MODELLING OF TWO STRATEGIES IN INVENTORY CONTROL SYSTEM WITH RANDOM LEAD TIME AND DEMAND

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1 art I. robobabilystic Moels Computer Moelling an New echnologies 27 Vol. No. 2-3 ransport an elecommunication Institute omonosova iga V-9 atvia MOEING OF WO AEGIE IN INVENOY CONO YEM WIH ANOM EA IME AN EMAN Eugene Kopytov eoni Greenglaz 2 Aivar Muravyov Evin uzinevich ransport an elecommunication Institute omonosova tr. iga V-9 atvia h.: fa: iga International chool of Economics an Business Aministration Meza tr. buil. 2 iga V-48 atvia he paper consiers two multiple perio single-prouct inventory control moels with ranom parameters. hese moels are of interest because they illustrate real situations of the business. he first moel is a moel with fie reorer point an fie orer quantity. he secon moel is the moel with fie perio of time between the moments of placing neighbouring orers. Orer quantity is etermine as ference between the fie stoc level an quantity of goos in the moment of orering. he consiere moels are realize using analytical an simulation approaches. he numerical eamples of problem solving are presente. Keywors: inventory control eman lea time orer quantity reorer point analytical moel simulation. Introuction Most inventory control situations of signicance are comple. ecision-maer s nee to unerstan this compleity epens on his role within the business an the way he chooses to solve the problems. Mathematical moels can provie a escription of business situations that are ficult to eamine in any other way. he search of the effective solutions of stoc control in transport company shoul be base on a number of economic social an technical characteristics [4]. In practice we have to investigate the stochastic moels for ferent situations characterizing inventory control systems a set of stochastic moels are available to solve the inventory control problem [ 5]. In the given paper two multiple perio single-prouct inventory control moels with ranom eman an lea time are consiere. he first moel is a moel with fie reorer point an fie orer quantity. his moel escribes epenency of average epenses for goos holing orering an losses from eficit per time unit on two control parameters the orer quantity an reorer point. he escription of this moel an analytical metho of problem solving are eamine in the previous authors wor [2]. We have solve this problem using regenerative approach. he secon moel is a moel with fie time interval between the moments of placing neighbouring orers. In this moel the orer quantity is etermine as ference between the fie stoc level an quantity of goos in the moment of orering. he analytical escription of the secon moel is consiere in the given paper. Note that in the secon moel we have use the same economical criteria minimum of average total cost in inventory system. o we have two inventory control moels with continuously review inventory position permanent stoc level monitoring. he strategy of each moel selection is base on the real conitions of the business. hus the first moel can be use for the system with arbitrary time moment of placing the orer this situation taes place in inventory system use own means of transportation for orer elivery. he secon moel is suggeste for the system with fie moment of placing the orers where the orer transportation epens on scheule of transport eparture. he consiere moels can be realize using analytical an simulation methos. As it was shown in the previous wors of these authors the analytical moels are fairly comple. An alternative to solution by mathematical manipulation is simulation [3]. In the given paper analytical an simulation approaches are investigate. he numerical results of problem solving are obtaine in simulation pacage Eten. 2

2 art I. robobabilystic Moels 2. escription of the Moels 2.. Moel We consier a single-prouct stochastic inventory control moel uner following conitions. he eman for goos is a oisson process with intensity λ. In the moment of time when the stoc level falls till certain level a new orer is place see Figure. he quantity is calle as reorer point. he orer quantity Q is constant. We suppose that Q.he lea time time between placing an orer an receiving it has a normal istribution with a mean μ an a stanar eviationσ. here is the possible situation of eficit when eman uring lea time ecees the value of reorer point. We suppose that in case of eficit the last cannot be covere by the epecte orer. ϕ t Q t Figure. ynamics of inventory level uring one cycle for Moel enote as the quantity of goos in stoc in the time moment immeiately after receiving of orer. We can etermine this quantity of goos as function of eman uring lea time : Q Q. Formula is basic. It allows epressing ferent economical inees of the consiere process. et is the uration of a cycle. ength of the cycle consists of two parts: time between receiving the goos an placing a new orer an lea time i.e.. We suppose that net economic parameters of the moel are nown: the orering cost C is nown function of the orer quantity Q i.e. C C Q the holing cost is proportional to quantity of goos in stoc an holing time with coefficient of proportionality C H the losses from eficit are proportional to quantity of eficit with coefficient of proportionality C H. et us enote as eman for goos within perio of time. rincipal aim of the consiere moel is to efine the optimal values of orer quantity Q an reorer point which are control parameters of the moel. Criteria of optimization are minimum of average total epenses costs per time unit. We solve this problem using regenerative approach [5] Moel 2 et us consier the Moel 2 with fie time of the cycle i.e. with fie time between neighbouring moments of placing the orers. It is a single-prouct stochastic inventory control moel uner the following conitions. he eman for goos is a oisson process with intensity λ. he lea 22

