![unbalanced assignment problem meaning unbalanced assignment problem meaning](https://cbom.atozmath.com/IMAGES1/backtotop.png) Balanced and Unbalanced Transportation Problems![unbalanced assignment problem meaning Class Registration Banner](https://cdn1.byjus.com/wp-content/uploads/2023/06/Dark-BG-mobile-top.webp) The two categories of transportation problems are balanced and unbalanced transportation problems . As we all know, a transportation problem is a type of Linear Programming Problem (LPP) in which items are carried from a set of sources to a set of destinations based on the supply and demand of the sources and destinations, with the goal of minimizing the total transportation cost. It is also known as the Hitchcock problem. Introduction to Balanced and Unbalanced Transportation ProblemsBalanced transportation problem. The problem is considered to be a balanced transportation problem when both supplies and demands are equal. Unbalanced Transportation Problem Unbalanced transportation problem is defined as a situation in which supply and demand are not equal. A dummy row or a dummy column is added to this type of problem, depending on the necessity, to make it a balanced problem. The problem can then be addressed in the same way as the balanced problem. Methods of Solving Transportation ProblemsThere are three ways for determining the initial basic feasible solution. They are 1. NorthWest Corner Cell Method. 2. Vogel’s Approximation Method (VAM). 3. Least Call Cell Method. The following is the basic framework of the balanced transportation problem: ![unbalanced assignment problem meaning Basic Structure of Balanced Transportation Problem](https://cdn1.byjus.com/wp-content/uploads/2022/04/basic-structure-of-balanced-transportation-problem.png) The destinations D1, D2, D3, and D4 in the above table are where the products/goods will be transported from various sources O1, O2, O3, and O4. The supply from the source Oi is represented by S i . The demand for the destination Dj is d j . If a product is delivered from source Si to destination Dj, then the cost is called C ij . Let us now explore the process of solving the balanced transportation problem using one of the ways known as the NorthWest Corner Method in this article. ![](//ortec.site/777/templates/cheerup1/res/banner1.gif) Solving Balanced Transportation problem by Northwest Corner MethodConsider this scenario: ![unbalanced assignment problem meaning Balanced Transportation Problem -1](https://cdn1.byjus.com/wp-content/uploads/2022/04/balanced-transportation-problem-1.png) With three sources (O1, O2, and O3) and four destinations (D1, D2, D3, and D4), what is the best way to solve this problem? The supply for the sources O1, O2, and O3 are 300, 400, and 500, respectively. Demands for the destination D1, D2, D3, and D4 are 250, 350, 400, and 200, respectively. The starting point for the North West Corner technique is (O1, D1), which is the table’s northwest corner. The cost of transportation is calculated for each value in the cell. As indicated in the diagram, compare the demand for column D1 with the supply from source O1 and assign a minimum of two to the cell (O1, D1). Column D1’s demand has been met, hence the entire column will be canceled. The supply from the source O1 is still 300 – 250 = 50. ![unbalanced assignment problem meaning Balanced Transportation Problem - 2](https://cdn1.byjus.com/wp-content/uploads/2022/04/balanced-transportation-problem-2.png) Analyze the northwest corner, i.e. (O1, D2), of the remaining table, excluding column D1, and assign the lowest among the supply for the appropriate column and rows. Because the supply from O1 is 50 and the demand for D2 is 350, allocate 50 to the cell (O1, D2). Now, row O1 is canceled because the supply from row O1 has been completed. Hence, the demand for Column D2 has become 350 – 50 = 50. ![unbalanced assignment problem meaning Balanced Transportation Problem - 3](https://cdn1.byjus.com/wp-content/uploads/2022/04/balanced-transportation-problem-3.png) The northwest corner cell in the remaining table is (O2, D2). The shortest supply from source O2 (400) and the demand for column D2 (300) is 300, thus putting 300 in the cell (O2, D2). Because the