write the mathematical formulation of an assignment problem

Operations Research - Definition and formulation of Assignment Problem | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research, definition and formulation of assignment problem.

Definition and formulation

Consider the problem of assigning n jobs to n machines (one job to one machine). Let C ij be the cost of assigning i th job to the j th machine and x ij represents the assignment of i th job to the j th machine.

 x ij is missing in any cell means that no assignment is made between the pair of job and machine.( i.e ) x ij = 0.

x ij presents in any cell means that an assignment is made their.In such cases x ij = 1

The assignment model can be written in LPP as follows

Subject to the constrains

The optimum assignment schedule remains unaltered if we add or subtract a constant from all the elements of the row or column of the assignment cost matrix.

If for an assignment problem all C ij > 0 then an assignment schedule (x ij ) which satisfies ∑ C ij x ij   = 0 must be optimal.

Related Topics

Privacy Policy , Terms and Conditions , DMCA Policy and Compliant

Copyright © 2018-2024 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.

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:

• In the above asignment problem, we have to provide assignments such that there is one to one assignments and the overall cost is minimized.

How Assignment Problem is related to LPP? OR Write mathematical formulation of Assignment Model.

→ Assignment Model is a special application of Linear Programming (LP).

→ The mathematical formulation for Assignment Model is given below:

→ Let, C i j \text {C}_{ij} C ij ​ denotes the cost of resources 'i' to the task 'j' ; such that

→ Now assignment problems are of the Minimization type. So, our objective function is to minimize the overall cost.

→ Subjected to constraint;

(i) For all j t h j^{th} j t h task, only one i t h i^{th} i t h resource is possible:

(ii) For all i t h i^{th} i t h resource, there is only one j t h j^{th} j t h task possible;

(iii) x i j x_{ij} x ij ​ is '0' or '1'.

Types of Assignment Problem:

(i) balanced assignment problem.

• It consist of a suqare matrix (n x n).
• Number of rows = Number of columns

(ii) Unbalanced Assignment Problem

• It consist of a Non-square matrix.
• Number of rows ≠ \not=  = Number of columns

Methods to solve Assignment Model:

(i) integer programming method:.

In assignment problem, either allocation is done to the cell or not.

So this can be formulated using 0 or 1 integer.

While using this method, we will have n x n decision varables, and n+n equalities.

So even for 4 x 4 matrix problem, it will have 16 decision variables and 8 equalities.

So this method becomes very lengthy and difficult to solve.

(ii) Transportation Methods:

As assignment problem is a special case of transportation problem, it can also be solved using transportation methods.

In transportation methods ( NWCM , LCM & VAM), the total number of allocations will be (m+n-1) and the solution is known as non-degenerated. (For eg: for 3 x 3 matrix, there will be 3+3-1 = 5 allocations)

But, here in assignment problems, the matrix is a square matrix (m=n).

So total allocations should be (n+n-1), i.e. for 3 x 3 matrix, it should be (3+3-1) = 5

But, we know that in 3 x 3 assignment problem, maximum possible possible assignments are 3 only.

So, if are we will use transportation methods, then the solution will be degenerated as it does not satisfy the condition of (m+n-1) allocations.

So, the method becomes lengthy and time consuming.

(iii) Enumeration Method:

It is a simple trail and error type method.

Consider a 3 x 3 assignment problem. Here the assignments are done randomly and the total cost is found out.

For 3 x 3 matrix, the total possible trails are 3! So total 3! = 3 x 2 x 1 = 6 trails are possible.

The assignments which gives minimum cost is selected as optimal solution.

But, such trail and error becomes very difficult and lengthy.

If there are more number of rows and columns, ( For eg: For 6 x 6 matrix, there will be 6! trails. So 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 trails possible) then such methods can't be applied for solving assignments problems.

(iv) Hungarian Method:

It was developed by two mathematicians of Hungary. So, it is known as Hungarian Method.

It is also know as Reduced matrix method or Flood's technique.

There are two main conditions for applying Hungarian Method:

(1) Square Matrix (n x n). (2) Problem should be of minimization type.

