## Unbalanced Assignment Problem: Definition, Formulation, and Solution Methods

Are you familiar with the assignment problem in Operations Research (OR)? This problem deals with assigning tasks to workers in a way that minimizes the total cost or time needed to complete the tasks. But what if the number of tasks and workers is not equal? In this case, we face the Unbalanced Assignment Problem (UAP). This blog will help you understand what the UAP is, how to formulate it, and how to solve it.

## What is the Unbalanced Assignment Problem?

The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has additional constraints that make it more complex than the traditional assignment problem.

## Formulation of the Unbalanced Assignment Problem

To formulate the UAP, we start with a matrix that represents the cost or time required to assign each task to each worker. If the matrix is square, we can use the Hungarian algorithm to solve the problem. But when the matrix is not square, we need to add dummy tasks or workers to balance the matrix. These dummy tasks or workers have zero costs and are used to make the matrix square.

Once we have a square matrix, we can apply the Hungarian algorithm to find the optimal assignment. However, we need to be careful in interpreting the results, as the assignment may include dummy tasks or workers that are not actually assigned to anything.

## Solutions for the Unbalanced Assignment Problem

Besides the Hungarian algorithm, there are other methods to solve the UAP, such as the transportation algorithm and the auction algorithm. The transportation algorithm is based on transforming the UAP into a transportation problem, which can be solved with the transportation simplex method. The auction algorithm is an iterative method that simulates a bidding process between the tasks and workers to find the optimal assignment.

In summary, the Unbalanced Assignment Problem is a variant of the traditional Assignment Problem in OR that deals with assigning tasks to workers when the number of tasks and workers is not equal. To solve the UAP, we need to balance the matrix by adding dummy tasks or workers and then apply algorithms such as the Hungarian algorithm, the transportation algorithm, or the auction algorithm. Understanding the UAP can help businesses and organizations optimize their resource allocation and improve their operational efficiency.

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## Operations Research

1 Operations Research-An Overview

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

2 Linear Programming: Formulation and Graphical Method

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

3 Linear Programming-Simplex Method

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

4 Transportation Problem

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

5 Assignment Problem

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

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

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

8 Integer Programming

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

9 Dynamic Programming

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

10 Non-Linear Programming

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

11 Introduction to game theory and its Applications

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

12 Monte Carlo Simulation

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

13 Queueing Models

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

## 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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## Double-debiased machine learning with unbalanced treatment assignment

Updated on 8.11.2023 after correction of a bug.

## Introduction

In the realm of highly unbalanced classification problems, training machine learning models to predict the minority class can be a daunting task. Interestingly, this issue also extends to causal inference. The popular double-debiased machine learning approach for estimating causal effects is sensitive to unbalanced treatment assignment. In this blog post, I will explore how to address this concern and adjust for unbalanced treatment assignment when employing the double-debiased machine learning approach.

I’ll start with a concise review of the double-debiased estimator, highlighting its core principles. Then, I’ll delve into different methods for adjusting unbalanced treatment assignment, with a focus on a new approach based on undersampling. Remarkably, this method preserves the favorable asymptotic properties of the double-debiased approach.

To illustrate the impact of unbalanced treatment assignment, I’ll present the findings of a small simulation study. This study will underscore the sensitivity of the double-debiased estimator and showcase how the proposed adjustment can mitigate bias effectively.

Furthermore, I’ll extend the simulation to evaluate the performance of these adjustment in a decision-making scenarios where we try to find an optimal policy to assign treatments.

The code for this blog post is available on GitHub .

## Double-debiased ML

In many fields we are interested in measuring the expected effect that a specific policy $D$ has on an outcome variable $Y$. This quantity is called the average treatment effect (ATE) and defined as $E[Y(1)-Y(0)]$, where $Y(1)$ is the potential outcome if we would be subject to the policy (treated), and $Y(0)$ if we would not be (non-treated). For example, we might be interested to know the effect that a specific drug has on the recovery of a patient. In this case, the drug is the policy and the recovery is the outcome.

