THE USE OF CLUSTER ANALYSIS AND THE THEORY OF MATHEMATICAL RECORDS IN THE PROCESS OF PLANNING THE PRODUCTION-WAREHOUSE FLOW

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THE USE OF CLUSTER ANALYSIS AND THE THEORY OF MATHEMATICAL RECORDS IN THE PROCESS OF PLANNING THE PRODUCTION-WAREHOUSE FLOW Anrzej Bujak*, Katarzyna Topolska** an Mariusz Topolski*** WSB University

1 THE USE OF CLUSTER ANALYSIS AND THE THEORY OF MATHEMATICAL RECORDS IN THE PROCESS OF PLANNING THE PRODUCTION-WAREHOUSE FLOW Anrzej Bujak*, Katarzyna Topolska** an Mariusz Topolski*** WSB University in Wrocław, Fabryczna 29/31, Wrocław, Polan * ** *** Abstract The paper proposes a new approach to the agglomeration of ata in cluster analysis. These approach faces the problem of ata classification where uner the same conitions ifferent conclusions are raw. Such problems occur in many areas of aily life: meicine, technology, an economic an social sciences an economics. The new approach assumes that events like the harvest are attribute to the cumulative probability of their occurrence at the same time. Such approaches will not be foun in probability. Thanks to the mathematical theory of recors fairly accurate classification of the object can be provie. The paper presents a metho of agglomeration of measurement ata using the mathematical theory of recors. This is the metho which can be use in the cluster analysis by agglomeration. Paper type: Research Paper Publishe online: 16 July 2016 Vol. 6, No. 3, pp DOI: /j ISSN (Print) ISSN (Online) 2016 Poznan University of Technology. All rights reserve. Keywors: cluster analysis, warehousing, logistics flows

2 260 A. Bujak, K. Topolska an M. Topolski 1. INTRODUCTION A contemporary economy is base on knowlege. Each ay new algorithms are create as well as methos aiming to support the solution of ecision problems. Due to high complexity of the area of classifie objects, the use is computer is necessary to make necessary calculations. One of the biggest problems which researchers face is the elaboration of methos which will precisely classify objects uner uncertain conitions an in view of incomplete information. However, it is limite to estimation of probability of one incient an not a group of incients happening at one time. Often in everyay life we eal with situations when various ecisions have to be mae on the basis of premises. This problem may be ealt with by the theory of mathematical recors. Whereas the problem of classifying similar objects may be solve by the cluster analysis. The secon point of this paper is evote to this analysis. The research aims to look for a computer algorithm assigning an object to a certain class uner uncertain conitions an uncertainty of measure information in particular with the same premises but ifferent ecisions. 2. CLUSTER ANALYSIS One of the problem contemporary researchers face is organization an grouping ata in sensible structures. The term cluster analysis was introuce by Tryon in The cluster analysis is also calle as an exploratory ata analysis aiming to assign ata to specific structures of highest similarities insie (Kurzyński, Woźniak, & Żołnierek, 2007). At the same time combining objects with other objects shoul be the smallest. The above classification is very often use for classifying animals into specific species, transmitting sources of contamination, evaluating characteristic features of amages to mechanical elements etc. Apart from the cluster analysis use for evaluating the number of factors affecting the measurement of a specific feature, methos of factor analysis or multiple regression are use. These methos are preominantly use in the fiel of technical sciences. We apply cluster analysis when we want to istinguish groups of similar objects if the objects are escribe by more than one feature. Cluster analysis is a metho istinguishing clusters of similar objects when new objects are escribe by more than one feature. The application of cluster analysis is very comprehensive. It is use by search engines for creating thematic groups, in psychology an meicine for aggregating psychological an somatic symptoms. So clustering or a group of a class shall be use to escribe a set of objects where a similarity between a pair of objects is greater than a similarity of any object belonging to a class an an object not belonging to it (Gatnar & Walesiak, 2004). There are numerous methos escribing istances between clusters i.e. the nearest

