Multi-Item Multi-Objective Inventory Model with Fuzzy Estimated Price dependent Demand, Fuzzy Deterioration and Possible Constraints

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1 Advaces Fuzzy Mathematcs. ISSN XVolume 11, Number (016), pp Research Ida Publcatos Mult-Item Mult-Objectve Ivetory Model wth Fuzzy Estmated Prce depedet Demad, Fuzzy Deterorato ad Possble Costrats Nrmal Kumar Duar & Trpt Chakrabart Departmet of appled Mathematcs, Uversty of Calcutta, 9, A P C Road, Kolkata , Ida. E-mal: abu1985@gmal.com, trptchakrabart@gmal.com Abstract Ths paper presets a mathematcal model of vetory cotrol problem for determg the mmum total cost of mult-tem mult-objectve vetory model wth fuzzy estmated prce depedet demad, fuzzy deterorato ad possble costrats. Here, three costrats we have take; they are warehouse space costrat, vestmet amout costrat ad the thrd costrat s the percetage of utlzato of volume of the warehouse space. Warehouse mateace s oe of the essetal parts of servce operato. The warehouse space the sellg stores plays a mportat role stockg the goods. I the proposed model, the warehouse space the sellg store s cosdered volume. The model s llustrated wth a umercal example ad provdes sestvty aalyss of dfferet parameters wth the help of Lgo. Keywords: Membershp fucto, tragular fuzzy umber, volume of the warehouse INTRODUCTION The lteral meag of the vetory s the stock of goods for future use (producto or sales). The cotrol of vetores of physcal goods s a problem commo to all eterprses ay sector of a ecoomy. I ay dustry, the vetores are essetal but they mea lockup of captal. The excess vetores are udesrable, whch calls for cotrollg the vetores the most proftable way. The dfferet types of costs (Purchasg cost, Setup cost, holdg cost, deterorato cost, shortages cost, etc.) volved vetory problems are affect the effcecy of a vetory cotrol problem. Warehouse space avalable the sellg store plays a mportat role vetory model. Warehouse space ca be cosdered terms of area ad/or volume,

2 158 Nrmal Kumar Duar & Trpt Chakrabart but most of the researchers cosder oly the area of the warehouse space. Here, the warehouse space the sellg store s cosdered volume. The classcal vetory problem s desged by cosderg that the demad rate of a tem s costat ad determstc ad that the ut prce of a tem s cosdered to be costat ad depedet ature [1]. But practcal stuato, ut prce ad demad rate of a tem may be related to each other. Whe the demad of a tem s hgh, a tem s produced large umbers ad fxed costs of producto are spread over a large umber of tems. Hece the ut prce of a tem versely relates to the demad of that tem. Itally, the fuzzy set theory s used the decso-makg problem []. The objectves are troduced as fuzzy goals over the α-cut of a fuzzy costrat set [3]. Ad, the cocept of solvg a mult-objectve lear programmg problem s troduced [4]. Now, the fuzzy set theory has made a etry to the vetory cotrol systems. The EOQ formula s examed the fuzzy set theoretc perspectve assocatg the fuzzess wth the cost data [5]. Mult objectve fuzzy vetory model s formulated wth three costrats ad solved by usg geometrc programmg method [6]. Mult-tem stochastc ad fuzzy stochastc vetory models formulated uder mprecse goal ad chace costrats [7]. Saat et al. [9, 10] solved the posblstc lear programmg problem based o a-cut. May authors use the membershp fuctos of the fuzzy objectve fucto ad fuzzy costrats but they do ot cosder the membershp fuctos of fuzzy coeffcets [11, 1, ad 13]. I most of the above artcles, the warehouse space avalable the sellg store s take terms of area. If the warehouse space s take terms of volume the less percetage of volume of the warehouse space wll be cosumed. Cosequetly, the maxmum of the volume of the warehouse space ca be utlzed effectvely. Deterorato s defed as decay, damage or spolage. Food tems, photographc flms, drugs, pharmaceutcal, chemcals, electrocs compouds ad radoactve substace are some example of tems whch suffcet deterorato may occur durg the ormal storage perod of the uts ad cosequetly ths loss must be take to accout whle aalyzg the vetory system. Aggarwal S. P. ad Ja [14] cosder shortages. But Chakrabort, T., Gr, B. C., Chaudhur, K. S. [15], Dev, M., ad Chaudhur, K. [16], Dave, U., [17], Goswam, A. ad Chaudhur, K. S. [18], ad Jala, A. Gr, R. R., Chaudhur, K. S. [19], Kudu. A., Chakrabort. T. [0] they cosder deterorato ad Shortages. To optmze the total expedture of the orgazato by usg mult objectve fuzzy vetory model ad warehouse locato problem, the avalable warehouse space the sellg store has bee take terms of area [8]. I ths paper, mult objectve fuzzy vetory model wth fuzzy deterorato s developed uder three costrats such as warehouse space costrat, vestmet amout costrat ad the thrd costrat s the percetage of utlzato of volume of the warehouse space. Here, the volumes of the ut tems are take for calculatos. Ad, the demad s depedet o ut cost. The ut cost s take fuzzy evromet. The ut cost ad lot sze are the decso varables.

