International Journal of Supply and Operations Management. Mathematical modeling for EOQ inventory system with advance payment and fuzzy Parameters

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1 nternational Journal of Supply and Operations Management JSOM November 0, Volume, ssue 3, pp SSN-Print: SSN-Online: Mathematial modeling for EOQ inventory system with advane payment and fuzzy Parameters S. Priyan a, M. Palanivel *a,. Uthayakumar a a Department of Mathematis, The Gandhigram ural nstitute - Deemed University, Gandhigram, Dindigul, Tamil Nadu, ndia Abstrat This study onsiders an EOQ inventory model with advane payment poliy in a fuzzy situation by employing two types of fuzzy numbers that are trapezoidal and triangular. Two fuzzy models are developed here. n the first model the ost parameters are fuzzified, but the demand rate is treated as risp onstant. n the seond model, the demand rate is fuzzified but the ost parameters are treated as risp onstants. For eah fuzzy model, we use signed distane method to defuzzify the fuzzy total ost and obtain an estimate of the total ost in the fuzzy sense. Numerial example is provided to asertain the sensitiveness in the deision variables about fuzziness in the omponents. n pratial situations, osts may be dependent on some foreign monetary unit. n suh a ase, due to a hange in the exhange rates, the osts are often not known preisely. The first model an be used in this situation. n atual appliations, demand is unertain and must be predited. Aordingly, the deision maker faes a fuzzy environment rather than a stohasti one in these ases. The seond model an be used in this situation. Moreover, the proposed models an be expended for imperfet prodution proess. Keywords: nventory osts; Advane payment; Fuzzy numbers; Signed distane method.. ntrodution An advane payment is a sum of money that is provided in advane of the date that it is ontratually due, or in advane of delivery of the produts or servies that the payment is intended to pay for. Suh payments may be strutured into a ontrat or offered to address a speifi adversity situation. n some ases, the full amount due will be paid in advane, while in * Corresponding author address: palanimathsgri@gmail.om 60

2 nt J Supply Oper Manage (JSOM) others, part of the money will be offered in advane and the other part will be paid later. A ommon reason to offer an advane payment is beause someone is in immediate need of funds that will be due even though the person has not yet earned them. Many authors are offered advanes on their work to provide them with money to pay off debts and to live on while they finish their books. ikewise, it is not unommon for an insurane ompany to provide part of a settlement in advane to help the ustomer reover more quikly. People an also arrange advanes on payheks and salary through their employers to address troubled eonomi irumstanes. Some everyday examples of advane payments are prepaid ell phones, or simply prepaying your rent or utilities as many people do now. n Chinese automobile industry, advane payment to the manufaturer is a ommon pratie for the S (sale, spare-part, servie, and survey) stores who are required to pay all of the payment before delivery. n Chinese steel industry, huge steel plants also require advane payment, espeially for small buyers. Sometimes, the buyer is offered a prie disount if he/she gives advane payment even not required by the seller, suh ases an be found in briks and tile fatories in ndia (Zhang et al. (0)). Eonomi order quantity-based inventory models minimize the sum of mainly two osts, that are the holding and the ordering osts. These models assume that the input parameters as ost and demand are desribed as risp values or having risp statistial distributions where their total inventory ost funtions are minimized without ambiguity in the results. But exat values of the ost harateristis are rarely attained as they may be vague and impreise to ertain extent in pratial situations. For instane, the shortage in an inventory system may our due to various auses, viz. sudden inrease of demand, transportation problems, hike in wages, delayed prodution, et. Shortage brings loss of goodwill and it is diffiult to measure the exat amount of shortage ost. The same problem is also experiened in the ase of the ordering and holding osts. Similarly, the demand beomes extremely variable beause of shorter produt life yles in the highly ompetitive market. Thus in inventory system, the deision maker may allow some flexibility in the ost parameter values to takle the unertainties that always fit the real situations. Statistial treatment of the ost harateristis is ineffiient for these models, beause of the lak of random observations. Therefore, the pratitioner should be more areful in aounting flexibility in the ost omponents and demand. n this paper, an EOQ inventory model with advane payment poliy is desribed by establishing fuzziness in the ost and demand parameters. The main ontribution of this paper is to establish the mathematial model and propose a solving approah for the EOQ inventory problem with advane payment poliy in fuzzy random environment. Our main objetive is to study the impat and sensitiveness of the impreiseness of ost omponents in the deision variables and the total ost. This researh may be very useful for Chinese automobile and steel industries. The next setion reviews the relevant literature on EOQ inventory models.. iterature review Some vendors offer their buyers a delay period in purhasing payment. Therefore, before the end of delay period, buyers an sell the goods and aumulate revenue and earn interest. t is a very 6

