assignment method operations management

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:

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.

\(\begin{array}{l}\begin{bmatrix}J/M & I & II & III & IV & V \\1 & 9 & 22 & 58 & 11 & 19 \\2 & 43 & 78 & 72 & 50 & 63 \\3 & 41 & 28 & 91 & 37 & 45 \\4 & 74 & 42 & 27 & 49 & 39 \\5 & 36 & 11 & 57 & 22 & 25 \\\end{bmatrix}\end{array} \)

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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.

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Prof. Charles H. Fine

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Sloan School of Management

As Taught In

Operations Management
Supply Chain Management

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Assignments, written assignments.

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Procedure, Example Solved Problem | Operations Research - Solution of assignment problems (Hungarian Method) | 12th Business Maths and Statistics : Chapter 10 : Operations Research

Chapter: 12th business maths and statistics : chapter 10 : operations research.

Solution of assignment problems (Hungarian Method)

First check whether the number of rows is equal to the numbers of columns, if it is so, the assignment problem is said to be balanced.

Step :1 Choose the least element in each row and subtract it from all the elements of that row.

Step :2 Choose the least element in each column and subtract it from all the elements of that column. Step 2 has to be performed from the table obtained in step 1.

Step:3 Check whether there is atleast one zero in each row and each column and make an assignment as follows.

Step :4 If each row and each column contains exactly one assignment, then the solution is optimal.

Example 10.7

Solve the following assignment problem. Cell values represent cost of assigning job A, B, C and D to the machines I, II, III and IV.

Here the number of rows and columns are equal.

∴ The given assignment problem is balanced. Now let us find the solution.

Step 1: Select a smallest element in each row and subtract this from all the elements in its row.

Look for atleast one zero in each row and each column.Otherwise go to step 2.

Step 2: Select the smallest element in each column and subtract this from all the elements in its column.

Since each row and column contains atleast one zero, assignments can be made.

Step 3 (Assignment):

Thus all the four assignments have been made. The optimal assignment schedule and total cost is

The optimal assignment (minimum) cost

Example 10.8

Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment schedule.

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

Column 3 contains no zero. Go to Step 2.

Thus all the five assignments have been made. The Optimal assignment schedule and total cost is

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

Since the number of columns is less than the number of rows, given assignment problem is unbalanced one. To balance it , introduce a dummy column with all the entries zero. The revised assignment problem is

Here only 3 tasks can be assigned to 3 men.

Step 1: is not necessary, since each row contains zero entry. Go to Step 2.

Step 3 (Assignment) :

Since each row and each columncontains exactly one assignment,all the three men have been assigned a task. But task S is not assigned to any Man. The optimal assignment schedule and total cost is

The optimal assignment (minimum) cost = ₹ 35

Assignment Method

Assignment Problem is a special type of linear programming problem where the objective is to minimize the cost or time of completing a number of jobs by a number of persons. The assignment problem in the general form can be stated as follows: “Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimized (Maximized or Minimized).”Several problems of management have a structure identical with the assignment problem. Example I: A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each person is given. His objective is to assign each person to one and only one job in such a way that the total cost of assignment is minimized. Example II: A manager has four operators for four separate jobs and the time of completion of each job by each operator is given. His objective is to assign each operator to one and only one job in such a way that the total time of completion is minimized. Example III: A tourist car operator has four cars in each of the four cities and four customers in four different cities. The distance between different cities is given. His objective is to assign each car to one and only one customer in such a way that the total distance covered is minimized.

Hungarian Method Although an assignment problem can be formulated as a linear programming problem, it is solved by a special method known as Hungarian Method because of its special structure. If the time of completion or the costs corresponding to every assignment is written down in a matrix form, it is referred to as a Cost matrix.

The Hungarian Method is based on the principle that if a constant is added to every element of a row and/or a column of cost matrix, the optimum solution of the resulting assignment problem is the same as the original problem and vice versa. The original cost matrix can be reduced to another cost matrix by adding constants to the elements of rows and columns where the total cost or the total completion time of an assignment is zero. Since the optimum solution remains unchanged after this reduction, this assignment is also the optimum solution of the original problem. If the objective is to maximize the effectiveness through Assignment, Method can be applied to a revised cost matrix obtained from the original matrix.