The estimation of this quantity is not trivial. In fact, we generally do not observe both potential outcomes for the same subject. Moreover, in many situations we do not have data from a controlled experiment. In order to estimate the ATE, we need to make four assumptions. The first one is the stable unit treatment value assumption (SUTVA). It states that the potential outcome of a subject does not depend on the treatment assignment of other subjects and that definition of “treatment” is consistent across subjects. The second assumption that we need to make is the unconfoundedness assumption. Under this assumption, the treatment assignment is as good as random, conditional on the covariates (i.e. a set of observable characteristics). Third, we need to assume that the treatment assignment is not deterministic. And finally, we assume that the treatment assignment does not affect the covariates (exogenity assumption).

Under these assumptions, we can estimate the ATE using the following formula:

where $e(X)$ is the probability of being treated (propensity score) and $\mu_{(d)}(X)=E[Y|X, D=d]$. The right-hand side of the equation is called the augmented inverse probability weighting (AIPW) estimator or double-robust estimator. Ideally we would like to estimate the nuisance functions $\mu_1(X)$, $\mu_0(X)$ and $e(X)$ using modern machine learning methods while still have a consistent and asymptotically normally distributed estimator. With two crucial ingredients, this is indeed possible. First, the following conditions need to be satisfied 1 :

• Overlap: $\eta < e(x) < 1-\eta$ for $\eta>0$ and for all $x \in \mathcal{X}$.
• Consistency: the machine learning methods are sup-norm consistent
• Risk decay: the machine learning methods have a risk decay rate that satisfies

Second, we estimate the ATE using a so-called cross-fitting approach: we split the data randomly into two folds, $\mathcal{I}_1$ and $\mathcal{I}_2$. We then train the machine learning models on $\mathcal{I}_1$ and use them to compute the following quantity on the other fold $\mathcal{I}_2$:

We then repeat the same procedure flipping the folds. We obtain the final estimate of the ATE by averaging the $\hat{\tau}_i$:

In practice, we can use more than two folds, and extend this procedure to $K$ folds.

For a good explanation of the proof of why this estimation approach is in fact consistent and asymptotically normally distributed, I recommend the lecture notes of Stefan Wager’s course “STATS 361: Causal Inference” 1 . For a more technical explanation, you can directly refer to the paper on double-debiased machine learning by Chernozhukov et al. (2018). 2

## Unbalanced treatment assignment

In many real-world applications, the treatment assignment is not balanced. This means that only very few subjects are treated. This problem arises, for example, in medical applications where the treatment assignment might be very expensive. In this case, the machine learning model will have a hard time to correctly estimate the propensity scores $e(X)$. In fact, the more extreme the unbalancedness, the more the model will tend to predict a probability of being treated close to zero for all subjects. This is because the model will try to minimize the loss function, which will be dominated by the non-treated subjects. Since the propensity scores appear in the denominator of the doubly-robust estimator, we will obtain very large values for the term:

This will lead to a very high variance of the ATE estimator and in finite-samples to a considerable bias.

Luckily, this is a well known issue in machine learning classification problems. A common approach to address this issue is undersampling . This means that we will randomly select a subset of the non-treated subjects and use this subset to train the machine learning models and estimate the treatment effect. This will lead to a more balanced dataset. However, undersampling the data results in effectively using less data to estimate $\mu_0(X)$, $e(X)$, and most importantly $\theta$.

Here I will explore an alternative approach which only partially undersamples the dataset. This approach is based on the idea that we can undersample only the data used to estimate the machine learning models, while still using all the data to estimate the treatment effect. This requires however an adjustment of the propensity scores. In what follows, I will describe the adjustment proposed by Dal Pozzolo et al. (2015). 3 While they apply their method to the problem of unbalanced classification, it can be easily adapted to the problem of estimating propensity scores.

To formalize the concept of undersampling, we can define a random variable $S_i$ which equals 1 if the observation is part of the undersampled data and 0 otherwise. It then follows that $P(S_i=1|D_i=1)=1$ since we keep all treated observations. For the non-treated once, we instead have $P(S_i = 1 | D_i=0) = \gamma < 1$. Notice that since the undersampling technique is not dependent on the covariates $X$, we have $P(S_i=1|D_i=d, X_i)=P(S_i=1|D_i=d)$. So how does undersampling affect the propensity score? This can be easily derived using Bayes’ rule:

So when we are using an undersampled dataset to estimate the propensity score, we are in fact not estimating the population propensity score $e(X_i)$, but the propensity score of a balanced dataset $e_S(X_i)$. This is also the reason why we have to estimate the treatment effect on the undersampled dataset: the machine learner will be sup-consistent for the propensity score on the undersampled population.

