Multisection Technique to Solve Interval-valued Purchasing Inventory Models without Shortages

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1 ISSN Englnd UK Journl of Informtion nd Computing Science Vol. 5 No pp Multisection Technique to Solve Intervl-vlued Purchsing Inventory Models without Shortges Susovn Chkrortty Mdhumngl Pl Prsun Kumr Nyk 3 Deprtment of pplied Mthemtics with Ocenology nd Computer ProgrmmingVidysgr University Midnpore-7 0 Indi 3 Bnkur Christin College Bnkur 7 0 Indi (eceived My 4 00 ccepted July 00) strct. This pper investigtes n intervl vlued economic order quntity (EO) prolem without shortge. Since it is lmost impossile to find n nlytic method to solve the proposed model n optimiztion lgorithm is designed. First rief survey of the existing works on compring nd rnking ny two intervl numers on the rel line is presented. Finlly the effectiveness of the designed lgorithm is illustrted y numericl exmple. Key words: Inventory Intervl Numer Demnd Production Simultion. Introduction The economic order quntity (EO) model is first introduced y F.Hrris [4]. Inventory control is n importnt field in supply chin mngement since it cn help compnies rech the gol of ensuring delivery voiding shortges helping sles t competitive prices nd so forth. proper control of inventory cn significntly enhnce compny's profit. To control n inventory system one cnnot ignored demnd monitoring since inventory is prtilly driven y demnd nd s suggested y u nd u [] in mny cses smll chnge in the demnd pttern my result in lrge chnge in optiml inventory decisions. mnger of compny hs to investigte the fctors tht influence demnd pttern ecuse customers' purchsing ehvior my e ffected y fctors such s selling price inventory level sesonlity nd so on. lrge numer of cdemic ppers (for review see []) hve een pulished descriing numerous vritions of the sic EO model. The ody of the reserch ssumes tht the prmeters involved in the EO model such s the demnd nd the purchsing cost re crisp vlues or rndom vriles. However in relity the demnd nd the cost of the items often chnge slightly from one cycle to nother. For exmple inventory crrying cost my e different in riny seson compred to summer or winter sesons (costs of tking proper ction to prevent deteriortions of items in different sesons nd lso the lour chrges in different sesons re different). Ordering cost eing dependent on the trnsporttion fcilities my lso vry from seson to seson. Chnges in the price of fuels miling chrges telephonic chrges my lso mke the ordering cost fluctuting. Unit purchse cost is highly dependent on the costs of rw mterils nd lour chrges which my fluctute over time. To solve the prolem with such imprecise numers stochstic fuzzy nd fuzzy-stochstic pproches [ ] my e used. In stochstic pproch the prmeters re ssumed to e rndom vriles with known proility distriution. In fuzzy pproch the prmeters constrints nd gols re considered s fuzzy sets with known memership functions. On the other hnd in fuzzy-stochstic pproch some prmeters re viewed s fuzzy sets nd others s rndom vriles. However if the memership function of the fuzzy vrile is complex for exmple when trpezoidl fuzzy numer nd Gussin fuzzy numer coexist in model it is hrd to otin the memership function of the totl cost. Therefore these memership functions ply significnt role in the method. However in prctice one my not e le to get exct memership function for fuzzy vlues nd proility distriution for stochstic vrile. Since precise Corresponding uthor. E-mil ddress: susovn_chkrortty@ymil.com E-mil ddress: mmplvu@gmil.com 3 E-mil ddress: nyk_ Pulished y World cdemic Press World cdemic Union

