Advances in Operations Research

Volume 2018, Article ID 8958393, 19 pages

https://doi.org/10.1155/2018/8958393

## An Assignment Problem and Its Application in Education Domain: A Review and Potential Path

^{1}Decision Science Department, School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, Malaysia^{2}Institute of Strategic Industrial Decision Modeling (ISIDM), School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, Malaysia

Correspondence should be addressed to Syariza Abdul-Rahman; ym.ude.muu@azirays

Received 11 November 2017; Accepted 18 March 2018; Published 17 May 2018

Academic Editor: Yi-Kuei Lin

Copyright © 2018 Syakinah Faudzi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a review pertaining to assignment problem within the education domain, besides looking into the applications of the present research trend, developments, and publications. Assignment problem arises in diverse situations, where one needs to determine an optimal way to assign subjects to subjects in the best possible way. With that, this paper classified assignment problems into two, which are timetabling problem and allocation problem. The timetabling problem is further classified into examination, course, and school timetabling problems, while the allocation problem is divided into student-project allocation, new student allocation, and space allocation problems. Furthermore, the constraints, which are of hard and soft constraints, involved in the said problems are briefly elaborated. In addition, this paper presents various approaches to address various types of assignment problem. Moreover, direction and potential paths of problem solving based on the latest trend of approaches are also highlighted. As such, this review summarizes and records a comprehensive survey regarding assignment problem within education domain, which enhances one’s understanding concerning the varied types of assignment problems, along with various approaches that serve as solution.

#### 1. Introduction

Problems related to assignment arise in a range of fields, for example, healthcare, transportation, education, and sports. In fact, this is a well-studied topic in combinatorial optimization problems under optimization or operations research branches. Besides, problem regarding assignment is an important subject that has been employed to solve many problems worldwide [1]. This problem has been commonly encountered in many educational activities all over the world. Within the education domain, this review classified the assignment problem into two: timetabling problem and allocation problem.

Assignment problem refers to the analysis on how to assign objects to objects in the best possible way (optimal way) [2, 3]. The two components of assignment problem are the assignments and the objective function. The assignment signifies underlying combinatorial structure, while the objective function reflects the desires to be optimized as much as possible. Nonetheless, the question is, “how to carry out an assignment with optimal objective, and at the same time, fulfilling all the related constraints?” In order to address the question, several diverse methods have been proposed [1, 2], such as the exact method [4], the heuristics method [5], the local search-based [6], the population search-based [7], and the hybrid algorithm [8].

The aim of looking into assignment problem is to discover an assignment among two or more sets of elements, which could minimize the total cost of all matched pairs. Relying on the specific structure of the matched sets and the cost function form, the allocation problems can be categorised into quadratic, bottleneck, linear, and multidimensional groups [9]. Hence, every assignment problem has a table or matrix. Normally, the rows are comprised of objects or people to assign, while the columns consist of the things or tasks to be assigned. Meanwhile, the numbers in the table refer to the costs related to every particular assignment. With that, this study presents a review of assignment problem within educational activities, where the problems were classified into timetabling and allocation problems. In fact, studies within this area have commonly displayed substantial progress with diverse methodologies.

The organization of this paper is given as follows: Section 2 discusses the definition and the mathematical formulation of general assignment problem. Next, the types of assignment problem within the education domain, along with their approaches, are presented in Section 3. In fact, this section is divided into subsections that elaborate in detail the two types of problem: (i) timetabling problem and (ii) allocation problem. Finally, the conclusion, future direction, and potential path of solution approach are presented in Section 4.

#### 2. Definition and Mathematical Formulation of General Assignment Problem

The general aim of assignment problem is to optimize the allocation of resources to demand points where both resources and demand point share equal number [1]. The problem, hence, can be mathematically presented as follows:

Optimize

subject towhere is the cost or effectiveness of assigning th resource to th demand, is 0 or 1 (as presented in (4)), and is the number of resources or demands. The constraints of the problem are defined as (2) and (3). Equation (2) indicates that each resource needs to be assigned to only one demand , while (3) shows that each demand needs to be assigned to only one resource .

