We analyze the approximation behavior of some of the best-known polynomial-time approximation algorithms for bin-packing under an approximation criterion, called differential ratio, informally the ratio (n − λ(I))/(n − β(I)), where n is the size of the input list, λ(I) is the size of the solution provided by an approximation algorithm and β(I) is the size of the optimal one. This measure has or...

We consider a one dimensional storage system where each container can store a bounded amount of capacity as well as a bounded number of items k ≥ 2. This defines the (standard) bin packing problem with cardinality constraints which is an important version of bin packing, introduced by Krause, Shen and Schwetman already in 1975. Following previous work on the unbounded space online problem, we e...

In this paper we present a dual approximation scheme for the class constrained shelf bin packing problem. In this problem, we are given bins of capacity 1, and n items of Q different classes, each item e with class ce and size se. The problem is to pack the items into bins, such that two items of different classes packed in a same bin must be in different shelves. Items in a same shelf are pack...

The determination of the required sample size is always an important step during planning quality research for a quantitative study. researchers need to have sound prerequisite knowledge inferential statistics concept determining minimum requirement and its estimation because insufficient may not be able answer questions large wastage resources, add complexity, take more time Both situations ar...

Motivated by the problem of packing Virtual Machines on physical servers in the cloud, we study the problem of one-dimensional online stochastic bin packing. Items with sizes sampled independent and identically (i.i.d.) from a distribution with integral support arrive as a stream and must be packed on arrival in bins of size B, also an integer. The size of an item is known when it arrives and t...