demand for column D2 has been met, the column can be deleted, and the remaining supply from source O2 is 400 – 300 = 100. ![unbalanced assignment problem meaning Balanced Transportation Problem - 4](https://cdn1.byjus.com/wp-content/uploads/2022/04/balanced-transportation-problem-4.png) Again, find the northwest corner of the table, i.e. (O2, D3), and compare the O2 supply (i.e. 100) to the D2 demand (i.e. 400) and assign the smaller (i.e. 100) to the cell (O2, D2). Row O2 has been canceled because the supply from O2 has been completed. Column D3 has a leftover demand of 400 – 100 = 300. ![unbalanced assignment problem meaning Balanced Transportation Problem -5](https://cdn1.byjus.com/wp-content/uploads/2022/04/balanced-transportation-problem-5.png) Continuing in the same manner, the final cell values will be: ![unbalanced assignment problem meaning Balanced Transportation Problem - 6](https://cdn1.byjus.com/wp-content/uploads/2022/04/balanced-transportation-problem-6.png) It should be observed that the demand for the relevant columns and rows is equal in the last remaining cell, which was cell (O3, D4). In this situation, the supply from O3 was 200, and the demand for D4 was 200, therefore this cell was assigned to it. Nothing was left for any row or column at the end. To achieve the basic solution, multiply the allotted value by the respective cell value (i.e. the cost) and add them all together. I.e., (250 × 3) + (50 × 1) + (300 × 6) + (100 × 5) + (300 × 3) + (200 × 2) = 4400. Solving Unbalanced Transportation ProblemAn unbalanced transportation problem is provided below. Because the sum of all the supplies, O1, O2, O3, and O4, does not equal the sum of all the demands, D1, D2, D3, D4, and D5, the situation is unbalanced. ![unbalanced assignment problem meaning Unbalanced Transportation Problem - 1](https://cdn1.byjus.com/wp-content/uploads/2022/04/unbalanced-transportation-problem-1.png) The idea of a dummy row or dummy column will be applied in this type of scenario. Because the supply is more than the demand in this situation, a fake demand column will be inserted, with a demand of (total supply – total demand), i.e. 117 – 95 = 22, as seen in the image below. A fake supply row would have been introduced if demand was greater than supply. ![unbalanced assignment problem meaning Unbalanced Transportation Problem - 2](https://cdn1.byjus.com/wp-content/uploads/2022/04/unbalanced-transportation-problem-2.png) Now this problem has been changed to a balanced transportation problem, and it can be addressed using any of the ways listed below to solve a balanced transportation problem, such as the northwest corner method mentioned earlier. Frequently Asked Questions on Balanced and Unbalanced Transportation ProblemsWhat is meant by balanced and unbalanced transportation problems. The problem is referred to as a balanced transportation problem when both supplies and demands are equal. Unbalanced transportation is defined as a situation where supply and demand are not equal. What is called a transportation problem?The transportation problem is a type of Linear Programming Problem in which commodities are carried from a set of sources to a set of destinations while taking into account the supply and demand of the sources and destinations, respectively, in order to reduce the total cost of transportation. What are the different methods to solve transportation problems?The following are three approaches to solve the transportation issue: - NorthWest Corner Cell Method.
- Least Call Cell Method.
- Vogel’s Approximation Method (VAM).
Leave a Comment Cancel replyYour Mobile number and Email id will not be published. Required fields are marked * Request OTP on Voice Call Post My Comment ![unbalanced assignment problem meaning unbalanced assignment problem meaning](https://cdn1.byjus.com/wp-content/uploads/2022/12/Vector-2219-2.png) Register with BYJU'S & Download Free PDFsRegister with byju's & watch live videos. Nash Balanced Assignment Problem- Conference paper
- First Online: 21 November 2022
- Cite this conference paper
![unbalanced assignment problem meaning unbalanced assignment problem meaning](https://media.springernature.com/w72/springer-static/cover-hires/book/978-3-031-18530-4?as=webp) - Minh Hieu Nguyen 11 ,
