computer science assignment problem

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Course info, instructors.

• Prof. Eric Grimson
• Prof. John Guttag

Departments

• Electrical Engineering and Computer Science

As Taught In

• Programming Languages

Introduction to Computer Science and Programming

Assignments.

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AP®︎/College Computer Science Principles

Welcome learners, unit 1: digital information, unit 2: the internet, unit 3: programming, unit 4: algorithms, unit 5: data analysis, unit 6: simulations, unit 7: online data security, unit 8: computing innovations, unit 9: exam preparation, unit 10: ap®︎ csp standards mappings.

CS50: Introduction to Computer Science

An introduction to the intellectual enterprises of computer science and the art of programming.

Associated Schools

Harvard School of Engineering and Applied Sciences

What you'll learn.

A broad and robust understanding of computer science and programming

How to think algorithmically and solve programming problems efficiently

Concepts like abstraction, algorithms, data structures, encapsulation, resource management, security, software engineering, and web development

Familiarity with a number of languages, including C, Python, SQL, and JavaScript plus CSS and HTML

How to engage with a vibrant community of like-minded learners from all levels of experience

How to develop and present a final programming project to your peers

Course description

This is CS50x , Harvard University's introduction to the intellectual enterprises of computer science and the art of programming for majors and non-majors alike, with or without prior programming experience. An entry-level course taught by David J. Malan, CS50x teaches students how to think algorithmically and solve problems efficiently. Topics include abstraction, algorithms, data structures, encapsulation, resource management, security, software engineering, and web development. Languages include C, Python, SQL, and JavaScript plus CSS and HTML. Problem sets inspired by real-world domains of biology, cryptography, finance, forensics, and gaming. The on-campus version of CS50x , CS50, is Harvard's largest course. 

Students who earn a satisfactory score on 9 problem sets (i.e., programming assignments) and a final project are eligible for a certificate. This is a self-paced course–you may take CS50x on your own schedule.

Instructors

David J. Malan

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CS50 for Lawyers

This course is a variant of Harvard University's introduction to computer science, CS50, designed especially for lawyers (and law students).

How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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Operations Research

1 Operations Research-An Overview

• History of O.R.
• Approach, Techniques and Tools
• Phases and Processes of O.R. Study
• Typical Applications of O.R
• Limitations of Operations Research
• Models in Operations Research
• O.R. in real world

2 Linear Programming: Formulation and Graphical Method

• General formulation of Linear Programming Problem
• Optimisation Models
• Basics of Graphic Method
• Important steps to draw graph
• Multiple, Unbounded Solution and Infeasible Problems
• Solving Linear Programming Graphically Using Computer
• Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

• Principle of Simplex Method
• Computational aspect of Simplex Method
• Simplex Method with several Decision Variables
• Two Phase and M-method
• Multiple Solution, Unbounded Solution and Infeasible Problem
• Sensitivity Analysis
• Dual Linear Programming Problem

4 Transportation Problem

• Basic Feasible Solution of a Transportation Problem
• Modified Distribution Method
• Stepping Stone Method
• Unbalanced Transportation Problem
• Degenerate Transportation Problem
• Transhipment Problem
• Maximisation in a Transportation Problem

5 Assignment Problem

• Solution of the Assignment Problem
• Unbalanced Assignment Problem
• Problem with some Infeasible Assignments
• Maximisation in an Assignment Problem
• Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

• Concepts of goal programming
• Goal programming model formulation
• Graphical method of goal programming
• The simplex method of goal programming
• Using Excel Solver to Solve Goal Programming Models
• Application areas of goal programming

8 Integer Programming

• Some Integer Programming Formulation Techniques
• Binary Representation of General Integer Variables
• Unimodularity
• Cutting Plane Method
• Branch and Bound Method
• Solver Solution

9 Dynamic Programming

• Dynamic Programming Methodology: An Example
• Definitions and Notations
• Dynamic Programming Applications