3 Priyan, Palanivel and Uthayakumar ommon poliy used by vendors to promote their ommodities. The buyer s inventory poliies under delayed payment have been widely addressed in reent deades. n literature, Goyal (985) was the first to onsider the vendor s delayed payment in analyzing the buyer s EOQ. ater, lots of papers extended the work of Goyal (985) under various pratial situations. eviews of literature on inventory systems with trade redit are available in Seifert et al. (03). Although the influenes of delayed payment on inventory poliies have attrated great attention, the advane payment and its influenes on inventory deisions are rarely disussed in the literature. Maiti et al. (009) onsidered the advane payment and developed an inventory model for one item in stohasti environment with prie-dependent demand over a finite time horizon. They assumed that the buyer s prourement prie dependent on the fration of the advane payment. Gupta et al. (009) further extended the model by onsidering the impreise information of the ost. Thangam (0) inorporated the advane payment sheme and two-ehelon trade redits into an EOQ model for perishable items. eently, Zhang et al. (0) developed an EOQ model with two types of advane payment, i.e., all payment paid in advane and partial advane payment. Fuzzy set theory, introdued by Zadeh (965), has been reeiving onsiderable attention amongst researhers in prodution and inventory management. Several authors have applied the fuzzy set onepts to deal with the inventory ontrol problems. shii and Konno (998) fuzzified the shortage ost to an -fuzzy number in the lassial newsboy problem. A bakorder inventory model whih fuzzifies the order quantity as triangular and trapezoidal fuzzy numbers and keeps the shortage ost as a risp parameter was developed by Yao and ee (999). Yao and Chiang (003) developed an EOQ model with the total demand and the unit arrying ost being triangular fuzzy numbers. They used the signed distane and the entroid as defuzzifiation methods. Vijayan and Kumaran (008) onsidered inventory models with partial bakorders and lost sales and fuzzy stok-out periods and used signed distane method to defuzzify. Bjrk (009) addressed an EOQ inventory model with bakorders, where the demand and the lead times are kept fuzzy (general triangular fuzzy numbers). Sadjadi et al. (00) propose a priing and marketing model with fuzzy parameters. n their model, there are three elastiity, namely, selling prie, marketing expenditure, and lot size are assumed to be fuzzy numbers. iu (0) developed a solution for fuzzy integrated prodution and marketing planning based on extension priniple method. eently, Mahata and Goswami (03) developed inventory models for items with imperfet quality and shortage bakordering in fuzzy environments by employing two types of fuzzy numbers suh as trapezoidal and triangular. From the authors literature, none of the authors developed EOQ inventory model with advane payment poliy and fuzziness in the ost and demand parameters. Therefore, this paper intends to fill this remarkable gap in the literature. n this study, we represent the ost parameters suh as ordering ost A, holding ost h and purhasing ost p, and demand rate D by two types of fuzzy numbers whih are trapezoidal and triangular. Two fuzzy models are developed. n the first model the ost parameters are fuzzified while the demand rate is treated as risp onstant. n the seond model the demand rate is fuzzified but the ost parameters are treated as risp onstants. For eah fuzzy model, a method of defuzzifiation, namely the signed distane, is employed to find the 6

