This research is an optimal allocation of assignment schedule in science subjects. Two methods of assignment problem were used, namely; the Hungarian method (Algorithm) and Linear interactive & discrete optimization (LINGO) technique. The results from both methods yielded the same optimal outcome. It reveals that the total minimum opportunity that will maximize the educational quality is 84 and the total maximum effectiveness that will maximize the educational quality is 416. Therefore, we recommend to the school management to adopt these methods in staff-subject allocation schedule to science, management and arts subjects for optimal benefits. Accordingly, the government should also apply the findings of this research to other schools.
There is a well-known class of linear programming problems which are commonly referred to as the “assignment family”. The assignment family is a large class of problems which can generally be defined as an application used to assign “individuals” to “tasks” (or tasks to machines, etc.) subject to restrictions with the objective of minimizing the total cost of the assignment made (Kaye, 1985) [
The assignment problem (AP) is used worldwide in solving real world problems. An assignment problem plays an important role in industry and other applications. It is one of the well-studied optimization problems in management science and has wide applications in both manufacturing and service system (Srinivas and Genesan, 2015) [
Therefore, assignment problem is nothing rather than a balanced transportation problem in which all the supplies and demands are equal to one and involves efficient assignment of: jobs to machine, people to projects, contract to bidders, salespeople to territories, vehicles to routes, product to factories, development engineering to several construction site, etc. The objective is to minimize total cost or total time of performing the tasks at hand. One important characteristic of assignment problem is that only one job or worker is assigned to one machine or project. Each assignment problem has associated with a table or matrix, generally the rows contains the objects or people we wish to assign and the column comprise the things or tasks we want to assign to. The numbers in the table or entries in the matrix are costs associated with a particular assignment.
The specific problem in view is to model the staff-subjects allocation of Dutse Model International School of Jigawa State as an assignment problem and to find the optimal assignment (allocation) that will maximize the educational quality. The school existed for so many years as Dutse Capital School which later was renamed as Duste Model International School due to the advancement and the societal needs of the people in 2001. The Dutse Model International School was officially commissioned by his excellency General Ibrahim Badamasi Babangida (GCFR) the former president of the Federal Republic of Nigeria on Sunday 26th August, 2012 AD (9 Shawwal, 1433 AH). The school has a total enrollment of 1365 students comprising boys and girls. The school participated and received a numerous number of Mathematical medals and awards among which includes:
Ø National Cowbell Mathematics Competition Senior Category 2nd Position (2015).
Ø National Quiz Mathematics Competition on Nigerian Capital Market 2nd Position (2014).
The research is of paramount importance as it seeks to find the optimal assignment schedule of staff-subjects allocation that will helps in promoting Teaching-Learning Processes, thereby improving the educational quality and national development in general. The research covers Dutse Model International Secondary School of Jigawa State. Therefore, the broad aim of this research is to determine the optimal assignment (allocation) schedule of five subjects to five different staffs (Teachers), on one-to-one basis. By applying the Hungarian method and determining the best way of allocating subject-staff, it will, to a large extent enhance the number of mathematical medals and awards.
The paper is structured into six parts: part one cover the introduction; the second part dealt with literature review; the third part dwells on material and methods; the fourth part captures the analysis and results; fifth part concludes the paper.
Linear Programming (LP) can be defined as a mathematical method of determining an optimum assignment of interdependent activities, given the availability of resources (Loomba, 1964) [
The assignment problem as a class of linear programming problem was developed and published by Harold Kuhn in 1955, who gave the name “Hungarian Method” because the algorithm was largely based on the earlier work of two Hungarian mathematicians: Denes Koning and Jeno Egervary. However, Simon (2012) [
Moreover, Thiruppathi and Iranian (2015) [
Besides, Kotwal and Dhope (2015) [
Similarly, Maxon and Bhadury (2001) [
However, Sarin et al., (2010) [
The limitations of previous works: Thiruppathi and Iranian (2015) [
The study covers data gathered from the period of two academic sessions (2014/15-2015/16). One of the most widely used methods for solving assignments is called the “Hungarian Method” developed and published by H. W. Kuhn in 1955 (see Kuhn, 1955) [
The Hungarian method (also known as Flood’s Technique or the reduced matrix method) of assignment problem, it provides us with an efficient means of finding the optimal solutions without having to make direct comparison of every option. It operate on a principle of matrix reduction which means that by subtracting and adding appropriate numbers in the cost table or matrix, we can reduce the problem to a matrix of opportunity cost (opportunity cost show the relative penalties associated with assigning any workers to a job or least cost assignment). If we can reduce the matrix to a point where there is one zero element in each row and column, then it is possible to make optimal assignment, for example, assignment schedule are made in such a way that the opportunity cost for assignment is zero. The Hungarian method of assignment problem is typically minimization case (Singh et al., 2012) [
The assignment problem is a special type of transportation problem which is also a resources allocation problem. Here, we have n-jobs to be performed by n-machines and the problem is how to distribute the job to the different machines involved. This problem has extensive applications in allocation problem where different entities are allocated or assigned to other entities. The objective function in assigning different jobs to different machines is to find the optimal assignment (allocation) that will minimize the total cost or time taken to finish all the jobs by the machines.
