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A new linearizing method is presented for globally solving sum of linear ratios problem with coefficients. By using the linearizing method, linear relaxation programming (LRP) of the sum of linear ratios problem with coefficients is established, which can provide the reliable lower bound of the optimal value of the initial problem. Thus, a branch and bound algorithm for solving the sum of linear ratios problem with coefficients is put forward. By successively partitioning the linear relaxation of the feasible region and solving a series of the LRP, the proposed algorithm is convergent to the global optimal solution of the initial problem. Compared with the known methods, numerical experimental results show that the proposed method has the higher computational efficiency in finding the global optimum of the sum of linear ratios problem with coefficients.

In this paper, we consider the following sum of linear ratios problem with coefficients:

The sum of linear ratios problem with coefficients (SLRC) has attracted the interest of researchers and practitioners for many years. In part, this is because the problem (SLRC) and its special case have broad applications in many practical problems, for example, profit rates and pricing decisions problem [

During the past 20 years, many algorithms have been presented for globally solving special cases of the problem (SLRC), which are only developed for solving the sum of linear ratios problem with the assumption that all coefficients

The purpose of this paper is to develop a new linearizing method for globally solving the sum of linear ratios problem with coefficients (SLRC). The proposed new linearizing method uses more information of the objective function of the problem (SLRC); by using the method, the linear relaxation programming (LRP) of the problem (SLRC) is established, which can provide a tighter lower bound of the global optimal value of the problem (SLRC) than the previous linearizing method in branch and bound algorithm and which can be used to reduce rapidly the growth of branching tree in the branch and bound algorithm for solving the problem (SLRC); therefore it can improve the computational efficiency of the algorithm. By successively subdividing feasible region of the problem (SLRC) and solving a series of the LRP, the proposed algorithm is convergent to the global optimum of the problem (SLRC). Finally, compared with the known methods, numerical experimental results imply that the proposed new linearizing method can be used to globally solve the problem (SLRC) with the higher computational efficiency.

The remainder sections of this paper are organized as follows. In Section

To globally solve the problem (SLRC), by solving

By the characteristic of fractional function

The important structure in the establishment of a branch and bound procedure to globally solve the problem (SLRC) is the calculation of lower bounds for this problem and for its subproblems. A lower bound of the global optimal value of the problem (SLRC) and its subproblems can be computed by solving a sequence of linear relaxation programming problems. The proposed new linearizing method for establishing linear relaxation programming of the problem (SLRC) is to underestimate or overestimate each function

Let

By the above inequality (

By the above inequalities (

By the characteristic of the convex function, we can derive its linear underestimating function

Based on the above discussion, for each

Similarly, for each convex function

According to the above discussion, for each

By (

Let

According to the above linearizing method, for

According to the construction method of the linear relaxation programming (LRP), for

The following theorem ensures that the linear function

For all

Let

Let

Then, we have

Let

Since

Since

Therefore, we have

By the above proof, we have

Next, we consider the difference

Thus,

Let

Since

Since

Therefore, we have

Since

By the above proof, we can follow that

Therefore,

By the above discussion, it is obvious that the conclusion is followed.

In this section, based on the former new linearizing method, we present an effective branch and bound algorithm for globally solving the sum of linear ratios problem with coefficients (SLRC). The critical construction in ensuring that the proposed branch and bound algorithm is convergent to the global optimum of the problem (SLRC) is the selection of a reasonable branching rule. In this paper, we use a standard branching rule which is called bisection. The selected branching rule is described as follows.

Assume that the hyperrectangle

Assume that

Set the initial convergence tolerance

Calculate

Subdivide hyperrectangle

For every

Let

The above proposed algorithm either stops finitely with the global optimal solution for the problem (SLRC) or generates an infinite sequence of iterations

If the proposed algorithm stops finitely at iteration

If the proposed algorithm generates an infinite sequence of partitioned subrectangles

To verify the reliability and effectiveness of the proposed new linearizing method, several test examples that appeared in the recent literatures are implemented on an Intel(R) Core(TM)2 Duo CPU (1.58 GHZ) microcomputer. The proposed algorithm using the new linearizing method is coded in C++ and every linear relaxation programming problem is solved by simplex method. These test examples and their numerical results are described as follows.

We have the following:

With

Using the method in [

We have the following:

With

But, using the method in [

We have the following:

With

Using the method in [

We have the following:

With

Using the method in [

By substituting verification, we know that the global optimal solutions of Examples

From numerical results for Examples

The new linearizing method used for solving the problem (SLRC) can be extended to seek a global optimal solution of the sum of linear ratios problem with coefficients whose domain is not linear. Some extensions are given as follows.

When domain

When constraint functions of the problem (SLRC) are also sum of linear fractional functions with coefficients, whose mathematical modeling can be reformulated as follows:

Using the linearizing method proposed in Section

Using the same algorithm step in Section

We have the following:

With

We have the following:

With

From numerical results for Examples

It should also be noted that our approach could be extended to solve more general generalized linear fractional programming problems; this will constitute a subject for future research.

In this paper, by utilizing the linear approximation of exponential and logarithmic functions, a new linearizing method is presented. Combining the linearizing method within the branch and bound scheme, a branch and bound algorithm is constructed for solving the problem (SLRC). By subsequently partitioning linear relaxation of the feasible region and solving a series of linear programming problems, the proposed algorithm is convergent to a global optimal solution of the problem (SLRC). Compared with the known methods, numerical experimental results show that the proposed new linearizing method can be used to globally solve the problem (SLRC) with the higher computational efficiency.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper is supported by the National Natural Science Foundation of China under Grant no. 11171094 and the Science and Technology Key Project of Education Department of Henan Province no. 14A110024.