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 __________.
 steppingstone 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.
Search MBA MCQ.com
Hungarian Method
The Hungarian method is a computational optimization technique that addresses the assignment problem in polynomial time and foreshadows following primaldual alternatives. In 1955, Harold Kuhn used the term “Hungarian method” to honour two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry. Let’s go through the steps of the Hungarian method with the help of a solved example.
Hungarian Method to Solve Assignment Problems
The Hungarian method is a simple way to solve assignment problems. Let us first discuss the assignment problems before moving on to learning the Hungarian method.
What is an Assignment Problem?
A transportation problem is a type of assignment problem. The goal is to allocate an equal amount of resources to the same number of activities. As a result, the overall cost of allocation is minimised or the total profit is maximised.
Because available resources such as workers, machines, and other resources have varying degrees of efficiency for executing different activities, and hence the cost, profit, or loss of conducting such activities varies.
Assume we have ‘n’ jobs to do on ‘m’ machines (i.e., one job to one machine). Our goal is to assign jobs to machines for the least amount of money possible (or maximum profit). Based on the notion that each machine can accomplish each task, but at variable levels of efficiency.
Hungarian Method Steps
Check to see if the number of rows and columns are equal; if they are, the assignment problem is considered to be balanced. Then go to step 1. If it is not balanced, it should be balanced before the algorithm is applied.
Step 1 – In the given cost matrix, subtract the least cost element of each row from all the entries in that row. Make sure that each row has at least one zero.
Step 2 – In the resultant cost matrix produced in step 1, subtract the least cost element in each column from all the components in that column, ensuring that each column contains at least one zero.
Step 3 – Assign zeros
 Analyse the rows one by one until you find a row with precisely one unmarked zero. Encircle this lonely unmarked zero and assign it a task. All other zeros in the column of this circular zero should be crossed out because they will not be used in any future assignments. Continue in this manner until you’ve gone through all of the rows.
 Examine the columns one by one until you find one with precisely one unmarked zero. Encircle this single unmarked zero and cross any other zero in its row to make an assignment to it. Continue until you’ve gone through all of the columns.
Step 4 – Perform the Optimal Test
 The present assignment is optimal if each row and column has exactly one encircled zero.
 The present assignment is not optimal if at least one row or column is missing an assignment (i.e., if at least one row or column is missing one encircled zero). Continue to step 5. Subtract the least cost element from all the entries in each column of the final cost matrix created in step 1 and ensure that each column has at least one zero.
Step 5 – Draw the least number of straight lines to cover all of the zeros as follows:
(a) Highlight the rows that aren’t assigned.
(b) Label the columns with zeros in marked rows (if they haven’t already been marked).
(c) Highlight the rows that have assignments in indicated columns (if they haven’t previously been marked).
(d) Continue with (b) and (c) until no further marking is needed.
(f) Simply draw the lines through all rows and columns that are not marked. If the number of these lines equals the order of the matrix, then the solution is optimal; otherwise, it is not.
Step 6 – Find the lowest cost factor that is not covered by the straight lines. Subtract this leastcost component from all the uncovered elements and add it to all the elements that are at the intersection of these straight lines, but leave the rest of the elements alone.
Step 7 – Continue with steps 1 – 6 until you’ve found the highest suitable assignment.


Hungarian Method Example
Use the Hungarian method to solve the given assignment problem stated in the table. The entries in the matrix represent each man’s processing time in hours.
\(\begin{array}{l}\begin{bmatrix} & I & II & III & IV & V \\1 & 20 & 15 & 18 & 20 & 25 \\2 & 18 & 20 & 12 & 14 & 15 \\3 & 21 & 23 & 25 & 27 & 25 \\4 & 17 & 18 & 21 & 23 & 20 \\5 & 18 & 18 & 16 & 19 & 20 \\\end{bmatrix}\end{array} \)
With 5 jobs and 5 men, the stated problem is balanced.
