assignment problem mathematical formulation

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.

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Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

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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.

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OPERATIONS RESEARCH

Lesson 8. introduction and mathematical formulation.

Current course

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.

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Assignment problem in linear programming : introduction and assignment model.

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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.

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

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Assignment Model | Linear Programming Problem (LPP) | Introduction

What is assignment model.

→ Assignment model is a special application of Linear Programming Problem (LPP) , in which the main objective is to assign the work or task to a group of individuals such that;

i) There is only one assignment.

ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).

→ In assignment problem, the cost of performing each task by each individual is known. → It is desired to find out the best assignments, such that overall cost of assigning the work is minimized.

For example:

Suppose there are 'n' tasks, which are required to be performed using 'n' resources.

The cost of performing each task by each resource is also known (shown in cells of matrix)

• 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:

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Assignment Problem

• Feb 20, 2024

The assignment problem is a special case of transportation problem.

Suppose there are $n$ jobs to be performed and $n$ persons are available for these jobs. Assume that each person can do each job at a time, though with varying degree of efficiency.

Let $c_{ij}$ be the cost if the $i^{th}$ person is assigned the $j^{th}$ job. Then the problem is to find an assignment so that the total cost of performing all jobs is minimum. (i.e., which job should be assigned to which person with minimum cost). Such problem is called an Assignment Problem (AP).

The tabular form of the assignment problem is as follows

The above table is called the $n\times n$ cost-matrix, where $c_{ij}$ are real numbers.

Thus the objective in the Assignment Problem is to assign a number of jobs to the equal number of persons at a minimum cost or maximum profit. e.g. assigning men to offices, classes to rooms, drivers to trucks, problems to research teams etc.

Mathematical Formulation of AP

Mathematically, the AP can be stated as $$ \begin{equation}\label{eq2.4} \min z = \sum_{i=1}^n\sum_{j=1}^n x_{ij} c_{ij}. \end{equation} $$ subject to $$ \begin{equation}\label{eq2.5} \sum_{j=1}^n x_{ij} =1,; \text{ for } i=1,2,\ldots, n \end{equation} $$

$$ \begin{equation}\label{eq2.6} \sum_{i=1}^n x_{ij} =1,; \text{ for } j=1,2,\ldots,n \end{equation} $$ where $$ \begin{equation*} x_{ij}=\left{ \begin{array}{ll} 1, & \hbox{if $i^{th}$ person is assigned $j^{th}$ job;} \ 0, & \hbox{otherwise.} \end{array} \right. \end{equation*} $$ Constraint \eqref{eq2.5} indicate that one job is done by $i^{th}$ person $i=1,2,\ldots, n$ and constraint \eqref{eq2.6} indicate that one person should be assigned $j^{th}$ job $j=1,2,\ldots, n$.

It may be observed from the above formulation that AP is a special type of Linear programming problem.

Unbalanced Assignment Problem

If the cost matrix of an assignment problem is not a square matrix, the assignment problem is called an Unbalanced Assignment Problem . In such a case, add dummy row or dummy column with zero cost in the cost matrix so as to form a square matrix. Then apply the usual assignment method to find the optimal solution.

Maximal Assignment Problem

Sometimes, the assignment problem deals with the maximization of an objective function instead of minimization.. For example, it may be required to assign persons to jobs in such a way that the expected profit is maximum. In such a case, first convert the problem of maximization to minimization and apply the usual procedure of assignment.

The assignment problem of maximization type can be converted to minimization type by subtracting all the elements of the given profit matrix from the largest element.

Restrictions on Assignment

Sometimes due to technical or other difficulties do not permit the assignment of a particular facility to a particular job. In such a situation, the difficulty can be overcome by assigning a very high cost (say, infinity i.e. $\infty$) to the corresponding cell.

Because of assigning an infinite penalty to such a cell the activity will be automatically excluded from the optimal solution. Then apply the usual procedure to find the optimal assignment.