Fortunately, the above formula does not only show that $e_S(X_i) \neq e(X_i)$, but also directly suggests an adjustment of the estimated propensity score to recover the population propensity score:

The propensity score $e(X_i)$ can therefore be estimated by plugging in the estimated propensity score $e_S(X_i)$ from the undersampled dataset and an estimate of $\gamma$:

Notice that here we can estimated $\gamma$ using the entire dataset, and not a cross-fitting approach. In fact, as I show below, the ATE estimator obtained by following this approach is still asymptotically normal and consistent. Here a short summary of the estimator:

Undersampled-calibrated doubly-robust estimator (UC-DR) Compute $\hat\gamma = \frac{\sum_{i=1}^N D_i}{\sum_{i=1}^N 1-D_i}$ Split the data randomly into two folds, $\mathcal{I}_1$ and $\mathcal{I}_2$ Undersample $\mathcal{I}_1$ to obtain a balanced sample $\mathcal{I}_1^S$ Train the machine learning models on $\mathcal{I}_1^S$ to obtain $\hat{\mu}_0$, $\hat{\mu}_1$ and $\hat{e}_S$. Calibrated the propensity score: $$\hat{e}(X_i) = \frac{\hat\gamma \hat{e}_S(X_i)}{\hat\gamma \hat{e}_S(X_i) + 1-\hat{e}_S(X_i)}$$ Compute the following quantity on the other fold $\mathcal{I}_2$: $$\hat{\tau}_i = \hat{\mu}_1(X_i) - \hat{\mu}_0(X_i) + \frac{D_i(Y_i-\hat{\mu}_1(X_i))}{\hat{e}(X_i)} - \frac{(1-D_i)(Y_i-\hat{\mu}_0(X_i))}{1-\hat{e}(X_i)}.$$ Repeat the same procedure flipping the folds. Obtain the final estimate of the ATE by averaging the $\hat{\tau}_i$: $$\hat{\theta} = \frac{1}{N} \sum_{i=1}^N \hat{\tau}_i$$

You can find a proof of the asymptotic normality of this estimator at the end of this post.

## A small simulation study

To better understand how the undersampling-calibrated doubly-robust estimator (UC-DR) performs, I run a small simulation study. I use a very similar set-up as in the extensive simulation study of Gabriel Okasa 4 (2022). If you are not interested in all the details about the data generating process, you can skip to the next section. In brief, I generate data such that the average treatment effect equals 1, only very few observations are actually treated (~2% and ~12%), the potential outcomes and propensity score are highly non-linear, and only four out of the 100 features are actually relevant. The machine learner of choice is a random forest with 200 trees for both classification and regression tasks. The code for the simulation study can be found in my GitHub repository .

## Data generating process

The outcome variable is generated as follows:

The features $X_i$ are a 100-dimensional vector drawn independently from a uniform distribution on $[0,1]$, that is $X_i \sim \mathcal{U}([0,1]^{100})$. The propensity score is given by:

where $f(x) = \sin(\pi \cdot x_1 \cdot x_2 \cdot x_3 \cdot x_4)$ and $\beta_{2,4}$ is the beta cumulative distribution function with parameter 2 and 4. I set the parameter $k$ such that only either 2% or 12% of the observations are treated. Finally, I define the potential outcomes as follows:

where the function $\eta(x)$ is defined as:

Before presenting the results, I will briefly explain how I evaluate the performance of the UC-DR learner with that of the baseline double-robust estimator (DR) and with its undersampled version (U-DR). First, I compute the average bias of the estimator. This measure will give me an idea of how close the estimator is to the true value of the average treatment effect. Second, I compute the root mean squared error (RMSE) of the estimator. This measure, combined with the bias, will tell me how much the estimator varies across different samples.

