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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:

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Job Sequencing with Deadline

Job scheduling algorithm.

Job scheduling algorithm is applied to schedule the jobs on a single processor to maximize the profits.

The greedy approach of the job scheduling algorithm states that, “Given ‘n’ number of jobs with a starting time and ending time, they need to be scheduled in such a way that maximum profit is received within the maximum deadline”.

Set of jobs with deadlines and profits are taken as an input with the job scheduling algorithm and scheduled subset of jobs with maximum profit are obtained as the final output.

Consider the following tasks with their deadlines and profits. Schedule the tasks in such a way that they produce maximum profit after being executed −

Find the maximum deadline value, dm, from the deadlines given.

Arrange the jobs in descending order of their profits.

The maximum deadline, d m , is 4. Therefore, all the tasks must end before 4.

Choose the job with highest profit, J4. It takes up 3 parts of the maximum deadline.

Therefore, the next job must have the time period 1.

Total Profit = 100.

The next job with highest profit is J5. But the time taken by J5 is 4, which exceeds the deadline by 3. Therefore, it cannot be added to the output set.

The next job with highest profit is J2. The time taken by J5 is 2, which also exceeds the deadline by 1. Therefore, it cannot be added to the output set.

The next job with higher profit is J3. The time taken by J3 is 1, which does not exceed the given deadline. Therefore, J3 is added to the output set.

Since, the maximum deadline is met, the algorithm comes to an end. The output set of jobs scheduled within the deadline are {J4, J3} with the maximum profit of 140 .

Following is the final implementation of Job sequencing Algorithm using Greedy Approach −

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Java Collection: Exercises, Practice, Solution

Java collection exercises [126 exercises with solution].

Java Collection refers to a framework provided by Java to store and manipulate groups of objects. It offers a set of interfaces (like List, Set, Queue, etc.) and classes (like ArrayList, HashSet, PriorityQueue, etc.) that provide different ways to organize and work with collections of elements. This framework simplifies common operations such as adding, removing, and accessing elements. It offers a wide range of data structures to suit various programming needs.

List of Java Collection Exercises :

• ArrayList Exercises [22 exercises with solution]
• LinkedList Exercises [26 exercises with solution]
• HashSet Exercises [12 exercises with solution]
• TreeSet Exercises [16 exercises with solution]
• PriorityQueue Exercises [12 exercises with solution]
• HashMap Exercises [12 exercises with solution]
• TreeMap Exercises [26 exercises with solution]

In Java, an ArrayList is a resizable array implementation of the List interface provided by the Java Collections Framework. It's part of the java.util package. Unlike arrays, which have a fixed size, ArrayList can dynamically grow and shrink in size as elements are added or removed. This makes it a more flexible data structure for handling collections of objects.

Key features and characteristics of ArrayList:

• Dynamic Size: As mentioned, an ArrayList can dynamically increase or decrease its size based on the number of elements it contains.
• Random Access: Elements within an ArrayList can be accessed using their index position. This allows for fast retrieval of elements based on their position in the list.
• Generics: ArrayList supports generics, which means it can hold elements of a specified type. This ensures type safety and avoids explicit type casting when retrieving elements from the list.
• Iterable: ArrayList implements the Iterable interface, which means it can be easily traversed using iterators or enhanced for loops.
• Not Synchronized: This class is roughly equivalent to Vector, except it is unsynchronized.

LinkedList:

In Java, LinkedList is another implementation of the List interface provided by the Java Collections Framework. It's part of the java.util package. Unlike ArrayList, which is backed by an array, LinkedList is implemented as a doubly-linked list.

Key features and characteristics of LinkedList:

• Internal Structure: LinkedList utilizes a doubly linked list internally for element storage, facilitating efficient insertion and deletion operations.
• Data Manipulation: LinkedList excels in data manipulation tasks due to its efficient insertion and removal operations.
• List and Queue Operations: The LinkedList class can serve as both a list and a queue, as it implements both the List and Deque interfaces.
• Performance Advantage: Manipulating elements in a LinkedList tends to be faster compared to ArrayList due to its employment of a doubly linked list structure, eliminating the need for bit shifting in memory operations.
• Optimal Use Cases: LinkedList is ideal for scenarios where frequent addition and removal of items occurs at the beginning or middle of the list, and where random access to elements is not essential.

