assignment problem qt

Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey

Get full access to Quantitative Techniques: Theory and Problems and 60K+ other titles, with a free 10-day trial of O'Reilly.

There are also live events, courses curated by job role, and more.

WHAT IS ASSIGNMENT PROBLEM

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.

The assignment problem in the general form can be stated as follows:

“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

Get Quantitative Techniques: Theory and Problems now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.

Don’t leave empty-handed

Get Mark Richards’s Software Architecture Patterns ebook to better understand how to design components—and how they should interact.

It’s yours, free.

Check it out now on O’Reilly

Dive in for free with a 10-day trial of the O’Reilly learning platform—then explore all the other resources our members count on to build skills and solve problems every day.

Linear assignment with non-perfect matching

Dec 8, 2020

The linear assignment problem (or simply assignment problem) is the problem of finding a matching between two sets that minimizes the sum of pair-wise assignment costs. This can be expressed as finding a matching (or independent edge set) in a bipartite graph $G = (U, V, E)$ that minimizes the sum of edge weights. The edge weights may be positive or negative and the bipartite graph does not need to be complete : if there is no edge between two vertices then they cannot be associated. Note that a maximum-weight assignment can be obtained by negating the weights and finding a minimum-weight assignment.

The simplest form of the assignment problem assumes that the bipartite graph is balanced (the two sets of vertices are the same size) and that there exists a perfect matching (in which every vertex has a match). Let $n$ be the number of elements in each set and let $C$ be a square matrix of size $n \times n$ that contains the edge weights. Missing edges are represented by $\infty$, such that $C_{i j} < \infty \Leftrightarrow (i, j) \in E$. The assignment problem can then be clearly expressed as an integer linear program : (note that the problem is not actually solved using a general-purpose ILP solver, it is just a convenient framework in which to express the problem)

The constraint that the sum of each row and column is equal to one ensures that each element has exactly one match. However, what happens when the graph is not balanced or does not contain a perfect matching? We cannot enforce the sums to be equal to one. Which problem should be solved?

I recommend reading the tech report “On Minimum-Cost Assignments in Unbalanced Bipartite Graphs” by Lyle Ramshaw and Robert E. Tarjan. I will summarize some of the main points here.

Let us consider a more general, rectangular problem of size $r \times n$ and assume (without loss of generality) that $r \le n$. If $r = n$ then the problem is balanced, if $r < n$ it is unbalanced.

Clearly an unbalanced probem cannot have a perfect matching, since there will be at least $n - r$ unmatched elements in the larger set. However, it may be possible to find a matching in which every vertex in the smaller set has a match. This is referred to as a one-sided perfect matching and the optimization problem can be expressed:

The inequality constraints enforce that each element in the smaller set has exactly one match while each element in the larger set has at most one match. Ramshaw and Tarjan outline a method to reduce from an unbalanced problem to a balanced problem while preserving sparsity. A simple alternative is to add $n - r$ rows of zeros and then exclude these edges from the eventual solution. Most libraries for the assignment problem solve either the balanced or unbalanced version of this problem (see the later section).

However, whether balanced or unbalanced, it may still occur that the constraint set is infeasible, meaning that there does not exist a (one-sided) perfect matching. Let $\nu(W) \le r$ denote the size of the largest matching in the graph. If $\nu(W) < r$, then there does not exist a one-sided perfect matching and all possible matchings are imperfect.

Ramshaw and Tarjan actually outline three different versions of the assignment problem:

perfect matching
imperfect matching
minimum-weight matching

The imperfect matching problem is to find the matching of size $\nu(G)$ with the minimum cost. The minimum-weight matching problem is to find the matching of any size with the minimum cost. If $\nu(G) = r = n$, then perfect and imperfect matching coincide. Otherwise, when $\nu(G) < n$, there does not exist a perfect matching.

The imperfect matching problem can be expressed

and the minimum-weight matching problem can be expressed

Ramshaw and Tarjan show that both of these problems can be reduced to finding a perfect matching in a balanced graph. When using linear assignment, we should carefully consider which of the three problems we actually want to solve.

In support of minimum-weight matching

The minimum-weight matching problem is often the most natural choice, since it puts no constraint on the size of the matching. To illustrate the difference between this and the other problems, consider the following balanced problem:

The solution to perfect (or imperfect) matching is to choose -1 and -2 for a total score of -3 and a cardinality of 2. The solution to minimum-weight matching is to choose -4 with a cardinality of 1.

Minimum-weight matching will never select an edge with positive cost: it is better to simply leave it unselected. Edges with zero cost have no impact.

It may be more natural to consider the maximization of positive weights than the minimization of negative costs.

Min-cost imperfect matching with positive weights

Be careful when solving imperfect matching problems with positive edge weights! I would avoid this situation altogether due to the tension that exists between maximizing the number of matches and minimizing the sum of (positive) costs. This may result in the unexpected behaviour that adding an edge to the graph increases the minimum cost. For example, compare the following two problems:

Quick and dirty transformations

Ramshaw and Tarjan above describes some clever techniques to transform imperfect and minimum-weight matching problems into perfect matching problems while preserving sparsity. Here we describe some quick and dirty alternatives.

We can always transform an unbalanced (and therefore imperfect) problem into a balanced problem by adding $n - r$ rows of zeros. The resulting balanced graph has a perfect matching if and only if the original unbalanced graph had a matching of size $r$ (in which every vertex in the smaller set is matched).

If we need to solve imperfect matching but we only have a solver for perfect matching, it suffices to replace the infinite edge weights with a large, finite cost (e.g. larger than the total absolute value of all weights). The resulting graph must contain a perfect matching since it is a complete bipartite graph, and each high-cost edge is worth more than all original edges combined. The high-cost edges can be excluded at the end.

Most existing packages either solve perfect, one-sided perfect or imperfect matching. To use one of these solvers for the minimum-weight matching problem, it suffices to replace all positive edges (including infinite edges) with zero. If using a solver that leverages sparsity, it is better to use the technique described by Ramshaw and Tarjan.

Python packages

The table below outlines the different behaviour of several popular packages. The code that was used to determine the behaviour is available as a Jupyter notebook .

Module	Function	Behaviour
		Requires that problem is balanced (or else raises an exception). Requires that a perfect matching exists (or else returns infinite cost).
		Supports unbalanced problems (with ; although check issues , ). Requires that a one-sided perfect matching exists (or else returns infinite cost).
		Supports unbalanced problems. Requires that a one-sided perfect matching exists (or else raises an exception).
		Supports unbalanced problems. Supports imperfect matching.
		Requires problem is balanced. Requires that a perfect matching exists (or else raises an exception). Requires that costs are integer-valued.