\(\begin{array}{l}A = \begin{bmatrix}20 & 15 & 18 & 20 & 25 \\18 & 20 & 12 & 14 & 15 \\21 & 23 & 25 & 27 & 25 \\17 & 18 & 21 & 23 & 20 \\18 & 18 & 16 & 19 & 20 \\\end{bmatrix}\end{array} \)
Subtract the lowest cost element in each row from all of the elements in the given cost matrix’s row. Make sure that each row has at least one zero.
\(\begin{array}{l}A = \begin{bmatrix}5 & 0 & 3 & 5 & 10 \\6 & 8 & 0 & 2 & 3 \\0 & 2 & 4 & 6 & 4 \\0 & 1 & 4 & 6 & 3 \\2 & 2 & 0 & 3 & 4 \\\end{bmatrix}\end{array} \)
Subtract the least cost element in each Column from all of the components in the given cost matrix’s Column. Check to see if each column has at least one zero.
\(\begin{array}{l}A = \begin{bmatrix}5 & 0 & 3 & 3 & 7 \\6 & 8 & 0 & 0 & 0 \\0 & 2 & 4 & 4 & 1 \\0 & 1 & 4 & 4 & 0 \\2 & 2 & 0 & 1 & 1 \\\end{bmatrix}\end{array} \)
When the zeros are assigned, we get the following:
The present assignment is optimal because each row and column contain precisely one encircled zero.
Where 1 to II, 2 to IV, 3 to I, 4 to V, and 5 to III are the best assignments.
Hence, z = 15 + 14 + 21 + 20 + 16 = 86 hours is the optimal time.
Practice Question on Hungarian Method
Use the Hungarian method to solve the following assignment problem shown in table. The matrix entries represent the time it takes for each job to be processed by each machine in hours.
\(\begin{array}{l}\begin{bmatrix}J/M & I & II & III & IV & V \\1 & 9 & 22 & 58 & 11 & 19 \\2 & 43 & 78 & 72 & 50 & 63 \\3 & 41 & 28 & 91 & 37 & 45 \\4 & 74 & 42 & 27 & 49 & 39 \\5 & 36 & 11 & 57 & 22 & 25 \\\end{bmatrix}\end{array} \)
Stay tuned to BYJU’S – The Learning App and download the app to explore all Mathsrelated topics.
Frequently Asked Questions on Hungarian Method
What is hungarian method.
The Hungarian method is defined as a combinatorial optimization technique that solves the assignment problems in polynomial time and foreshadowed subsequent primal–dual approaches.
What are the steps involved in Hungarian method?
The following is a quick overview of the Hungarian method: Step 1: Subtract the row minima. Step 2: Subtract the column minimums. Step 3: Use a limited number of lines to cover all zeros. Step 4: Add some more zeros to the equation.
What is the purpose of the Hungarian method?
When workers are assigned to certain activities based on cost, the Hungarian method is beneficial for identifying minimum costs.
MATHS Related Links  
Leave a Comment Cancel reply
Your Mobile number and Email id will not be published. Required fields are marked *
Request OTP on Voice Call
Post My Comment
Register with BYJU'S & Download Free PDFs
Register with byju's & watch live videos.
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 costeffective or profitmaximizing assignment.
Solution of the Assignment Problem using the Hungarian Method
The Hungarian method is an algorithm that uses a stepbystep 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 reallife situations.
Applications in Computer Science
The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.
Applications in Economics
The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.
Applications in Logistics
The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.
Applications in Management
The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.
Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:
Task 1  Task 2  Task 3  

Emp 1  5  7  6 
Emp 2  6  4  5 
Emp 3  8  5  3 
The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.
Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:
Task 1  Task 2  Task 3  

Emp 1  0  2  1 
Emp 2  2  0  1 
Emp 3  5  2  0 
Next, we subtract the smallest entry in each column from all the entries of the column:
Task 1  Task 2  Task 3  

Emp 1  0  2  1 
Emp 2  2  0  1 
Emp 3  5  2  0 
0  0  0 
We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:
Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:
 Emp 1 to Task 3
 Emp 2 to Task 2
 Emp 3 to Task 1
This assignment results in a total time of 9 units.
I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.
Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.
How useful was this post?
Click on a star to rate it!
Average rating 0 / 5. Vote count: 0
No votes so far! Be the first to rate this post.
We are sorry that this post was not useful for you! 😔
Let us improve this post!
Tell us how we can improve this post?
Operations Research
1 Operations ResearchAn 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 ProgrammingSimplex Method
 Principle of Simplex Method
 Computational aspect of Simplex Method
 Simplex Method with several Decision Variables
 Two Phase and Mmethod
 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 NonLinear Programming
 Solution of a Nonlinear Programming Problem
 Convex and Concave Functions
 KuhnTucker 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 handcomputed simulation
 Computer simulation
13 Queueing Models
 Characteristics of a queueing model
 Notations and Symbols
 Statistical methods in queueing
 The M/M/I System
 The M/M/C System
 The M/Ek/I System
 Decision problems in queueing
Assignment Problem: Meaning, Methods and Variations  Operations Research
After reading this article you will learn about: 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.
Meaning of Assignment Problem:
An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.
The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.
Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.
Definition of Assignment Problem:
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 ith person is assigned to the jth job. The problem is to find an assignment (which job should be assigned to which person one onone 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:
Check it out now on O’Reilly
Dive in for free with a 10day 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.
 Mathematics
Ones assignment method for solving assignment problems
 January 2012
 Shahid Chamran University of Ahvaz
Discover the world's research
 25+ million members
 160+ million publication pages
 2.3+ billion citations
 R. Murugesan
 S. Thamarai Selvi
 T. ESAKKIAMMAL
 Research Scholar
 Tanuja S Dhope
 J. Boopalan
 Hamdy A. Taha
 J OPER RES SOC
 John J. Jarvis
 Shayle R. Searle
 Tapadar Rudrajit
 Recruit researchers
 Join for free
 Login Email Tip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? Keep me logged in Log in or Continue with Google Welcome back! Please log in. Email · Hint Tip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? Keep me logged in Log in or Continue with Google No account? Sign up
In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation.
Generalized Assignment Problem
 Reference work entry
 pp 1153–1162
 Cite this reference work entry
 O. Erhun Kundakcioglu 3 &
 Saed Alizamir 3
2934 Accesses
15 Citations
Article Outline
Introduction
MultipleResource Generalized Assignment Problem
Multilevel Generalized Assignment Problem
Dynamic Generalized Assignment Problem
Bottleneck Generalized Assignment Problem
Generalized Assignment Problem with Special Ordered Set
Stochastic Generalized Assignment Problem
BiObjective Generalized Assignment Problem
Generalized MultiAssignment Problem
Exact Algorithms
Heuristics
Conclusions
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
 Available as PDF
 Read on any device
 Instant download
 Own it forever
 Durable hardcover edition
 Dispatched in 3 to 5 business days
 Free shipping worldwide  see info
Tax calculation will be finalised at checkout
Purchases are for personal use only
Institutional subscriptions
Similar content being viewed by others
Some results on an assignment problem variant
Combinatorial clustering: literature review, methods, examples.
