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Solve an assignment problem online

Fill in the cost matrix of an assignment problem and click on 'Solve'. The optimal assignment will be determined and a step by step explanation of the hungarian algorithm will be given.

Fill in the cost matrix ( random cost matrix ):

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• 8 puzzle Problem using Branch And Bound

Job Assignment Problem using Branch And Bound

• N Queen Problem using Branch And Bound
• Traveling Salesman Problem using Branch And Bound

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 D 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 the implementation of the above approach:

Time Complexity: O(M*N). This is because the algorithm uses a double for loop to iterate through the M x N matrix.  Auxiliary Space: O(M+N). This is because it uses two arrays of size M and N to track the applicants and jobs.

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Solving an Assignment Problem

This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver.

In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3). Note that there is one more worker than in the example in the Overview .

The costs of assigning workers to tasks are shown in the following table.

The problem is to assign each worker to at most one task, with no two workers performing the same task, while minimizing the total cost. Since there are more workers than tasks, one worker will not be assigned a task.

MIP solution

The following sections describe how to solve the problem using the MPSolver wrapper .

Import the libraries

The following code imports the required libraries.

Create the data

The following code creates the data for the problem.

The costs array corresponds to the table of costs for assigning workers to tasks, shown above.

Declare the MIP solver

The following code declares the MIP solver.

Create the variables

The following code creates binary integer variables for the problem.

Create the constraints

Create the objective function.

The following code creates the objective function for the problem.

The value of the objective function is the total cost over all variables that are assigned the value 1 by the solver.

Invoke the solver

The following code invokes the solver.

Print the solution

The following code prints the solution to the problem.

Here is the output of the program.

Complete programs

Here are the complete programs for the MIP solution.

CP SAT solution

The following sections describe how to solve the problem using the CP-SAT solver.

Declare the model

The following code declares the CP-SAT model.

The following code sets up the data for the problem.

The following code creates the constraints for the problem.

Here are the complete programs for the CP-SAT solution.

Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.

Last updated 2023-01-02 UTC.

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These calculators will solve three types of 'work' word problems . Also, it will provide a detailed explanation.

First worker can do the job in Days Hours Minutes and the second worker takes Days Hours Minutes . How long would it take the two workers together to finish the job?

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A swimming pool has 2 inlet pipes. One fills the pool in hours, the other in hours. The outlet pipe empties the pool in hours. If the pool is at first empty and all three pipes are open, how many hours will it take to fill up the pool?

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Use this calculator to find out the grade of a course based on weighted averages. This calculator accepts both numerical as well as letter grades. It also can calculate the grade needed for the remaining assignments in order to get a desired grade for an ongoing course.

Final Grade Calculator

Use this calculator to find out the grade needed on the final exam in order to get a desired grade in a course. It accepts letter grades, percentage grades, and other numerical inputs.

Related GPA Calculator

The calculators above use the following letter grades and their typical corresponding numerical equivalents based on grade points.

Brief history of different grading systems

In 1785, students at Yale were ranked based on "optimi" being the highest rank, followed by second optimi, inferiore (lower), and pejores (worse). At William and Mary, students were ranked as either No. 1, or No. 2, where No. 1 represented students that were first in their class, while No. 2 represented those who were "orderly, correct and attentive." Meanwhile at Harvard, students were graded based on a numerical system from 1-200 (except for math and philosophy where 1-100 was used). Later, shortly after 1883, Harvard used a system of "Classes" where students were either Class I, II, III, IV, or V, with V representing a failing grade. All of these examples show the subjective, arbitrary, and inconsistent nature with which different institutions graded their students, demonstrating the need for a more standardized, albeit equally arbitrary grading system.

In 1887, Mount Holyoke College became the first college to use letter grades similar to those commonly used today. The college used a grading scale with the letters A, B, C, D, and E, where E represented a failing grade. This grading system however, was far stricter than those commonly used today, with a failing grade being defined as anything below 75%. The college later re-defined their grading system, adding the letter F for a failing grade (still below 75%). This system of using a letter grading scale became increasingly popular within colleges and high schools, eventually leading to the letter grading systems typically used today. However, there is still significant variation regarding what may constitute an A, or whether a system uses plusses or minuses (i.e. A+ or B-), among other differences.

An alternative to the letter grading system

Letter grades provide an easy means to generalize a student's performance. They can be more effective than qualitative evaluations in situations where "right" or "wrong" answers can be easily quantified, such as an algebra exam, but alone may not provide a student with enough feedback in regards to an assessment like a written paper (which is much more subjective).