Finally, I use the different estimators in what is commonly called a “policy learning” problem. The idea is that I want to optimally assign the treatment to new observations. This means that, for a group of new individuals, I have to decide who should be treated, taking into account that the treatment has a certain cost. For simplicity, I assume that treating an individual bears a constant const equal to one. If the potential outcomes of each individual would be known, it would be easy to compute the optimal assignment. I would simply assign the treatment to the individuals where the effect of treatment $Y(1)-Y(0)$ exceeds the cost. However, in practice, the potential outcomes are not known. Athey and Wager 5 (2021) show that it is possible to solve this problem by training a classification model using the $\hat\tau_i$ obtained from a doubly-robust estimator. I skip here all the details of this procedure, but you can find more information either directly in their paper or in Micheal Knaus’ slides available in his GitHub repo . I will report the regret of the different estimators. The regret is defined as the difference between the average outcome of the optimal assignment and the average outcome of the assignment obtained by using the estimated $\hat\tau_i$ for a new random sample of 10’000 individuals. An estimator will perform well in this task if it is able to capture the heterogeneity in the treatment effect.

All the results are summarized in the table below. They show that:

In small samples or when the unbalancedness is high, the UC-DR estimators perform best. The undersampling approach combined with a calibration of the propensity score allows to deal with the unbalanced sample while still estimating the ATE with all the data. This leads to a lower bias and a lower RMSE.

The DR estimator performs best in large samples. In this case, the DR estimator has the lowest RMSE. This is because the additional data used by the UC-DR estimator cannot offset the variance related with the estimation of $\hat\gamma$. However, the UC-DR estimator still performs well in this case.

So which estimator should we use in practice when dealing with unbalanced treatment assignment? As in many cases: it depends. If the sample is not very large and (highly) unbalanced, I would recommend to undersample the data used to train the machine learning models and then adjust the propensity score predictions. This will allow to use more data to estimate the ATE or to train other models on the $\hat\tau_i$. This could be particularly useful when we are interested in conditional treatment effects or in solving a policy learning problem. In any case, the baseline doubly-robust estimator should be avoided in highly unbalanced samples, unless we have a very large sample size.

## Proof of the asymptotic properties

The proof of the asymptotic properties of the UC-DR estimator is very similar to the proof of the classical doubly-robust estimator as outlined in Stefan Wager’s script 1 . In the following I will therefore only cover the differences in the proofs to try to keep this part as short as possible.

Notice that the estimator of $\gamma$ is simply a maximum likelihood estimator for which we have that $|\hat\gamma - \gamma|= o_p(n^{-1/2})$. Moreover, I assume that $\epsilon < \gamma < 1 - \epsilon$ for $\epsilon>0$.

I will assume that the previously mentioned conditions hold for the undersampled machine learner:

• Overlap: $\eta < e_S(x) < 1-\eta$ for $\eta>0$ and for all $x \in \mathcal{X}$.
• Consistency:
• Risk decay:

Now by noticing that

we can conclude that $\hat{e}_S(x)\hat\gamma$ is sup-norm consistent and therefore, thanks to the overlap assumption, $\hat{e}(X)$ is also sup-norm consistent.

From here I follow the proof in Wager’s script. I will focus on an estimator for $\theta_1=E[Y(1)]$. Extending this proof to the ATE estimator is straight forward since $\theta=\theta_1 - \theta_0$. First, if we would know the true functions $\mu_1(X)$ and $e(X)$, the oracle estimator:

would simply be an average of independent random variables and by the central limit theorem we would have that $\sqrt{N}(\widetilde\theta_1 - \theta_1) \xrightarrow{d} \mathcal{N}(0, V)$. Next, if we can show that $\sqrt{N}(\widetilde\theta_1 - \hat\theta_1)=o_p(1)$, we can conclude that our estimator converges to the same distribution as the oracle estimator.

Since I use cross-fitting for the estimation, I can rewrite the estimator as follows:

So it is sufficient to show that $\sqrt{N}(\widetilde\theta_1^{\mathcal{I}_1} - \hat\theta_1^{\mathcal{I}_1})=o_p(1)$. Stefan wager shows how we can decompose the difference into three terms:

We therefore have to show that each of these three components converge to zero at rate $N^{-1/2}$. First, (A) does not dependent on the estimation of the propensity score, and we can therefore use the same argument as in Stefan Wager’s script, which is why I will skip this part of the proof. Second, I compute the squared $L_2$-norm of (B):

