Assignment Model | Linear Programming Problem (LPP) | Introduction

What is assignment model.

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

i) There is only one assignment.

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

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

For example:

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

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

Fig 1-assigment model intro

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

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

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

→ The mathematical formulation for Assignment Model is given below:

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

assignment model balanced

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

→ Subjected to constraint;

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

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

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

Types of Assignment Problem:

(i) balanced assignment problem.

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

(ii) Unbalanced Assignment Problem

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

Methods to solve Assignment Model:

(i) integer programming method:.

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

So this can be formulated using 0 or 1 integer.

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

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

So this method becomes very lengthy and difficult to solve.

(ii) Transportation Methods:

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

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

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

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

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

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

So, the method becomes lengthy and time consuming.

(iii) Enumeration Method:

It is a simple trail and error type method.

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

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

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

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

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

(iv) Hungarian Method:

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

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

There are two main conditions for applying Hungarian Method:

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

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The Assignment Model

The linear programming formulation of the assignment model is similar to the formulation of the transportation model, except all the supply values for each source equal one, and all the demand values at each destination equal one. Thus, our example is formulated as follows :

assignment model balanced

This is a balanced assignment model. An unbalanced model exists when supply exceeds demand or demand exceeds supply.

assignment model balanced

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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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How to build a balanced assessment system

How to build a balanced assessment system

As my colleague Chase Nordengren said recently, teaching and learning have been transformed by COVID-19 school closures—and they’re unlikely to return to what we were used to anytime soon, if ever. They’ll also have a big impact on what most children are ready for in the fall.

Student learning differences are not a new challenge for educators. However, the scope and learning variance that students will display this fall is likely to be fairly significant. This moment in time is an opportunity to revisit and rebalance your assessment practices. In this post, I offer up a mental model for how a balanced assessment system—built on formative assessment practices—can guide instruction to meet the needs of your students. 

Make data easier to understand and use

There is a saying that schools can be data rich, but information poor. This means that you can have many sources of data on students but lack the coherent information you need to make effective decisions. It’s helpful to consult many sources of formal and informal data to inform your instructional design, of course, but without an intentional, well-thought-out plan for how all the sources of data fit together, it will be hard to make decisions well. A coherent approach to assessment practices can streamline decision-making and improve learning.

One way to achieve this coherence is by developing a balanced assessment system. A balanced assessment system intentionally makes use of formative, interim, and summative assessment practices—with the most emphasis placed on formative assessment. This type of system is at the heart of a Multi-Tiered System of Support (MTSS), which uses a decision-tree approach to assist in streamlining decisions, as shown below.

MTSS decision tree

Strike a balance

To create a balanced assessment system, there are two major domains that teachers need to consider:

  • The standards-based core instruction domain that aligns to grade-level or advanced content
  • The intervention domain for students who are not yet achieving standards and need additional support

Formative assessment plays a key role in both domains and should always be the starting point. It begins as a universal screening process for all students. Universal screening can take many forms, such as an early literacy probe, behavioral data, attendance patterns, grades, and even MAP® Growth™ or MAP® Reading Fluency™. The purpose, just like when doctors take your blood pressure and weight during an annual checkup, is to look for signs that something might be off track. Following the administration of a universal screening process, educators face a decision point that affects which of the two domains come into focus for teaching and learning.

For students who are more or less on track with the universal screening measures, teachers should proceed with business as usual in the core instruction domain, using formative assessment practices to connect to and activate prior knowledge in ways that guide the relationship of teaching and learning, check for understanding along the way, and assess mastery against grade-level outcomes to determine if future adjustments need to be made.

This moment in time is an opportunity to revisit and rebalance your assessment practices. […] [A] balanced assessment system—built on formative assessment practices—can guide instruction to meet the needs of your students. 

If the universal screener indicates that the learning or social-emotional well-being of a student is at risk, then the best course of action for teachers is to employ formative assessment practices that diagnose and pinpoint what support is needed within the intervention domain, monitor progress on a learning progression, and assess mastery of prerequisite learning.

How to move forward with core instruction

All students should experience teaching and learning that supports their success in the core instruction domain. This begins with teachers reviewing the scope and sequence of standards-aligned content, establishing clear learning targets, and using formative assessment data to develop responsive plans for lessons and units. The figure below illustrates three key assessment practices within core instruction: activate prior knowledge, check for understanding, and check for mastery and adjust instruction as needed.

