Development/Application of Dynamic Programming Model for Students’ Academic Planning and Performance in the Universities

Dynamic programming is a member of non-linear programming; it makes use of optimality and recursive relationship to arrive at an 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 forms 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 starts from the last stage and back to the first stage. It has diverse applications, especially in those areas where most of other non-linear and linear optimization cannot be applied. 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, one of its greatest shortcomings is the restriction imposed on its applications to large problems; which depicts the characteristics of real life problems. This restriction is the “curse of dimensionality”. 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 sub-problems. Dynamic programming is used when the sub-problem are not independent. Therefore, Dynamic programming solves each sub-problem just once and stores 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 sub-solutions, then the problem exhibits optimal substructure. By overlapping sub-problems, we mean when a recursive algorithm would visit the same sub-problems repeatedly, then the problem has overlapping sub-problem. If a problem has optimal sub-structure, then we can recursively define an optimal solution. If a problem has overlapping sub-problems, then we can improve on a recursive implementation by computing each sub-problem 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’s, second year’s, etc performance. If follows the optimal substructure and overlapping sub-problems. So, the academic performance of students in the universities, which is measured by CGPA to determine the various classes each students made, adequately fits the dynamic programming model.

Careful and pre-planning 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 vary 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 well-being of individuals determine to a large extent their academic performance but this is not always the case because some empirical studies have 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 leaves other variables constant. We seek to determine the best study policy that will enhance academic performance of students in the Nigerian universities.

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 sub-problems. The key to the process is the principle of optimality.

Amuji et al (2017), observed that dynamic programming method of solution to non-linear programming problems decomposes a multistage decision problem as a sequence of single-stage 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 multi-objective dynamic programming to improve their design and operational strategies. The researchers aimed at adapting Dynamic programming-based metaheuristic to solve Optimization problem and to apply it to the multi-objective unit commitment problem (MO-UCP). They were of the opinion that their model overcomes the poor performance of standard evolutionary operators on such a heavily-constrained 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 sub-problems to find an optimal subsequence of decisions.

Fernandez-Villaverde 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 real-world 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 literature; 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 student’s academic planning and performance.

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 student’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 (n-1) 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 students’ 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

(1)

Consider investment in activities 1 and 2;

if is the return from a total amount when is invested in activity 2, we have

(2)

But we want to maximize the return on and must choose is maximized, i.e.

R₂(x) = Max[ r₂(x, y₂)] (3)

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

(4)

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

(6)

since we defined as the maximum return obtainable from a sumwith 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

(7)

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 ≥2, we have

(8)

as the return function and as the corresponding decision function.

Finally, we have

(9)

as the maximum return with the sum available; therefore, equation (9) is the Dynamic programming model for students’ optimal academic planning and performance in the university.

Dynamic programming model in equation (9) can formally be written as in equation (10) below

(10)

3.3 Optimal Decision Policy

Let S_i be the state variables, i = 1, . . , n ; y_i be the decision variables and k_i be the decision at each stage, then we can state the optimal allocation policy as follows:

S_i=S_i*; y_i* = k_i ………………. for the first year

S_i-1 = (S_i- k_i)* = k_i-1; y_i-1*= k_i-1 ………….. for the second year

S_i-2= (S_i- k_i-1)* = k_{i-2 ;} y_i-2* = k_i-2 …………… for the third year

…………………………………………………………………….

S_{i
– i+1}= (S_i- k_i-i+1)* = k_{i-i+1 ;} (y_i-i+1)* = k_i-i+1 ……………… for the final year

Hence, allocation of the resources in the order (k_i , k_i-1, k_i-2, . . . , k_i-i+1) will yield an optimal value of S_i*(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
Students	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	3.03

Source: 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 students’ 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 (R_n)	Stage (Y_i)
State (R_n)	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

4.2 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 students’ academic performance.

From equation (10), let R_n be the number of the students available and Y_i 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	5

Applying 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	3

Table 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	1

Table 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	1

Table 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	0

4.3 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 one-fifth of their efforts (time or resources) in second year, third year, fifth year and two-fifth 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 one-fifth or two-fifth 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 students’ 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. Applying 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 put no much effort in their studies in the first year but subsequently put one-fifth of the required efforts (time) in second year, third year, fifth year and two-fifth 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.

5.2. 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 students’ career after school depends on it. For this reason, we tried as much as possible to optimize the students’ 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

Amuji H O, Ugwuanyim G U, Ogbonna C J, Iwu H C, Okechukwu B N (2017). The Usefulness of Dynamic Programming in Course Allocation in the Nigerian Universities, Open Journal of Optimization, 6: 176 -186

Amuji H O, Onwuegbuchunam D E, Aponjolosun M O, Okeke K O, Mbachu J C and Ojutalayo J F (2022). The dynamic programming model for optimal allocation of laden containers to Nigerian seaports. Journal of Sustainable Development of Transport and Logistics, 7(2): 69-79.

De Farias D P and Van Roy B (2003). The Linear Programming Approach to Approximate Dynamic Programming, Operations Research, 51(6): 850-865.

Doraszelski U and Judd K L (2012). Avoiding the Curse of Dimensionality in Dynamic Stochastic Games, University of Pennsylvania.

Fernandez-Villaverde J, Nuno, G, Sorg-Langhans G and Vogler M (2020). Solving High-Dimensional Dynamic Programming Problems using Deep Learning, University of Pennsylvania.

Peter S (1989). Dynamic Programming in Action, Palgrave Macmillan Journals 40(9): 779 - 787.

Sophie J, Laetitia J and El-Ghazali T (2016). A multi-objective dynamic programming-based metaheuristic to solve a biobjective unit commitment problem using a multi-objective decoder, Int. J. Metaheuristics, 5(1): 3 -30.

ARTICLE INFO	ABSTRACT
*Article No.:* 030123023 *Type: Research* *Full Text:* *PDF, HTML, PHP, EPUB* *DOI:* *10.5281/zenodo.7765105*	In this study, we developed dynamic programming model for students’ 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 Science Laboratory Technology (SLT) to obtain the proportion of the resources (time) needed for the student to achieve the maximum CGPA in the university. Applying the optimal decision policy, we found that the allocation of available resources (time) yielded the following results, (0, 1, 1, 2, 1). This means that even if the students put no much effort in their studies in the first year but subsequently put one-fifth of the required efforts (time) in the second year, third year, fifth year and two-fifth of their efforts (time) in the 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:* 03/03/2023 *Published:* 23/03/2023
Corresponding Author Harrison O. Amuji* *E-mail:* 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*