3 Mult-Item Mult-Objectve Ivetory Model wth Fuzzy Estmated Prce 159 NOTATIONS We use the followg otatos proposed model: = umber of tems I = Total vestmet cost for repleshmet L - Isde legth of the warehouse B - Isde breadth of the warehouse M - Maxmum heght of the shelf V Volume of the warehouse space For th tem: ( = 1, ) D = D(p) demad rate (fucto of cost prce ) Q1 = lot sze ( a decso varable ) Q = shortages level S = set up cost per cycle ɵ =deterorato rate per tem ch = holdg cost per ut tem cd =deterorato cost per ut tem cs = shortages cost per ut tem p = prce per ut tem (a fuzzy decso varable) l - Legth of the ut tem b - Breadth of the ut tem h - Heght of the ut tem v -Volume of the ut tem Vw - Percetage of utlzato of volume of the warehouse MATHEMATICAL MODEL AND ASSUMPTIONS The basc assumptos about the model are: 1. Repleshmet s stataeous. Shortage s allowed 3. Lead tme s zero 4. Demad s related to the ut prce as: c D e P Where C (>0) ad e (0 < e < 1) are costats ad real umbers selected to provde the best ft of the estmated prce fucto. Whle C >0 s a obvous codto sce both D ad p must be o-egatve. 5. Deterorato rate s ɵ for the tem, cosder to be costat. Volume of the ut tem s defed by v l b h Where l, b ad h are the legth, breath ad heght of the ut tem.

4 160 Nrmal Kumar Duar & Trpt Chakrabart To calculate the volume of the warehouse space, multply the legths of the dmesos of the sde of the warehouse, that s, multply the sde legth, sde breadth ad maxmum shelf heght..e., Volume of the warehouse space s defed by V = L B M Fgure1: The vetory level for th tem Now, the dfferet costs are: a) Purchasg cost: PC = D p, for th tem D s b) Set up cost: SC =, for th tem Q c) Holdg cost: HC = 1 Q c d) Deterorato cost: DC = e) Shortage cost: SC = 1 h Q cs, for th tem Q c 1 d, for th tem, for th tem The total cost s the sum of all the above costs..e. sum of purchasg cost, set up cost, holdg cost, deterorato cost ad shortage cost. Therefore, total cost s TC = D p + D Q s 1 + Q1 ch + Q c 1 d + Q cs, for th tem There are some restrctos o avalable resources vetory problems that caot be gored to derve the optmal total cost. The restrctos are as the lmtato o the avalable warehouse space the store, the upper lmt of the total amout vestmet ad Percetage of utlzato of volume of the warehouse.

