- Department of Mathematics

# Department of Mathematics

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Item Unsteady MHD Flow of Nanofluid with Variable Properties over a Stretching Sheet in the Presence of Thermal Radiation and Chemical Reaction(International Journal of Mathematics and Mathematical Sciences, 2019-05-02) Mjankwi, MusaShow more he unsteady magnetohydrodynamics (MHD) flow of nanofluid with variable fluid properties over an inclined stretching sheet in the presence of thermal radiation and chemical reaction is studied taking into account the effect of variable fluid properties in thermal conductivity and diffusion coefficient. The governing partial differential equations are transformed into ordinary differential equations by using similarity transformation. The numerical solutions of the problem are obtained by using the fourth order Runge-Kutta method in line with the shooting technique. It is found that the increase in both thermal conductivity and radiative heat flux decreases the heat transfer rate but increases the skin friction and mass transfer rates. It is further observed that the increase in porosity parameter and magnetic field reduces the skin friction, heat, and mass transfer rates.Show more Item Dynamics of breast cancer under different rates of chemoradiotherapy(Computational and Mathematical Methods in Medicine, 2019-09-11) Mango, SaraShow more A type of cancer which originates from the breast tissue is referred to as breast cancer. Globally, it is the most common cause of death in women. Treatments such as radiotherapy, chemotherapy, hormone therapy, immunotherapy, and gene therapy are the main strategies in the fight against breast cancer. The present study aims at investigating the effects of the combined radiotherapy and chemotherapy as a way to treat breast cancer, and different treatment approaches are incorporated into the model. Also, the model is fitted to data on patients with breast cancer in Tanzania. We determine new treatment strategies, and finally, we show that when sufficient amount of chemotherapy and radiotherapy with a low decay rate is used, the drug will be significantly more effective in combating the disease while health cells remain above the threshold.Show more Item Common fixed points of a pair of multivalued non-self mappings in partial metric spaces(Malaya Journal of Matematik, 2018, 2018-07-01) Kumar, Santosh; Rugumisa, TerentiusShow more In this paper, we utilize the concept of the partial Hausdorff metric, first introduced by Aydi et al.[4] for partial metric space, to consider a pair of multivalued mappings which are non-self almost contractions on metrically convex partial metric spaces. We establish the existence of fixed point in such mappings.Show more Item Fixed points for non-self mappings in multiplicative metric spaces(Malaya Journal of Matematik, Vol. 6, No. 4, 800-806, 2018, 2018-07-01) Kumar, Santosh; Rugumisa, TerentiusShow more In this paper, we develop a theorem for a pair of non-self mappings. For this purpose, we define a multiplicative convex metric space and state the condition for a mapping in such space to have a fixed point. We explain the procedure of locating the fixed point. We also provide an illustrative example on the use of the theorem.Show more Item FIXED POINTS FOR HYBRID MAPPINGS SATISFYING AN IMPLICIT RELATION IN PARTIAL METRIC SPACES(Jnanabha-Vijñāna Parishad of India, 2018-12-01) Kumar, Santosh; Rugumisa, TerentiusShow more In this paper, we prove a fi xed point theorem for hybrid mappings in partial metric spaces. The theorem contains an altering distance function and involves an implicit relation satisfying the (E.A) - property. In doing so, we generalize a theorem by Popa and Patriciu.Show more Item Fixed Point Theorem for F-contraction Mappings, in Partial Metric Spaces(Pleiades Publishing, Ltd., 2019-03-01) Luambano, Sholastica; Kumar, Santosh; Kakiko, GraysonShow more The purpose of this paper is to establish a fixed point theorem for F-contraction mappings in partial metric spaces. Also, as a consequence, a fixed point theorem for a pair of F-contraction mappings having a unique common fixed point is obtained. In particular, the main results in this paper generalize and extend a fixed point theorem due to Wardowski 2012 in which F-contraction was introduced as a generalization of Banach Contraction Principle. An illustrative example is provided to validate the results.Show more Item Projecting Tanzania pension fund system(African Journal of Applied Statistics, 2017) Andongwisye, John; Larsson, Tobjorn; Singull, Martin; Mushi, AllenShow more A mandatory Tanzania pension fund with a nal salary de ned bene t is analyzed. This fund is a contributory pay-as-you-go de ned bene t pension system which is much a ected by the change in demography. Two kinds of pension bene t, a commuted (at