Core instruction domain

Before core instruction: Activate prior knowledge

Lessons and units should start with formative assessment practices in the form of a pre-assessment or a process of activating prior knowledge. This serves the purpose of illustrating what students already know and assists teachers and students in understanding the learning path that students will need to take to reach the learning target.

Formative assessment at the beginning of a lesson or unit can take many forms, such as entrance tickets , K-W-L chart activities , Venn diagrams , think-pair-share , and more. No matter the type, a formative assessment activity at the beginning of a lesson or unit will create the context for helping you know how to adapt core instruction by adding more scaffolding for students who may struggle; adapting content to adjust for key background knowledge that the whole class may need to be successful; or developing differentiated paths for advanced students who may wish to go deeper with their learning in the particular content area.

During core instruction: Check for understanding

Formative assessment practices should take the form of checking for understanding. In a lesson, for example, this may occur when you monitor small group conversations, review students’ quick writing assignments, or listen to how students report out on jigsaw activities . Over the course of a unit, formative assessment should be occurring throughout, even incorporating more formal interim assessments , quizzes, and longer-term assignments.

All students should experience teaching and learning that supports their success in the core instruction domain.

What makes these practices formative is using them to adjust instruction to keep learning progressing. If the activities are used for grading or there’s no change to the long-term instructional trajectory, they no longer serve a formative purpose and swing over into the arena of summative assessment.

After core instruction: Check for mastery and adjust

At the end of a lesson or unit, a balanced assessment system will make use of purposeful summative assessment. If the learning targets were clear from the beginning, a summative assessment will focus solely on the success criteria by which students demonstrate that they have learned what was expected. It is often common practice that end-of-unit summative assessments do not serve a formative purpose. However, if you intend to reteach the content or proceed to a new unit that builds on the previous one, summative assessment can be utilized in a formative manner if there is an intentional effort made to adjust teaching and learning based on the degree to which students mastered the success criteria.

Tackling the intervention domain

In the intervention domain, assessment practices often take on different terminology and more formal designs, but they represent similar ideas to the core instruction domain and are guided by the principles of formative assessment. When students are identified by a universal screener as being at risk, adopt the MTSS sequence illustrated below: diagnose learning needs; monitor progress; and check for mastery and adjust.

Intervention domain

Before intervention: Diagnose learning needs

In elementary schools, educators often make the mistake of making intervention about the content of the universal screener. For example, an early literacy screener might emphasize reading fluency, so some teachers will make intervention about fluency. Without diagnosis, the teacher may not uncover that the root cause of the student’s poor fluency performance is an underlying issue with phonics.

By implementing a clear plan for diagnosis before intervening, you stay true to the idea of formative assessment by gaining the information you need to pinpoint the best starting point for teaching and learning. In early literacy, there are diagnostic assessments for phonemic awareness, phonics, comprehension, and more. In high school, a mathematics teacher may engage in diagnostic assessment by assessing students on a spectrum of math standards from lower grade levels. Regardless, the purpose of diagnostic assessment has the long-term learning trajectory in mind and can be matched with short-term success criteria that students can demonstrate to show their learning is on track. This creates the connection between diagnostic assessments and progress monitoring.

During intervention: Monitor progress

Once you pinpoint the entry level for intervention, instruction and a progress-monitoring plan are needed. For example, an eighth-grade algebra teacher may diagnose that a student has strengths in many areas but is struggling because they have not yet learned to identify when two expressions are equivalent (a sixth-grade standard). This means that during intervention, instruction would begin at this level, and a learning path would slowly build toward eighth-grade standards. Formative assessment would occur in the form of progress monitoring that is broken out to measure the success criteria of each step needed to meet the related eighth-grade math standards.

Similarly, in early literacy, when students have mastered their basic phonics skills but still need support working on automaticity, accuracy, and prosody (i.e., fluency), a teacher might choose to use the progress monitoring for oral reading test within MAP Reading Fluency as a progress-monitoring tool.

By implementing a clear plan for diagnosis before intervening, you stay true to the idea of formative assessment by gaining the information you need to pinpoint the best starting point for teaching and learning.

Similarly, in early literacy, if a third-grade student is identified as struggling with variant vowels (a first-grade skill), intervention would build from variant vowels and measure student progress toward mastery of this and successive phonics skills until the student demonstrates grade-appropriate success criteria with word reading.