In Java, HashSet is an implementation of the Set interface provided by the Java Collections Framework. It's part of the java.util package.

Key features and characteristics of HashSet:

• Support for Null Values: HashSet permits null values within its collection.
• Hashing Mechanism: Elements in a HashSet are organized and stored based on a hashing mechanism, optimizing insertion, deletion, and search operations.
• Non-Synchronized Implementation: HashSet is a non-synchronized class, meaning it's not inherently thread-safe and requires external synchronization for concurrent access.
• Hash Code-based Element Management: Elements within a HashSet are inserted and identified using their respective hash codes, facilitating efficient retrieval and manipulation.
• Unique Element Constraint: HashSet enforces uniqueness among its elements, ensuring no duplicate elements within the collection.
• Primarily Suited for Search Operations: HashSet is particularly advantageous for search operations due to its constant-time complexity for basic operations, such as retrieval and verification of element existence.
• Default Capacity and Load Factor: By default, HashSet initializes with a capacity of 16 and a load factor of 0.75, which can be adjusted as needed to optimize performance and memory usage.
• Hash Table Data Structure: HashSet utilizes the hash table data structure internally to manage its elements efficiently, supporting rapid access and modification operations.

In Java, TreeSet is an implementation of the SortedSet interface provided by the Java Collections Framework. It's part of the java.util package.

Here are the key features and characteristics of TreeSet:

• Uniqueness of Elements: TreeSet ensures that only unique elements are stored within its collection, discarding duplicate elements.
• Absence of Insertion Order Preservation: Unlike certain collection types, such as lists, TreeSet does not maintain the order in which elements are inserted.
• Ascending Order Sorting: Elements within a TreeSet are automatically sorted in ascending order according to their natural ordering or a custom comparator.
• Lack of Thread Safety: TreeSet is not inherently thread-safe, meaning it does not provide built-in mechanisms to handle concurrent access by multiple threads. External synchronization is required for thread safety.

PriorityQueue:

In Java, PriorityQueue is an implementation of the Queue interface provided by the Java Collections Framework. It's part of the java.util package.

Here are the key features and characteristics of PriorityQueue:

• Priority-Based Ordering: Elements are dequeued based on their priority, not in the order they were inserted.
• Heap-Based Data Structure: Internally uses a binary heap for efficient element management.
• No Guaranteed Order: Order of elements with equal priority is not guaranteed.
• Element Priority: Determined by natural ordering or a specified comparator.
• Not Synchronized: Requires external synchronization for thread safety.
• Performance: Offers logarithmic time complexity for insertion and removal operations.
• Usage: Commonly used in algorithms requiring prioritized element processing, like graph algorithms and scheduling.

In Java, HashMap is an implementation of the Map interface provided by the Java Collections Framework. It's part of the java.util package.

Here are the key features and characteristics of HashMap:

• Key-Value Storage: Stores data in key-value pairs for efficient retrieval.
• Unordered Collection: Does not maintain the order of elements.
• Unique Keys: Each key must be unique; duplicate keys are not allowed.
• Null Keys and Values: Allows null keys and multiple null values.
• Hash Table Structure: Uses a hash table internally for fast retrieval, insertion, and deletion.
• Not Synchronized: Not inherently thread-safe; external synchronization is needed for concurrent access.
• Good Performance: Offers constant-time performance for most operations.
• Iteration Order: The order of iteration is not guaranteed.

In Java, TreeMap is an implementation of the SortedMap interface provided by the Java Collections Framework. It's part of the java.util package.

Here are the key features and characteristics of TreeMap:

• Sorted Collection: Maintains elements in sorted order based on keys.
• Balanced Binary Search Tree: Uses a Red-Black Tree internally for efficient organization.
• Unique Keys: Each key must be unique; duplicates are prohibited.
• Null Keys: Does not allow null keys.
• Null Values: Allows null values to be associated with keys.
• Not Synchronized: Not inherently thread-safe; external synchronization needed for concurrent access.
• Performance: Offers guaranteed logarithmic time complexity for most operations.
• Navigable Map Operations: Provides additional operations for navigating elements based on their order.

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Constraint Satisfaction Problems (CSP) in Artificial Intelligence

Finding a solution that meets a set of constraints is the goal of constraint satisfaction problems (CSPs), a type of AI issue. Finding values for a group of variables that fulfill a set of restrictions or rules is the aim of constraint satisfaction problems. For tasks including resource allocation, planning, scheduling, and decision-making, CSPs are frequently employed in AI.