2 74 Susovn Chkrortty et l: Multisection Technique to Solve Intervl-vlued Purchsing Inventory Models without Shortges informtion is required the lck of ccurcy will ffect the qulity of the solution otined. For these resons we hve represented the imprecise numer y intervl numers [3 3]. Thus the intervl numer theory rther thn the trditionl proility theory nd fuzzy set theory is well suited to the inventory prolem. ccording to the decision mker's point of view under chngele conditions we my replce the rel numers y the intervl vlued numers to formulte the prolems more ppropritely. We orgnize the pper s follows : In section we give some sic definitions nottions nd comprison on intervl numers. In section 3 we give the model formultion nd the solution procedure.. Intervl numer n intervl numer proposed y Moore [3] is considered s n extension of rel numer nd s rel suset of the rel line. Definition. n intervl numer is closed intervl defined y = [ ] = { x : x ; e the set of ll rel numers}. The numers re clled respectively the lower nd upper limits of the intervl. n intervl numer lterntively represented in men-width or center-rdius form s = m( ) w( ) = { x : m( ) w( ) x m( ) w( )} () where m( ) = ( ) nd w( ) = ( ) re the mid-point nd hlf-width of the intervl. ctully ech rel numer cn e regrded s n intervl such s for ll x x cn e written s n intervl [ x x] which hs zero length. The set of ll intervl numers in is denoted y I ()... Bsic intervl rithmetic et = [ ] = m w nd B = [ ] = m w I( ) e two intervl numers then B = [ ]; B = m m w w. The multipliction of n intervl y rel numer c 0 is defined s c = [ c c ]; if c 0 ndc = [ c c ]; if c < 0. c = cm w = cm c w. The difference of these two intervl numers is B = [ ]. The product of these two distinct intervl numers is given y B = min{ } mx{ }. The division of these two intervl numers with 0 B is given y \ B = min{ } mx{.. Comprison etween intervl numers et = [ ] = m w B = [ ] = m w e two intervl numers within I (). These two intervls my e one of the following types:. Two intervls re completely disoint (non-overlpping).. Two intervls re nested (fully overlpping). 3. Intervls re prtilly overlpping. rief comprison on different intervl orders is given in [ ]. }. JIC emil for contriution: editor@ic.org.uk

3 Journl of Informtion nd Computing Science Vol. 5 (00) No. 3 pp Cse (Disoint suintervls): Moore [3] defined trnsitive order reltions over intervls s : Figure : Disoint suintervls is strictly less thn B if nd only if < nd this is denoted y < B. This reltion is n extension of `< ' on the rel line. This reltion seems to e strict order reltion tht is smller thn B. Cse (Nested suintervls) : et = [ ] B = [ ] I( ) e such tht <. Figure : Nested suintervls Then B is contined in nd it is denoted y B which is the extension of the concept of the set inclusion [3]. The extension of the set inclusion here only descries the condition tht B is nested in ut it cn not order nd B in terms of vlue. et nd B e two cost intervls nd minimum cost intervl is to e chosen. If the decision mker (DM) is optimistic then he/she will prefer the intervl with mximum width long with the risk of more uncertinty giving less importnce. gin if the DM is pessimistic then he/she will py more ttention on more uncertinty i.e. on the right end points of the intervls nd will choose the intervl with minimum width. The cse will e reverse when nd B represent profit intervls. In this cse we define the rnking order of nd B s = if the plyer is optimistic B B if the plyer is pessimistic. The nottion B represents the mximum mong the intervl numers nd B. Similrly B = if the plyer is optimistic B if the plyer is pessimistic. The nottion B represents the minimum mong the intervl numers nd B. Cse 3 (Prtilly overlpping suintervls) : The ove mentioned order reltions introduced y Moore [3] cn not explin rnking etween two overlpping closed intervls. Figure 3: Prtilly overlpping suintervls We define n cceptility index to compre nd order ny two intervl numers on the rel line in terms of vlue s in [ ] which re used throughout the pper. Definition For m m nd w w 0 the vlue udgement index or cceptility index (I) of the premise B is defined y m I( B) = w which is the vlue udgement y which is inferior to B ( B is superior to ) in terms of vlue. Here `inferior to' `superior to' re nlogous to `less thn' `greter thn' respectively. Here ` ' e n extended order reltion etween the intervls nd B on the rel line. For ny sort m w JIC emil for suscription: pulishing@wu.org.uk

