Search results for: Kolmogorov-Chentsov continuity theorem

Authors: Zhaopeng Zhu, Xianzhi Song, Gensheng Li

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Transports of solid particles through the drill pipe, drill string-hole annulus and hydraulically generated fractures are important dynamic processes encountered in oil and gas well drilling and completion operations. Different from particle transport in infinite space, the transports of cuttings, proppants and formation sand are hindered by a finite boundary. Therefore, an accurate description of the particle transport behavior under the bounded wall conditions encountered in drilling and hydraulic fracturing operations is needed to improve drilling safety and efficiency. In this study, the particle settling experiments were carried out to investigate the particle settling behavior in the pipe, annulus and between the parallel plates filled with power-law fluids. Experimental conditions simulated the particle Reynolds number ranges of 0.01-123.87, the dimensionless diameter ranges of 0.20-0.80 and the fluid flow behavior index ranges of 0.48-0.69. Firstly, the wall effect of the annulus is revealed by analyzing the settling process of the particles in the annular geometry with variable inner pipe diameter. Then, the geometric continuity among the pipe, annulus and parallel plates was determined by introducing the ratio of inner diameter to an outer diameter of the annulus. Further, a unified dimensionless diameter was defined to confirm the relationship between the three different geometry in terms of the wall effect. In addition, a dimensionless term independent from the settling velocity was introduced to establish a unified explicit settling velocity model applicable to pipes, annulus and fractures with a mean relative error of 8.71%. An example case study was provided to demonstrate the application of the unified model for predicting particle settling velocity. This paper is the first study of annulus wall effects based on the geometric continuity concept and the unified model presented here will provide theoretical guidance for improved hydraulic design of cuttings transport, proppant placement and sand management operations.

Keywords: wall effect, particle settling velocity, cuttings transport, proppant transport in fracture

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Authors: Ying Yi Wu

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In this paper, we present a proof of the fact that a finite morphism is proper. We show that a finite morphism is universally closed and of finite type, which are the conditions for properness. Our proof is based on the theory of schemes and involves the use of the projection formula and the base change theorem. We first show that a finite morphism is of finite type and then proceed to show that it is universally closed. We use the fact that a finite morphism is also an affine morphism, which allows us to use the theory of coherent sheaves and their modules. We then show that the map induced by a finite morphism is flat and that the module it induces is of finite type. We use these facts to show that a finite morphism is universally closed. Our proof is constructive, and we provide details for each step of the argument.

Keywords: finite, morphism, schemes, projection.

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Authors: K. Vigneshwaran, Manoj Pandey

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In this paper, breathing crack is considered for the non linear dynamic analysis. The stiffness of the cracked beam is found out by using influence coefficients. The influence coefficients are calculated by using Castigliano’s theorem and strain energy release rate (SERR). The equation of motion of the beam was derived by using Hamilton’s principle. The stiffness and natural frequencies for the cracked beam has been calculated using XFEM and Eigen approach. It is seen that due to presence of cracks, the stiffness and natural frequency changes. The mode shapes and the FRF for the uncracked and breathing cracked cantilever beam also obtained and compared.

Keywords: breathing crack, XFEM, mode shape, FRF, non linear analysis

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Authors: Jyh-Yang Wu, Sheng-Gwo Chen

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In this paper, we present a simple effective numerical geometric method to estimate the divergence of a vector field over a curved surface. The conservation law is an important principle in physics and mathematics. However, many well-known numerical methods for solving diffusion equations do not obey conservation laws. Our presented method in this paper combines the divergence theorem with a generalized finite difference method and obeys the conservation law on discrete closed surfaces. We use the similar method to solve the Cahn-Hilliard equations on evolving spherical surfaces and observe stability results in our numerical simulations.

Keywords: conservation laws, diffusion equations, Cahn-Hilliard equations, evolving surfaces

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Authors: P. K. Rahul Krishna, S. Sandeep Kumar, Jayanthi Sunder Raj

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The article explores one of the important relations between numbers-the Pythagorean triples (triplets) which finds its application in distance measurement, construction of roads, towers, buildings and wherever Pythagoras theorem finds its application. The Pythagorean triples are numbers, that satisfy the condition “In a given set of three natural numbers, the sum of squares of two natural numbers is equal to the square of the other natural number”. There are numerous methods and equations to obtain the triplets, which have their own merits and demerits. Here, quadratic approach for generating triples uses the hypotenuse leg difference method. The advantage is that variables are few and finally only three independent variables are present.