Introduction to Optimisation
AlbaredaSambola M, van der Vlerk MH, Fernandez E (2006) Exact solutions to a class of stochastic generalized assignment problems. Eur J Oper Res 173:465–487
Article MATH Google Scholar
Amini MM, Racer M (1994) A rigorous computational comparison of alternative solution methods for the generalized assignment problem. Manag Sci 40(7):868–890
Amini MM, Racer M (1995) A hybrid heuristic for the generalized assignment problem. Eur J Oper Res 87(2):343–348
Asahiro Y, Ishibashi M, Yamashita M (2003) Independent and cooperative parallel search methods for the generalized assignment problem. Optim Method Softw 18:129–141
Article MathSciNet MATH Google Scholar
Balachandran V (1976) An integer generalized transportation model for optimal job assignment in computer networks. Oper Res 24(4):742–759
Barnhart C, Johnson EL, Nemhauser GL, Savelsbergh MWP, Vance PH (1998) Branchandprice: column generation for solving huge integer programs. Oper Res 46(3):316–329
Beasley JE (1993) Lagrangean heuristics for location problems. Eur J Oper Res 65:383–399
Cario MC, Clifford JJ, Hill RR, Yang J, Yang K, Reilly CH (2002) An investigation of the relationship between problem characteristics and algorithm performance: a case study of the gap. IIE Trans 34:297–313
Google Scholar
Cattrysse DG, Salomon M, Van LN Wassenhove (1994) A set partitioning heuristic for the generalized assignment problem. Eur J Oper Res 72:167–174
Cattrysse DG, Van LN Wassenhove (1992) A survey of algorithms for the generalized assignment problem. Eur J Oper Res 60:260–272
Ceselli A, Righini G (2006) A branchandprice algorithm for the multilevel generalized assignment problem. Oper Res 54:1172–1184
Chalmet L, Gelders L (1976) Lagrangean relaxation for a generalized assignment type problem. In: Advances in OR. EURO, North Holland, Amsterdam, pp 103–109
Chu EC, Beasley JE (1997) A genetic algorithm for the generalized assignment problem. Comput Oper Res 24:17–23
Cohen R, Katzir L, Raz D (2006) An efficient approximation for the generalized assignment problem. Inf Process Lett 100:162–166
de Farias Jr, Johnson EL, Nemhauser GL (2000) A generalized assignment problem with special ordered sets: a polyhedral approach. Math Program, Ser A 89:187–203
de Farias Jr, Nemhauser GL (2001) A family of inequalities for the generalized assignment polytope. Oper Res Lett 29:49–55
DeMaio A, Roveda C (1971) An all zeroone algorithm for a class of transportation problems. Oper Res 19:1406–1418
Diaz JA, Fernandez E (2001) A tabu search heuristic for the generalized assignment problem. Eur J Oper Res 132:22–38
Drexl A (1991) Scheduling of project networks by job assignment. Manag Sci 37:1590–1602
Dyer M, Frieze A (1992) Probabilistic analysis of the generalised assignment problem. Math Program 55:169–181
Article MathSciNet Google Scholar
Feltl H, Raidl GR (2004) An improved hybrid genetic algorithm for the generalized assignment problem. In: SAC '04; Proceedings of the 2004 ACM symposium on Applied computing. ACM Press, New York, pp 990–995
Chapter Google Scholar
Fisher ML, Jaikumar R (1981) A generalized assignment heuristic for vehicle routing. Netw 11:109–124
Fisher ML, Jaikumar R, van Wassenhove LN (1986) A multiplier adjustment method for the generalized assignment problem. Manag Sci 32:1095–1103
Fleischer L, Goemans MX, Mirrokni VS, Sviridenko M (2006) Tight approximation algorithms for maximum general assignment problems. In SODA '06: Proceedings of the seventeenth annual ACMSIAM symposium on Discrete algorithm. ACM Press, New York, pp 611–620
Book Google Scholar
Freling R, Romeijn HE, Morales DR, Wagelmans APM (2003) A branchandprice algorithm for the multiperiod singlesourcing problem. Oper Res 51(6):922–939
French AP, Wilson JM (2002) Heuristic solution methods for the multilevel generalized assignment problem. J Heuristics 8:143–153