4 76 Susovn Chkrortty et l: Multisection Technique to Solve Intervl-vlued Purchsing Inventory Models without Shortges of vlue udgement the I index consistently stisfies the decision mker i.e. if B I( ) then either I( B) > 0 or I( B ) > 0 or I( B) = I( B ) = 0 holds. Thus on the sis of comprtive position of men nd width of intervls B the vlues of I( B ) of the premise B re given y ( i) I( B) when m < m nd which refer to Cse ; ( ii) 0 < I( B) < when m < m nd > ; ( iii) I( B) = 0 when m = m. Using the I index we hve presented the ordering for closed intervls B I( ) reflecting decision mker's preference s (i) When I( B) we hve m < m nd. In this cse is preferred over B (i.e. is less thn B ) with cceptility index greter thn or equl to nd so the decision mker (DM) ccepts it with solute stisfction. (ii) When 0 < I( B) < we hve m < m nd > then is preferred over B with different grdes of stisfction lying etween 0 nd excluding 0 nd. (iii) If I( B) = 0 then oviously m = m. Now if w = w then there is no question of comprison s is identicl with B. But if w = w then the intervls nd B re non-inferior to ech other i.e. cceptility index ecomes insignificnt. In this cse DM hs to negotite with the widths of nd B. et nd B e two cost intervls nd minimum cost intervl is to e chosen. If the DM is optimistic then he/she will prefer the intervl with mximum width long with the risk of more uncertinty giving less importnce. gin if the DM is pessimistic then he/she will py more ttention on more uncertinty i.e. on the right end points of the intervls nd will choose the intervl with minimum width. The cse will e reverse when nd B represent profit intervls. Ex. et = [030] = 55 nd B = [3438] = 36 e two intervls. Then 36 5 I( B) = =.57 >. Hence the DM ccept the decision tht ` is less thn B ' with full 5 stisfction. Ex. et = [06] = 33 nd B = [48] = 6 then 6 3 I( B) = = 0.6 <. Here 3 ` is less thn B ' with grde of stisfction 0.6. Ex 3. et = [86] = 4 nd B = [68] = 6 e two intervls. Here m = m = nd < w w nd so I( B ) = P 0. Hence oth the intervls re non-inferior to ech other. In this cse from the optimistic point of view the DM will prefer the intervl B insted of. Becuse if nd B re oth the profit intervls then DM will py more ttention on the highest possile profit of 8 unit ignoring the risk of minimum profit of 6 unit. In the sme mnner if nd B re cost intervls then the DM will py his ttention on the minimum cost of 6 units i.e. the left end points of oth the cost intervl nd B nd select B insted of s 6 < 8. gin when oth the intervls re profit intervls then the DM with pessimistic outlook will prefer the profit intervl ecuse his ttention will e drwn to the fct tht the minimum profit of 8 unit will never e decresed wheres his choice of might cost him the loss of unit profit nd this pprehension will determine from selecting the intervl B. Similr is the explntion when nd B re cost intervls. The ove oservtions cn e put into compct form s follows JIC emil for contriution: editor@ic.org.uk

5 Journl of Informtion nd Computing Science Vol. 5 (00) No. 3 pp B B = B if I( B) > 0 if I( P B) = 0 nd w if I( B) = 0 nd w P < w < w nd DM is pessimistic nd DM is optimistic. Similrly if m m nd w w then there lso exist strict preference reltion etween nd B. Thus similr oservtions cn e put into compct form s B if I( B ) > 0 B = if I( B P ) = 0 nd w > w nd DM is pessimistic B if I( B P ) = 0 nd w > w nd DM is optimistic. n unified lgorithm involving the dominnce of intervl numers: Two intervl numers = m w nd B = m w re sid to e non-dominting if ( i ) m = m nd ( ii) w w. The following function computes the minimum etween two intervl numers. Function