Application of the Combinatorial Sequencing Theory for Decreasing of the Logistic Costs

APPLICATION OF THE COMBINATORIAL SEQUENCING THEORY FOR DECREASING OF THE LOGISTIC COSTS IN PRODUCTION NETWORKS

Oresta Bandyrska1, Marta Talan2, Volodymyr Riznyk1,3

1Institution of Computer Engineering

Lviv National Polytechnic University

12 S.Bandera Str., 79013 Lviv

UKRAINE

2Department of Economical Analysis and Logistics

European College of Polish and Ukrainian Universities

5 Maria Curie-Skladowskiej Str., 20-029 Lublin

POLAND

3Department of Telecommunications and Electrical Engineeing

University of Technology and Agriculture

7 Kaliskiego, 85-796 Bydgoszcz

POLAND

Abstract: - This paper is concerned with the reduction of the logistics costs in production networks based on Combinatorial Sequencing Theory, namely the concept of Ideal Numerical Rings (INR), which can be used for configure hihg performance manufacturing based on logistics. Research into the underlying mathematical principles relating to the favourable distribution of structural elements (parameters) in production networks (e.g. production flows) using conventional logistic chains.

Key-Words: - Combinatorial Sequencing Theory, Ideal Numerical Ring, Favourable distribution, Production network, Manufacturing goods, Logistic cost.

1 Introduction

Combionatorial technique, is known, to be of basic for finding optimal solutions for wide classes of technological problems in business and economics. An important part for this relates to research of combinatorial models such as one- and multidimensional Ideal Numerical Rings [1] based on the appropriate algebraic constructions. These research results can be used for improving cooperation among the enterprises by conventional logistic ring sequences based on combinatorial technique. Such approach make it possible to decrease logistic costs because the enterprises need very little time for appropriate management and the dynamical reconfiguration.

2Existing Theories and Work

The "ordered chain" approach to the study of elements and events is well known to be of widespread applicability, and has been extremely effective when applied to the problem of finding the optimum ordered arrangement of structural elements in a distributed technological system and process engineering [1,2]. So, quality of manufacturing goods in enterprise is known to be of very dependents from positioning of elements and bonds in production network. The theoretical problem of optimal distributed production networks, is known, to be of very important for finding the smallest possible number of consecutive elements and bonds of the system while maintaining or improving on smart, dynamic organization and management. This problem is like to the optimal placement of structural elements (manufactured goods and distributed processing) in in order to meet the requirement like “the desired product at the right time, and at the right place”. The approach provides finding optimal solution underlying problems, based on the development of combinatorial principle and appropriate mathematical models.

3Problem Solution

We say the method of optimal structural proportions based on idea of “Ideal Numerical Ring” [1].

For example, 3-stage (n=3) chain sequence {1,3,2} is the Ideal Numerical Ring, because the sequence allows an enumeration of all numbers from 1 to Sn = 6 exactly once: 1=1, 2=2, 3=3, 4=1+3, 5=3+2, 6=1+3+2. These combinatorial constructions are named “Ideal Golomb Rulers”.

Let us calculate all sums of the terms in the numerical n-stage chain sequence of distinct positive integers Cn= k1,k2,…,kn, where we require all terms in each sum to be consecutive elements of the sequence.

Clearly the maximum number of distinct sums is

Sn = 1+2+…+n = n(n + 1)/2 (1)

The Golomb rulers are well useful in applications to problems of signal design for radar, sonar, and data communications [3]. At the same time, it is known, that Ideal Golomb Rulers of n 4 not exist.

Let us consider ringlike combinatorial construction, namely the Ideal Numerical Ring (INR), based on the and perfect multidimensional combinatorial constructions may be used for advanced design too.

If we regard the chain sequence Cnas being cyclic, so that knis followed by k1, we call this a ring sequence. A sum of consecutive terms in the ring sequence can have any of the n terms as its starting point, anf can be of any length (number of terms) from 1 to n-1. In addition, there is the sum T of all n terms, which is the same independent of the starting point. Hence the maximum number of distinct sums S of consecutive terms of the ring sequence is given by

S=n(n-1)+1. (2)

Let us consider an n-stage ring sequence Cn of natural numbers for which the set of all circular sums consists of the numbers from 1 to S.

For example, to see 4-stage sequence {1,2,6,4}, as below (Fig.1),

4 2

Fig.1. 1D-INR {1,2,6,4}

we observe:

1=1, 2=2, 3=1+2, 4=4,

5=4+1, 6=6, 7=4+1+2, 8=2+6,

9=1+2+6, 10=6+4, 11=6+4+1, 12=2+6+4, 13=1+2+6+4.

