Solution Times For Problems Research Articles

A Computational Study of the Effects of Problem Dimensions on Solution Times for Transportation Problems

Abstract : The paper presents an indepth study of the influence of problem structure on the computational efficiency of the primal simplex transportation algorithm. The input for the study included over 1000 randomly generated problems with 185 different combinations of the number of sources, the number of destinations and the number of variables. Objective function coefficients were generated using three different probability distributions to study the effects of variance and skewness in these parameters. Every problem was solved using three different starting procedures, and the following data were collected for each problem: (1) Time required to obtain an optimal solution; (2) time required to obtain an initial basic solution; (3) number of artificial variables in the initial basic solution; (4) number of basis changes; (5) average time to perform a change of basis; (6) average number of basic variables in the 'stepping stone path' in each change of basis; (7) average number of variables considered before selecting one to enter the basis. (Modified author abstract)

Benefit-Cost Analysis of Coding Techniques for the Primal Transportation Algorithm

A computer code for the transportation problem that is even more efficient than the primal-dual method is developed. The code uses the well-known (primal) MODI method and is developed by a benefit-cost investigation of the possible strategies for finding an initial solution, choosing the pivot element, finding the stepping-stone tour, etc. A modified Row Minimum Start Rule, the Row Most Negative Rule for choice of pivot, and a modified form of the Predecessor Index Method for locating stepping-stone tours were found to perform best among the strategies examined. Efficient methods are devised for the relabeling that is involved in moving from one solution to another. The 1971 version of this transportation code solves both 100 × 100 assignment and transportation problems in about 1.9 seconds on the Univac 1108 Computer, which is approximately the same time as that required by the Hungarian method for 100 × 100 assignment problems. An investigation of the effect on mean solution time of the number of significant digits used for the parameters of the problem indicates that the cost parameters have a more significant effect than the rim parameters and that the solution time “saturates” as the number of significant digits is increased. The Minimum Cost Effect, i.e. the fact that total solution time asymptotically tends to the time for finding the initial solution as the problem size is increased (keeping the number of significant digits for the cost entries constant), is illustrated and explained. Detailed breakup of the solution times for both transportation and assignment problems of different sizes is provided. The paper concludes with a study of rectangular shaped problems.

Design of an accurate sampled-data simulator using field-effect transistors

The paper describes the design of an analogue simulator suitable for sampled-data systems. The basic unit, the zero-order sample and hold, uses field-effect transistors and employs methods which allow a single value of the holding capacitor to be chosen to cover a variation in sample frequencies of 1:6 × 105, consistent with a sample time ts≥ 1 μs and a hold/sample ratio not greater than 3 × 107, while maintaining an accuracy of within 0.1%. The minimum sample time of the technique described is about 30ns, which compares favourably with a gallium-arsenide-diode bridge technique of greater complexity and having an inferior holding performance. The simulator uses a parallel sequential method of delay and is capable of operating continuously over the sample frequency range 20kHz to 30s per cycle, maintaining an overall accuracy of about 0.37% without recourse to any form of switching. A method of selecting and measuring output levels from the simulator is described which, when used in conjunction with a pen recorder, could produce a problem-solution time of the same order as that produced by a digital-computer program with a graphical output routine. Practical tests are described which show clearly the extent and nature of the small errors which occur during a general simulation.

Computer for automatizing network analyzer operation

THE recent activity and interest in the development of computer methods for speeding up the solution time for power-system transient-stability problems has prompted the reporting of some work done on the Massachusetts Institute of Technology (MIT) network analyzer. The idea of coupling a computing device to a network analyzer for solving the electromechanical differential equations of the synchronous machine rotor positions following a fault or other large disturbances to the system is certainly not a new one. Many people have considered the idea and have concluded that the problem is not one of concept but of constructing and instrumenting a computing system which would fulfill all of the operational and economic requirements of the network analyzer service.

Modern concepts for digital computer input-output philosophy

The trend among users and builders of automatic digital computing machines is to rely more and more on the computer itself to assist with the compilation of a computing routine. Experience has shown that use of the so-called interpretive subroutines often reduces the overall cost of acquiring a solution to a particular problem. It is also true that, disregarding problem solution time, interpretive subroutines could be written which would permit machines with the most elementary logic (i.e. an instruction list of plus, minus, compare, in, and out) to compare favorably with machines containing an elegant command list. However, though there has been an increased use in computers assembling their own routines, the complement of instructions built into such computers has also been increasing. This is typified by the advent of built-in floating point operations, base registers, etc. Unfortunately the trend of providing more special internal circuitry does not appear to have developed with respect to input-output, wherein negligible increase in machine flexibility other than through programming has been incorporated into newly released computers.