linear programming faculty assignment problem

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A linear programming solution to the faculty assignment problem

1976, Socio-economic Planning Sciences

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elrond.informatik.tu-freiberg.de

Manar Hosny

In this paper we describe a heuristic-based algorithm for the faculty assignment problem. The algorithm was designed and implemented as part of a graduation project in the IT department of King Saud University, Riyadh, Saudi Arabia. The purpose of the project was to automate the manual tedious process of assigning TAs (Teaching Assistants) to suitable lab sessions; a task that needs to be done every semester in the IT department. The algorithm was designed to fulfill the assignment task while taking into consideration a number of constraints pertaining to both the TAs (e.g. personal preference and available hours) and the IT department (e.g. maximum allowed workload). The problem definition, the algorithm, and the experimental results are thoroughly explained in this paper. The computational experimentation indicate the efficiency of the proposed algorithm in assigning courses that conform to the preferences of the TAs and the rules set forth by the IT department.

Journal of Computer Science IJCSIS

Resource planning in university is a very hard management science problem. Faculty members are expensive resource that a university needs to utilize them efficiently and deploy them effectively for courses that they can teach. In this paper, we focus on one of the most important problems in the universities – the academic calendar which comprised of faculty-course assignment, course scheduling and timetabling. We propose an innovative two-steps approach to solve the problem using mathematical models to optimize the resource allocation while satisfying the faculty preferences. We also showcase using a real-world example how this problem is solved easily and solution improves the productivity of the staff and enhances the satisfaction of faculty.

Branka Marasovic , ivana Tadić

Contemporary organisations are surrounded with various constrains as well as limited number of different resources essential for their performance. Their managers seek to find an optimal resource allocation to a number of tasks, for optimizing objectives due to given constraints. Consequently, optimizing human resource allocation in order to fulfil organisational goals, primary to maximise profit and at the same time to minimize total costs or loss of time, represents one of the most demanding managerial decision making. The main aim of this paper is to optimise human resource allocation within Croatian higher education system in terms of minimising employees' costs. The paper suggests usage of integer linear programing model in order to determine the required number of teaching and researching faculty staff to fulfil all their duties (related to students) and at the same time to minimise their salaries expenditures. The model will be applied on the chosen example of Croatian faculty, due to the fact that Croatian teaching and researching staff is lately constrained in advancement and promotion by insufficient investments and is confronted with significant cutting costs. Besides minimising salaries expenditures, as its primary goal, model has to offer optimal solution which will provide the best possible quality for students within higher education system. Suggested model will be applicable to any Croatian faculty. It will secure optimal level of teaching and researching positions whose objective is to evolve and progress on individual level, delivering quality to their students as well as recognition and competitive advantage to their institution. Keywords: higher educational system, human resource allocation, human resource planning and programming, integer linear programming model 1 INTRODUCTION Many Croatian faculties nowadays are facing with problems of employing a great number of professors at the highest teaching and researching positions which correspond to high salaries expenditures, while on the other hand is evident shortage of the overall teaching stuff due to a limited number of new employments. Moreover, two main reasons for employing the large number of full professors at their young age are: the existence of relatively low criteria which scholars had to meet in order to be promoted and the fact that scholars who meet the required criteria for their promotion were assigned the new status, regardless of other factors such as salary, organisational structure, etc. The main question of this research is whether a different approach to human resource allocation within chosen Faculty would yield a higher efficiency with the same or even lower financial burden. Therefore, this paper will suggest optimal human resources allocation using human resource programming on the example of chosen Croatian Faculty. Furthermore, the results obtained from presented model will be compared to current structure of teaching and researching staff upon the chosen example. 2 THEORETICAL ASPECTS

European Journal of Operational Research

Bruno Domenech

American Scientific Research Journal for Engineering, Technology, and Sciences

Demy Gabriel

Faculty course scheduling optimization is the second of the three stages of the University Course Timetable Problem optimization. The optimization process was modeled using genetic algorithms, binary integer programming, and linear programming. There are four simple problems and four difficult problems that were used in the study. Linear programming had the highest total rating but infeasible because it produced fractional timetable values. Since the output of both genetic algorithms and binary integer programming were feasible and the total rating of binary integer programming was higher, it was considered as the best model. The binary integer programming model gives the optimal solution for as long as formulation of the needed functions and constraints is possible and the solver can process them. An alternative model is the genetic algorithms that is capable of giving feasible solutions even in very complicated scheduling conditions. The linear programming model is the basis of th...

mdpi Preprints

Frank Werner

University course scheduling (UCS) is one of the most important and time-consuming issues that all educational institutions face yearly. Most of the existing techniques to model and solve UCS problems have applied approximate methods, which are different in terms of efficiency, performance, and optimization speed. Accordingly, this research aims to apply an exact optimization method to provide an optimal solution to the course scheduling problem. In other words, in this research, an integer programming model is presented to solve the USC problem. In this model, hard and soft constraints include the facilities of classrooms, courses of different levels and compression of students' curriculum, courses outside the faculty and planning for them, and the limited time allocated to the professors. The objective is to maximize the weighted sum of allocating available times to professors based on their preferences in all periods. To evaluate the presented model's feasibility, it is implemented using the GAMS software. Finally, the presented model is solved in a larger dimension using a real data set from a college in China and compared with the current program in the same college. The obtained results show that considering the mathematical model's constraints and objective function, the faculty courses' timetable is reduced from 4 days a week to 3 working days. Moreover, master courses are planned in two days, and the courses in the educational groups do not interfere with each other. Furthermore, by implementing the proposed model for the real case study, the maximum teaching hours of the professors are significantly reduced. The results demonstrate the efficiency of the proposed model and solution method in terms of optimization speed and solution accuracy.