Keywords: arithmetic progression, hypotenuse leg difference method, natural numbers, Pythagorean triplets, quadratic equation

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Authors: Catarina Seco Matos, Miguel Amaral

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The developed world is facing falling birth rates and rising life expectancies. As a result, the overall demographic structure of societies is becoming markedly older. This leads to an economic and political pressure towards the extension of individuals’ working lives. On the other hand, evidence shows that some older workers choose to stay in the labour force as employees, whereas others choose to pursue a more entrepreneurial occupational path. Thus, entrepreneurship or self-employment may be an option for socioeconomic participation of older individuals. Previous research on senior entrepreneurship is scarce and it focuses mainly on entrepreneurship determinants and individuals’ intentions. The fact that entrepreneurship is perceived as a voluntary or involuntary decision or as a positive or a negative outcome by older individuals is, to the best of our knowledge, still unexplored in the literature. In order to analyse the determinants of entrepreneurial satisfaction among older individuals, primary data were obtained from a unique questionnaire survey, which was sent to Portuguese senior entrepreneurs who have launched their company aged 50 and over (N=181). Portugal is one of the countries in the world with the with the largest ageing population and with a high proportion of older individuals who remain active after their official retirement age – which makes it an extremely relevant case study on senior entrepreneurship. Findings suggest that non pecuniary factors (rather than pecuniary) are the main driver for entrepreneurship at older ages. Specifically, results show that the will to remain active is the main motivation of older individuals to become entrepreneurs. This is line with the activity and continuity theories. Furthermore, senior entrepreneurs tend to have had an active working life (using their professional experience as a proxy) and, thus, want to keep the same lifestyle at an older age (in line with theory of continuity). Finally, results show that even though older individuals’ companies may not show the best financial performance that does not seem to affect their satisfaction with the company and with entrepreneurship in general. The present study aims at exploring, discussing and bring new research on senior entrepreneurship to the fore, rather than assuming purely deductive approach; hence, further confirmatory analyses with larger sets from different countries of data are required.

Keywords: active ageing, entrepreneurship, older entrepreneur, Portugal, satisfaction, senior entrepreneur

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Authors: Zhifu Li, Lei Li, Wanting Zhou, Yuanhang He

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This paper proposed a lightweight certificate-less authentication and key exchange protocol (Light-CL-PKC) based on elliptic curve cryptography and the Chinese Remainder Theorem for smart home scenarios. Light-CL-PKC can efficiently reduce the computational cost of both sides of authentication by forgoing time-consuming bilinear pair operations and making full use of point-addition and point-multiplication operations on elliptic curves. The authentication and key exchange processes in this system are also completed in a a single round of communication between the two parties. The analysis result demonstrates that it can significantly minimize the communication overhead of more than 32.14% compared with the referenced protocols, while the runtime for both authentication and key exchange have also been significantly reduced.

Keywords: authentication, key exchange, certificateless public key cryptography, elliptic curve cryptography

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Authors: Anandhi Santhosh

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In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, impulsive differential equations has been used to describe the system model. In this article, the problem of approximate controllability for nonlinear impulsive integrodifferential equations with state-dependent delay is investigated. We study the approximate controllability for nonlinear impulsive integrodifferential system under the assumption that the corresponding linear control system is approximately controllable. Using methods of functional analysis and semigroup theory, sufficient conditions are formulated and proved. Finally, an example is provided to illustrate the proposed theory.

Keywords: approximate controllability, impulsive differential system, fixed point theorem, state-dependent delay

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Authors: Laskri Yamina, Rehouma Abdel Hamid

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In this paper we present a new class named polar of monic orthogonal polynomials with respect to the area measure supported on G, where G is a bounded simply-connected domain in the complex planeℂ. We analyze some open questions and discuss some ideas properties related to solving asymptotic behavior of polar Bergman polynomials over domains with corners and asymptotic behavior of modified Bergman polynomials by affine transforms in variable and polar modified Bergman polynomials by affine transforms in variable. We show that uniform asymptotic of Bergman polynomials over domains with corners and by Pritsker's theorem imply uniform asymptotic for all their derivatives.

