Operations Research/Transportation and Assignment Problem
The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first.
Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money which depends on several factors and varies for each choice of factory and outlet. The total amount of the product a particular factory makes is fixed and so is the total amount a particular outlet can store. The problem is to decide how much of the product should be supplied from each factory to each outlet so that the total cost is minimum.
Let us consider an example.
Suppose an auto company has three plants in cities A, B and C and two major distribution centers in D and E. The capacities of the three plants during the next quarter are 1000, 1500 and 1200 cars. The quarterly demands of the two distribution centers are 2300 and 1400 cars. The transportation costs (which depend on the mileage, transport company etc) between the plants and the distribution centers is as follows:
Which plant should supply how many cars to which outlet so that the total cost is minimum?
The problem can be formulated as a LP model:
The whole model is:
subject to,
The problem can now be solved using the simplex method. A convenient procedure is discussed in the next section.
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Assignment Problem
5.1 introduction.
The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY (1953), hence the method is named Hungarian.
5.2 GENERAL MODEL OF THE ASSIGNMENT PROBLEM
Consider n jobs and n persons. Assume that each job can be done only by one person and the time a person required for completing the i th job (i = 1,2,...n) by the j th person (j = 1,2,...n) is denoted by a real number C ij . On the whole this model deals with the assignment of n candidates to n jobs ...
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Transportation Problem | Set 1 (Introduction)
Transportation problem is a special kind of Linear Programming Problem (LPP) in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the sources and destination respectively such that the total cost of transportation is minimized. It is also sometimes called as Hitchcock problem.
Types of Transportation problems: Balanced: When both supplies and demands are equal then the problem is said to be a balanced transportation problem.
Unbalanced: When the supply and demand are not equal then it is said to be an unbalanced transportation problem. In this type of problem, either a dummy row or a dummy column is added according to the requirement to make it a balanced problem. Then it can be solved similar to the balanced problem.
Methods to Solve: To find the initial basic feasible solution there are three methods:
- NorthWest Corner Cell Method.
- Least Cost Method.
- Vogel’s Approximation Method (VAM).
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The spatial organization of transportation and mobility
Traffic Assignment Problem
Traffic assignment problems usually consider two dimensions.
- Generation and attraction . A place of origin generates movements that are bound (attracted) to a place of destination. The relationship between traffic generation and attraction is commonly labeled as spatial interaction. The above example considers one origin/generation and destination/attraction, but the majority of traffic assignment problems consider several origins and destinations.
- Path selection . Traffic assignment considers which paths are to be selected and the amount of traffic using these paths (if more than one unit). For simple problems, a single path will be selected, while for complex problems, several paths could be used. Factors behind the choice of traffic assignment may include cost, time, or the number of connections.
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A Comprehensive Literature Review on Transportation Problems
- Review Article
- Published: 24 September 2021
- Volume 7 , article number 206 , ( 2021 )
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- Yadvendra Kacher 1 &
- Pitam Singh 1
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A systematic and organized overview of various existing transportation problems and their extensions developed by different researchers is offered in the review article. The article has gone through different research papers and books available in Google scholar, Sciencedirect, Z-library Asia, Springer.com, Research-gate, shodhganga, and many other E-learning platforms. The main purpose of the review paper is to recapitulate the existing form of various types of transportation problems and their systematic developments for the guidance of future researchers to help them classify the varieties of problems to be solved and select the criteria to be optimized.
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First author (Yadvendra Kacher) acknowledges the financial support as Junior research fellowship (JRF) received from CSIR (Govt. of India) through HRDG(CSIR) senction Letter No./File No.: 09/1032(0019)/2019-EMR-I.
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Kacher, Y., Singh, P. A Comprehensive Literature Review on Transportation Problems. Int. J. Appl. Comput. Math 7 , 206 (2021). https://doi.org/10.1007/s40819-021-01134-y
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The assignment problem is a special case of the transportation problem, which is a special case of the minimum cost flow problem, which in turn is a special case of a linear program. While it is possible to solve any of these problems using the simplex algorithm , each specialization has a smaller solution space and thus more efficient ...
problems, the Transportation and Assignment Problems. Both of these problems can be solved by the simplex algorithm, but the process would result in very large simplex tableaux and numerous simplex iterations. Because of the special characteristics of each problem, however, alternative solution methods requiring signi cantly less mathematical ...
Prasad A Y, Dept of CSE, ACSCE, B'lore-74. Page 33. Module 4: Transportation Problem and Assignment problem. This means that programmer 1 is assigned programme C, programmer 2 is assigned programme A, and so on. The minimum time taken in developing the programmes is = 80 + 80 + 100 + 90 = 350 min.