Suggested Notes:

Modified Distribution Method (MODI) | Transportation Problem | Transportation Model

Stepping Stone | Transportation Problem | Transportation Model

Vogel’s Approximation Method (VAM) | Method to Solve Transportation Problem | Transportation Model

Transportation Model - Introduction

North West Corner Method | Method to Solve Transportation Problem | Transportation Model

Least Cost Method | Method to Solve Transportation Problem | Transportation Model

Tie in selecting row and column (Vogel's Approximation Method - VAM) | Numerical | Solving Transportation Problem | Transportation Model

Crashing Special Case - Multiple (Parallel) Critical Paths

Crashing Special Case - Indirect cost less than Crash Cost

Basics of Program Evaluation and Review Technique (PERT)

Numerical on PERT (Program Evaluation and Review Technique)

Network Analysis - Dealing with Network Construction Basics

Construct a project network with predecessor relationship | Operation Research | Numerical

Graphical Method | Methods to solve LPP | Linear Programming

Basics of Linear Programming

Linear Programming Problem (LPP) Formulation with Numericals

All comments that you add will await moderation. We'll publish all comments that are topic related, and adhere to our Code of Conduct .

Want to tell us something privately? Contact Us

Post comment

Education Lessons

Stay in touch, [notes] operation research, [notes] dynamics of machinery, [notes] maths, [notes] science, [notes] computer aided design.

www.springer.com The European Mathematical Society

• StatProb Collection
• Recent changes
• Current events
• Random page
• Project talk
• Request account
• What links here
• Related changes
• Special pages
• Printable version
• Permanent link
• Page information
• View source

Assignment problem

The problem of optimally assigning $ m $ individuals to $ m $ jobs. It can be formulated as a linear programming problem that is a special case of the transport problem :

maximize $ \sum _ {i,j } c _ {ij } x _ {ij } $

$$ \sum _ { j } x _ {ij } = a _ {i} , i = 1 \dots m $$

(origins or supply),

$$ \sum _ { i } x _ {ij } = b _ {j} , j = 1 \dots n $$

(destinations or demand), where $ x _ {ij } \geq 0 $ and $ \sum a _ {i} = \sum b _ {j} $, which is called the balance condition. The assignment problem arises when $ m = n $ and all $ a _ {i} $ and $ b _ {j} $ are $ 1 $.

If all $ a _ {i} $ and $ b _ {j} $ in the transposed problem are integers, then there is an optimal solution for which all $ x _ {ij } $ are integers (Dantzig's theorem on integral solutions of the transport problem).

In the assignment problem, for such a solution $ x _ {ij } $ is either zero or one; $ x _ {ij } = 1 $ means that person $ i $ is assigned to job $ j $; the weight $ c _ {ij } $ is the utility of person $ i $ assigned to job $ j $.

The special structure of the transport problem and the assignment problem makes it possible to use algorithms that are more efficient than the simplex method . Some of these use the Hungarian method (see, e.g., [a5] , [a1] , Chapt. 7), which is based on the König–Egervary theorem (see König theorem ), the method of potentials (see [a1] , [a2] ), the out-of-kilter algorithm (see, e.g., [a3] ) or the transportation simplex method.

In turn, the transportation problem is a special case of the network optimization problem.

A totally different assignment problem is the pole assignment problem in control theory.

• This page was last edited on 5 April 2020, at 18:48.
• Privacy policy
• About Encyclopedia of Mathematics
• Disclaimers
• Impressum-Legal

Your Article Library

Assignment problem in linear programming : introduction and assignment model.

ADVERTISEMENTS:

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.

1. 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,

1. 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.

2. 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.

3. 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.

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:

(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.

5. 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.

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.

7. Repeat the procedure from step (3) until the number of assignments becomes equal to the number of rows or number of columns.

Related Articles:

• Two Phase Methods of Problem Solving in Linear Programming: First and Second Phase
• Linear Programming: Applications, Definitions and Problems

No comments yet.

Leave a reply click here to cancel reply..

You must be logged in to post a comment.

Assignment problem

1 formulation of the problem, 2 variants of the problem, 3 algorithms for solving the problem, 4 references.

