Mathematics and Statistics

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Mathematical programming formulations for the examinations timetable problem: The case of the University of Dar es Salaam
(African Journal of Science and Technology, 2004) Mushi, Allen R.
Examinations Timetabling Problem (ETP) is the problem of assigning courses to be examined and candidates to time periods and examination rooms while satisfying a set of constraints. Every University has a different set of constraints and structure of examinations. Thus there is no general ETP model for all Universities around the world [1]. ETP is NP-Hard [2] and therefore no optimal algorithm is known for this problem which can solve a general problem within reasonable time. However, exact methods can be used to provide a benchmark for the heuristic methods. There is no general model for University Timetabling Problems because the problem feature differs from one University to another. In this paper we focus in the formulation of the ETP for the University of Dar as salaam. We formulate, test and compare three Integer Programming models. It is concluded that, although exact methods cannot give a solution to a real-size problem, these models give a good benchmark for testing the performance of other approaches. This paper also gives a direction for better exact models for the University of Dar es salaam’s ETP.
Optimization model for solid waste management at Ilala Municipal, Tanzania
(Journal of Informatics and Virtual Education, 2011) Lyeme, Halidi A; Mujuni, Egbert; Mushi, Allen R.
The existing solid waste management system at Ilala Municipal suffers from the lack of a real plan for collection centre locations and vehicle routes. In this study, a proposed mathematical model for municipal solid waste management process for Ilala municipality is presented. It includes the use of the concept of collection centres. Operational research methodology particularly Mixed Integer Programming is used to model the problem. The problem is solved to optimality which provides the best distribution of collection centres and their capacities. The solution shows a leastcost transportation plan with a cost saving of 38.3% per day compared to the current system.
Two phase heuristic algorithm for the university course timetabling problem: The case of University of Dar es Salaam
(Tanzania Journal of Science, 2011) Mushi, Allen R.
University course timetabling is the problem of scheduling resources such as lecturers, courses, and rooms to a number of timeslots over a planning horizon, normally a week, while satisfying a number of problem-specific constraints. Since timetabling problems differ from one institution to another, this paper investigated the case of the University of Dar Es salaam, based on the combination of Simulated Annealing (SA), and steepest descent in a two-phase approach. Solutions have been generated which greatly outperform the manually generated ones. Furthermore, the method compares well with previous work on Tabu Search but with faster execution time and higher quality on rooms allocation. It is concluded that the approach gives good results given a careful selection of parameters.
Optimal portfolio management when stocks are driven by Mean-Reverting Processes
(University of Dar es Salaam, 2012) Mbigili, Lusungu Julius
In this work, we present and solve the problem of portfolio optimization within the context of continuous-time stochastic model of financial variables. We consider an investment problem where an investor has two assets, namely, risk-free assets (e.g. bonds) and risky assets (e.g. stocks) to invest on and tries to maximize the expected utility of the wealth at some future time. The evolution of the risk-free asset is described deterministically while the dynamics of the risky asset is described by the geometric mean reversion (GMR) model. The controlled wealth stochastic deferential equation (SDE) and the portfolio problem are formulated. The portfolio optimization problem is then successfully formulated and solved with the help of the theory of stochastic control technique where the dynamic programming principle (DPP) and the HJB theory were used. We obtained very interesting results which are the solution of the non-linear second order partial deferential equation and the optimal policy which is the optimal control strategy for the investment process. So far we have considered utility functions which are members of hyperbolic absolute risk aversion (HARA) family, called power and exponential utility. In both cases, the optimal control (investment strategy) has explicit form and is wealth dependent, in the sense that, as the investor becomes more rich, the less he invests on the risky assets. Linearization of the logarithmic term in the portfolio problem was necessary to be undertaken for making the work of obtaining the explicit form of the optimal control much simple than it was expected.