After intervention: Check for mastery and adjust

Following instruction, student learning should be verified through a summative assessment that measures whether or not a student has mastered the goals that have been set within their learning progression. A summative approach could even be the same diagnostic assessment tool that was used to identify the student’s learning needs. If this is the case, the purpose changes from a formative, diagnostic use to a summative checkpoint that assesses mastery.

Tying it all together

Here’s a visual representation of the sequence and relationship between formative, interim, and summative assessment and the relevant assessment approaches that are most helpful in the core instruction domain and the intervention domain.

Balanced assessment system

Want to be sure you’re engaging in formative assessment every step of the way? Here’s how:

  • Use the information you glean about students before instruction to plan core instruction and intervention
  • Take what you learn during instruction to respond to students’ needs and adjust what comes next in your lesson or unit plans
  • Put summative assessment gathered after instruction to use guiding how you will reteach content or adjust your plans for the next unit

To explore this topic further—on your own or with your colleagues—try the following discussion questions:

Questions for teachers

  • What are ways to activate learning in your classroom?
  • During core instruction, how are you checking for understanding during the lesson?
  • How does instruction in the domain of intervention differ from the domain of core instruction?
  • How can you ensure your classroom has a balanced assessment system in place? In what ways do all of your assessment practices inform each other?
  • How have you determined the progression of learning that your students need?
  • How are you diagnosing or pinpointing student intervention needs within a learning progression?

Questions for leaders

  • What processes do you have in place to monitor school-wide data and reflect on improvements that are needed for teaching, learning, and leading?
  • How can you ensure there is a balanced assessment system in place system-wide? To what extent does your school have a systematic approach where different types of assessments inform each other?
  • How can you support teachers in identifying effective learning progressions and developing responsive plans that move students forward along a progression?
  • Does your school’s schedule assure there is sufficient time for both core instruction and intervention? 

This is the third in a series on formative assessment. Read the previous post. And read the entire series in our e-book .

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Solving Assignment Problem using Linear Programming in Python

Learn how to use Python PuLP to solve Assignment problems using Linear Programming.

In earlier articles, we have seen various applications of Linear programming such as transportation, transshipment problem, Cargo Loading problem, and shift-scheduling problem. Now In this tutorial, we will focus on another model that comes under the class of linear programming model known as the Assignment problem. Its objective function is similar to transportation problems. Here we minimize the objective function time or cost of manufacturing the products by allocating one job to one machine.

If we want to solve the maximization problem assignment problem then we subtract all the elements of the matrix from the highest element in the matrix or multiply the entire matrix by –1 and continue with the procedure. For solving the assignment problem, we use the Assignment technique or Hungarian method, or Flood’s technique.

The transportation problem is a special case of the linear programming model and the assignment problem is a special case of transportation problem, therefore it is also a special case of the linear programming problem.

In this tutorial, we are going to cover the following topics:

Assignment Problem

A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming.

For example, an operation manager needs to assign four jobs to four machines. The project manager needs to assign four projects to four staff members. Similarly, the marketing manager needs to assign the 4 salespersons to 4 territories. The manager’s goal is to minimize the total time or cost.

Problem Formulation

A manager has prepared a table that shows the cost of performing each of four jobs by each of four employees. The manager has stated his goal is to develop a set of job assignments that will minimize the total cost of getting all 4 jobs.  

Assignment Problem

Initialize LP Model

In this step, we will import all the classes and functions of pulp module and create a Minimization LP problem using LpProblem class.

Define Decision Variable

In this step, we will define the decision variables. In our problem, we have two variable lists: workers and jobs. Let’s create them using  LpVariable.dicts()  class.  LpVariable.dicts()  used with Python’s list comprehension.  LpVariable.dicts()  will take the following four values:

  • First, prefix name of what this variable represents.
  • Second is the list of all the variables.
  • Third is the lower bound on this variable.
  • Fourth variable is the upper bound.
  • Fourth is essentially the type of data (discrete or continuous). The options for the fourth parameter are  LpContinuous  or  LpInteger .

Let’s first create a list route for the route between warehouse and project site and create the decision variables using LpVariable.dicts() the method.

Define Objective Function

In this step, we will define the minimum objective function by adding it to the LpProblem  object. lpSum(vector)is used here to define multiple linear expressions. It also used list comprehension to add multiple variables.

Define the Constraints

Here, we are adding two types of constraints: Each job can be assigned to only one employee constraint and Each employee can be assigned to only one job. We have added the 2 constraints defined in the problem by adding them to the LpProblem  object.

