the procedure used to solve assignment problem where one reduces the original

Operations Research

191. The large negative opportunity cost value in an unused cell in a transportation table is chosen to improve the current solution because

• It represents per unit cost reduction
• It represents per unit cost improvement
• It ensure no rim requirement violation
• None of the above

Correct answer: (A) It represents per unit cost reduction

192. The method of finding an initial solution based upon opportunity costs is called __________.

• the northwest corner rule
• Vogel's approximation
• Johanson's theorem
• Flood's technique
• Hungarian method

Correct answer: (B) Vogel's approximation

193. The net cost of shipping one unit on a route not used in the current transportation problem solution is called the __________.

• change index
• Improvement index

Correct answer: (E) Improvement index

194. The objective function and constraints are functions of two types of variables, __________ variables and __________ variables.

• Positive and negative
• Controllable and uncontrollable
• Strong and weak

Correct answer: (B) Controllable and uncontrollable

195. The objective function for a minimization problem is given by z = 2 x1 - 5 x2 + 3 x3 The hyperplane for the objective function cuts a bounded feasible region in the space (x1,x2,x3). Find the direction vector d, where a finite optimal solution can be reached.

• d(-2,-5,-3)

Correct answer: (B) d(-2,5,-3)

196. The occurrence of degeneracy while solving a transportation problem means that

• Total supply equals total demand
• The solution so obtained is not feasible
• The few allocations become negative

Correct answer: (B) The solution so obtained is not feasible

197. The only restriction we place on the initial solution of a transportation problem is that: we must have nonzero quantities in a majority of the boxes.

• all constraints must be satisfied.
• demand must equal supply.
• we must have a number (equal to the number of rows plus the number of columns minus one) of boxes which contain nonzero quantities.

Correct answer: (A) all constraints must be satisfied.

198. The Operations research technique which helps in minimizing total waiting and service costs is

• Queuing Theory
• Decision Theory
• Both A and B

Correct answer: (A) Queuing Theory

199. The procedure used to solve assignment problems wherein one reduces the original assignment costs to a table of opportunity costs is called __________.

• stepping-stone method
• matrix reduction
• MODI method
• northwest reduction
• simplex reduction

Correct answer: (B) matrix reduction

200. The purpose of a dummy source or dummy destination in a transportation problem is to

• prevent the solution from becoming degenerate.
• obtain a balance between total supply and total demand.
• make certain that the total cost does not exceed some specified figure.
• provide a means of representing a dummy problem.

Correct answer: (B) obtain a balance between total supply and total demand.

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How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

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

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

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

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

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

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

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Operations Research

1 Operations Research-An Overview

• History of O.R.
• Approach, Techniques and Tools
• Phases and Processes of O.R. Study
• Typical Applications of O.R
• Limitations of Operations Research
• Models in Operations Research
• O.R. in real world

2 Linear Programming: Formulation and Graphical Method

• General formulation of Linear Programming Problem
• Optimisation Models
• Basics of Graphic Method
• Important steps to draw graph
• Multiple, Unbounded Solution and Infeasible Problems
• Solving Linear Programming Graphically Using Computer
• Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

• Principle of Simplex Method
• Computational aspect of Simplex Method
• Simplex Method with several Decision Variables
• Two Phase and M-method
• Multiple Solution, Unbounded Solution and Infeasible Problem
• Sensitivity Analysis
• Dual Linear Programming Problem

4 Transportation Problem

• Basic Feasible Solution of a Transportation Problem
• Modified Distribution Method
• Stepping Stone Method
• Unbalanced Transportation Problem
• Degenerate Transportation Problem
• Transhipment Problem
• Maximisation in a Transportation Problem

5 Assignment Problem

• Solution of the Assignment Problem
• Unbalanced Assignment Problem
• Problem with some Infeasible Assignments
• Maximisation in an Assignment Problem
• Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

• Concepts of goal programming
• Goal programming model formulation
• Graphical method of goal programming
• The simplex method of goal programming
• Using Excel Solver to Solve Goal Programming Models
• Application areas of goal programming

8 Integer Programming

• Some Integer Programming Formulation Techniques
• Binary Representation of General Integer Variables
• Unimodularity
• Cutting Plane Method
• Branch and Bound Method
• Solver Solution

9 Dynamic Programming

• Dynamic Programming Methodology: An Example
• Definitions and Notations
• Dynamic Programming Applications

10 Non-Linear Programming

• Solution of a Non-linear Programming Problem
• Convex and Concave Functions
• Kuhn-Tucker Conditions for Constrained Optimisation
• Quadratic Programming
• Separable Programming
• NLP Models with Solver