Pricing barrier options when the dynamics of the prices are driven by the Mean Reverting Process
(University of Dar es Salaam, 2013) Komunte, Masoud
This dissertation considers a problem of pricing barrier options when the dynamics of the asset prices (X(t)) are driven by the mean reverting process, the market/asset price X(t) is obtain from mean reversion model and a Black-Scholes PDE model for pricing barrier options under mean reversion model is obtained upon using It ˆ o formula. Through the Homotopy Analysis Method (HAM) the price of the chosen barrier option (up􀀀and􀀀out European call) that satisfies the Black-Scholes PDE model was determined. Thus, through HAM we can determine approximated prices of barrier options when the dynamics of the prices are driven by the mean reverting process (Liao, 2004). Lastly the analysis is conducted to observe the behaviour of the option price when value of one parameter increases while the value of the other two parameters remain constant. The analysis shows that the option price tends to increase with the increase of the value of the parameter for the case of volatility and degree of mean reversion while for interest rate the option price decreases when interest rate increases. In all cases it is observed that early exercise is better than late exercise to owner of the option since the option price tends to decrease as time increases also to minimize risk owner of the option should exercise the option when the volatility of the market become large. It is recommended that in future, areas of interest for research related to this study are; first, finding the option price by using direct integration after obtaining a reflection principle which is useful in determining the joint distribution of the It ˆ o integral and secondly, finding the price which is a closed form solution by using Laplace transform.
A model for predicting food security status among households in developing countries
(International Journal of Development and Sustainability, 2013) Mbukwa, Justine N.
Food security prediction has been challenging aspects in developing countries particularly in African countries such as Tanzania. Consequently, government lack proper stimulated information that is necessary in making decision on efforts required for stabilizing food situation and status in their countries. Scientifically it has been observed in research and practical that this is caused by lack of proper mechanisms, tools and approach suitable for modeling and predicting food status among households. This paper proposes a logistic regression based model for analysis and prediction of food security status. The proposed model is empirically test using practical data collected from one district in Tanzania.
Some Aspects of Correlation of Physical Capital and Infrastructures on Household Food Security: Evidence from Rural Tanzania
(Journal of Economics and Sustainable Development, 2014) Mbukwa, Justine N.
To achieve the first Millennium Development Goals is still a challenge. The problem of poverty in the context of hunger still persists in Tanzania. Household’s members have not sustainable access to enough and quality of food. The major objective of this study is to ascertain whether exists some aspects of correlation between physical capital and infrastructures on the households` food security. This study was carried out in rural part of Tanzania, in Mvomero district covering three villages selected randomly (a total sample 0f 382 households). Data analysis was done using SPSS version 15.0. Chi-square test was adopted for plausible analysis assesses the extent to which some correlation exists between food security status of the households against independent variables (physical capital and infrastructures of the households). 2 χ = 6.963; − p value Based on the data analyzed empirically, it is remarkable by evidence that variables such as pesticides ( = ( 2 χ = 13.343; − p value = food security in the study area. 0.008 ), tractors ( 2 χ = 0.000 10.024; − p value = 0.002 ) and electricity ) were found to be statistically significant correlated with household’s In view of these findings, there is a need to pay attention supporting rural farmers’ to be able to access farm inputs because of existing some correlation with the household food security status. Finally, this study recommends further study to be carried by incorporating advanced statistical model such cluster analysis, principal components and factor analysis which deals with large data for plausible and interpretable findings.
Statistical Analytic Tool for Dimensionalities in Big Data: The role of Hierarchical Cluster Analysis
(Research India Publications, 2016) Mbukwa, Justine N.; Tabita, G Neeha; Anjaneyulu, GVSR; Rajasekharam, OV
An interest for presenting this paper rose because of massive increase information with a very high dimensional from different sources in this era of globalization. Data are produced continuously and are unstructured (1). This paper is confined to literature review search for big data issue and challenges of several scopes in data. It brings a detailed discussion on the problem on these data and analysis done using the effective multivariate statistical tool namely clustering analysis technique as a data reduction technique. It is used as a base for discussion for existing challenge of multi-dimensionalities of data. The findings indicated that, the world is noisy due to massive flow of information continuously. Findings revealed that data emanating from face book, you tube and twitter can be used to predict the epidemic of influenza and even market trend (2 and 3). With face book data is used to predict the people`s interest. However, data from different sources have been proved to be useful in decision making efficiently and effectively for public as well as private sector. Cluster analysis technique sorts data/alike things into groups, to see if there a high natural degree association among members of the same group and low between members of different groups. Finally, this technique has proved failure to handle such heap of data with varied sources. With regards to data stored, it remains to be a challenge in terms of analysis among researchers and scientists. Therefore, it calls for advanced statistical software to cater for such an existing challenges.
Application of K-Means and Partitioning Around Mediods (PAM) clustering techniques on Maize and Beans yield in Tanzania
(KY Publications, 2016) Mbukwa, Justine Nkundwe; Anjaneyulu, G.V.S.R.