I skipped the last steps since, thanks to the sup-consistency of the calibrated propensity score estimator, they coincide with the proof of the double-robust estimator. The main difference to the usual proof is the fact that I had to condition not only on the (undersampled) estimation sample, but also on the treatment assignments of the entire sample $D_i$, $i=1,\dots, N$. This step is necessary, since the calibrated propensity score depends on $\hat\gamma$ which is estimated over the entire sample. Despite this, the elements in the sum are still uncorrelated (forth equality):

by the law of iterated expectations and the fact that the observations are independent. Lastly, we can focus on the (C):

where the first the first term in the last step is $o(N^{-1/2})$ by the risk-decay assumption and the second term is $o(N^{-1/2})$ by the convergence rate of $\hat\gamma$ and sup-consistency of $\hat\mu_1$. The positive constants $c_1$ and $c_2$ come from the boundedness of $e_S$ and $\gamma$ (and that of their respective estimators). Combined with the law of large numbers, we can conclude that the term (C) is $o_p(N^{-1/2})$. This concludes the proof of the theorem as all three terms (A), (B) and (C) are $o_p(N^{-1/2})$.

Stefan Wager (2020), STATS 361: Causal Inference, Retrieved from Wager’s webpage .  ↩   ↩ 2   ↩ 3

Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W. and Robins, J. (2018), Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal , 21: C1-C68. doi:10.1111/ectj.12097 .  ↩

A. D. Pozzolo, O. Caelen, R. A. Johnson and G. Bontempi (2015), Calibrating Probability with Undersampling for Unbalanced Classification. IEEE Symposium Series on Computational Intelligence , pp. 159-166, doi: 10.1109/SSCI.2015.33 .  ↩

G. Okasa (2022), Meta-Learners for Estimation of Causal Effects: Finite Sample Cross-Fit Performance. arXiv working paper , arXiv:2201.12692v1   ↩

Athey, S. and Wager, S. (2021), Policy Learning With Observational Data. Econometrica, 89: 133-161. https://doi.org/10.3982/ECTA15732   ↩

## Daniele Ballinari

Statistic's enthusiast

• Español – América Latina
• Português – Brasil
• Tiếng Việt

## 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.

Last updated 2023-01-02 UTC.

## A Comparative Analysis of Assignment Problem

• Conference paper
• First Online: 06 June 2023
• Cite this conference paper

• Shahriar Tanvir Alam   ORCID: orcid.org/0000-0002-0567-3381 5 ,
• Eshfar Sagor 5 ,
• Tanjeel Ahmed 5 ,
• Tabassum Haque 5 ,
• Md Shoaib Mahmud 5 ,
• Salman Ibrahim 5 ,
• Ononya Shahjahan 5 &
• Mubtasim Rubaet 5

Part of the book series: EAI/Springer Innovations in Communication and Computing ((EAISICC))

Included in the following conference series:

• International Conference on Big Data Innovation for Sustainable Cognitive Computing

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The aim of a supply chain team is to formulate a network layout that minimizes the total cost. In this research, the lowest production cost of the final product has been determined using a generalized plant location model. Furthermore, it is anticipated that units have been set up appropriately so that one unit of input from a source of supply results in one unit of output. The assignment problem is equivalent to distributing a job to the appropriate machine in order to meet customer demand. This study concentrates on reducing the cost of fulfilling the overall customer demand. Many studies have been conducted, and various algorithms have been proposed to achieve the best possible result. The purpose of this study is to present an appropriate model for exploring the solution to the assignment problem using the “Hungarian Method.” To find a feasible output of the assignment problem, this study conducted a detailed case study. The computational results indicate that the “Hungarian Method” provides an optimum solution for both balanced and unbalanced assignment problems. Moreover, decision-makers can use the study’s findings as a reference to mitigate production costs and adopt any sustainable market policy.

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## Author information

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Shahriar Tanvir Alam, Eshfar Sagor, Tanjeel Ahmed, Tabassum Haque, Md Shoaib Mahmud, Salman Ibrahim, Ononya Shahjahan & Mubtasim Rubaet

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Correspondence to Shahriar Tanvir Alam .

## Editor information

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Anandakumar Haldorai

Department of Computer Science and Engineering, CMR University, Bengaluru, Karnataka, India

Arulmurugan Ramu

Sri Eshwar College of Engineering, Coimbatore, Tamil Nadu, India

Sudha Mohanram

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Cite this paper.

