Soluton of perodc revew nventory model wth general constrans Soluton of perodc revew nventory model wth general constrans Prof Dr J Benkő SZIU Gödöllő Summary Reasons for presence of nventory (stock of goods) are physcal and economcal constrants n busness However we would lke accordng to our present knowledge nventory can not be elmnated from producton and dstrbuton of goods at most we can reduce and mnmze t The name of the scence that deals wth determnaton of the optmal lot sze s nventory theory Inventory theory formulates mathematcal models to help decson makers One groups of nventory models s called perodc revew that allows to vary the requred amounts from perod to perod Ths study deals wth such a model n that maxmum quantty produced or ordered for perod maxmum nventory after chargng and mnmum nventory at the end of perod are gven and takes also a suggeston for the soluton Introducton The presence of stocks n the economy s caused by physcal and economc constrants Even f we would lke to obtan stocks from the producton and dstrbuton processes accordng to our present knowledge the best we can do s to decrease or mnmze stocks The scence that determnes the optmal sze of stocks s called stockng theory that helps the people to make decsons by usng mathematcal models Prelmnares The frst classcal stock managng model that became known as Optmzng Economc Order Quantty (EOQ) model was publshed n 95 [3] The orgnal usablty condtons of the model or formula that s able to compute the optmal order quantty are so strct that they can hardly be realzed n lfe but n spte of ths t s stll the most often mentoned and used model The reason for t's wde use s not only because t can be used by mathematcally less traned people but because the soluton s not so senstve to the accuracy of the nput parameters The new models developed from the orgnal form are the contnuous stock watchng models [] These models are able to consder the shortage the lmted refllng capacty the orderng expenses that s dependng on the sze of the amount but they stll assume that the rates of consumpton or the decrement are equal n every perod The contnuous stock watchng models n the frst place can be used at companes that produce great amounts of products n almost equal rate wthout hurtng the startng condtons But these condtons are not full-flled n case of producton that show seasonal fuctuaton or n case of producton that follows the changes of orders In ths case f we do not want to make too bg mstake we are forced to use models that are more complcated need more preparng and more work One group of these knds of models s the Perodc Revew Inventory Models that are able to wrte down the predctable changng demands over perods The frst model was publshed by Wagner and Wthn n 958 who used dynamc programmng to solve the problem Later Wthn hmself developed the algorthm [4] In the end of the 50's and n the begnnng of the 60's the relatve backwardness of computer scence forced the mathematcans and researchers to develop algorthms that use the less memory and countng n computers Ths ambton can be dscovered n the Wagner-Wthn algorthm too where the creators have made the calculatons easer but the components and
2 Hungaran Agrcultural Engneerng N 5/2002 results became less punctual Ths means that they have set up bounds and cost functon that needed less calculatons In the past tme the developng of computer technology that was showed n the ncrease of the speed besdes others the dffcultes have passed off Now there s the possblty to create models that are more accurate and closer to realty Descrpton of the problem Ths study s also engaged n usng and solvng the developed Wagner-Wthn model The great leap forward n the new model s that n ths calculaton the upper and lower bounds that are needed n practce can be prescrbed and the concept of stock n the cost functon s closer to realty The constrants are the followng: the perodcal purchase or producton and the stock after refllng can not exceed a lmt the stock can't fall below the prescrbed amount Let us frst examne the cost unt consdered n the model Let K be the constant cost that occurs at the begnnng of the perods (= 2 n) at the startng of orderng or producton We call ths orderng or settng cost The cost of purchase or producton (c) per pece s generally constant n the new model t can change through perods or t may be based on the amount of the order The amount of stock per pece (h) can also be constant or changng through perod or based on the amount of the stock In the orgnal model they consdered the left amount at the end of the perod as the stock that generates the stock keepng cost In the developed model assumng lnear changes we consder stock as the average of the countable amount n the begnnng and at the end of the perod The amount can be optonal or lmted Lmtng the mnmum amount of stock n practce s usually because of safety reasons and that means settng a mnmal amount (s) and the stock can never go below ths lmt The upper lmt of the stock can be determned as the amount that can be purchased or produced (Z) or the accessble capacty of the warehouse (8) It s obvous that the amount that s left at the end of perod (stock) s the startng stock of the next perod (+) We assume that the startng stock s known n the begnnng of the frst perod and the stock decreases to 0 to the end of the perod n In the model we descrbe the change of stock by the demand of the perod : r (= 2 n) nstead of the rate of consumpton (the changng of stock per tme unt) The am of solvng the stock