Solve Model

In this step, we will solve the LP problem by calling solve() method. We can print the final value by using the following for loop.

From the above results, we can infer that Worker-1 will be assigned to Job-1, Worker-2 will be assigned to job-3, Worker-3 will be assigned to Job-2, and Worker-4 will assign with job-4.

In this article, we have learned about Assignment problems, Problem Formulation, and implementation using the python PuLp library. We have solved the Assignment problem using a Linear programming problem in Python. Of course, this is just a simple case study, we can add more constraints to it and make it more complicated. You can also run other case studies on Cargo Loading problems , Staff scheduling problems . In upcoming articles, we will write more on different optimization problems such as transshipment problem, balanced diet problem. You can revise the basics of mathematical concepts in  this article  and learn about Linear Programming  in this article .

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Assignment Problem: Meaning, Methods and Variations | Operations Research

assignment model balanced

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:

assignment model balanced

Balanced Transportation and Assignment Model

Jul 27, 2014

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November 5, 2012 AGEC 352-R. Keeney. Balanced Transportation and Assignment Model. ‘Balanced’ Transportation. Recall With 2000 total units (maximum) at harbor and 2000 units (minimum) demanded at assembly plants it is not possible for slack constraints Supply <=2000 Demand >=2000

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Presentation Transcript

November 5, 2012 AGEC 352-R. Keeney Balanced Transportation and Assignment Model

‘Balanced’ Transportation • Recall • With 2000 total units (maximum) at harbor and 2000 units (minimum) demanded at assembly plants it is not possible for slack constraints • Supply <=2000 • Demand >=2000 • Supply = Demand • Total movement of 2000 motors is the only feasible combination, leading all constraints to bind • One binding constraint is trivial

Balanced Transportation: Trivia • Transportation problems do not have to be balanced • Real world problems are rarely balanced • If you have an unbalanced model, might want to balance it with other activities • If Supply > Demand introduce a storage destination that takes up the excess • What is the cost of holding excess supply? • Storage costs or waste/spoil • If Demand > Supply introduce a penalty source that deals with the imbalance • What is the cost of shipping less than required? • Lost customers or contract penalties

Balanced Transportation:Important Facts • If the constraints have integers on RHS the optimal solution will have transport quantities in integers • This can be shown mathematically • Convenient for solving smaller problems by hand • Choose a route to enter the model, then keep adding until you hit the supply or demand constraint • In a balanced problem, one constraint is mathematically redundant • This is the trivial constraint and it is the one with the constraint that binds (LHS=RHS) but has a zero shadow price

Assignment Problem • The assignment problem is the mathematical allocation of ‘n’ agents or objects to ‘n’ tasks • The agents or objects are indivisible • Each can be assigned to one task only • Example using Autopower Company: • Auditing the Assembly Plants @ • Leipzig, Nancy, Liege, Tilburg • A VP is assigned to visit and spend two weeks conducting the audit • VP’s of Finance, Marketing, Operations, Personnel • Considerations… • Expertise to problem areas at plants • Time demands on VP • Language ability

Estimated Opportunity Costs(Objective Coefficient Matrix)

Assignment Problem-Costs Data • How do you get those costs? • Clearly when you are talking about opportunity costs and the additional cost of having someone out of their specialty or who is not a native speaker being assigned the problem a solution is heavily dependent on how reliable the opportunity cost information is • Perhaps the cost of having a full-time translator or additional support staff for a VP who is dealing with a lot of problems that are not her specialty • Other ways--think of skill/aptitude tests • ASVAB

Solving a Small Problem • Enumeration is a way of solving a small problem by hand • Enumeration means check all possible combinations… • Combinations for an ‘n’ valued assignment problem are just n factorial (n!) • n = 4  n!=4*3*2*1=24 • That’s still a lot to check • There are other tempting methods • Start with the lowest costs and work your way up?