11 Introduction to game theory and its Applications

• Important terms in Game Theory
• Saddle points
• Mixed strategies: Games without saddle points
• 2 x n games
• Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

• Reasons for using simulation
• Monte Carlo simulation
• Limitations of simulation
• Steps in the simulation process
• Some practical applications of simulation
• Two typical examples of hand-computed simulation
• Computer simulation

13 Queueing Models

• Characteristics of a queueing model
• Notations and Symbols
• Statistical methods in queueing
• The M/M/I System
• The M/M/C System
• The M/Ek/I System
• Decision problems in queueing

The assignment problem revisited

• Original Paper
• Published: 16 August 2021
• Volume 16 , pages 1531–1548, ( 2022 )

Cite this article

• Carlos A. Alfaro   ORCID: orcid.org/0000-0001-9783-8587 1 ,
• Sergio L. Perez 2 ,
• Carlos E. Valencia 3 &
• Marcos C. Vargas 1  

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First, we give a detailed review of two algorithms that solve the minimization case of the assignment problem, the Bertsekas auction algorithm and the Goldberg & Kennedy algorithm. It was previously alluded that both algorithms are equivalent. We give a detailed proof that these algorithms are equivalent. Also, we perform experimental results comparing the performance of three algorithms for the assignment problem: the \(\epsilon \) - scaling auction algorithm , the Hungarian algorithm and the FlowAssign algorithm . The experiment shows that the auction algorithm still performs and scales better in practice than the other algorithms which are harder to implement and have better theoretical time complexity.

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Some results on an assignment problem variant

Integer Programming

A Full Description of Polytopes Related to the Index of the Lowest Nonzero Row of an Assignment Matrix

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Alfaro, C.A., Perez, S.L., Valencia, C.E. et al. The assignment problem revisited. Optim Lett 16 , 1531–1548 (2022). https://doi.org/10.1007/s11590-021-01791-4

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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:

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:

REVISED ONES ASSIGNMENT METHOD FOR SOLVING ASSIGNMENT PROBLEM

The assignment problem (AP) is a special case of the transportation problem, in which the objective is to assign a number of re- sources to the equal number of activities at a minimum cost (or maximum profit). It has great significance subject discussed in real physical world for e.g. production planning, particular job tasks, economic etc. We endeavor in this paper to introduce a new approach to assignment problem namely “Revised Ones Assignment Method (ROA)” for solving wide range of problem. In ROA method first we define assignment matrix, then reduced matrix till it has at least one in each row and column. The new method is based on creating some ones in the assignment matrix and to complete exact assignment to their ones, we have added new step to ROA algorithm can be utilized for all types of AP with maximize or minimize objective functions and at last we have illustrate some numerical exam- ples.

Related Papers

Hogpodrat Pangkum

Assignment problem is a special case of Transportation problem. It is actually a minimizing model that assigns numbers of people with equal number of jobs, henceforth, minimizing the corresponding costs. In this paper an introduction is given to " New Improved Ones Assignment " which is a path to making an assignment problem. Earlier H. Gamel also brought to light the drawbacks of One assignment method. Our improvement to the Ones assignment method, leads to comparatively brief computation time and more convenient and strong codes. It also overcomes the drawbacks as mentioned previously

Trisna Darmawansyah

Assignment problem is an important subject discussed in real physical world. We endeavor in this paper to introduce a new approach to assignment problem namely, ones assignment method, for solving a wide rang of such problems. This method offers significant advantages over similar methods, in the process, first we define the assignment matrix, then by using determinant representation we obtain a reduced matrix which has at least one 1 in each row and columns. Then by using the new method, we obtain an optimal solution for assignment problem by assigning ones to each row and each column. The new method is based on creating some ones in the assignment matrix and then try to find a complete assignment to there ones. The proposed method is a systematic procedure, easy to apply and can be utilized for all types of assignment problem with maximize or minimize objective functions. At the end, this method is illustrated with some numerical examples.

Assignment problems arise in different situation where we have to find an optimal way to assign n-objects to mother objects in an injective fashion. The assignment problems are a well studied topic in combinatorial optimization. These problems find numerous application in production planning, telecommunication VLSI design, economic etc. The assignment problems is a special case of Transportation problem. Depending on the objective we want to optimize, we obtain the typical assignment problems. Assignment problem is an important subject discussed in real physical world we endeavor in this paper to introduce a new approach to assignment problem namely, matrix ones assignment method or MOA-method for solving wide range of problem. An example using matrix ones assignment methods and the existing Hungarian method have been solved and compared it graphically. Also some of the variations and some special cases in assignment problem and its applications have been discussed in the paper.

