By Amuji, HO; Onwuegbuchunam, DE; Okoroji, LI; Nwachi, CC; Mbachu, JC (2023). Greener Journal of Science, Engineering and Technological Research, 12(1): 2025.
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Greener Journal of Science, Engineering and Technological Research
ISSN: 22767835
Vol. 12(1), pp. 2025, 2023
Copyright Â©2023, the copyright of this article is retained by the author(s)
DOI: https://doi.org/10.5281/zenodo.7765106
Development/Application of Dynamic Programming Model for Studentsâ€™ Academic Planning and Good Performance in the Universities
Harrison O. Amuji*1; Donatus E. Onwuegbuchunam2; Lazerus I. Okoroji3; Christy C. Nwachi 4; Justice C. Mbachu5
1Department of Statistics, Federal University of Technology, Owerri, Nigeria.
2,5Department of Maritime Management Technology, Federal University of Technology, Owerri Nigeria.
3Department of Transport Management Technology, Federal University of Technology Owerri, Nigeria
4Department of Urban and Regional Planning, Federal University of Technology, Owerri Nigeria.
Emails: 1harrison.amuji @ futo.edu.ng, 2don @ futo.edu.ng, 3okoroji_ @ yahoo.com 4christynwachi @ gmail.com, 5justice.mbachu @ futo.edu.ng
ARTICLE INFO
ABSTRACT
Article No.: 030123023Type: Research
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In this study, we developed dynamic programming model for studentsâ€™ academic planning and performance in the university. Our focus was on the decision policy that will help to enhance the CGPA of students. We applied the developed model on the CGPA data collected from the department of SLT to obtain the proportion of the resources (time) needed for the student to achieve the maximum CGPA in the university. Appling the optimal decision policy, we obtained that the allocation of available resources (time) yield the following results, (0, 1, 1, 2, 1). This means that even if the students puts no much effort in their studies in the first year but subsequently put onefifth of the required efforts (time) in second year, third year, fifth year and twofifth of their efforts (time) in fourth year, they would have made the maximum obtainable CGPA in the university. This optimal decision shows that students do not put the needed efforts (time) in their studies thereby making them to perform poorly in their overall CGPA.
Accepted: 03/03/2023Published: 23/03/2023 *Corresponding Author
Harrison O. Amuji
Email: harrison.amuji@ futo.edu.ng
Phone: +234 803 647 8180 Keywords: Dynamic programming, Students academic performance, Cumulative great point average, Optimal allocation policy, Investment of time in studies
1. INTRODUCTION
Dynamic programming is a member of nonlinear programming; it makes use of optimality and recursive relationship to arrive at optimal decision. It divides a set of problem into different stages, and each stage has an independent decision though the stages are not independent of the other but linked by recursive relationship, see Amuji et al (2022). One stage form the basis for the next stage and at the end, the optimal solution for the entire problem is reached. It uses backward approach in attaining optimal solution, that is, the solution to the problem stats from the last stage and back to the first stage. It has diverse applications, especially in those areas where most of other nonlinear and linear optimization cannot be applies. It is mostly used to provide solutions and models for those problems that cannot fit into known optimization models. Though dynamic programming has many advantages but one of its greatest shortcomings is the restriction imposed on its applications to large problem; which depicts the character by real life problems. This restriction is the â€ścurse of dimensionalityâ€ť. The curse of dimensionality is the problem caused by the exponential increase in volume associated with adding extra dimensions to Euclidean space. The curse of dimensionality occurs when the complexity increases rapidly which is caused by the increasing number of possible combinations of inputs.
There are two attributes that a problem must possess before qualifying to be modeled by dynamic programming. They are optimal substructure and overlapping subproblems. Dynamic programming is used when the subproblem are not independent. Therefore, Dynamic programming solves each subproblem just once and store the result in a table so that it can be repeatedly retrieved if need be. By optimal structure, we mean that if an optimal solution contains optimal sunsolutions, then the problem exhibits optimal substructure. By overlapping subproblems, we mean when a recursive algorithm would visit the same subproblems repeatedly, then the problem has overlapping subproblem. If a problem has optimal substructure, then we can recursively define an optimal solution. If a problem has overlapping subproblems, then we can improve on a recursive implementation by computing each subproblem only once.
The above attributes apply to the relationship that exists among the performance of students from year one to final year in the university. The final year CGPA (Cumulative Grade Point Average) is dependent on the first year, second year, etc performance. If follows the optimal substructure and overlapping subproblems. So, the academic performance of students in the universities which is measured by CGPA to determine the various classes each students made adequately fit dynamic programming model.