- Mourad Baiou 11 &
- Viet Hung Nguyen 11
Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13526)) Included in the following conference series: - International Symposium on Combinatorial Optimization
373 Accesses 2 Citations In this paper, we consider a variant of the classic Assignment Problem (AP), called the Balanced Assignment Problem (BAP) [ 2 ]. The BAP seeks to find an assignment solution which has the smallest value of max-min distance : the difference between the maximum assignment cost and the minimum one. However, by minimizing only the max-min distance, the total cost of the BAP solution is neglected and it may lead to an inefficient solution in terms of total cost. Hence, we propose a fair way based on Nash equilibrium [ 1 , 3 , 4 ] to inject the total cost into the objective function of the BAP for finding assignment solutions having a better trade-off between the two objectives: the first aims at minimizing the total cost and the second aims at minimizing the max-min distance. For this purpose, we introduce the concept of Nash Fairness (NF) solutions based on the definition of proportional-fair scheduling adapted in the context of the AP: a transfer of utilities between the total cost and the max-min distance is considered to be fair if the percentage increase in the total cost is smaller than the percentage decrease in the max-min distance and vice versa. We first show the existence of a NF solution for the AP which is exactly the optimal solution minimizing the product of the total cost and the max-min distance. However, finding such a solution may be difficult as it requires to minimize a concave function. The main result of this paper is to show that finding all NF solutions can be done in polynomial time. For that, we propose a Newton-based iterative algorithm converging to NF solutions in polynomial time. It consists in optimizing a sequence of linear combinations of the two objective based on Weighted Sum Method [ 5 ]. Computational results on various instances of the AP are presented and commented. This is a preview of subscription content, log in via an institution to check access. Access this chapter- Available as PDF
- Read on any device
- Instant download
- Own it forever
- Available as EPUB and PDF
- Compact, lightweight edition
- Dispatched in 3 to 5 business days
- Free shipping worldwide - see info
Tax calculation will be finalised at checkout Purchases are for personal use only Institutional subscriptions Bertsimas, D., Farias, V.F., Trichakis, N.: The price of fairness. Oper. Res. January–February 59 (1), 17–31 (2011) MathSciNet MATH Google Scholar Martello, S., Pulleyblank, W.R., Toth, P., De Werra, D.: Balanced optimization problems. Oper. Res. Lett. 3 (5), 275–278 (1984) Article MathSciNet MATH Google Scholar Kelly, F.P., Maullo, A.K., Tan, D.K.H.: Rate control for communication networks: shadow prices, proportional fairness and stability. J. Oper. Res. Soc. 49 (3), 237–252 (1997). https://doi.org/10.1057/palgrave.jors.2600523 Article Google Scholar Ogryczak, W., Luss, H., Pioro, M., Nace, D., Tomaszewski, A.: Fair optimization and networks: a survey. J. Appl. Math. 2014 , 1–26 (2014) Marler, R.T., Arora, J.S.: The weighted sum method for multi-objective optimization: new insights. Struct. Multi. Optim. 41 (6), 853–862 (2010) Heller, I., Tompkins, C.B.: An extension of a theorem of Dantzig’s. Ann. Math. Stud. (38), 247–254 (1956) Google Scholar Kuhn, H.W.: The Hungarian method for assignment problem. Naval Res. Logist. Q. 2 (1–2), 83–97 (1955) Martello, S.: Most and least uniform spanning trees. Discrete Appl. Math. 15 (2), 181–197 (1986) Beasley, J.E.: Linear programming on Clay supercomputer. J. Oper. Res. Soc. 41 , 133–139 (1990) Nguyen, M.H, Baiou, M., Nguyen, V.H., Vo, T.Q.T.: Nash fairness solutions for balanced TSP. In: International Network Optimization Conference (INOC2022) (2022) Download references Author informationAuthors and affiliations. INP Clermont Auvergne, Univ Clermont Auvergne, Mines Saint-Etienne, CNRS, UMR 6158 LIMOS, 1 Rue de la Chebarde, Aubiere Cedex, France Minh Hieu Nguyen, Mourad Baiou & Viet Hung Nguyen You can also search for this author in PubMed Google