Balanced Assignment Problem Balanced Assignment Problem is an assignment problem where the number of facilities is equal to the number of jobs. Unbalanced Assignment Problem Unbalanced Assignment problem is an assignment problem where the number of facilities is not equal to the number of jobs. To make unbalanced assignment problem, a balanced one, a dummy facility(s) or a dummy job(s) (as the case may be) is introduced with zero cost or time. Dummy Job/Facility A dummy job or facility is an imaginary job/facility with zero cost or time introduced to make an unbalanced assignment problem balanced. An Infeasible Assignment An Infeasible Assignment occurs in the cell (i, j) of the assignment cost matrix if ith person is unable to perform jth job. It is sometimes possible that a particular person is incapable of doing certain work or a specific job cannot be performed on a particular machine. The solution of the assignment problem should take into account these restrictions so that the infeasible assignments can be avoided. This can be achieved by assigning a very high cost to the cells where assignments are prohibited. Practical Steps Involved In Solving Minimization Problems Step 1: See whether Number of Rows are equal to Number of Column. If yes, problem is balanced one; if not, then add a Dummy Row or Column to make the problem a balanced one by allotting zero value or specific value (if any given) to each cell of the Dummy Row or Column, as the case may be. Step 2: Row Subtraction: Subtract the minimum element of each row from all elements of that row. Note: If there is zero in each row, there is no need for row subtraction. Step 3: Column Subtraction: Subtract the minimum element of each column from all elements of that column. Note: If there is zero in each column, there is no need for column subtraction. Step 4: Draw minimum number of Horizontal and/or Vertical Lines to cover all zeros.

To draw minimum number of lines the following procedure may be followed: 1. Select a row containing exactly one uncovered zero and draw a vertical line through the column containing this zero and repeat the process till no such row is left.

2. Select a column containing exactly one uncovered zero and draw a horizontal line through the row containing the zero and repeat the process till no such column is left. Step 5: If the total lines covering all zeros are equal to the size of the matrix of the Table, we have got the optimal solution; if not, subtract the minimum uncovered element from all uncovered elements and add this element to all elements at the intersection point of the lines covering zeros. Step 6: Repeat Steps 4 and 5 till minimum number of lines covering all zeros is equal to the size of the matrix of the Table. Step 7: Assignment: Select a row containing exactly one unmarked zero and surround it by , ‘and draw a vertical line through the column containing this zero. Repeat this process till no such row is left; then select a column containing exactly one unmarked zero and surround it by, and draw a horizontal line through the row containing this zero and repeat this process till no such column is left. Note: If there is more than one unmarked zero in any row or column, it indicates that an alternative solution exists. In this case, select anyone arbitrarily and pass two lines horizontally and vertically. Step 8: Add up the value attributable to the allocation, which shall be the minimum value. Step 9 : Alternate Solution: If there are more than one unmarked zero in any row or column, select the other one (i.e., other than the one selected in Step 7) and pass two lines horizontally and vertically. Add up the value attributable to the allocation, which shall be the minimum value.

Minimization Problem Carefree Corporation has four plants each of which can manufacture anyone of the four products. Product costs differ from one plant to another as follow:

Solution Part (a) Step 1 : Row Deduction: Subtracting the minimum element of each row from all the elements of that row:

Step 2 : Column Deduction: Subtracting the minimum element of each column of above matrix from all the elements of that column and then drawing the minimum number of lines (whether horizontal/vertical) to cover all the zeros:

Since number of lines drawn =3 and order of matrix =4, we will have to take the step to increase the number of zeros. Step 3 ; Subtracting the minimum uncovered element (1 in this case) from all the uncovered elements and adding to the elements at intersection points, and then drawing the minimum numbers of lines to cover all zeros.

VARIATIONS IN THE ASSIGNMENT PROBLEM

The Maximal Assignment Problem Sometimes, the assignment problem deals with the maximization of an objective function rather than to, minimize it. For example, it may be required to assign persons to jobs in such a way that the expected profit is maximum. Such problem may be solved easily by first converting it to a minimization problem and then applying the usual procedure of assignment algorithm.