Careful and preplanning before embarking on educational venture is rare, and that is why poor academic performance is at alarming rate in the Nigerian universities today. Many people apply the rule of thumb in academic planning and execution. In the real sense, this rule does not always apply. Though we know that if a person devotes more time to studies, there is likelihood that the person will perform better than he/she would perform if no such efforts are applied but this is not always the case as assimilation and retention capacities varies from one individual to another. There are people that give their best output at a minimal time while others do the same at a very long time of their studies. The state of mind also follow suit and not necessarily the time put in studies. The wellbeing of individual determine to a large extent their academic performance but this is not always the case because some empirical studies has shown that those that do well in their academics are not necessarily the children of the rich, but in many cases the poor or the middle class who may not be able to afford all that life required. In this study, we intend to develop a model that will help to determine the pattern that should be followed for optimal academic performance. The paper specifically concentrates on the time invested in the studies and leaving other variables constant. We seek to determine the best study policy that will enhance academic performance of students in the Nigerian universities.
2. LITERATURE REVIEW
In this chapter, we review some related works done by other researchers in this area. Though there is no direct work done on the dynamic programming modelling of academic performance of students but there are some related works.
Doraszelski and Judd (2012), developed an algorithm for a discrete discounted cost dynamic programming problem which the author was of the opinion that it can be obtained from the complementary slackness theorem of linear programming. The author observed that the policy improvement procedure for solving such a problem coincides with the Simplex method solution to a linear program. According to the researchers, this observation does not seem to have appeared in the literature. An optimal solution to the dynamic programming problem can also be found by solving the linear program.
Peter (1989), observed that many writers on dynamic programming bemoan the lack of practical applications of the technique. But the increasingly powerful computing facilities now available mean that the solution of many hitherto intractable problems is becoming a reality. This study shows how dynamic programming could be used to help authorities arrive at an optimal budgeting strategy. The paper illustrates the stages involved in constructing a satisfactory dynamic programming model. He observed that the cost of the dynamic programming formulation is the computer memory requirement. But one of the virtues of the dynamic programming approach is the freedom it allows concerning functional forms.
De Farias and Van Roy (2003), observed that dynamic programming has been proposed as an effective method of solving combinatorial problems of a sequential nature. It is considered to be computationally advantageous to use dynamic programming since the concept can provide convergence to an optimum solution without a total enumeration. In the development of dynamic programming recursion formulae, the problem is decomposed into stages which are evaluated independently, given a set of environmental conditions (states). By combining the solutions to the smaller problems, they obtain the solution to the whole problem. Since the new state gives rise to a new decision, it is possible to link together the subproblems. The key to the process is the principle of optimality.
Amuji et al (2017), observed that dynamic programming method of solution to nonlinear programming problems decomposes a multistage decision problem as a sequence of singlestage decision problems with each stage being independent of the other but connected by recursive relationship from where the final decision on the entire problem can be made. They used dynamic programming to determine the optimal course allocation to lecturers in the Nigerian universities.
Sophie et al (2016), used multiobjective dynamic programming to improve their design and operational strategies. The researchers aimed at adapting Dynamic programmingbased metaheuristic to solve Optimization problem and to apply it to the multiobjective unit commitment problem (MOUCP). They were of the opinion that their model overcomes the poor performance of standard evolutionary operators on such a heavilyconstrained problem. In dynamic programming, instead of looking at the optimization problem as a single unified task to find the optimal sequence of decisions, the problem is separated into subproblems to find an optimal subsequence of decisions.
FernandezVillaverde et al (2020), observed that the curse of dimensionality is the problem caused by the exponential increase in volume associated with adding extra dimensions to Euclidean space. A higher number of dimensions theoretically allow more information to be stored, but practically it rarely helps due to the higher possibility of noise and redundancy in the realworld data. Gathering a huge number of data may lead to the dimensionality problem where highly noisy dimensions with fewer pieces of information and without significant benefit can be obtained due to the large data.
We build on the existing knowledge in the above literatures and acknowledging that curse of dimensionality is still a problem in the application of dynamic programming. For this reason, we restricted the data set for this paper to a manageable size as we demonstrate the application of dynamic programming to studentâ€™s academic planning and performance.
3. MATERIALS AND METHODS
In this section the methodology is divided into two sections corresponding to the steps involved in the development of dynamic programming model and the determination of the solution to the academic problem. Our interest in this paper is to develop a dynamic programming model for optimal academic performance. Our focus is on studentâ€™s academic performance as returns on the efforts invested in their studies. We can observe that the returns on the efforts investment into study depend on the efforts made in the past and the efforts made at the current time. So this relationship between the efforts invested into studies and returns on the efforts invested are recursive. That is to say that the current gains (performance) depend on the previous efforts of the students on their studies (investment) in the past. In developing the model, we applied the principle of optimality and recursive relationship between the investment in the (n1) stage and the returns from the investment in the (n+1) stages to develop the Dynamic programming model. With the model, we can determine the optimal allocation policy, that is, the best way to invest time in education to obtain the maximum returns (good performance), this is the second stage of the model development.