Scholar Corresponding authorCorrespondence to Viet Hung Nguyen . Editor informationEditors and affiliations. ESSEC Business School of Paris, Cergy Pontoise Cedex, France Ivana Ljubić IBM TJ Watson Research Center, Yorktown Heights, NY, USA Francisco Barahona Georgia Institute of Technology, Atlanta, GA, USA Santanu S. Dey Université Paris-Dauphine, Paris, France A. Ridha Mahjoub Proposition 1 . There may be more than one NF solution for the AP. Let us illustrate this by an instance of the AP having the following cost matrix By verifying all feasible assignment solutions in this instance, we obtain easily three assignment solutions \((1-1, 2-2, 3-3), (1-2, 2-3, 3-1)\) , \((1-3, 2-2, 3-1)\) and \((1-3, 2-1, 3-2)\) corresponding to 4 NF solutions (280, 36), (320, 32), (340, 30) and (364, 28). Note that \(i-j\) where \(1 \le i,j \le 3\) represents the assignment between worker i and job j in the solution of this instance. \(\square \) We recall below the proofs of some recent results that we have published in [ 10 ]. They are needed to prove the new results presented in this paper. Theorem 2 [ 10 ] . \((P^{*},Q^{*}) = {{\,\mathrm{arg\,min}\,}}_{(P,Q) \in \mathcal {S}} PQ\) is a NF solution. Obviously, there always exists a solution \((P^{*},Q^{*}) \in \mathcal {S}\) such that Now \(\forall (P',Q') \in \mathcal {S}\) we have \(P'Q' \ge P^{*}Q^{*}\) . Then The first inequality holds by the Cauchy-Schwarz inequality. Hence, \((P^{*},Q^{*})\) is a NF solution. \(\square \) Theorem 3 [ 10 ] . \((P^{*},Q^{*}) \in \mathcal {S}\) is a NF solution if and only if \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P(\alpha ^{*})}\) where \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) . Firstly, let \((P^{*},Q^{*})\) be a NF solution and \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) . We will show that \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P(\alpha ^{*})}\) . Since \((P^{*},Q^{*})\) is a NF solution, we have Since \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) , we have \(\alpha ^{*}P^{*}+Q^{*} = 2Q^{*}\) . Dividing two sides of ( 6 ) by \(P^{*} > 0\) we obtain So we deduce from ( 7 ) Hence, \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P}(\alpha ^{*})\) . Now suppose \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) and \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P}(\alpha ^{*})\) , we show that \((P^{*},Q^{*})\) is a NF solution. If \((P^{*},Q^{*})\) is not a NF solution, there exists a solution \((P',Q') \in \mathcal {S}\) such that We have then which contradicts the optimality of \((P^{*},Q^{*})\) . \(\square \) Lemma 3 [ 10 ] . Let \(\alpha , \alpha ' \in \mathbb {R}_+\) and \((P_{\alpha }, Q_{\alpha })\) , \((P_{\alpha '}, Q_{\alpha '})\) be the optimal solutions of \(\mathcal {P(\alpha )}\) and \(\mathcal {P(\alpha ')}\) respectively, if \(\alpha \le \alpha '\) then \(P_{\alpha } \ge P_{\alpha '}\) and \(Q_{\alpha } \le Q_{\alpha '}\) . The optimality of \((P_{\alpha }, Q_{\alpha })\) and \((P_{\alpha '}, Q_{\alpha '})\) gives By adding both sides of ( 8a ) and ( 8b ), we obtain \((\alpha - \alpha ') (P_{\alpha } - P_{\alpha '}) \le 0\) . Since \(\alpha \le \alpha '\) , it follows that \(P_{\alpha } \ge P_{\alpha '}\) . On the other hand, inequality ( 8a ) implies \(Q_{\alpha '} - Q_{\alpha } \ge \alpha (P_{\alpha } - P_{\alpha '}) \ge 0\) that leads to \(Q_{\alpha } \le Q_{\alpha '}\) . \(\square \) Lemma 4 [ 10 ] . During the execution of Procedure Find ( \(\alpha _{0})\) in Algorithm 1 , \(\alpha _{i} \in [0,1], \, \forall i \ge 1\) . Moreover, if \(T_{0} \ge 0\) then the sequence \(\{\alpha _i\}\) is non-increasing and \(T_{i} \ge 0, \, \forall i \ge 0\) . Otherwise, if \(T_{0} \le 0\) then the sequence \(\{\alpha _i\}\) is non-decreasing and \(T_{i} \le 0, \, \forall i \ge 0\) . Since \(P \ge Q \ge 0, \, \forall (P, Q) \in \mathcal {S}\) , it follows that \(\alpha _{i+1} = \frac{Q_i}{P_i} \in [0,1], \, \forall i \ge 0\) . We first consider \(T_{0} \ge 0\) . We proof \(\alpha _i \ge \alpha _{i+1}, \, \forall i \ge 0\) by induction on i . For \(i = 0\) , we have \(T_{0} = \alpha _{0} P_{0} - Q_{0} = P_{0}(\alpha _{0}-\alpha _{1}) \ge 0\) , it follows that \(\alpha _{0} \ge \alpha _{1}\) . Suppose that