Easy to see, that each number in the ring sequence occurs exactly once. So, our construction is called an "One-dimensional Ideal Numerical Ring " (1D-INR).

Comparing the equations (1) and (2), we see that the number of sums Sfor consecutive terms in the ring topology is nearly double the number of sums S in the daisy-chaintopology, for the same sequence Cn of n terms.

An n-stage sequence Cn={k1,k2,...,kn} of natural numbers for which the set of all S circular sums consists of the numbers from 1 to S =n(n-1)+1=n2 - n +1 (each number occuring exactly once) is called an "Ideal Numerical Ring" (INR).

Next we regard two-dimensional ring construction {(1,1),(1,2),(1,4),(1,3)} with n=4 and S={(4),(5)}, where 4 is modulo for the first component, and 5 is modulo for the second its component (Fig.2).

(1,1)

(1,3) (1,2)

(1,4)

Fig.2. 2D-INR {(1,1),(1,2),(1,4),(1,3)}

Here we obtain circular sums taking modulo 4, and modulo 5 for correspondent components as follows:

(1,1)=(1,1) (1,2)=(1,2) (1,3)=(1,3) (1,4)=(1,4)

(2,1)=(1,2)+(1,4) (2,2)=(1,4)+(1,3) (2,3)=(1,1)+(1,2) (2,4)=(1,3)+(1,1)

(3,1)=(1,3)+(1,1)+(1,2) (3,2)=(1,1)+(1,2)+(1,4) (3,3)=(1,4)+(1,3)+(1,1) (3,4)=(1,2)+(1,4)+(1,3)

The result of calculation forms 3  4 - matrix which consists of all circular 2D vector-sums on the matrix exactly once. We say that this 2D INR has parameters n=4, and L1  L2 = 3  4, where L1 and L2 - sizes of the matrix.

So, two-dimensional INR has been defined as an algebraic object which consists of both a carrying ordered set (ringlike sequence of elements) and an operation of addition by modulos, therefore the addition on the carrying set is realised by going round the elements one after another successively in this set, besides the result of the calculation forms 2-vector matrix which exhausts of all circular 2-vector-sums.

There are exist a lot of two- and multi-dimensional IRBs, moreover a sum of consecutive terms in IRB can be of any long length (number of terms).

The t-dimensional (t>2) INR of order n can be represented as n-stage ringlike sequence {(K11,K21,..,Kt1),(K12,K22,...,Kt2),..,(K1n,K2n,...,Ktn)} which give us a set of circular t-vector-sums on the sequence as M1  M2 ...Mt- matrix exactly R times [1].

A symbolic-form model of a vector data logistic management for the production network, based on the 2-D INR {(1,1),(1,2),(1,4),(1,3)} is shown in Fig.3.

Set of incoming Set ofoutput

flows: (A,B,C,D) flows: (A,B,C,D)

Fig. 3. A symbolic-form INR model of 2-D logistic management for the production network, based on the INR{(1,1),(1,2),(1,4),(1,3)}

Here is a circle with four (n=4) incoming flows (A,B,C,D) and four output flows (A,B,C,D) as being arranged non-uniformly, so that cyclic relationship between points of its placement on the circle correspondents to the INR symbolizes a concurrent enterprising program. The model provides any of cyclic operating or process as being 2-D parameters, starting from programmed (one of four) place-time point, and finishing in one of the other its point. In general case both any of incoming flows (A,B,C,D) and any of engineering output flows (A,B,C,D) can be at any manner as well as at any time with the smallest possible number of the points.

4 Conclusion

Underlying research provides the innovative approach to configure the production netwotk based on Combinatorial Sequencing Theory, namely the concept of Ideal Numerical Rings (INR)s. The approach make it possible to configure high performance production networks with fewer logistic costs and total expenses for manufacturing process due to favourable distribution of structural elements (parameters) and bonds of manufacturing processes for in-line production networks. Research results can be used for further developing the INR-idea production networks in the high perfrmance collaborative economy with respect to productivity and logistic management.

References:

[1] V.V.Riznyk (1998) Multidimensional Systems Based on Perfect Combinatorial Models, IEE, Multidimensional Systems: Problems and Solutions, No 225, London, pp.5/1-5/4.

[2] M.Riznyk, V.Riznyk (2000) Manufacturing system Based on Perfect Distribution Phenomenon, Proceedings of the SPIE, Vol.4192, Boston, pp.364-367.

[3] S.W. Golomb "Application of combinatorial Mathematics to Communication Signal Design", Proceedings of the IAM Conference on Applications of Combinatorial Mathematics, London, U.K., 1995.