3 The use of cluster analysis an the theory of mathematical recors 261 neighbour, the furthest neighbourhoo, a meian, a group mean, a centre of gravity or War (Grabiński, 1988). In cluster analysis methos of clustering are very important an are ivie into two categories: hierarchical an non-hierarchical. The hierarchical metho groups objects in a iterative way into smaller an smaller clusters. In non-hierarchical moels you transfer objects between clusters looking for the best solution proceeing accoring to the establishe criteria. The most common cluster methos are an agglomeration metho an K-mean metho. Hierarchical moels create a hierarchy of classifications for a set of objects starting with the set in which every object creates an inepenent cluster an ening with a ivision of objects making up a cluster. In cluster analysis the following agglomeration methos may be ifferentiate: single-linkage clustering, complete-linkage clustering, average linkage clustering, weighte average clustering, centroi linkage, meians, War s methos. Distance measurements are as follows: Eucliean, city, Chebychev, power, percent isagreement, 1-r Pearson s. 3. ELEMENTS OF THE THEORY OF MATHEMATICAL RECORDS The theory of mathematical recors is also calle the theory of belief functions an Dempster Shafer. The theory allows us to create moels of uncertainty ealing with accepting numerous values of a particular attribute at one time. It allows us to treat plausibility in a subjective way. When classifying an object on the basis of features, it is classifie to a certain class, where, an is the number of classes. Certain prior knowlege about classes is represente by a prior function of probability allocation (Stanisz, 2007).

4 262 A. Bujak, K. Topolska an M. Topolski DEFINITION 1. A prior function of probability allocation, etermine on the basis of subjective jugment (e.g. a learning set) correspons to prior probability of classes an can be written own as follows (Topolski, 2008): (1) Where set is a focal element for Bel belief function meeting conition. In the Beys classifier knowlege is represente by conitional an unconitional probability. However, in reasoning by means of the theory of mathematical recors we assign sentences with Bel value being a belief egree. The probability allocation function (1) is a component of the Bel belief function accoring to the theory of Dempster-Shafer. The Bel belief function is the basis for reasoning an may be classifie as follows: DEFINITION 2. A belief function for a certain fuzzy set Bel(Y) is referre to the function which results from base istribution function of probability allocation a priori class with the function of probability allocation m(y*) of the fuzzy set Y* satisfying the epenency (Topolski, 2008): (2) (3) where Y, Y*, Y** = for Beys belief function. In efinition 2 there is formula 3 which is calle the rule of Dempster s combination. This function enables to make inepenent convictions an upates. 4. CASE STUDY OF A WAREHOUSING PROCESS THE MODEL OF CLUSTER ANALYSIS MERGER AND THE THEORY OF MATHEMATICAL RECORDS In warehousing we have to eal with four main processes such as receiving, storing, picking an releasing. When accepting, materials or goos orere by an

5 The use of cluster analysis an the theory of mathematical recors 263 enterprise shoul be checke for the quantity an quality before recoring them into stocks. By the acceptance we unerstan activities such as: accepting the shipment from a carrier or a supplier (unpacking, counting, weighing or measuring an comparing the quantity with shipping ocuments from the supplier), checking if the shipment is not amage or if it oes not have any efects cause uring transportation or by the supplier, checking the quality of the shipment (Duziński, 2011). Goos an materials accepte to a warehouse must come from external eliveries or internal supplies of an enterprise. External eliveries inclue goos an materials purchase in a ifferent enterprise, returne as unaccepte by a recipient or from external processing. Internal supplies of an enterprise comprise materials not use by prouction epartments, returne, transferre between warehouses, eposits accepte for keeping or finishe goos receive by warehouses from prouction epartments (Duziński, 2011). Regarless of the way of accepting materials an goos coming into a warehouse, the warehouse shoul be notifie about kins of shipments being elivere to it. It enables people to organize a place in a warehouse necessary for receiving the cargo. The following epartments cooperate with the warehouse when accepting goos: procurement epartments checking compliance of the shipment with the orer, transport an loaing teams, a quality team assessing the quality of elivere materials, a team of warehouse employees unpacking an putting goos into esignate places (Duziński, 2011). The moel propose below may be applie in a task of etecting association rules for the purposes of establishing what proucts customers most often buy together. The moel may be use to evaluate the sequence of occurring events e.g. the orer of places visite by tourists. The metho may also be applie to the analysis of customers ata an to infer about possibilities of oing shopping. When classifying clusters on the basis of object features an object is classifie to a certain class (cluster) where, an is the number of classes (clusters). Certain knowlege about probabilities a priori of classes is represente by a prior probability allocation function: Using the metho of the nearest neighbour accoring to the theory of cluster analysis as a istance (between two objects x i an x k ) we are going to use the Eucliean istance expresse by the formulas: p ( xi, xk ) ik ( xij xkj j1 ) 2 (4)