Suppose that there are [math]n[/math] agents and [math]m[/math] tasks, which can be distributed between these agents. Only one task can be assigned to each agent, and each task can be assigned to only one agent. The cost of assignment of the [math]i[/math] -th task to the [math]j[/math] -th agent is [math]c(i, j)[/math] . If [math]c(i, j) = \infty[/math] , then the [math]i[/math] -th task cannot be assigned to the [math]j[/math] -th agent.

The assignment problem : find a feasible set of assignments [math]A = \{ (i_1, j_1), \ldots, (i_k, j_k) \}[/math] , [math]k = \min \{m, n\}[/math] having the maximum total cost:

If [math]m = n[/math] , then we say of the linear assignment problem : each agent is assigned to perform exactly one task, and each task is assigned to exactly one agent.

In the case of unit weights, we have to find a maximum matching in a bipartite graph, and the problem reduces to assigning as much tasks as possible.

• The Hungarian Method [1] [2] [3] for the linear problem. The complexity is [math]O(n^4)[/math] (and can be reduced [4] to [math]O(n^3)[/math] );
• the auction algorithm [5] [6] ;
• the Hopcroft-Karp algorithm [7] for the problem with unit weights. The complexity is [math]O(m \sqrt{n})[/math] .
• ↑ Kuhn, H W. “The Hungarian Method for the Assignment Problem.” Naval Research Logistics Quarterly 2, no. 1 (March 1955): 83–97. doi:10.1002/nav.3800020109.
• ↑ Kuhn, H W. “Variants of the Hungarian Method for Assignment Problems.” Naval Research Logistics Quarterly 3, no. 4 (December 1956): 253–58. doi:10.1002/nav.3800030404.
• ↑ Munkres, James. “Algorithms for the Assignment and Transportation Problems.” Journal of the Society for Industrial and Applied Mathematics 5, no. 1 (March 1957): 32–38. doi:10.1137/0105003.
• ↑ Tomizawa, N. “On Some Techniques Useful for Solution of Transportation Network Problems.” Networks 1, no. 2 (1971): 173–94. doi:10.1002/net.3230010206.
• ↑ Bertsekas, Dimitri P. “Auction Algorithms for Network Flow Problems: a Tutorial Introduction.” Computational Optimization and Applications 1 (1992): 7–66.
• ↑ Zavlanos, Michael M, Leonid Spesivtsev, and George J Pappas. “A Distributed Auction Algorithm for the Assignment Problem,” Proceedings of IEEE CDC'08, 1212–17, IEEE, 2008. doi:10.1109/CDC.2008.4739098.
• ↑ Hopcroft, John E, and Richard M Karp. “An $N^{5/2} $ Algorithm for Maximum Matchings in Bipartite Graphs.” SIAM Journal on Computing 2, no. 4 (1973): 225–31. doi:10.1137/0202019.
• Problem level
• Articles in progress

Navigation menu

Personal tools.

• Create account
• View source
• View history
• Recent changes

File storage

• Upload file
• What links here
• Related changes
• Special pages
• Printable version
• Permanent link
• Page information
• This page was last edited on 6 March 2018, at 17:08.
• Content is available under Creative Commons Attribution unless otherwise noted.
• Privacy policy
• About Algowiki
• Disclaimers

Please enable JavaScript to pass antispam protection! Here are the instructions how to enable JavaScript in your web browser http://www.enable-javascript.com . Antispam by CleanTalk.

Assignment Problem: Linear Programming

The assignment problem is a special type of transportation problem , where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.

In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem.

The model's primary usefulness is for planning. The assignment problem also encompasses an important sub-class of so-called shortest- (or longest-) route models. The assignment model is useful in solving problems such as, assignment of machines to jobs, assignment of salesmen to sales territories, travelling salesman problem, etc.

It may be noted that with n facilities and n jobs, there are n! possible assignments. One way of finding an optimal assignment is to write all the n! possible arrangements, evaluate their total cost, and select the assignment with minimum cost. But, due to heavy computational burden this method is not suitable. This chapter concentrates on an efficient method for solving assignment problems that was developed by a Hungarian mathematician D.Konig.