Quadratic Assignment Problems

Reference work entry
First Online: 01 January 2013
pp 2741–2814
Cite this reference work entry

Rainer E. burkard 4

8329 Accesses

33 Citations

This chapter introduces quadratic assignment problems (QAP) as models for finding an optimal assignment among two sets of interrelated objects. References to many applications of QAPs, like in location theory, are given. Moreover, several classical combinatorial optimization problems are identified as special cases of the QAP, like the traveling salesman problem, graph partitioning, max clique problem, minimum-weight feedback arc set problem, (FASP) and packing problems in graphs. These subproblems show that the QAP is $\mathcal{N}\mathcal{P}$ -hard.Several different formulations of the QAP are presented in detail as being basic for different solution approaches. Special emphasis is laid to the different possibilities for describing QAPs as (mixed-)integer linear programs. Moreover, QAP polyhedra are discussed.Crucial for exact solution approaches are lower bounding procedures. The paradigm of admissible transformations is introduced for describing several variants of Gilmore–Lawler type bounds. Moreover, bounds based on eigenvalues and bounds found by semidefinite programming are discussed in detail. Among exact solution methods, special emphasis is laid on branch-and-bound approaches. With respect to heuristics, not only simple construction heuristics and improvement heuristics are described, but also several metaheuristics like simulated annealing, tabu search, greedy randomized search, genetic algorithms, and ant systems are discussed. Links to available computer programs are delivered.As QAPs are $\mathcal{N}\mathcal{P}$ -hard problems, there is a special interest in polynomially solvable special cases. A comprehensive survey of results in this direction is given. Moreover, QAPs show an interesting asymptotic behavior, as the ratio of best and worst solution value tends to one in probability. This phenomenon is thoroughly discussed and implications of this strange behavior are shown.Finally, related problems are shortly treated. So, the quadratic bottleneck assignment problem is introduced and lower bounds, efficiently solvable special cases, as well as asymptotic results are reported. Moreover, cubic assignment problems, biquadratic assignment problems, and the quadratic semi-assignment problem are surveyed. An extensive bibliography provides the most important links to the literature in this field.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save.

Get 10 units per month
Download Article/Chapter or eBook
1 Unit = 1 Article or 1 Chapter
Cancel anytime
Available as PDF
Read on any device
Instant download
Own it forever
Available as EPUB and PDF
Durable hardcover edition
Dispatched in 3 to 5 business days
Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

The Quadratic Assignment Problem

A comprehensive review of quadratic assignment problem: variants, hybrids and applications.

Location Problems with Multiple Criteria

Author information

Authors and affiliations.

Institute of Optimization and Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010, Graz, Austria

Rainer E. burkard

You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rainer E. burkard .

Editor information

Editors and affiliations.

Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

Panos M. Pardalos

Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

Ding-Zhu Du

Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

Ronald L. Graham

Rights and permissions

Reprints and permissions

Copyright information

About this entry

Cite this entry.

burkard, R.E. (2013). Quadratic Assignment Problems. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_22

Download citation

DOI : https://doi.org/10.1007/978-1-4419-7997-1_22

Published : 26 July 2013

Publisher Name : Springer, New York, NY

Print ISBN : 978-1-4419-7996-4

Online ISBN : 978-1-4419-7997-1

eBook Packages : Mathematics and Statistics Reference Module Computer Science and Engineering

Share this entry

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Publish with us

Policies and ethics

Find a journal
Track your research

Quadratic assignment problem

Author: Thomas Kueny, Eric Miller, Natasha Rice, Joseph Szczerba, David Wittmann (SysEn 5800 Fall 2020)

1 Introduction
2.1 Koopmans-Beckman Mathematical Formulation
2.2.1 Parameters
2.3.1 Optimization Problem
2.4 Computational Complexity
2.5 Algorithmic Discussions
2.6 Branch and Bound Procedures
2.7 Linearizations
3.1 QAP with 3 Facilities
4.1 Inter-plant Transportation Problem
4.2 The Backboard Wiring Problem
4.3 Hospital Layout
4.4 Exam Scheduling System
5 Conclusion
6 References

Introduction

The Quadratic Assignment Problem (QAP), discovered by Koopmans and Beckmann in 1957 [1] , is a mathematical optimization module created to describe the location of invisible economic activities. An NP-Complete problem, this model can be applied to many other optimization problems outside of the field of economics. It has been used to optimize backboards, inter-plant transportation, hospital transportation, exam scheduling, along with many other applications not described within this page.

Theory, Methodology, and/or Algorithmic Discussions

Koopmans-beckman mathematical formulation.

Economists Koopmans and Beckman began their investigation of the QAP to ascertain the optimal method of locating important economic resources in a given area. The Koopmans-Beckman formulation of the QAP aims to achieve the objective of assigning facilities to locations in order to minimize the overall cost. Below is the Koopmans-Beckman formulation of the QAP as described by neos-guide.org.

Quadratic Assignment Problem Formulation

$F=(F_{ij})$

Inner Product

$A,B$

Note: The true objective cost function only requires summing entries above the diagonal in the matrix comprised of elements

$F_{i,j}(X_{\phi }DX_{\phi }^{T})_{i,j}$

Since this matrix is symmetric with zeroes on the diagonal, dividing by 2 removes the double count of each element to give the correct cost value. See the Numerical Example section for an example of this note.

Optimization Problem

With all of this information, the QAP can be summarized as:

$\min _{X\in P}\langle F,XDX^{T}\rangle$

Computational Complexity

QAP belongs to the classification of problems known as NP-complete, thus being a computationally complex problem. QAP’s NP-completeness was proven by Sahni and Gonzalez in 1976, who states that of all combinatorial optimization problems, QAP is the “hardest of the hard”. [2]

Algorithmic Discussions

While an algorithm that can solve QAP in polynomial time is unlikely to exist, there are three primary methods for acquiring the optimal solution to a QAP problem:

Dynamic Program
Cutting Plane

Branch and Bound Procedures

The third method has been proven to be the most effective in solving QAP, although when n > 15, QAP begins to become virtually unsolvable.

The Branch and Bound method was first proposed by Ailsa Land and Alison Doig in 1960 and is the most commonly used tool for solving NP-hard optimization problems.

A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores branches of this tree, which represent subsets of the solution set. Before one lists all of the candidate solutions of a branch, the branch is checked against upper and lower estimated bounds on the optimal solution, and the branch is eliminated if it cannot produce a better solution than the best one found so far by the algorithm.

Linearizations

The first attempts to solve the QAP eliminated the quadratic term in the objective function of

$min\sum _{i=1}^{n}\sum _{j=1}^{n}c{_{\phi (i)\phi (j)}}+\sum _{i=1}^{n}b{_{\phi (i)}}$

in order to transform the problem into a (mixed) 0-1 linear program. The objective function is usually linearized by introducing new variables and new linear (and binary) constraints. Then existing methods for (mixed) linear integer programming (MILP) can be applied. The very large number of new variables and constraints, however, usually poses an obstacle for efficiently solving the resulting linear integer programs. MILP formulations provide LP relaxations of the problem which can be used to compute lower bounds.

Numerical Example

Qap with 3 facilities.

$D={\begin{bmatrix}0&5&6\\5&0&3.6\\6&3.6&0\end{bmatrix}}$

Cost for Each Permutation in
Permutation	Cost
(123)	91.4
	99.8
	98.4
	86.5
	103.3
	90

$\phi _{4}=(13)$

Applications

Inter-plant transportation problem.

The QAP was first introduced by Koopmans and Beckmann to address how economic decisions could be made to optimize the transportation costs of goods between both manufacturing plants and locations. [1] Factoring in the location of each of the manufacturing plants as well as the volume of goods between locations to maximize revenue is what distinguishes this from other linear programming assignment problems like the Knapsack Problem.

