Optimization of the Document Placement in the RFID Cabinet

The study is devoted to the issue of optimization of the document placement in a single RFID cabinet. It has been assumed that the optimization problem means the reduction of archivization time with respect to the information on all documents with RFID tags. Since the explicit form of the criterion function remains unknown, for the purpose of its approximation, the regression analysis method has been used. The method uses data from a computer simulation of the process of archiving data about documents. To solve the optimization problem, the modified gradient projection method has been used.


Introduction
One of the important problems related to the use of the document archivization system with the RFID (Radiofrequency identification) technology [1,2], is the issue of such placement of the set of documents on particular shelves of the RFID cabinet that minimize the length of the time interval of registering the data about these documents in a database of an information system. To solve this problem, the standard optimization approach may be applied [3,4]. One of the issues, which need to be solved by using such approach, is the problem of collecting data necessary for approximation of the criterion function. Since the process of data collection is time-consuming and laborious, to improve it, the stochastic simulation method has been applied [5][6][7]. It is based on generating pseudorandom real numbers, whose values are interpreted as the length of time intervals of archiving the information about the RFID-tagged documents. Thanks to such features as the ability to quickly generate a wide variety of data and flexibility in planning and performing various experiments, computer simulation has become in recent years a kind of modus operandi of many scientific specialties [8]. On the basis of the simulation data, local approximations of the criterion function are performed using the regression analysis method [9][10][11]. Thanks to applying the complete two-level plan of experiments [12,13], the computational complexity of the estimation process of the local linear parameters of the regression functions was reduced. Section 2 formulates the optimization problem. Section 3 describes the method for solving the problem. In particular, section 3.1 presents the method for estimation the parameters of the local linear regression functions, whereas section 3.2 describes an algorithm for solving the optimization problem reflecting the modified gradient projection method [14,15].

Formulation of the optimization problem
where:  , ,g 0

Method for solving the optimization problem
Problem (1) is a nonlinear problem of the mathematical programming with linear constraints. If the mapping f : S o \ \ is differentiable and set 0 0 is closed and convex, the problem may be solved by the modified gradient projection method. This method consists of generating a sequence of approximations: where: 0 0 x 0 is a predefined initial point, 0 k W ! is the kth step size, the value of which can be determined by the Armijo rule, k S d \ is the direction, in which the minimum of the criterion function f is searched. Since the form of this function is unknown, it is necessary to conduct a simulation experiment aimed at collecting statistical data, based on which it will be possible to perform approximation of the function f in the neighborhood ^, The approximation may be performed on the basis of the regression analysis method.

Local linear approximation of the criterion function
Let us assume that the probabilistic model for the process of changing the number of documents on the shelf s is a  [16] consists in the planning of experiments in such a manner that the mean-square error of the model with the vector parameter ȕ is minimized.
Let us assume that in the neighborhood where: 1, For that purpose, it is essential to come up with a plan of the simulation experiments allowing to collect the statistical data, based on which the estimation of this vector will be possible. Let where: n y is the observed length of time intervals of archiving the information about the RFID-tagged documents from set located on the shelves of the RFID cabinet in a manner determined by the value of vector We are looking for the linear regression model of mappingf k (6) in the form of where > @ 0 1 , , , Let us rewrite equations (6) in the following matrix form: 1 2 , , , On the other hand, the equations (7) Problem (8) is a linear squares issue [9][10][11]. Let us write the minimized sum of squared errors in the form of q 2 .   T  k  k  k  T  T  T T  T T  k  k  k  k  T  T T  T T  k  k  k            b  y Zb  y Zb   y

y y Zb b Z y b Z Zb y y b Z y b Z Zb
When we differentiate function q with respect to the vector k b , we obtain: According to the above formula, we obtain the system of the normal equation: If the matrix T Z Z is not singular, then the solution of problem (8) is the estimator in the form of To facilitate the estimation of parameter vector ˆk b (9), the complete two-level plan of experiments are often used [13]. According to this method, it is assumed that the number of focal points N of plan k P (5) is 2 S N , and the components of the plan of experiment assume the values from the set {-1,1}. Let 1 ij N S u u ª º ¬ ¼ U be the plan of experiment, whose components ij u , i=1,…,N, j=1,…,S+1, assume the values from this set. Figure 1 presents matrixU for S=3.