"A mathematician is a device for turning coffee into theorems." -Paul Erdos

Formulation of an assignment problem

Suppose a company has n persons of different capacities available for performing each different job in the concern, and there are the same number of jobs of different types. One person can be given one and only one job. The objective of this assignment problem is to assign n persons to n jobs, so as to minimize the total assignment cost. The cost matrix for this problem is given below:

The structure of an assignment problem is identical to that of a transportation problem.

To formulate the assignment problem in mathematical programming terms , we define the activity variables as

for i = 1, 2, ..., n and j = 1, 2, ..., n

In the above table, c ij is the cost of performing jth job by ith worker.

Generalized Form of an Assignment Problem

The optimization model is

Minimize c 11 x 11 + c 12 x 12 + ------- + c nn x nn

subject to x i1 + x i2 +..........+ x in = 1          i = 1, 2,......., n x 1j + x 2j +..........+ x nj = 1          j = 1, 2,......., n

x ij = 0 or 1

In Σ Sigma notation

x ij = 0 or 1 for all i and j

An assignment problem can be solved by transportation methods, but due to high degree of degeneracy the usual computational techniques of a transportation problem become very inefficient. Therefore, a special method is available for solving such type of problems in a more efficient way.

Assumptions in Assignment Problem

• Number of jobs is equal to the number of machines or persons.
• Each man or machine is assigned only one job.
• Each man or machine is independently capable of handling any job to be done.
• Assigning criteria is clearly specified (minimizing cost or maximizing profit).

Share this article with your friends

Operations Research Simplified Back Next

Goal programming Linear programming Simplex Method Transportation Problem

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

Inner Product

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

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:

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

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.

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

as well as a set of r points

In his paper he derives the below formula:

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:

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]

• ↑ 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.

Navigation menu

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 Programming Problem and Its Mathematical Formulation

Sometimes one seeks to optimize (maximize or minimize) a known function (could be profit/loss or any output), subject to a set of linear constraints on the function. Linear Programming Problems (LPP) provide the method of finding such an optimized function along with/or the values which would optimize the required function accordingly.

Browse more Topics under Linear Programming

• Different Types of Linear Programming Problems
• Graphical Method of Solving Linear Programming Problems

It is one of the most important Operations Research tools. It is widely used as a decision making aid in almost all industries. There can be various fields of application of LPP, in the areas of Economics, Computer Sciences, Mathematics , etc. Since it is a very important topic with numerous practical applications , we will proceed slowly in building up this topic and making it very clear to you. So, let’s begin!

Suggested Videos on Linear Programming

.embed-container { position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden; max-width: 100%; } .embed-container iframe, .embed-container object, .embed-container embed { position: absolute; top: 0; left: 0; width: 100%; height: 100%; }

Mathematical Formulation

Formulation of an LPP refers to translating the real-world problem into the form of mathematical equations which could be solved. It usually requires a thorough understanding of the problem.

Steps towards formulating a Linear Programming problem:

• Step 1:   Identify the ‘n’ number of decision variables which govern the behaviour of the objective function (which needs to be optimized).
• Step 2:  Identify the set of constraints on the decision variables and express them in the form of linear equations / inequations . This will set up our region in the n- dimensional space within which the objective function needs to be optimized. Don’t forget to impose the condition of non-negativity on the decision variables i.e. all of them must be positive since the problem might represent a physical scenario, and such variables can’t be negative.
• Step 3:  Express the objective function in the form of a linear equation in the decision variables.
• Step 4:  Optimize the objective function either graphically or mathematically.

Now let us look at an example aimed at enabling you to learn how to formulate a problem in Linear Programming!

Download Linear Programming Problem Cheat Sheet

Solved Examples for You

Question 1: A calculator company produces a handheld calculator and a scientific calculator. Long-term projections indicate an expected demand of at least 150 scientific and 100 handheld calculators each day. Because of limitations on production capacity, no more than 250 scientific and 200 handheld calculators can be made daily.

To satisfy a shipping contract, a minimum of 250 calculators must be shipped each day. If each scientific calculator sold, results in a 20 rupees loss, but each handheld calculator produces a 50 rupees profit; then how many of each type should be manufactured daily to maximize the net profit?

