algorithm assignment scheduling problem

Looking at this question made me think about this problem, and whether genetic algorithms would be usable in this case. However it would be pretty hard to mutate possibilities while maintaining the sanity check rules. Also it's not clear to me how to distinguish incompatible requirements.

A small addendum to better specify the problem. This is applied to Italian school style classrooms where all students are associated in different classes (for example: year 1 section A) and the teachers move between classes. All students of the same class have the same schedule, and have no choice over which lessons to attend.

1 This is an awfully specific problem to find independently proved NP-hard. You might have better luck looking for a less constrained problem that might be proved NP-hard. Preferences play no part in its complexity unless you can assign some metric (like teachers always get their preferred day off). – David Thornley Oct 11, 2010 at 21:40

10 Answers 10

I am one of the developer that works on the scheduler part of a student information system. During our original approach of the scheduling problem, we researched genetic algorithms to solve constraint satisfaction problems, and even though we were successful initially, we realized that there was a less complicated solution to the problem (after attending a school scheduling workshop)

Our current implementation works great, and uses brute force with smart heuristics to get a valid schedule in a short amount of time. The master schedule (assignment of the classes to the teachers) is first built, taking in consideration all the constraints that each teacher has while minimizing the possibility of conflicts for the students (based of their course requests). The students are then scheduled in the classes using the same method.

Doing this allows you to have the machine build a master schedule first, and then have a human tweak it if needed.

The scheduler current implementation is written in perl, but other options we visited early on were Prolog and CLIPS (expert system)

Brute force usually hits the wall at about n=10 (depends on the use case). The thing is, even if you double your processing power, the wall hardly moves. Smart forms of brute force (backtracking, pruning) etc may move the wall to n=20 but no further. In my experience, Tabu Search and Late Acceptance works best (much better than Genetic Algorithms). – Geoffrey De Smet Jun 20, 2013 at 10:12

I think you might be missing some constraints.

One would prefer where possible to have teachers scheduled to classes for which they are certified.

One would suspect that the classes that are requested, and the expected headcount in each would be significant.

I think the place to start would be to list all of your constraints, figure out a data structure to represent them.

Then create some sort of engine to that builds a trial solution, then evaluates it for fitness according to the constraints.

You could then throw the fun genetic algorithms part at it, and see if you can get the fitness to increase over time as the genes mix.

Start with a small set of constraints, and increase them as you see success (if you see success)

There might be a way to take the constraints and shoehorn them together with something like a linear programming algorithm.

I agree. It sounds like a fun challenge

This is a mapping problem: you need to map to every hour in a week and every teacher an activity (teach to a certain class or free hour ).

Split the problem:

In steps 2 and 3 show each iteration to the user: items left in the list, positions on the map and the next computed move and let the user intervene.

I did not try this, but this would be my initial approach.

It's basically a brute force method like Laurenty suggested, but the approach involves applying constraints in an efficient order to cause the infeasible solutions to fail fast - to minimise the computation required.

Googling "Constraint Based Reasoning" brings up a lot of resources on the technique and its application to scheduling problems.

I don't agree. I think that a hybrid between CP-OR and then maybe AI. I have seen good academic generic solutions with CP-OR mix but this research field never gets any money so it seems that we are doomed. – Jonke Oct 26, 2008 at 0:34

Answering my own question:

The FET project mentioned by gnud uses this algorithm:

Some words about the algorithm: FET uses a heuristical algorithm, placing the activities in turn, starting with the most difficult ones. If it cannot find a solution it points you to the potential impossible activities, so you can correct errors. The algorithm swaps activities recursively if that is possible in order to make space for a new activity, or, in extreme cases, backtracks and switches order of evaluation. The important code is in src/engine/generate.cpp. Please e-mail me for details or join the mailing list. The algorithm mimics the operation of a human timetabler, I think.

