job assignment problem c

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Or check our popular categories..., branch and bound | set 4 (job assignment problem).

Let there be N workers and N jobs. Any worker can be assigned to perform any job, incurring some cost that may vary depending on the work-job assignment. It is required to perform all jobs by assigning exactly one worker to each job and exactly one job to each agent in such a way that the total cost of the assignment is minimized.

Let us explore all approaches for this problem.

Solution 1: Brute Force We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O(n!).

Solution 2: Hungarian Algorithm The optimal assignment can be found using the Hungarian algorithm. The Hungarian algorithm has worst case run-time complexity of O(n^3).

Solution 3: DFS/BFS on state space tree A state space tree is a N-ary tree with property that any path from root to leaf node holds one of many solutions to given problem. We can perform depth-first search on state space tree and but successive moves can take us away from the goal rather than bringing closer. The search of state space tree follows leftmost path from the root regardless of initial state. An answer node may never be found in this approach. We can also perform a Breadth-first search on state space tree. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS.

Solution 4: Finding Optimal Solution using Branch and Bound The selection rule for the next node in BFS and DFS is “blind”. i.e. the selection rule does not give any preference to a node that has a very good chance of getting the search to an answer node quickly. The search for an optimal solution can often be speeded by using an “intelligent” ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an optimal solution. It is similar to BFS-like search but with one major optimization. Instead of following FIFO order, we choose a live node with least cost. We may not get optimal solution by following node with least promising cost, but it will provide very good chance of getting the search to an answer node quickly.

There are two approaches to calculate the cost function:

• For each worker, we choose job with minimum cost from list of unassigned jobs (take minimum entry from each row).
• For each job, we choose a worker with lowest cost for that job from list of unassigned workers (take minimum entry from each column).

In this article, the first approach is followed.

Let’s take below example and try to calculate promising cost when Job 2 is assigned to worker A.

Since Job 2 is assigned to worker A (marked in green), cost becomes 2 and Job 2 and worker A becomes unavailable (marked in red).

Now we assign job 3 to worker B as it has minimum cost from list of unassigned jobs. Cost becomes 2 + 3 = 5 and Job 3 and worker B also becomes unavailable.

Finally, job 1 gets assigned to worker C as it has minimum cost among unassigned jobs and job 4 gets assigned to worker C as it is only Job left. Total cost becomes 2 + 3 + 5 + 4 = 14.

Below diagram shows complete search space diagram showing optimal solution path in green.

Complete Algorithm:

Below is its C++ implementation.

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Venkatesan Prabu

Wikitechy Founder, Author, International Speaker, and Job Consultant. My role as the CEO of Wikitechy, I help businesses build their next generation digital platforms and help with their product innovation and growth strategy. I'm a frequent speaker at tech conferences and events.

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Solved the Job Assignment Problem using a branch and bound algorithm

Valor-boop/Job-Assignment-Problem

Folders and files, repository files navigation, job-assignment-problem.

Solved the Job Assignment Problem using both brute force as well as branch and bound. The code contains 5 functions:

job_assignment(cost_matrix): Find an optimal solution to the job assignment problem using branch and bound. Input: an nxn matrix where a row represents a person and a column represents the cost each person takes to complete the jobs. Return: the minimal cost, an optimal solution, and the number of partial or full solutions evaluated.

get_csf(cost_matrix, partial_solution): Input: an nxn cost matrix and a partial solution. Return: the partial solution's Cost So Far (CSF). A partial solution is represented as a list of n elements, where an undecided element is denoted by a -1. For instance, a partial solution [2, -1, -1] represents assigning job 2 to person 0 and leaving the assignments of person 1 and person 2 undecided.

get_gfc(cost_matrix, partial_solution): Input: an nxn cost matrix and a partial solution. Return: the partial solution's Guaranteed Future Cost (GFC).

get_ffc(cost_matrix, partial_solution): Input: an nxn cost matrix and a partial solution. Return: the partial solution's Feasible Futre Cost (FFC)

brute_force(cost_matrix): This function finds an optimal solution for the job assignment problem using brute force. Input: an nxn cost matrix. Return: the minimal total cost, an optimal solution, and the number of full solutions investigated.

• Python 100.0%

Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

Mathematical Formulation of the Assignment Problem:

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AI-fueled cheating—and how to stop students from doing it—has become a major concern for educators.

But how prevalent is it? Newly released data from a popular plagiarism-detection company is shedding some light on the problem.

And it may not be as bad as educators think it is.