This paper contributes to the application of k-means and k-mediods multivariate statistical methods for the purpose of revealing optimal clusters and assessing the consistency of individual districts within the group. Data used were extracted from united republic of Tanzania (Ministry of Agriculture, Livestock and Fisheries(MALF) (2003/12) consisted a total of (n=36 districts) with both maize and beans yield. The R-statistical computing version (3.1.1) was used. The study findings revealed that 36 districts were grouped into six clusters using k-means algorithm. Using the k-mediods, it was revealed that only 11 districts were found to very well structured since their silhouette width (Si) is above 0.5. Nevertheless, the clusters validation was done in such a way that individual district whose silhouette width (Si) close to 1 was regarded as highly consistency clustered whereas districts with (Si) greater or equal to 0.25 were said to be somehow well clustered and otherwise. The paper concludes that few districts that are very consistent given the threshold margin should to be monitored and evaluated effectively to ascertain productivity. The study recommends that the government should pay attention on allocating the scarce resources to the consistency clusters along with policy review in favour of smallholder farmers through access and timely for all important farm inputs in future.
On the use of Sparse Principal Component Analysis and Robust: Selection Features of Maize Yield in Rural Tanzania
(MANECH Pblications, 2017) Mbukwa, Justine N.; Anjaneyulu, GVSR
This paper has been motivated as a result of an existence of high dimensionality problem in maize yield. This means that an application of the Sparse Principal Component Analysis (SPCA) pattern recognition technique is unknown in selecting few consistent features and easier interpretation as opposed to classical PCA. This paper fulfills the existing knowledge gap in the context of Tanzania. A structure questionnaire was used to collect primary data from Mbozi and Mvomero Districts among small farming household in rural areas. The study was designed on the basis of hierarchical random sampling. The breakdown of facts was made by R-Statistical computing (version 3.3.2) whereas the findings were depicted using graphs and tables. The statistical estimates like percentage, mean and variance were also used. In line with SPCA, PCA and Robust PCA were also fitted for comparison purpose. Results showed 19 variables were condensed to six components explaining 63.7 per cent variations under PCA. Contrary to these findings, there were great improvements of the loadings, consistent and easier to interpret in each PC of the modified model (SPCA). However, the paper discovered that the Robust PCA condensed the p-variable to two PCs such that PC1 explained (81.0 per cent) variances. The study recommends the Sparse and Robustness as the best filtering techniques with reliable results as contrasted to the ordinary PCA.
Implementation of a goal programming model for solid waste management: A case study of Dar es Salaam – Tanzania
(EDP Sciences, 2017) Lyeme, Halidi Ally; Mushi, Allen R.; Nkansah-Gyekye, Yaw
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 f inal 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.
Projecting Tanzania pension fund system
(Statistics and Probability African Society, 2017) John, Andongwisye; Torbjörn, Larsson; Singull, Martin; Mushi, Allen R.
A mandatory Tanzania pension fund with a final salary defined benefit is analyzed. This fund is a contributory pay-as-you-go defined benefit pension system which is much affected by the change in demography. Two kinds of pension benefit, 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, benefits, 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 benefit 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 insufficient to cover benefit payouts. Un fonds de pension obligataire en Tanzanie avec un salaire final défini est défini. Ce fonds est un système de retraite à prestations déterminées contributif et payant qui est très affecté par les changements démographiques. Deux types de prestations de retraite, une retraite de rachat (à la retraite) et une pension mensuelle (vieillesse) sont considérés. Un facteur décisif dans l'analyse est l'augmentation de l'espérance de vie des participants au fonds. La projection des futurs membres et des retraités du fonds se fait à l'aide des taux de mortalité attendus de la population active et de la longévité espérée. Les contributions futures, les avantages, la valeur de l'actif et du passif sont analysés. Nos projections montrent que le fonds ne sera pas pleinement durable à long terme, en raison de l'augmentation de l'espérance de vie de membres. Les contributions ne couvriront pas les paiements de prestations et la valeur de l'actif ne couvrira pas entièrement le passif. Une évaluation de certaines réformes possibles du fonds montre qu'ils ne peuvent garantir une viabilité à long terme. Un rendement plus élevé de la valeur de l'actif améliorera le ratio de financement, mais les contributions restent insuffisantes pour couvrir les paiements des prestations.