Following up the "Constraint Based Reasoning" lead by Stringent Software on Wikipedia lead me to these pages which have an interesting paragraph:

Solving a constraint satisfaction problem on a finite domain is an NP-complete problem in general. Research has shown a number of polynomial-time subcases, mostly obtained by restricting either the allowed domains or constraints or the way constraints can be placed over the variables. Research has also established relationship of the constraint satisfaction problem with problems in other areas such as finite model theory and databases.

This reminds me of this blog post about scheduling a conference , with a video explanation here .

How I would do it:

Have the population include two things:

This way we can't have conflicts (a teacher in 2 places, or a class having two subjects at the same time).

The fitness function would include:

Maybe take the standard deviation for all of them since they should be balanced.

Genetic Algorithms are very well suited to problems such as this. Once you come up with a decent representation of the chromosome (in this case, probably a vector representing all of the available class slots) you're most of the way there.

Don't worry about maintaining sanity checks during the mutation phase. Mutation is random. Sanity and preference checks both belong in the selection phase. A failed sanity check would drastically lower the fitness of an individual, while a failed preference would only mildly lower the fitness.

Incompatible requirements are a different problem altogether. If they're completely incompatible, you'll get a population that doesn't converge on anything useful.

Good luck. Being the son of a father with this sort of problem is what took me to the research group that I ended up in ...

When I was a kid my father scheduled match officials in a local sports league, this had a similarly long list of constraints and I tried to write something to help. When I got to University I even used it as my final year project eventually settling on a Monte Carlo implementation (using a Simulated Annealing model).

The basic idea is that if it's not NP, it's pretty close, so rather than assuming there is a solution, I would set out to find the best within a given timeframe. I would weight all the constraints with costs for breaking them: sanity checks would have huge costs, the preferences would have lower costs (but increasing for more breaks, so breaking it once would be less than half the cost of breaking it twice).

The basic idea is that I started with a 'random' solution and costed it; then made changes by swapping a small number of assignments, re-valued it and then, probalistically accepted or declined the change.

After thousands of iterations you inch closer to an acceptable solution.

Believe me, though, that this class of problems has research groups churning out PhDs so you're in very good company.

You might also find some interest in the Linear Programming arena, e.g. simplex and so on.

Yes, I think this is NP complete - or at least to find the optimal solution is NP complete.

I worked on a similar problem in college when i told a friend's father (who was a teacher) that I could solve his scheduling problems for him if he did not find a suitable program for it (this was back in 1990 or so)

I had no idea what I got myself into. Luckily for us all I had to do was find one solution that fit the constraints. But in my testing I was always worried about determining IF there was a solution at all. He never had too many constraints and the program used different heuristics and back tracking. It was a lot of fun.

I think Bill Gates also worked on a system like this in high school or college for his high school. Not sure though.

Good luck. All my notes are gone and I never got around to implementing a solution that I could market. It was a specialty project that I re-coded as I learned new languages (Basic, Scheme, C, VB, C++)

Have fun with it

i see that this problem can be solved by Prolog program by connecting it to a database and the program can make the schedule given a set of constraints read abt "Constraint satisfaction Problem prolog"

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Jan 10, 2021

Classroom scheduling using heuristics and genetic algorithm

This article is written based on the assumption that the reader has an understanding of genetic algorithms. The code is available on https://github.com/taytzehao/School_scheduling/tree/main

Problem Statement

This problem is centered around making a school schedule for my secondary school using heuristic and genetic algorithm methods.

Unlike scheduling for universities, this school schedule is packed throughout the day with a fixed break in the middle of the morning. The format would be as of the following diagram

Where lower form refers to grades 7–9 and upper form refers to grades 10-12. Recess for the upper form students is at the 6th period while the lower form is at the 5th period. Upper form students have extra classes on Wednesday and Thursday.

The schedule has to conform to the following rules:

Initializing the classes

A form (aka grade) can have multiple classrooms that have a different class name. Therefore, the classroom inherits from From. Each classroom contain would have multiple subjects that have to be taught while each subject can be taught in multiple classes. The same interaction goes for the teacher and the classroom.