4 nt J Supply Oper Manage (JSOM) estimate of total ost in the fuzzy sense, and then the orresponding optimal yle time is derived to minimize the total ost. Numerial example is arried out to investigate the behavior of our proposed models, and the results are ompared with those obtained from the risp model. The artile is designed as follows: Setion 3 desribes the researh methodology to demonstrate the nature of the researh. Setion outlines preliminary onepts that have been used for model building purposes. n Setion 5, notations and assumptions are given. The formulations of both models are disussed in Setion 6 and 7, respetively. Then we use an example to illustrate the idea of this paper in Setion 8. Finally, some onlusions are drawn from the disussion in Setion esearh methodology The methodology used in this paper is the quantitative method, whih is based on the priniples of Operational esearh and Management Siene. The models are built on mathematially oriented inventory theory. Solving models annot be meaningfully disussed until the formulation proess is ompleted. n this study, we shall onentrate on showing how mathematial analysis an be used to help operating rules for ontrolling EOQ inventory system. t is neessary to desribe mathematially the system to be studied when mathematis is applied to the solution of inventory problems. Suh a desription is often related to as a mathematial model. Mathematial models are idealized representations, but that are expressed in terms of mathematial symbols and notations. A mathematial model of a real life problem inludes the system of equations that desribe the essene of the problem. This study is based upon mathematial modeling. Mathematial modeling is to desribe a real world problem with tratable mathematial formulations. These mathematial formulations should not only produe values of performane measures, but also provide insights, and a deeper understanding for the phenomenon studied. The purpose of modeling inventory situations is to derive an operating dotrine. For this, the following four simple steps are involved. Examining the inventory situation, listing harateristis and assumptions onerning the situation. Finding the total annual relevant ost equation in narrative. Transforming the total annual ost equation from narrative into the shorthand logi of mathematis. Optimizing the ost equation, finding the optimum for when to order (yle time). n this paper, the methodology followed to find the optimal order quantity is listed below:. Mathematial model formulation. Solution proedure 3. Numerial example. Preliminaries The fuzzy set theory was introdued to deal with problems in whih fuzzy phenomena exist. n a 63

5 Priyan, Palanivel and Uthayakumar universe of disourse X, a fuzzy subset a of X is defined by the membership funtion ( x) whih maps eah element x in X to a real number in the interval [0, ]. The funtion a value of ( x) denotes the grade of membership. a Definition : Fuzzy onvex (Vijayan and Kumaran (008)): A fuzzy set a on X is onvex if ( ( ) ) min( ( ), ( x a x x a x a )) for all x, x in X and for [0, ], where min denotes the minimum operator. Definition : Fuzzy point (Pu and iu (980)): et a be a fuzzy set on (, ). t is alled a fuzzy point if its membership funtion is, x a a ( x) () 0, x a Definition 3: Fuzzy number (Vijayan and Kumaran (008)): A fuzzy number is a fuzzy subset of the real line whih is both normal and onvex. For a fuzzy number A, its membership funtion an be denoted by l( x), x m ( x), m x n () A u( x), x n where l (x) is upper semi ontinuous, stritly inreasing for x m and there exists m m suh that l ( x) 0 for x m, u(x) is ontinuous, stritly dereasing funtion for x n and there exists n n suh that u ( x) 0 for x n, l (x) and u (x) are alled the left and right referene funtions, respetively. Definition : Trapezoidal fuzzy number (Zimmerman (99)): The fuzzy number A is said to be a trapezoidal fuzzy number if it is fully determined by a, a, a, ) of risp numbers suh that ( 3 a a a a3 a, whose membership funtion, representing a trapezoid, an be denoted by 6

6 nt J Supply Oper Manage (JSOM) x a, a x a a a, a x a3 ( x) (3) A x a, a3 x a a3 a 0, otherwise where a, a, a3, a are the lower limit, lower mode, upper mode and upper limit, respetively of the fuzzy number A. The interval a, ] is alled the support of the fuzzy number and it gives [ a the range of all possible values of A that are at least marginally possible or plausible. The interval a, ] orresponds to the ore of fuzzy number and gives the range of most plausible [ a3 values. The intervals a, ] and a, ] are alled penumbra of the fuzzy number A. [ a [ 3 a et A ( a, a, a, ), A ( a, a, a, ) be two trapezoidal fuzzy numbers, then 3 a 3 a A A ( a a, a a, a a, a ) and, for all b 0, 3 3 a ba ( ba, ba, ba3, ba ). Definition 5: Triangular Fuzzy Numbers (Chiang et al. (005)): et A ( a, a, a3), a a a3, be a fuzzy set on (, ). t is alled a triangular fuzzy number, if its membership funtion is x a, a x a a a a3 x ( x), a x a A 3 () a3 a 0, otherwise A( ) x : ( x), where [0,] is alled the ut of A. ( ) A A is a nonempty The set bounded losed interval ontained in the set of real numbers and it an be denoted by A( ) A ( ), A (. A ( ) and A ( ) are respetively the left and right limits of A ( ) ) and are usually known as the left and right uts of A. ( ) a ( a ) and A a 65