Starting with the Lowest Cost • Tempting and seems logical but does not guarantee you an optimal solution • for a small problem we can find the best solution using tradeoffs • Think of the destinations as demanding VP with the lowest cost VP being the preference • Leipzig prefers Personnel • Nancy prefers Finance • Liege prefers Marketing • Tilburg prefers Finance • Two locations have Finance as a first preference, this is the only thing that makes this problem interesting

Tradeoffs (Spreadsheet) • Tradeoff 1: 1000 improvement • Tradeoff 2: 6000 worse • Tradeoff 3: 2000 improvement • Hopefully this convinces you that LP might be easier for solving these types of problems than wrangling all of the potential tradeoffs that occur

LP Setup • Setup is the same as the Balanced Transportation problem from last week • Destinations are the locations or assignments with >=1 constraints • Sources are the persons or objects to be assigned with <= 1 constraints • What is different? • Number of rows and columns are the same (i.e. square and balanced) • Not the case for transportation problems

Standard Algebraic Form for an Assignment Problem

Interpretation • Recall the problem from Monday’s lecture of assigning VP’s to plants to be audited • Objective • We want to minimize the cost of sending Vice Presidents to assembly plants given the per unit costs matrix C(i,j) • Assignees • The sources • For any assignee i, that assignee can be placed in a maximum of one assignment • Assignments • The destinations • For any destination j, the assignment requires that at least one assignee be put in place • Non-negativity • Decision variables must be zero or positive

Assignment Problems • Since the problem is balanced and assignments are 1 to 1 (1 person to 1 place) • Decision variables will all have an ending value of either 1 or 0 • Recall that balanced transport problems have integer solutions if RHS are integer values • In general, the assignment model can be formulated as a transportation model in which supply at each source and demand at each destination is equal to one

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Quantitative Techniques: Theory and Problems by P. C. Tulsian, Vishal Pandey

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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.

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assignment model balanced

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.

assignment model balanced

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.

assignment model balanced

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.

assignment model balanced

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.

assignment model balanced

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

Step 3 (Assignment):

assignment model balanced

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

assignment model balanced

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.

assignment model balanced

∴ The given assignment problem is balanced.

Now let us find the solution.

The cost matrix of the given assignment problem is

assignment model balanced

Column 3 contains no zero. Go to Step 2.

assignment model balanced

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

assignment model balanced

The optimal assignment (minimum) cost = ` 9

Example 10.9

Solve the following assignment problem.

assignment model balanced

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

assignment model balanced

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.

assignment model balanced

Step 3 (Assignment) :

assignment model balanced

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

assignment model balanced

The optimal assignment (minimum) cost = ₹ 35

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In: Advanced Math

Explain what it means for an assignment model to be balanced. Give detailed mathematical examples.

Expert Solution

What is Balanced assignment problem :

assignment model balanced

Mathematical Formulation of the above problem is:

assignment model balanced

1.Balance assignment problem:

A company has four machines that are used for four jobs. Each job can be assigned to one and only one machine. The cost of each job on each machine is given in the following Table.

assignment model balanced

2.Unbalance assignment problem:

A company has four machines that are used for three jobs. Each job can be assigned to one and only one machine. The cost of each job on each machine is given in the following Table.

assignment model balanced

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  1. Assignment model

    Assignment model - Example 1 (Balanced) maxus knowledge 27K subscribers Subscribe Subscribed 114 Share 45K views 10 years ago Assignment model In this video you will learn how to solve an...

  2. Assignment problem

    This is a balanced assignment problem. Its solution is whichever combination of taxis and customers results in the least total cost. Now, suppose that there are four taxis available, but still only three customers. This is an unbalanced assignment problem.

  3. Assignment Model

    (i) Balanced Assignment Problem It consist of a suqare matrix (n x n). Number of rows = Number of columns (ii) Unbalanced Assignment Problem It consist of a Non-square matrix. Number of rows = Number of columns

  4. PDF The Assignment Models

    total cost. The situation is known as the assignment problem. An assignment problem is a balanced transportation problem in which all supplies and demands equal to 1. In this model jobs represent supplies and machines represent demands. In the following table we can see the general representation of the assignment problem. Table 5.5.1 Jobs

  5. PDF Chapter 5: Linear Programming: Transportation and Assignment Models

    the resulting formulation is called a balanced transportation model, otherwise it is unbalanced. The balanced model differs from the unbalanced model only in the fact that all constraints are equations. That is Minimize: ij m i n j ∑∑Cij x = =1 1 Subject to: i n j ∑xij =a =1 i = 1, 2, ….,m (units supplied)

  6. The Assignment Model

    This is a balanced assignment model. An unbalanced model exists when supply exceeds demand or demand exceeds supply. Previous page Table of content Next page Introduction to Management Science (10th Edition) ISBN: 0136064361 EAN: 2147483647 Year: 2006 Pages: 358 Authors: Bernard W. Taylor BUY ON AMAZON

  7. Solving an Assignment Problem

    Print the solution. The following code prints the solution to the problem. Here is the output of the program. Total cost = 265.0 Worker 0 assigned to task 3. Cost = 70 Worker 1 assigned to task 2. Cost = 55 Worker 2 assigned to task 1. Cost = 95 Worker 3 assigned to task 0. Cost = 45.