The Backboard Wiring Problem

As the QAP is focused on minimizing the cost of traveling from one location to another, it is an ideal approach to determining the placement of components in many modern electronics. Leon Steinberg proposed a QAP solution to optimize the layout of elements on a blackboard by minimizing the total amount of wiring required. [4]

When defining the problem Steinberg states that we have a set of n elements

$E=\left\{E_{1},E_{2},...,E_{n}\right\}$

as well as a set of r points

$P_{1},P_{2},...,P_{r}$

In his paper he derives the below formula:

$min\sum _{1\leq i\leq j\leq n}^{}C_{ij}(d_{s(i)s(j))})$

In his paper Steinberg a backboard with a 9 by 4 array, allowing for 36 potential positions for the 34 components that needed to be placed on the backboard. For the calculation, he selected a random initial placement of s1 and chose a random family of 25 unconnected sets.

The initial placement of components is shown below:

After the initial placement of elements, it took an additional 35 iterations to get us to our final optimized backboard layout. Leading to a total of 59 iterations and a final wire length of 4,969.440.

Hospital Layout

Building new hospitals was a common event in 1977 when Alealid N Elshafei wrote his paper on "Hospital Layouts as a Quadratic Assignment Problem". [5] With the high initial cost to construct the hospital and to staff it, it is important to ensure that it is operating as efficiently as possible. Elshafei's paper was commissioned to create an optimization formula to locate clinics within a building in such a way that minimizes the total distance that a patient travels within the hospital throughout the year. When doing a study of a major hospital in Cairo he determined that the Outpatient ward was acting as a bottleneck in the hospital and focused his efforts on optimizing the 17 departments there.

Elshafei identified the following QAP to determine where clinics should be placed:

$min\sum _{i,j}\sum _{k,q}f_{ik}d_{jq}y_{ij}y_{kq}$

For the Cairo hospital with 17 clinics, and one receiving and recording room bringing us to a total of 18 facilities. By running the above optimization Elshafei was able to get the total distance per year down to 11,281,887 from a distance of 13,973,298 based on the original hospital layout.

Exam Scheduling System

The scheduling system uses matrices for Exams, Time Slots, and Rooms with the goal of reducing the rate of schedule conflicts. To accomplish this goal, the “examination with the highest cross faculty student is been prioritized in the schedule after which the examination with the highest number of cross-program is considered and finally with the highest number of repeating student, at each stage group with the highest number of student are prioritized.” [6]

$n!$

↑ 1.0 1.1 1.2 Koopmans, T., & Beckmann, M. (1957). Assignment Problems and the Location of Economic Activities. Econometrica, 25(1), 53-76. doi:10.2307/1907742
↑ 2.0 2.1 Quadratic Assignment Problem. (2020). Retrieved December 14, 2020, from https://neos-guide.org/content/quadratic-assignment-problem
↑ 3.0 3.1 3.2 Burkard, R. E., Çela, E., Pardalos, P. M., & Pitsoulis, L. S. (2013). The Quadratic Assignment Problem. https://www.opt.math.tugraz.at/~cela/papers/qap_bericht.pdf .
↑ 4.0 4.1 Leon Steinberg. The Backboard Wiring Problem: A Placement Algorithm. SIAM Review . 1961;3(1):37.
↑ 5.0 5.1 Alwalid N. Elshafei. Hospital Layout as a Quadratic Assignment Problem. Operational Research Quarterly (1970-1977) . 1977;28(1):167. doi:10.2307/300878
↑ 6.0 6.1 Muktar, D., & Ahmad, Z.M. (2014). Examination Scheduling System Based On Quadratic Assignment.

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:

	Task 1	Task 2	Task 3
Emp 1	5	7	6
Emp 2	6	4	5
Emp 3	8	5	3

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:

	Task 1	Task 2	Task 3
Emp 1	0	2	1
Emp 2	2	0	1
Emp 3	5	2	0

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

	Task 1	Task 2	Task 3
Emp 1	0	2	1
Emp 2	2	0	1
Emp 3	5	2	0
	0	0	0

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.

How useful was this post?

Click on a star to rate it!

Average rating 0 / 5. Vote count: 0

No votes so far! Be the first to rate this post.

We are sorry that this post was not useful for you! 😔

Let us improve this post!

Tell us how we can improve this post?

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

Data Structures
Linked List
Binary Tree
Binary Search Tree
Segment Tree
Disjoint Set Union
Fenwick Tree
Red-Black Tree
Advanced Data Structures

Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

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

Please Login to comment...

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

assignment problems in qt

Quantitative techniques: theory and problems by p. c. tulsian, vishal pandey.

Get full access to Quantitative Techniques: Theory and Problems and 60K+ other titles, with a free 10-day trial of O'Reilly.

There are also live events, courses curated by job role, and more.

WHAT IS ASSIGNMENT PROBLEM

Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.

The assignment problem in the general form can be stated as follows:

Several problems of management has a structure identical with the assignment problem.

Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...

Get Quantitative Techniques: Theory and Problems now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.

Don’t leave empty-handed

Get Mark Richards’s Software Architecture Patterns ebook to better understand how to design components—and how they should interact.

It’s yours, free.

Check it out now on O’Reilly

Dive in for free with a 10-day trial of the O’Reilly learning platform—then explore all the other resources our members count on to build skills and solve problems every day.

Saved searches

Use saved searches to filter your results more quickly.

To see all available qualifiers, see our documentation .

Notifications You must be signed in to change notification settings

C++/QT application for solving the assignment problem

Kazutano/NETB507-Assignment-problem

Folders and files.

Name		Name
5 Commits

Repository files navigation

Netb507-assignment-problem.

Analysis of Algorithms
Backtracking
Dynamic Programming
Divide and Conquer
Geometric Algorithms
Mathematical Algorithms
Pattern Searching
Bitwise Algorithms
Branch & Bound
Randomized Algorithms

Quadratic Assignment Problem (QAP)

The Quadratic Assignment Problem (QAP) is an optimization problem that deals with assigning a set of facilities to a set of locations, considering the pairwise distances and flows between them.

The problem is to find the assignment that minimizes the total cost or distance, taking into account both the distances and the flows.

The distance matrix and flow matrix, as well as restrictions to ensure each facility is assigned to exactly one location and each location is assigned to exactly one facility, can be used to formulate the QAP as a quadratic objective function.

The QAP is a well-known example of an NP-hard problem , which means that for larger cases, computing the best solution might be difficult. As a result, many algorithms and heuristics have been created to quickly identify approximations of answers.

There are various types of algorithms for different problem structures, such as:

Precise algorithms
Approximation algorithms
Metaheuristics like genetic algorithms and simulated annealing
Specialized algorithms

Example: Given four facilities (F1, F2, F3, F4) and four locations (L1, L2, L3, L4). We have a cost matrix that represents the pairwise distances or costs between facilities. Additionally, we have a flow matrix that represents the interaction or flow between locations. Find the assignment that minimizes the total cost based on the interactions between facilities and locations. Each facility must be assigned to exactly one location, and each location can only accommodate one facility.

Facilities cost matrix:

	L1	L2	L3	L4
F1	0	2	3	1
F2	2	0	1	4
F3	3	1	0	2
F4	1	4	2	0

Flow matrix:

	F1	F2	F3	F4
L1	0	1	2	3
L2	1	0	4	2
L3	2	4	0	1
L4	3	2	1	0

To solve the QAP, various optimization techniques can be used, such as mathematical programming, heuristics, or metaheuristics. These techniques aim to explore the search space and find the optimal or near-optimal solution.

The solution to the QAP will provide an assignment of facilities to locations that minimizes the overall cost.

