This paper is intended to present a simple and quick method for obtaining approximate solutions to large scale zero-one programming problems. The method does not use enumeration. Instead, it assigns measures of preferability to zero-one variables that change the values of the variables from zero to one. The method yields very good approximate solutions to zero-one programming problems in dramatically short computation time. Even for problems involving more than a thousand zero-one variables the computation time is of little concern. The method is applicable not only to those problems associated with obtaining the optimal package of variables with the value one but also to a great variety of binary choice ("Yes-No") problems.

There are many decision making problems in which one seeks to choose the optimal package from a large number of indivisible independent proposals. For instance, jobbing firms have often to choose the most profitable package of orders from hundreds of potential ones under a great many restrictions on available resources, such as working time of different facilities, number of specialists, materials, etc. This article is intended to present a simple approach to obtaining approximate solutions for such problems. The fundamental concept is to make some ordinal scales among proposals. Steps of calculation are illustrated by examples of choosing the optimal mix of orders, one of which involves 60 candidate proposals with 30 restricting conditions. This method may be of great help when (1) the number of candidates and restricting conditions are large; (2) the estimated or raw data on required resources for proposals and their incremental profits contain some errors; and (3) the distribution of incremental profits and required resources of candidates differs greatly, say, week by week, and the limits on resources can be extended in a practical manner by carrying over an inventory of profitable backlog orders, reducing, in effect, the remaining capacity in the future week.

We study the covariant entropy bound in the context of gravitational collapse. First, we discuss critically the heuristic arguments advanced by Bousso. Then we solve the problem through an exact model: a Tolman-Bondi dust shell collapsing into a Schwarzschild black hole. After the collapse, a new black hole with a larger mass is formed. The horizon, L, of the old black hole then terminates at the singularity. We show that the entropy crossing L does not exceed a quarter of the area of the old horizon. Therefore, the covariant entropy bound is satisfied in this process.

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