Multi-objective optimization model formulation for solid waste management in Dar es Salaam, Tanzania
(Asian Journal of Mathematics and Applications, 2017) Lyeme, Halidi Ally; Mushi, Allen R.; Nkansah-Gyekye, Yaw
Solid waste management is a challenging problem in developing nations. The health and environmental negative implications associated with solid waste management are very serious particularly in the developing nations where a large percent of waste is dumped into open areas. These implications are essentially on climate change and global warming due to environmental problems. In this paper, a multiobjective optimization model is developed to address the conflicting objectives of cost minimization, minimization of final waste disposal to the landfill, and environmental impact minimization. The model follows a mixed-integer programming formulation and tested by data from selected wards in Dar es Salaam city. The output is the best location of recycling plants, separating plants, composting plants, incinerating plants, landfill and waste flow allocation between them. The solution shows a high reduction of the amount of waste to the landfill and greenhouse gas emissions by 76% and 55.2% respectively compared to the current system.
Probabilistic weather forecasting using Bayesian Model averaging: the case of Sagcot Regions
(Mzumbe University, 2018) Joel, Emmanuel
Over the past decade Tanzania has experienced spontaneous population increase (1.556 mil annual). But the number is estimated to further increase by 2050 to 2.982 mil annual, thus Tanzania is estimated to have population of 137 million people in 2050 (UN, 2015). The fast growing population is mainly depending on rainfed agriculture, which contribute 29 percent of the country GDP and providing employment to 65.5 percent of Tanzanians (Deloitte, 2016). The diversity in climatic and weather activities has posed a challenge in rainfed agriculture especially on when to plant seeds. Therefore, in order to promote agricultural activities, stable and reliable weather information are crucial in order for production to match with population increase. This study explores the challenge facing the Numerical Weather Prediction (NWP) namely WRF-ARW, by creating the system of equation (ensembles) from WRF-ARW resulting from the use of different initial conditions. Ensemble allow for probabilistic forecast to take the form of predictive probability function (PDF). But, raw ensemble forecast system are finite hence they only capture some of the uncertainty of the NWP. Thus, this study used Bayesian Model Averaging (BMA) methods of post processing ensemble forecast to maximize the sharpness of the parameter and calibration. The findings show BMA method successively removes most of under disersion showed by raw ensembles. Thus, calibrated and sharp results of BMA approach resolves a number of the weaknesses of the ensemble forecasts including their under dispersion and the discrepancy between forecasts and observations. Therefore, BMA can be used to attain higher consistency in the probabilistic forecasts of an operational model.
Asset liability management for Tanzania: Pension funds by stochastic programming
(Afrika Statistika, 2018) John, Andongwisye; Larsson, Torbjörn; Singull, Martin; Mushi, Allen R.
We present a long-termmodel of asset liability management for Tanzania pension funds. The pension system is pay-as-you-go where contributions are used to pay current benefits. The pension plan is a final salary defined benefit. Two kinds of pension benefits, a commuted (at retirement) and a monthly (old age) pension are considered. A decisive factor for a long-term asset liability management is that, Tanzania pension funds face an increase of their members’ life expectancy, which will cause the retirees to contributors dependence ratio to increase. We present a stochastic programming approach which allocates assets with the best return to raise the asset value closer to the level of liabilities. The model is based on work by Kouwenberg in 2001, with features from Tanzania pension system. In contrast to most asset liability management models for pension funds by stochastic programming, liabilities are modeled by using number of years of life expectancy for monthly benefit. Scenario trees are generated by using Monte Carlo simulation. Numerical results suggest that, in order to improve the long-term sustainability of the Tanzania pension fund system, it is necessary to make reforms concerning the contribution rate, investment guidelines and formulate target funding ratios to characterize the pension funds’ solvency situation.
Irrigation water allocation optimization using Multi-Objective Evolutionary Algorithm (MOEA): A review
(EDP Sciences, 2018) Fanuel, Ibrahim Mwita; Mushi, Allen R.; Kajunguri, Damian
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.
Mathematical model for Tanzania population growth
(Tanzania Journal of Science, 2019) Mwakisisile, Andongwisye; Mushi, Allen R.