retirement) and a monthly (old age) pension are considered. A decisive factor in the analysis is the increased life expectancy of members of the fund. The projection of the fund's future members and retirees is done using expected mortality rates of working population and expected longevity. The future contributions, bene ts, asset values and liabilities are analyzed. The projection shows that the fund will not be fully sustainable on a long term due to the increase in life expectancy of its members. The contributions will not cover the bene t payouts and the asset value will not fully cover liabilities. Evaluation of some possible reforms of the fund shows that they cannot guarantee a long-term sustainability. Higher returns on asset value will improve the funding ratio, but contributions are still insu cient to cover bene t payouts.Show more Item Implementation of a goal programming model for solid waste management: a case study of Dar es Salaam – Tanzania(International Journal for Simulation and Multidisciplinary Design Optimization (IJSMDO), 2017) Lyeme, Halid; Mushi, Allen; Yaw, GyekyeShow more In this research article, the multi-objective optimization model for solid waste management problem is solved by the goal programming method. The model has three objectives: total cost minimization, minimization of final waste disposal to the landfill, and environmental impact minimization. First, the model is solved for the higher priority goal, and then its value is never allowed to deteriorate. The model is solved for the next priority goal and so on until the problem is solved. The model was tested with real data for solid waste management system from Dar es Salaam city. The results determine the best locations for recycling plants, separating plants, composting plants, incinerating plants, landfill and waste flow allocation between them. Furthermore, the solution shows a high reduction of the amount of waste to the landfill and greenhouse gas emissions by 78% and 57.5% respectively if fully implemented compared to the current system.Show more Item Optimizing Schedules for School Bus Routing Problem: the case of Dar Es Salaam Schools(International Journal of Advanced Research in Computer Science, 2015-02) Ngonyani, Beatha; Mujuni, Egbert; Mushi, AllenShow more The School Bus Routing Problem (SBRP) deals with transportation of students to and from their schools. Given a set of fleet of buses of a school, a set of bus stops, the time matrix and the number of students at each stop, the task is to determine the schedule of buses that minimizes amount of time students spend in the buses on the way to and from school. The school bus routing problem is a special case of the Vehicle Routing Problem (VRP) and is known to be NP-hard. This NP-hardness implies that it is very unlikely that the problem can be solved in polynomial time. The common methods used to solve NP-hard problems are heuristic algorithms which gives quick and good solutions without guarantee that the solution obtained is optimal. In this paper a Tabu search based heuristic for SBRP is developed. The algorithm has been implemented using Borland C++ 4.5 programming language and tested using data from Tusiime Nursery and Primary School in Dar es salaam, Tanzania. The proposed implementation results in reduction of students’ travelling time by 19.24%.Show more Item Review of Multi-Objective Optimization Models for Solid Waste Management Systems with Environmental Considerations(Journal of Mathematics and Computer Science (JMCS), 2017) Lyeme, Halid; Mushi, Allen; Yaw, GyekyeShow more This paper analyzes more than 50 papers with a limited area in the field of solid waste management systems and supply chain management, extending over mathematical models that include economic factors, as well as environmental and/or waste flow allocation. The review finds that there are a number of limitations to the current research in sustainable solid waste management systems. The narrow scope of environmental factors as constraints in the current models means that there is a need to go further and include new environmental metrics. The effective inclusion of environmental objectives in models with improved multi-objective approaches is a gap that needs to be filled. Furthermore, there are significant gaps in sensitivity analysis of models limiting the general applicability of the models. The paper concludes with promising new avenues of research that demand effective inclusion of sustainability into solid waste management system models.Show more Item Optimal Solution Strategy for University Course Timetabling Problem(International Journal of Advanced Research in Computer Science, 2013-01) Chacha, Stephen; Mushi, AllenShow more This paper describes formulations of the University Course Timetabling Problem as used at Mkwawa University College of Education. University