The solution generates all possible permutations of the assignment and calculates the total cost for each assignment. The optimal assignment is the one that results in the minimum total cost.

To calculate the total cost, we look at each pair of facilities in (i, j) and their respective locations (location1, location2). We then multiply the cost of assigning facility1 to facility2 (facilities[facility1][facility2]) with the flow from location1 to location2 (locations[location1][location2]). This process is done for all pairs of facilities in the assignment, and the costs are summed up.

Overall, the output tells us that assigning facilities to locations as F1->L1, F3->L2, F2->L3, and F4->L4 results in the minimum total cost of 44. This means that Facility 1 is assigned to Location 1, Facility 3 is assigned to Location 2, Facility 2 is assigned to Location 3, and Facility 4 is assigned to Location 4, yielding the lowest cost based on the given cost and flow matrices.This example demonstrates the process of finding the optimal assignment by considering the costs and flows associated with each facility and location. The objective is to find the assignment that minimizes the total cost, taking into account the interactions between facilities and locations.

Applications of the QAP include facility location, logistics, scheduling, and network architecture, all of which require effective resource allocation and arrangement.

Please Login to comment...

Similar reads, improve your coding skills with practice.

What kind of Experience do you want to share?

All the dots are connected / it's not rocket science.

Transportation and assignment problems with r.

In the previous post “ Linear Programming with R ” we examined the approach to solve general linear programming problems with “Rglpk” and “lpSolve” packages. Today, let’s explore “lpSolve” package in depth with two specific problems of linear programming: transportation and assignment.

1. Transportation problem

Code & Output:

The solution is shown as lptrans$solution and the total cost is 20500 as lptrans$objval.

2. Assignment problem

Similarly, the solution and the total cost are shown as lpassign$solution and lpassign$objval respectively.

This article was really helpful, but I am facing issue while solving unbalanced transportation problem when there is excess demand. Could you please guide me on what has to be done in this case.

Hello sir, this article was really helpful. But, I am facing issue while solving unbalanced transportation problem when there is excess demand, it gives solution as no feasible solution. works perfectly fine for balanced and excess supply problems. Could you please guide me on why this issue is occurring and a possible solution for the same. Thank you.

Popular Courses
More classes
More Courses
Start a discussion
Unanswered
Report Abuse

Doubt in assignment problems- qt

chandrakala akkireddy (article assistant) (38 Points)

In CA Final QT... there are steps to be followed while solving Assignment problems. In Step 3, we have to do row minimisation and then column minimisation. What will be the effect if we do Column minimisation first? Why we should not do in this way?

4 Replies

Maithili (1) (476 Points) --> Replied 27 January 2015

The Answer/Solution u obtain will be the same irrespective of whether u perform Row Operation or Column Operation first

Online classes for CA CS CMA. Professional courses for GST, Tally, Others & Books

sandhya (CA Final) (185 Points) --> Replied 27 January 2015

Do you have any steps for solving the crashing problem in QT..I am not able to understand them from the module.

Amar Kumar (CA Final) (178 Points) --> Replied 27 January 2015

For Assignment Problem, you've to follow the given flowchart

You don't need to go into any further detail.Just follow the steps.

There can be cases when performing column operations before row operations will not impact the solution. My guess is that it'll be when then is a zero in each row & column and so the matrix remains unchanged. So ,My advice is that you by heart the steps and follow it.

Originally posted by :
	Do you have any steps for solving the crashing problem in QT..I am not able to understand them from the module.

Try to understand the concept behind crashing.

In examination, you must study every chapter in AMA. So that you have a choice of not opting Simplex,Crashing and a degenerate Transportation problem.

If you want I can explain the concept of crashing and I can also brief you on how to solve it, but you don't need to overburden yourself just to understand crashing. Finish all other topics in AMA and then tackle these three topics.

For exams, try to avoid Simplex & Crashing unless you've no other choice.

Your are not logged in . Please login to post replies Click here to Login / Register

Related Threads

Popular discussion.

Trending Online Classes

3 Days Certification Course on Tax Audit Under Income Tax Act 1961

Intricacies of Sec 44AD, 44ADA, 44AE while filing ITR

Certification Course on Chat GPT and AI Tools for Professionals

Subscribe to the latest topics :

Search forum:.

Whatsapp Groups

India's largest network for finance professionals

Alternatively, you can log in using:

Troubleshooting Qt Creator project issues

This page contains information on how to resolve common problems with Qt Creator projects. Consider searching this page using Ctrl+F (Command+F on Mac) to jump quickly to the text matching the error message or issue you are having. You may also want to check the [Guide to Common Build/Run/Debug problems][commonissues] if for issues compiling or running your program.

How to "re-initialize" a Qt Creator project

A large number of Qt Creator ills can be resolved by following these steps to re-initialize your project. (Please note that this requires deleting some files from your project. Be careful not to delete your assignment solution code. Make frequent backups of your files.)

Quit Qt Creator.
Navigate to the folder where your project is stored. Delete the file with the extension . pro . user , such as Life . pro . user Take care : Delete only the file with exactly the pro . user extension , not the other files such as the Life . pro ( screenshot )
Also delete the "build" folder for your project. This is located in the parent directory of your project's directory, and has a long name like build - Life - Desktop_Qt_5_x_x_kit_xxbit - Debug . Delete the entire build directory. ( screenshot )
Re-start Qt Creator. Choose menu item "File" -> "Open File or Project…", navigate to your project folder and open its . pro file. Qt should ask you to "Configure Project", just as if you were opening for the first time.
Now try to build and run to see if things work any better.

Installation woes

Opening a project, permission problems.

Another thing that can cause the "permission denied" problem is anti-virus software. In particular, McAfee Anti-Virus places strong locks on all . exe files that prevents you from modifying or deleting them. You may need to disable or uninstall McAfee to make this behavior go away.

Pardon Our Interruption

As you were browsing something about your browser made us think you were a bot. There are a few reasons this might happen:

You've disabled JavaScript in your web browser.
You're a power user moving through this website with super-human speed.
You've disabled cookies in your web browser.
A third-party browser plugin, such as Ghostery or NoScript, is preventing JavaScript from running. Additional information is available in this support article .

To regain access, please make sure that cookies and JavaScript are enabled before reloading the page.

SI SWIMSUIT
SI SPORTSBOOK

Exciting Seattle Mariners' Reliever Gets Timeline For When He'll Start Rehab Assignment

Brady farkas | jun 29, 2024.

Chicago White Sox relief pitcher Gregory Santos (60) celebrates a win against the Chicago Cubs at Wrigley Field in 2023.

Seattle Mariners

Seattle Mariners ' relief pitcher Gregory Santos took another major step forward in his recovery on Friday afternoon and it's led to some exciting news.

After throwing a 21-pitch live session at T-Mobile Park, we now know when Santos will begin his rehab assignment. He's been out all year with a lat issue.

Per @LookoutLanding on social media:

Justin Hollander says Gregory Santos will start a rehab assignment with Tacoma July 2nd

Justin Hollander says Gregory Santos will start a rehab assignment with Tacoma July 2nd — Lookout Landing (@LookoutLanding) June 28, 2024

Considering that Santos hasn't played in any games at all this year (he was injured in spring training), it's surprising to see him instantly come out of the gate at Triple-A, but Single-A Everett is on the road that day, so maybe this is just more about reducing travel than level of competition. Tacoma is home that day against Salt Lake City.

