Multiattribute electronic procurement using goal programming

One of the key challenges of current day electronic procurement systems is to enable procurement decisions transcend beyond a single attribute such as cost. Consequently, multiattribute procurement have emerged as an important research direction. In this paper, we develop a multiattribute e-procurement system for procuring large volume of a single item. Our system is motivated by an industrial procurement scenario for procuring raw material. The procurement scenario demands multiattribute bids, volume discount cost functions, inclusion of business constraints, and consideration of multiple criteria in bid evaluation. We develop a generic framework for an e-procurement system that meets the above requirements. The bid evaluation problem is formulated as a mixed linear integer multiple criteria optimization problem and goal programming is used as the solution technique. We present a case study for which we illustrate the proposed approach and a heuristic is proposed to handle the computational complexity arising out of the cost functions used in the bids

Analysis of straightening formula

The straightening formula has been an essential part of a proof showing that the set of standard bitableaux (or the set of standard monomials in minors) gives a free basis for a polynomial ring in a matrix of indeterminates over a field. The straightening formula expresses a nonstandard bitableau as an integral linear cobmbination of standard bitableaux. In this paper we analyse the exchanges in the process of straightening a nonstandard pure tableau of depth two. We give precisely the number of steps required to straighten a given violation of a nonstandard tableau. We also characterise the violation which is eliminated in a single step

On Hilbertian ideals

Abhyankar defined the index of a monomial in a matrix of indeterminates X to be the maximal size of any minor of X whose principal diagonal divides the given monomial. Using this concept, he characterized a free basis for general type of determinantal ideals formed by the minors coming from a saturated subset of X. In this paper, to a monomial in X of index p we associate a combinatorial object called a superskeleton of latitude p, which can loosely be described as a p-tuple of "almost nonintersecting paths" in a rectangular lattice of points. Using this map, we prove that the ideal generated by the p by p minors of a saturated set in X is hilbertian, i.e., the Hilbert polynomial of this ideal coincides with its Hilbert function for all nonnegative integers

Direct Message Delivery Among Processors with Limited Capacity

Processors with finite storage capacities send and receive messages among one another, with each message going directly from its sender to its recipient. Messages that have been sent need no longer be stored, but all in-coming messages must be stored. This paper studies the question of when it is possible to send all the messages, and how to determine e#ciently the sequence in which to send them. A graph model involving Euler tours leads to a complete solution in the case in which all messages have the same length and it is not desired to send messages in parallel. More complicated situations, however, give rise to problems that are NP-complete. Mathematics Subject Classification (1991): primary: 90B12; secondary: 05C45, 68Q25, 68R10. Key words: communications network, Euler tour, NP-complete, message-passing, parallel computation. Running head: Direct Message Delivery Among Processors. This paper appears in Graph Theory, Combinatorics, and Applications, Vol. I, Y. Alavi and A. J. Sch..

On the Minors of an Incidence Matrix and its Smith Normal Form

Consider the vertex-edge incidence matrix of an arbitrary undirected, loopless graph. We completely determine the possible minors of such a matrix. These depend on the maximum number of vertex-disjoint odd cycles (i.e., the odd tulgeity) of the graph. The problem of determining this number is shown to be NP-hard. Turning to maximal minors, we determine the rank of the incidence matrix. This depends on the number of components of the graph containing no odd cycle. We then determine the maximum and minimum absolute values of the maximal minors of the incidence matrix, as well as its Smith normal form. These results are used to obtain sufficient conditions for relaxing the integrality constraints in integer linear programming problems related to undirected graphs. Finally, we give a sufficient condition for a system of equations (whose coefficient matrix is an incidence matrix) to admit an integer solution

Generalized matrix tree theorem for mixed graphs

In this article we provide a combinatorial description of an arbitrary minor of the Laplacian matrix (L) of a mixed graph (a graph with some oriented and some unoriented edges). This is a generalized Matrix Tree Theorem. We also characterize the non-singular substructures of a mixed graph. The sign attached to a nonsingular substructure is described in terms of labeling and the number of unoriented edges included in certain paths. Nonsingular substructures may be viewed as generalized matchings, because in the case of disjoint vertex sets corresponding to the rows and columns of a minor of L, our generalized Matrix Tree Theorem provides a signed count over matchings between those vertex sets. A mixed graph is called quasi bipartite if it does not contain a non singular cycle (a cycle containing an odd number of un-oriented edges). We give several characterizations of quasi-bipartite graphs

Edge version of the matrix tree theorem for trees

We provide a combinatorial description of all the minors of the edge version of the Laplacian matrix of a mixed tree. The description involves the common SDR's for the forests obtained by deleting from the tree the edge sets corresponding to the row and column indices of the minor