in an assignment problem mcq

Operations Research

121. The transportation problem is balanced, if ______________.

• total demand and total supply are equal and the number of sources equals the number of destinations.
• none of the routes is prohibited
• total demand equals total supply irrespective of the number of sources and destinations
• number of sources matches with number of destinations

122. In an assignment problem involving 5 workers and 5 jobs, total number of assignments possible are ______________.

123. All of the following are assumptions of the EOQ model except ______________

• the usage rate is reasonably constant
• replenishment is not instantaneous
• only one product is involved
• there are no quantity discount price

124. Average number of trains spent in the yard is denoted by ______________.

125. Graphical method of linear programming is useful when the number of decision variable are ______________

126. The cost of a surplus variable is ______________.

127. The dual of the dual is ______________.

• dual-primal
• primal-dual

128. Solution of a Linear Programming Problem when permitted to be infinitely large is called ______________.

• optimum solution
• no solution

129. When the total demand is not equal to supply then it is said to be ______________.

• maximization
• minimization

130. All equality constraints can be replaced equivalently by ______________ inequalities

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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

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:

• For each row of the matrix, find the smallest element and subtract it from every element in its row.
• Do the same (as step 1) for all columns.
• Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
• Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, and must proceed to step 5.
• Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.

Try it before moving to see the solution

Explanation for above simple example:

  An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.

Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).

Space complexity :   O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.

In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY

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Assignment Problems — Quiz It!

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20 questions

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The following are descriptions of Assignment Problems EXCEPT:

Assignment problems are an optimization technique which comprises of linear constraints and a linear objective function.

Assignment problems (AP) are a unique form of linear programming issues that are focused on the assigning of different types of merchandise.

The goal of assignment problems are to find the optimal assignment, minimizing the expenses and the like.

Assignment problems emerge on the grounds that accessible assets, for example, men, machines and so on.

Assignment problems (AP) are applied and seen in which of the following fields:

Dietary Plans

Transportation

Distribution of tasks

Which of the following is NOT considered as an assumption in an assignment problem?

The number of assignees and number of task are equal (this number is denoted by n ).

Each assignee is to be assigned to perform exactly one task.

Each task is to be performed by exactly one assignee.

There is no cost or profit associated with assignees performing different task.

What is an assignment table?

The assignment table does not display the data in relation to the problems.

The data displayed within the table are combined within the rows and columns.

The assignment table is a table which displays the data in relation to the problems.

These data reflect what we call their respective "consequences."

What is a balanced assignment problem?

A given assignment problem is considered to be "balanced" if the supply and demand are not equal to one another.

A given assignment problem is considered to be "balanced" if the number of rows are equal to the number of columns.

A given assignment problem is considered to be "balanced" if the number of rows are not equal to the number of columns.

A given assignment problem is considered to be "balanced" if the supply and demand are equal to one another.

If there are 5 machines in a given assignment problem, how many people must be assigned per machine?

If there are 5 tasks and 4 workers, there would be a need to include a:

Dummy contract

Dummy worker

Which of the following is the method used in solving assignment problems?

North-West Corner Rule

Hungarian Method

Vogel's Approximation Method

Corner-Point Maximization Method

Which of the following is NOT a step in the Hungarian Method process?

Identify the feasible region of the assignment problem.

Identify the largest value or number present within the original cost table.

Revise present cost table.

Find opportunity cost table.

To consider a solution in the assignment problem as the optimal one, which of the following conditions must be satisfied:

The lines drawn to cover the zeroes must be less than the number of rows or columns.

The lines drawn to cover the zeroes must be more than the number of rows or columns.

There are at least two zeroes per row.

The lines drawn to cover the zeroes must be equal to either the number of rows or columns.

DLSU has reached the finals of the Track and Field Men's Relay. Before sending off the players on to the track, the head coach had decided to carry out a strategy as to who would be the first, second, third and fourth runner. Here, the head coach had tabulated the total time each runner would take to accomplish their station as based on which among the four places they are placed in.

What is the head coach's goal in this problem?

Minimization

Maximization

What is the first step in solving this assignment problem?

Identify the largest number within each row of the original table.

Subtract the largest number within each cell or value of the original table.

Identify the smallest number within each row of the original table.

Add the identified smallest number to the other values contained within the same row.

Upon subtracting the smallest number to every other number contained within the same column, can optimal assignments be carried out?

Runner C must be which among the four places?

First Runner

Second Runner

Third Runner

Fourth Runner

Runner D must be which among the four places?

A company is looking for sub-contractors to provide aid in several construction projects across the NCR. Here, JGC, DMCI, PowerSteel and DCI have made their respective bids in the different projects in Manila, Makati, BGC and Quezon City, as seen below.

What is the company's goal in assigning the companies to their respective projects?

What is the first step in this assignment problem?

Conduct column scanning.

Identify the largest value present in the current assignment table.

Conduct row scanning.

Identify the smallest value present in the current assignment table.

After carrying out the first step of the Hungarian Method, what is the value of PowerSteel's bid for the BGC project?

To maximize the profit, which city project must be assigned to JGC?

Quezon City

To maximize the profit, which city project must be assigned to DMCI?

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The assignment problem

The assignment problem.

A. Requires that only one activity be assigned to each resource

B. Is a special case of transportation problem

C. Can be used to maximize resources

D. All of the above

Answer: Option D

Solution(By Examveda Team)

This Question Belongs to Management >> Operations Research

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Related Questions on Operations Research

The use of decision models.

A. Is possible when the variables value is known

B. Reduces the scope of judgement & intuition known with certainty in decision-making

C. Require the use of computer software

D. None of the above

Every mathematical model.

A. Must be deterministic

B. Requires computer aid for its solution

C. Represents data in numerical form

A physical model is example of.

A. An iconic model

B. An analogue model

C. A verbal model

D. A mathematical model

The qualitative approach to decision analysis relies on.

A. Experience

B. Judgement

C. Intuition

More Related Questions on Operations Research

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Related MCQs

• Every basic feasible solution of a general assignment problem having a square pay-off matrix of order n should have assignments equal to___________.
• The similarity between assignment problem and transportation problem is _______.
• Maximization assignment problem is transformed into a minimization problem by_________.
• The assignment problem is a special case of transportation problem in which ______.
• To proceed with the MODI algorithm for solving an assignment problem, the number of dummy allocations need to be added are___________.
• An optimal assignment requires that the maximum number of lines which can be drawn through squares with zero opportunity cost be equal to the number of______.
• In marking assignments, which of the following should be preferred?
• While solving an assignment problem, an activity is assigned to a resource through a square with zero opportunity cost because the objective is to_________.
• An assignment problem can be solved by______.
• The Hungarian method used for finding the solution of the assignment problem is also called ___________.