The teach object would be a class that associates teacher, classroom, and subject altogether. Every time a class is assigned, the teach object would be appended to the teacher and classroom object to easily identify the information of the activity while pointing to the same object without creating duplicate data in separate classes.

A brief introduction to Genetic algorithm

Genetic algorithm is a metaheuristic method to solve optimization problems. It resembles the genetics to allow the solution to evolve, mutate, and improve over generations, much like a living organism.

It starts off with the step of initializing random sets of solutions called “population” with each solution called “individual”. The better individuals are firstly being selected to eliminate solutions that are bad. The individuals in the population then cross-breed with each other by mixing their solutions together. Then, the individuals mutate by randomly changing some of the genes. If the population solution fulfills the fitness criteria, the process ends there. However, if the fitness criteria is not fulfilled, the process would be reiterated back to the selection step.

Genetic algorithm solution

In the genetic algorithm solution, the problem is being broken down into 2 sections i.e.

a) The assignment of teachers to each subject of each class since each subject of each class can only be taught by 1 teacher as in Rule 5.

b)The assignment of classes to each class.

This is meant to reduce the problem space that the genetic algorithm needs to solve. This is because of the previous iteration of solutions that merged both the problems lead to premature convergence.

Teacher assignment

Initialization

Each teacher is provided a unique ID number to be encoded into the gene. Each gene is initialized with a random teacher. The length of the individual is equivalent to the sum of subjects of all classes. Eg, class A has 10 subjects and class B has 11 subjects, the individual would have 21 genes. Each position of the gene represents the teacher assigned to the subject. The population is being initialized to 300 individuals per population.

Fitness function

The fitness is decreased as to the following 2 criteria:

a) Subjects that require specialized skills that are assigned teachers who are specialized in it would decrease fitness score by 1.

b) Teachers that are assigned more than the maximum number of teaching hours would decrease the fitness by the number of exceeding hours.

As this is a regular encoding problem, the following operators are used for solving this problem

a) Uniform crossover

It selects 2 parents and randomly exchanges the element of both parents from the same position as of below. As compared to 1 or 2 points crossover, this method better promotes exploration.

b) Random mutation

A random teacher ID is selected and switched to another random teacher ID.

The Roulette method is used for selecting individuals from each population.

Class assignment

Encoding method

The classes are being encoded as a permutation type problem here with each ID representing a set of classes. This means for a subject that has “side_by_side=True” and 5 periods allocated to it, the subject would be assigned class sets of 2,2,1 with 3 unique IDs. So, each ID would contain 2 information, the teaching activity and the number of periods for the class set.

The position of each gene in the chromosome represents its location on the schedule. The gene of the next class is at the position after the last position of the former class. The diagram below shows an example of 2 classes that have 10 and 11 class sets

This encoding method is developed after rounds of iteration in order to prevent premature convergence of the population and reduces the problem space significantly by limiting the permutation to involve classes within the classroom.

The fitness function is also designed to maximize the fitness score. The fitness score is decreased as of the following conditions:

a) Each morning period that is assigned after recess decreases the fitness score by 1

b) Each period that intersects with the teaching schedule of the teacher decreases the fitness score by 1.

c) Decrease fitness for every subject that exceeds the limit of its classes per day by the number of excess periods.

d) Decrease the fitness by 1 for every class set that has 2 periods that do not have consecutive periods. For example, the Biology class in the diagram below is interrupted by the Recess would have the fitness decreased by 1.

All operations used in this problem are designed to only include the gene which is of the same classroom. The operators used for this are

a) Ordered Crossover. It selects a substring from parent 1 and maintains its position in the offspring. The remaining elements that are not in the substring would utilize the position of the rest of the elements from parent 2 as in the diagram below. The modified version of it is done for each class.

