Application of the theory of mass service in logistic management of services 2

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1 Mchał Tymńsk Jan ochanowsk Unversty n elce Jerzy Tymńsk Applcaton of the theory of mass servce n logstc management of servces Customer tme servce effectveness The customer evaluates the effectveness of the receved servce through the qualty but also, or even more mportantly, through the tme they have to spend n a queue watng to be served. Every mnute spent queung s the tme lost for the customer. Queues are a part of our realty. They are present vrtually everywhere: n cultural lfe (cnema, theatre etc.) n means of publc transport, and also n other spheres of our everyday lfe n trade or caterng, where the problem of queues s partcularly mportant as every customer would lke to eat quckly and well. Ths aspect of effectve actvtes of servce provders s mportant not only for customers, for whom tme s money but also for organzatons provdng the servce, for whom tme s proft. Accordng to R.C. Larson 3 mllons Amercans spend half an hour daly queung whch makes approx. 37 bllon hours of the tme lost queung every year. The tme whch could have been spent on effectve relaxaton s lost and thus t decreases the level of welfare n the socety. Optmum control of queung does not only mean shortenng the tme of awatng the servce (customer satsfacton) but also ncrease n effectveness of company management, and consequently an ncrease n company proftablty. Decson-makng for the purpose of logstc management s a complex process 4. rofessonal and effectve managng the process requres not only nterdscplnary knowledge but also expertse n methodology. Ths s connected wth masterng a wde range of numercal, statstcal, econometrc and smulaton methods and technques n a comprehensve approach but also other specalst methods used n the theory of decson- makng n the area of rsk and uncertanty. The theory of mass servce s one of them. Fundamentals of the theory of mass servce (Queung Theory) The theory of mass servce was devsed by Erlang at the begnnng of the prevous century and the ratonale behnd the theory was the need to modfy telecommuncatons systems 5. The essence of the theory of mass servce (also known as the Queung Theory) can be reduced to the followng basc assumptons: the system of mass servce can be explctly satsfed f a set of objects (or subjects) and a set of servce devces are known n arbtrarly defned tme nterval the number of applcatons s uncorrelated (ndependent) wth the number of applcatons n another tme nterval possblty of occurrence of more than one applcaton at the same moment of tme s excluded, whch means that the applcatons are ndependent of one another the probablty of occurrence of a sutable number of applcatons n a defnte tme nterval depends only on the length of the nterval, and t does not depend on ether the begnnng or the end of the nterval. From the above assumptons t follows that f applcatons arrve at the moments of tme dffcult to foresee then the way of arrval (measurement) can be descrbed by sutable dstrbutons of probablty. It s assumed that f the tme whch elapses between two successve applcatons s longer than t, then the dstrbuton of probablty of such events occurrng can be descrbed wth the use of exponental functon of the form: at ( t) e. In other words ths s the osson dstrbuton whose characterstc s that probablty of event has a property that n the t tme the n applcatons are observed. Ths can be expressed by the followng formula: ( at) n ( t) n! n e at () Dr M. Tymńsk, Jan ochanowsk Unversty n elce, otrków Trybunalsk Campus, Faculty of Socal Scences, Department of Natonal Securty and hlologcal-hstorcal Faculty, Internatonal Relatons Insttut. Revewed paper.. Mude, A. Cottam, Usług. Zarządzane marketng. Warszawa 998, p. 8. D. sperska-moroń, odstawy decyzj logstycznych w przedsęborstwe, Wyd. Akadem Ekonomcznej, atowce Wł. Radzkowsk, Badana operacyjne w zarządzanu przedsęborstwem, Toruńska Szkoła Zarządzana, Toruń 997, p Logstyka 6/4 3878