Keywords: Bergman orthogonal polynomials, polar rthogonal polynomials, asymptotic behavior, Faber polynomials

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Authors: Jiri Barta

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This paper deals with the prevention and management of emergencies. It focuses on the protection of assets of the critical infrastructure entities that are important to preventing, preparing for and management of emergencies and crisis situations. The paper defines assets and specifies their use and place in the process of crisis management and planning. Critical assets that are protected from the negative effects of emergency or crisis situation we can use in crisis management and response. This basic rule applies mainly to the substantial assets used in the protection of critical infrastructure processes.

Keywords: asset, continuity, critical infrastructure, crisis management process

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Authors: Jiri Barta, Jiří F. Urbanek

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This paper deals with a protection of the national and European infrastructure. It is issue nowadays. The paper deals with the perspectives and possibilities of "smart solutions" to critical infrastructure protection. The research project deals with computers aided technologies are used from the perspective of new, better protection of selected infrastructure objects. Protection is focused on communication and information channels. These communication and information channels are very important for the functioning of the system of protection of critical infrastructure elements.

Keywords: interoperability, communication systems, controlling proces, critical infrastructure, crisis workplaces, continuity

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Authors: Wieslaw Marszalek, Zdzislaw Trzaska

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Interesting properties of various one-period loops of singularly perturbed memristive circuits with mixed-mode oscillations (MMOs) are analyzed in this paper. The analysis is mixed, both analytical and numerical and focused on the properties of pinched hysteresis of the memristive element and other one-period loops formed by pairs of time-series solutions for various circuits' variables. The memristive element is the only nonlinear element in the two circuits. A theorem on periods of mixed-mode oscillations of the circuits is formulated and proved. Replacements of memristors by parallel G-C or series R-L circuits for a MMO response with equivalent RMS values is also discussed.

Keywords: mixed-mode oscillations, memristive circuits, pinched hysteresis, one-period loops, singularly perturbed circuits

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Authors: Jiri Barta

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This paper deals with a simulation programs and technologies using in the educational process for members of the crisis management. Risk analysis, simulation, preparation and planning are among the main activities of workers of crisis management. Made correctly simulation of emergency defines the extent of the danger. On this basis, it is possible to effectively prepare and plan measures to minimize damage. The paper is focused on simulation programs that are trained at the University of Defence. Implementation of the outputs from simulation programs in decision-making processes of crisis staffs is one of the main tasks of the research project.

Keywords: crisis management, continuity, critical infrastructure, dangerous substance, education, flood, simulation programs

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Authors: H. D. Ibrahim, H. C. Chinwenyi, T. Danjuma

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An option is defined as a financial contract that provides the holder the right but not the obligation to buy or sell a specified quantity of an underlying asset in the future at a fixed price (called a strike price) on or before the expiration date of the option. This paper examined two approaches for derivation of Partial Differential Equation (PDE) options price valuation formula for the Heston stochastic volatility model. We obtained various PDE option price valuation formulas using the riskless portfolio method and the application of Feynman-Kac theorem respectively. From the results obtained, we see that the two derived PDEs for Heston model are distinct and non-unique. This establishes the fact of incompleteness in the model for option price valuation.

Keywords: Black-Scholes partial differential equations, Ito process, option price valuation, partial differential equations

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Authors: John K. Avor, Choong-Koo Chang

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The transfer of electrical loads at power generation stations from Standby Auxiliary Transformer (SAT) to Unit Auxiliary Transformer (UAT) and vice versa is through a fast bus transfer scheme. Fast bus transfer is a time-critical application where the transfer process depends on various parameters, thus transfer schemes apply advance algorithms to ensure power supply reliability and continuity. In a nuclear power generation station, supply continuity is essential, especially for critical class 1E electrical loads. Bus transfers must, therefore, be executed accurately within 4 to 10 cycles in order to achieve safety system requirements. However, the main problem is that there are instances where transfer schemes scrambled due to inaccurate interpretation of key parameters; and consequently, have failed to transfer several critical loads from UAT to the SAT during main generator trip event. Although several techniques have been adopted to develop robust transfer schemes, a combination of Artificial Neural Network and Fuzzy Systems (Neuro-Fuzzy) has not been extensively used. In this paper, we apply the concept of Neuro-Fuzzy to determine plant operating mode and dynamic prediction of the appropriate bus transfer algorithm to be selected based on the first cycle of voltage information. The performance of Sequential Fast Transfer and Residual Bus Transfer schemes was evaluated through simulation and integration of the Neuro-Fuzzy system. The objective for adopting Neuro-Fuzzy approach in the bus transfer scheme is to utilize the signal validation capabilities of artificial neural network, specifically the back-propagation algorithm which is very accurate in learning completely new systems. This research presents a combined effect of artificial neural network and fuzzy systems to accurately interpret key bus transfer parameters such as magnitude of the residual voltage, decay time, and the associated phase angle of the residual voltage in order to determine the possibility of high speed bus transfer for a particular bus and the corresponding transfer algorithm. This demonstrates potential for general applicability to improve reliability of the auxiliary power distribution system. The performance of the scheme is implemented on APR1400 nuclear power plant auxiliary system.