5 Mult-Item Mult-Objectve Ivetory Model wth Fuzzy Estmated Prce 161 Therefore, we get the problem for all the tems as C C s 1 1 M TC = P C h CdQ e e 1 Cs Q for =1,.. 1 P P Q1 Subject to v Q V p Q I Where Where V Vw 1 Where v Q 100 Q Q 1 Q Q Q 1 Q Q Q 1 Q Fuzzy model by cosderg fuzzy demad ad fuzzy deterorato: We cosder the tal demad rate c D e P, whch p s fuzzy, partcular let t be a tragular fuzzy Number, whose membershp fucto s defed by, p p1 f p1 p p p p1 p3 p ~ p ( p ) f p p p3 p3 p 0, Other wse Where a tragular fuzzy umber (TFN) ~ p s specfed by the trplet (p1, p, p3) ad s defed by ts cotuous membershp fucto ( p ) F 0,1 : ~ p Alpha-cut: It s a set cosstg of elemets p of the uversal set A, whose membershp values are ether greater tha or equal to the value of α. It s deoted by the symbol A p ) ad s defed as ( A ( p ) = {p / µa(p) α} The we have ~ p ( p ), 0 α 1 p p1 p3 p The, ad p p p p 1 p p1 p p1 3 p p3 p3 p.e. Ad So, a α-cut of p~ ca be expressed by the followg terval ~ p ( ) [ p ( p p ), p ( p p )], η [0,1 ] where

6 16 Nrmal Kumar Duar & Trpt Chakrabart p p p ad p p p p 1 1 upper cut respectvely. p 3 3 are kow as lower cut ad I a smlar way we cosder the deterorato Ѳ as fuzzy let t be tragular fuzzy umber ad correspodg membershp fucto s gve by 1 f A f other wse Is defed by ( ) / ~, the we have, ~ p ( p ) 0 α 1 A The ad 1 1 lower ad upper cut respectvely. 3 3 are the correspodg Ivetory Model Fuzzy Evromet: Whe p~ s fuzzy decso varable ad s fuzzy deterorato the the sad crsp vetory model s trasformed fuzzy evromet, therefore C ~ C s 1 ~ 1 M TC = P h d e ~ C C Q C e 1 s Q 1 P P Q 1 for =1,.. Subject to v Q V Where Q Q 1 Q ~ p Q I Where Q Q 1 Q V Vw 1 v Q 100 Where Q Q 1 Q Where p~ ~, ad Q>0 (=1,,..., ), 0 Vw 100 (Here cap ~ deotes the fuzzfcato of the parameters) Therefore the lower ad upper cut of the total cost fuctos are gve by M TC M TC, p M TC M TC, p

7 Mult-Item Mult-Objectve Ivetory Model wth Fuzzy Estmated Prce 163 Subject to v Q V Where Q Q 1 Q ~ p Q I Where Q Q 1 Q V Vw 1 Where v Q 100 Q Q 1 Q Where p p1 p p1, p p3 p3 p ad, 1 1 ad upper cuts of p~ ad ~ respectvely. 3 3 are the correspodg lower Thus the problem s a mult-objectve fucto. Therefore to optmze the problem we eed to help a fucto that optmze the problem. We use here Global Crtera method to optmze the above problem to be dscussed later the umercal example secto Numercal examples: The model s llustrated for oe tem ( = 1) ad also the commo parametrcal values assumed for the gve model are =1, C1=100, S1=$100, Ch1=5, Cd1=4, Cs1=3, ɵ1=0.3, L=10, B=1, M=30, I=4000, e=0. the p1=15, Q1 = , Q=66.666, Vw=100, ths mples for ths example that the total space s fully occuped ad Total cost= Global Crtera Method: The fuzzy model s a mult-objectve model whch s solved by Global Crtera (GC) Method wth the help of Geeralzed Reducto Gradet techque. The Mult- Objectve No Lear Iteger Programmg (MONLIP) problems are solved by Global Crtera Method covertg t to a sgle objectve optmzato problem. The soluto procedure s as follows: Step-1: Solve the mult-objectve programmg problem as a sgle objectve problem usg oe objectve at a tme gorg the other. Step-: From the result of Step-1, determe the deal objectve vector, m (say m max TC, TC ) ad (say max TC, TC ). Here the deal objectve vector s use as a referece pot. The problem s the to solve the auxlary problem: TC M (GC) =Mmze {( TC TC TC max TC TC TC TC max m p ) ( m max m ) p } 1 p

8 164 Nrmal Kumar Duar & Trpt Chakrabart Where 1 p<. As usual value of p s. The method s also sometmes called compromse Programmg. Hece TC = $ TC = $ max m TC = $ TC = $ max m TC = $ TC = $ Ad usg the Global Crtera method we have the requred Total Cost wth p=, s $ Sestvty Aalyss: Sestvty aalyss of Total cost wth parameter o Demad ad Deterorato s gve by the followg table ad graphcally: Table 1. ɵ1 Q1 Q Vw % TC Fgure. Shows the varato of TC wth deterorato crsp evromet

9 Mult-Item Mult-Objectve Ivetory Model wth Fuzzy Estmated Prce 165 Table. O total cost wth deterorato fuzzy: ~ 1 Q1 TC LC TC UC GC Fgure 3. Shows the varato of TC wth deterorato fuzzy evromet Table 3. O total cost wth demad parameters: e Q1 TC -m TC +m GC

10 166 Nrmal Kumar Duar & Trpt Chakrabart Fgure 4. Shows the varato of TC wth demdd parameter e fuzzy evromet Table 4. O total cost wth demad parameters: p Q1 TC -m TC +m GC Fgure 5. Shows the varato of TC lower cut wth demdd parameter p fuzzy evromet

11 Mult-Item Mult-Objectve Ivetory Model wth Fuzzy Estmated Prce 167 Fgure 6. Shows the varato of TC upper cut wth demdd parameter p fuzzy evromet Fgure 7. Shows the varato of GC wth demdd parameter p fuzzy evromet

12 168 Nrmal Kumar Duar & Trpt Chakrabart Observatos: Deterorato creases mply the crease of total cost both the cases crsp ad fuzzy sese. That s deterorato s more sestve ths model. Total cost creases whe prce hke estmated prce per ut ad demad (lot sze) falls but ot always arse that for the demad parameter e. That demad parameters are also sestve. CONCLUSIONS Fuzzy set theoretc approach of solvg a vetory cotrol problem s realstc as there s othg lke fully rgd the world. Here, the fuzzy vetory model s take wth three costrats, partcularly; volumes of the ut tems are take the warehouse space costrat. By solvg the above fuzzy vetory model, the optmal result wll be calculated. The result reveals the mmum expected aual total cost of the vetory model ad also the optmal percetage of utlzato of the volume of the warehouse. I the result, the percetage of utlzato of the volume of the warehouse space s folly occuped ths partcular gve example; t ca be decreased by chagg the values of volume parameters, vestmet cost, etc. We may develop the proposed model wth may lmtatos, such as the shortage level, umber of orders, etc. REFERENCES [1] E.A. Slver, R. Peterso, Decso Systems for Ivetory Maagemet ad Producto Plag. New York: Joh Wley, [] R.E. Bellma, L.A. Zadeh, Decso-makg a fuzzy evromet, Maagemet Scece, 17(4), B141-B164, 1970.

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14 170 Nrmal Kumar Duar & Trpt Chakrabart [0] Kudu, A., Chakrabort, T., 009. A EOQ model for tems wth Expoetal Icreasg demad ad Webull dstrbuto deterorato ad Shortages, Joural of Mathematcs & system scece, vol. 5(1), pp