French AP, Wilson JM (2007) An lpbased heuristic procedure for the generalized assignment problem with special ordered sets. Comput Oper Res 34:2359–2369
Garey MR, Johnson DS (1990) Computers and Intractability; A Guide to the Theory of NPCompleteness. Freeman, New York
Gavish B, Pirkul H (1991) Algorithms for the multiresource generalized assignment problem. Manag Sci 37:695–713
Geoffrion AM, Graves GW (1974) Multicommodity distribution system design by benders decomposition. Manag Sci 20(5):822–844
Glover F, Hultz J, Klingman D (1979) Improved computer based planning techniques, part ii. Interfaces 4:17–24
Gottlieb ES, Rao MR (1990) \( (1,k) \) configuration facets for the generalized assignment problem. Math Program 46(1):53–60
Gottlieb ES, Rao MR (1990) The generalized assignment problem: Valid inequalities and facets. Math Stat 46:31–52
MathSciNet MATH Google Scholar
Guignard M, Rosenwein MB (1989) An improved dual based algorithm for the generalized assignment problem. Oper Res 37(4):658–663
Haddadi S (1999) Lagrangian decomposition based heuristic for the generalized assignment problem. Inf Syst Oper Res 37:392–402
Haddadi S, Ouzia H (2004) Effective algorithm and heuristic for the generalized assignment problem. Eur J Oper Res 153:184–190
HajriGabouj S (2003) A fuzzy genetic multiobjective optimization algorithm for a multilevel generalized assignment problem. IEEE Trans Syst 33:214–224
Janak SL, Taylor MS, Floudas CA, Burka M, Mountziaris TJ (2006) Novel and effective integer optimization approach for the nsf panelassignment problem: a multiresource and preferenceconstrained generalized assignment problem. Ind Eng Chem Res 45:258–265
Article Google Scholar
Jörnsten K, Nasberg M (1986) A new lagrangian relaxation approach to the generalized assignment problem. Eur J Oper Res 27:313–323
Jörnsten KO, Varbrand P (1990) Relaxation techniques and valid inequalities applied to the generalized assignment problem. AsiaP J Oper Res 7(2):172–189
Klastorin TD (1979) An effective subgradient algorithm for the generalized assignment problem. Comp Oper Res 6:155–164
Klastorin TD (1979) On the maximal covering location problem and the generalized assignment problem. Manag Sci 25(1):107–112
Kogan K, Khmelnitsky E, Ibaraki T (2005) Dynamic generalized assignment problems with stochastic demands and multiple agent task relationships. J Glob Optim 31:17–43
Kogan K, Shtub A, Levit VE (1997) Dgap – the dynamic generalized assignment problem. Ann Oper Res 69:227–239
Kuhn H (1995) A heuristic algorithm for the loading problem in flexible manufacturing systems. Int J Flex Manuf Syst 7:229–254
Laguna M, Kelly JP, GonzfilezVelarde JL, Glover F (1995) Tabu search for the multilevel generalized assignment problem. Eur J Oper Res 82:176–189
Lawler E (1976) Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, Winston, New York
MATH Google Scholar
Lin BMT, Huang YS, Yu HK (2001) On the variabledepthsearch heuristic for the linearcost generalized assignment problem. Int J Comput Math 77:535–544
Lorena LAN, Narciso MG (1996) Relaxation heuristics for a generalized assignment problem. Eur J Oper Res 91:600–610
Lorena LAN, Narciso MG, Beasley JE (2003) A constructive genetic algorithm for the generalized assignment problem. J Evol Optim
Lourenço HR, Serra D (1998) Adaptive approach heuristics for the generalized assignment problem. Technical Report 288, Department of Economics and Business, Universitat Pompeu Fabra, Barcelona
Lourenço HR, Serra D (2002) Adaptive search heuristics for the generalized assignment problem. Mathw Soft Comput 9(2–3):209–234
Martello S, Toth P (1981) An algorithm for the generalized assignment problem. In: Brans JP (ed) Operational Research '81, 9th IFORS Conference, NorthHolland, Amsterdam, pp 589–603
Martello S, Toth P (1990) Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York