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Mathematical Formulation of the Problem

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Mathematical formulation.

The link between formulation & design is investigated. The establishment of variable definitions and the functional relationships of the variables needed to model the issue under study are the most important components of mathematical formulation . Creating an effective mathematical formulation is similar to other engineering design tasks. It necessitates a thorough understanding of the problem at hand and the capacity to devise new approaches to the problem. The following are the design considerations of mathematical formulation mentioned in this paper:

• The mathematical formulation of the problem.
• Definitions and relationships for a preliminary design.
• Redesign to decrease the number of unknowns, variables, & constraints.
• Refining of the redesign.
• Generalisation to a multi-project scheduling issue.

Only algebraic manipulations are used, despite the complicated symbology.

A mathematical formulation of the problem

Let x and y represent the number of cabinets of kind 1 and 2 that he must produce. Non-negative constraints are non-negative limitations.

The company is allowed to invest 540 hours of labour and must build up to 50 cabinets. Hence,

15x + 9y <= 540

x + y <= 50

Linear constraints are defined as the above two equations.

Let Z be the profit he makes from producing x and y pieces of type 1 and 2 cabinets. Thus,

Z = 5000x + 3000

The goal here is to maximise Z. As a result, Z is referred to as the objective function. We employ graphs to obtain the answer to this question, which is referred to as the graphical approach of answering LPP. 

A mathematical formulation of a linear programming problem:

Linear programming, also known as linear optimisation, obtains the best possible solution to a mathematical model or maths formulation by taking into account specific linear relationships. LPP is another name for it. It encompasses issues such as maximising revenues, decreasing expenditures, and making the most efficient use of resources. The simplex approach or the graphical method can tackle these problems.

The phases in the technique for mathematically formulating a linear programming problem are as follows.

• Determine the variables that will be used to make a choice.
• Identify the goal function that must be maximised or minimised and describe it as a linear function of choice factors.
• Identify the collection of constraint conditions and describe them in terms of the decision variables as linear inequalities or equations.

A mathematical formulation of assignment problem

The assignment problem is a linear programming problem that entails allocating resources to individual tasks. It does so that the process’s cost or time is kept to a minimum while the profit or sale is maximised. Though similar problems can be handled using simplex or transportation methods, the assignment model provides a more straightforward solution.

The problem is formed on a one-to-one basis. A supervisor at a workplace might have six workers available & six tasks to fire. He will have to decide which work should be assigned to which job. This is a difficulty with an assignment.

Problem-solving strategy

Consider the minimisation type objective function. To solve this Assignment problem, you must take the following steps:

• Beginning with the first row, find the smallest cost element in each row in the given cost table. This lowest element is now subtracted from each row’s element. As a result, each new table row will have at least one zero.
• After constructing the table (like in step 1), take the table’s columns. Starting with the first column, find the cheapest element in each column. Subtract this smallest element from each column’s element. After completing steps 1 and 2, we should have at least one zero in each column of the reduced cost table.
• The allocations for the smaller table are now made in the following manner:

(I) Rows are inspected one by one until the row with exactly one (zero) zero is discovered. This single zero is assigned by putting a square around it. All other zeros in the corresponding column are crossed out (x) because they will not be utilised to create additional assignments. For each row, a step is taken. 

(II) Now do Step 3 (I) on the columns as follows:

Columns are inspected one by one until one with exactly one zero is discovered. Now, by placing the square around this one zero, all other zeros in the associated rows are crossed out (x); a step is performed for each column.

(III) Steps 3 (I) and 3 (II) are repeated until all zeros have been crossed out or noted. The ideal solution has been found when the number of marked zeros or made assignments equals the number of rows or columns. There shall be exactly one assignment in each or no assignment in any of the columns. We’ll go to step 4 in this scenario.

• At this point, draw the smallest number of lines (horizontal and vertical) required to cover all zeros in the matrices obtained in steps 3 and 4.