The assignment problem can be formally stated as follows: Given n-tasks and n-machines, the problem is to assign or allocate the tasks and machine such that each task is assigned to exactly one machine and each machine to only one task. There is a cost of allocating ith task to jth machine which is denoted by [cij] and the objective is to minimize the total cost or time of this allocation.
Consider the problem of assigning n-jobs to n-machines, job i when assigned to machine j incur a cost of Cij for i = 1 , 2 , 3 , ⋯ , n and j = 1 , 2 , 3 , ⋯ , n and the objective function is to assign (allocate) jobs to machines on one-to-one basis (i.e. one machine one job), so that the total cost is minimum.
The binary decision variable in assignment problem model holds only for the binary digits.
X i j = { 1 , if job i is assignedtomachine j 0 , otherwise
The function to be minimize subject to the given constraints is called the objective function and denoted by letter Z. Consider the allocation of n-jobs to n-machines which is associated with a cost Cij and let Xij be a binary decision variable for this assignment. Since we assumed that the total cost function is linear. Therefore, the total cost function of this assignment is given by Cij × Xij. The objective function to be minimized is given by:
Minimized Z = ∑ i = 1 n ∑ j = 1 n C i j X i j .
The constraints are conditions that force the machine assigned to meet the demand of the job. However, an assignment problem is a special case of transportation problem model in which the workers (machine) represent the source origin and the tasks (jobs) represent the destination. In assignment problem the machines and the jobs constraint are unity (Ragsdale, 2010) [
1) Machine constraint:
∑ j = 1 n X i j = 1 , j = 1 , 2 , 3 , ⋯ , n
2) Job constraints:
∑ i = 1 n X i j = 1 , i = 1 , 2 , 3 , ⋯ , n
3) Non-negativity:
X i j ≥ 0 for all i = 1 , 2 , 3 , ⋯ , n and j = 1 , 2 , 3 , ⋯ , n
The condition that X i j ≥ 0 is satisfied since X i j = 0 or 1 for i, j.
Let Xij denote the assignment of ith job to jth machine and Cij denote the time or cost of effective assignment ith job to jth machine for i = 1 , 2 , 3 , ⋯ , n and j = 1 , 2 , 3 , ⋯ , n .
Then, the assignment problem can be formulated below:
Minimize Z = ∑ i = 1 n ∑ j = 1 n C i j X i j (3.1)
Subject to
∑ i = 1 n X i j = 1 , i = 1 , 2 , 3 , ⋯ , n (3.2)
∑ j = 1 n X i j = 1 , j = 1 , 2 , 3 , ⋯ , n (3.3)
X i j ≥ 0 for all i = 1 , 2 , 3 , ⋯ , n and j = 1 , 2 , 3 , ⋯ , n
where;
X i j = { 1 , if job i is assignedtomachine j 0 , otherwise
C i j = The cost or time of assigning job i to machine j (monetary unit, time unit)
n = Equal number of machines and jobs available
Each assignment problem is associated with a table or a matrix called the cost or effectiveness matrix. Generally, the rows contain the workers or machines we wish to assign and the column comprise jobs or tasks we want to assign to them. Consider the problem of assigning n-jobs to n-machines and the overall objective function is to minimize the cost or time in such a way that each machine is associated with only one and only one job. The cost matrix [Cij] is given in the
The cost matrix of the assignment problem is the same as that of a transportation problem except that the availability at each resources and the requirement at each destination is 1.
The assignment problem is to be balanced if the cost matrix has equal number of rows and columns or equivalently, if the given problem is a square matrix (i.e. n × n matrix). Otherwise is said to be unbalanced, a dummy variable must be added to balance the problem.
J 1 | J 2 | J 3 | ⋯ | J n | Availability | |
---|---|---|---|---|---|---|
M 1 | C 11 | C 12 | C 13 | ⋯ | C 1 n | 1 |
M 2 | C 21 | C 22 | C 23 | ⋯ | C 2 n | 1 |
M 3 | C 31 | C 32 | C 33 | ⋯ | C 3 n | 1 |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |
M n | C n 1 | C n 2 | C n 3 | ⋯ | C n n | 1 |
Requirement | 1 | 1 | 1 | ⋯ | 1 |
There are numerous methods for solving linear assignment problems from the general simplex algorithm to complex techniques. The most famous specific method for solving assignment problem is called the “Hungarian Method”. The method provides us with an efficient and effective way to get an optimal assignment that will minimize the cost or time.