Martello S, Toth P (1992) Generalized assignment problems. Lect Notes Comput Sci 650:351–369
MathSciNet Google Scholar
Martello S, Toth P (1995) The bottleneck generalized assignment problem. Eur J Oper Res 83:621–638
Mazzola JB, Neebe AW (1988) Bottleneck generalized assignment problems. Eng Costs Prod Econ 14(1):61–65
Mazzola JB, Wilcox SP (2001) Heuristics for the multiresource generalized assignment problem. Nav Res Logist 48(6):468–483
Monfared MAS, Etemadi M (2006) The impact of energy function structure on solving generalized assignment problem using hopfield neural network. Eur J Oper Res 168:645–654
Morales DR, Romeijn HE (2005) Handbook of Combinatorial Optimization, supplement vol B. In: Du DZ, Pardalos PM (eds) The Generalized Assignment Problem and extensions. Springer, New York, pp 259–311
Narciso MG, Lorena LAN (1999) Lagrangean/surrogate relaxation for generalized assignment problems. Eur J Oper Res 114:165–177
Nauss RM (2003) Solving the generalized assignment problem: an optimizing and heuristic approach. INFORMS J Comput 15(3):249–266
Nauss RM (2005) The elastic generalized assignment problem. J Oper Res Soc 55:1333–1341
Nowakovski J, Schwarzler W, Triesch E (1999) Using the generalized assignment problem in scheduling the rosat space telescope. Eur J Oper Res 112:531–541
Nutov Z, Beniaminy I, Yuster R (2006) A \( (11/e) \) ‐approximation algorithm for the generalized assignment problem. Oper Res Lett 34:283–288
Park JS, Lim BH, Lee Y (1998) A lagrangian dualbased branchandbound algorithm for the generalized multiassignment problem. Manag Sci 44(12S):271–275
Pigatti A, de Aragao MP, Uchoa E (2005) Stabilized branchandcutandprice for the generalized assignment problem. In: Electronic Notes in Discrete Mathematics, vol 19 of 2nd Brazilian Symposium on Graphs, Algorithms and Combinatorics, pp 385–395,
Osman IH (1995) Heuristics for the generalized assignment problem: simulated annealing and tabu search approaches. ORSpektrum 17:211–225
Racer M, Amini MM (1994) A robust heuristic for the generalized assignment problem. Ann Oper Res 50(1):487–503
Romeijn HE, Morales DR (2000) A class of greedy algorithms for the generalized assignment problem. Discret Appl Math 103:209–235
Romeijn HE, Morales DR (2001) Generating experimental data for the generalized assignment problem. Oper Res 49(6):866–878
Romeijn HE, Piersma N (2000) A probabilistic feasibility and value analysis of the generalized assignment problem. J Comb Optim 4:325–355
Ronen D (1992) Allocation of trips to trucks operating from a single terminal. Comput Oper Res 19(5):445–451
Ross GT, Soland RM (1975) A branch and bound algorithm for the generalized assignment problem. Math Program 8:91–103
Ross GT, Soland RM (1977) Modeling facility location problems as generalized assignment problems. Manag Sci 24:345–357
Ross GT, Zoltners AA (1979) Weighted assignment models and their application. Manag Sci 25(7):683–696
Savelsbergh M (1997) A branchandprice algorithm for the generalized assignment problem. Oper Res 45:831–841
Shmoys DB, Tardos E (1993) An approximation algorithm for the generalized assignment problem. Math Program 62:461–474
Shtub A (1989) Modelling group technology cell formation as a generalized assignment problem. Int J Prod Res 27:775–782
Srinivasan V, Thompson GL (1973) An algorithm for assigning uses to sources in a special class of transportation problems. Oper Res 21(1):284–295
Stützle T, Hoos H (1999) The MaxMin Ant System and Local Search for Combinatorial Optimization Problems. In: Voss S, Martello S, Osman IH, Roucairol C (eds) Metaheuristics; Advances and trends in local search paradigms for optimization. Kluwer, Boston, pp 313–329
Toktas B, Yen JW, Zabinsky ZB (2006) Addressing capacity uncertainty in resourceconstrained assignment problems. Comput Oper Res 33:724–745