our hypothesis is true until \(i = k \ge 0\) , we will prove that it is also true with \(i = k+1\) . Indeed, we have The inductive hypothesis gives \(\alpha _k \ge \alpha _{k+1}\) that implies \(P_{k+1} \ge P_k > 0\) and \(Q_{k} \ge Q_{k+1} \ge 0\) according to Lemma 3 . It leads to \(Q_{k}P_{k+1} - P_{k}Q_{k+1} \ge 0\) and then \(\alpha _{k+1} - \alpha _{k+2} \ge 0\) . Hence, we have \(\alpha _{i} \ge \alpha _{i+1}, \, \forall i \ge 0\) . Consequently, \(T_{i} = \alpha _{i}P_{i} - Q_{i} = P_{i}(\alpha _{i}-\alpha _{i+1}) \ge 0, \, \forall i \ge 0\) . Similarly, if \(T_{0} \le 0\) we obtain that the sequence \(\{\alpha _i\}\) is non-decreasing and \(T_{i} \le 0, \, \forall i \ge 0\) . That concludes the proof. \(\square \) Lemma 5 [ 10 ] . From each \(\alpha _{0} \in [0,1]\) , Procedure Find \((\alpha _{0})\) converges to a coefficient \(\alpha _{k} \in \mathcal {C}_{0}\) satisfying \(\alpha _{k}\) is the unique element \(\in \mathcal {C}_{0}\) between \(\alpha _{0}\) and \(\alpha _{k}\) . As a consequence of Lemma 4 , Procedure \(\textit{Find}(\alpha _{0})\) converges to a coefficient \(\alpha _{k} \in [0,1], \forall \alpha _{0} \in [0,1]\) . By the stopping criteria of Procedure Find \((\alpha _{0})\) , when \(T_{k} = \alpha _{k} P_{k} - Q_{k} = 0\) we obtain \(\alpha _{k} \in C_{0}\) and \((P_{k},Q_{k})\) is a NF solution. (Theorem 3 ) If \(T_{0} = 0\) then obviously \(\alpha _{k} = \alpha _{0}\) . We consider \(T_{0} > 0\) and the sequence \(\{\alpha _i\}\) is now non-negative, non-increasing. We will show that \([\alpha _{k},\alpha _{0}] \cap \mathcal {C}_{0} = \alpha _{k}\) . Suppose that we have \(\alpha \in (\alpha _{k},\alpha _{0}]\) and \(\alpha \in \mathcal {C}_{0}\) corresponding to a NF solution ( P , Q ). Then there exists \(1 \le i \le k\) such that \(\alpha \in (\alpha _{i}, \alpha _{i-1}]\) . Since \(\alpha \le \alpha _{i-1}\) , \(P \ge P_{i-1}\) and \(Q \le Q_{i-1}\) due to Lemma 3 . Thus, we get By the definitions of \(\alpha \) and \(\alpha _{i}\) , inequality ( 9 ) is equivalent to \(\alpha \le \alpha _{i}\) which leads to a contradiction. By repeating the same argument for \(T_{0} < 0\) , we also have a contradiction. \(\square \) Rights and permissionsReprints and permissions Copyright information© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG About this paperCite this paper. Nguyen, M.H., Baiou, M., Nguyen, V.H. (2022). Nash Balanced Assignment Problem. In: Ljubić, I., Barahona, F., Dey, S.S., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2022. Lecture Notes in Computer Science, vol 13526. Springer, Cham. https://doi.org/10.1007/978-3-031-18530-4_13 Download citationDOI : https://doi.org/10.1007/978-3-031-18530-4_13 Published : 21 November 2022 Publisher Name : Springer, Cham Print ISBN : 978-3-031-18529-8 Online ISBN : 978-3-031-18530-4 eBook Packages : Computer Science Computer Science (R0) Share this paperAnyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative Policies and ethics - Find a journal
- Track your research
![unbalanced assignment problem meaning MBA Notes](https://mbahub.in/wp-content/uploads/sites/4/2023/10/MBA-Hub-Logo.png) How to Solve the Assignment Problem: A Complete GuideTable of Contents Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem. Understanding the Assignment ProblemBefore we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource. Solving the Assignment ProblemThere are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method. Step 1: Set up the cost matrixThe first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively. Step 2: Subtract the smallest element from each row and columnTo simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction. Step 3: Cover all zeros with the minimum number of linesThe next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering. Step 4: Test for optimality and adjust the matrixTo test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution. Step 5: Assign the tasks to the agentsThe final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment. Solution of the Assignment Problem using the Hungarian MethodThe Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps: - Subtract the smallest entry in each row from all the entries of the row.