Keywords: auxiliary power system, bus transfer scheme, fuzzy logic, neural networks, reliability

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Authors: Xia Yang, Zheng Yang, Haiyong Sun, Yan Fang, Jingyu Liu, Jia Song

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The smart contract in Ethereum is a unique program deployed on the Ethereum Virtual Machine (EVM) to help manage cryptocurrency. The security of this smart contract is critical to Ethereum’s operation and highly sensitive. In this paper, we present a formal model for smart contract, using the separated term-obligation (STO) strategy to formalize and verify the smart contract. We use the IBM smart sponsor contract (SSC) as an example to elaborate the detail of the formalizing process. We also propose a formal smart sponsor contract model (FSSCM) and verify SSC’s security properties with an interactive theorem prover Coq. We found the 'Unchecked-Send' vulnerability in the SSC, using our formal model and verification method. Finally, we demonstrate how we can formalize and verify other smart contracts with this approach, and our work indicates that this formal verification can effectively verify the correctness and security of smart contracts.

Keywords: smart contract, formal verification, Ethereum, Coq

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Authors: Vijay Kumar Kukreja, Ravneet Kaur

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In this paper, one dimensional advection diffusion model is analyzed using finite difference method based on Crank-Nicolson scheme. A practical problem of filter cake washing of chemical engineering is analyzed. The model is converted into dimensionless form. For the grid Ω × ω = [0, 1] × [0, T], the Crank-Nicolson spatial derivative scheme is used in space domain and forward difference scheme is used in time domain. The scheme is found to be unconditionally convergent, stable, first order accurate in time and second order accurate in space domain. For a test problem, numerical results are compared with the analytical ones for different values of parameter.

Keywords: Crank-Nicolson scheme, Lax-Richtmyer theorem, stability, consistency, Peclet number, Greschgorin circle

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Authors: Salisu ibrahim, Abdalla Rababah

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In this research, we use Weighted G2-Multi-degree reduction of Bezier curve of degree n to a Bezier curve of degree m, m < n. The degree reduction of Bezier curves is used to represent a given Bezier curve of n by a Bezier curve of degree m, m < n. Exact degree reduction is not possible, and degree reduction is approximate process in nature. We derive a weighted degree reducing method that is geometrically continuous at the end points. Different norms will be considered, several error minimizations will be given. The proposed methods produce error function that are less than the errors of existing methods.

Keywords: Bezier curves, multiple degree reduction, geometric continuity, error function

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Authors: Saadat Ali, Rashid Abdulla Ahmed Alshehhi

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Cable faults are always probable and common during or after commissioning, causing significant delays and disrupting power distribution or transmission network, which is intolerable for the utilities&service providers being their reliability and business continuity measures. Therefore, the adoption of rapid localization & rectification methodology is the main concern for them. This paper explores the present techniques available for high voltage cable localization & rectification and which is preferable with regards to easier, faster, and also less harmful to cables. It also provides insight experience of high resistance fault locating by utilization of the Very Low Frequency (VLF) method.

Keywords: faults, VLF, real cases, cables

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Authors: Michael Airton

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This case study examines the development of a new service delivery model for prisons that focuses on using NGO’s to provide more effective case management and post release support functions. The model includes the co-design of the service delivery model and innovative commercial agreements that encourage embedded service providers within the prison and continuity of services post release with outcomes based payment mechanisms. The collaboration of prison staff, probation and parole officers and NGO’s is critical to the success of the model and its ability to deliver value and positive outcomes in relation to desistance from offending.