3 art I. robobabilystic Moels time has a normal istribution with a mean μ an a stanar eviationσ. We suppose that lea time is essentially less as time of the cycle: μ 3σ. here eists the possible situation of eficit when the eman uring time between neighbouring moments of receiving of orer ecees the quantity of goos in stoc in the time moment immeiately after receiving of orer. Analogy Moel we suppose that in case of eficit the last cannot be covere by epecte orer. In Figure 2 the cycle with number is presente. et is the rest of goos in stoc at the start of the -th perio an is the rest of goos at the en of -th cycle or the rest at start of cycle with number. We enote as the goos quantity which is neee ieally for one perio an it equals to the sum where is the average eman for cycle time is the safety stoc. In the given sentence we suppose that ieally gives us in future the minimum of total epeniture for orering holing an loses from eficit per unit of time. ϕ t Q Q t Figure 2. ynamics of inventory level uring -th cycle for Moel 2 o in the suggeste moel perio of time an stoc level are control parameters. he orer quantity Q is the ference Q. 2 We suppose that in the moment of time when a new orer has to be place it may be situation when the stoc level is so big that a new orering oesn t occur. However for generality of moel we ll eep the conception of lea time an quantity of goos at the time moment immeiately after receiving of orer in such case too. It correspons to real situation when the customer uses the transport means which epart at the fie moments of time not epening on eistence of the orer an which have the ranom lea time for eample transportation by trailers which epart each first an fteenth ay of each month. aing into account that in case of eficit it can t be covere by the epecte orer we can obtain the epression for goos quantity at the moment of time immeiately after receiving of orer Q 3 Q. an using 2 we have: 4. he rest at the start of the -th perio an the goos quantity at the moment of time immeiately after receiving of orer tae values from interval [ ]: 23

4 art I. robobabilystic Moels in the previous cycle the eman uring the time between the receiving of orer an placing of the new orer is more or equal i.e. in the previous cycle is equal an there isn t the eman uring the time perio i.e.. the rest to the moment of orering is i.e. orer quantity Q is an eman uring lea time is more or equal i.e. the rest to the moment of orering is or eman uring lea time is absent i.e.. In the net section we shoul etermine the average total cost per cycle for the fie rest of prouct in the moment of orering. 3. Analytical approach to creation of the moels he analytical escription of the Moel is presente in the previous paper of the authors [2]. In the given section we consier a etaile creation of Moel 2 with fie perio of time between the moments of placing the neighbouring orers. 3.. istribution of eman uring ea ime As eman for goos is a oisson flow with intensity λ we can etermine istribution for eman within fie perio of time i λ -λ i e i i! If f is a ensity function for lea time then istribution for eman within time can be calculate by formula i i f. 6 In the case of normal istribution for we obtain the formula 2 2 -μ -μ i - i - λ 2 2 -λ 2σ λ i -λ 2σ i e e! e e. 7 i σ 2π i! σ 2π 3.2. Holing Cost uring One Cycle Calculation process of the holing cost uring one cycle is ivie in two stages: calculation for lea time an calculation for time between receiving the goos an placing a new orer. et is the length of time from the last orering an. If the eman uring the time equals i then the holing cost uring the time interval is C i C i 8 H H an epecte holing cost uring the lea time is E CH CH f i i 9 i where i is efine by formula 5. et consier the epecte holing cost uring the time. If is the goos quantity at the moment of time immeiately after receiving of orer is time interval after the receiving of orer an 24