  8. How to solve Assignment Modelling Exercises

    In this video, you will learn- meaning of Assignment model.- balanced and unbalanced Assignment model.-Methods of solving Assignment Model problems.-how to s...

  9. What is Assignment Problem

    Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons. The assignment problem in the general form can be stated as follows: "Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to ...

  10. How to build a balanced assessment system

    To create a balanced assessment system, there are two major domains that teachers need to consider: The standards-based core instruction domain that aligns to grade-level or advanced content. The intervention domain for students who are not yet achieving standards and need additional support. Formative assessment plays a key role in both ...

  11. Solving Assignment Problem using Linear Programming in Python

    Assignment Problem. A problem that requires pairing two sets of items given a set of paired costs or profit in such a way that the total cost of the pairings is minimized or maximized. The assignment problem is a special case of linear programming. For example, an operation manager needs to assign four jobs to four machines.

  12. Question: Explain what it means for an assignment model to be balanced

    Explain what it means for an assignment model to be balanced. Give detailed mathematical examples. There are 2 steps to solve this one. Expert-verified 100% (1 rating) Share Share Step 1 Explanation: In the context of assignment models, "balanced" refers to a situation where the number of resources ... View the full answer Step 2 Unlock Answer

  13. Assignment Problem

    Assignment Problem - Balanced HAMEEDA MATHTUBER 17.6K subscribers Subscribe Subscribed 4.5K 286K views 3 years ago INDIA #assingmentproblem #minimization #balancedassingmentproblem ...more...

  14. Assignment Problem: Meaning, Methods and Variations

    Step 1: Develop the Cost Table from the given Problem: ADVERTISEMENTS: If the no of rows are not equal to the no of columns and vice versa, a dummy row or dummy column must be added. The assignment cost for dummy cells are always zero. Step 2: Find the Opportunity Cost Table:

  15. Hungarian Method for Balanced Assignment Problem-example

    Balanced Assignment Problem Example Five jobs are to assigned to five people, each person will do one job only. The expected times (in hour) required for each person to complete each job have been estimated and are shown in the following table. Use the Hungarian method to determine the optimal assignments. Solution

  16. Project Manager Assignment Model

    To achieve a balanced mix of perspectives, we selected six experts from different professions: researchers, consultants, and practitioners. The purpose of the mix was to help minimize the impact of prejudice, if any (Kocaoglu, 1983). ... These propositions are organized as a model of project manager assignment, and include assignment process ...

  17. Assignment model, Part-5 : Unbalanced assignment problems

    Assignment model, Part-5 : Unbalanced assignment problems QNTS formulation 629 subscribers Subscribe Subscribed 663 64K views 5 years ago Before applying Hungarian method, form a balanced /...

  18. Balanced Transportation and Assignment Model

    Assignment Problems • Since the problem is balanced and assignments are 1 to 1 (1 person to 1 place) • Decision variables will all have an ending value of either 1 or 0 • Recall that balanced transport problems have integer solutions if RHS are integer values • In general, the assignment model can be formulated as a transportation model ...

  19. 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. Get Quantitative Techniques: Theory and Problems now with the O ...

  20. Assignment model

    ASSIGNMENT MODEL • Assigning of jobs to factors (men or machine) to get most optimum output or get least cost. • Hungarian method is the mostly used method of solving assignment problems. • Types of Assignment problems: (i) Balanced (ii) Unbalanced. 4. OBJECTIVES (i) No workers is more than one job.

  21. Solution of assignment problems (Hungarian Method)

    Determine the optimum assignment schedule. Solution: 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. The cost matrix of the given assignment problem is. Column 3 contains no ...

  22. Assignment model

    In this video, you will learn how to solve an unbalanced assignment model problem.

  23. Explain what it means for an assignment model to be balanced. Give

    1.Balance assignment problem: A company has four machines that are used for four jobs. Each job can be assigned to one and only one machine. The cost of each job on each machine is given in the following Table. observe that there are 4 machines and 4 jobs that means each job goes to one machine or each machine assigns one job.