7 Priyan, Palanivel and Uthayakumar ( ) a ( a ) for a trapezoidal fuzzy number A ( a, a, a, ) and A a3 3 a ( ) a ( a ) and ( ) a ( a ) for triangular fuzzy number A a A ( a, a, ). a3 A 3 3 a Definition 6: evel Fuzzy nterval (Chiang et al. (005)): et [ a, b, ] be a fuzzy set on (, ). t is alled a level fuzzy interval, 0, a b, if its membership funtion is (, a x b [ a, b; ] x) (5) 0, otherwise Next, as in Yao and Wu (000), we introdue the onept of the signed distane whih will be needed later. We first onsider the signed distane on.. Signed distane method The signed distane between the real numbers a and 0, denoted by d (,0) 0 a is given by d0 ( a,0) a. Hene the signed distane of A ( ) and A ( ) measured from 0 are d ( A ( ),0) A ( ) and d ( A ( ),0) A ( ), respetively. 0 0 The signed distane of the interval ( A ( ), A ( )) measured from the origin 0 by d ( A ( ),0) d ( A ( ),0) A ( ) A ( ) d 0 (( A ( ), A ( )),0) 0 0 (6) where A ( ) and A ( ) exist and are integrable for [0,]. For eah [0,], the risp interval A ( ), A ( ) and the level fuzzy interval ( ), A ( )]; [ A are in one to one orrespondene. The signed distane from [ ( ), A ( )]; [ A ( ), A ( )];, 0 d0(( A ( ), A ( )),0) A ( ) A ( ) A to 0 (where 0 is the level fuzzy point whih maps to the origin) is (7) The signed distane of A measured from 0 defined as d( A, 0) A ( ) A ( ) d (8) 0 66

8 nt J Supply Oper Manage (JSOM) emma : et A i, i,,3... N be N fuzzy numbers and b i, i,,3... N are real risp onstants. Then N N d bi Ai, 0 bi d( A, 0) (9) i i Aording to Yao and Wu (000), the signed distane formula in Eq. (8) is onsidered when ranking fuzzy numbers. Then the fuzzy numbers Aand A ranked as A A if d A, 0) d(,0) (0) ( A 5. Notations and Assumptions We adopt the following notations and assumptions in order to develop the proposed model. 5.. Notations T time interval between suessive orders (deision variable) D demand rate A ordering ost per order h unit stok-holding ost per item per unit time exluding interest harges p unit purhase ost in $ t prie disount fator for advane payment length of advane payment interest harges per $ investment in stoks per year 5.. Assumptions. The vendor offers prie disount for the buyer if all payment is paid in advane.. eplenishments are instantaneous and the shortages are not allowed. 3. Time horizon is infinite. 6. Crisp mathematial modeling of the EOQ problem with advane payment n this study the vendor gives all the payment paid in advane payment term similar to Zhang et al. (0). n the proposed senario, the buyer pays a purhase ost DTp and inurs an ordering ost A at time zero. The inventory level before arrival of a prourement is zero. The purhase ost has to be finaned at interest rate, and the loan interest ost equals 67

9 Priyan, Palanivel and Uthayakumar DTp t Dp t during this period. During the stok period, that is, from time t to T timet T, the buyer makes payment to the interest-bearing aount immediately after the selling of the goods. As the loan is being paid bak, the interest payable is dereasing. On the last day of stok period, the buyer pays the remaining balane. Hene, the average outstanding amount of the loan is DTp, and the interest ost is DT p from time t to time t T in one yle. The physial holding ost is the same as that of the traditional eonomi order quantity inventory model and is not influened by the payment terms. The behavior of inventory for this model is depited in Fig.. Based on the assumptions desribed above with Fig., the buyer's total ost per unit time an be obtained as Zhang et al. (0) is A DTh Dp T ( T) Dp t () T where the first term is ordering ost, the seond term is holding ost ((exluding interest harge), the third term is ost of interest harges at the time of advane aount payment and the last term is ost of interest harges when the goods are kept in stok. Figure : Time-weighted inventory when all the payment paid in advane 6.. Solution proedure We proved the onvexity of the total ost (T ) in emma based on lassial differential alulus optimization tehnique. emma. The total ost (T ) is a onvex funtion of yle time T. Proof. Taking the first and seond partial derivatives of (T ) with respet to T, we have 68