Alam, S.T. et al. (2023). A Comparative Analysis of Assignment Problem. In: Haldorai, A., Ramu, A., Mohanram, S. (eds) 5th EAI International Conference on Big Data Innovation for Sustainable Cognitive Computing. BDCC 2022. EAI/Springer Innovations in Communication and Computing. Springer, Cham. https://doi.org/10.1007/978-3-031-28324-6_11

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## Assignment Model | Linear Programming Problem (LPP) | Introduction

What is assignment model.

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

i) There is only one assignment.

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

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

## For example:

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

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

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

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

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

→ The mathematical formulation for Assignment Model is given below:

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

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

→ Subjected to constraint;

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

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

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

## Types of Assignment Problem:

(i) balanced assignment problem.

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

## (ii) Unbalanced Assignment Problem

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

## Methods to solve Assignment Model:

(i) integer programming method:.

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

So this can be formulated using 0 or 1 integer.

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

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

So this method becomes very lengthy and difficult to solve.

## (ii) Transportation Methods:

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

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

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

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

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

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

So, the method becomes lengthy and time consuming.

## (iii) Enumeration Method:

It is a simple trail and error type method.

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

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

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

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

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

## (iv) Hungarian Method:

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

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

There are two main conditions for applying Hungarian Method:

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

## Linear Programming Problem (LPP) Formulation with Numericals

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## Education Lessons

Stay in touch, [notes] operation research, [notes] dynamics of machinery, [notes] maths, [notes] science, [notes] computer aided design.

## 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 :

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

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## Unbalanced Assignment Problem by Using Modified Approach

— The assignment problem is one of the main problems while assigning task to the worker. It is an important problem in mathematics and is also discuss in real physical world. It is a combinatorial optimization problem in the field of operational research. In a normal case of transportation problem where the objective is to assign the available resources to the activity going on so as to get the minimum cost or maximize total benefits of allocation. In this paper we proposed modified assignment model for the solution of assignment problem. Here in this paper with the help of numerical examples or problem is solved to show its efficiency and also its comparison with Hungarian method is shown. A new cost is achieved by using unbalanced assignment problem.

## Related Papers

archana pandey

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

1 í µí°·í µí±í µí±í µí±í µí±í µí±¡í µí±í µí±í µí±í µí±¡í µí±í µí±í µí±í µí±í µí±¡ í µí±í µí±í µí±í µí±í µí±¡í µí±í µí±í µí± ,í µí° ¶í µí±í µí±í µí±í µí±í µí±í µí±í µí±í µí±í µí±í µí±£í µí±í µí±í µí± í µí±í µí±¡í µí±¦ ,í µí°µí µí±í µí±í µí±í µí±í µí±í µí±í µí±í µí± í µí± 2 í µí°·í µí±í µí±í µí±í µí±í µí±¡í µí±í µí±í µí±í µí±¡í µí±í µí±í µí±í µí±í µí±¡ í µí±í µí±í µí±í µí±í µí±¡í µí±í µí±í µí± ,í µí° ¶í µí±í µí±í µí±í µí±í µí±í µí±í µí±í µí±í µí±í µí±£í µí±í µí±í µí± í µí±í µí±¡í µí±¦ ,í µí°µí µí±í µí±í µí±í µí±í µí±í µí±í µí±í µí± í µí± Abstract: Assignment problem is an important problem in mathematics and is also discuss in real physical world. In this paper we attempt to introduce a new proposed approach for solving assignment problem with algorithm and solution steps. We examine a numerical example by using new method and compute by existing two methods. Also we compare the optimal solutions among this new method and two existing method .The proposed method isa systematic procedure, easy to apply for solving assignment problem.

International Journal of Mathematical Archive ( …

Surapati Pramanik, Ph. D.

This paper presents solution methodology for assignment problem with fuzzy cost. The fuzzy costs are considered as trapezoidal fuzzy numbers. Ranking method (introduced by S. Abbasbandy and T. Hajjary, 2009) has been used for ranking the trapezoidal fuzzy ...

IOSR Journals publish within 3 days

In this paper a ground reality the entries of the cost matrix are not always crisp. In many application this parameters are uncertain and this uncertain parameters are represented by interval. In this contribution we proposed a new Interval Hungarian Method to solve the mid values of each interval in the cost matrix with four different cases of the assignment problem and finally we conclude that same optimum solution.