Given that Santos has been out all year, he'll require a lengthy rehab assignment that will have many more boxes to check. He'll need to show that he can throw all of his pitches, work on back-to-back days and show the ability to recover without discomfort. There has been hope that he'd be able to join the M's in July and that looks possible still at this point if everything goes well.

If and when Santos is able to join the team, he'll add another dimension to the back end of the bullpen for manager Scott Servais. He features an upper-90s fastball and has a ton of movement.

Here's what Ryan Bliss had to say after facing Bliss on Friday, per Curtis Crabtree:

Ryan Bliss faced Santos both in Tampa and here today and spoke highly of his stuff, especially his sinker. “It’s 98 with splitter movement. You just don’t see that. The ball drops out of nowhere. You don’t really see it. It’s something unique and it’s a really good pitch.”

Ryan Bliss faced Santos both in Tampa and here today and spoke highly of his stuff, especially his sinker. “It’s 98 with splitter movement. You just don’t see that. The ball drops out of nowhere. You don’t really see it. It’s something unique and it’s a really good pitch.” https://t.co/YY9OKLFh2V — Curtis Crabtree (@Curtis_Crabtree) June 28, 2024

The Mariners will play the Twins again on Saturday night at 7:10 p.m. PT.

Follow Inside the Mariners on social media

Continue to follow our Inside the Mariners coverage on social media by liking us on Facebook and by following Brady on "X" @ wdevradiobrady

Today’s Teenagers Have Invented a Language That Captures the World Perfectly

An illustration of a man with an open book and a pencil, sweating as a teenager stands behind him using a pointer stick to point to the word “cringe,” written on a large paper pad on the wall. They are surrounded by stacks of books.

By Stephen Marche

Mr. Marche is the author, most recently, of “The Next Civil War.”

My son just completed high school and when he leaves for college in the fall my life will change in ways I’m still struggling to contemplate. Among the things I’ll miss most are his lessons in teenage slang. My son has always been generous with me, and I’ve found the slang of his generation to be so much better and more useful than any that I’ve ever used. His slang has also offered me an accidental and useful portrait of how he and his generation see the world.

The primary value of slang has been to create linguistic shibboleths, a way to differentiate yourself quickly from other people. Sometimes the distinction was generational, sometimes it was racial, and sometimes it was ideological, but the slang itself was ultimately a form of social etiquette. From one generation to the next, the terms changed, but the meanings typically didn’t. New words were routinely adopted to express familiar concepts: one generation’s “cool” becomes another’s “dope” and so on.

Members of my son’s generation have a vastly superior approach to slang. They’ve devised a language that responds to the new and distinct reality they face.

Anyone with children, especially ones on the cusp of adulthood, has to reckon with the shameful fact that the world we’re leaving them is so much worse than the one we brought them into. My son’s slang reflects that: It’s a distinct language created for a society that’s characterized, online and off, by collapsing institutions, erosions in trust and a loss of faith in a shared sense of meaning.

“Mid” is an obvious example. I don’t think it even qualifies as teenage slang anymore — it’s too useful and, by now, too widespread. In my son’s usage, things that are mid are things that are essentially average or slightly below. You can’t really complain about them, but they produce no joy. They’re often the result of the refinement of market research to the exact level that tepid consumer acceptance is achieved. Everything in Starbucks falls into the category of “mid.” So does everything in an airport. It’s a brilliant, precise word for a world full of mild disappointments, where the corner bakery that used to do some things well and other things poorly has been reliably replaced by yet another Le Pain Quotidien.

“Glazed” has a similarly impressive precision. When my son describes something as glazed, it’s meant to signify not lying, exactly, or even exaggerating, but the act of positively spinning a judgment. “Glazed” indicates a gilding of information; sports commentary, for example, is 90 percent glaze. When Stephen A. Smith, the quintessential glazer, likens Anthony Edwards to Michael Jordan , a proper response might be “The Ant glazing is crazy.” But glaze is also the perfect description of the way social media works: The world you encounter online is perpetually glazed, with everything taking on an artificially positive, unreal and not entirely trustworthy gloss.

We are having trouble retrieving the article content.

Please enable JavaScript in your browser settings.

Thank you for your patience while we verify access. If you are in Reader mode please exit and log into your Times account, or subscribe for all of The Times.

Thank you for your patience while we verify access.

Already a subscriber? Log in .

Want all of The Times? Subscribe .

the intact one

Read MBA, BBA, B.COM Notes

Restriction in Assignment

It is sometimes possible that a particular person is incapable of doing certain work or a specific job cannot be performed on a particular machine. The solution of the assignment problem should take into account these restrictions so that the restricted (infeasible) assignment can be avoided. This can be achieved by assigning a very high cost (say ∞ or M)to the cells where assignments are prohibited, thereby restricting the entry of this pair of job-machine or resource-activity into the final solution.

Example : Five jobs are to be assigned to five men. The cost (in Rs.) of performing the jobs by each man is given in the matrix. The assignment has restrictions that Job 4 cannot be performed by Man 1 and Job 3 cannot be performed by Man 4 Find the optimal assignment of job and its cost involved.

Assignment Problem

Solution: Assign large value to the restricted combinations or introduce ‘M’, see table.

Large Value Assignment to Restricted Combinations

Reducing the matrix row-wise

Reducing the matrix column-wise

Draw minimum number of lines to cover all zeros, see Table.

All Zeros Covered

Now, number of lines drawn = Order of matrix, hence optimality is reached. Allocating Jobs to Men.

Job Allocation to Men

Assignment Schedule and Cost

As per the restriction conditions given in the problem, Man 1 and Man 4 are not assigned to Job 4 and Job 3 respectively.

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:

ADVERTISEMENTS:

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:

Code & Output:

The solution is shown as lptrans$solution and the total cost is 20500 as lptrans$objval.

2. Assignment problem

Similarly, the solution and the total cost are shown as lpassign$solution and lpassign$objval respectively.

This article was really helpful, but I am facing issue while solving unbalanced transportation problem when there is excess demand. Could you please guide me on what has to be done in this case.

the intact one

Read MBA, BBA, B.COM Notes

Assignment Problem: Hungarian Assignment Methods

Assignment problem is a special type of linear programming problem which deals with the allocation of the various resources to the various activities on one to one basis. It does it in such a way that the cost or time involved in the process is minimum and profit or sale is maximum. Though there problems can be solved by simplex method or by transportation method but assignment model gives a simpler approach for these problems.

In a factory, a supervisor may have six workers available and six jobs to fire. He will have to take decision regarding which job should be given to which worker. Problem forms one to one basis. This is an assignment problem.

ASSIGNMENT MODEL

Suppose there are n facilitates and n jobs it is clear that in this case, there will be n assignments. Each facility or say worker can perform each job, one at a time. But there should be certain procedure by which assignment should be made so that the profit is maximized or the cost or time is minimized.

In the table, Co ij is defined as the cost when j th job is assigned to i th worker. It maybe noted here that this is a special case of transportation problem when the number of rows is equal to number of columns.

Mathematical Formulation

Any basic feasible solution of an Assignment problem consists (2n – 1) variables of which the (n – 1) variables are zero, n is number of jobs or number of facilities. Due to this high degeneracy, if we solve the problem by usual transportation method, it will be a complex and time consuming work. Thus a separate technique is derived for it. Before going to the absolute method it is very important to formulate the problem.