(I) Mark all of the rows that have no assignment.

(ii) Tick all the columns in the ticked rows with a zero value.

(iii) Tick all the rows that haven’t been marked yet and have an assignment in the designated columns.

(iv) Repeat steps (4(i), 4(ii), 4(iii) until there are no more rows or columns to mark.

(v) Now, draw straight lines that cross all unmarked rows and columns. It’s also worth noting that in a n x n matrix, if there is no solution among them, always less than ‘n’ lines will cover all the zeros.

• If the number of lines produced in step 4 equals n or the number of rows, the solution is optimal; otherwise, proceed to step 6.
• Among all the uncovered elements, choose the smallest element. This element is now deducted from all the elements that were previously uncovered & added to the intersection of two lines. For new assignments, this is the matrix.
• Steps 3–7 should be repeated until the total number of assignments equals the total number of rows or columns.

A known function (maybe profit/loss or any output) is sometimes sought to optimise (maximise or minimise) under a set of linear restrictions. Linear programming problems (LPP) describe how to locate such an optimal function and the values that would optimise the needed function.

It’s one of the most crucial instruments in operations research. In almost all industries, it is widely used as a decision-making tool. LPP can be used in various domains, including economics, computer science, mathematics, and so on.

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Assignment Problems

Definition of the assignment problem, mathematical formulation of the assignment problem, hungarian method for solving assignment problem, flow chart of hungarian method.

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.

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.

An assignment problem can be mathematically formulated as follows:

Minimise the total cost

\(x_{ij} =1\), if \(i^{th}\) person is assigned to the \(j^{th}\) job \(x_{ij}=0\), if \(i^{th}\) person is that assigned to the \(j^{th}\) job

subject to the constraints

i) \(\sum_{i=1}^n x_{ij} = 1, j=1, 2, \cdots n\)

which means that only one job is done by the \(i^{th}\) person, \(i= 1, 2, \cdots, n\)

ii) \(\sum_{i=1}^n x_{ij} = 1, j=1, 2, \cdots n\)

which means that only one person should be assigned to the \(j^{th}\) person, \(j= 1, 2, \cdots, n\)

The Hungarian method of assignment provides us with an efficient method of finding the optimal solution without having to make a-direct comparison of every solution. It works on the principle of reducing the given cost matrix to a matrix of opportunity costs.

Opportunity cost show the relative penalties associated with assigning resources to an activity as opposed to making the best or least cost assignment. If we can reduce the cost matrix to the extent of having at least one zero in each row and column, it will be possible to make optimal assignment.

The Hungarian method can be summarized in the following steps:

Step 1: Develop the Cost Table from the given Problem

If the no of rows are not equal to the no of columns and vice versa, a dummy row or dummy column must be added. The assignment cost for dummy cells are always zero.

Step 2: Find the Opportunity Cost Table

(a) Locate the smallest element in each row of the given cost table and then subtract that from each element of that row, and

(b) In the reduced matrix obtained from 2 (a) locate the smallest element in each column and then subtract that from each element. Each row and column now have at least one zero value.

Step 3: Make Assignment in the Opportunity Cost Matrix

The procedure of making assignment is as follows:

(a) Examine rows successively until a row with exactly one unmarked zero is obtained. Make an assignment single zero by making a square around it.

(b) For each zero value that becomes assigned, eliminate (Strike off) all other zeros in the same row and/ or column

(c) Repeat step 3 (a) and 3 (b) for each column also with exactly single zero value all that has not been assigned.

(d) If a row and/or column has two or more unmarked zeros and one cannot be chosen by inspection, then choose the assigned zero cell arbitrarily.

(e) Continue this process until all zeros in row column are either enclosed (Assigned) or struck off (x)

Step 4: Optimality Criterion

If the member of assigned cells is equal to the numbers of rows column then it is optimal solution. The total cost associated with this solution is obtained by adding original cost figures in the occupied cells.