10 nt J Supply Oper Manage (JSOM) ( T) A Dh Dp T T and () ( T) A 0. 3 T T Therefore (T ) is a onvex funtion T. This ompletes the proof of emma. Hene, the unique optimal value of the buyer's optimal replenishment yle T an be obtain by setting Eq. () to zero as T * 0 A D( h p ) (3) The buyer's optimal replenishment yle is not related to the length of the advane payment. However, the prie disount assoiated with advane payment makes the apital ost of inventory smaller whih leads to a larger replenishment yle. Hene, the optimal order quantity is larger than that of the traditional EOQ model. When the supplier is huge and the buyer is small, suh as the automobile industry in China, the supplier may ask the buyer to pay in advane without any disount, that is. n this situation, the buyer s optimal replenishment yle is the same as the outome of the EOQ model Zhang et al. (0). 7. Fuzzy mathematial modeling of the EOQ problem with advane payment There are two types of fuzzy models that developed in this setion. n the first, the ost parameters are fuzzified while the demand rate is treated as risp onstant. n the seond model the demand rate is fuzzified but the ost parameters are treated as risp onstants. 7. The EOQ advane payment model with all ost omponents fuzzy Considering that the ost omponents, ordering ost A, holding ost h and purhase ost p are all fuzzy but the demand rate D is risp onstant. We represent ost omponents A, h and p by trapezoidal fuzzy numbers as given below A ( A, A, A 3, A ) h ( h 5, h 6, h 7, h 8) p ( p, p, p, p ) 9 0 () where, l,,3...,, l are arbitrary positive numbers under the following restritions 69

11 Priyan, Palanivel and Uthayakumar S, 3 ; A 5 6, 7 8 ; C b 9 0,. Then the left and right limits of uts of A ( ) A ( ) 0, h ( ) h 5( 5 6) 0, p ( ) p ( ) 0, A, h and p are given below A ( ) A ( 3) 0 h ( ) h 8( 8 7) 0 p ( ) p ( ) 0 (5) Aordingly, when the osts A, h and p in Eq. () are fuzzified to be A, h and p as expressed in Eq. (), we an obtain the total ost funtion in the fuzzy sense is given by A DTh ( ) Dp T T Dp t (6) T From Eq. (5), we an obtain the left and right hand side of the - ut, ( 0 ), of ( T ) by the following form A ( ) DTh ( ) Dp ( ) T ( T) ( ) ( ) Dp t (7) T A ( ) DTh ( ) Dp ( ) T ( T) ( ) ( ) Dp t (8) T respetively. Hene, when the osts desribe with trapezoidal number, using Eqs. (8), (9), (7) and (8), the defuzzified value of ( T ) (as expressed in Eq. (6)) is given by H DTH DH 3 T d( ( T), 0) DH 3 t (9) T where H A ( 3 ) 0, H h ( ) 0, H 3 p ( 0 9) 0. The defuzzified value d( ( T), 0) (as expressed in Eq. (9)) is taken as the estimate of fuzzy ost funtion in Eq. (5), denoted by E ( ( T)), when the osts represented by trapezoidal number. 70

12 nt J Supply Oper Manage (JSOM) The estimate in Eq. (9) is a onvex funtion of T similar to Eq. (). Then a unique minimum of E ( ( T)) is obtained by equating to zero the first order partial derivatives of E ( ( T)) with respet to T. That is E ( ( T)) 0 T Solving Eq. (0), we obtain * H T () D( H H ) 3 On the other hand, if we treat the ost omponents A, h and p are the triangular fuzzy numbers as given below. A ( A, A, A ), 0 A,0 h ( h 3, h, h ), 0 3 h,0 p ( p, p, p ), 0 p,0 5 6 Then we obtain the left and right limits of uts of A ( ) A 0, h ( ) h 3 3 0, p ( ) p 0, A ( ) A 0 h ( ) h 0 p ( ) p A, h and p as follows. Hene, when the osts A, h and p in Eq. () are desribed with triangular numbers A, h and p as expressed in Eq. (), following similar steps as in the trapezoidal ase and using Eq. (3), the signed distane value (defuzzified value) of total ost is established as G DTG DG 3 T d( ( Q,, n), 0) DG 3 t () T Where G A ( ) 0, G h ( 3) 0, G 3 p ( 6 5) 0. Similarly, the defuzzified value d( ( T), 0) (as given in Eq. ()) is taken as the estimate of fuzzy ost funtion in Eq. (), denoted by E ( ( T)), when the osts are represented by triangular number. The estimate in Eq. () is a onvex funtion of T similar to Eq. (9). Then a (0) () (3) 7

