Sushil Verma

A methodology for developing a web-based factory simulator for manufacturing education

Historically, manufacturing engineering education has focused on teaching mathematical models using simplifying assumptions that can mask the realities of complex manufacturing systems. Recent pedagogical approaches to manufacturing education have focused on developing a more holistic view of the manufacturing enterprise. In this paper, we describe the contents and development methodology of a Virtual Factory Teaching System (VFTS) whose aim is to provide a workspace that illustrates the concepts of factory management and design for complex manufacturing systems. The VFTS is unique in its integration of four domains: web-based simulations, engineering education, the Internet, and virtual factories. Evolutionary development of the VFTS is accomplished by separating the simulation model from the graphical interface and user interaction.

Flowshop scheduling with identical jobs and uniform parallel machines

The single-stage scheduling problem to minimize the makespan of identical jobs on uniform parallel machines is known to be solvable in polynomial-time. We extend this work to consider multi-stage systems with flowshop configuration. We show that the 2-stage problem may also be solved in polynomial-time and for the number of stages greater than two, the problem is shown to be NP-hard. We present a branch and bound procedure which provides an optimal solution to the 3-stage problem, and a fast heuristic procedure that is shown to provide good approximate solutions on sample problems. This heuristic is a natural extension of the 2-stage polynomial-time procedure. We develop theoretical bounds showing that the maximum deviation between the solution derived by the heuristic procedure and the optimal solution is bounded by the maximum processing time of a machine at the second stage, independent of the number of jobs and the processing times at the first and third stages. We also show that the heuristic provides an approximate solution bounded by a ratio of 1.75 to the optimal solution.

Single-Machine Scheduling of Unit-Time Jobs with Earliness and Tardiness Penalties

The problem of determining a schedule of jobs with unit-time lengths on a single machine that minimizes the total weighted earliness and tardiness penalties with respect to arbitrary rational due-dates is formulated as an integer programming problem. We show that if the penalties meet a certain criterion, called the Dominance Condition, then there exists an extremal optimal solution to the LP-relaxation that is integral, leading to a polynomial-time solution procedure. The general weighted symmetric penalty structure is one cost structure that satisfies the Dominance Condition; we point out other commonly found penalty structures that also fall into this category.

MULTISTAGE HYBRID FLOWSHOP SCHEDULING WITH IDENTICAL JOBS AND UNIFORM PARALLEL MACHINES

SUMMARY The single-stage scheduling problem to minimize the makespan of identical jobs with general release times on uniform parallel machines is known to be solvable in polynomial time using the latest start time (LST) rule to determine the optimal schedule. In this paper, we first show that the two-stage problem is NP-hard using a known result that the three-stage problem with equal job release times is NP-hard. The rest of the paper consists of two parts. In the first paper, we compare the extension of the LST rule to the general multistage problem with other heuristics. We compare the heuristics based on the theoretically determined worst-case absolute error bound and on the experimentally determined average deviation from a developed lower bound. The analysis shows that in the presence of many stages, the LST rule does not perform as well as the other heuristics even though they are suboptimal for the single-stage problem. In the second part of this paper, we present a (2#e)-approximation algorithm for the multi-stage problem. Copyright ( 1999 John Wiley & Sons, Ltd.

A note on the strong polynomiality of convex quadratic programming

We prove that a general convex quadratic program (QP) can be reduced to the problem of finding the nearest point on a simplicial cone inO(n3 +n logL) steps, wheren andL are, respectively, the dimension and the encoding length of QP. The proof is quite simple and uses duality and repeated perturbation. The implication, however, is nontrivial since the problem of finding the nearest point on a simplicial cone has been considered a simpler problem to solve in the practical sense due to its special structure. Also we show that, theoretically, this reduction implies that (i) if an algorithm solves QP in a polynomial number of elementary arithmetic operations that is independent of the encoding length of data in the objective function then it can be used to solve QP in strongly polynomial time, and (ii) ifL is bounded by a ‘first order’ exponential function ofn then (i) can be stated even in stronger terms: to solve QP in strongly polynomial time, it suffices to find an algorithm running in polynomial time that is independent of the encoding length of the quadratic term matrix or constraint matrix. Finally, based on these results, we propose a conjecture.