Of the more than 200 million writing assignments reviewed by Turnitin’s AI detection tool over the past year, some AI use was detected in about 1 out of 10 assignments, while only 3 out of every 100 assignments were generated mostly by AI.

These numbers have not changed much from when Turnitin released data in August of 2023 about the first three months of the use of its detection tool, said the company’s chief product officer, Annie Chechitelli.

“We hit a steady state, and it hasn’t changed dramatically since then,” she said. “There are students who are leaning on AI too much. But it’s not pervasive. It wasn’t this, ‘the sky is falling.’”

The fact that the number of students using AI to complete their schoolwork hasn’t skyrocketed in the past year dovetails with survey findings from Stanford University that were released in December. Researchers there polled students in 40 different high schools and found that the percentage of students who admitted to cheating has remained flat since the advent of ChatGPT and other readily available generative AI tools. For years before the release of ChatGPT, between 60 and 70 percent of students admitted to cheating, and that remained the same in the 2023 surveys, the researchers said.

Turnitin’s latest data release shows that in 11 percent of assignments run through its AI detection tool that at least 20 percent of each assignment had evidence of AI use in the writing. In 3 percent of the assignments, each assignment was made up of 80 percent or more of AI writing, which tracks closely with what the company was seeing just 3 months after it launched its AI detection tool .

Experts warn against fixating on cheating and plagiarism

However, a separate survey of educators has found that AI detection tools are becoming more popular with teachers, a trend that worries some experts.

The survey of middle and high school teachers by the Center for Democracy and Technology, a nonprofit focused on technology policy and consumer rights, found that 68 percent have used an AI detection tool, up substantially from the previous year. Teachers also reported in the same survey that students are increasingly getting in trouble for using AI to complete assignments. In the 2023-24 school year, 63 percent of teachers said students had gotten in trouble for being accused of using generative AI in their schoolwork, up from 48 percent last school year.

Despite scant evidence that AI is fueling a wave in cheating, half of teachers reported in the Center for Democracy and Technology survey that generative AI has made them more distrustful that their students are turning in original work.

Some experts warn that fixating on plagiarism and cheating is the wrong focus.

This creates an environment where students are afraid to talk with their teachers about AI tools because they might get in trouble, said Tara Nattrass, the managing director of innovation and strategy at ISTE+ASCD, a nonprofit that offers content and professional development on educational technology and curriculum.

“We need to reframe the conversation and engage with students around the ways in which AI can support them in their learning and the ways in which it may be detrimental to their learning,” she said in an email to Education Week. “We want students to know that activities like using AI to write essays and pass them off as their own is harmful to their learning while using AI to break down difficult topics to strengthen understanding can help them in their learning.”

Shift the focus to teaching AI literacy, crafting better policies

Students said in the Stanford survey that is generally how they think AI should be used: as an aid to understanding concepts rather than a fancy plagiarism tool.

Nattrass said schools should be teaching AI literacy while including students in drafting clear AI guidelines.

Nattrass also recommends against schools using AI detection tools. They are too unreliable to authenticate students’ work, she said, and false positives can be devastating to individual students and breed a larger environment of mistrust. Some research has found that AI detection tools are especially weak at identifying the original writing of English learners from AI-driven prose.

“Students are using AI and will continue to do so with or without educator guidance,” Nattrass said. “Teaching students about safe and ethical AI use is a part of our responsibility to help them become contributing digital citizens.”

AI detection software actually uses AI to function: these tools are trained on large amounts of machine- and human-created writing so that the software can ideally recognize differences between the two.

Turnitin claims that its AI detector is 99 percent accurate at determining whether a document was written with AI, specifically ChatGPT, as long as the document was composed with at least 20 percent of AI writing, according to the company’s website.

Chechitelli pointed out that no detector or test—whether it’s a fire alarm or medical test—is 100 percent accurate.

While she said teachers should not rely solely on AI detectors to determine if a student is using AI to cheat, she makes the case that detection tools can provide teachers with valuable data.

“It is not definitive proof,” she said. “It’s a signal that taken with other signals can be used to start a conversation with a student.”

As educators become more comfortable with generative AI, Chechitelli said she predicts the focus will shift from detection to transparency: how should students cite or communicate the ways they’ve used AI? When should educators encourage students to use AI in assignments? And do schools have clear policies around AI use and what, exactly, constitutes plagiarism or cheating?

“What the feedback we’re hearing now from students is: ‘I’m gonna use it. I would love a little bit more guidance on how and when so I don’t get in trouble,” but still use it to learn, Chechitelli said.

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