Nasruddin Hassan

A mathematical programming model is built to optimize the allocation of students into academic programs of a department. The mathematical programming model takes into account the limits of space capacity, financial allocation, the number of instructors and affirmative action quotas as goal constraints that are required to be fulfilled. Each constraint has a priority level and a weight attached. This goal programming model is then applied to the School of Mathematical Sciences, Universiti Kebangsaan Malaysia. The results of the preemptive goal programming model are then compared to that of the weighted non-preemptive goal programming model and current allocation using the weighted mean absolute percentage error. The successful application demonstrates the ability of the mathematical programming model to comply with the student intake requirement and goal constraints of the academic programs.

A goal programming model is built to optimize the allocation of students into academic departments of a faculty. The goal programming model takes into account the limits of space capacity, financial allocation, the number of instructors and affirmative action quotas as goal constraints that are required to be fulfilled. Each constraint has a priority level and a weight attached. This goal programming model is then applied to the Faculty of Science and Technology, Universiti Kebangsaan Malaysia. The results of the preemptive goal programming model are then compared to that of the current allocation using the weighted mean absolute percentage error. The successful application demonstrates the ability of the goal programming model to comply with the student intake requirement and goal constraints of the academic departments.

Joseph McGarrity

Administrators have little guidance from the academic literature when they attempt to spend resources efficiently. This paper seeks to help fill this need. It applies McGarrity’s optimization model to show the optimal allocation of faculty lines across departments in Colleges of Business at five universities.

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An Integer Programming Formulation for University Course Timetabling

• First Online: 26 January 2020

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• Gabriella Colajanni 11  

Part of the book series: AIRO Springer Series ((AIROSS,volume 3))

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The university timetabling problem is defined as the process of assigning lessons of university courses to specific time periods throughout the five working days of the week and to specific classrooms suitable for the number of students registered and the needs of each course. A university timetabling problem is modeled, in this paper, as an optimization problem using 0-1 decision variables and other auxiliary variables. The model provides constraints for a large number of different rules and regulations that exist in academic environments, ensuring the absence of collisions between courses, teachers and classrooms. The real case of a Department and some instances from the literature are presented along with its solution as resulted from the presented ILP formulation.

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Acknowledgement

This work has been supported by the Università degli Studi di Catania, “Piano della Ricerca 2016/2018 Linea di intervento 2”.

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Gabriella Colajanni

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Colajanni, G. (2019). An Integer Programming Formulation for University Course Timetabling. In: Paolucci, M., Sciomachen, A., Uberti, P. (eds) Advances in Optimization and Decision Science for Society, Services and Enterprises. AIRO Springer Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-34960-8_20

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Types of Linear Programming Problems

Linear programming is a mathematical technique for optimizing operations under a given set of constraints. The basic goal of linear programming is to maximize or minimize the total numerical value. It is regarded as one of the most essential strategies for determining optimum resource utilization. Linear programming challenges include a variety of problems involving cost minimization and profit maximization, among others. They will be briefly discussed in this article.

The purpose of this article is to provide students with a comprehensive understanding of the different types of linear programming problems and their solutions.

What is Linear Programming?

Linear programming (LP) is a mathematical optimization technique used to maximize or minimize a linear objective function, subject to a set of linear equality and inequality constraints. It is widely used in various fields such as economics, engineering, operations research, and management science to find the best possible outcome given limited resources.

Components of Linear Programming

Components of linear programming include:

• Objective Function: This is a linear function that needs to be optimized (maximized or minimized). It represents the quantity to be maximized or minimized, such as profit, cost, time, etc.
• Decision Variables: These are the variables that represent the choices or decisions to be made. They are the unknown quantities that the optimization process seeks to determine. Decision variables must be continuous and can take any real value within a specified range.
• Constraints: These are restrictions or limitations on the decision variables that must be satisfied. Constraints can be expressed as linear equations or inequalities. They represent the limitations imposed by available resources, capacity constraints, demand requirements, etc.
• Feasible Region: The feasible region is the set of all possible combinations of decision variables that satisfy all constraints. It is defined by the intersection of the constraint boundaries.
• Optimal Solution: This is the best possible solution that maximizes or minimizes the objective function while satisfying all constraints. In graphical terms, it is the point within the feasible region that maximizes or minimizes the objective function.

Linear programming provides a systematic and efficient approach to decision-making in situations where resources are limited and objectives need to be optimized.

Different Types of Linear Programming Problems

The following are the types of linear programming problems:

• Manufacturing problems
• Diet problems
• Transportation problems
• Optimal alignment problem

Let’s discuss more about each of them.