min( B) if = B then minimum = ; if = m w nd B = m w re not non-dominting then if (( B) or ( B)) then minimum = ; minimum = B ; if < ) w then ( w P if the decision mker is optimistic minimum = B ; if the decision mker is pessimistic minimum = ; return(minimum); End Function. Similrly in the following we hve given nother function mx which determines the mximum etween two intervl numers. Function mx( B) if = B then mximum = ; if = m w nd B = m w re not non-dominting then if (( B) or ( P B)) then mximum = B ; mximum = ; if > ) w then ( w if the decision mker is optimistic mximum = ; if the decision mker is pessimistic mximum = B ; JIC emil for suscription: pulishing@wu.org.uk

6 78 Susovn Chkrortty et l: Multisection Technique to Solve Intervl-vlued Purchsing Inventory Models without Shortges return(mximum); End Function. 3. Model formultion nd nlysis The purpose of the EO model is to find the optiml order quntity of inventory items t ech time such tht the sum of the order cost nd the crrying cost i.e. totl cost is miniml. In the clssicl EO model without shortge n instntneous replenishment is ssumed to tke plce when the inventory level drops to zero nd the stock items re exhusted with fixed demnd rte. Moreover the setup cost in ech replenishment re ssumed to e deterministic. But in rel situtions the setup cost is usully ffected y vrious uncontrollle fctors nd often show some fluctution. Similrly lso for demnd. In most cses these re descried y "lies etween nd ''. It is more resonle therefore to chrcterize these s intervl numers. Nottions : For the ske of clrity the following nottions re used throughout the pper. T length of one cycle; D = [ d l d r ] demnd rte; order quntity per cycle; = ( T ) = = [ C C ] = [ C C ] C totl cost in the pln period; C the inventory crrying cost per unit item per unit time; C the ordering or setup cost/ unit item ssumptions : We hve the following ssumptions:. No shortges re llowed.. ed time is zero. 3. The inventory plnning horizon is infinite nd the inventory system involves only one item nd one stocking point. 4. Only single order will e plced t the eginning of ech nd the entire lot is delivered in one tch. 5. The quntities C C 3 nd D re ssumed to e intervl numer elongs to I () typicl ehvior of the EO lot size model with uniform demnd nd without shortge is depicted in Fig 4. B t = 0 t = T t = T t Figure 4: EO model without shortge et us ssume tht n enterprize purchses n mount of units of item t time t = 0. This mount will e depleted to meet up the customers demnd. Ultimtely the stock level reches to zero t time t = T. The totl demnd D in pln period [0T ] cn e expressed s = DT. JIC emil for contriution: editor@ic.org.uk

7 Journl of Informtion nd Computing Science Vol. 5 (00) No. 3 pp The inventory crrying cost for the entire cycle T is given y C re of OB = C T =. C.. T nd the ordering cost for tht cycle T is C 3. Hence the totl cost in the pln period [0T ] cn e expressed s X = C3. C.. T. X Therefore totl verge cost C () is given y C ( ) = i.e. T C. D C ( ) = 3 C. C C3 or C( T ) = 3. C. =... T. C D () T T By using clculus we optimize C () nd we get optimum vlues of T nd C s. C3. D. C3 = T = C =. C. C3. D. C C. D Usully in mthemticl progrmming we del with the rel numers which re ssumed to e fixed in vlue. In usul models- Crrying cost ( C ) set up cost ( C 3) demnd (D) re lwys fixed in vlue. But in rel life usiness cnnot e properly formulted in this wy due to uncertinty. Becuse the demnd of customers cn never e fixed similrly the other costs lso never e fixed in vlue. In such cses demnd nd other costs re ssumed to e intervl vlued. But in intervl oriented system we cnnot use the clculus method for optimiztion. 3.. Intervl vlued EO model et us ssume intervl vlued demnd y D = [ d d ] crrying cost y C = [ c c ] nd set up cost y C = [ c 3 3 c 3 ] where first term within the rcket denote lower limit nd nd term within the rcket denote the upper limit of the vrile. eplcing D y [ d d ] C y [ c c ] in eqution () we hve C( T ) = [ c3 c3 ]..