Heuristic mutation-Each individual would randomly select gamma amount of genes and permutate their position. The permutation with the best arrangement would be selected. The modified version of this mutation separately selects the genes from each class and permutates the arrangement for each class.

Tournament selection. Individuals are randomly selected from the population based on the tournament size. Out of this batch, the best individual is selected for the offspring. This process is repeated until the offspring size reaches the population size. The smaller the tournament size, the better the exploration of the offspring. This is because this gives a higher probability of 2 low fitness individuals to be selected, and thus improving the diversity of the population.

Migration ring. The migration ring emigrates k amount of individuals that are selected using the tournament method stated above to replace random individuals in its destination population. This can help to increase population diversity and reduce the likelihood of premature convergence. This is performed at the end of each iteration.

Heuristics solution

As the class assignment sequence diagram describes, the main goal of creating the process is to minimize the possibility of assigning classes to times that are not viable for the subject. The more clashes there are, the more computing power is required in order to swap blank periods with periods that are already assigned classes.

Every sequence would go through the following processes:

a) Loop through every class

b) Loop through every subject

c) Select the subject that fulfills the criteria as in the diagram

d) Select period coordinates that suit all requirements. If no period is viable, the time of the other classes would be swapped with the period with blank classes. This would significantly increase the computing time.

e) Randomly select the coordinate and assign the period the classes

Morning classes are being assigned first as there is only 1 type of subject for each class that requires the classes to be in the morning. Plus, the requirement of having them in the morning reduces the number of viable periods to assign them to by half.

Next, double period classes are being assigned first before single period classes. This is because single period classes have a higher probability of taking away consecutive blank periods that would be used by the double period classes. Plus, single period classes have higher flexibility in terms of the timing as it does not need 2 consecutive blank classes.

Assessing the results

From the 2 performance graphs above, it is clear the using genetic algorithm utilizes significantly more computing power to solve the problem. Plus, it does not scale well as the number of classes increases as its time required grows exponentially with the number of classrooms.

This is mainly because of the random nature of genetic algorithm that would usually cause some collision with some other constraint while correcting the problem at hand.

It is concluded that the heuristic method would be suited and more scalable to arrange the school schedule. Some other points that could be used to improve this project for the future include:

a) Scale it up to more classes

b) Use simulated annealing algorithm method

c) Include teacher performance into assigning the teachers into each class. This is to ensure all classes are assigned a balanced amount of high and low-performing teachers.

d) Assign classes based on facilities availability according to subjects.

e) Minimization of walking distance by teachers

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Solving Job Scheduling Problem Using Genetic Algorithm

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Part of the Lecture Notes in Networks and Systems book series (LNNS,volume 227)

The efficient scheduling of independent computational jobs in a computing environment is an important problem where there are some deadlines for each job to become complete. Finding optimal schedules for such an environment is (in general) an NP-complete problem, and so heuristic approaches must be used. Genetic algorithms are known to give the best solutions to such problems. The purpose of this paper is to propound a solution to a job scheduling problem using genetic algorithms. The experimental results show that the most important factor on the time complexity of the algorithm is the size of the population and the number of generations.

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Gheisari, S., Rezaee, A., Hajati, F. (2021). Solving Job Scheduling Problem Using Genetic Algorithm. In: Barolli, L., Woungang, I., Enokido, T. (eds) Advanced Information Networking and Applications. AINA 2021. Lecture Notes in Networks and Systems, vol 227. Springer, Cham. https://doi.org/10.1007/978-3-030-75078-7_53

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Get Started with OR-Tools for Python

The following sections will get you started with OR-Tools for Python:

What is an optimization problem?

Solving an optimization problem in python, more python examples, identifying the type of problem you wish to solve.

The goal of optimization is to find the best solution to a problem out of a large set of possible solutions. (Sometimes you'll be satisfied with finding any feasible solution; OR-Tools can do that as well.)