10 Non-Linear Programming

• Solution of a Non-linear Programming Problem
• Convex and Concave Functions
• Kuhn-Tucker Conditions for Constrained Optimisation
• Quadratic Programming
• Separable Programming
• NLP Models with Solver

11 Introduction to game theory and its Applications

• Important terms in Game Theory
• Saddle points
• Mixed strategies: Games without saddle points
• 2 x n games
• Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

• Reasons for using simulation
• Monte Carlo simulation
• Limitations of simulation
• Steps in the simulation process
• Some practical applications of simulation
• Two typical examples of hand-computed simulation
• Computer simulation

13 Queueing Models

• Characteristics of a queueing model
• Notations and Symbols
• Statistical methods in queueing
• The M/M/I System
• The M/M/C System
• The M/Ek/I System
• Decision problems in queueing

Introduction to Scientific Computing and Problem Solving

Welcome to CS4, Introduction to Scientific Computing and Problem Solving . CS4 provides an introduction to using computers to solve STEM (Science, Technology, Engineering, and Mathematics) data analysis, visualization, simulation, and numerical analysis problems. The course begins with an introduction to the basics of programming, accompanied by several applications of fundamental coding elements and concepts. As we do this we will explore some of the breadth of Computer Science as a discipline. The first part of the course (which runs until Spring Break) will be taught in Python. Following this, we will explore more specialized topics related to scientific computing and mathematics that will allow students to access and analyze a number of "real world" problems. The later portion of the course will be taught in MATLAB.

Requirements: No prior programming experience is required to take this course (Python and MATLAB are easy and fun to use!). A calculus course (perhaps in high school) is highly recommended.

If you have any feedback about assignments or the course in general, please fill out the Anonymous Feedback Form to let the course staff know. If you have any academic/SEAS accommodations that we should be aware of, please fill out this form so that the course staff can best support you. We understand that being a student can be stressful and that certain circumstances can affect your performance in the course. Please refer to the syllabus for more information about receiving academic support for CS0040 or email the HTAs or Prof. Gaudette for more information regarding accommodations. You can also find some resources the CS department has compiled for students here and a helpful message from your TAs here

COURSE DOCUMENTS

Quick links, working from home, how do ta hours work, diversity and inclusion goals.

Jason Gaudette (jegaudet)

Team: Puppies

Welcome to CS4! I am happy to be your instructor this semester. As a research engineer with loads of real-world experience in scientific computing, my goal is to teach you the art and joy of the field. Outside of my main job, I coach kids in FIRST LEGO robotics, volunteer for the IEEE, teach programming and electronics courses, sail a Hobie Cat, and run with my dog Luna."

E-mail us at [email protected] if you have administrative or private questions.

Griffin Kao (gk16)

Team: Babies

Hi there, I’m a junior from Philadelphia studying computer science and engineering. A fun fact about me is that Kobe Bryant went to my high school!

Hersh Gupta (hgupta1)

Hey everybody! My name is Hersh and I’m a junior concentrating in chemistry and computational biology. I’m a big sports fan and love going on Netflix binges but never actually finishing the show.

Joy Bestourous (jbestour)

Hi hello! I'm a junior studying computer science while filling pre-med requirements (and maybe pre-law! Who knows!). I like New York pizza, iced coffee, going to the gym, dancing, and singing Disney songs. Loudly.

E-mail us at [email protected] or post a question (privately!) on Piazza if you have any questions about the course.

Annie He (ahe6)

Hey y’all! I’m a junior from Dallas, TX concentrating in computational biology. If you don’t see me in the scili, I’m either chilling in my pjs or exploring new restaurants! I also love to rock climb, ski, bungee jump, basically anything adventurous! :)

Alex Liu (aliu31)

I am a junior concentrating in Applied Mathematics and Computer Science. I’m from Eugene, Oregon and outside of class I am involved with the Socially Responsible Investment Fund, I play golf whenever I get the chance, and I try my best to keep up on practicing the Piano. I also run Brown Data Science, a club that, much like this class, brings educational opportunities related to Data Science to undergraduates of all academic backgrounds.