IAEME Publication

This paper presents multi-objective assignment problem with traveling time and territory control of the mail carrier from the central post office Bandung in delivering the package to the destination location, where all the objectives are optimized by using Hungarian method. Sensitivity analysis against data changes that may occur was also conducted. The sampled data in this study are the territory control and traveling time of 10 mail carriers who will be assigned to deliver mail package to 10 post office delivery centers in Bandung. The result of this research is the combination of traveling time and territory control optimal from 10 mail carriers as follows: mail carrier 1 to Soreang, mail carrier 2 to Dayeuh Kolot, mail carrier 3 to Ujung Berung, mail carrier 4 to Padalarang, mail carrier 5 to Situ Saeur, mail carrier 6 to Cipedes, mail carrier 7 to Cimahi, mail carrier 8 to Asia-Afrika, mail carrier 9 to Cikutra, mail carrier 10 to Cikeruh. Based on this result, manager of the central post office Bandung can make optimal decisions to assign tasks to their mail carriers.

Bhausaheb G Kore

In this paper I have proposed a new approach to solve an unbalanced assignment problem (UBAP). This approach includes two parts. First is to obtain an initial basic feasible solution (IBFS) and second part is to test optimality of an IBFS. I have proposed two new methods Row Penalty Assignment Method (RPAM) and Column Penalty Assignment Method (CPAM) to obtain an IBFS of an UBAP. Also I have proposed a new method Non-basic Smallest Effectiveness Method (NBSEM) to test optimality of an IBFS. We can solve an assignment problem of maximization type using this new approach in opposite sense. By this new approach, we achieve the goal with less number of computations and steps. Further we illustrate the new approach by suitable examples. INTRODUCTION The assignment problem is a special case of the transportation problem where the resources are being allocated to the activities on a one-to-one basis. Thus, each resource (e.g. an employee, machine or time slot) is to be assigned uniquely to a particular activity (e.g. a task, site or event). In assignment problems, supply in each row represents the availability of a resource such as a man, machine, vehicle, product, salesman, etc. and demand in each column represents different activities to be performed such as jobs, routes, factories, areas, etc. for each of which only one man or vehicle or product or salesman respectively is required. Entries in the square being costs, times or distances. The assignment method is a special linear programming technique for solving problems like choosing the right man for the right job when more than one choice is possible and when each man can perform all of the jobs. The ultimate objective is to assign a number of tasks to an equal number of facilities at minimum cost (or maximum profit) or some other specific goal. Let there be 'm' resources and 'n' activities. Let c ij be the effectiveness (in terms of cost, profit, time, etc.) of assigning resource i to activity j (i = 1, 2, …., m; j = 1, 2,…., n). Let x ij = 0, if resource i is not assigned to activity j and x ij = 1, if resource i is assigned to activity j. Then the objective is to determine x ij 's that will optimize the total effectiveness (Z) satisfying all the resource constraints and activity constraints. 1. Mathematical Formulation Let number of rows = m and number of columns = n. If m = n then an AP is said to be BAP otherwise it is said to be UBAP. A) Case 1: If m < n then mathematically the UBAP can be stated as follows:

Trisna Darmawansyah

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

Hogpodrat Pangkum

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

IJAR Indexing

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

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1. Unbalanced Assignment Problem

Here is the video about unbalanced Assignment problem using Hungarian method,In this video we have seen how to solve unbalanced assignment problem using step...

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

Before applying Hungarian method, form a balanced / square matrix..Add dummy rows ..Add dummy columns ..

3. Assignment model

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

4. Unbalanced Assignment Problem: Definition, Formulation, and Solution

The Unbalanced Assignment Problem is an extension of the Assignment Problem in OR, where the number of tasks and workers is not equal. In the UAP, some tasks may remain unassigned, while some workers may not be assigned any task. The objective is still to minimize the total cost or time required to complete the assigned tasks, but the UAP has ...

5. Assignment problem

The assignment problem consists of finding, in a weighted bipartite graph, a matching of a given size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment. Otherwise, it is called unbalanced assignment. [1] If the total cost of the assignment for all ...

6. 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 ...

7. Solving the Unbalanced Assignment Problem: Simpler Is Better

A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM. Bhausaheb G Kore. In this paper I have proposed a new approach to solve an unbalanced assignment problem (UBAP). This approach includes two parts. First is to obtain an initial basic feasible solution (IBFS) and second part is to test optimality of an IBFS.