Trick M (1992) A linear relaxation heuristic for the generalized assignment problem. Nav Res Logist 39:137–151
Trick MA (1994) Scheduling multiple variablespeed machines. Oper Res 42(2):234–248
Wilson JM (1997) A genetic algorithm for the generalised assignment problem. J Oper Res Soc 48:804–809
Wilson JM (2005) An algorithm for the generalized assignment problem with special ordered sets. J Heuristics 11:337–350
Yagiura M, Ibaraki T, Glover F (2004) An ejection chain approach for the generalized assignment problem. INFORMS J Comput 16:133–151
Yagiura M, Ibaraki T, Glover F (2006) A path relinking approach with ejection chains for the generalized assignment problem. Eur J Oper Res 169:548–569
Yagiura M, Yamaguchi T, Ibaraki T (1998) A variable depth search algorithm with branching search for the generalized assignment problem. Optim Method Softw 10:419–441
Yagiura M, Yamaguchi T, Ibaraki T (1999) A variable depth search algorithm for the generalized assignment problem. In: Voss S, Martello S, Osman IH, Roucairol C (eds) Metaheuristics; Advances and Trends in Local Search paradigms for Optimization, Kluwer, Boston, pp 459–471
Zhang CW, Ong HL (2007) An efficient solution to biobjective generalized assignment problem. Adv Eng Softw 38:50–58
Zimokha VA, Rubinshtein MI (1988) R & d planning and the generalized assignment problem. Autom Remote Control 49:484–492
Download references
Author information
Authors and affiliations.
Department of Industrial and Systems Engineering, University of Florida, Gainesville, USA
O. Erhun Kundakcioglu & Saed Alizamir
You can also search for this author in PubMed Google Scholar
Editor information
Editors and affiliations.
Department of Chemical Engineering, Princeton University, Princeton, NJ, 085445263, USA
Christodoulos A. Floudas
Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 326116595, USA
Panos M. Pardalos
Rights and permissions
Reprints and permissions
Copyright information
© 2008 SpringerVerlag
About this entry
Cite this entry.
Kundakcioglu, O.E., Alizamir, S. (2008). Generalized Assignment Problem . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/9780387747590_200
Download citation
DOI : https://doi.org/10.1007/9780387747590_200
Publisher Name : Springer, Boston, MA
Print ISBN : 9780387747583
Online ISBN : 9780387747590
eBook Packages : Mathematics and Statistics Reference Module Computer Science and Engineering
Share this entry
Anyone you share the following link with will be able to read this content:
Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt contentsharing initiative
 Publish with us
Policies and ethics
 Find a journal
 Track your research
IMAGES
VIDEO
COMMENTS
The procedure used to solve assignment problems wherein one reduces the original assignment costs to a table of opportunity costs is called _____. steppingstone method; matrix reduction; MODI method; northwest reduction; simplex reduction; View answer. Correct answer: (B)
The Hungarian method is a computational optimization technique that addresses the assignment problem in polynomial time and foreshadows following primaldual alternatives. In 1955, Harold Kuhn used the term "Hungarian method" to honour two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry. Let's go through the steps of the Hungarian method with the help of a solved example.
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.
This is an unbalanced assignment problem. One way to solve it is to invent a fourth dummy task, perhaps called "sitting still doing nothing", with a cost of 0 for the taxi assigned to it. This reduces the problem to a balanced assignment problem, which can then be solved in the usual way and still give the best solution to the problem.
The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. In an assignment problem , we must find a maximum matching that has the minimum weight in a weighted bipartite graph .