The data for this paper is a secondary data collected from studentsâ€™ CGPA from year one to year five in the department of Science Laboratory Technology (SLT), Federal university of Technology Owerri. And because of complexity that arises in expanding the dimension of the problem (curse of dimensionality), we restricted the size of the problem to a sample of five to demonstrate the empirical application of the developed model. A code to take care of unrestricted number of students is recommended as further studies in this area.
3.2 Stage 1: Development of Dynamic Programming Model
Definitions:
Let = investment on educational activities i; (1, n)
Let = return on investment
Let = total sum of resources available (time)
Let
Let = maximum return obtained from investing the sum on activities 1 to i inclusive
Let Provided is an increasing function
Hence:
Suppose we have a sum of to be invested in various activities, i; (1, n), we want is maximized.
Since is an increasing function, then
Consider investment in activities 1 and 2;
if is the return from a total amount when is invested in activity 2, we have
But we want to maximize the return on and must choose is maximized, i.e.
where is joint return on investment,
takes value between 0 and , and applying the Principle of Optimality, that is, we must also use the amount invested in activity 1 in the final stage, we have
For each value of , we can find the value of ,
If we now permit investment in activity 3, investing in this from a total available, we have the return as
in activity 1 and 2. (5)
using the Principle of Optimality, we have
since we defined as the maximum return obtainable from a sum with investment in activities 1 and 2.
We still have to ensure that the is chosen optimally, and we do this as before, by letting it take all possible values (i.e. in the range 0, ) and chose that particular which maximizes the return, i.e we compute
This gives us the maximum return from three activities, and we may again compute this for all in the range (0, ), at the same time we automatically generate the corresponding decision function , denoting the amount which is invested in activity 3 when is available for activities 1, 2 and 3.
We can proceed in this manner and for all i â‰Ą2, we have
as the return function and as the corresponding decision function.
Finally, we have
as the maximum return with the sum available; therefore, equation (9) is the Dynamic programming model for studentsâ€™ optimal academic planning and performance in the university.
Dynamic programming model in equation (9) can formally be written as in equation (10) below
3.3 Optimal Decision Policy
Let Si be the state variables, i = 1, . . , n ; yi be the decision variables and ki be the decision at each stage, then we can state the optimal allocation policy as follows:
Si=Si*; yi* = ki â€¦â€¦â€¦â€¦â€¦â€¦. for the first year
Si1 = (Si– ki)* = ki1; yi1* = ki1 â€¦â€¦â€¦â€¦.. for the second year
Si2 = (Si– ki1)* = ki2 ; yi2* = ki2 â€¦â€¦â€¦â€¦â€¦ for the third year
â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.
Si â€“ i+1 = (Si– kii+1)* = kii+1 ; (yii+1)* = kii+1 â€¦â€¦â€¦â€¦â€¦â€¦ for the final year
Hence, allocation of the resources in the order (ki , ki1, ki2, . . . , kii+1) will yield an optimal value of Si*(CGPA). This policy helps us to determine how resources (time) should be allocated to enhance the academic performance of the students in the Nigerian universities.
4. Data Presentation and Analysis
4.1. Data Presentation
Table 4.1 CGP of Five Students from 100 to 500 Levels
Students Stage/ C.G.P.A 1 2 3 4 5 1 2.16 2.21 2.62 2.75 2.81 2 2.00 2.24 2.67 2.80 2.68 3 3.34 3.21 3.09 3.06 3.43 4 2.02 1.59 1.28 2.41 2.55 5 2.45 3.04 3.30 2.64 3.03Source: SLT Department 2022
Table 4.1 presents the CGPA of five students from SLT department, FUTO from first year (100Level) to fifth year (500Level). We applied our developed dynamic programming model to determine the optimal decision that would optimize the studentsâ€™ performance in the university.
Putting Table 4.1 in a dynamic programming format, we have Table 4.2 as presented below
Table 4.2 CGP of Five Students from 100 to 500 Levels
State (Rn) Stage (Yi) 1 2 3 4 5 0 0 0 0 0 0 1 2.16 2.21 2.62 2.75 2.81 2 2.00 2.24 2.67 2.80 2.68 3 3.34 3.21 3.09 3.06 3.43 4 2.02 1.59 1.28 2.41 2.55 5 2.45 3.04 3.30 2.64 3.03
 Data Analysis
In this section, we analyze the problem in Table 4.2 using the methods described in section three; we use the Dynamic programming model presented in equation (10) and Optimal decision policy presented in section 3.3 to arrive at the optimal decision that will help to improve studentsâ€™ academic performance.