If a zero cell was chosen arbitrarily in step (3), there exists an alternative optimal solution. But if no optimal solution is found, then go to step (5).

Step 5: Revise the Opportunity Cost Table

Draw a set of horizontal and vertical lines to cover all the zeros in the revised cost table obtained from step (3), by using the following procedure:

(a) For each row in which no assignment was made, mark a tick (√)

(b) Examine the marked rows. If any zero occurs in those columns, tick the respective rows that contain those assigned zeros.

(c) Repeat this process until no more rows or columns can be marked.

If a no of lines drawn is equal to the no of (or columns) the current solution is the optimal solution, otherwise go to step 6.

Step 6: Develop the New Revised Opportunity Cost Table

(a) From among the cells not covered by any line, choose the smallest element, call this value K

(b) Subtract K from every element in the cell not covered by line.

(c) Add K to very element in the cell covered by the two lines, i.e., intersection of two lines.

(d) Elements in cells covered by one line remain unchanged.

Step 7: Repeat Step 3 to 6 Unlit an Optimal Solution is Obtained

The flow chart of steps in the Hungarian method for solving an assignment problem is shown in following figures:

In a factory there are five operator \(O_1\), \(O_2\), \(O_3\), \(O_4\), \(O_5\) and five machine \(M_1\), \(M_2\), \(M_3\), \(M_4\), \(M_5\). The operating costs are given when the \(O_i\) operator operates the \(M_j\) machine \((i,j=1,2,..,5)\). But there is a restriction that \(O_3\) cannot be allowed to operate the third machine \(M_3\) and \(O_2\) cannot be allowed to operate the fifth machine \(M_5\). The cost matrix is given above. Find the optional assignment and the optimal assignment cost also.

[2018, 15M]

2) Solve the following assignment problem to maximize the sales: \(\begin{array}{|c|c|c|c|c|c|} \hline {} & {I} & {II} & {III} & {IV} & {V} \\ \hline {A} & {3} & {4} & {5} & {6} & {7} \\ \hline {B} & {4} & {15} & {13} & {7} & {6} \\ \hline {C} & {6} & {13} & {12} & {5} & {11} \\ \hline {D} & {7} & {12} & {15} & {8} & {5} \\ \hline {E} & {8} & {13} & {10} & {6} & {9} \\ \hline \end{array}\)

where \(I\), \(II\), \(III\), \(IV\) and \(V\) are Territories; \(A\), \(B\), \(C\), \(D\), \(E\) are Salesmen.

[2015, 10M]

3) Solve the minimum time assignment problem: \(\begin{array}{|c|c|c|c|c|} \hline { } & {I} & {II} & {III} & {IV} \\ \hline {A} & {3} & {12} & {5} & {4} \\ \hline {B} & {7} & {9} & {8} & {12} \\ \hline {C} & {5} & {11} & {10} & {12} \\ \hline {D} & {6} & {14} & {4} & {11} \\ \hline \end{array}\)

where \(I\), \(II\), \(III\) and \(IV\) are Machines; \(A\), \(B\), \(C\) and \(D\) are Jobs.

[2013, 15M]

4) A travelling salesman has to visit 5 cities. He wishes to start from a particular city, visit each city once and then return to his starting point. Cost of going from one city to another is given below:

You are required to find the least cost route.

[2004, 15M]

5) Find the optimal solution for the assignment problem with the following cost matrix: \(\begin{bmatrix}{6} & {1} & {9} & {11} & {12} \\ {2} & {8} & {17} & {2} & {5} \\ {11} & {8} & {3} & {3} & {3} \\ {4} & {10} & {8} & {6} & {11} \\ {8} & {10} & {11} & {5} & {13}\end{bmatrix}\)

Indicate clearly the rule you apply to arrive at the complete assignment.