6 264 A. Bujak, K. Topolska an M. Topolski where x ij the value of object x i in terms of a feature j, whereas p is the number of these features. Having a matrix n of objects an p of variables we esign a matrix of istances between particular objects: D ik, i,k=1,..n (5) Matrix D is use to calculate the total of all istances. ( x p ik j1 ij x kj ) 2 (6) The function of probability allocation for each istance is establishe on the basis of the formula: m ik ik (7) Meeting the conition m 1 an m 0. ik Recognition of similar objects is an essential element of classification. When analysing object 6, we compare its probability with 5 other objects. We might have a situation that the classifie object in its vicinity has 3 objects from cluster 1 an two from cluster 2. When the ifference between clusters is 1, the classification is flawe. Thus the istance between the two points is moifie to the set: m ik1, ik2,..., ikn ik ik1, ik2,..., The next step moifies each element of matrix using the rule: Finally the affiliation to the cluster is establishe by maximizing a belief function with the formula: m m Y Bel Y min m m min (9) Y Y 1 m m ikn (8)

7 The use of cluster analysis an the theory of mathematical recors 265 Storing material goos in a warehouse is relate to a basic function of the warehouse namely temporary storage. Storing is a set of activities precee by receiving goos to the warehouse. Material goos, usually in the form of loaable units, are put in a systematic way in the warehouse. For the purpose of the most effective use of warehousing space special machines are use (pallet-trucks, fork lifts, cranes an other equipment ajuste to loa units an ways of storing materials) (Fertsch, 2006). Methos of storing martials epen on the type of a warehousing builing, its surface an its layout as well as other factors. Goos may be for instance store in a esignate place, stacke in multiple layers epening on the specificity of a loa unit. The most frequent metho of storing materials is a high-bay warehouse (mainly close warehouses). It reuces the amount of space occupie by a warehouse an is quite comfortable as each loa unit has a location of its own. Storage conitions are essential parameters characterizing each store material. The parameters inclue a temperature, humiity, air cleanness, requirements for limite access or fire regulations. It is vital to put each material in a place in a warehouse to secure appropriate storing conitions (Fertsch, 2006). The next activity involves picking which is one of the basic stages in a warehousing process compiling store items accoring to external requirements mae in line with customers orers. Due to the fact that orer picking involves making ecisions about moving the assortment line from a supplier to a recipient, it generates the highest execution costs as it is threatene with a possibility of mistakes an faults. This situation makes us perceive picking as a critical element of entire warehousing an if optimize may improve effectiveness of the entire process which may be achieve from the process as well infrastructural perspective. The other aspect refers to a technical sie of the issue an highly automate technologies use for picking. Automation of the picking process is characteristic for the so calle ynamic variant where store assortment is move from the storing area (by means of automate trucks, cranes, carriers) to special picking stans where it is picke up by the staff an put on the orere loa unit. Depening on the storage, the assortment is elivere to stans in single packages or entire units where it is store. In the latter case a loa unit returns to the storage place after the right number of materials is taken. Because of the way goos are being picke it is calle goos-to-man picking system. Finally, goos are release which is the last warehousing stage. It involves taking the right number of goos from their storage place, checking their weight an type requirements with an issuance orer, picking an packaging an transferring them to an authorize person (a river, a carrier or a customer s representative) (Duziński, 2011). The basis for releasing goos form the warehouse is constitute by an orer issue by relevant organizational units in charge of the stock. These orers are in the form of goos issue notes (issue internally), limit cars (issue mainly by prouction companies applying the rule of limiting issue materials), goos ispatche