Suppose x jj is a variable which is defined as

1 if the i th job is assigned to j th machine or facility

0 if the i th job is not assigned to j th machine or facility.

Now as the problem forms one to one basis or one job is to be assigned to one facility or machine.

The total assignment cost will be given by

The above definition can be developed into mathematical model as follows:

Determine x ij > 0 (i, j = 1,2, 3…n) in order to

Subjected to constraints

and x ij is either zero or one.

Method to solve Problem (Hungarian Technique)

Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem,

Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row. So, we will be getting at least one zero in each row of this new table.
Having constructed the table (as by step-1) take the columns of the table. Starting from first column locate the smallest cost element in each column. Now subtract this smallest element from each element of that column. Having performed the step 1 and step 2, we will be getting at least one zero in each column in the reduced cost table.
Now, the assignments are made for the reduced table in following manner.

(i) Rows are examined successively, until the row with exactly single (one) zero is found. Assignment is made to this single zero by putting square around it and in the corresponding column, all other zeros are crossed out (x) because these will not be used to make any other assignment in this column. Step is conducted for each row.

(ii) Step 3 (i) in now performed on the columns as follow:- columns are examined successively till a column with exactly one zero is found. Now , assignment is made to this single zero by putting the square around it and at the same time, all other zeros in the corresponding rows are crossed out (x) step is conducted for each column.

(iii) Step 3, (i) and 3 (ii) are repeated till all the zeros are either marked or crossed out. Now, if the number of marked zeros or the assignments made are equal to number of rows or columns, optimum solution has been achieved. There will be exactly single assignment in each or columns without any assignment. In this case, we will go to step 4.

At this stage, draw the minimum number of lines (horizontal and vertical) necessary to cover all zeros in the matrix obtained in step 3, Following procedure is adopted:

(i) Tick mark () all rows that do not have any assignment.

(ii) Now tick mark() all these columns that have zero in the tick marked rows.

(iii) Now tick mark all the rows that are not already marked and that have assignment in the marked columns.

(iv) All the steps i.e. (4(i), 4(ii), 4(iii) are repeated until no more rows or columns can be marked.

(v) Now draw straight lines which pass through all the un marked rows and marked columns. It can also be noticed that in an n x n matrix, always less than ‘n’ lines will cover all the zeros if there is no solution among them.

In step 4, if the number of lines drawn are equal to n or the number of rows, then it is the optimum solution if not, then go to step 6.
Select the smallest element among all the uncovered elements. Now, this element is subtracted from all the uncovered elements and added to the element which lies at the intersection of two lines. This is the matrix for fresh assignments.
Repeat the procedure from step (3) until the number of assignments becomes equal to the number of rows or number of columns.

Harris' border work was on 'root causes' of migration; she wasn't in charge | Fact check

The claim: Kamala Harris was 'put in charge of the border'

A July 21 Instagram post ( direct link , archive link ) by Donald Trump Jr. blames Vice President Kamala Harris for the country's immigration problems.

"She was put in charge of the border and we saw the worst invasion of illegals in our history!!!" reads part of the post, which is a screenshot of a post from X, formerly Twitter.

Similar posts on Threads have described Harris as the Biden administration's "border czar."

The Instagram post was liked more than 200,000 times in a day.

More from the Fact-Check Team: How we pick and research claims | Email newsletter | Facebook page

Our rating: False

The post exaggerates the vice president's role in addressing migration at the southern border. Harris was never put in charge of the border or made "border czar," immigration experts said. President Joe Biden tasked Harris with leading the administration's diplomatic efforts addressing the "root causes" of migration in El Salvador, Guatemala and Honduras.

Harris led effort addressing 'root causes' of migration in Central America

Early in his presidency, Biden tasked Harris with addressing the “root causes” of migration in Central America. The assignment came out of an executive order Biden issued in February 2021 that sought to reduce migration from the Northern Triangle countries of El Salvador, Guatemala and Honduras, where gang violence, trafficking networks and economic insecurity have caused people to flee.

But the vice president’s role was more limited than being put in charge of the southern border, or being named a so-called “border czar,” immigration experts said.

"VP Harris was never made the border czar or charged with managing the border," Andrew Salee , president of the Migration Policy Institute , said in an email. "That role has always been held by the secretary of Homeland Security . She was asked to be the chief diplomatic officer with Central American countries at a time when most of the increase in unauthorized immigration was coming from three countries in Central America and to help lead a private investment strategy in the region."

Homeland Security Secretary Alejandro Mayorkas himself noted the different responsibilities between himself and Harris in June 2021 comments at the El Paso, Texas, border.

"The vice president is leading our nation’s efforts to address the root causes – that fundamental question of why people leave their homes," Mayorkas said. "And it is my responsibility as the secretary of Homeland Security to address the security and management of our border."

In March 2021, Biden announced Harris would lead the administration's diplomatic efforts with the Northern Triangle countries to stem migration to the U.S. southern border and work with these nations to enhance migration enforcement at their borders. Harris said at the time that the administration "must address the root causes that – that cause people to make the trek, as the president has described, to come here."

Aaron Reichlin-Melnick , policy director at the American Immigration Council , said the "root causes" work Harris took on is distinct from border policy because it focuses on different problems and targets.

"Border policy focuses on individuals who have already made the decision to leave home and have made it to the U.S.-Mexico border and aims to either prevent them or to quickly process them for humanitarian relief or deportation once they cross," Reichlin-Melnick said in an email. "By contrast, 'root causes' policy focuses on individuals who have not left their homes yet, and aims to convince them to stay in their home countries either through economic development – which discourages migration for economic opportunities – or through reduction of violence and persecution that forces people to seek protection elsewhere."

The White House released the administration's " Root Causes Strategy " in July 2021. Its implementation was ongoing as of March when the vice president and the Partnership for Central America , a non-governmental organization, jointly announced $1 billion in new private-sector commitments to address the underlying conditions leading to migration in Guatemala, El Salvador and Honduras. The public-private partnership has generated more than $5.2 billion since May 2021 , the White House said.

Fact check : Joe Biden dropped out of presidential race but is finishing term

Elina Treyger , a senior political scientist at the RAND Corporation whose research includes migration and immigration enforcement, also said Harris' diplomatic role with the Central American countries "is in no way a 'border czar'-like position." Treyger said border policy involves many other issues such as enforcement policies, how to process migrants expressing fear of prosecution or torture and how to allocate resources at the border.

U.S. Border Patrol encounters with migrants at the southern border have soared under the Biden administration . Illegal crossings at the U.S.-Mexico border hit a record high of 2.2 million in 2022, and the number of people taken into custody by U.S. Border Patrol has reached the highest levels in the agency's history under Biden, the Washington Post reported .

After a bipartisan border security bill failed to advance in Congress, Biden issued a directive in June to turn away migrants who do not enter the country through legal ports of entry when the number of crossings is high.

Trump, the son of former President Donald Trump, did not immediately respond to a request for comment.