- Subtract the smallest entry in each column from all the entries of the column.
- Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
- Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.
The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. Applications of the Assignment ProblemThe assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations. Applications in Computer ScienceThe assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors. Applications in EconomicsThe assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors. Applications in LogisticsThe assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers. Applications in ManagementThe assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments. Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below: | Task 1 | Task 2 | Task 3 | Emp 1 | 5 | 7 | 6 | Emp 2 | 6 | 4 | 5 | Emp 3 | 8 | 5 | 3 | The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog. Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row: | Task 1 | Task 2 | Task 3 | Emp 1 | 0 | 2 | 1 | Emp 2 | 2 | 0 | 1 | Emp 3 | 5 | 2 | 0 | Next, we subtract the smallest entry in each column from all the entries of the column: | Task 1 | Task 2 | Task 3 | Emp 1 | 0 | 2 | 1 | Emp 2 | 2 | 0 | 1 | Emp 3 | 5 | 2 | 0 | | 0 | 0 | 0 | We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three: Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are: - Emp 1 to Task 3
- Emp 2 to Task 2
- Emp 3 to Task 1
This assignment results in a total time of 9 units. I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method. Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way. How useful was this post? Click on a star to rate it! Average rating 0 / 5. Vote count: 0 No votes so far! Be the first to rate this post. We are sorry that this post was not useful for you! 😔 Let us improve this post! Tell us how we can improve this post? Operations Research1 Operations Research-An Overview - History of O.R.
- Approach, Techniques and Tools
- Phases and Processes of O.R. Study
- Typical Applications of O.R
- Limitations of Operations Research
- Models in Operations Research
- O.R. in real world
2 Linear Programming: Formulation and Graphical Method - General formulation of Linear Programming Problem
- Optimisation Models
- Basics of Graphic Method
- Important steps to draw graph
- Multiple, Unbounded Solution and Infeasible Problems
- Solving Linear Programming Graphically Using Computer
- Application of Linear Programming in Business and Industry
3 Linear Programming-Simplex Method - Principle of Simplex Method
- Computational aspect of Simplex Method
- Simplex Method with several Decision Variables
- Two Phase and M-method
- Multiple Solution, Unbounded Solution and Infeasible Problem
- Sensitivity Analysis
- Dual Linear Programming Problem
4 Transportation Problem - Basic Feasible Solution of a Transportation Problem
- Modified Distribution Method
- Stepping Stone Method
- Unbalanced Transportation Problem
- Degenerate Transportation Problem
- Transhipment Problem
- Maximisation in a Transportation Problem
5 Assignment Problem - Solution of the Assignment Problem
- Unbalanced Assignment Problem
- Problem with some Infeasible Assignments
- Maximisation in an Assignment Problem
- Crew Assignment Problem
6 Application of Excel Solver to Solve LPP - Building Excel model for solving LP: An Illustrative Example
7 Goal Programming - Concepts of goal programming
- Goal programming model formulation
- Graphical method of goal programming
- The simplex method of goal programming
- Using Excel Solver to Solve Goal Programming Models
- Application areas of goal programming
8 Integer Programming - Some Integer Programming Formulation Techniques
- Binary Representation of General Integer Variables
- Unimodularity
- Cutting Plane Method
- Branch and Bound Method
- Solver Solution
9 Dynamic Programming - Dynamic Programming Methodology: An Example
- Definitions and Notations
- Dynamic Programming Applications
10 Non-Linear Programming - Solution of a Non-linear Programming Problem
- Convex and Concave Functions
- Kuhn-Tucker Conditions for Constrained Optimisation
- Quadratic Programming
- Separable Programming
- NLP Models with Solver
11 Introduction to game theory and its Applications - Important terms in Game Theory
- Saddle points
- Mixed strategies: Games without saddle points
- 2 x n games
- Exploiting an opponent’s mistakes
12 Monte Carlo Simulation - Reasons for using simulation
- Monte Carlo simulation
- Limitations of simulation
- Steps in the simulation process
- Some practical applications of simulation
- Two typical examples of hand-computed simulation
- Computer simulation
13 Queueing Models - Characteristics of a queueing model
- Notations and Symbols
- Statistical methods in queueing
- The M/M/I System
- The M/M/C System
- The M/Ek/I System
- Decision problems in queueing
Stack Exchange NetworkStack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Balancing an Unbalanced Assignment problem using optimization techniquesHow can I balance the following assignment problem (where machines are to be assigned the jobs in optimal way such that the profit is maximized). Cost matrix is given in the problem. ![unbalanced assignment problem meaning Cost matrix is given in problem](https://i.sstatic.net/ohSTC.png) First step is to convert it into minimization problem by subtracting all the entries in the matrix from maximum value in the matrix. In next step, We try to balance the unbalanced problem i.e. adding dummy machines in this case. How to do this step? What should be row entries for dummy machines in matrix. - convex-optimization
- linear-programming
- discrete-optimization
![unbalanced assignment problem meaning Anoop Kumar's user avatar](https://www.gravatar.com/avatar/6b887cc8c181c7fee73a7f16a09d8673?s=64&d=identicon&r=PG&f=y&so-version=2) You must log in to answer this question.Browse other questions tagged convex-optimization linear-programming discrete-optimization .. - Featured on Meta
- Upcoming sign-up experiments related to tags
Hot Network Questions- How should I end a campaign only the passive players are enjoying?