Such word problems can be very tricky to solve if not formulated properly. Hence, let us approach it in a step by step manner as discussed above.

Step 1: The decision variables : Since the question has asked for an optimum number of calculators, that’s what our decision variables in this problem would be. Let,

x = number of scientific calculators produced y =  number of handheld calculators produced

Therefore, we have 2 decision variables in this problem, namely ‘ x’  and ‘ y’.

Step 2: The constraints: Since the company can’t produce a negative number of calculators in a day, a natural constraint would be:

x ≥ 0 y ≥ 0

However, a lower bound for the company to sell calculators is already supplied in the problem. We can note it down as:

x ≥ 150 y ≥ 100

We have also been given an upper bound for these variables, owing to the limitations on production by the company. We can write as follows:

x ≤ 250 y ≤ 200

Besides, we also have a joint constraint on the values of  ‘x’ and  ‘y’ due to the minimum order on a shipping consignment; given as:

x + y ≥ 250

Step 3:  Objective Function: Clearly, here we need to optimize the Net Profit function. The Net Profit Function is given as:

P = -20x + 50y

Step 4:  Solving the problem: the system here –

Maximization of  P = -20x + 50y, subject to: 150 ≤ x ≤ 250 100 ≤ y ≤ 200 x + y ≥ 250

We can solve it graphically or mathematically as per convenience. We will discuss it in later sections. This completes the discussion on the mathematical formulation of a Linear Programming problem !

Question 2: What is meant by LPP?

Answer: The full form of LPP is Linear Programming Problems. This method helps in achieving the best outcome in a mathematical model. The best outcome could be maximum profit or the lowest cost or the best possible price. The representation of this model’s requirements is by linear relationships.

Question 3: Explain how one can calculate LPP?

Answer: In order to calculate LPP, one must follow the following steps:

• Formulate the LP problem.
• Construct a graph and then plot the various constraint lines.
• Ascertain the valid side of all constraint lines.
• Identify the region of feasible solution.
• Plot the objective function.
• Finally, find out the optimum point.

Question 4: What are the various advantages of LPP?

Answer: The various advantages of LPP are as follows:

• Utilized for analyzing numerous military, social, economic social, and industrial problems.
• Linear programming is appropriate for solving complex problems.
• It assists in productive management of an organization for better outcomes.

Question 5: Who is credited with the development of LPP?

Customize your course in 30 seconds

Which class are you in.

Linear Programming

• Types of Linear Programming Problems

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Download the App

• 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)

• Channel Assignment Problem
• Assignment Operators In C++
• Solidity - Assignment Operators
• Job Assignment Problem using Branch And Bound
• Range Minimum Query with Range Assignment
• Transportation Problem | Set 1 (Introduction)
• Transportation Problem | Set 2 (NorthWest Corner Method)
• Transportation Problem | Set 3 (Least Cost Cell Method)
• Transportation Problem | Set 4 (Vogel's Approximation Method)
• Assignment Operators in C
• QA - Placement Quizzes | Profit and Loss | Question 7
• QA - Placement Quizzes | Profit and Loss | Question 4
• QA - Placement Quizzes | Profit and Loss | Question 12
• QA - Placement Quizzes | Profit and Loss | Question 10
• QA - Placement Quizzes | Profit and Loss | Question 8
• QA - Placement Quizzes | Age | Question 4
• QA - Placement Quizzes | Profit and Loss | Question 9
• QA - Placement Quizzes | Profit and Loss | Question 11
• IBM Placement Paper | Quantitative Analysis Set - 5
• Must Do Coding Questions for Companies like Amazon, Microsoft, Adobe, ...
• Top 50 Array Coding Problems for Interviews
• Introduction to Recursion - Data Structure and Algorithm Tutorials
• Difference between BFS and DFS
• Must Do Coding Questions for Product Based Companies
• A* Search Algorithm
• Top 10 Algorithms in Interview Questions
• Sliding Window Technique
• How to write a Pseudo Code?
• Asymptotic Notation and Analysis (Based on input size) in Complexity Analysis of Algorithms

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:

Flow matrix:

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?