A GENERAL ITERATIVE SOLVER FOR UNBALANCED INCONSISTENT TRANSPORTATION PROBLEMS

The transportation problem, as a particular case of a linear programme, has probably the highest relative frequency with which appears in applications. At least in its classical formulation, it involves demands and supplies. When, for practical reasons, the total demand cannot satisfy the total supply, the problem becomes unbalanced and inconsistent, and must be reformulated as e.g. finding a least squares solution of an inconsistent system of linear inequalities. A general iterative solver for this class of problems has been proposed by S. P. Han in his 1980 original paper. The drawback of Han’s algorithm consists in the fact that it uses in each iteration the computation of the Moore-Penrose pseudoinverse numerical solution of a subsystem of the initial one, which for bigger dimensions can cause serious computational troubles. In order to overcome these difficulties we propose in this paper a general projection-based minimal norm solution approximant to be used within Han-type algorithms for approximating least squares solutions of inconsistent systems of linear inequalities. Numerical experiments and comparisons on some inconsistent transport model problems are presented.


Introduction
The classical transportation problem involves sources for the transportation of one unit from source i S to destination j D become the entries of the : C n m  cost matrix (see Table 1).
If we denote by , = 1, , , = 1, , The problem is called balanced if the total supply equals the total demand (i.e. , the linear program (1.1) is consistent and well known methods (including Simplex-type algorithms) are available (see Koopmans and Beckmann, 1957;Popa et al., 2013).We will consider in this paper the unbalanced case for which the linear program (1.1) becomes inconsistent (i.e. the set of feasible solutions is empty).
In the paper (Han, 1980) the author showed that the problem (1.1) can be written as an inconsistent system of linear inequalities In Han (1980), (see also Carp et al., 2015), Han presented an iterative method for solving (1.3).For this, he transformed (1.3) in an optimization problem of the form: find and for , mm yy    is defined by Step 2. Compute k   as the smallest minimizer of the function The existence of the smaller minimizer for the convex function  from (1.7) was explained in Han (1980) and an algorithmic procedure to find it was given in Popa et al. (2013).The following result was proved in Han (1980).
x  be the sequence generated by the algorithm H.
-Either it exists an integer then, for some * x solution of (1.4), -The algorithm H produces from any starting point 0 x , a solution for (1.4), in a finite number of iterations (in exact arithmetics).For the approximate computation of the minimal norm solution from Step 1 (1.6), we proposed in Carp et al. (2015) the Kaczmarz Extended (KE) algorithm from Popa (1998).The obtained algorithm is the HKE below.
Algorithm HKE Let 0 n x  be arbitrary fixed; for = 0,1, k do: Step 1. Find = ( ) The replacement of the original Han's pseudoinverse solver (1.6) with the iterative approximation given by the projection-based KE algorithm from Popa (1998) has already been analyzed in Carp et al. (2015).But, the KE algorithm is not always oriented to the sparsity structure of the system matrix A from (1.3), which may slow down the convergence speed.In the present paper we propose the replacement of the iterative solver KE from Step 1 of the algorithm HKE with a sparsity oriented method, algorithm DWE from Popa (2010).For this, we firstly introduce in Step 1 a more general projection-type method (denoted by ALG).In section 2 we prove a result as in Theorem 2 above for the Han-type algorithm obtained, whereas in section 3 we present some numerical experiments and comparisons on some inconsistent transport model problems, involving as iterative solvers the algorithms HKE and HDWE.

Algorithm Han with a general projectionbased minimal norm solution approximant
be real matrices satisfying the (main) assumptions where || || Q denotes the spectral norm of the matrix Q .In the paper Nicola et al. ( 2011) we proposed the following algorithm.

Algorithm General Projections with Correction (GPC)
and, by a recursive argument,  and the proof is complete.

Remark 1
The general method GPC covers many standard projection based algorithms : Kaczmarz Extended, Cimmino Extended, Projection Jacobi Extended, DW Extended, etc. (see Popa, 2012).The advantage of this consists in the case the system matrix B is sparse more efficient algorithms as DW Extended from Popa ( 2010) can be used (see also our Numerical Experiments section).

Numerical experiments
We will consider in our numerical experiments an unbalanced and inconsistent transport problem P described by (1.1) and ii ii sd  ), so will be the system of linear inequalities equivalent with solving it in a least squares sense Ax b  , which motivates us to solve it with Han type algorithms.In order to have a comparative analysis, we also applied the Simplex algorithm on the problem.All the computations were made in Matlab R2010a, a widely used software for mathematical, science and engineering applications.We made use of the build-in Matlab implementation of Simplex algorithm, linprog, whereas the Han-type algorithm was programmed as user-defined function.
All runs are started with the initial approximations 00 = ( ,0) The problem P has the restrictions (1.1: (*) (**))  corresponding to the relations (3.2).   4 and 5 indicate the solutions obtained for the inconsistent transportation problem P. We observe that HKE or HDWE algorithm solution is more reliable (for a practical view point).

Conclusions
In the present paper we considered some iterative solvers for inconsistent systems of linear inequalities.We started with the original Han algorithm from Han (1980), and the iterative projection-based approximation with Kaczmarz Extended (KE) algorithm from Carp et. ( 2015).Then we proposed and theoretically analyzed the replacement of the KE method with a general projection-based algorithm.This allowed us finally to also consider, beside KE, the more sparsity oriented DWE algorithm from Popa, 2010.We then performed numerical experiments and comparisons with both projection-based solvers, KE and DWE, on an unbalanced and inconsistent transport model problem.
get the following mathematical model of the (classical) transportation problem:   will be the Euclidean scalar product).Concerning the matrix involved in (2.1) we will suppose for the whole paper that it has nonzero rows We will consider the general linear least squares problem: find n x  such that:

Table 3 ;
HKE is HGPC algorithm with KE solver in Step 1, whereas HDWE has as iterative solver in Step 1 the DWE algorithm from Popa (2010).We notice that DWE, being sparsity pattern oriented, is more efficient (see Popa, 2010, and numerical experiments therein); moreover, it is a fully parallelizable method.Tables

Table 3 :
Results for problem P