1 í µí°·í µí±’í µí±í µí±Ží µí±Ÿí µí±¡í µí±ší µí±’í µí±›í µí±¡í µí±œí µí±“í µí±€í µí±Ží µí±¡ í µí±•í µí±’í µí±ší µí±Ží µí±¡í µí±–í µí±í µí± ,í µí° ¶í µí±œí µí±ší µí±–í µí±™í µí±™í µí±Ží µí±ˆí µí±›í µí±–í µí±£í µí±’í µí±Ÿí µí± í µí±–í µí±¡í µí±¦ ,í µí°µí µí±Ží µí±›í µí±”í µí±™í µí±Ží µí±‘í µí±’í µí± í µí±• 2 í µí°·í µí±’í µí±í µí±Ží µí±Ÿí µí±¡í µí±ší µí±’í µí±›í µí±¡í µí±œí µí±“í µí±€í µí±Ží µí±¡ í µí±•í µí±’í µí±ší µí±Ží µí±¡í µí±–í µí±í µí± ,í µí° ¶í µí±œí µí±ší µí±–í µí±™í µí±™í µí±Ží µí±ˆí µí±›í µí±–í µí±£í µí±’í µí±Ÿí µí± í µí±–í µí±¡í µí±¦ ,í µí°µí µí±Ží µí±›í µí±”í µí±™í µí±Ží µí±‘í µí±’í µí± í µí±• Abstract: Assignment problem is an important problem in mathematics and is also discuss in real physical world. In this paper we attempt to introduce a new proposed approach for solving assignment problem with algorithm and solution steps. We examine a numerical example by using new method and compute by existing two methods. Also we compare the optimal solutions among this new method and two existing method .The proposed method isa systematic procedure, easy to apply for solving assignment problem.

IAEME Publication

Generalized Assignment Problem is a very popular topic which is used to solve different engineering and management problems. It becomes more realistic when used under fuzzy environment since in practical situation it is difficult to get precise data. In this paper cost for assigning the j-th job to the i-th person is taken as trapezoidal fuzzy numbers (TrFNs) which are more realistic and general in nature. To solve the real life problems restrictions have been put on both job cost and person cost depending on his/her efficiency/qualification. Yager’s Ranking Method[1] has been used for ranking the fuzzy numbers. The fuzzy generalized assignment problem (FGAP) has been solved by Modified Extremum Difference Method (EDM) for initial Basic Feasible Solution (BFS). To test the optimality MODI method is used. Again the solution by the proposed method has been verified by LINGO 9.0

IJAR Indexing

Assignment problems deal with the question how to assign n objects to m other objects in an injective fashion in the best possible way. An assignment problem is completely specified by its two components the assignments, which represent the underlying combinatorial structure, and the objective function to be optimized, which models \\\\\\\"the best possible way\\\\\\\". The assignment problem refers to another special class of linear programming problem where the objective is to assign a number of resources to an equal number of activities on a one to one basis so as to minimize total costs of performing the tasks at hand or maximize total profit of allocation. In this paper we introduce a new technique to solve assignment problems namely, Divide Row Minima and Subtract Column Minima .For the validity and comparison study we consider an example and solved by using our technique and the existing Hungarian (HA) and matrix ones assignment method(MOA) and compare optimum result shown graphically.

Different situations give rise to the assignment problem, where we must discover an optimal way to assign 'n' objects to 'm' in an bijective function. We have, in this research, propose the possibility of solving exactly the Linear Assignment Problem with a method that would be more efficient than the Hungarian method of exact solution. This method is based on applying a series of pairwise interchanges of assignments to a starting heuristically generated feasible solution, wherein each pairwise interchange is guaranteed to improve the objective function value of the feasible solution.It seems that our algorithm finds the optimal solution which is the same as one found by the Hungarian method, but in much simpler. 7980 M. Khalid et al.

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Solving an Assignment Problem

This section presents an example that shows how to solve an assignment problem using both the MIP solver and the CP-SAT solver.

In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3). Note that there is one more worker than in the example in the Overview .

The costs of assigning workers to tasks are shown in the following table.

The problem is to assign each worker to at most one task, with no two workers performing the same task, while minimizing the total cost. Since there are more workers than tasks, one worker will not be assigned a task.

MIP solution

The following sections describe how to solve the problem using the MPSolver wrapper .

Import the libraries

The following code imports the required libraries.

Create the data

The following code creates the data for the problem.

The costs array corresponds to the table of costs for assigning workers to tasks, shown above.