Although a written analysis of each individual student's work may be a more effective form of feedback, there exists the argument that students and parents are unlikely to read the feedback, and that teachers do not have the time to write such an analysis. There is precedence for this type of evaluation system however, in Saint Ann's School in New York City, an arts-oriented private school that does not have a letter grading system. Instead, teachers write anecdotal reports for each student. This method of evaluation focuses on promoting learning and improvement, rather than the pursuit of a certain letter grade in a course. For better or for worse however, these types of programs constitute a minority in the United States, and though the experience may be better for the student, most institutions still use a fairly standard letter grading system that students will have to adjust to. The time investment that this type of evaluation method requires of teachers/professors is likely not viable on university campuses with hundreds of students per course. As such, although there are other high schools such as Sanborn High School that approach grading in a more qualitative way, it remains to be seen whether such grading methods can be scalable. Until then, more generalized forms of grading like the letter grading system are unlikely to be entirely replaced. However, many educators already try to create an environment that limits the role that grades play in motivating students. One could argue that a combination of these two systems would likely be the most realistic, and effective way to provide a more standardized evaluation of students, while promoting learning.

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Optimal assignment problem.

Find the amount of electricity a company must send from its four power plants to five cities so as to maximize profit and minimize cost while meeting the cities' peak demands.

This example demonstrates how LinearFractionalOptimization may be used to minimize the ratio of cost to profit within given constraints. Use of a matrix-valued variable makes the modeling relatively simple.

As an example, here is the cost of transporting one million kilowatt hours (kWh) of electricity from four plants to five cities.

The profit that each power plant generates by selling 1 million kWh to each city is shown here.

The cities have a peak demand of 45, 20, 30, 30 and 40 million kWh, respectively, and a minimum demand of 5 million kWh.

The power plants can supply a minimum of 35, 50, 40 and 40 million kWh of electricity, respectively.

The optimal amount of electricity to send each city by each plant can be found by minimizing the ratio of cost to profit.

The breakdown of electricity supplied is shown.

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Number line.

• \mathrm{Lauren's\:age\:is\:half\:of\:Joe's\:age.\:Emma\:is\:four\:years\:older\:than\:Joe.\:The\:sum\:of\:Lauren,\:Emma,\:and\:Joe's\:age\:is\:54.\:How\:old\:is\:Joe?}
• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
• Is there a calculator that can solve word problems?
• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

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Final Grade Calculator

Enter Final Info

My final exam is worth:

I want (at least) this in the class:

Enter Class Grades

Calculator Instructions

• In the top part of the form, enter how much your final exam is worth and the grade that you would like to get in the class. For example, your final test might be worth 20% of your overall grade and you want to get at least a 93% in the class. You would enter these numbers into the form.
• In the bottom half of the form, enter a description (optional) of the classwork, the grade received for that classwork, and the weight of the classwork. Most class grades are made up of several components such as homework assignments, tests, exams, quizzes, class participation, attendance, etc. For example, a class exam might be worth 10% of your grade and you received a 95% on the test. You would enter those values into the form.
• If you need more than four rows, press the "Add Row" button to add an additional line. You can add as many rows as you need.
• Once you have finished entering your grades, press the "Calculate" button and the grade you need on the final exam will be displayed.

Final Grade Formula

final grade = ((g wanted x w total ) - gw) / w final

w total = w 1 + w 2 + w 3 + ... + w final

w 1 = weight of assignment #1

w final = weight of final exam

gw = g 1 x w 1 + g 2 x w 2 + g 3 x w 3 + ...

g 1 = grade for assignment #1

g wanted = grade wanted in the class

Example Calculation

Let's say your class has the following grading plan.

Now let's assume you received the following grades on your classwork.

Finally, let's assume that you want to get a 90% in the class. To determine what you need to get on your final exam in order to get a 90% in the class, let's do some math using the formula above.

First add the weight of all the class assignments together including your final:

w total = 10% + 10% + 20% + 20% + 20% = 100%

Next, multiple the grade you received on each assignment by the weight of the assignment.

gw = (91% x 10%) + (85% x 10%) + (75% x 20%) + (95% x 20%) + (97% x 20%) = 7100%

Now, calculate what you need on the final exam:

final exam grade = ((90% x 100%) - 7100%) / 20% = 95%

This is how you manually calculate your final grade. Of course, you can make your life a little easier using the calculator above!

What if my class grade is based on points rather than percentages?

Let's assume you have the following class syllabus that is based on points.

Let's assume you received the following grades.

To enter these grades in the calculator above, you first need to calculate your grade percentage for each assignment using the following formula:

grade percentage = points earned / possible points x 100

So taking your mid-term test grade as an example, we get the following:

mid-term test = 190 points earned / 200 possible points x 100 = 95%

In the weight column of the calculator, you would enter the possible points for each assignment.

Assuming you wanted to get at least a 90% in the class and your final exam is worth 250 points (i.e.the weight), you would enter the following information into the calculator.

In this example, you would need to get a 93.6% on your final in order to get a 90% in the class.

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