2 The tme of servce s an nseparable component of the osson process. It s also frequently assumed that t s a random value. Thus, the assumptons we make for the tme factor are smlar to the ones we make dstrbuton of applcatons whch are characterstc for the osson dstrbuton 6. We assume that the process of servce s characterzed by the exponental dstrbuton of the servce tme and the osson dstrbuton of the number of applcatons n the unt of tme. Ths s a typcal osson process of servce whch s characterzed by the two basc stages 7. () contnuous regstraton of frequency of applcatons and tme of servce, () estmaton of parameters: λ (the average number of applcatons n one unt of tme) and μ (the average number of probable servces n a gven tme unt); that s such a selecton of parameters λ and μ, so as to receve possbly the most probable model of the nvestgated phenomenon whch would enable to assume the hypothess on the exponental dstrbuton of breaks between successve applcatons, and also on the exponental dstrbuton of servce tme. The assumptons ncluded n the second stage can be expressed n a formal way as: λ rate of customers arrval (applcatons). It s defned as the nverse of the average tme whch separates two successve applcatons (two successve customers): λ t where: t average tme between successve applcatons or customers μ rate of servce understood as the average number of applcatons or customers served n one unt of tme. In the analyss of stablty of the servce system we use the ntensty parameter ρ whch expresses the degree of use of the channel of servce. Ths can be expressed mathematcally by the followng equaton: λ ρ µ The ntroduced parameters are used to conduct a fnal analyss of the stablty of the servce system: λ < μ the system s n the state of balance (under the assumpton that both rates are stable); λ μ the system s unstable, and lengthenng of queue s certan. Examples of applcatons of elements of the theory of mass servce n management of logstc processes Food ndustry company called Alfa has a mantenance- repar base whch specalzes n reparng means of transport used for food transportaton. The management of the base ams to streamlne the organzaton of the servce (repar) base. Therefore, t conducts systematc analyses of effcency of organzaton of customer servce.e. logstc unts. From the analyss t follows that: the average occupaton tme of repar pont by a transport unt s equal to 67 hours the repar base s open hours a day durng the season the base has only one servce pont the dstrbuton of applcatons for servce s presented as follows: Tab.. Emprcal dstrbuton of stream of applcatons (customer flow). Number of customer flow daly Number of nvestgated cases > Σ Source: Author s own elaboraton. 6 W. Sadowsk: Teora podejmowana decyzj, Warszawa 973, p Ibdem, p , Logstyka 6/4 3879

3 The example can be presented n a smplfed form usng the followng graphc representaton (Fg. ). λ... Servce pont (µ*) applcatons served queue k of applcatons Fg.. Dagram of the system of servce n a logstc system approach roblem soluton. Evaluaton of flow of applcatons. Average monthly number of applcatons + 6 4, ,5 + 3, ,5 + 3,5 λ,65 applcatons If we assume that the dstrbuton of applcatons s of the type of the osson dstrbuton then we can also assume that the average of the dstrbuton has the value of applcatons monthly. In ths case the probablty of monthly applcatons takes the form: (! λt) λt e and! e. (),,, 3,8 4,9 5,38 6,63 7,9 8,3 9,5,5,4,95 3,73 4,5 5,35 6, 7,3 8,7 9,4,,, The calculated probablty s grouped by classes correspondng to the dstrbuton of the flow of customers. Ths wll enable to compare the number of theoretcal cases wth the number of cases whch occurred n the real lfe... Tab.. Number of theoretcal and real cases Monthly number of customers flowng to the servce blty n cumulated classes n partcular classes n partcular classes Calculated theoretcal proba- Number of theoretcal cases Number of real cases system () of customer flow ( ) ( N ) ( N ) ,,8,453,334,76,7,,8 45,3 33,4 7,6, Total,, Source: Author s own elaboraton. In order to verfy the hypothess on compatblty of the observed frequences (flow of customers) wth the osson dstrbuton we wll use the χ test. To smplfy further calculatons we wll cumulate ntervals nto three classes of numercal values. 388 Logstyka 6/4