[ c c ].[ d d ]. T (3) T ddition nd other composition rules (seen in the section. in this pper) on intervl numers re used in this eqution. But in intervl oriented system we cnnot use the clculus method for optimiztion of C ( T ). If we tke T [ T T ] then the expression (3) ecomes = C( T ) = [ c3 c3 ]..[ c c ].[ d d ].[ T T ]. (4) T T Since the vlue of C ( T ) is intervl vlued we cnnot use the clculus method for optimiztion. In the next section we hve presented new method dependent on intervl computing technique to solve the unconstrined optimiztion prolems. By using multi-section method we re to find T = [ T T ] for which C ( T ) hve the optiml (minimum) vlue. Multi-section method nd solution procedure of the system: Here we use the multi-section lgorithm s descried y Mhto nd Bhuni [4]. The ide of multisection comes out from the concept of multiple isection where more thn one isection re done t single itertion cycle. The sis of this method is the comprison of intervls ccording to the DM's point of view. et us consider ound unconstrined optimiztion (mximiztion or minimiztion) prolem with fixed coefficients s follows: JIC emil for suscription: pulishing@wu.org.uk

8 80 Susovn Chkrortty et l: Multisection Technique to Solve Intervl-vlued Purchsing Inventory Models without Shortges z = f ( x) l x u where x = ( x x xn) l = ( l l ln) u = ( u u un) n represents the numer of decision vriles the th decision vrile x ; ( = n) lies in the prescried intervl [ l u ]. Hence the serch spce of the ove prolem is s follows: n D = x : l x u = n. Suppose firm divides the sles seson into periods. Now our oect is to split the ccepted region(reduced)region (for the first time it is the given serch spce or ssumed if the serch spce is not given )into finite numer of distinct equl suregions to select the suregion contining the est function vlue. et f ( i ) = [ fi fi ]; i = e the intervl vlued oective function f (x) in the i th suregion i where f i fi denote the upper nd lower ounds of f (x) in i computed y the ppliction of finite intervl rithmetic. Now compring ll the intervl-vlued vlues of oective function f (x) in i ( i = ) with the help of intervl order reltions mentioned in erlier section the suregion contining est oective function vlue is ccepted. gin this ccepted suregion is divided into other smller distinct suregions i ( i = ) y the foresid process nd pplying the sme cceptnce criteri we get the reduced suregion. This process is terminted fter reching the desired degree of ccurcy nd finlly we get the est vlue of the oective function nd the corresponding vlues of the decision vriles in the form of closed intervls with negligile width. lgorithm : To solve the prolem (4) the optiml solution or n pproximtion of it hs een otined y pplying the following lgorithm. Step : Initilize n (here n = for T ) l (lower ounds) nd u (upper ounds) where = n. Step : Divide the ccepted region X into equl suregions where i i= i = such tht = X. i Step 3 : Using intervl rithmetic find the intervl vlue F ( i ) = [ fi fi ] of the oective function in the suregions i for i =. Step 4 : pplying pessimistic order reltions (defined in the section.) opt etween ny two intervl numers choose the suregion mong i ( i = ) which hs etter oective function vlue y compring the intervl vlues F( i ) i = 3 to ech other. Step 5 : Compute the widths w = ( u l ) = 3 n. Step 6 : If w < pre-ssigned very smll positive numer for = 3 n go to next step; otherwise go to step 3. Step 7 : Print the vlues of the vriles nd of the oective function in the form of closed intervls with negligile width. Step 8 : Stop. 3.. Numericl exmple Numericl exmple hs een crried out to test the performnce of the proposed pproch descried in this pper. To illustrte the developed model n exmple with the following dt hs een considered. Consider intervl vlued EO inventory system without shortge in which the crrying cost JIC emil for contriution: editor@ic.org.uk