Sufficient matrices belong to L

In this paper, we establish a significant matrix class inclusion that seems to have been overlooked in the literature of the linear complementarity problem. We show that P*, the class of sufficient matrices, is a subclass of L. In the course of demonstrating this inclusion, we introduce other new matrix classes that forge interesting new connections between known matrix classes.

Combinatorial complexity of the central curve

Combinatorial Complexity of the Central Curve Peter A. Beling* Sushil Vermat We study certain combhatorial properties of the centraJ curve szsociated with interior point methods for linear optimization. The central curve forms the basis for a large class of interior point algorithms, most of which follow the central curve in discrete steps. In the context of convex quadratic programming, the number of discrete steps needed to follow the central curve to its conclusion is known to be bounded by a polynomial in the size of the problem data. Rather than studying a discretization scheme, we examine the central curve directly. A measure of complexity for the curve is defined in terms of the number of turns it makes with respect to an arbitrary coordinate system. An average case analysis of this measure is performed for a large class of linear complement arity problems. We show that under a mild transversity condition the expected number of turns taken by the central curve is bounded by nz – n, where the expectation is taken with respect to a sign-invariant probability distribution on the problem data. As an alternative messure of complexity, we also consider the number of times the central curve intersects with a wide class of algebraic hypersurfaces, including such objects ss spheres and boxes. As an example of the results obtained, we show that the primal and dual variables in each coordinate of the central curve cross each other at most once, on average. As a further example, we show that the central curve intersects any sphere centered at the ●Department of Systems Engineering, University of Virginia, Charlottesville, VA 22903. EmaiL pb3aQvirginia.edu tDepartment of Industrial Engineering and Systems Engineering, University of Southern California, Los Angeles, CA 90210. Email: verrna@rcf.usc.edu origin at most once, on average.

A probabilistic analysis of a measure of combinatorial complexity for the central curve

Abstract.We investigate certain combinatorial properties of the central curve associated with interior point methods for linear optimization. We define a measure of complexity for the curve in terms of the number of turns, or changes of direction, that it makes in a geometric sense, and then perform an average case analysis of this measure for P-matrix linear complementarity problems. We show that the expected number of nondegenerate turns taken by the central curve is bounded by n2-n, where the expectation is taken with respect to a sign-invariant probability distribution on the problem data. As an alternative measure of complexity, we also consider the number of times the central curve intersects with a wide class of algebraic hypersurfaces, including such objects as spheres and boxes. As an example of the results obtained, we show that the primal and dual variables in each coordinate of the central curve cross each other at most once, on average. As a further example, we show that the central curve intersects any sphere centered at the origin at most twice, on average.

A New Parameterization Algorithm for the Linear Complementarity Problem

We study a new parameterization algorithm for the P-matrix restriction of the linear complementarity problem. This parameterization can be considered similar to a path-following penalty method, and introduces to this class of algorithms a primal-dual symmetry similar to that seen in the highly successful path-following logarithmic barrier methods. The trajectory associated with the parameterization is distinguished by a naturally defined starting point and by a piecewise characterization as a fractional polynomial function of a single parameter. The trajectory can be followed exactly using root isolation and a simplex-like pivoting scheme. Convergence is guaranteed for a weakly-regular subclass of P-matrix LCPs. We show that under a weakly-regular and sign-invariant distribution for the input matrices and vectors, the average number of pieces in the trajectory is O(n 2), where n is the dimension of the problem space, and hence that the path-following algorithm has average-case polynomial running time.

A fast symmetric penalty algorithm for the linear complementarity problem

In an earlier paper, the authors presented a new parameterization algorithm for the the linear complementarity problem. The trajectory associated with this parameterization is distinguished by a naturally defined starting point and by a piecewise characterization as a fractional polynomial function of a single parameter. In order to follow this trajectory, however one needs to isolate roots of polynomials-a computationally expensive operation. In this paper, we present an algorithm in which we parameterize one dimension at a time. This results in polynomials of degree one, which allows trivial root isolation. The algorithm stays unaffected in other key respects. For example, the average number of pieces in the trajectory is still O(n/sup 2/), where n is the dimension of the problem space. This implies that our algorithm is competitive, in an average sense, to Lemkes method.