Declare the MIP solver

The following code declares the MIP solver.

Create the variables

The following code creates binary integer variables for the problem.

Create the constraints

Create the objective function.

The following code creates the objective function for the problem.

The value of the objective function is the total cost over all variables that are assigned the value 1 by the solver.

Invoke the solver

The following code invokes the solver.

Print the solution

The following code prints the solution to the problem.

Here is the output of the program.

Complete programs

Here are the complete programs for the MIP solution.

CP SAT solution

The following sections describe how to solve the problem using the CP-SAT solver.

Declare the model

The following code declares the CP-SAT model.

The following code sets up the data for the problem.

The following code creates the constraints for the problem.

Here are the complete programs for the CP-SAT solution.

Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License , and code samples are licensed under the Apache 2.0 License . For details, see the Google Developers Site Policies . Java is a registered trademark of Oracle and/or its affiliates.

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What is the difference between minimal cost network flows and transportation

The minimal cost network flows are special cases of transportation problems

The transportation problems are special cases of the minimal cost network flows

There is no difference

The transportation problems are formulated in terms of tableaus, while the minimal

cost network flows are formulated in terms of graphs

With the transportation technique, the initial solution can be generated in any fashion

one chooses. The only restriction is that

the edge constraints for supply and demand are satisfied.

the solution is not degenerate.

the solution must be optimal.

one must use the northwest-corner method.

The purpose of the stepping-stone method is to

develop the initial solution to the transportation problem.

assist one in moving from an initial feasible solution to the optimal solution.

determine whether a given solution is feasible or not.

identify the relevant costs in a transportation problem.

The purpose of a dummy source or dummy destination in a transportation problem is

prevent the solution from becoming degenerate.

obtain a balance between total supply and total demand.

make certain that the total cost does not exceed some specified figure.

provide a means of representing a dummy problem.

Which of the following is NOT needed to use the transportation model?

the cost of shipping one unit from each origin to each destination

the destination points and the demand per period at each

the origin points and the capacity or supply per period at each

Which of the following is a method for improving an initial solution in a

transportation problem?

northwest-corner

intuitive lowest-cost

southeast-corner rule

stepping-stone

The transportation method assumes that

there are no economies of scale if large quantities are shipped from one source to

one destination.

the number of occupied squares in any solution must be equal to the number of rows

in the table plus the number of columns in the table plus 1.

there is only one optimal solution for each problem.

the number of dummy sources equals the number of dummy destinations.

Which of these statements about the stepping-stone method is best?

A dummy source and destination must be added if the number of rows plus columns

minus 1 is not equal to the number of filled squares.

Only squares containing assigned shipments can be used to trace a path back to

an empty square.

An improvement index that is a net positive means that the initial solution can be

Only empty squares can be used to trace a path back to a square containing an

assigned shipment

In a transportation problem, we must make the number of __________ and

__________ equal.

destinations; sources

units supplied; units demanded

columns; rows

positive cost coefficients; negative cost coefficients

warehouses; suppliers

__________ or __________ are used to "balance" an assignment or transportation

Destinations; sources

Units supplied; units demanded

Dummy rows; dummy columns

Large cost coefficients; small cost coefficients

Optimal solution of an assignment problem can be obtained only if

Each row & column has only one zero element

Each row & column has at least one zero element

The data is arrangement in a square matrix

None of the above

In assignment problem of maximization, the objective is to maximise

optimization

Which of the following is used to come up with a solution to the assignment problem?

MODI method

northwest corner method

stepping-stone method

Hungarian method

The procedure used to solve assignment problems wherein one reduces the original

assignment costs to a table of opportunity costs is called __________.

matrix reduction

northwest reduction

The occurrence of degeneracy while solving a transportation problem means that

Total supply equals total demand

The solution so obtained is not feasible

The few allocations become negative

An alternative optimal solution to a minimization transportation problem exists

whenever opportunity cost corresponding to unused route of transportation is:

Positive & greater than zero

Positive with at least one equal to zero

Negative with at least one equal to zero

The solution to a transportation problem with ‘m’ rows (supplies) & ‘n’ columns

(destination) is feasible if number of positive allocations are

If an opportunity cost value is used for an unused cell to test optimality, it should be

Equal to zero

Most negative number

Most positive number

The degeneracy in the transportation problem indicates that

Dummy allocation(s) needs to be added

The problem has no feasible solution

The multiple optimal solution exist

a & b but not c

In a transportation problem, when the number of occupied routes is less than the

number of rows plus the number of columns -1, we say that the solution is:

Unbalanced.

Infeasible.

impossible.

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