There are mainly three basic components in the constraint satisfaction problem:

Variables: The things that need to be determined are variables. Variables in a CSP are the objects that must have values assigned to them in order to satisfy a particular set of constraints. Boolean, integer, and categorical variables are just a few examples of the various types of variables, for instance, could stand in for the many puzzle cells that need to be filled with numbers in a sudoku puzzle.

Domains: The range of potential values that a variable can have is represented by domains. Depending on the issue, a domain may be finite or limitless. For instance, in Sudoku, the set of numbers from 1 to 9 can serve as the domain of a variable representing a problem cell.

Constraints: The guidelines that control how variables relate to one another are known as constraints. Constraints in a CSP define the ranges of possible values for variables. Unary constraints, binary constraints, and higher-order constraints are only a few examples of the various sorts of constraints. For instance, in a sudoku problem, the restrictions might be that each row, column, and 3×3 box can only have one instance of each number from 1 to 9.

Constraint Satisfaction Problems (CSP) representation:

• The finite set of variables V 1 , V 2 , V 3 ……………..V n .
• Non-empty domain for every single variable D 1 , D 2 , D 3 …………..D n .
• e.g., V 1 ≠ V 2
• Example: <(V 1 , V 2 ), V 1 not equal to V 2 > 
• Scope = set of variables that participate in constraint. 
• There might be a clear list of permitted combinations. Perhaps a relation that is abstract and that allows for membership testing and listing.

Constraint Satisfaction Problems (CSP) algorithms:

• The backtracking algorithm is a depth-first search algorithm that methodically investigates the search space of potential solutions up until a solution is discovered that satisfies all the restrictions. The method begins by choosing a variable and giving it a value before repeatedly attempting to give values to the other variables. The method returns to the prior variable and tries a different value if at any time a variable cannot be given a value that fulfills the requirements. Once all assignments have been tried or a solution that satisfies all constraints has been discovered, the algorithm ends.
• The forward-checking algorithm is a variation of the backtracking algorithm that condenses the search space using a type of local consistency. For each unassigned variable, the method keeps a list of remaining values and applies local constraints to eliminate inconsistent values from these sets. The algorithm examines a variable’s neighbors after it is given a value to see whether any of its remaining values become inconsistent and removes them from the sets if they do. The algorithm goes backward if, after forward checking, a variable has no more values.
• Algorithms for propagating constraints are a class that uses local consistency and inference to condense the search space. These algorithms operate by propagating restrictions between variables and removing inconsistent values from the variable domains using the information obtained.

Implementations code for Constraint Satisfaction Problems (CSP):

Implement constraint satisfaction problems algorithms with code, define the problem .

Here we are solving Sudoku Puzzle with Constraint Satisfaction Problems algorithms

Define Variables for the Constraint Satisfaction Problem

Define the domains for constraint satisfaction problem, define the constraint for constraint satisfaction problem, find the solution to the above sudaku problem, full code :, real-world constraint satisfaction problems (csp):.

• Scheduling: A fundamental CSP problem is how to efficiently and effectively schedule resources like personnel, equipment, and facilities. The constraints in this domain specify the availability and capacity of each resource, whereas the variables indicate the time slots or resources.
• Vehicle routing: Another example of a CSP problem is the issue of minimizing travel time or distance by optimizing a fleet of vehicles’ routes. In this domain, the constraints specify each vehicle’s capacity, delivery locations, and time windows, while the variables indicate the routes taken by the vehicles.
• Assignment: Another typical CSP issue is how to optimally assign assignments or jobs to humans or machines. In this field, the variables stand in for the tasks, while the constraints specify the knowledge, capacity, and workload of each person or machine.
• Sudoku: The well-known puzzle game Sudoku can be modeled as a CSP problem, where the variables stand in for the grid’s cells and the constraints specify the game’s rules, such as prohibiting the repetition of the same number in a row, column, or area.
• Constraint-based image segmentation: The segmentation of an image into areas with various qualities (such as color, texture, or shape) can be treated as a CSP issue in computer vision, where the variables represent the regions and the constraints specify how similar or unlike neighboring regions are to one another.

Constraint Satisfaction Problems (CSP) benefits:

• conventional representation patterns
• generic successor and goal functions
• Standard heuristics (no domain-specific expertise).

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