Here's a typical optimization problem. Suppose that a shipping company delivers packages to its customers using a fleet of trucks. Every day, the company must assign packages to trucks, and then choose a route for each truck to deliver its packages. Each possible assignment of packages and routes has a cost, based on the total travel distance for the trucks, and possibly other factors as well. The problem is to choose the assignments of packages and routes that has the least cost.

Like all optimization problems, this problem has the following elements:

The objective —the quantity you want to optimize. In the example above, the objective is to minimize cost. To set up an optimization problem, you need to define a function that calculates the value of the objective for any possible solution. This is called the objective function . In the preceding example, the objective function would calculate the total cost of any assignment of packages and routes.

An optimal solution is one for which the value of the objective function is the best. ("Best" can be either a maximum or a minimum.)

The constraints —restrictions on the set of possible solutions, based on the specific requirements of the problem. For example, if the shipping company can't assign packages above a given weight to trucks, this would impose a constraint on the solutions.

A feasible solution is one that satisfies all the given constraints for the problem, without necessarily being optimal.

The first step in solving an optimization problem is identifying the objective and constraints.

Next, we give an example of an optimization problem, and show how to set up and solve it in Python.

A linear optimization example

One of the oldest and most widely-used areas of optimization is linear optimization (or linear programming ), in which the objective function and the constraints can be written as linear expressions. Here's a simple example of this type of problem.

Maximize 3x + y subject to the following constraints:

The objective function in this example is 3x + y . Both the objective function and the constraints are given by linear expressions, which makes this a linear problem.

Main steps in solving the problem

For each language, the basic steps for setting up and solving a problem are the same:

Python program

This section walks through a Python program that sets up and solves the problem.

Here are the steps:

Complete program

The complete program is shown below.

Running the program

You can run the program above as follows:

The program returns the values of x and y that maximize the objective function:

For more Python examples that illustrate how to solve various types of optimization problems, see Examples .

There are many different types of optimization problems in the world. For each type of problem, there are different approaches and algorithms for finding an optimal solution.

Before you can start writing a program to solve an optimization problem, you need to identify what type of problem you are dealing with, and then choose an appropriate solver — an algorithm for finding an optimal solution.

Below you will find a brief overview of the types of problems that OR-Tools solves, and links to the sections in this guide that explain how to solve each problem type.

Linear optimization

Constraint optimization, mixed-integer optimization, network flows.

As you learned in the previous section , a linear optimization problem is one in which the objective function and the constraints are linear expressions in the variables.

The primary solver in OR-Tools for this type of problem is the linear optimization solver, which is actually a wrapper for several different libraries for linear and mixed-integer optimization , including third-party libraries.

Learn more about linear optimization

A mixed integer optimization problem is one in which some or all of the variables are required to be integers. An example is the assignment problem , in which a group of workers needs be assigned to a set of tasks. For each worker and task, you define a variable whose value is 1 if the given worker is assigned to the given task, and 0 otherwise. In this case, the variables can only take on the values 0 or 1.

Learn more about mixed-integer optimization

Constraint optimization, or constraint programming (CP), identifies feasible solutions out of a very large set of candidates, where the problem can be modeled in terms of arbitrary constraints. CP is based on feasibility (finding a feasible solution) rather than optimization (finding an optimal solution) and focuses on the constraints and variables rather than the objective function. However, CP can be used to solve optimization problems, simply by comparing the values of the objective function for all feasible solutions.

Learn more about constraint optimization

Assignment problems involve assigning a group of agents (say, workers or machines) to a set of tasks, where there is a fixed cost for assigning each agent to a specific task. The problem is to find the assignment with the least total cost. Assignment problems are actually a special case of network flow problems .

Learn more about assignment

Bin packing is the problem of packing a set of objects of different sizes into containers with different capacities. The goal is to pack as many of the objects as possible, subject to the capacities of the containers. A special case of this is the Knapsack problem , in which there is just one container.