4 Tab. 3. Number of theoretcal and real cases n the dstngushed classes Numercal ntervals (classes) () Calculated theoretcal probablty n cumulated classes ( ) Number of theoretcal cases n partcular classes ( N ) Number of real cases n partcular classes ( N ) ,3,787,83 3, 78,7 8,3 8 8 Total, Source: Author s own elaboraton. ' Table 3 contans theoretcal numbers ( N ) and the observed (emprcal) values of the flow of customers. It facltates conductng the procedure of determnng values of measure When we take from the table values χ. χ at the sgnfcance level equal to,5 and V (number of degrees of freedom) we get the value, and 5,99. The calculated value χ,49 s wthn the range of percentage values from the table. There s no ground to reject the hypothess that the dstrbuton of the flow of applcatons has the osson dstrbuton. That s why we accept as true the hypothess that the frequency of customer flow (frequency of applcatons) has got the osson dstrbuton wth a parameter λ. Analyss of the servce process At the observed value of the average tme of servce of transport unts whch amounts to hours / tem, the monthly 6 output of one pont of servce equals t: μ 9 transport unts. 67 Tab. 4. Determnng crtcal value χ. Numercal ntervals (classes) () Total Calculated numercal values of cases theoretcal 3, 78,7 8,3 ' N real N 8 8 Dfference ' N N ) ( 5-3,3 -,7 Input values for the test Dfference squared ' N N ) ( 5,89,89 χ Quotent of dfference ' squared ( N N ) / N,93,38,348 χ, x x,49 Source: Author s own elaboraton. From the table t follows that the analyzed process s characterzed by the osson flow of applcatons (parameter value λ ). We assume that the servce tme has got the exponental dstrbuton ( μ 9 ). Thus the ntensty of movement (flow) takes the value: λ ρ, µ 9 The dstrbuton of the servce tme was examned emprcally so t was assumed accordng to the model descrpton that t s the exponental dstrbuton. Comparng the value of both parameters we conclude that: λ > μ or the rate of arrvals s hgher than the rate of servce ρ > from whch t follows that the nvestgated system s unstable.e. at t the length of queue s growng to nfnty. The state of balance can only be reached by changng decson makng condtons: shortenng the average servce tme by ncrease n productvty establshng the second pont of servce Let us consder the frst decson- makng stuaton n the system of logstc servce. (3) Logstyka 6/4 388

5 The mprovement n the organzaton (purchase of better equpment, more effectve use of the workng tme etc.) led to 6 shortenng the average servce tme from: 67 to 48 hours /unt (then: μ, 5 ). Thus: ρ, 8, whch allows 48, 5 us to state that n the mantenance repar base the stuaton has mproved, as λ < μ and ρ <. The system s relatvely stable whle the probablty of long queues s decreasng. The followng characterstcs of the servce process can be observed: The probablty that there are k transport unts n the system wll be equal to:,5 ( ρ) ( ) ( ) ( ),5,5,5,5 ρ (4) e.g. for the monthly number of applcatons (from the - nterval) the probablty wll amount to: (,8) (,8),8 whle for k,,. Ths means that the system of servce does not have even one unt of equpment to serve customers so t s completely useless. It also means that on average the system (channel of servce) provdes servce to:,5 (,) unts of equpment durng a month. The expected (average) number of applcatons n the whole system of the logstc servce (.e. n a queue and n the pont of servce) wll amount to: E ρ /( ρ),8/, 4 transport unts (5) Whereas the average number of applcatons awatng servce (.e. the expected length of the queue) s equal to: E ρ /( ρ),8 /, 3, transport unts (6) The expected tme of stay n the whole system of servce amounts to: E 4,4 µ λ,5 of day.e. 4 hours (ce.,4 * hours) (7) robablty that there s at least one applcaton n the queue (.e. the pont of servce s occuped) s equal to:,8 robablty that no fewer than of applcatons (equpment to be served) wll be watng n the queue wll amount to (e.g. 3 applcatons): + 3+ ( > ) ρ (,8),496 (9) whle for 6 applcatons t wll be equal to: 6+ etc. () (6 + > 6) (,8), From the above consderatons t follows that the fears that the queue may ncrease endlessly (ad nfntum) are groundless. The probablty of the event when the customer of Alfa company wll be watng n the queue longer than t t unts of tme s equal to (e.g.. for t hour) ( t > t (8) t ( µ λ ) (,5 ),5 ) ρ e,8e,8e,66 () Therefore, we can say that the probablty that the customer of the company loses more than 3,96 mn (,66 * 6) watng n a queue s very small. Abstract The am of the paper s to present the applcaton of basc elements of the theory of mass servce n the analyss of effcency of organzaton of logstc customer servce. The applcaton of the theory s presented usng an example of the company actve n the Lodz area of food ndustry. The obtaned characterstcs of the analyzed process of logstc servce wth the use of elements of the theory of mass servce show that the optmal organzaton of the mantenance repar base wll satsfy requrements of effcent organzaton when some mprovements are made. The mprovements whch guarantee ncrease n productvty of the pont of servce wll reduce the assumed tme of servce from 67 mn. to 48 mn.. Ths wll create a smooth, qute strongly synchronzed system 388 Logstyka 6/4

6 of servce: λ μ, whch ratonalzes the way of organzaton of the mantenance repar base and whch ncreases ts capacty whle smultaneously savng the tme of customers watng n a queue. Zastosowane teor masowej obsług w logstycznym zarządzanu usługam Streszczene Celem artykułu jest zaprezentowane zastosowana podstawowych elementów teor masowej obsług w organzacj usług logstycznych. Zastosowane teor zaprezentowano na przykładze przedsęborstwa prowadzącego dzałalność w województwe łódzkm w przemyśle spożywczym. Otrzymane charakterystyk analzowanego procesu logstycznej obsług klenta przy wykorzystanu elementów teor masowej obsług wskazują, że optymalna organzacja bazy konserwacyjno-naprawczej będze spełnała wymog racjonalnej organzacj, gdy dokonane zostaną usprawnena gwarantujące wzrost wydajnośc stanowska obsługowego powodujący obnżene przejętego czasu obsług z 67 do 48 mnut. Stworzy sę tym samym płynny, zsynchronzowany układ systemu obsług λ μ, racjonalzujący organzację dzałana bazy obsługowej, zwększający jej przepustowość, a równocześne oszczędzający czas oczekwana klentów w kolejce. BIBLIOGRAHY. Byśko J., Cygan Z., Dzadykewcz L. (985), Sterowane, zarządzane eksploatacja systemów techncznych, WN, Warszawa.. aplńsk O. (97), erowane procesam transportowo-montażowym w budownctwe, roblemy Organzacj nr sperska-moroń D. (), odstawy decyzj logstycznych w przedsęborstwe, Wyd. Akadem Ekonomcznej, atowce. 4. Mazur T., Małek A. (979), Zarządzane eksploatacją systemów techncznych, WNT. Warszawa. 5. Mszczyńska D., Mszczyńsk M. (), Wybrane metody badań operacyjnych, Wyd. WSE-H w Skernewcach, Skernewce. 6. Mude., Cottam A. (998), Usług. Zarządzane marketng, WN, Warszawa. 7. Okręglck W., Łopuszyńsk B. (98), Użytkowane urządzeń mechancznych, WNT, Warszawa. 8. Ostasewcz S., Rusnak Z., Sedlecka U. (99), Statystyka. Elementy teor zadana, Wyd. Akadem Ekonomcznej, Wrocław. 9. Radzkowsk W. (997), Badana operacyjne w zarządzanu przedsęborstwem, Toruńska Szkoła Zarządzana, Toruń.. Sadowsk W. (969), Teora podejmowana decyzj, WE, Warszawa.. Tymńsk J. (), Elementy teor nezawodnośc budynków meszkalnych, Zeszyty Naukowe WSG Zeszyt I, utno.. Tymńsk J., Tymńsk M. (), rzebeg procesów utraty właścwośc użytkowych obektu meszkalnego. Momenty rozpoczęca remontu, Zeszyty Naukowe WSG tom II, utno. Logstyka 6/4 3883