This conversion can be very easily done by subtracting from the highest element, all the elements of the given profit matrix; or equivalently, by placing minus sign before each element of the profit-matrix in order to make it cost-matrix. Following examples will make the procedure clear.

Example 1. (Maximization Problem)

A company has 5jobs to be done. The following matrix shows the return in rupees on assigning I th (i = 1,2,3,4,5) machine to the jth job (j =A, B, C, D, E).

Assign the five jobs to the five machines so as to maximize the total expected profit.

Step 1. Converting from Maximization to Minimization: Since the highest element is ]4, so subtracting all the elements from ]4, the following reduced cost (opportunity loss of maximum profit) matrix is obtained.

Step 2. Now following the usual procedure of solving an assignment problem, an optimal assignment is obtained in the following table:

Example 2. (Maximization Problem). A company has four territories open, and four salesmen available for assignment. The territories are not equally rich in their sales potential.. It is estimated that a typical salesman operating in each territory would bring in the following annual sales:

If the criterion is maximum expected total sales, then intuitive answer is to assign the best salesman to the richest territory, the next best salesman to the second richest, and so on. Verify this answer by the assignment technique. Step 1. To construct the effectiveness matrix: In order to avoid the fractional values of annual sales of each salesman in each territory, it will be rather convenient to consider the sales for 21 years (the sum of proportions: 7 + 5 + 5 + 4 =21), taking Rs. 10,000 as one unit. Divide the individual sales in each territory by 21, if the annual sales by salesman are required. Thus, the sales matrix for maximization is obtained as follows:

Step 2. (To convert ‘the maximum sales matrix’ to ‘minimum sales matrix’.) The problem of ‘maximization’ can be converted to ‘minimization’ one, by simply multiplying each element of given matrix (Table) by -I.

Thus resulting matrix becomes:

Step 3. Subtracting the smallest element in each row from every element in that row, we get the reduced matrix.

Step 4. Subtract the smallest element in each column from every element in that column to get the second reduced matrix. Step 5. Since all zeros in Table can be covered by minimum number of lines.(L1 ,L2) , which is less than 4 (the number of rows in the matrix), the optimal assignment is not possible at this stage.

In Table select the minimum element’ l’ among all uncovered elements. Then subtract this value 1 from each uncovered element, and add 1 at the intersection of two lines L1,L2 . Thus, the revised matrix is obtained as Table:

Step 6. Again repeat Step 5. Since the minimum number of lines (L1, L2) in Table to cover all zeros is less than 4 (the number of rows/columns), subtract the min. element 1 from all uncovered elements and add 1 at the intersection of lines (L1 ,L2) and (L1 ,L2). Then find the optimal assignment as explained in Step 7.

Step 7. To find an optimal assignment. Since there is a single zero element in row 1and column 4 only, make the zero assignment by putting ‘0 ‘ around these two zeros and cross-out other zeros in column 1 and row 4. Other zero-assignments are quite obvious from the following tables:

Thus, two possible solutions are: (i) A-I, B-II, C-III, D-IV; (ii) A-I, B-III, C-II, D-IV. Both the solutions show that the best salesman A is assigned to the richest territory I, the worst salesman D to the poorest territory IV. Salesman B and C being equally good, so they may be assigned to either II or III. This verifies the answer.

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Please note you do not have access to teaching notes, operations management research methodologies using quantitative modeling.

International Journal of Operations & Production Management

ISSN : 0144-3577

Article publication date: 1 February 2002

Gives an overview of quantitative model‐based research in operations management, focusing on research methodology. Distinguishes between empirical and axiomatic research, and furthermore between descriptive and normative research. Presents guidelines for doing quantitative model‐based research in operations management. In constructing arguments, builds on learnings from operations research and operations management research from the past decades and on research from a selected number of other academic disciplines. Concludes that the methodology of quantitative model‐driven empirical research offers a great opportunity for operations management researchers to further advance theory.

Operations management
Methodology
Operational research
Quantitative modelling

Will M. Bertrand, J. and Fransoo, J.C. (2002), "Operations management research methodologies using quantitative modeling", International Journal of Operations & Production Management , Vol. 22 No. 2, pp. 241-264. https://doi.org/10.1108/01443570210414338

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What Is Operations Management (OM)?