8 266 A. Bujak, K. Topolska an M. Topolski note (releasing goos externally e.g. stales), orers an invoices (use by commercial companies mainly instea goos ispatche notes), warehouse transfer sheets (Duziński, 2011). Most release objects are loa units which originate from picking. Others are loa units which are release as they were elivere to the warehouse or as they were store. When necessary, materials are package an forme into transport units. Then iniviual requirements are agree with the customer or relate to expecte transport conitions. Materials reay for release must be checke. In case of external orers fulfilment, the result of the check-up leas to loaing them into an external transport means. Similarly to goos receiving, an internal transport means is use an loa hanling evices. If goos are release internally then the internal transportations means is use (Fertsch, 2006). A warehouse manager shoul remember about the principle of releasing goos from the warehouse. Goos which came first shoul first go (FIFO). He shoul receive an orer for releasing goos in avance so that he ha time for preparing the orer. It shortens waiting for the goos an eliminates elays. When releasing the goos the warehouse manager shoul carefully check the number, the type an the quality of release goos. He cannot sell things or accept money (Duziński, 2011). Analysing the warehousing process in a real enterprise which operates on the basis of Total Quality Management System (TQM). TQM in an enterprise combines traitional values with effective planning an quality check in eight stages efine by PDCA techniques. At present goos are store accoring to two guielines. The first one refers to the origin of goos. Does the component come from the European Union member state, Asia or Turkey? The secon one refers to information about which factory goos got to. Each factory has its iniviual suppliers so store goos cannot be mixe up with goos from other factories an each has a esignate space. In this case a specific rack an lines. Table 1 Avantages of using cluster analysis an the theory of mathematical recors in a warehousing process; Source: Elaboration of one s own Warehousing process Receiving Picking Storing Releasing Avantages of changing guielines for storing goos Optimize process of receiving eliveries, Freeom in configuring accepte eliveries, Enhance capacities of accepte pallets, Faster orer picking (especially in case of fast rotating goos being put in front racks), Faster location of pallets in racks, Safe placement of goos in high bay warehouse, Visible istinction of goos in the release area, Faster location of particular goos in the release area.

9 The use of cluster analysis an the theory of mathematical recors 267 Thanks to the use of cluster analysis an the theory of mathematical recors in the process of planning warehouse flows, wastage may be eliminate an the flow of materials orere throughout all warehouse activities. Aitionally a perioic ABC analysis may be applie an fast rotating goos may be put in areas closest to the shipment. The table 1 presents avantages of using cluster analysis an the theory of mathematical recors for the purposes of facilitating the entire warehousing process. 5. CONCLUSIONS Very often researchers face the problem of classifying objects to specific groups. One of the metho solving this problem is cluster analysis. The paper has presente a metho of agglomerating measure ata by means of the theory of mathematical recors. This is the metho which can be applie to the cluster analysis using the agglomeration metho. Thanks to the application of the merger of two classifiers: knn algorithm (k nearest neighbours) an a belief function, a moel has been rawn up which seems to iscriminate the space of objects. The propose moel may be use for classifying goos in a warehouse or for analysing similar spaces of the ata. REFERENCES Duziński Z., (2011), Vaemecum organizacji gospoarki magazynowej, Wy. ODDK, Gańsk. Gatnar E. & M. Walesiak, (2004), Metoy statystycznej analizy wielowymiarowej w baaniach marketingowych, Wrocław, Wy. Akaemii Ekonomicznej. Grabiński T., (1988), Metoy taksonometrii, Kraków, Wy. Akaemii Ekonomicznej. Kurzyński M., Woźniak M., & Żołnierek A., (2007), Analiza przyatności wybranych meto rozpoznawania sekwencyjnego la problemu z zakresu iagnostyki meycznej, Inteligentne Wyobywanie Informacji, Technologie Informacyjne: Diagnostyka. PWNT Gańsk. Stanisz A., (2007), Przystępny kurs statystyki z zastosowaniem STATISTICA PL na przykłaach z meycyny, Tom 2. Moele liniowe i nieliniowe, Kraków. Topolski M., (2008), Komputerowe algorytmy rozpoznawania sekwencyjnego łączące teorię zbio-rów rozmytych z teorią ewiencji matematycznej, Raport PRE 1/08, Politechnika Wrocławska. Fertsch M. (e.), Postawy logistyki, Instytut Logistyki i Magazynowania, Poznań 2006.

10 268 A. Bujak, K. Topolska an M. Topolski

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