transportation assignment problem

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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Traffic Assignments to Transportation Networks

• First Online: 01 January 2014

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• Dietmar P. F. Möller 3  

Part of the book series: Simulation Foundations, Methods and Applications ((SFMA))

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This chapter begins with a brief overview of traffic assignment in transportation systems. 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 on the assumption that cost does not depend on traffic flow. Section 3.3 introduces the topic of traffic assignment and congested models based on assumptions from traffic flow modeling, e.g., each vehicle is traveling at the legal velocity, v , and each vehicle driver is following the preceding vehicle at a legal safe velocity. Section 3.4 covers the important topic of equilibrium assignment which can be expressed by the so-called fixed-point models where origin to destination (O-D) demands are fixed, representing systems of nonlinear equations or variational inequalities. Equilibrium models are also used to predict traffic patterns in transportation networks that are subject to congestion phenomena. Section 3.5 presents the topic of multiclass assignment, which is based on the assumption that travel demand can be allocated as a number of distinct classes which share behavioral characteristics. In Sect. 3.6, dynamic traffic assignment is introduced which allows the simultaneous determination of a traveler’s choice of departure time and path. With this approach, phenomenon such as peak spreading in response to congestion dynamics or time-varying tolls can be directly analyzed. In Sect. 3.7, transportation network synthesis is introduced which focuses on the modification of a transportation road network to fit a required demand. Section 3.8 covers a case study involving a diverging diamond interchange (DDI), an interchange in which the two directions of traffic on a nonfreeway road cross to the opposite side on both sides of a freeway overpass. The DDI requires traffic on the freeway overpass (or underpass) to briefly drive on the opposite side of the road. Section 3.9 contains comprehensive questions from the transportation system area. A final section includes references and suggestions for further reading.

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References and Further Readings

Bando M, Hasebe K, Nakayama A, Shibata A, Sugiyama Y (1995) Dynamic model of traffic congestion and numerical simulation. Phys Rev E 51(2):1035–1042

Article   Google Scholar  

Bliemer MCJ (2001) Analytical dynamic traffic assignment with interacting user-classes: theoretical advances and applications using a variational inequality approach. PhD thesis, Delft University of Technology, The Netherlands

Google Scholar  

Bliemer MCJ, Castenmiller RJ, Bovy PHL (2002) Analytical multiclass dynamic traffic assignment using a dynamic network loading procedure. In: Proceedings of the 9th meeting EURO Working Group on Transportation. Tayler & Francis Publication, pp 473–477

Cascetta E (2009) Transportation systems analysis: models and application. Springer Science + Business Media, LLV, New York

Book   Google Scholar  

Chiu YC, Bottom J, Mahut M, Paz A, Balakrishna R, Waller T, Hicks J (2011) Dynamic Traffic Assignment, A Primer for the Transportation Network Modeling Committee, Transportation Research Circular, Number E-C153, June 2011

Chlewicki G (2003) New interchange and intersection designs: the synchronized split-phasing intersection and the diverging diamond interchange. In: Proceedings of the 2nd urban street symposium, Anaheim

Correa ER, Stier-Moses NE (2010) Wardrop equilibria. In: Cochran JJ (ed) Encyclopedia of operations research and management science. Wiley, Hoboken

Dafermos SC, Sparrow FT (1969) The traffic assignment problem for a general network. J Res US Nat Bur Stand 73B:91–118

Article   MathSciNet   Google Scholar  

Dubois D, Bel G, Llibre M (1979) A set of methods in transportation network synthesis and analysis. J Opl Res Soc 30(9):797–808

Florian M (1999) Untangling traffic congestion: application of network equilibrium models in transportation planning. OR/MS Today 26(2):52–57

MathSciNet   Google Scholar  

Florian M, Hearn DW (2008) Traffic assignment: equilibrium models. In: Optimization and its applications, vol 17. Springer Publ., pp 571–592

Hughes W, Jagannathan R (2010) Double crossover diamond interchange. TECHBRIEF FHWA-HRT-09-054, U.S. Department of Transportation, Federal Highway Administration, Washington, DC, FHWA contact: J. Bared, 202-493-3314

Inman V, Williams J, Cartwright R, Wallick B, Chou P, Baumgartner M (2010) Drivers’ evaluation of the diverging diamond interchange. TECHBRIEF FHWA-HRT-07-048, U.S. Department of Transportation, Federal Highway Administration, Washington, DC. FHWA contact: J. Bared, 202-493-3314

Knight FH (1924) Some fallacies in the interpretation of social cost. Q J Econ 38:582–606

Larsson T, Patriksson M (1999) Side constrained traffic equilibrium models—analysis, computation and applications. Transport Res 33B:233–264

Lozovanu D, Solomon J (1995) The problem of the synthesis of a transport network with a single source and the algorithm for its solution. Comput Sci J Moldova 3(2(8)):161–167

MathSciNet   MATH   Google Scholar  

ProcessModel (1999) Users Manual, ProcessModel Corporation, Provo, UT

Rodrigus J-P (2013) The geography of transportation systems. Taylor & Francis, Routledge

Steinmetz K (2011) How it works, traffic gem, diverging-diamond interchanges can save time and lives. Time Magazine, pp. 54–55, 7 Feb 2011

Wardrop JG (1952) Some theoretical aspects of road traffic research. In: Proceedings of the institute of civil engineers, Part II, vol 1, ICE Virtual Library, Thomas Telford Limited, pp 325–378

Wilson AG (1967) A statistical theory of spatial distribution models. Transport Res 1:253–269

Yang H, Huang H-J (2004) The multi-class multi-criteria traffic network equilibrium and systems optimum problem. Transport Res Part B 38:1–15

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Möller, D.P.F. (2014). Traffic Assignments to Transportation Networks. In: Introduction to Transportation Analysis, Modeling and Simulation. Simulation Foundations, Methods and Applications. Springer, London. https://doi.org/10.1007/978-1-4471-5637-6_3

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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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Balanced and Unbalanced Transportation Problems

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 Problems

Balanced 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 Problems

There 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:

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.

Solving Balanced Transportation problem by Northwest Corner Method

Consider this scenario:

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.

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.

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.

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.

Continuing in the same manner, the final cell values will be:

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 Problem

An 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.

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.

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 Problems

What 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).

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