Keywords: collaborative service delivery, desistance, non-government organisations, post release support services

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Authors: C. Li, G. Coates, N. Johnson, M. Mc Guinness

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In the UK, flooding is responsible for significant losses to the economy due to the impact on businesses, the vast majority of which are Small and Medium Enterprises (SMEs). Businesses of this nature tend to lack formal plans to aid their response to and recovery from disruptive events such as flooding. This paper reports on work on how an agent-based model (ABM) is being developed based on interview data gathered from SMEs at-risk of flooding and/or have direct experience of flooding. The ABM will enable simulations to be performed allowing investigations of different response strategies which SMEs may employ to lessen the impact of flooding, thus strengthening their resilience.

Keywords: ABM, flood response, SMEs, business continuity

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Authors: Syukri Himran, Erwin Eka Putra, Nanang Roni

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This study explains the natural convection of viscous fluid flowing on semi-infinite vertical plate. A set of the governing equations describing the continuity, momentum and energy, have been reduced to dimensionless forms by introducing the references variables. To solve the problems, the equations are formulated by explicit finite-difference in time dependent form and computations are performed by Fortran program. The results describe velocity, temperature profiles both in transient and steady state conditions. An approximate value of heat transfer coefficient and the effects of Pr on convection flow are also presented.

Keywords: natural convection, vertical plate, velocity and temperature profiles, steady and unsteady

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Authors: Alexandra Andrade Brandão Soares, Paulo Batista Gonçalves

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Using a modal expansion that satisfies the boundary and continuity conditions and expresses the modal couplings characteristic of cylindrical shells in the nonlinear regime, the equations of motion are discretized using the Galerkin method. The resulting algebraic equations are solved by the Newton-Raphson method, thus obtaining the nonlinear frequency-amplitude relation. Finally, a parametric analysis is conducted to study the influence of the geometry of the shell, the gradient of the functional material and vibration modes on the degree and type of nonlinearity of the cylindrical shell, which is the main contribution of this research work.

Keywords: cylindrical shells, dynamics, functionally graded material, nonlinear vibrations

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Authors: František Včelař, Zuzana Pátíková

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Recently a new type of very general relational structures, the so called (L-)complete propelattices, was introduced. These significantly generalize complete lattices and completely lattice L-ordered sets, because they do not assume the technically very strong property of transitivity. For these structures also the main part of the original Tarski’s fixed point theorem holds for (L-fuzzy) isotone maps, i.e., the part which concerns the existence of fixed points and the structure of their set. In this paper, fundamental properties of (L-)complete propelattices are recalled and the so called L-fuzzy relatively isotone maps are introduced. For these maps it is proved that they also have fixed points in L-complete propelattices, even if their set does not have to be of an awaited analogous structure of a complete propelattice.

Keywords: fixed point, L-complete propelattice, L-fuzzy (relatively) isotone map, residuated lattice, transitivity

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Authors: Eithan Hourie, Miki Malul, Raphael Bar-El

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To investigate the relationship between various welfare policies and the value of job security, we conducted a study with 201 people regarding their assessments of the value of job security with respect to three elements: income stability, assurance of continuity of employment, and security in the job. The experiment simulated different welfare policy scenarios, such as the amount and duration of unemployment benefits, workfare, and basic income. The participants evaluated the value of job security in various situations. We found that the value of job security is approximately 22% of the starting salary, which is distributed as follows: 13% reflects income security, 8.7% reflects job security, and about 0.3% is for being able to keep their current employment in the future. To the best of our knowledge, this article is one of the pioneers in trying to quantify the value of job security in different market scenarios and at varying levels of welfare policy. Our conclusions may help decision-makers when deciding on a welfare policy.

Keywords: job security value, employment protection legislation, status quo bias, expanding welfare policy

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Authors: Viet-Hung Truong, Seung-Eock Kim

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Rain–wind induced vibration of inclined cable is a special aerodynamic phenomenon because it is easily influenced by many factors, especially the distribution of rivulet and wind velocity. This paper proposes a new 3D model of inclined cable, based on single degree-of-freedom model. Aerodynamic forces are firstly established and verified with the existing results from a 2D model. The 3D model of inclined cable is developed. The 3D model is then applied to assess the effects of wind velocity distribution and the continuity of rivulets on the cable. Finally, an inclined cable model with small sag is investigated.

