Mathematical Algorithms Applied To Transport Problems

Mathematical algorithms applied to transport problems

Introduction

Transport problems consists in deciding how many units move from certain points of origin (silver, cities, etc.) At certain destination points (distribution centers, cities, etc.) in order to minimize transport costs, given supply and demand at these points. Unitary transport costs, demand requirements and available supply are supposed to be known. The main objectives of a transport model are the satisfaction of all the requirements established by the destinations and of course is the minimization of the costs related to the plan determined by the routes chosen.

Developing

 The context in which the transport model is applied is wide and can generate solutions at aim to the area of ​​operations, inventory and allocation of elements. Any transport model consists of units of a good to be distributed, origins, destinations, resources at the origin, demands in the destinations and distribution costs per unit. Additionally, there are several assumptions. Requirements assumption: Each origin has a fixed supply of units that must be distributed completely among the destinations. Cost assumption: The cost of distributing units of an origin to any destination is directly proportional to the number of distributed units.

Property of feasible solutions: A transport problem has feasible solutions only if the sum of resources in the origins is equal to the sum of demands in destinations.

Property of entire solutions. In cases where both resources and demands take an entire value, all basic variables (assignments), of any of the basic feasible solutions (including the optimal solution), also assume entire values. The first thing to do is formulate the problem in terms of linear programming for this, it is necessary to identify the activities and requirements of the problem to in this way.

It should be used to formulate it as a linear programming problem. After formulating the problem, the next step is to obtain a basic feasible solution, which can be obtained from any of the following 3 criteria:

•  Northwest corner rule.
• Preferred Route Method.
• Vogel approach method.

Northwest corner rule: the first election X11, that is, the assignment for the northwest table of table begins. Then move to the right column if there are still resources at that origin. Otherwise it moves to the region below to make all the assignments.

Preferred Route Method: It is based on the allocation from the minimum cost of distributing a unit. First, this cost is identified the maximum possible resources allocation and then the following cost is identified by carrying out the same procedure until all allocations. Vogel assignment method: For each line and column, its difference is calculated, which is defined as the arithmetic difference between the smallest unit cost and the lower cost that follows in that line or column. In the line or column with the biggest difference, the lower unit cost is assigned. 

The transport or distribution problem is a special network problem in linear programming that is based on the need to carry units of a specific point called source or origin to another specific point called destination. The main objectives of a transport model are the satisfaction of all the requirements established by the destinations, and of course, the minimization of the costs related to the plan determined by the routes chosen. The context in which the transport model is applied is wide and can generate solutions at aim to the area of ​​operations, inventory and allocation of elements.

The resolution procedure of a transport model can be carried out by common linear programming, however its structure allows the creation of multiple solution alternatives such as the allocation structure or the most popular heuristic methods such as Vogel, northwest or minimum corner Costs. Transportation or distribution problems are one of the most applied in the current economy, leaving how to anticipate multiple cases of global success that stimulate their apprehension. A finite group of organized operations is called algorithm in a logical and orderly manner that allows a certain problem to be solved.

It is a series of established instructions or rules that, through a succession of steps, allow to reach a result or solution. According to mathematics experts, algorithms allow us to work from a basic or initial state and, after following the proposed steps, reaching a solution. It should be noted that, although algorithms are usually associated with the mathematical field (since they allow to cite specific cases, find out the quotient between a couple of digits. Determine which is the maximum common divisor between two figures belonging to the group of integers), although they do not always imply the presence of numbers

In addition to all the above, in the mathematical field, and when we are determined to carry out the description of one of those algorithms, it must be taken into account that it can be done through three levels. Thus, first, we find the high level, which is the formal description and finally the implementation task. The transport model is a particular case of problems referring to linear programming. Treat situations for sending products from places called points of origin (supply sources) to destination points (consumption sources), being its objective, determining the optimal amounts of shipping of the sources.

The supply to consumption sources minimize the total cost of transport, while satisfying both the limits of the supply and the demand requirements. The transport algorithm organizes calculations in a more comfortable way taking advantage of the advantage of the special transport model structure. Stop this follows the same steps as the simplex method, however, instead of using the Singlex Normal Table, the advantage of the special transport model structure is used to organize calculations in a more comfortable way. It should be added that the special transport algorithm. 

This was first developed when the norm were the calculations by hand and solutions were needed with abbreviated method. Today we have computing programs that support us in the solution of the problems that arise in the investigation of operations, however, the algorithm in addition to its historical importance allows a perspective to have a primal-dual theoretical relationships to reach A practical result, to improve hand calculations. Another important detail is that the transport algorithm is based on the hypothesis that the model is balanced and that means that the total demand is equal to the total supply.

conclusion

If the model is unbalanced, it can always be increased with a fictitious source or fictional destination to restore balance or balance. The steps of the transport algorithm are exactly the same to those of the simplex algorithm. In the first step a basic feasible starting solution is determined to help us continue in step two. In the second step the optimality condition of the simplex method is used to determine the input variable between all basic variables.  In the third step the feasibility condition of the simplex method is used to determine the output variable and thus obtain the new solution. Subsequently return to step two.