According to Simon (2012) [
1) Balanced Assignment: Assignment problem is said to be balanced if there is equal number of rows and column or equivalently, if the given problem is a square matrix (i.e. n × n matrix).
2) Unbalanced Assignment: The assignment problem is said to be unbalanced if it is not balanced.
The algorithms for Hungarian method of assignment problem are summarized below:
Step 1: Find the cost matrix if it not given.
Step 2: Find the minimum cost of allocation in each row and column.
Step 3: Perform row reduction process that will create an equivalent cost matrix that has zero element(s) in every row by subtracting the smallest cost element(s).
Step 4: Perform a column reduction process if in the cost matrix has any column without zero elements by subtracting the smallest element in that column.
Step 5: If a single machine is assigned to more than one jobs, then go to step 6 else go to step 7.
Step 6: Carryout crossing of zeros by drawing minimum number of lines to cross all zeros. Then, select the smallest of uncrossed elements and subtract it from every other uncrossed element and add it to every element at the intersection of two lines.
Step 7: Make job allocation to machine.
The head of the staff has the problem of allocating teachers to five different subjects which includes: Mathematics, English, Physics, Chemistry and Biology offered by science class in his school. He has availability of five staffs named A, B, C, D, and E graduates teachers who have handled or taught at least one of these subjects in the past two academic sessions and have been evaluated with a score from 0 to 100 (Basic rating = 100) regarding the ability of each staff to each subject respectively. The scores are shown in the
The problem now is how the head should assign his staff to the subject on one-to-one basis so as to maximize the educational quality of his school in sciences. If he decide to apply the Hungarian method as a criterion for judging who should teach each subject. Since the assignment problem (Hungarian method) deals with minimization problems, the above maximization problem can be reduced to minimization problem by finding the opportunity or regret matrix, as shown in the
STAFFS | SUBJECTS | ||||
---|---|---|---|---|---|
MATHEMATICS | ENGLISH | PHYSICS | CHEMISTRY | BIOLOGY | |
A B C D E | 77 85 86 67 74 | 83 69 72 83 78 | 85 73 78 86 85 | 73 66 75 81 80 | 85 81 82 81 84 |
STAFFS | SUBJECTS | ||||
---|---|---|---|---|---|
MATHEMATICS | ENGLISH | PHYSICS | CHEMISTRY | BIOLOGY | |
A B C D E | 23 15 14 33 26 | 17 31 28 17 22 | 15 27 22 14 15 | 27 34 25 19 20 | 15 19 18 19 16 |
The problem can be modeled below as:
Minimize: Z = ∑ i = 0 n ∑ j = 0 n S i j X i j (4.1)
Subject to:
∑ i = 0 n X i j = 1 (4.2)
∑ j = 0 n X i j = 1 (4.3)
Xij ≥ 0 for all i = 1, 2, 3, 4, 5 and j = 1, 2, 3, 4, 5
where;
Sij = the opportunity score of assigning teacher i to subject j
X i j = { 1 , if teacher i isassignedtomeetthedemandofsubject j 0 , otherwise
n = equal number of staffs and subjects
Hence, the problem becomes (as shown in
Minimize: Z = 23 X 11 + 17 X 12 + 15 X 13 + 27 X 14 + 15 X 15 + 15 X 21 + 31 X 22 + 27 X 23 + 34 X 24 + 19 X 25 + 14 X 31 + 28 X 32 + 22 X 33 + 25 X 34 + 18 X 35 + 33 X 41 + 17 X 42 + 14 X 43 + 19 X 44 + 19 X 45 + 26 X 51 + 22 X 52 + 15 X 53 + 20 X 54 + 16 X 55
Subject to:
X 11 + X 12 + X 13 + X 14 + X 15 = 1
X 21 + X 22 + X 23 + X 24 + X 25 = 1
X 31 + X 32 + X 33 + X 34 + X 35 = 1
X 41 + X 42 + X 43 + X 44 + X 45 = 1
X 51 + X 52 + X 53 + X 54 + X 55 = 1
X 11 + X 21 + X 31 + X 41 + X 51 = 1
X 12 + X 22 + X 32 + X 42 + X 52 = 1
X 13 + X 23 + X 33 + X 43 + X 53 = 1
X 14 + X 24 + X 34 + X 44 + X 54 = 1
X 15 + X 25 + X 35 + X 45 + X 55 = 1
This result is not optimal, so we cannot make assignment. We compute the next step in the algorithm as shown in
Since, more than one subject is assigned to one teacher. Then we carry out crossing of zeros by drawing minimum number of vertical and horizontal lines to cover all zeros as shown in
STAFFS | SUBJECTS | ||||
---|---|---|---|---|---|
