Katta G. Murty

An Algorithm for the Traveling Salesman Problem

A “branch and bound” algorithm is presented for solving the traveling salesman problem. The set of all tours feasible solutions is broken up into increasingly small subsets by a procedure called branching. For each subset a lower bound on the length of the tours therein is calculated. Eventually, a subset is found that contains a single tour whose length is less than or equal to some lower bound for every tour. The motivation of the branching and the calculation of the lower bounds are based on ideas frequently used in solving assignment problems. Computationally, the algorithm extends the size of problem that can reasonably be solved without using methods special to the particular problem.

Some NP-complete problems in quadratic and nonlinear programming

AbstractIn continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether(a)a given feasible solution is not a local minimum, and(b)the objective function is not bounded below on the set of feasible solutions. We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.

Storage space allocation in container terminals

Container terminals are essential intermodal interfaces in the global transportation network. Efficient container handling at terminals is important in reducing transportation costs and keeping shipping schedules. In this paper, we study the storage space allocation problem in the storage yards of terminals. This problem is related to all the resources in terminal operations, including quay cranes, yard cranes, storage space, and internal trucks. We solve the problem using a rolling-horizon approach. For each planning horizon, the problem is decomposed into two levels and each level is formulated as a mathematical programming model. At the first level, the total number of containers to be placed in each storage block in each time period of the planning horizon is set to balance two types of workloads among blocks. The second level determines the number of containers associated with each vessel that constitutes the total number of containers in each block in each period, in order to minimize the total distance to transport the containers between their storage blocks and the vessel berthing locations. Numerical runs show that with short computation time the method significantly reduces the workload imbalance in the yard, avoiding possible bottlenecks in terminal operations.

Letter to the Editor—An Algorithm for Ranking all the Assignments in Order of Increasing Cost

The Hungarian method gives an efficient algorithm for finding the minimal cost assignment. However, in some cases it may be useful to determine the second minimal assignment (i.e., the best assignment after excluding the minimal cost assignment) and in general the kth minimal assignment for k = 1, 2, …. These things can easily be determined if all the assignments can be arranged as a sequence in increasing order of cost. This paper describes an efficient algorithm for such a ranking of all the assignments. The maximum computational effort required to generate an additional assignment in the sequence is that of solving at most (n − 1) different assignment problems, one each of sizes 2, 3, …, n.

A decision support system for operations in a container terminal

We describe a variety of inter-related decisions made during daily operations at a container terminal. The ultimate goal of these decisions is to minimize the berthing time of vessels, the resources needed for handling the workload, the waiting time of customer trucks, and the congestion on the roads and at the storage blocks and docks inside the terminal; and to make the best use of the storage space. Given the scale and complexity of these decisions, it is essential to use decision support tools to make them. This paper reports on work to develop such a decision support system (DSS). We discuss the mathematical models and algorithms used in designing the DSS, the reasons for using these approaches, and some experimental results.

SOLVING THE FIXED CHARGE PROBLEM BY RANKING THE EXTREME POINTS

An algorithm for ranking the basic feasible solutions corresponding to a linear programming problem in increasing order of the linear objective function is described. An application of this algorithm for obtaining the minimal cost solution to a fixed charge problem is given. This algorithm can be applied in general to solve any fixed charge problem. However, the algorithm works efficiently when the problem is nondegenerate and the range in the values of the variable costs is large compared to the fixed charges. This algorithm can also be applied when the fixed charge part of the cost function is replaced by a concave function.

On the Number of Solutions to the Complementarity Problem and Spanning Properties of Complementary Cones

Abstract The relationship between the number of solutions to the complementarity problem, w = Mz + q , w⩾0, z⩾0, w T z=0 , the right-hand constant vector q and the matrix M are explored. The main results proved in this work are summarized below. The number of solutions to the complementarity problem is finite for all q ϵ R n if and only if all the principal subdeterminants of M are nonzero. The necessary and sufficient condition for this solution to be unique for each q ϵ R n is that all principal subdeterminants of M are strictly positive. When M ⩾0, there is at least one complementary feasible solution for each q ϵ R n if and only if all the diagonal elements of M are strictly positive; and, in this case, the number of these solutions is an odd number whenever q is nondegenerate. If all principal subdeterminants of M are nonzero, then the number of complementary feasible solutions has the same parity (odd or even) for all q ϵ R n which are nondegenerate. Also, if the number of complementary feasible solutions is a constant for each q ϵ R n , then that constant is equal to one and M is a P -matrix. In the cartesian system of coordinates for R n , an orthant is a convex cone generated by a set of n -column vectors in R n , {A. 1 ,…,A. n }, where for each j = 1 to n , A . j is either the j th column vector of the unit matrix of order n (denoted by I. j ) or its negative - I. j . There are thus 2 n orthants in R n , and they partition the whole space. It is interesting to know what properties these orthants possess if we obtain them after replacing - I. j by some given column vector - M. j for j = 1 to n . Orthants obtained in this manner are called complementary cones , and their spanning properties are studied.

Computational complexity of parametric linear programming

We establish that in the worst case, the computational effort required for solving a parametric linear program is not bounded above by a polynomial in the size of the problem.

Effect of block length and yard crane deployment systems on overall performance at a seaport container transshipment terminal

As more and more container terminals open up all over the world, terminal operators are discovering that they must increase quay crane work rates in order to remain competitive. In this paper we present a simulation study that shows how a terminals long-run average quay crane rate depends on (1) the length of the storage blocks in the terminals container yard and (2) the system that deploys yard cranes among blocks in the same zone. Several different block lengths and yard crane deployment systems are evaluated by a fully dynamic, discrete event simulation model that considers the detailed movement of individual containers passing through a vessel-to-vessel transshipment terminal over a several week period. Experiments consider four container terminal scenarios that are designed to reproduce the multi-objective, stochastic, real-time environment at a multiple-berth facility. Results indicate that a block length between 56 and 72 (20-ft) slots yields the highest quay crane work rate, and that a yard crane deployment system that restricts crane movement yields a higher quay crane work rate than a system that allows greater yard crane mobility. Interestingly, a block length of 56-72 slots is somewhat longer than the average block in use today. The experiments provide the first direct connection in the literature between block length and long-run performance at a seaport container terminal. The simulator can be suitably customized to real, pure-transshipment ports and adequately tuned to get an appreciable prescriptive power.

Rubber tired gantry crane deployment for container yard operation

Container terminals competitiveness is generally measured by the vessel discharging and loading time. The shore crane operation has attracted a number of research works. However, yard operation management has been very much experience based and did not receive much attention until last decade. This paper presents an algorithm and a mathematical model for the optimal yard crane deployment. The potential of the model in optimizing yard crane deployment was tested with a set of real operation data extracted from a major container yard terminal.