- "comfortable", but in the conceptual sense
- Is there any reason to keep old checks?
- What does 'bean honey' refer to, in Dorothy L. Sayers' 1928 story
- how do you read this chord?
- Infinitary logics and the axiom of choice
- What was the first modern chess piece?
- I'm a web developer but I am being asked to automate testing in Selenium
- Non-standard alignment of multiline equation
- Why is the Newcomb problem confusing?
- Are there any precautions I should take if I plan on storing something very heavy near my foundation?
- Going around in circles
- John, in his spaceship traveling at relativistic speed, is crossing the Milky Way in 500 years. How many supernovae explosions would he experience?
- "Could" at the beginning of a non-question sentence
- How do I perform pandas cumsum while skipping rows that are duplicated in another field?
- In "Romeo and Juliet", why is Juliet the "sun"?
- Are there several types of mind-independence?
- Why did the UNIVAC 1100-series Exec-8 O/S call the @ character "master space?"
- What was Jessica and the Bene Gesserit's game plan if Paul failed the test?
- Was Croatia the first country to recognize the sovereignity of the USA? Was Croatia expecting military help from USA that didn't come?
- Are many figures in a paper considered bad practice?
- Article that plagiarized our paper is still available and gets cited - what to do?
- How do lee waves form?
- Why don't they put more spare gyros in expensive space telescopes?
![unbalanced assignment problem meaning](https://math.stackexchange.com/posts/2353890/ivc/a24b?prg=24504236-37d8-4bdc-ba54-106d5e28fa3a) ![unbalanced assignment problem meaning BMS | Bachelor of Management Studies Unofficial Portal](https://www.bms.co.in/wp-content/uploads/2011/01/bms.co_.in_1.jpg) FYBMS, SYBMS, TYBMS and beyond BMS What is Balanced or Unbalanced Assignment problem?Operations Research ![unbalanced assignment problem meaning Score Tutorial](https://www.gravatar.com/avatar/1a7f4a304ff3b207c2380795c0ff256c?s=66&r=g&d=mm) The Assignment problem can be Balanced or Unbalanced problem. A Balanced problem means the no. of rows and no. of columns in the problem are equal. E. g. if the problem contains 4 workers and 4 jobs, then it is balanced. Where as, an Unbalanced problem means the no. of rows and no. of columns are not equal. E. g. if the problem contains 4 workers and 3 jobs it is not balanced. Then first we need to balance the problem by taking a Dummy job (imaginary job). Like it? Share with your friends!![unbalanced assignment problem meaning Score Tutorial](https://www.gravatar.com/avatar/1a7f4a304ff3b207c2380795c0ff256c?s=186&r=g&d=mm) Posted by Score TutorialCancel reply. You must be logged in to post a comment. Facebook comments:Forgot password. This Website Is For Sale. Email us an offer we cannot refuse on [email protected] :) ![unbalanced assignment problem meaning zombify-logo](https://www.bms.co.in/wp-content/plugins/zombify/assets/images/zombify-logo.png) ![](//ortec.site/777/templates/cheerup1/res/banner1.gif) |
IMAGES
VIDEO
COMMENTS
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 ...