5 an the eman equals C art I. robobabilystic Moels uring this time equals i then the holing cost uring the time > i C i. H H et s note that the time is i an taes values from interval. o epecte holing cost uring H C H f i i r i E C where conition > is equivalent to conition r. 2 Average holing cost E C within cycle is the sum of the corresponing aenums: H E C E C E C. 3 H H H 3.3. osses from eficit imilar to previous point the calculation process of the losses from eficit uring one cycle is ivie into two stages: calculation for lea time an calculation for time between receiving the goos an placing a new orer. If within lea time the eman ecees the value of reorer point then eficit of goos is present. et i an i > then losses from eficit are CH i. o average shortage cost within lea time E C. 4 H CH i i i et eman for time equals i i an the goos quantity at the moment of time immeiately after receiving of orer is an i >. hen losses from eficit are C H i. hus an average shortage cost uring the time is E C where E C H C i i H 5 i i i f an probability is calculate by formula 2. An average shortage cost E C within cycle is the sum of the corresponing aenums: H H E C E C. 6 H H Finally an average total cost for a cycle is E C E C E C C 7 H H where E CH an E CH is calculate by formulas 3 an 6 accoringly an average total cost per time unit in inventory system is E C E AC. 8 25

6 art I. robobabilystic Moels 26 Using the nown istributions of eman an lea time an formula 4 applying recurrence metho we can fin the conitional istribution of the rest of prouct at the en of -th cycle start of cycle with number for the nown value of rest : 9 an combining epressions 4 an 9 we have. 2 In accorance with 9 we can calculate probability of event for conition that the rest at the beginning of cycle equals to. As it is evient from 2 the rest taes values from interval [ ]. In particular eman uring perio of cycle is absent i.e.. At first let s consier the case where >. Accoring to conition of the tas we can write. et an eman uring time equals to an. In this case an the request > is equivalent to the conition accoringly the first line of the formula 2. hen probability of event that the rest of prouct at the en of cycle equals to where > uner the conition that an is calculate by the formula: / f. 2 Accoringly f /. 22 imilarly an where > then an request > is equivalent to conition the probability / > is calculate by formula f > /. 23 Finally > then f. 24 easoning by analogy it can be shown that then f > >. 25 For analytical solving of the consiere problem we have create a comple of programs realize on the base of programming system EHI. For calculation there were use stanar quantitative methos.

7 4. imulation Approach art I. robobabilystic Moels As it was shown in the previous section the analytical inventory control moel is rather comple. As alternative to analytical approach the authors have use simulation moels realize in the simulation pacage Eten [3]. 4.. Moel et us consier the moel with two fie control parameters: reorer point an orer quantity Q. he schema of the tas simulation is shown in Figure Orer quantity 8 eman Mae orer ischarge erminal ea time Cost of elivery Cost of elivery Inventory cost Inventory cost toc evel Mae orer 7 4 toc evel ischarge erminal eorer point toc evel 6 ea time 9 eman eorer point Figure 3. imulation moel overview: inventory control with fie reorer point an fie orer quantity et us consier the main blocs of the simulation schema. In the bloc # the ecision of a new orering Mae Orer is generate using ata about eorer point bloc # an quantity of goos in stoc toc level. As the result variable Mae Orer taes value it is transmitte to connector of bloc #2 an a new goos orering is eecute. In bloc #5 the process of orer elivery is simulate. he value of ranom lea time is generate in bloc #4 Input anom Number using parameters μ an σ of normal istribution. he eman for goos is generate in bloc #9 as ranom value with oisson istribution an nown parameter λ. he warehouse is realize in hierarchical bloc #8 which schema is shown in Figure 4. rocess of goos realization is simulate in bloc #. Bloc #2 ummy source of goos an bloc #3 et Attribute are use for goo eficit calculation. he results of simulation are printe out in tet file bloc #7 an on the screen bloc #6. Figure 4. Warehouse simulation moel overview 27