Understanding operations management (om), operations and supply chain management (oscm), what operations managers do.

Frequently Asked Questions

The Bottom Line

Business Essentials

Operations Management: Understanding and Using It

Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and behavioral finance. Adam received his master's in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.

Operations management (OM) is the administration of business practices to create the highest level of efficiency possible within an organization. It is concerned with converting materials and labor into goods and services as efficiently as possible to maximize the profit of an organization.

Operations management teams attempt to balance costs with revenue to achieve the highest net operating profit possible.

Key Takeaways

Operations management (OM) is the administration of business practices to create the highest level of efficiency possible within an organization.
Operations management is concerned with converting materials and labor into goods and services as efficiently as possible.
Corporate operations management professionals try to balance costs with revenue to maximize net operating profit.

Katie Kerpel / Investopedia

Operations management involves utilizing resources from staff, materials, equipment, and technology. Operations managers acquire, develop, and deliver goods to clients based on client needs and the abilities of the company.

Operations management handles various strategic issues, including determining the size of manufacturing plants and project management methods and implementing the structure of information technology networks. Other operational issues include the management of inventory levels, including work-in-process levels and raw materials acquisition, quality control, materials handling, and maintenance policies.

Operations management entails studying the use of raw materials and ensuring that minimal waste occurs. Operations managers use numerous formulas, such as the economic order quantity formula, to determine when and how large an inventory order to process and how much inventory to hold on hand.

The combination of understanding and coordinating the work of a company is central to becoming a successful operations manager.

A critical function of operations management relates to the management of inventory through the supply chain. This process is known as operations and supply chain management (OSCM). To be an effective operations management professional, one must be able to understand the processes that are essential to what a company does and get them to flow and work together seamlessly. The coordination involved in setting up business processes in an efficient way requires a solid understanding of logistics .

An operations management professional understands local and global trends, customer demand, and available resources for production. Operations management approaches the acquisition of materials and the use of labor in a timely, cost-effective manner to deliver customer expectations. Inventory levels are monitored to ensure that excessive quantities are on hand. Operations management is responsible for finding vendors that supply the appropriate goods at reasonable prices and have the ability to deliver the product when needed.

Another large facet of operations management involves the delivery of goods to customers. This includes ensuring that products are delivered within the agreed time commitment. Operations management also typically follows up with customers to ensure that the products meet quality and functionality needs. Finally, operations management takes the feedback received and distributes the relevant information to each department to use in process improvement.

Operations managers are involved in coordinating and developing new processes while reevaluating current structures. Organization and productivity are two key drivers of being an operations manager, and the work often requires versatility and innovation. As part of their daily responsibilities, operations managers must possess a variety of skill sets, including:

Technical expertise in areas such as production automation, data entry, budget tracking, and design.
Organizational ability and attention to detail to include keeping track of project files, employee reports, budgets, schedules, and other details related to company processes.
Motivational prowess in the form of strong leadership skills that provide the expertise to motivate others, inspire ideas, and foster a supportive and diverse team.
Analytical aptitude , including skill in risk analysis and mitigation when initiating new projects. Operations managers also must analyze processes to identify challenges and offer solutions in the event that negative situations develop.
Decision-making proficiency , especially under stress when there is very little time to assess all factors.
Ability to maintain quality standards , including as they relate to raw materials, machinery, manufacturing procedures, packaging, delivery processes, and the finished product.

A master of business administration (MBA) degree in operations management can provide global perspective on industry trends and an awareness of financial regulations and political uncertainties that can affect an organization. It also provides a solid grasp of the inherent complexities and the tools needed to respond well to change.

What is the purpose of operations management (OM)?

Operations management (OM) is concerned with controlling the production process and business operations in the most efficient manner possible. OM professionals attempt to balance operating costs with revenue to maximize net operating profit.

What are some systems of operations management?