Learn more about bin packing

Scheduling problems involve assigning resources to perform a set of tasks at specific times. An important example is the job shop problem , in which multiple jobs are processed on several machines. Each job consists of a sequence of tasks, which must be performed in a given order, and each task must be processed on a specific machine. The problem is to assign a schedule so that all jobs are completed in as short an interval of time as possible.

Learn more about scheduling

Routing problems involve finding the optimal routes for a fleet of vehicles to traverse a network, defined by a directed graph. The problem of assigning packages to delivery trucks, described in What is an optimization problem ? , is one example of a routing problem. Another is the traveling salesperson problem .

Learn more about routing

Many optimization problems can be represented by a directed graph consisting of nodes and directed arcs between them. For example, transportation problems, in which goods are shipped across a railway network, can be represented by a graph in which the arcs are rail lines and the nodes are distribution centers.

In the maximum flow problem , each arc has a maximum capacity that can be transported across it. The problem is to assign the amount of goods to be shipped across each arc so that the total quantity being transported is as large as possible.

Learn more about network flows

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Multi-objective task scheduling optimization in spatial crowdsourcing.

1. Introduction

4.1. The Performance of the Ranking Strategy Algorithm

4.2. Performance of the MOTSO Model

4.2.1. maximizing the number of completed tasks, 4.2.2. minimizing the total travel costs (ttcs), 4.2.3. minimizing the standard deviation of the workload balance, 5. conclusions, author contributions, data availability statement, conflicts of interest, abbreviations.

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Alabbadi, A.A.; Abulkhair, M.F. Multi-Objective Task Scheduling Optimization in Spatial Crowdsourcing. Algorithms 2021 , 14 , 77. https://doi.org/10.3390/a14030077

Alabbadi AA, Abulkhair MF. Multi-Objective Task Scheduling Optimization in Spatial Crowdsourcing. Algorithms . 2021; 14(3):77. https://doi.org/10.3390/a14030077

Alabbadi, Afra A., and Maysoon F. Abulkhair. 2021. "Multi-Objective Task Scheduling Optimization in Spatial Crowdsourcing" Algorithms 14, no. 3: 77. https://doi.org/10.3390/a14030077

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Is there a greedy algorithm to solve the assignment problem?

The assignment problem is defined as:

There are n people who need to be assigned to n jobs, one person per job. The cost that would accrue if the ith person is assigned to the jth job is a known quantity C[i,j] for each pair i, j = 1, 2, ..., n. The problem is to find an assignment with the minimum total cost.

There is a question asking to design a greedy algorithm to solve the problem. It also asks if the greedy algorithm always yields an optimal solution and for the performance class of the algorithm. Here is my attempt at designing an algorithm:

Am I correct in saying that my algorithm is of O(n^2)? Am I also correct in saying that a greedy algorithm does not always yield an optimal solution? I used my algorithm on the following cost matrix and it is clearly not the optimal solution. Did I Do something wrong?

Applying greedy algorithm to cost matrix

$\begingroup$ Algorithms for maximum weight bipartite maximum matching are unfortunately more complicated than that. They don't really follow the greedy paradigm. $\endgroup$ – Yuval Filmus Apr 7, 2017 at 6:05

The answer of your post question (already given in Yuval comment) is that there is no greedy techniques providing you the optimal answer to an assignment problem.

The commonly used solution is the Hungarian algorithm, see

Harold W. Kuhn, "The Hungarian Method for the assignment problem", Naval Research Logistics Quarterly, 2: 83–97, 1955

for the original paper.

Otherwise your solution seems correct to me, and as usually with greedy algorithms, it will provide you a feasible solution which you may hope to not be "too far" from the global optimal solution.

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Is there a greedy algorithm to solve the assignment problem?
2. The answer of your post question (already given in Yuval comment) is that there is no greedy techniques providing you the optimal answer to an assignment problem. The commonly used solution is the Hungarian algorithm, see. Harold W. Kuhn, "The Hungarian Method for the assignment problem", Naval Research Logistics Quarterly, 2: 83-97, 1955.