In this paper, a mathematical model for Tanzania population growth is presented. The model is developed by using exponential and logistic population growth models. Real data from censuses conducted by Tanzania National Bureau of Statistics (NBS) are used. The Tanzania growth rate is obtained by using data of 1967 and 2012 censuses. The prediction of population for the period of 2013 to 2035 is done. Numerical results show that the population grows at the rate of 2.88%. In 2035 the population is expected to be 87,538,767 by exponential model and 85,102,504 by logistic model. The carrying capacity is 2,976,857,550, which implies that the population will still grow faster since it is far from its limiting value. Comparisons of the models with real data from the five censuses are done. Also NBS projections are compared with populations predicted by the two models. Both comparisons show that the exponential model is performing better than logistic model. Also, the projection up to 2050 gives the population of 135,244,161 by exponential model and 131,261,794 by logistic model.
Vector autoregressive approach after first differencing: A time series analysis of inflation and its determinants in Tanzania
(Oradea Journal of Business and Economics, 2021) Cheti, Rachel R. ; Ilembo, Bahati
The objective of the study was to examine the trend of inflation and its key determinants in Tanzania. We used secondary time series data observed annually from January 1970 to 2020 which are inflation rate, GDP, Exchange rate and money supply. The vector autoregressive (VAR) model was employed for modeling. Augmented Dickey-Fuller test (ADF) found that inflation rate, Gross Domestic Product (GDP), exchange rate and Money supply (M3) were initially non-stationary but they became stationary after first differencing so as to proceed with the analysis. Preliminary tests before obtaining vector auto regressive model were carried out before determining the relationship between the variables. Diagnostic test such as serial correlation, heteroscedasticity, stability and normality were also important to evaluate the model assumptions and investigate whether or not there are observations with a large, undue influence on the analysis. We used Granger causality test (GCT) to determine causal- effect relationship between the variables. The results show that, there is a long run relationship between the variables, also the results showed that exchange rate and money supply (M3) both have a positive impact on inflation rate while gross domestic product (GDP) revealed a negative impact on inflation rate. Finally, the forecast of inflation rate for 15 years ahead was performed. The study recommends that the government should pursue both contractionary monetary policy and fiscal policy in order to control inflation in the country
Utilization of non-financial business support services to aid development of Micro, Small and Medium Enterprises (MSMEs) in Tanzania
(AECA (Asociación Española de Contabilidad y Administración de Empresas) Universidad Politécnica de Cartagena, 2021) Lwesya, Francis; Mwakalobo, Adam Beni Swebe; Mbukwa, Justine N.
A variety of factors inhibit the development of MSMEs in African countries, which in turn affects entrepreneurship, job creation and economic transformation. Using cross-sectional data from 250 MSMEs in the Dar es Salaam region, we find in most of the examined variables the positive relationship between the use of non-financial business support services (BDS) and the development of MSMEs in the Dar es Salaam region. However, contrary to expectations, building business linkages and mentoring programs recorded negative relationships with MSMEs development. This is related to restricted capacities stemming from the size of MSMEs compared to large companies and deficiencies in the content of mentoring programs. In addition, the discrepancy between BDS demand and supply as well as the low adoption rate of BDS are associated with the inadequate adaptation of BDS to the needs of MSMEs, high service costs and a lack of qualified service providers. Thus, we argue that the provision of BDS to MSMEs should be demand-driven and that institutions should build on the pre-eminent characteristics of MSMEs when designing business support programs. On the other hand, Government efforts to nurture the development of MSMEs through policies and programs should extend to promoting business linkages between MSMEs and large enterprises.
Solving the University course timetabling problem using bat inspired algorithm
(Tanzania Journal of Science, 2021) Limota, Ushindi; Mujuni, Egbert; Mushi, Allen R.
Many mathematical optimization problems from real-life applications are NP-hard, and hence no algorithm that solves them to optimality within a reasonable time is known. For this reason, metaheuristic methods are mostly preferred when their size is big. Many meta-heuristic methods have been proposed to solve various combinatorial optimization problems. One of the newly introduced metaheuristic methods is a bat-inspired algorithm, which is based on the echolocation behaviour of microbats. Bat algorithm (BA) and its variants have been used successfully to solve several optimization problems. However, from the No-free Lunch Theorem, it is known that there is no universal metaheuristic method that can solve efficiently all optimization problems. Thus, this study work focused on investigating the usefulness of BA in solving an optimization problem called Course Teaching Problem (CTP). Since BA was originally designed to solve continuous problems, and CTP is a combinatorial optimization problem, a discrete version of BA for CPT has been proposed and tested using real-life data from the Dar es Salaam University College of Education (DUCE). The algorithm has produced promising results, as in each execution test, it generated a timetable in which all hard constraints were met and soft constraints were significantly satisfied within a few iterations.

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