10 Feb 2019. Whenever the cost matrix of an assignment problem is not a square matrix, that is, whenever the number of sources is not equal to the number of destinations, the assignment problem is called an unbalanced assignment problem. In such problems, dummy rows (or columns) are added in the matrix so as to complete it to form a square matrix.
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 ...
Most libraries for the assignment problem solve either the balanced or unbalanced version of this problem (see the later section). However, whether balanced or unbalanced, it may still occur that the constraint set is infeasible, meaning that there does not exist a (one-sided) perfect matching.
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 ...
Example: Unbalanced Assignment Problem. Solution. Since the number of persons is less than the number of jobs, we introduce a dummy person (D) with zero values. The revised assignment problem is given below: Table. Now use the Hungarian method to obtain the optimal solution yourself. Ans. = 20 + 17 + 17 + 0 = 54.
Unbalanced Assignment Problem: Unbalanced Assignment problem is an assignment problem where the number of facilities is not equal to the number of jobs. To make unbalanced assignment problem, a balanced one, a dummy facility(s) or a dummy job(s) (as the case may be) is introduced with zero cost or time. Dummy Job/Facility: ...
unbalanced assignment problems is based on the assumption that some jobs should be assigned to pseudo or dummy machines, but these jobs are left unexecuted by the dummy machines in the Hungarian method. However, it is sometimes impractical in real-world situations. Moreover, Lampang, Boonjing, and Chanvarasuth introduced a new space- ...
The typical textbook solution to the balanced assignment problem is then found using Kuhn's [3] Hungarian method. Problems in which there are more jobs than machines and more than one job can be ...
For unbalanced assignment problem where the number of jobs is more than the number of machines, the existing approaches assign some jobs to a dummy machine which means these jobs are left without ...
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 ...
Unbalanced Assignment Problem. If the cost matrix of an assignment problem is not a square matrix, the assignment problem is called an Unbalanced Assignment Problem. In such a case, add dummy row or dummy column with zero cost in the cost matrix so as to form a square matrix. Then apply the usual assignment method to find the optimal solution.
7. PDF. 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 ...
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.
4. Unbalanced Assignment Problem. Unbalanced Assignment Problem If number of rows is not equal to number of columns then it is called Unbalanced Assignment Problem. So to solve this problem, we have to add dummy rows or columns with cost 0, to make it a square matrix. ExampleFind Solution of Assignment problem using Hungarian method (MIN case)
Unbalanced assignment problem, Hungarian method, Optimal solution. Introduction The assignment problem is a combinatorial optimization problem in the field of operations research. It is a special case and completely degenerate form of a transportation problem, which occurs when each supply is 1 and
Problem definition: unbalanced assignment problem with multiple jobs. The Unbalanced Assignment Problem deals with processing n tasks to m agents (n and m are integers and different) where each task is exactly affected to only one agent and each agent needs to carry out at least one task (and no more than q tasks). The main objective is to ...
Unbalanced transportation problem is defined as a situation in which supply and demand are not equal. A dummy row or a dummy column is added to this type of problem, depending on the necessity, to make it a balanced problem. The problem can then be addressed in the same way as the balanced problem.
The Assignment Problem (AP) is a fundamental combinatorial optimization problem. It can be formally defined as follows. Given a set n workers, a set of n jobs and a \(n \times n\) cost matrix whose elements are positive representing the assignment of any worker to any job, the AP aims at finding an one-to-one worker-job assignment (i.e., a bipartite perfect matching) that minimizes certain ...
Step 1: Set up the cost matrix. The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.
How can I balance the following assignment problem (where machines are to be assigned the jobs in optimal way such that the profit is maximized). Cost matrix is given in the problem. First step is to convert it into minimization problem by subtracting all the entries in the matrix from maximum value in the matrix.
Unit 8: Assignment Problem - Unbalanced. When an assignment problem has more than one solution, then it is Notes (a) Multiple Optimal solution (b) The problem is unbalanced (c) Maximization problem (d) Balanced problem. 8 Unbalanced Assignment Problem. If the given matrix is not a square matrix, the assignment problem is called an unbalanced ...
The Assignment problem can be Balanced or Unbalanced problem.. A Balanced problem means the no. of rows and no. of columns in the problem are equal. E. g. if the problem contains 4 workers and 4 jobs, then it is balanced. Where as, an Unbalanced problem means the no. of rows and no. of columns are not equal.E. g. if the problem contains 4 workers and 3 jobs it is not balanced.