In relation to every assignment problem, there is a matrix named cost or effectiveness matrix , where is the assigning cost of th resource to th demand. In this paper, it is called an assignment matrix, where every resource can be assigned to only one demand and signify it, as given in the following:

Moreover, in solving assignment problem, some constraints need to be fulfilled under certain conditions. These constraints are recognized as “hard” constraints as they must adhere to any condition, in which satisfying the condition(s) could generate feasible solution. On the other hand, “soft” constraints are considered as needed, but not crucial. In reality, it is very rare to fulfil all the soft constraints. Usually, the violated soft constraints assessed the solution quality as the objective function (cost function or “fitness” or “penalty”) [10]. When the soft constraints are adhered, they will not affect the solution feasibility, but they have to be fulfilled as much as possible to gain a solution with high quality. Besides, handling a great range of constraints is a pretty tough task. Each added constraint will increase the difficulty and the complexity of the problem, thus making the solution extra resource consuming. Some instances of constraints involved within assignment problem are as follows: the number of students in a room may not surpass the capacity of the room, and a lecturer may prefer to teach in a specific room or an exam should take place in a specific building.

#### 3. Types of Assignment Problems and Their Approaches

In this review, assignment problem within the education domain is classified into two problems, which are timetabling and allocation problems. As such, this section discusses these two problems, along with the approaches on solving the problems. Furthermore, varied methods have been used in prior studies to solve assignment problem. In fact, there are countless number and a diverse of complex problems that appear in real-life applications that need to be solved. Eventually, this has served as motivation to encourage the development of well-organized procedures to produce good solutions, even if not optimal. Therefore, choosing an appropriate solution is the key of success factor to achieve optimized results. The discussion on approaches in allocation problem is divided into exact method, heuristic and metaheuristic (local search- and population search-based), hybrid, and other techniques.

In fact, according to Martí et al. [11], exact methods assure to provide an optimum solution, while heuristic methods simply try to produce a good but not certainly optimum solution. Nevertheless, the duration to find an optimum solution of a complex problem of exact method is more complicated than that of the heuristic (due to incorporation of many irrelevant cases). Besides, there are several exact methods, for example, dynamic programing, integer programming, and linear programming. Moreover, the exact method guarantees to produce an optimum solution and assesses its optimality [12]. Heuristics and metaheuristics, on the other hand, are often used when the problem becomes too large for exact methods. Heuristic methods attempt to produce a good but not certainly optimal solution. Meanwhile, metaheuristic can be categorised into two, which are local search-based and population search-based techniques. The local search techniques iteratively employ single candidate solution for improvement and the examples are simulated annealing (SA), Tabu search (TS), and great deluge (GD), whereas the population-based techniques employ a population of candidate solutions throughout the iterative search process for further improvement, such as Fly Algorithm, genetic algorithm (GA), and Ant Colony Optimization. Furthermore, hybrid algorithms have also been applied in solving assignment problem, where it unites a few algorithms to solve a problem. The hybridization can be done by either selecting one or switching the algorithms. The next subsections discuss these problems.

##### 3.1. Timetabling Problem

Timetabling problem is considered as a type of assignment problem. A timetable usually provides information about the time for particular events to occur and eventually relates to the resources allocation [13]. Besides, timetabling is described as an assignment of events to a limited number of timeslots and rooms subject to prescribed constraints [12]. In reality, allocation of resources at a specified time is indeed necessary. The problem is challenging because one has to schedule a large number of entities and satisfy a number of constraints and preferences. According to Burke et al. [14], “a timetabling problem is a problem with four parameters: , a finite set of times; , a finite set of resources; , a finite set of meetings; and , a finite set of constraints. The problem is to assign time and resources to the meetings so as to satisfy the constraints as much as possible.” As such, timetabling problem is classified into three subproblems, which are examination timetabling problem (ETP), course timetabling problem (CTP), and school timetabling problem (STP). They are further discussed in the following subsections.

###### 3.1.1. Examination Timetabling Problem

The examination timetabling problem (ETP) is defined as an assignment of a set of examinations to a set of timeslots while simultaneously satisfying several problem constraints. According to Carter and Laporte [15], ETP is defined as a process of assigning examinations to a limited number of timeslots with the aim of producing high quality timetable subject to constraints. In fact, the main objective of this problem is to produce timetable that optimizes certain objective functions. A set of examinations, , needs to be assigned to a limited number of timeslots, , subject to certain restricted constraints. In fact, Even et al. [16] claimed that ETP is considered as an -hard real-world problem, which is rich and diverse, besides involving some significant levels of information from the connected problems. Meanwhile, according to McCollum [17], the complexity of the problem in recent years is related to the increasing number of student enrolments, course flexibility, and various preferences. The manual solution proposed by institution is usually time-consuming and lacks feasibility, thus requiring advanced techniques so as to satisfy both institutional and personal preferences.

Hence, in considering the problem solution, the hard constraints have to be strictly obeyed under any condition to ascertain solution feasibility. On the other hand, although the soft constraints do not affect the solution feasibility, they must be satisfied as much as possible in order to produce a solution with high quality. In assessing timetable quality, both hard and soft constraints in ETP are described in Table 1.