An Application of Lagrange Multiplier for integer Linear Programming in Production- Transportation with flexible transportation cost

The Transportation problem is a classic Operations Research problem where the objective is to determine the schedule for transporting goods from source to destination in a way that minimizes the shipping cost while satisfying supply and demand constraints. Although it can be solving as a linear programming problem. Linear programming makes use of the simplex method, an algorithm invented to solve a linear program by progressing from one extreme point of the feasible polyhedron to an adjacent one. The algorithm contains tactics like pricing and pivoting. For a transportation problem, a simplified version of the regular simplex method can be used, known as the transportation simplex method. In this paper will discuss the functionality of both of these algorithms, and compared their optimized values with non-linear method called the Lagrange Multiplier Method. Lagrange Multiplier is an algorithm that uses different mechanisms to choose the best optimal solutions. This method based on transforming the linear structure transportation problem into the nonlinear structure and solved it directly, by the techniques. The objective of the study was to find out how these algorithms behave in terms of accuracy and speed when a large-scale problem is being solved. Transportation Problem The transportation problem is classic Operation Research problem. Therefore; arises frequently in planning for the distribution of goods and services from several supply locations (Sources) to several demand locations (Destinations) in a way that minimize the shipping cost, while satisfying supply and demand constraints. Typically, the quantity of goods available at each supply location is limited, and the quantity of goods needed at each of several demand locations is known. Therefore; a typical Transportation Problem has the following elements: 1. Source(s) 2. Destination(s) 3. Weighted edge(s) The objective of a Transportation Problem is to determine the Number Journal of Kirkuk University –Scientific Studies , vol.5, No.1,2009 111 Supply Points of units to be transported from source ( i ) to destination ( j ) so that the total transportation cost is minimum (Phillips et al.,1976 ; Gupta & Hira, 2002). A simple transportation network is shown in figure 1. Depots 1, 2, and 3 are the source nodes, Stations 1,2, ..., 5 are the destination nodes, the arcs between nodes represent the existence of a path. The supply and demand requirements appear beside the nodes. For example, depot 1 is capable of producing 89945 cubic-meters (M 3 ) of Gasoline and station 1 needs at least 3489 M 3 as shown in Figure 1 and 2.


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The Transportation problem is a classic Operations Research problem where the objective is to determine the schedule for transporting goods from source to destination in a way that minimizes the shipping cost while satisfying supply and demand constraints. Although it can be solving as a linear programming problem. Linear programming makes use of the simplex method, an algorithm invented to solve a linear program by progressing from one extreme point of the feasible polyhedron to an adjacent one. The algorithm contains tactics like pricing and pivoting. For a transportation problem, a simplified version of the regular simplex method can be used, known as the transportation simplex method.
In this paper will discuss the functionality of both of these algorithms, and compared their optimized values with non-linear method called the Lagrange Multiplier Method. Lagrange Multiplier is an algorithm that uses different mechanisms to choose the best optimal solutions. This method based on transforming the linear structure transportation problem into the nonlinear structure and solved it directly, by the techniques. The objective of the study was to find out how these algorithms behave in terms of accuracy and speed when a large-scale problem is being solved.

Transportation Problem
The transportation problem is classic Operation Research problem. Therefore; arises frequently in planning for the distribution of goods and services from several supply locations (Sources) to several demand locations (Destinations) in a way that minimize the shipping cost, while satisfying supply and demand constraints. Typically, the quantity of goods available at each supply location is limited, and the quantity of goods needed at each of several demand locations is known. Therefore; a typical Transportation Problem has the following elements: 1. Source(s) 2. Destination(s) 3. Weighted edge(s) The objective of a Transportation Problem is to determine the Number

Supply Points
of units to be transported from source ( i ) to destination ( j ) so that the total transportation cost is minimum (Phillips et al.,1976 ;Gupta & Hira, 2002).
A simple transportation network is shown in figure 1. Depots 1, 2, and 3 are the source nodes, Stations 1,2, …, 5 are the destination nodes, the arcs between nodes represent the existence of a path. The supply and demand requirements appear beside the nodes. For example, depot 1 is capable of producing 89945 cubic-meters (M 3 ) of Gasoline and station 1 needs at least 3489 M 3 as shown in Figure 1 and 2.

Figure 1
All the information can also be stored in a tableau form. Figure 2 shows the tableau form  This problem is simple and can be solved without much effort. However, practical transportation networks are much more complicated and intelligent techniques are needed to solve them efficiently. One way to solve Transportation Problems is through a method called Linear Programming. The next section will briefly discuss how this can be done.

Mathematical Form of Linear Programming Problem
Linear Programming is the mechanism of maximizing or minimizing a linear function over a convex polyhedron. Linear Programming applies to optimization problems in which the objective and constraint functions are strictly linear. Therefore; the general standard form of the linear primal is defined as:

Solving a Transportation Problem as a LP
Transportation Problem can be modeled as a Linear Programming problem. Such that the amount of supply at source i is i S and the demand at destination j is j D . The unit transportation cost between source i and destination j is ij C . Let ij X represent the quantities of the goods to be shipped from source ( i ) to destination ( j ); then the LP model representing the transportation problem is given generally as:

Simplex method
The most general technique of LP is called the Simplex Method it uses a tableau form of representing the numbers. The algorithm has two basic parts: (1) Find an initial basic feasible solution, (2) Obtain an optimal solution by making successive improvements to initial basic feasible solution until no further decrease in the transportation cost is possible. The Simplex Method is a greedy algorithm. It obtains basic feasible solutions by making the most improvement from the previous solution in every iteration (Sharrma, 1988).

