Power Strip Packing of Malleable Demands in Smart Grid

We consider a problem of supplying electricity to a set of $\mathcal{N}$ customers in a smart-grid framework. Each customer requires a certain amount of electrical energy which has to be supplied during the time interval $[0,1]$. We assume that each demand has to be supplied without interruption, with possible duration between $\ell$ and $r$, which are given system parameters ($\ell\le r$). At each moment of time, the power of the grid is the sum of all the consumption rates for the demands being supplied at that moment. Our goal is to find an assignment that minimizes the {\it power peak} - maximal power over $[0,1]$ - while satisfying all the demands. To do this first we find the lower bound of optimal power peak. We show that the problem depends on whether or not the pair $\ell, r$ belongs to a "good" region $\mathcal{G}$. If it does - then an optimal assignment almost perfectly "fills" the rectangle $time \times power = [0,1] \times [0, A]$ with $A$ being the sum of all the energy demands - thus achieving an optimal power peak $A$. Conversely, if $\ell, r$ do not belong to $\mathcal{G}$, we identify the lower bound $\bar{A} >A$ on the optimal value of power peak and introduce a simple linear time algorithm that almost perfectly arranges all the demands in a rectangle $[0, A /\bar{A}] \times [0, \bar{A}]$ and show that it is asymptotically optimal

Hybrid next-fit algorithm for the two-dimensional rectangle bin-packing problem

We present a new approximation algorithm for the two-dimensional bin-packing problem. The algorithm is based on two one-dimensional bin-packing algorithms. Since the algorithm is of next-fit type it can also be used for those cases where the output is required to be on-line (e. g. if we open an new bin we have no possibility to pack elements into the earlier opened bins). We give a tight bound for its worst-case and show that this bound is a parameter of the maximal sizes of the items to be packed. Moreover, we also present a probabilistic analysis of this algorithm.worst-case analysis;probabilistic analysis;bin-packing;heuristic algorithm;on-line algorithm;two-dimensional packing

Heuristic methods for cost-oriented assembly line balancing: a survey

This paper is concerned with cost-oriented assembly line balancing. This problem occurs especially in the final assembly of automotives, consumer durables or personal computers, where production is still very labour- intensive, and where the wage rates depend on the requirements and qualifications to fulfil the work. First a short problem description is presented. After that a classi"cation of existent and new heuristic methods for solving this problem is given. The heuristic methods presented in this paper are described in detail. A new priority rule called ` best change of idle cost a is proposed. This priority rule di!ers from the existent priority rules because it is the only one which considers that production cost are the result of both, production time and cost rates. Furthermore a new sophisticated method called ` exact solution of sliding problem windows a is presented. The solution process is illustrated by an example, showing how this metaheuristic works together with an exact method.Assembly line balancing; Cost-oriented production planning; Heuristic methods