problem wrtten above s to determne the amount of products that must be ordered or produced n the begnnng of a certan perod (z (=2 n)) and to mnmze the total cost at one tme We wll use dynamc programmng to solve ths problem that's varates are the followng n connecton wth storng The phase s dentfed as the perod and the condtons are ordered to answer the possble stock that step nto perod ndcated by x (=2 n) The amount of the x stock s known at the begnnng of the frst perod and at the end of the last perod the same s O n other words: x n+ l =0 Let the decson varable be the ordered or produced amount (z ) n the begnnng of perod And the demand n perod s r The soluton of the problem The cost n perod s B (x z ) that s functon of the startng stock (x ) and the produced amount (z ) As we ndcated before we consder the constant or settng cost (K) the c purchasng or producng cost that s proportonal wth the produced peces and the stock-keepng cost h The amount of the stock n perod s the average of the stock after the refllng x +z and the remanng stock x + =x +z r that s:
Soluton of perodc revew nventory model wth general constrans 3 usng ths the cost of perod : x 2 K c ( z ) z h ( x / 2) ha a z 0 B ( x z ) h( x / 2) ha a z 0 In the B (x z ) cost functon the c purchasng and producton and the h stock-keepng cost can change from one perod to another furthermore B (x z ) s not necessarly must be a lnear functon For example t s very common n lfe that we get dscount f we order bgger amount of a certan product that means the prce per pece (c ) s dependng on the orderng amount (z ) Because the nature of the problem the possble constrants are the followng: The maxmum of purchasng or producng s lmted: Z z The stock after producton or refllng s maxmzed: S x We do not allow shortage: x 0 The stock s mnmzed at the end of the perod: s x Because we chose z as decson varable we can arrange the constrants as t follows: z Z z S x z x x s / 2 Fgure A three perods model If we transform the last constrant from Fgure we can wrte: from ths equaton: x x x Substtutng nto the last constrant and arrangng t we get: Contractng the condtons: x r z s z s
4 Hungaran Agrcultural Engneerng N 5/2002 mn{ Z ( x s )( S )} max{ r } Let C (x z ) the total costs of the subpolces from the begnnng of the perod to the end of perod n that are as approprate dependng on the ngong stock and the produced amount Furthermore n case of x startng stock we ndcate the mnmal value of the C (x z ) set wth C * ( x ) Consderng the lmtatons the best subpolces can be calculated wth the followng recurrent formula n every = 2 n perod C ( x ) * mn z mn{ Z ( x r s )( S )} z max{ r } { C ( x z )} * where C n s defntonally equal zero and: Further condtons of the soluton: mn z mn{ Z ( x r s )( S )} z max{ r } x + =x +z n nz { B ( x z ) C * ( x )} That means that the sum of the products that can be purchased or produced and the amount of the startng stock can not be less than the total demand And the demand of the frst perod can not be more than the sum of the startng stock of the frst perod and the maxmum amount of the products that can be purchased or produced n a sngle perod: r Z x At the end of the plannng horzon the x n+ =0 condton can only be true f: s r n that means that the mnmal stock can not be more than the demand of the last perod The applcaton of the results To show the computer program based on the algorthm (the screen that shows the nput of the data can be seen n Fgure 2) let us examne the followng example: Table Perod () 2 3 4 5 6 Purchasng prce (c ) [thousand Fornts /tons] 8 3 7 20 0 Demand (r ) [tonn] 8 5 3 2 7 4 Stock keepng cost (h ) [thousand Fornts/tons] A company's Materal Supply Department must ensure the basc materal for producng n the amount (r ) that s gven n Table to guarantee t's annual producng plan The purchasng prce (c ) changes perodcally and the capacty of the warehouse s lmted: S=9 tons The startng stock n the frst perod s: x =2 tons The orderng cost s K=2 thousand Fornts per order the specfc stock keepng costs are equal n every perod h= thousand Fornts per tons Let us determne the optmal stockng polcy! The queston s what amounts (z) do we have to order n the certan perods f we do not lmt the stock and after ths we examne what s the cost ncrease f we change the mnmal stock to tons
Soluton of perodc revew nventory model wth general constrans 5 After runnng the program the results we get: the startng stocks of the perods (x) the purchased amounts n the perods (z) and the stock after refll (x+z) at zero mnmal stock level can be seen on Fgure 2 and at tons mnmal stock level can be seen on Fgure 3 In the frst case the total purchasng and stockng cost s 3955 thousand Fornts Ths amount grows to 445 thousand Fornts when ncreasng the mnmal stock level to ton Ths means that the ton ncrease n the mnmal stock level drops the costs by 9 thousand Fornts Fgure 2 The nput data wth results at 0 mnmal stock level Fgure 3 The nput data wth results at mnmal stock level
6 Hungaran Agrcultural Engneerng N 5/2002 IRODALOM BenkőJ: Logsztka tervezés Dnaszta Kadó Budapest 2000 2 Chkán A (szerk): Készletezés modellek Közgazdaság és Jog Könyvkadó Budapest 983 3 Harrs F: Operatons and Cost AW Shaw Co Chcago 95 4 Hadley G-Wthn T M: Analyss of Inventory Systems Prentce-Hall 963 5 Tersne R J: Prncples of Inventory and Materals Management North-Holland Amsterdam 988 6 Wagner H M-Wthn T M: Dynamc Verson of the Economc Lot Sze Model Management Scence 5 958 89-96 old Publkálva: Hungaran Agrcultural Engneerng N 5/2002 62-63 p