Feasible Solution
A set of non-negative values , that satisfies the constraints is called a feasible solution to the transportation problem (Gupta & Hira, 2002).

Basic Feasible Solution
A feasible solution that contains no more than ( 1   n m ) non-negative allocations is called a basic feasible solution to the transportation problem (Gupta & Hira, 2002).
For example, the basic feasible solution for the problem in figure 2 by Vogel method is:

Transportation Simplex method
The Transportation Simplex Method is a special version of the Simplex Method used to solve Transportation Problem. Although it has the basic steps as Simplex Method, it has a much more compact tableau form. This compact form takes less memory, therefore might be faster. For detailed steps of the Simplex and Transportation Simplex Method, see (Sharrma, 1988;Winston,1987)

Computational Difficulties
Any Linear Programming Problem can be solved by using the Simplex Algorithm. With the Transportation Simplex method, Transportation Problem can be solved accurately. However, it requires too much found by the algorithm that eventually gets rejected. Therefore, it takes too long for these algorithms to solve TSPs. Since Lagrange Multiplier is a mathematical tool for solving constrained optimization of differentiable functions. As known, the Lagrange multipliers can be used to solve nonlinear programming in which all the constraints are equality constraints. But hence, we try to use it to find optimality solution for linear transportation problem.
In this research project, we shall discuss the Lagrange's multipliers method which provides a necessary condition for an optimum when constraints are equations.

Lagrange Multipliers
In this research will cover how to find the maximum and minimum points on a function subject to constraint. This technique is called Lagrange's Method. Suppose we wish to find an optimal solution of a differentiable function. We consider linear programming of the transportation problem: Minimize: whose variables are subject to the ( ) constraints …(1) ; for and ; for ; for all i and j Since the transportation problem has ( n m  ) equality constraints and ( mn ) inequality (non-negativity) constraints, then the Lagrange formulation must have one Lagrange multiplier for each of these constraints. These will now be defined. Let the Lagrange multiplier ( Here (3) The following theorem gives conditions, which imply that any point Theorem: (Boyd & Vandenberghe, 2004) Suppose that: Minimize:

The Kuhn-Tucker Condition in Transportation Problem
In Lagrange multiplier method, it becomes difficult to solve the system of ( mn ) simultaneous equations. This difficulty can be removed by using Kuhn-Tuker Conditions. This research is concerned with developing the necessary and sufficient conditions for identifying the basic feasible solution of the general inequality constrained optimization transportation problems. These conditions are called the Kuhn-Tucker conditions. The development is based mainly on Lagrange method.
The second theorem give conditions that are necessary for a point (  (Dano,1975). Now, since the transportation problem is minimization problem, the Kuhn-tucker conditions are thus given by: (because negative difference implies decrease in cost) for one or more zero shipment cells, then it would be better to reduce the cost more by allocating as much as possible to the cell with the largest negative (smallest) value of ij  . This way, it is possible to improve the basic feasible solution successively for reduced cost till the optimal solution is obtained for which 0  ij  for each zero shipment (empty cell). It is well known that in pure transportation problem, there will be at most (m+n-1) positive source-destination flows in an optimal solution to the cost-minimizing problem. Thus, the system of Lagrange constraints which hold with equality will have (m+n-1) equations in (m+n+mn) unknowns. One of the Lagrange multipliers can be normalized to zero, with all other Lagrange multipliers calculated relative to this anchor point. This result also implies that there will be most (m-1) sources, which supply more than one destination.
We noted that, the method of Lagrange multipliers reproduces exactly the same iterations as the multipliers method, but it reduces the time in searching the optimal solution. The main difference occurs in the manner in which the multipliers method is based on duality theory. The development of the Lagrange multipliers method is based on solving the problem with non-linear structure.

Solving Transportation Problems Using Lagrange Multiplier Algorithm
To solve Transportation Problems using this Lagrange method, the following steps are needed: 1. Forming the Fitness Lagrange Function. Forming the Fitness Lagrange Function is generated by using the equation 2. For example, the objective function for problem in figure 2 is: Forming the Lagrange function L, for the Gasoline product, we begin with basic feasible solution by using Vogel method in figure 3, then:   for each nonbasic variable. Thus the optimal solution for transportation problem is finding after 5 th iterations.

Conclusions
In comparison to the existing methods, Lagrange algorithm proves to be more efficient as the size of the problem becomes greater for a problem. The non-linear method is much faster, easier and shorter than the simplex and transportation simplex method. This solution is not computationally very fast. We are interested in identifying the conditions under which the objective function associated with an optimal solution of the transportation problem is equal to the bound given by maximizing (minimizing) the expected overlap.