8 art I. robobabilystic Moels Using the create simulation moel we can fin the optimal solution for inventory control problem with two control parameters reorer point an orer quantity Q see Eample. Eample. et eman for goos is a oisson process with intensity units per ay lea time has a normal istribution with a mean ays an a stanar eviation 35 orering cost C equals to 2 EU holing cost C H equals to 2 EU per unit per year losses from eficit C H equals to 8 EU per unit unit time is year. he perio of simulation is one year an a number of realizations are. he results of simulation are shown in able an in Figure 5. Note that for the given steps of the control parameters changing the best result is achieve at the point Q 95 units an 5 units where for realizations an average total cost for one year perio equals EU. ABE. Average total cost per year in inventory system with fie reorer point an orer quantity Moel eorer point units Orer quantity units Moel 2. et us consier secon strategy of inventory control with fie perio of time between the moments of placing neighbouring orers. Note that in the suggeste moel perio of time an require stoc level are control parameters. Figure 5. Average total cost per year in inventory system with fie reorer point an fie orer quantity For simulation of inventory control process we have create the schema shown in Figure 6. et us consier the main blocs of schema. Bloc # generates the transactions in the fie moments of time these transactions are use for simulation of goos orering uring the consiere time perio. Bloc #2 calculates the Orer quantity using ata about toc level in the moment of orering an equire stoc level quantity of goos which is neee ieally for one perio this result is save in bloc #3 et Attribute. Bloc #4 etermines the moment of orer elivery using the value of lea time generate in bloc #5 Input anom Number as ranom variable with normal istribution an nown parameters. he eman for goos is generate in bloc # as ranom value with oisson istribution an nown parameter. rocess of goos realization is simulate in bloc #. Blocs #8 an #9 are use for goos eficit calculation. he results of simulation are printe out in tet file an are shown on the screen. 28

9 art I. robobabilystic Moels Figure 6. imulation moel overview: inventory control with fie time interval between placing neighbouring orers Eample 2. et us consier another strategy of inventory control accoringly Moel 2 using initial ata from Eample. For problem solving we have use the simulation moel shown in Figure 6. he results of simulation are shown in able 2 an in Figure 7. For the given steps of control parameters changing the best result is achieve at the point 9 units of goos an 75 ays where for realization an average total cost for one year perio equals 9659 EU. ABE 2. Average total cost per year in inventory system with fie time interval between placing neighbouring orers Moel 2 ime interval between placing neighbouring orers ays evel up to orer units Figure 7. An average total cost per year in inventory system with fie time interval between placing neighbouring orers 29

10 Conclusions art I. robobabilystic Moels rincipal aim of the propose moels is to efine the eact orer quantity an time of the orering to achieve minimum epenses for holing orering goos an losses from eficit per time unit for transport companies. wo consiere moels of inventory control base on ferent principles of orering give the closely relate results near optimum solution. he main avantages of the consiere methos of solving the inventory control problems for the suggeste moels are as follows: simulation approach gives - the clearness of the presentation of results firstly it touches the case of analysis of total epenses epenence on one control parameter with fiing others - the possibility of fining optimum solution of an inventory problem in the case when realization of analytical moel is rather ficult analytical approach gives - the mathematical moel of situation - the various possibilities of analysis - universality of usage. In the eamine paper single-prouct inventory control moels are consiere. In the present research the authors investigate multi-prouct moel with ranom correlate emans for ferent goos. In this research we use the simulation moelling in inventory system with a fie moment of placing the orer. In particular the ranom eman vector is generate using eman statistics an Holecy ecomposition of correlation matri. eferences. Chopra. an. Meinl. upply Chain Management. onon: rentice Hall Kopytov E. an. Greenglaz. On a tas of optimal inventory control. In: roceeings of XXIV International eminar on tability roblems for tochastic Moels ept Jurmala atvia. iga: ransport an elecommunication Institute 24 pp Kopytov E. Muravov A. Greenglaz. an Buraov G. Investigation of two strategies in inventory control system with ranom parameters. In: roceeings of the 2st European Conference on Moelling an imulation ECM 27. June rague Chec epublic. homas Bata University in lin 27 pp Kopytov E. issen F. Greenglaz. Inventory Control Moel for the ypical ailways Company. In: roceeings of the International Conference eliability an tatistics in ransportation an Communication eltat'3 6-7 October 23 iga atvia iga: ransport an elecommunication Institute Vol pp oss h. M. Applie robability Moels with Optimization Applications. New Yor: over ublications Inc

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