Modern operations management revolves around four theories:

Business process redesign (BPR) , which is focused on analyzing and designing workflow and business processes within a company. The goal of BPR is to help companies dramatically restructure the organization by designing the business process from the ground up.
Reconfigurable manufacturing systems , designed to incorporate accelerated change in structure, hardware, and software components. This allows systems to adjust rapidly to the capacity to which they can continue production and how efficiently they function in response to market or intrinsic system changes.
Six Sigma , an approach that focuses on quality. The word “six” references the control limits, which are placed at six standard deviations from the normal distribution mean. Tools used within the Six Sigma process include trending charts, potential defect calculations, and other ratios.
Lean manufacturing , which is the systematic elimination of waste within the manufacturing process. This theory sees resource use for any reason other than value creation for customers as wasteful and seeks to eliminate wasteful resource expenditures as much as possible.

What is an example of operations management?

Operations management is prevalent in the healthcare sector . The current healthcare system overuses expensive, technological, and emergency-based treatment. High costs from care often remain uncompensated due to uninsured patients. A prevalence of services in expensive settings creates a burden on taxpayers, health insurance holders, and healthcare institutions.

In simple terms, operations management (OM) is the process of prioritizing and employing business practices designed to achieve maximum efficiency as a means to achieve maximum profitability. Within the process, operations managers utilize organization and productivity to achieve their primary goals.

Balancing the efficient use of resources, including staff, materials, equipment, and technology, is key to a successful OM process and, by extension, to the success of the company.

University at Buffalo, School of Management. “ Operations and Supply Chain Management .”

Indeed. “ 15 Essential Operational Management Skills .”

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Operations Management Assignment

Assignment method operations management

Assignment method operations management.

They have developed the holistic Decision-making process to handle the operation and the designs at the process and the product level. Toyota also believed in providing top quality products and services which was evident in their production systems. In basic terms, the Monte Carlo approach or Monte Carlo simulation can be utilized to explain any strategy that estimates services to quantitative issues through analytical tasting.

They have been employing procedures to simplify the procedures, eliminate the wastage and speed up the production. Sample Assignment aims to provide operations management homework help through years of hands on experience and expertise.

Toyota Motor Kyushu Inc. It involves manipulation of raw materials, labour and energy to produce goods and services. So, the may not be as large as in the previous years, still the database of customers for the company is increasing and so the company would need to have proper technology support to handle the customers.

Apart from the automobile sector, Toyota is also involved in housing sector, Financial services sector, biotechnology sector and afforestation, marine industry and e-business sector, etc. But after they have started using the multi machine and multi process handling.

Johnston, There are also some products and services where the demand is seasonal. Product Development Assignment Help Supply Chain Management Assignment Online Process Design and Management Homework Help Service Assignments are given to test your understanding of the subject matter as well as ability to test your skills at tackling various problems and also various qualities pertaining to management, inventiveness, and leadership.

It involves two components which include maintenance and improvement. So, there would be a need to have an appropriate customer-processing technology. The suppliers are made aware of the general systems of work and the work flow of the company so that they can match their supplies accordingly. In a fixed position layout, the product remains at a single place and the machines and processes that are required to produce the product move to the product.

Heizer and Render, 3. Any significant change in the inputs would certainly result in a change in the processes and also in the output, thus a total change in the operations management procedure. They have firstly concentrated on creating a continuous flow of operations with the help of standardisation. Because of this, the company also follows activity based costing. The current sales of the company has been up topping 1,2 and 3 million marks in the years , and respectively. The visibility ensures that the supplier and the manufacturers have a transparent relation and they have discussed their problems first and both of them are well informed about each other.

They use the one-piece continuous flow for production system. Due to a rise in the fuel prices and economic slowdown there was a downturn that was witnessed in the industry. Gupta and Boyd, Also, when the company is operating in an international arena, there are sometimes when the company has to make changes in the operations to adjust to the national culture.

However, as illustrated in the NLVlMI example see OM in Practice. Because its intro in World War II, Monte Carlo simulation has actually been utilized to design a range of conceptual and physical systems.

It can be employed to assign the number of salespersons to a given territory or territories. They have integrated the manufacturing technology with the processes and the systems. The Monte Carlo technique, likewise called Monte Carlo analysis, is a method of analytical examination of mathematical function s utilizing random samples. Numerous various approaches have actually been proposed to resolve direct programs issues, however simplex method has actually shown to be the most efficient.