Course Timetabling is the Problem of scheduling resources such as lectures, courses and rooms to a number of timeslots over a planning horizon, normally a week, while satisfying a number of problem-specific constraints. In this study, we have developed three models and tested using real data from the stated University. It has been possible to get optimal solution for real problem instances through reformulations of models which involve a mixture of binary and time-indexed variables.Show more Item Sensitivity analysis of multi-objective optimization for solid waste management(New Trends in Mathematical Sciences, 2017-11-08) Lyeme, Halid; Mushi, Allen; Yaw, GyekyeShow more In this study, a sensitivity analysis of a multi-objective optimization model for solid waste management (SWM) for Dar es Salaam city in Tanzania is considered. Our objectives were to identify the most sensitive parameters and effect of other input data to the model output. Five scenarios were considered by varying their associated parameter values. The results showed that the decrease of total cost for the SWMsystem in all scenarios was observed compared to the baseline solution when the single landfill was considered. Furthermore, the analysis shows that the variable cost parameter for the processing facilities is very sensitivity in such a way that if you increase the variable cost then, there is a rapid increase of total cost for the SWM system and the vice versa is true. The relevant suggestions to the decision makers were also discussed.Show more Item Great Deluge Algorithm for the Linear Ordering Problem(International Journal of Information Technology and Computer Science, 2015) Mathias, Amos; Mushi, AllenShow more Given a weighted complete digraph, the Linear Ordering Problem (LOP) consists of finding and acyclic tournament with maximum weight. It is sometimes referred to as triangulation problem or permutation problem depending on the context of its application. This study introduces an algorithm for LOP and applied for triangulation of Tanzanian Input-Output tables. The algorithm development process uses Great Deluge heuristic method. It is implemented using C++ programming language and tested on a personal computer with 2.40GHZ speed processor. The algorithm has been able to triangulate the Tanzanian input-output tables of size 79×79 within a reasonable time (1.17 seconds). It has been able to order the corresponding economic sectors in the linear order, with upper triangle weight increased from 585,481 to 839,842 giving the degree of linearity of 94.3%.Show more Item Late Acceptance Heuristic for University Course Timetabling Problem(International Journal of Advanced Research in Computer Science, 2013-02) Marwa, Yohana; Mushi, AllenShow more This paper describes a Late Acceptance Heuristic for University Course Timetabling Problem, using a case study of a University College in Tanzania. Late Acceptance is one of relatively new heuristic procedures that try to improve searching by delaying acceptance of latest solutions. The results are compared with an implementation on Simulated Annealing heuristic, which is a well documented and successful heuristic procedure for similar problems. It is shown that Late Acceptance Procedure is a good procedure for Course timetabling problem and compares well with Simulated Annealing.Show more Item Multi-Objective Optimization Model for Irrigation Water Allocation(ASIAN JOURNAL OF MATHEMATICS AND APPLICATIONS, 2018) Fanuel, Ibrahim; Mushi, AllenShow more In Tanzania, management of land and water resources is considered an ab- solutely strategic priority for agricultural development. In this study, a Multi-Objective optimization (MOO) model was formulated to utilize the available water by identifying the best crop patterns which maximize the farm total net bene t and minimize the total vari- able costs. The data for 6 crops collected from Nkoanrua region and FAO were used for the model analysis. The Subdivision Algorithm which is a set oriented numerical method was used to analyse the model. After 54 subdivision steps, the model proposed that, the amount of land allocated to crops which are less pro table depends on the minimum requirement constraints, while, for more pro table crops, the allocation is based on costs of production, minimum and maximum requirement constraints, Bene t, and water requirements. Lastly, the sensitivity analysis shows that, the price and production costs for carrots and maize has no impact on the model solutions.Show more Item A Simulated Annealing Heuristic for Tanzania High Schools Timetabling Problem(Global Journal of Computer Applications and Technology, 2011) Mushi, Allen; Batho, PeterShow more High School Course Timetabling Problem involves scheduling of lessons and teachers to timeslots within a week while satisfying a set of constraints. Like many other