From equation (10), let Rn be the number of the students available and Yi be the levels of the students, where (n, i) = 1, 2, . . , 5.
Table 4.3; for n = 5
R5 f*5 Y*5 0 0 0 1 2.81 1 2 2.68 2 3 3.43 3 4 2.55 4 5 3.03 5Applying equation (10), we obtain Tables 4. 4 to 4.7
Table 4.4; for n = 4
R4/Y4 0 1 2 3 4 5 f*4 Y*4 0 0 0 0 1 2.81 2.75 2.81 0 2 2.68 5.56 2.80 2.80 2 3 3.43 5.48 5.61 3.06 5.61 2 4 2.55 6.18 5.48 5.87 2.41 6.18 1 5 3.03 5.30 5.84 5.87 5.22 2.64 5.87 3Table 4.5; for n = 3
R3/Y3 0 1 2 3 4 5 f*3 Y*3 0 0 0 0 1 2.81 2.62 2.81 0 2 2.80 5.43 2.67 5.43 1 3 5.61 5.42 5.43 3.09 5.61 0 4 6.18 8.23 5.47 5.90 1.28 8.23 1 5 5.87 8.80 8.28 5.89 4.08 3.30 8.80 1Table 4.6; for n = 2
R2/Y2 0 1 2 3 4 5 f*2 Y*2 0 0 0 0 1 2.81 2.21 2.81 0 2 5.43 5.02 2.24 5.43 0 3 5.61 7.64 5.05 3.21 7.64 1 4 8.23 8.23 7.67 6.02 1.59 8.23 0 or 1 5 8.80 10.44 7.85 8.64 4.40 3.04 10.44 1Table 4.7; for n = 1
R1/Y1 0 1 2 3 4 5 f*1 Y*1 0 0 0 0 1 2.81 2.16 2.81 0 2 5.43 4.97 2.00 5.43 0 3 7.64 7.59 4.81 3.34 7.64 0 4 8.23 9.80 7.43 6.15 2.02 9.80 1 5 10.44 10.39 9.64 8.77 4.83 2.45 10.44 04.3 Optimal Decision Policy
Thus for the optimal solution, we proceed as follows:
4.3. Interpretation of Result
Thus the investment of efforts in this order, (0, 1, 1, 2, 1), would yield the maximum CGPA. That is, at the first year, the students put no efforts in their study. But if the students in question puts onefifth of their efforts (time or resources) in second year, third year, fifth year and twofifth of their efforts (time or resources) in fourth year, they would have made the maximum CGPA which is first class. This optimal decision shows that students do not put the needed efforts in their studies thereby making them to perform poorly in their overall CGPA; if they could put as little as onefifth or twofifth of the required efforts in their studies, they will come out with first class
5. SUMMARY AND CONCLUSION
5.1. Summary
In this study, we developed dynamic programming model for studentsâ€™ academic planning and performance in the university. Our focus was on the decision policy that will help to enhance the CGPA of students. We also developed the optimal decision policy which was applied to obtain the proportion of the resources (time) needed for the student to achieve the maximum CGPA in the university. Appling the optimal decision policy, we obtained (0, 1, 1, 2, 1) allocation of available resources (time). In this case, the resources of interest are more of time invested in their respective studies. That is, even if the students puts no much effort in their studies in the first year but subsequently put onefifth of the required efforts (time) in second year, third year, fifth year and twofifth of their efforts (time) in fourth year, they would have made the maximum obtainable CGPA in the university. This optimal decision shows that students do not put the needed efforts (time) in their studies thereby making them to perform poorly in their overall CGPA.

 Conclusion
In this research, we developed and applied Dynamic programming model to plan and make optimal decision that will enhance the academic performance of students in the university. CGPA is so vital that studentsâ€™ career after school depends on it. For this reason, we tried as much as possible to optimize the studentsâ€™ academic performance using their CGPA. From our study, we deduced that if the students can allocate their resources (time) in this order (0. 1, 1, 2 1), they will achieve the maximum CGPA obtainable in the university.
Competing Interests
There is no competing interest
Authors Contributions
Each author contributed in the preparation of the manuscript, Literature review analysis and presentation of the work.
REFERENCES
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Doraszelski U and Judd K L (2012). Avoiding the Curse of Dimensionality in Dynamic Stochastic Games, University of Pennsylvania.
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Cite this Article: Amuji, HO; Onwuegbuchunam, DE; Okoroji, LI; Nwachi, CC; Mbachu, JC (2023). Development/Application of Dynamic Programming Model for Studentsâ€™ Academic Planning and Good Performance in the Universities. Greener Journal of Science, Engineering and Technological Research, 12(1): 2025. https://doi.org/10.5281/zenodo.7765106PDF VIEWER
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