Our fact-check sources:

Aaron Reichlin-Melnick , July 22, Email exchange with USA TODAY
Andrew Salee , July 22, Email exchange with USA TODAY
Elina Treyger , July 22, Email Exchange with USA TODAY
White House, Feb. 2, 2021, Executive Order on Creating a Comprehensive Regional Framework to Address the Causes of Migration, to Manage Migration Throughout North and Central America, and to Provide Safe and Orderly Processing of Asylum Seekers at the United States Border
White House, Feb. 6, 2023, FACT SHEET: Vice President Harris Announces Public-Private Partnership Has Generated More than $4.2 Billion in Private Sector Commitments for Northern Central America
White House, March 24, 2021, Remarks by President Biden and Vice President Harris in a Meeting on Immigration
White House, June 25, 2021, Remarks by Vice President Harris, Secretary of Homeland Security Mayorkas, Chairman Durbin, and Representative Escobar in Press Gaggle
White House, July 29, 2021, FACT SHEET: Strategy to Address the Root Causes of Migration in Central America
White House, March 25, FACT SHEET: Vice President Harris Announces Public-Private Partnership Has Generated More Than $5.2 Billion in Private Sector Commitments for Northern Central America
White House, July 2021, U.S. Strategy for Addressing the Root Causes of Migration in Central America
Department of State, Aug. 1, 2023, Central America Forward
The Washington Post, Feb. 11, Trump vs. Biden on immigration: 12 charts comparing U.S. border security
U.S. Embassy in Honduras, March 25, FACT SHEET: UPDATE ON THE U.S. STRATEGY FOR ADDRESSING THE ROOT CAUSES OF MIGRATION IN CENTRAL AMERICA
USA TODAY, July 17, Border security takes center stage at RNC. Here's the actual data under Trump, Biden

Thank you for supporting our journalism. You can subscribe to our print edition, ad-free app or e-newspaper here .

USA TODAY is a verified signatory of the International Fact-Checking Network, which requires a demonstrated commitment to nonpartisanship, fairness and transparency. Our fact-check work is supported in part by a grant from Meta .

Trump zeroes in on 'border czar Harris' attack as her campaign pushes back

WASHINGTON — Donald Trump is going on offense by painting new presidential campaign rival Kamala Harris as the face of a chaotic U.S. border , seizing on an assignment President Joe Biden gave her in 2021 to work with Central American countries to tackle the “root causes” of migration.

Trump labeled the vice president the “border czar” no fewer than six times in a fiery rally speech Wednesday in North Carolina, centering his criticism of her on the overwhelmed asylum system. “Under border czar Harris, illegal aliens are pouring in by the millions and millions and millions,” Trump said, drawing jeers from the crowd as he called Harris “crazy.”

The term, which Republicans has used widely to criticize Harris, traces back to March 2021, when Harris was charged with addressing the surge of Central American migrants, who came mostly from the Northern Triangle countries of El Salvador, Guatemala and Honduras, where violence and organized crime have driven millions to flee the region. The terms “czar” and “border czar” did not appear in White House materials, but they caught on among critics.

Days later, Texas Gov. Gregg Abbott, a Republican, made one of the first prominent references to it in a letter to the White House asking Harris “to visit the border and see the crisis for yourself.”

“Now that President Biden has named you Border Czar in charge of the administration’s response, I want to express to you the threats and challenges caused by this administration’s open border policies,” he wrote in a letter.

Harris' assignment was misunderstood: It was a diplomatic task to devise a regional strategy to mitigate the need for migration, not a security task to oversee domestic border enforcement.

The White House immediately sought to clarify that Harris’ mandate was not “the border” and that it was narrowly focused on the forces driving migrants out of the Northern Triangle. But with a crisis unfolding that was fueled by those migrants, the title stuck.

Harris runs on 'freedom,' says little about migration

More than three years later, Harris is the de facto Democratic presidential nominee after Biden’s late decision to withdraw from the race, and her still-nascent campaign has not focused on immigration or border security. And her public work on addressing the root causes of migration has largely evaporated, an NBC News analysis showed, though the White House said she has helped make key investments in Central America and continues to lead on the issue.

Harris has said little about immigration in recent speeches. Her launch video Thursday presents her as the candidate of “freedom,” highlighting issues like reproductive rights, health care access and gun violence — without mentioning immigration.

Migration patterns have changed, too. The recent surge of border crossings has been driven by immigrants from Venezuela, Colombia, Ecuador and other countries outside the Northern Triangle.

Still, the issue of border security presents political pitfalls for Harris and the Democratic Party, as it motivates conservatives. Surveys have found voters trust the GOP more to handle it. Biden and Democrats in Congress recognized th at , crafting a bipartisan border bill with Republicans to address immigration at the southern border, but Trump and his allies in Congress killed it .

Trump campaign senior adviser Danielle Alvarez accused the Harris campaign of “scrambling to rewrite history on her failures.”

“Border Czar Harris owns the bloodbath at the southern border, including the rape, murder, and brutal assault of women like Rachel Morin and Laken Riley . Try as they might, Kamala and her allies can’t change reality: she is responsible for the flood of migrant crime and deadly fentanyl into our country, and Americans will hold her accountable when they vote for President Trump in November,” Alvarez said in a statement.

A memo Monday from the GOP Senate campaign arm provides talking points to candidates to go after Harris, saying: “Kamala Harris is Joe Biden’s border czar and the architect of his biggest failure.”

Harris campaign goes after Trump on 'ripping mothers from their children'

Harris campaign spokesperson Kevin Munoz noted that Trump torpedoed the Senate border security deal the Biden-Harris White House backed this year. Trump successfully pushed Republicans to vote down the compromise they reached with Democrats as he campaigns on the issue.

“The only ‘plan’ Donald Trump has to secure our border is ripping mothers from their children and a few xenophobic placards at the Republican National Convention. He tanked the bipartisan border security deal because, for Donald Trump, this has never been about solutions just running on a problem,” Munoz said in a statement. “Like everything with Donald Trump, it’s never been about helping the country, it’s only about helping himself. There’s only one candidate in this race who will fight for bipartisan solutions to strengthen border security, and that’s Vice President Harris.”

Rep. Veronica Escobar, D-Texas, scoffed at the GOP attacks.

“That was never a term designated to her. And the goal of her work was not necessarily the border or border policy but looking at root causes, which she successfully did,” she said. “And I think it’s always important to point out Republican obstruction at every turn when it came to immigration or border policy.”

Since the bill failed in the Senate, Biden has taken executive actions to crack down on asylum-seeking. Last month, unlawful border crossings fell to the lowest monthly number of his presidency, lower than they were in some months under Trump in 2019. Biden highlighted that in his speech to the country Wednesday, saying “border crossings are lower today” than they were in the Trump era.

Sahil Kapur is a senior national political reporter for NBC News.

Jane C. Timm is a senior reporter for NBC News.

