A primal-dual method for solving linear programming problems with fuzzy cost coefficients based on linear ranking functions and its applications

Chapter 16 - Variants of the simplex method

We present a parallel algorithm for solving the dual transshipment problem. The traditional dual simplex method does not offer much scope for parallelization, because it moves from one basic feasible solution to another, performing one pivot operation at a time. We present a new method called modified network dual simplex method which uses concurrent pivots. This departure from the traditional LP approach raises several issues such as the need to convert a non-basic feasible solution to a basic feasible solution. We present our strategies to handle these issues as well as the corresponding parallel algorithms. We also present results of testing this algorithm on large graphs to solve the integrated layout compaction and wire balancing problem. >

Our paper presents two new algorithms for solving linear programming problems. These algorithms are based on the convergent criss-cross method and on the idea of the “Hungarian Method”. Similarly to the primal-dual algorithm of Dantzig-Ford-Fulkerson, our algorithms improve a feasible (may be not basic) solution step by step, but we use Terlaky's convergent criss-cross method for solving the subproblems. Our algorithms solve linear programming problems in a finite number of steps (i.e. cycling cannot occur). We show that the primal and dual Simplex methods are special cases of our algorithms. In these Simplex methods we use Bland's pivoting rule only if the basic transformations are degenerate. By this we show how can one derive the primal or dual simplex method from Terlaky's criss-cross method.

A parallel primal–dual simplex algorithm

It is well known that the G. B. Dantzig’s simplex method is not a good algorithm due to the reason that it is not a polynomial algorithm for use to handle computational complexity; however, it is the popular basic, most effective and widely used way to solve linear programming problems so far. Herewith, various improvements on the simplex procedure have filled many volumes of works. There are two key works that deserve our attention: how to get an initial feasible basis in the form of a unit matrix; how to get the optimal solution as quickly as possible. The basic thought of the simplex method is aimed at the standard form of a linear programming problem:max { }, , , ,C X A X b X b m nT m n = ≥ ≤ (1)Improving the optimal value of the object function via a transform from a basic feasible solution to another basic feasible solution start off at the point where a basic feasible solution is given in its feasible region; and then one obtains the optimal solution or arrives at a judgment that no optimal solution exists. This thought is realized by using a basis transformation in reality. The basis transformation is an operation of iteration by choosing a pivot via determining an entrance variable and an extraction variable according to a given rule. In practical uses, we find that theAlso, assume that B is a basis, and the iteration coefficients A and b are A j( )ai , b = (bi ). By letting XN = 0 , we obtain the corresponding basic solution (Editorial Group of Textbook of Operations Research. 1990, J. L. Xue. 1992, G. H. Wei, J. L. Fu, etc. 1987, Y. L. Niu, 1994)X b b b B b= ⎡⎣ ⎤⎦ = ⎡ ⎣⎢⎤ ⎦⎥1 21 0 00 , , , , , ,…the value of the object function Z C B bB T 1 , andthe criterion σ σ σ[ ,σ , , ]1 2 n T .

In this research, a new and efficient approach of finding an initial basic feasible solution to transportation problems is proposed. The proposed approach is named “Inverse Coefficient of Variation Method (ICVM)”, and the method is illustrated with seven numerical examples. Six existing methods; North West Corner Method (NWCM), Column Minimum Method (CMM), Least Cost Method (LCM), Row Minimum Method (RMM),Vogel’s Approximation Method (VAM), and Allocation Table Method (ATM) were compared with the proposed approach. It can be said conclusively that the proposed Inverse Coefficient of Variation Method (ICVM) provides an improved Initial Basic Feasible Solution to all the transportation problems used in the experiment. Further, the new method leads to the optimal solution to many of the problems considered.

In this paper we develop a new algorithm for the initial fuzzy basic feasible solution to a solid transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers. Using parametric form of triangular fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem without changing to its crisp equivalent form. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.

In this paper, we develop a new algorithm for the initial fuzzy basic feasible solution to a transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers or trapezoidal fuzzy numbers or real numbers. By applying the efficient ranking function and arithmetic operations the solution is obtained without changing it into its crisp equivalent form. Using the parametric form of fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using a proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.

This paper studied the balanced transportation problem. Many algorithms were developed to solve optimally this problem. The most commonly used method is the modified distribution method (MODI). The MODI algorithm starts by generating an initial basic feasible solution using the northwest corner method (NCM). In general, the cost of the initial basic feasible solution generated by NCM is too high and far away from the optimal cost. For this reason, we propose in this paper a new method to generate an initial basic feasible solution to the balanced transportation problem to solve this issue. This method, named Global Minimum Method (GMM), was tested and compared with the NCM, the minimum cost method (MCM), the Vogel's approximation method (VAM), and the optimal cost obtained by MODI. The computational results proved that the new heuristic is promising for small size instances and outperforms all other methods for big size problems.

The Simplex primal and dual methods, for the solution of $$\max \left\{ {c^T x:Ax = b, x \geqslant 0} \right\},$$ were presented previously in terms of certain bases ℤ and\(\mathbb{Y}\) ofN(A) andR(A T ) respectively. In these implementations, called the ℤ-Simplex Algorithm and the\(\mathbb{Y}\)-Dual Method, the bases ℤ and\(\mathbb{Y}\) (giving the edges of the polyhedron in question at the given basic feasible solution) are updated at each iteration. In this paper we show that only partial updates of ℤ are needed in the ℤ-Simplex Algorithm, analogously to the partial updates in the Revised Simplex Algorithm. Similar results can be given for the\(\mathbb{Y}\)-Dual Method.

The idea of the simplex method is to proceed from one basic feasible solution (that is, one extreme point) of the constraint set of a problem in standard form to another, in such a way as to continually decrease the value of the objective function until a minimum is reached. The results of Chap. 2 assure us that it is sufficient to consider only basic feasible solutions in our search for an optimal feasible solution. This chapter demonstrates that an efficient method for moving among basic solutions to the minimum can be constructed.

In this paper an algorithm is given for solving linear fractional functionals programming problems. This method permits to replace in each iteration several basic variables by an equal number of non‐basic ones so as to obtain a new basic feasible solution with the new value of the objective function which is, at least, as good as the old one. To initiate the algorithm, we start with a basic feasible solution. The method of determining a new basic feasible solution, in each iteration, requires the optimum solution of some auxiliary linear fractional functionals program.

Chapter 18 - Bounded variable problems

Transportation of products from sources to destinations with minimal total cost plays an important role in logistics and supply chain management. The Uncapacitated Transportation Problem (UTP) is a special case of network flow optimization problem. The prime objective of this UTP is to minimize the total cost of transporting products from origins to destinations subject to the respective supply and demand requirements. The UTP consists of special network structure. Due to the special structure of this problem, the transportation algorithm is preferred to solve it. The transportation algorithm consists of two major steps: 1) Finding an Initial Feasible Solution (IFS) to TP and 2) Examining the optimality of this IFS. A better IFS generates a lesser number of iterations to obtain a Minimal Total Cost Solution (MTCS). Recently, Juman and Nawarathne (2019)’s Method was introduced to find an IFS to UTP. In this paper, the Juman and Nawarathne (2019)’s Method is improved to get a better IFS to a UTP. A comparative study on a set of benchmark instances illustrates that the new improved method provides better primal solutions compared to the Juman and Nawarathne (2019)’s Method. The proposed method is found to yield the minimal total cost solutions to all the benchmark instances.

Computational behavior of a feasible direction method for linear programming