9 Journl of Informtion nd Computing Science Vol. 5 (00) No. 3 pp ˆ ( C ) = [.64.70] nd the ordering or setup cost ( C 3) = [500000] the demnd quntity D = [ ]. The pproch for computing the est found vlue in ech suregion of the given serch region of the test prolem hs een coded in C progrmming lnguge. The solution is T = optiml cost C = [ ] nd the optimum = [ ]. Bsed on the numericl exmple considered ove we now study sensitivity of T C nd to chnges in the vlues of the system prmeters C C 3 nd D. The sensitivity nlysis is performed y chnging ech of the prmeters y 50% 5% 5% nd 50% ; tking one prmeter t time nd keeping the remining prmeters unchnged. The results re shown the following tle. of Tle: Effect of chnges in the vrious prmeters of the inventory model Mid vlue % chnge the prmeter m( C ) m( C 3 ) T % Chnge in m( C ) m( m( D) ) C. 3. D. From the tle it is seen tht. C is firly sensitive while. Ech of T C nd T is less sensitive while 4. Conclusion T nd re less sensitive to chnges in the vlue of the crrying cost re modertely sensitive to chnges in the vlue of the setup cost 3 C nd C. re firly sensitive to chnges in the vlue of the demnd rte In this pper we hve presented n inventory model without shortge where crrying cost the ordering or setup cost nd demnd re ssumed s intervl numers insted of crisp or proilistic in nture. We hve considered the nture of these quntities s intervl numers to mke the inventory model more relistic. t first we hve formulted solution procedure to optimize generl function with coefficients s intervl vlued numers using intervl rithmetic. Using multi-section technique we hve derived the solution of the model. The lgorithm hs een tested using numericl exmple. stly to study the effect of the determined quntities on chnges of different prmeters sensitivity nlysis is lso presented. 5. eferences JIC emil for suscription: pulishing@wu.org.uk

10 8 Susovn Chkrortty et l: Multisection Technique to Solve Intervl-vlued Purchsing Inventory Models without Shortges []. Sengupt nd T. K. Pl. On compring intervl numers. Europen Journl of Opertionl eserch (): []. H. u nd H. S. u. Effects of demnd-curve's shpe on the optiml solutions of multi-echelon inventory/pricing model. Europen Journl of Opertionl eserch : [3] E. Hnsen nd G.W. Wlster. Glol optimiztion using intervl nlysis. New York: Mrcel Dekker 00. [4] F. Hris. How mny prts to mke t once Fctory. The Mgzine of Mngement. 93 0(): [5] G.C. Mht nd. Goswmi. Fuzzy EO models for deteriorting items with stock dependent demnd nd nonliner holding costs. Interntionl Journl of pplied Mthemtics nd Computer Sciences (): [6] J. Kcprzyk P. Stniewski. ong term inventory policy mking through fuzzy decision mking models. Fuzzy Sets nd Systems. 98 8: 7-3. [7]..Zdeh. Fuzzy sets. Informtion nd Control : [8] K. J. Chung. n lgorithm for n inventory model with inventory-level-dependent demnd rte. Computtionl Opertion eserch : [9] M. Vuosevic D. Petrovic nd. Petrovic. EO formul when inventory cost is fuzzy. Interntionl Journl of Production Economics : [0] M. Gen Y. Tsuimur nd D. Zheng. n ppliction of fuzzy set theory to inventory contol models. Computers nd Industril Engineering : [] N. Brhimi S. Duzere-Peres nd. Nordli. Single item lot sizing prolems. Europen Journl of Opertionl eserch (): - 6. [] P.K. Nyk nd M. Pl. iner progrmming technique to solve two person mtrix gmes with intervl py-offs. si-pcific Journl of Opertionl eserch (): [3].E. Moore. Method nd ppliction of Intervl nlysis. SIM Phildelphi 979. [4] S.K. Mhto nd.k. Bhuni. Intervl-rithmetic oriented intervl computing technique for glol optimiztion. pplied Mthemtics eserch express. 006 pp.-9. [5] S. Chkrortty M. Pl nd P. K. Nyk. Solution of Intervl-vlued Mnufcturing Inventory Model With Shortges. Interntionl Journl of Engineering nd Physicl Sciences. 00 4:. JIC emil for contriution: editor@ic.org.uk