7. Identify the relationship between assignment problems and transportation problems. 8. Formulate a spreadsheet model for an assignment problem from a description of the problem. 9. Do the same for some variants of assignment problems. 10. Give the name of an algorithm that can solve huge assignment problems that are well
154 Chapter5. Thetransportationproblemandtheassignmentproblem min z = (8 , 6 , 10 , 10 , 4 , 9) x11 x12 x13 x21 x22 x23 subjectto
The assignment problem is a special case of the transportation problem where the supply from every source and the demand at every sink are equal to 1. Such a situation arises naturally in the setting of assigning workers to jobs, or of assigning workers to a time schedule.
The Transportation and Assignment problems deal with assigning sources and jobs to destinations and machines. We will discuss the transportation problem first. Suppose a company has m factories where it manufactures its product and n outlets from where the product is sold. Transporting the product from a factory to an outlet costs some money ...
The Transportation problem is closely related to the assignment problem (it is in fact easier than the assignment problem) The Transportation problem can be formulated as a ordinary linear constrained optimization problem (i.e.: LP) Example: Cost Matrix:
Assignment problems, which are special cases of transportation problems, pose difficulties for the transportation algorithm and require the development of an algorithm which takes advantage of the simpler nature of these problems. § 1. An Example; The Balanced Transportation Problem We begin with a typical example of a transportation problem ...
Identify the relationship between assignment problems and transportation problems. Formulate a spreadsheet model for an assignment problem from a description of the problem. Do the same for some variants of assignment problems. Give the name of an algorithm that can solve huge assignment problems that are well beyond the scope of Solver.
The Simplex Method for Transportation Problems. Illustrative Examples and a Note on Degeneracy. The Simplex Tableau Associated with a Transportation Tableau. The Assignment Problem: (Kuhn's) Hungarian Algorithm. Alternating Path Basis Algorithm for Assignment Problems. A Polynomial-Time Successive Shortest Path Approach for Assignment Problems
Transportation, Transshipment, and Assignment Problems Learning Objectives After completing this chapter, you should be able to: 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.
The assignment problem is one of the special type of transportation problem for which more efficient (less-time consuming) solution method has been devised by KUHN (1956) and FLOOD (1956). The justification of the steps leading to the solution is based on theorems proved by Hungarian mathematicians KONEIG (1950) and EGERVARY (1953), hence the ...
In this blog, we looked at the transportation problem and demonstrated the use of the IMSL C algorithm imsl_f_transport() to solve both balanced and unbalanced transportation problems. In our next blog, we'll look at two special cases of the transportation problem, the assignment problem, and the transshipment problem.
Transportation and assignment problems are traditional examples of linear programming problems. Although these problems are solvable by using the techniques of Chapters 2-4 directly, the solution procedure is cumbersome; hence, we develop much more efficient algorithms for handling these problems. In the case of transportation problems, the ...
Transportation problem is a special kind of Linear Programming Problem (LPP) in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the sources and destination respectively such that the total cost of transportation is minimized. It is also sometimes called as Hitchcock problem. Types of Transportation problems:
The transportation problem is a distribution-type linear programming problem, concerned with transferring goods between various origins and destinations. In case its main goal is to minimize the ...
for many problems other than the physical distribution of goods. For example, it has been used to efficiently place employees at certain jobs within an organization. (This application sometimes is called the assignment problem. ) We could set up a transportation problem and solve it using the simplex method as with any LP problem (see Using the ...
Section 3.1 introduces the assignment problem in transportation as the distribution of traffic in a network considering the demand between locations and the transport supply of the network. Four trip assignment models relevant to transportation are presented and characterized. Section 3.2 covers traffic assignment to uncongested networks based ...
The transportation problem with multiple objectives (TPMO) TPMO is an example of a transportation problem where multiple objectives need to be considered simultaneously, such as minimizing transportation costs while maximizing resource use or minimizing the environmental impact of transportation. 326 Peter Malacký et al. / Transportation ...
Transportation problems are used to find the minimum cost of transportation of goods from m source to n destination. In this article we will learn transportation problem, formulation, types and finally how it differs from assignment problem.
Traffic assignment considers which paths are to be selected and the amount of traffic using these paths (if more than one unit). For simple problems, a single path will be selected, while for complex problems, several paths could be used. Factors behind the choice of traffic assignment may include cost, time, or the number of connections.
The Frank-Wolfe algorithm is used to solve the lower-level model, with a penalty function introduced to transform the constrained traffic assignment problem (TAP) into an unconstrained TAP. The proposed method is applied using the data of Beijing urban road network and a sensitivity analysis is conducted to examine the impacts of critical ...
A systematic and organized overview of various existing transportation problems and their extensions developed by different researchers is offered in the review article. The article has gone through different research papers and books available in Google scholar, Sciencedirect, Z-library Asia, Springer.com, Research-gate, shodhganga, and many other E-learning platforms. The main purpose of the ...