13 Priyan, Palanivel and Uthayakumar unique minimum of E ( ( T)) is obtained by equating to zero the first order partial derivatives of E ( ( T)) with respet to Q. That is E ( ( T)) 0 T Solving Eq. (5), we obtain * G T (6) D( G G ) 3 (5) 7. The EOQ advane payment model with fuzzy demand n this ase we onsider that the demand rate D is fuzzy in Eq. () while the ost omponents are risp onstants. We represent demand rate D by trapezoidal and triangular fuzzy numbers as given below D ( D 3, D, D 5, D 6) where D 3, 5 6 And D ( D 3, D, D ) where 0 7 A, 0 8 respetively. Then the left and right limits of uts of D for trapezoidal and triangular fuzzy numbers are given by And respetively. D ( ) D 3( 3 ) 0, D ( ) D 5( 5 6) 0 D ( ) D 7 7 0, D ( ) D Hene, when the demand D in Eq. () are fuzzified to be D, we an obtain the total ost funtion in the fuzzy sense is given by A DTh Dp T ( T) Dp t (7) T Now, the left and right uts of the funtion in Eq. (7) for trapezoidal and triangular fuzzy number ases are given by A D ( ) Th D ( ) p T ( T) ( ) ( ) D p t (8) T 7

14 nt J Supply Oper Manage (JSOM) and A D ( ) Th D ( ) p T ( T) ( ) ( ) D p t (9) T respetively. Hene, using Eqs. (9), (0), (8) and (9), the estimate of total ost funtion for the trapezoidal and triangular number ases under fuzzy demand are expressed as A H Th H p T d( ( T), 0) H p t (30) T where H D ( 6 5 3) 0 and A GTh G p T d( ( T), 0) G p t (3) T where G D ( 8 7) 0 respetively. Similarly, the defuzzified values as given in Eqs. (30) and (3) are taken as the estimate of fuzzy ost funtions for the trapezoidal and triangular number ases in Eq. () denoted by ( ( T)) and ( ( T)), respetively. Again, similar to Eqs. (9) and (), the estimates in Eqs. (30) and E D E D (3) are onvex funtions of T. Then the unique minimum of ( ( T)) E D and ( ( T)) E D are obtained by equating to zero the first order partial derivatives of ( ( T)) E D and ( ( T)) with E D respet to T. Hene the optimum value of T is obtained for the trapezoidal and triangular number ases from the following expressions * A T (3) H ( h p ) and T * A G ( h p respetively. ) (33) 73

15 Priyan, Palanivel and Uthayakumar 8. Numerial analysis Extensive numerial analysis has been done to assess the impat of the level of fuzziness in the input parameters over the deision variable. et us onsider an EOQ inventory model with the following data: 0.9, t 0. and 0.. Besides, the risp osts A 30, p 5 and h 0 as well as the risp demand D 00. Zhang et al. (0)'s statistial data is more appropriate for our proposed model. Therefore, we use the same numerial data as in Zhang et al. (0) to verify the results obtained by this paper. We set some trapezoidal and triangular fuzzy numbers of the input parameters ( p, D, h and A ) in Tables (a, b) and (a, b), respetively, to represent the omponents of fuzzy models developed in Setion. For eah of these parameters, the variations in the values are arranged arbitrary and their defuzzified values are determined by applying the signed distane method. The signed distane value (defuzzified value) and the orresponding perentage differene under fuzzy ase (based on the defuzzified value) from the risp values denoted by ˆ ˆ pˆ, D, hˆ, Aˆ, E ( ( T)) and ˆ E ( ( T)) for the omponents p, D, h, A, E ( ( T)) and E ( ( T)), respetively, are also shown along with the fuzzy numbers in Tables (a, b) and (a, b). Note that fuzzy ost parameters hosen are in the ranking order indiated in Eq. (0). * Based on these values, the optimal replenishment yle T with the minimum total ost for the fuzzy model developed in Setion 5 are omputed for eah set of trapezoidal and triangular fuzzy numbers. Moreover, we use Setion for the risp EOQ inventory model. The results are summarized in Tables 3 and. p Table (a): Fuzzy trapezoidal values for the input parameters for p and D d ( p, 0) pˆ D d (D, 0) Dˆ (,,8,30) 6-36 (00,50,5,55) (5,7,3,0) -6 (0,00,55,505) 30-0 (,0,30,55) 9 +6 (0,90,630,780) (0,7,30,75) (80,30,680,800) (5,0,,03) (30,360,790,930) Table (b): Fuzzy trapezoidal values for the input parameters for h and A h d (h, 0) ĥ A d (A, 0) Â (,3,,3) -0 (,5,3,33) 8-0 (3,6,5,30) 6-0 (5,9,37,5) -0 (6,5,5,50) +0 (0,,50,60) (5,,5,70) 8 +0 (,0,37,99) +0 (0,5,37,98) (3,7,90,00)