8. Solving the Unbalanced Assignment Problem: Simpler Is Better

Recently, Yadaiah and Haragopal published in the American Journal of Operations Research a new approach to solving the unbalanced assignment problem. They also provide a numerical example which they solve with their approach and get a cost of 1550 which they claim is optimum. This approach might be of interest; however, their approach does not guarantee the optimal solution.

9. Modified Hungarian method for unbalanced assignment problem with

This purpose can be served by assigning multiple jobs to a single machine. The present paper proposes a Modified Hungarian Method for solving unbalanced assignment problems which gives the optimal policy of assignment of jobs to machines. A stepwise algorithm of proposed method is presented and the developed algorithm is also coded in Java SE 11.

10. Modified Hungarian method for unbalanced assignment problem with

The present paper proposes a Modified Hungarian Method for solving unbalanced assignment problems which gives the optimal policy of assignment of jobs to machines. A stepwise algorithm of proposed ...

11. An Alternative Approach for Solving Unbalanced Assignment Problems

An Alternative Approach for Solving Unbalanced Assignment Problems Abdur Rashid Department of Mathematics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh. Abstract This paper is devoted to present a new approach to make an unbalanced assignment problem into a balanced one and a comparison is carried out with the existing methods.

12. A modified method for solving the unbalanced assignment problems

The present paper suggests a modified method for solving the unbalanced assignment problems. The Hungarian method [1] gives us total assignment cost 870 along with the other three jobs assigned to dummy machine, in other words that these three jobs are ignored for further processing, while, when the original problem is divided in the sub ...

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In the realm of highly unbalanced classification problems, training machine learning models to predict the minority class can be a daunting task. Interestingly, this issue also extends to causal inference. The popular double-debiased machine learning approach for estimating causal effects is sensitive to unbalanced treatment assignment.

14. 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. Example. In the example there are five workers (numbered 0-4) and four tasks (numbered 0-3).

15. A Comparative Analysis of Assignment Problem

This part illustrates the existing literature relating to assignment problems. Section 2.1 discusses the literature on assignment problems, whereas Sect. 2.2 illustrates the recent study of the unbalanced assignment problem.. 2.1 Literature Related to the Assignment Problems. Many researchers and practitioners in the past implemented the Hungarian method to resolve assignment problems [8,9,10].

16. Assignment Model

There are two main conditions for applying Hungarian Method: (1) Square Matrix (n x n). (2) Problem should be of minimization type. 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.

17. The Assignment Model

The assignment model is a special form of a linear programming model that is similar to the transportation model. There are differences, however. In the assignment model, the supply at each source and the demand at each destination are each limited to one unit. ... An unbalanced model exists when supply exceeds demand or demand exceeds supply ...

18. Unbalanced Assignment Model

Differences of Assignment Model and Minimization Problem Assignment Model - It's characterized by the constraint that each task is assigned to exactly one resource, and each resource is assigned to exactly one task. The assignment model is a subset of minimization problems that focuses on a particular type of optimization scenario.

19. UNBALANCED ASSIGNMENT MODEL

steps to solve an unbalanced assignment problem using ms-excel

20. Unbalanced Assignment Problem by Using Modified Approach

Trisna Darmawansyah. — The assignment problem is one of the main problems while assigning task to the worker. It is an important problem in mathematics and is also discuss in real physical world. It is a combinatorial optimization problem in the field of operational research. In a normal case of transportation problem where the objective is ...

21. Transportation, Transshipment, and Assignment Problems

Describe the nature of transportation transshipment and assignment problems. Formulate a transportation problem as a linear programming model. Use the transportation method to solve problems with Excel. Solve maximization transportation problems, unbalanced problems, and problems with prohibited routes.

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Unmanned aerial vehicles (UAVs) can be used in swarms to achieve multiple tasks cooperatively. Multi-UAV and multi-target cooperative task assignments are difficult. To solve the problem of unbalanced, phased, cooperative assignment between UAVs and tasks, we establish an unbalanced, phased task assignment model that considers the constraints of task execution, time, and target task execution ...

23. Introduction to Assignment Problem Unbalanced Hungarian ...

Introduction to Umbalanced Assignment Problem Hungarian Meethod|Linear Programming|Dream MathsInstagram:- https://Instagram.com/dreammathsTelegram:-https://t...