COMMENTS

Assignment problem
The formal definition of the assignment problem (or linear assignment problem) is . Given two sets, A and T, together with a weight function C : A × T → R.Find a bijection f : A → T such that the cost function: (, ())is minimized. Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as: , The problem is "linear" because the cost ...
What is Assignment Problem
Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons. The assignment problem in the general form can be stated as follows: "Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to ...
A comprehensive review of quadratic assignment problem: variants
The quadratic assignment problem (QAP) has considered one of the most significant combinatorial optimization problems due to its variant and significant applications in real life such as scheduling, production, computer manufacture, chemistry, facility location, communication, and other fields. QAP is NP-hard problem that is impossible to be solved in polynomial time when the problem size ...
Linear assignment with non-perfect matching
Linear assignment with non-perfect matching. Dec 8, 2020. The linear assignment problem (or simply assignment problem) is the problem of finding a matching between two sets that minimizes the sum of pair-wise assignment costs. This can be expressed as finding a matching (or independent edge set) in a bipartite graph $G = (U, V, E)$ that minimizes the sum of edge weights.
Quadratic Assignment Problem (QAP)
The Quadratic Assignment Problem (QAP) is an optimization problem that deals with assigning a set of facilities to a set of locations, considering the pairwise distances and flows between them.. The problem is to find the assignment that minimizes the total cost or distance, taking into account both the distances and the flows.
Quadratic Assignment Problems
The quadratic assignment problem (QAP) was introduced by Koopmans and Beckmann in 1957 as a mathematical model for the location of indivisible economical activities [].Consider the problem of allocating n facilities to n locations, with the cost being a function of the distance and flow between the facilities plus costs associated with placing a facility at a certain location.
Quadratic assignment problem
The Quadratic Assignment Problem (QAP), discovered by Koopmans and Beckmann in 1957, is a mathematical optimization module created to describe the location of invisible economic activities. An NP-Complete problem, this model can be applied to many other optimization problems outside of the field of economics. It has been used to optimize ...
PDF the Quadratic Assignment Problem
The QAP problem was introduced by Koopmans and Beckmann in 1957 as a mathematical model for the assignment of a set of economic activities to a set of locations, with taking into account the distance and flow between the facilities and the costs of placing a facility in a specific location.
How to Solve the Assignment Problem: A Complete Guide
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.
Assignment Problems
This book provides a comprehensive treatment of assignment problems from their conceptual beginnings in the 1920s through present-day theoretical, algorithmic, and practical developments. The revised reprint provides details on a recent discovery related to one of Jacobi's results, new material on inverse assignment problems and quadratic ...
Unbalanced Assignment Problems
10 Feb 2019. Whenever the cost matrix of an assignment problem is not a square matrix, that is, whenever the number of sources is not equal to the number of destinations, the assignment problem is called an unbalanced assignment problem. In such problems, dummy rows (or columns) are added in the matrix so as to complete it to form a square matrix.
Job Assignment Problem using Branch And Bound
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.
Assignment Problem- drawing minimum nnumber of lines
Another reference Problem 👇https://youtu.be/Qr408_i3lFs
Hungarian Algorithm for Assignment Problem
The problem is to find the assignment that minimizes the total cost or distance, taking into account both the distances and the flows. The distance matrix a. 11 min read. Job Assignment 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 ...
PDF Transportation and Assignment Problems
tation and Assignment ProblemsThe transportation model is a special class of linear programs. It received t. is name because many of its applications involve determining how to optimally transport goods. However, some of its imp. ant applications (eg production scheduling) actua. ly have nothing to do with transportation. The second typ.
assignment problems in qt
the intact one. Read MBA, BBA, B.COM Notes. Unbalanced Assignment Problems. Whenever the cost matrix of an assignment problem is not a square matrix, that is, whenever the number
Assignment Problem: Meaning, Methods and Variations
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 ...
Transportation and Assignment problems with R
In the previous post "Linear Programming with R" we examined the approach to solve general linear programming problems with "Rglpk" and "lpSolve" packages. Today, let's explore "lpSolve" package in depth with two specific problems of linear programming: transportation and assignment. 1. Transportation problem
PDF Assignment 1: CS106L GraphViz
Assignment 1: CS106L GraphViz _____ Due Tuesday, October 14, 11:59 PM Introduction One of the most useful abstractions in computer science is the ... this problem is difficult. The position of each node influences the position of each other node, so picking a layout that balances the heuristics requires solving a large, complicated system of ...
Assignment Problem
#assingmentproblem #minimization #balancedassingmentproblem
Assignment Problem: Hungarian Assignment Methods
Method to solve Problem (Hungarian Technique) Consider the objective function of minimization type. Following steps are involved in solving this Assignment problem, Locate the smallest cost element in each row of the given cost table starting with the first row. Now, this smallest element is subtracted form each element of that row.
MCQ QT
An assignment problem can be solved by A. Simplex method B. Transportation method C. Both a & b D. None of the above. For a salesman who has to visit n cities which of the following are the ways of his tour plan A. n! B. (n+1)! C. (n-1)! D. n. The assignment problem A. Requires that only one activity be assigned to each resource B.
Kamala Harris owns the mess at the border, even if Democrats deny it
She didn't oversee the border. Harris was never the 'border czar,' media claims. To put it more bluntly, they claim that she didn't oversee the disaster that is the Biden-Harris border policy.
Potential Harris Running Mate Makes Trump The Butt Of Zinger ...
Then came the line that mocked both GOP nominee Trump and his running mate with one line: "The problem with JD Vance is he has no conviction, but I guess his running mate has 34."
No, Kamala Harris wasn't put in charge of the U.S. border
The assignment came out of an executive order Biden issued in February 2021 that sought to reduce migration from the Northern Triangle countries of El Salvador, Guatemala and Honduras, where gang ...
Trump zeroes in on 'border czar Harris' attack as her campaign pushes back
Donald Trump is aggressively going after Kamala Harris on an issue at the center of his campaign, seizing on a misunderstood assignment from 2021 that gave rise to the term. IE 11 is not supported.
Harris' immigration work comes under scrutiny as campaign takes ...
As the vice president's campaign takes shape and as immigration remains a top issue for voters, her team is forced to contend with an assignment that, sources say, has showed early success in ...

Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey

WHAT IS ASSIGNMENT PROBLEM

Don’t leave empty-handed

It’s yours, free.

Check it out now on O’Reilly

Linear assignment with non-perfect matching

In support of minimum-weight matching

Min-cost imperfect matching with positive weights

Quick and dirty transformations

Python packages

Quadratic Assignment Problems

Access this chapter

Similar content being viewed by others

The Quadratic Assignment Problem

Location Problems with Multiple Criteria

Author information

Corresponding author

Editor information

Rights and permissions

Copyright information

About this entry

Download citation

Share this entry

Quadratic assignment problem

Introduction

Theory, Methodology, and/or Algorithmic Discussions

Quadratic Assignment Problem Formulation

Inner Product

Optimization Problem

Computational Complexity

Algorithmic Discussions

Branch and Bound Procedures

Linearizations

Numerical Example

Applications

The Backboard Wiring Problem

Hospital Layout

Exam Scheduling System

Navigation menu

How to Solve the Assignment Problem: A Complete Guide

Understanding the Assignment Problem

Solving the Assignment Problem

Step 1: Set up the cost matrix

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

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

Step 4: Test for optimality and adjust the matrix

Step 5: Assign the tasks to the agents

Solution of the Assignment Problem using the Hungarian Method

Applications of the Assignment Problem

Applications in Computer Science

Applications in Economics

Applications in Logistics

Applications in Management

Operations Research

Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)

Please Login to comment...

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

assignment problems in qt

WHAT IS ASSIGNMENT PROBLEM

Don’t leave empty-handed

It’s yours, free.

Check it out now on O’Reilly

Navigation Menu

Saved searches

Kazutano/NETB507-Assignment-problem

Repository files navigation

Quadratic Assignment Problem (QAP)

Please Login to comment...

What kind of Experience do you want to share?

1. Transportation problem

2. Assignment problem

Doubt in assignment problems- qt

4 Replies

Online classes for CA CS CMA. Professional courses for GST, Tally, Others & Books

Leave a reply

Recent Topics

Related Threads

Trending Online Classes

Subscribe to the latest topics :