Manufacturing Problems

In these problems, we evaluate the number of units of various items that should be produced and sold by a company when each product requires a given number of workforce, machine hours, labour hours per unit of product, warehouse space per unit of output, and so on, to maximize profit.

Manufacturing problems involve maximizing the production rate or net profits of manufactured products, which might measure the available workspace, the number of workers, machine hours, packing materials used, raw materials required, the product’s market value, and other factors. These are commonly used in the industrial sector to anticipate a company’s future capital increase over time.

Diet Problems

In these challenges, we assess how many components or nutrients a diet should contain in order to lower the cost of the desired diet while guaranteeing the minimal amount of each vitamin.

As the name suggests, diet-related problems can be resolved by eating more particular foods that are rich in essential nutrients and can support the adoption of a particular diet plan. Finding a set of meals that will satisfy a set of daily nutritional demands for the least amount of money is the aim of a diet problem.

Transportation Problems

In these problems , we create a transportation schedule to discover the most cost-effective method of carrying a product from various plants/factories to various markets.

The study of transportation routes or how items from diverse production sources are transported to various markets to minimize the total transportation cost is linked to transportation difficulties. Analyzing such challenges is crucial for large firms with several production units and a broad customer base.

Optimal Assignment Problems

This problem addresses a company’s completion of a given task/assignment by selecting a specific number of employees to complete the assignment within the required timeframe, assuming that each person works on only one job. Event planning and management in major organizations, for example, are examples of such problems.

Constraints and Objective Function of Each Linear Programming Problem

Steps for solving linear programming problems.

Step 1: Identify the decision variables : The first step is to determine which choice factors control the behaviour of the objective function. A function that needs to be optimised is an objective function. Determine the decision variables and designate them with X, Y, and Z symbols.

Step 2: Form an objective function : Using the decision variables, write out an algebraic expression that displays the quantity we aim to maximize.

Step 3: Identify the constraints : Choose and write the given linear inequalities from the data.

Step 4: Draw the graph for the given data : Construct the graph by using constraints for solving the linear programming problem.

Step 5: Draw the feasible region : Every constraint on the problem is satisfied by this portion of the graph. Anywhere in the feasible zone is a viable solution for the objective function.

Step 6: Choosing the optimal point : Choose the point for which the given function has maximum or minimum values.

Solved Problems of Linear Programming Problems

Question 1. A factory manufactures two types of gadgets, regular and premium. Each gadget requires the use of two operations, assembly and finishing, and there are at most 12 hours available for each operation. A regular gadget requires 1 hour of assembly and 2 hours of finishing, while a premium gadget needs 2 hours of assembly and 1 hour of finishing. Due to other restrictions, the company can make at most 7 gadgets a day. If a profit of $20 is realized for each regular gadget and $30 for a premium gadget, how many of each should be manufactured to maximize profit?

We define our unknowns:

Let the number of regular gadgets manufactured each day = x

and the number of premium gadgets manufactured each day = y

The objective function is

P = 20x + 30y

We now write the constraints. The fourth sentence states that the company can make at most 7 gadgets a day. This translates as

Since the regular gadget requires one hour of assembly and the premium gadget requires two hours of assembly, and there are at most 12 hours available for this operation, we get

x + 2y ≤ 12

Similarly, the regular gadget requires two hours of finishing and the premium gadget one hour. Again, there are at most 12 hours available for finishing. This gives us the following constraint.

2x + y ≤ 12

The fact that x and y can never be negative is represented by the following two constraints:

x ≥ 0, and y ≥ 0.

We have formulated the problem as follows :

Maximize P=20x + 30y Subject to : x + y ≤ 7, x + 2y ≤ 122, x + y ≤ 12, x ≥ 0, y ≥ 0

In order to solve the problem, we next graph the constraints and feasible region.

Again, we have shaded the feasible region, where all constraints are satisfied.

Since the extreme value of the objective function always takes place at the vertices of the feasible region, we identify all the critical points. They are listed as (0, 0), (0, 6), (2, 5), (5, 2), and (6, 0). To maximize profit, we will substitute these points in the objective function to see which point gives us the maximum profit each day. The results are listed below.

FAQ on Linear programming

How many methods are there in lpp.

There are different methods to solve a linear programming problem. Such as Graphical method, Simplex method, Ellipsoid method, Interior point methods.

What are the four 4 special cases in linear programming?

Four special cases and difficulties arise at times when using the graphical approach to solving LP problems: (1) infeasibility, (2) unboundedness, (3) redundancy, and (4) alternate optimal solutions.

What are the 3 components of linear programming?

The basic components of the LP are as follows: Decision Variables. Constraints. Objective Functions.

What are the applications of LPP?

LPP applications may include production scheduling, inventory policies, investment portfolio, allocation of advertising budget, construction of warehouses, etc.

What are the limitations of LPP?

Constraints (limitations) should be expressed in mathematical form. Relationships between two or more variables should be linear. The values of the variables should always be non-negative or zero. There should always be finite and infinite inputs and output numbers.

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