Aryan Srivastava (asriva11)

I am a freshman concentrating in CS (probably). I love watching and playing basketball (1v1 me), reading and talking philosophy, and watching artisan videos on youtube! My favorite thing to eat at Brown is a warm Blue Room Chocolate Chip Muffin.

Ellen Ling (eling)

Hi, I’m a junior from Shanghai studying physics and CS. I like good movies and dance and babies!

Irene Rhee (irhee)

Sup dudes, my name is Irene and I’m a junior from Corvallis, OR studying computer science! I like to drink my coffee black, am part of a dance group called Daebak (come to our spring show!!), and enjoy dim sum, boba, and tagging people in memes (go like my post on subtle asian traits). My favorite word is lit.

Joseph Chen (jchen88)

Hey everyone! I’m a junior from California concentrating in Neuroscience and CS, and I’m a big fan of all things related to swimming, playing basketball, Game of Thrones, and stand-up comedy.

Jarrett Huddleston (jhuddle1)

Hey! I’m a sophomore from Eastern Massachusetts concentrating in computer science. Outside of classes I enjoy camping and music, and I’m always looking for a good book!

Milla Shin (mshin7)

I’m a junior from Tokyo studying computer science. I like going to the beach. I like food and cooking! I like snowboarding and scuba diving! I love sweet potatoes, avocado, and matcha! Yummy. I like puppies but I like babies too.

Pedro de Freitas (pfreitas)

Senior from Portland, Maine studying cognitive neuroscience. I’m a lyra performer/instructor who also enjoys painting.

Solomon Rueschemeyer-Bailey (sruesche)

I am a junior trying not to let school get in the way of college. Come to my hours or my section if you want to learn about: The Bucket Theorem, whether or not true Love exists, and the future of teleportation.

Tiffany Ding (tding5)

Hi! I’m a sophomore from upstate NY studying applied math, economics, and computer science. When I’m not in the CIT, you can find me going for a run or taking photos for the Brown Daily Herald. I also love puns, penguins, and podcasts!

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Solutions to In-Class questions, Problem Sets and Exams of MIT Mathematics for Computer Science 2015 (same as 2019 Open Learning Library)

spamegg1/Math-for-CS-solutions

Folders and files, repository files navigation.

It soon became clear that the unconscious instincts for logic and language which had enabled me to succeed were not shared by the large majority of my students. - Susanna Epp (author of Discrete Mathematics with Applications, 5th edition )

Please note that the above book by Epp is much better suited for beginners, whereas MIT Math for CS is much more difficult. (Solutions to that one are here .)

Bjarne Stroustrup on the importance of Mathematics and Computer Science fundamentals

Feel free to open a discussion if you find a mistake or a typo!

Solutions to Problem Sets and Exams of MIT Mathematics for Computer Science 2015 (same as 2019 Open Learning Library )

Contact me: spamegg1 on Discord, or on Slack , or on Matrix

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .

Computer Science

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• Python | Linear search on list or tuples
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• How to begin with Competitive Programming?
• Dynamic Programming (DP) on Grids
• Knuth's Optimization in Dynamic Programming

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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Title: multi-auv kinematic task assignment based on self-organizing map neural network and dubins path generator.

Abstract: To deal with the task assignment problem of multi-AUV systems under kinematic constraints, which means steering capability constraints for underactuated AUVs or other vehicles likely, an improved task assignment algorithm is proposed combining the Dubins Path algorithm with improved SOM neural network algorithm. At first, the aimed tasks are assigned to the AUVs by improved SOM neural network method based on workload balance and neighborhood function. When there exists kinematic constraints or obstacles which may cause failure of trajectory planning, task re-assignment will be implemented by change the weights of SOM neurals, until the AUVs can have paths to reach all the targets. Then, the Dubins paths are generated in several limited cases. AUV's yaw angle is limited, which result in new assignments to the targets. Computation flow is designed so that the algorithm in MATLAB and Python can realizes the path planning to multiple targets. Finally, simulation results prove that the proposed algorithm can effectively accomplish the task assignment task for multi-AUV system.

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