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 ...
5.1 INTRODUCTION. The assignment problem is one of the special type of transportation problem for which more efficient (lesstime consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY ...
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 is a special type of Linear Programming Problem based on the following assumptions: However, solving this task for increasing number of jobs and/or resources calls for…
Definition of Assignment Problem. The statement of the assignment problem is as follows: There are n men and n jobs, with a cost c, for assigning man i to job j. It is required to assign all men to jobs such that one and only one man is assigned to each job and the total cost of the assignments is minimal.
50. The procedure used to solve assignment problems wherein one reduces the original assignment costs to a table of opportunity costs is called _____. A. steppingstone method B. matrix reduction C. MODI method D. northwest reduction E. simplex reduction 51. The method of finding an initial solution based upon opportunity costs is called
At most one cell can depart an input at a time.! At most one cell can arrive at an output at a time.! Cell arrives at input x and must be routed to output y. x3 x2 x1 y1 y2 y3 inputs outputs 20 IputQueued Switching FIFO queueing. Each input x maintains one queue of cells to be routed. Headofline blocking (HOL).!
The assignment problem is a standard topic discussed in operations research textbooks [8] and [10]. It is an important subject, put forward immediately after the transportation problem, is the assignment problem. This is particularly important in the theory of decision making. The assignment problem is one of the earliest
Step 2: Add a Dummy source or dummy destination, so that the cost table becomes a square matrix. The cost entries of dummy source / destinations are always zero Go to Step 3. Step 3: Obtain the Total Opportunity Cost Table (TOCT). Optimal Solution for Assignment Problem by Average Total Opportunity Cost Method 23.
modified to solve the assignment problem [3],[4],[5]. Also the signature method for the assignment problem was presented by Balinski [6]. Kore [7] proposed a new approach to solve an unbalanced assignment problem without balancing it. Basirzadeh [8] developed a Hungarianlike method,
In this paper, a new and simple metho d was introduced for solving assignment. problem. This method can b e used for all kinds of assignment problems, whether maximize or minimize ob jective ...
Tables 2, 3, 4, and 5 present the steps required to determine the appropriate job assignment to the machine. Step 1 By taking the minimum element and subtracting it from all the other elements in each row, the new table will be: Table 2 represents the matrix after completing the 1st step. Table 1 Initial table of a.
The selected zeros correspond to the ideal assignment in the original matrix. Once you get used to the process, the Hungarian Algorithm is a cinch, so keep practicing. To unlock this lesson you ...
The procedure used to solve assignment problems by reducing the original assignment costs to a table of opportunity costs is commonly referred to as the Hungarian method or the reduced cost method. The Hungarian method simplifies an assignment problem by converting all the potential costs into opportunity costs, thereby making it easier to ...
Improvement index Q104  The procedure used to solve assignment problems wherein one reduces the original assignment costs to a table of opportunity costs is called _____. steppingstone method. matrix reduction. MODI method. northwest reduction Q105  The method of finding an initial solution based upon opportunity costs is called_____.
The procedure used to solve assignment problems wherein one reduces the original assignment costs to a table of opportunity costs is called _____. a. steppingstone method b. matrix reduction c. MODI method d. northwest reduction e. simplex reduction correct answer B. The method of finding an initial solution based upon opportunity costs is ...
(i.e., to find a route in the assignment problem table), thus the development of a costeffective and realistic solution is highly desirable to solve the assignment problem in linear programming. In 1955, The Hungarian method was developed by H.W. Kuhn [6] combining the thoughts of two mathematicians: D. König [7] and J. Egerváry [8].
The generalized assignment problem (GAP) seeks the minimum cost assignment of n tasks to m agents such that each task is assigned to precisely one agent subject to capacity restrictions on the agents. The formulation of the problem is: where \ ( c_ {ij} \) is the cost of assigning task j to agent i , \ ( a_ {ij} \) is the capacity used when ...