Keywords: 3D model, rain - wind induced vibration, rivulet, analytical model

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Authors: Lisandro Blas, Héctor Tamanini

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From earliest 90 to now, no many companies or organization can “really” implement a knowledge management (KM) system that works (no only viewed from a measurement model, but in this continuity). Which are the reasons of that? Some of the reason maybe could be embedded in how KM is demanded (usefulness, priority, experts, a definition of KM) vs the importance and resources that the organizations afford (budget, responsible of a specific area of KM, intangibility). Many organizations “claim” the importance of Knowledge Management but thhese demands are not reflecting these claims in their future actions. With another’s tools or managements ideas the organizations put the economics and human resources to work. Why it´s not occur in KM? This paper tray to explain some of this reasons and tray to deal with this situations through a survey done in 2011 for a IAPG (Argentinean Institute from Oil & Gas) Congress.

Keywords: knowledge management into organizations, new perspectives, failure in implementation, claim

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Authors: Sabri Arik

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In this paper, we study the global asymptotic robust stability of delayed neural networks with norm-bounded uncertainties. By employing the Lyapunov stability theory and Homeomorphic mapping theorem, we derive some new types of sufficient conditions ensuring the existence, uniqueness and global asymptotic stability of the equilibrium point for the class of neural networks with discrete time delays under parameter uncertainties and with respect to continuous and slopebounded activation functions. An important aspect of our results is their low computational complexity as the reported results can be verified by checking some properties symmetric matrices associated with the uncertainty sets of network parameters. The obtained results are shown to be generalization of some of the previously published corresponding results. Some comparative numerical examples are also constructed to compare our results with some closely related existing literature results.

Keywords: neural networks, delayed systems, lyapunov functionals, stability analysis

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Authors: Michele Aleandri, Matteo Colangeli, Davide Gabrielli

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We study a generalization to a continuous setting of the classical Markov chain tree theorem. In particular, we consider an irreducible diffusion process on a metric graph. The unique invariant measure has an atomic component on the vertices and an absolutely continuous part on the edges. We show that the corresponding density at x can be represented by a normalized superposition of the weights associated to metric arborescences oriented toward the point x. A metric arborescence is a metric tree oriented towards its root. The weight of each oriented metric arborescence is obtained by the product of the exponential of integrals of the form ∫a/b², where b is the drift and σ² is the diffusion coefficient, along the oriented edges, for a weight for each node determined by the local orientation of the arborescence around the node and for the inverse of the diffusion coefficient at x. The metric arborescences are obtained by cutting the original metric graph along some edges.

Keywords: diffusion processes, metric graphs, invariant measure, reversibility

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Authors: Oluwaseun Adeleke, Samuel O. Ikani, Fidelis Edet, Anthony Nwala, Mopelola Raji, Simeon Christian Chukwu

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Background: The Delivering Innovations in Selfcare (DISC) project aims to increase access to self-care options for women of reproductive age, starting with self-inject subcutaneous depot medroxyprogesterone acetate (DMPA-SC) contraception services. However, the project has faced challenges in ensuring the continuous availability of the commodity in health facilities. Although most states in the country rely on the federal ministry of Health for supplies, some are gradually funding the procurement of Family Planning (FP) commodities. This attempt is, however, often accompanied by procurement delays and purchases inadequate to meet demand. This dilemma was further exacerbated by the commencement of demand generation activities by the project in supported states which geometrically increased commodity utilization rates and resulted in receding stock and occasional service disruptions. Strategies: The project deployed various strategies were implemented to ensure the continuous availability of commodities. These include facilitating inter-facility transfer, monthly tracking of commodity utilization, and alerting relevant authorities when stock levels reach a minimum. And supporting state-level procurement of DMPA-SC commodities through catalytic interventions. Results: Effective monitoring of commodity inventory at the facility level and strategic engagement with federal and state-level logistics units have proven successful in mitigating stock-out of commodities. It has helped secure up to 13,000 units of DMPA-SC commodities from federal logistics units and enabled state units to prioritize supported sites. This has ensured the continuity of DMPA-SC services and an increasing trend in the practice of self-injection. Conclusion: A functional supply chain is crucial to achieving commodity security, and without it, health programs cannot succeed. Stakeholder engagement, stock management and catalytic interventions have provided both short- and long-term measures to mitigate stock-outs and ensured a consistent supply of commodities to clients.

Keywords: family planning, contraception, DMPA-SC, self-care, self-injection, commodities, stock-out

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