variants of timetabling problems, it is known to be NP- Hard, and therefore no optimal solution procedure is known to solve the problem in a reasonable time scale. Currently, very little work has been done in automating the high schools timetabling in Tanzania. Given the growing needs for educational resources in the country, optimization techniques for these resources cannot be avoided. In this paper, we present a heuristic algorithm based on Simulated Annealing. The algorithm has been tested with success on real data obtained from three high schools. A thorough experimentation has been done on six of cooling schedules. The results compare well with the previous work on the same data set and perform much better than manual system.Show more Item Irrigation water allocation optimization using multi-objective evolutionary algorithm (MOEA) a review(International Journal for Simulation and Multidisciplinary Design Optimization (IJSMDO), 2018-01-10) Mwita Fanuel, Ibrahim; Mushi, Allen; Kajunguri, DamianShow more This paper analyzes more than 40 papers with a restricted area of application of Multi-Objective Genetic Algorithm, Non-Dominated Sorting Genetic Algorithm-II and Multi-Objective Differential Evolution (MODE) to solve the multi-objective problem in agricultural water management. The paper focused on different application aspects which include water allocation, irrigation planning, crop pattern and allocation of available land. The performance and results of these techniques are discussed. The review finds that there is a potential to use MODE to analyzed the multi-objective problem, the application is more significance due to its advantage of being simple and powerful technique than any Evolutionary Algorithm. The paper concludes with the hopeful new trend of research that demand effective use of MODE; inclusion of benefits derived from farm byproducts and production costs into the model.Show more Item Non-Linear Great Deluge algorithm for Tanzanian High Schools Timetabling(International Journal of Advanced Research in Computer Science, 2011-07) Mushi, AllenShow more High school timetabling problem involves allocation of students, lessons, and teachers into timeslots while respecting constraints, both on students, teachers and other available resources. It is one of the Combinatorial Optimization Problems which are known to be NP-Hard and therefore no optimal algorithm is known for its solution. The problems differ from one institution to another depending on the educational system and administrative structures. In this paper, a Great Deluge Algorithm is developed based on an adaptation which employs a non-linear decay rate in the reduction of ‘water level’. This is a case study in the application of the algorithm to Tanzanian high schools. The algorithm is tested on three high school systems in Tanzania. Since no such work has been previously done in Tanzania, the algorithm is compared with the manually generated timetables for the same schools. It has been shown that, the algorithm performs very well and can be used to greatly improve timetabling at Tanzanian high schools.Show more Item Two-Phase Great Deluge Algorithm for Course Timetabling Problem(2011-09) Mushi, AllenShow more Academic course timetabling involve assigning resources such as lecturers, rooms and courses to a fixed time period, normally a week, while satisfying a number of problem-specific constraints. This study describes a Great Deluge Algorithm in two phases that creates timetables by heuristically minimizing penalties over infeasibilities. The algorithm is developed with special focus on the University of Dar-as-salaam and compares the results with a previous work on Tabu Search, and a manually generated solution. We conclude that Great Deluge gives a much more stable solution because it produces good solutions with less number of parameters for tuning compared to Tabu Search.Show more Item The Linear Ordering Problem: An Algorithm for the Optimal Solution(African Journal of Science and Technology (AJST), 2005-06) Mushi, AllenShow more In this paper we describe and implement an algorithm for the exact solution of the Linear Ordering problem. Linear Ordering is the problem of finding a linear order of the nodes of a graph such that the sum of the weights which are consistent with this order is as large as possible. It is an NP - Hard combinatorial optimisation problem with a large number of applications, including triangulation of input - output matrices in Economics, aggregation of individual preferences and ordering of teams in sports. We implement an algorithm for the exact solution using cutting plane and branch and bound procedures. The program developed is then applied to the triangulation problem for the input - output tables. We have been able to triangulate input - output matrices of size up to 41 x 41.Show more