MATHEMATICS | ENGLISH | PHYSICS | CHEMISTRY | BIOLOGY | |
A B C D E | 8 0 0 19 11 | 2 16 14 3 7 | 0 12 8 0 0 | 12 19 11 5 5 | 0 4 4 5 1 |
STAFFS | SUBJECTS | ||||
---|---|---|---|---|---|
MATHEMATICS | ENGLISH | PHYSICS | CHEMISTRY | BIOLOGY | |
A B C D E | 8 0 0 19 11 | 0 14 12 1 5 | 0 12 8 0 0 | 7 14 6 0 0 | 0 4 4 5 1 |
STAFFS | SUBJECTS | ||||
---|---|---|---|---|---|
MATHEMATICS | ENGLISH | PHYSICS | CHEMISTRY | BIOLOGY | |
A B C D E | 8 0 0 19 11 | 0 14 12 1 5 | 0 12 8 0 0 | 7 14 6 0 0 | 0 4 4 5 1 |
The result is not optimal, now select the least of uncrossed element and subtract it from every other uncrossed element and add it to every element at intersection of two lines as shown in
This result is optimal, so we can make the allocations.
(A, English) Þ Teacher A → English
(B, Mathematics) Þ Teacher B → Mathematics
(C, Biology) Þ Teacher C → Biology
(D, Physics) Þ Teacher D → Physics
(E, Chemistry) Þ Teacher E → Chemistry
Moreover, anywhere we see zero elements, it is possible to make allocation, i.e. we can allocate B Mathematics and find another possible space for C. We allocate only A to take English; B for Mathematics; C, we allocate Biology; staff D can be allocated either Physics or Chemistry; and staff E can be allocated either Chemistry or Physics.
The total minimum opportunity that will maximize the total educational quality is given below:
Total minimum Z = 17 + 15 + 18 + 14 + 20 = 84
Also, Z = 17 + 15 + 18 + 15 + 19 = 84
Using Equation (4.1) above we have: Z = 84. Therefore, the total maximum effectiveness that will maximize the educational quality is given in
Therefore the total maximum effectiveness = 416.
The result from both Hungarian algorithm and that of LINGO computer software package yielded the same optimal outcome. The computer solution by LINGO software shows that the total minimum opportunity that will maximize the educational quality is 84 and the total maximum effectiveness that will
STAFFS | SUBJECTS | |||||
---|---|---|---|---|---|---|
MATHEMATICS | ENGLISH | PHYSICS | CHEMISTRY | BIOLOGY | ||
A B C D E | 12 0 0 23 15 | 0 10 8 1 5 | 0 8 4 0 0 | 7 10 2 0 0 | 0 0 0 5 1 | |
Optimal Assignment | Effectiveness |
---|---|
Teacher A → English | 83 |
Teacher B → Mathematics | 85 |
Teacher C → Biology | 82 |
Teacher D → Physics | 86 |
Teacher E → Chemistry | 80 |
Total | 416 |
maximize the education quality is 416.
The staff-subject allocation problem of Dutse Model International School has been addressed. The Hungarian algorithm was used to solve the staff-subject allocation problem. The research focused on the use of the assignment problem for selection of staffs to a given subject in order to obtain the best quality results from a teacher. The Hungarian assignment algorithm and Linear Interactive & Discrete Optimization (LINGO) Software Package were used to maximize the total effectiveness of teaching-learning process of the school. The Hungarian algorithm and LINGO software package yielded the same optimal result. The research seeks to solve the real-life problem in the service (educational) industry of staff-subject allocation problem using the Hungarian algorithm. It was found that the solution that gave a minimum and maximum achievable result for assigning a teacher to a subject was 84 and 416 respectively.
The use of Hungarian algorithm approach gives a systematic and transparent solution as compared with LINGO computer software package which produced the same optimal allocation policy of the school staffs to various subjects to gives a better result. The management may benefit from the proposed approach for selection of staff to subject to guarantee optimal result from the staff. We therefore recommend that the assignment problem model should be adopted by the schools in staff-subject allocation so as to get better medal rewards in the future.
Kabiru, S., Saidu, B.M., Abdul, A.Z. and Ali, U.A. (2017) An Optimal Assignment Schedule of Staff- Subject Allocation. Journal of Mathematical Finance, 7, 805-820. https://doi.org/10.4236/jmf.2017.74042