16 nt J Supply Oper Manage (JSOM) Table (a): Fuzzy triangular values for the input parameters for p and D p d ( p, 0) pˆ D d (D, 0) Dˆ (3,5,7) 0-0 (70,00,0) 30-0 (7,5,3) - (0,00,0) 30-5 (9,5,3) 8 + (390,00,650) (,5,8) (380,00,70) (0,5,70) (380,00,900) Table (b): Fuzzy triangular values for the input parameters for h and A h d (h, 0) ĥ A d (A, 0) Â (,0,) 6-0 (3,30,33) -0 (,0,8) 8-0 (0,30,38) 7-0 (,0,36) +0 (8,30,5) (0,0,6) +0 (6,30,68) (5,0,65) (5,30,83) +30 pˆ ĥ Â Table 3: Optimal inventory poliy with fuzzy osts for trapezoidal and triangular fuzzy numbers Trapezoidal number ase * T E ( ( T)) ˆ E ( ( T)) pˆ ĥ Â Triangular number ase * T E ( ( T)) Eˆ ( ( T)) Table : Optimal inventory poliy with fuzzy demand for trapezoidal and triangular fuzzy numbers Trapezoidal number ase Dˆ * T E ( ( T)) Eˆ ( ( T)) Triangular number ase Dˆ * T E ( ( T)) Eˆ ( ( T))

17 Priyan, Palanivel and Uthayakumar 9. Conlusion Huge steel plants require advane payment in Chinese steel industry espeially for small buyers. n addition, in real world appliations, the input ost and other parameters in the EOQ inventory problem may not be known preisely or it may be unertain due to some unontrollable fators. Hene, approximate solution methodologies have been illustrated for the solution of a lass of realisti inventory problems. Fuzzy methodologies provide a useful way to model vagueness in human reognition and judgment. Unertainties haraterized by impreise expression an be represented by fuzzy sets. Moreover, fuzzy numbers are largely applied on data analysis, artifiial intelligene, and deision making. Fuzzy numbers are frequently used in appliations and they also make handling the realisti problem easier. This paper proposed two fuzzy models for an EOQ inventory model with advane payment option under fuzzy osts and demand environment. The fuzziness in the ost omponents and demand are represented by the trapezoidal and triangular fuzzy numbers. Numerous methods for total ordering of fuzzy numbers have been suggested in the literature. Eah method appears to have some advantages as well as disadvantages. n the ontext of eah appliation, some methods seem more appropriate than others. However, the issue of hoosing a proper ordering method in a given ontext is still a subjet of ative researh. But some existing studies show that the signed distane method is better for defuzzifiation than other methods. Therefore, we use signed distane method to defuzzify the fuzzy total ost and obtain an estimate of the total ost in the fuzzy sense. Classial differential alulus optimization tehnique is used to find the optimal solution of the models. Numerial example is provided to asertain the sensitiveness in the deision variables about fuzziness in the omponents. Our results indiate that the optimal solutions of the fuzzy model slightly flutuate from the solutions of the risp model (see Tables 3 and ). Hene, the researh reveals that in all the models, the deision variables and the total expeted ost are sensitive to the level of fuzziness in the ost omponents and demand. Hene the pratitioner is advised to be more areful in aounting flexibility in the ordering ost, holding ost, purhasing ost and demand rate. Aknowledgment The authors would like to thank anonymous referees and editors for their most valuable, onstrutive, innovative omments and suggestions that have enouraged the authors to make signifiant improvements in this paper. The first author is grateful to the University Grants Commission (UGC), Government of ndia, New Delhi, for providing finanial assistane in the form of Basi Siene esearh Fellowship. The seond author s researh work is supported by DST NSPE, Ministry of Siene and Tehnology, Government of ndia under the grant no. DST/NSPE Fellowship/0/3B dated

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