the probability of a long-term chaotic capture from a single encounter is generally low, the high density of co-orbital bodies in the primordial protoplanetary disk makes this outcome possible, if not probable. The early Venus was surrounded by a dusty gaseous disk of its own, which, coupled with the tidal dissipation of the kinetic energy in the moon and the planet, could shrink the initial orbit and stabilize the captured body within the Hill surface. The tidal torque from the moon, for which we use the historical name Neith, gradually brakes the prograde rotation of Venus, and then reverses it, while the orbit continues to decay. Neith eventually reaches the Roche radius and disintegrates, probably depositing most of its material on Venus’ surface. Our calculations show that surface density values of about 0.06 kg m−2 for the debris disk may be sufficient to stabilize the initial chaotic orbit of Neith and to bring it down within several radii of Venus, where tidal dissipation becomes more efficient.

Forward Security with Crash Recovery for Secure Logs

Logging is a key mechanism in the security of computer systems. Beyond supporting important forward security properties, it is critical that logging withstands both failures and intentional tampering to prevent subtle attacks leaving the system in an inconsistent state with inconclusive evidence. We propose new techniques combining forward security with crash recovery for secure log data storage. As the support of specifically forward integrity and the online nature of logging prevent the use of conventional coding, we propose and analyze a coding scheme resolving these unique design constraints. Specifically, our coding enables forward integrity, online encoding, and most importantly a constant number of operations per encoding. It adds a new log item by 𝖷𝖮𝖱 ing it to k cells of a table. If up to a certain threshold of cells is modified by the adversary, or lost due to a crash, we still guarantee recovery of all stored log items. The main advantage of the coding scheme is its efficiency and compatibility with forward integrity. The key contribution of the paper is the use of spectral graph theory techniques to prove that k is constant in the number n of all log items ever stored and small in practice, e.g., k = 5. Moreover, we prove that to cope with up to \(\sqrt {n}\) modified or lost log items, storage expansion is constant in n and small in practice. For k = 5, the size of the table is only 12% more than the simple concatenation of all n items. We propose and evaluate original techniques to scale the computation cost of recovery to several GBytes of security logs. We instantiate our scheme into an abstract data structure which allows to either detect adversarial modifications to log items or treat modifications like data loss in a system crash. The data structure can recover lost log items, thereby effectively reverting adversarial modifications.

Brillouin frequency shift extraction by bidirectional integration of Brillouin gain spectrum

A novel Brillouin frequency shift (BFS) extraction approach based on bidirectional integration is proposed and investigated for mitigating the fluctuations in Brillouin optical time domain reflectometer (BOTDR) system. The proposed method involves calculating the BFS through the forward and backward integrations of the original Brillouin gain spectrum (BGS) data. The findings from both simulations and experiments demonstrate that bidirectional integration achieves superior BFS accuracy compared to Lorentzian curve fitting. To further evaluate its performance, the root mean square error of BFS is measured under the different signal-to-noise ratios (SNRs) and frequency steps and the results show that bidirectional integration has higher tolerance for low SNR. Specifically, bidirectional integration achieves a BFS accuracy of 0.853 MHz under SNR of 5.22 dB and significantly reduces the measurement time by 85.86 %. Additionally, the introduced coefficient k characterizes the influence on the BGS compression, with bidirectional integration exhibiting a narrower full width at half maximum of BGS (52.74 MHz) compared to Lorentzian curve fitting (67.45 MHz).

IMPLEMENTASI METODE IFAS EFAS DALAM PENENTUAN STRATEGI BISNIS VI HIGH LABEL RECORD

This thesis examines the business strategy of VI High Label Record in facing the commercial advertising industry. The research involves the analysis of SWOT, IFAS, EFAS, and IE matrices to identify factors influencing the company. The analysis reveals that VI High Label Record's main strength lies in "Commercial music usage," with a weakness in "Lack of physical offline sales market." The most significant opportunity is the "Continuous growth of the digital music industry," while the primary threat is "Rapid changes in music trends." The results of the IFAS and EFAS analyses indicate that the company's strengths outweigh its weaknesses (3.0>1.4), and opportunities outweigh threats (2.5>1.0). VI High Label Record is positioned in IE matrix cell II, suggesting a "Grow" and "Build" strategy. Recommendations include enhancing product marketing, international expansion, developing recording infrastructure, collaborating with influencers and social media, as well as backward and forward integration. This research aids record labels in devising strategies to confront advertising industry challenges and maintain their positions. The findings offer practical guidance for VI High Label Record and contribute to the music industry's understanding of record label challenges.

A journey toward global value chain upgrading: Exploring the transition from backward to forward integration

Global value chains (GVCs) are embraced worldwide as a gateway to technological and economic upgrading. Countries integrate into backward, value-importing linkages with the aim of accumulating technological capabilities and transitioning towards creating their own forward, value-exporting chains and capturing a greater share of the value generated within GVCs. Existing knowledge, which is largely fragmented and descriptive, points to a number of uncertainties and complexities that make this process anything but linear. It is open question whether the deepening backward linkages facilitates forward integration in GVCs. Using data from 65 countries over two decades, we show that the impact of backward integration on forward integration in GVCs varies over time and that it is moderated by the level of development of the country, the diversity of the GVC partner network and the innovation conditions in the home market. The research adds a new perspective to the literature on GVC-driven upgrading.

Research on Secondary Innovation Design of Valve Products based on 3D Printing Technology and Reverse Engineering

This study will take simple valves as the research object, the data extraction of some parts of valves, explore the research idea of three-dimensional model reconstruction of valves based on reverse engineering technology using forward and reverse integration modeling method, and deeply analyze the application prospect of 3D printing technology and reverse engineering technology in the valve industry. For domestic and foreign valve product users to provide rapid prototyping, secondary innovation design can be used for reference ideas.

Analysis of IFAS, EFAS Strategies and Feasibility of Mangot Cultivation Business to Reduce Household Waste

This research aims to analyze the strategy and feasibility of the Renon Trash Maggot (RTM) Joint Business Group (KUBE) Maggot cultivation business located in Renon Village, Denpasar City. This business helps break down household waste, in its role of caring for sustainable development, using analytical methods. EFAS, EFAS, SWOT Matrix, QPSM where the results of the analysis found that the business of KUBE RTM is in the Growth and Build zone, and strategies that are suitable for this zone are intensive (market penetration, product development) or integration (backward integration, forward integration , and horizontal integration) and from the business feasibility financial aspect of the net present value (NPV) analysis the results are positive, the internal rate of return (IRR) results are 30%, the profitability index (PI) results are > 1, namely 1,2 times and the business payback period (PP) is 10 months, 28 days.