[2003, 15M]

Quadratic Assignment Problem

The objective of the Quadratic Assignment Problem (QAP) is to assign \(n\) facilities to \(n\) locations in such a way as to minimize the assignment cost. The assignment cost is the sum, over all pairs, of the flow between a pair of facilities multiplied by the distance between their assigned locations.

Problem Statement

The quadratic assignment problem (QAP) was introduced by Koopmans and Beckman in 1957 in the context of locating “indivisible economic activities”. The objective of the problem is to assign a set of facilities to a set of locations in such a way as to minimize the total assignment cost. The assignment cost for a pair of facilities is a function of the flow between the facilities and the distance between the locations of the facilities.

Consider a facility location problem with four facilities (and four locations). One possible assignment is shown in the figure below: facility 2 is assigned to location 1, facility 1 is assigned to location 2, facility 4 is assigned to location 3, and facility 4 is assigned to location 3. This assignment can be written as the permutation \(p = \{2, 1, 4, 3\}\), which means that facility 2 is assigned to location 1, facility 1 is assigned to location 2, facility 4 is assigned to location 3, and facility 4 is assigned to location 3. In the figure, the line between a pair of facilities indicates that there is required flow between the facilities, and the thickness of the line increases with the value of the flow.

To calculate the assignment cost of the permutation, the required flows between facilities and the distances between locations are needed.

Then, the assignment cost of the permutation can be computed as \(f(1,2) \cdot d(2,1) + f(1,4) \cdot d(2,3) + f(2,4) \cdot d(1,3) + f(3,4) \cdot d(3,4)\) = \(3 \cdot 22 + 2 \cdot 40 + 1 \cdot 53 + 4 \cdot 55\) = 419. Note that this permutation is not the optimal solution.

Mathematical Formulation

Here we present the Koopmans-Beckmann formulation of the QAP. Given a set of facilities and locations along with the flows between facilities and the distances between locations, the objective of the Quadratic Assignment Problem is to assign each facility to a location in such a way as to minimize the total cost.

Sets \(N = \{1, 2, \cdots, n\}\) \(S_n = \phi: N \rightarrow N\) is the set of all permutations

Parameters \(F = (f_{ij})\) is an \(n \times n\) matrix where \(f_{ij}\) is the required flow between facilities \(i\) and \(j\) \(D = (d_{ij})\) is an \(n \times n\) matrix where \(d_{ij}\) is the distance between locations \(i\) and \(j\)

Optimization Problem \(\text{min}_{\phi \in S_n} \sum_{i=1}^n \sum_{j=1}^n f_{ij} \cdot d_{\phi(i) \phi(j)}\)

The assignment of facilities to locations is represented by a permutation \(\phi\), where \(\phi(i)\) is the location to which facility \(i\) is assigned. Each individual product \(f_{ij} \cdot d_{\phi(i) \phi(j)}\) is the cost of assigning facility \(i\) to location \(\phi(i)\) and facility \(j\) to location \(\phi(j)\).

Solve some QAPs!

Follow the links below to test your skill at finding good solutions to QAPs of various sizes. Notice that as the problem size increases, it becomes much more difficult to find an optimal solution. As \(n\) increases beyond a small number, it becomes impossible to enumerate and evaluate all possible assignment vectors. Instead, advanced solution algorithms are required to solve larger instances.

QAP of size 4

Qap of size 5, qap of size 6, qap of size 7, qap of size 8, qap of size 9.

• Anstreicher, K.M. 2003. Recent advances in the solution of quadratic assignment problems. Mathematical Programming Series B 97 , 27 - 42.
• Çela, E. 1998. The Quadratic Assignment Problem: Theory and Algorithms . Kluwer Academic Publishers, Dordrecht.
• Koopmans, T. C. and M. J. Beckmann. 1957. Assignment problems and the location of economic activities. Econometrica 25 , 53 - 76.
• QAPLIB Home Page

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).

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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:

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.

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