DETERMINATION OF OPTIMAL ORDER QUANTITY OF INTEGRATED AN INVENTORY MODEL USING YAGER RANKING METHOD

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1 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) An Online International Journal Aailale at 03 Vol. 3 () January-March, pp.73-80/ritha and SSA DETERMINATION OF OPTIMAL ORDER QUANTITY OF INTEGRATED AN INVENTORY MODEL USING YAGER RANKING METHOD *W. Ritha and Sr. A. Sagayarani SSA Department of Mathematics, Holy Cross College, Tiruchirapalli, India Sacred Heart Girls High School, Bangalore, India *Author for Correspondence ABSTRACT The integrated endor uyer production inentory models are gaining much significance in recent times. Seeral methods such as classical differential calculus, cost difference rate comparison approach are employed to determine the optimal lot size with the assumption that the demand and the costs inoled are deterministic in nature, which is not possile in this world of uncertainity. This leads to the transformation of deterministic parameters to fuzzy parameters. In this paper the demand and the associated costs are taken as fuzzy ariales. To determine the optimal inentory policies the Yager s ranking methods for fuzzy numers is utilized. A set of numerical data is employed to analyse the characteristics of the proposed model. Key Words: Inentory, Integrated Vendor Buyer, Fuzzy Numer, Yager Ranking Method INTRODUCTION In the past decades the inentory models were deised for endor and the uyer separately, ut during the last few years the concept of integrated endor uyer inentory model has earned faour of the top management of the usiness concern as it minimize the total costs. As co-operation, the main ackone of the successful functioning of the production sector, the phenomena of integrated endor-uyer inentory model has great welcome among the enterprise manager.the researchers who hae wide range of interest in integrated models to determine the optimal order quantity. A rief literature reiew of their contriutions is gien y Chun-Jen Chung. To mention a few Yang et al., (007); Minner (007) and Teng (009). To these integrated models the concept of ackordering is merged to hae etter cost control of the inentory system. One common point that has to e noted in all the earlier models is that the nature of demand and the cost are deterministic which is quite impossile at all times. The reasons for the cause of such situations are inflation, sudden rise and fall in the economy of the mighty nations and so on. The effects are, the fluctuations of demand and costs which pae way for fuzzy parameters. In accordance to it integrated inentory prolems are addressed under fuzzy enironment to determine the optimal order quantity. For instance Park (987) discusses the EOQ model with fuzzy cost coefficients. Ishii and Kunno (998), Petroic et al., (996) and Kao and Hsu (00) inestigate the newsoy inentory model with fuzzy cost coefficients and demands respectiely. Roy and Maiti (997), Chang (009) construct a fuzzy EOQ model with fuzzy defectie rate and fuzzy demand. Most of the paper directly supposes the model parameters as a triangular fuzzy numer and then finds their optimal solutions. Yao and Chiang (003); Lee and Yao (998) and Chang and Yao (998) deelop the EOQ model with fuzzy ordering quantities. In this paper a fuzzy integrated endor uyer production model is discussed. The structure of this paper is organized as follows. In section, the preliminaries are gien. In Section 3, the total inentory cost of the integrated prolem under deterministic nature is discussed. In Section, fuzzy optimal ordering quantity is determined using Yager s ranking (98) method. Finally the characteristics of proposed models will e illustrated and some conclusions will e made. 73

2 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) An Online International Journal Aailale at 03 Vol. 3 () January-March, pp.73-80/ritha and SSA PRELIMINARIES Definition: Fuzzy Set A fuzzy set A is defined y A {(x, μ (x) A ) : x X, μ (x) A [0, ]}. In the pair {(x, μ (x) A )}, the first element x elong to the classical set A, the second element μ (x) A, elong to the interal [0, ], called memership function or grade of memership. The memership function is also a degree of compatiility or a degree of truth of x in A. α - Cut The set of elements that elong to the fuzzy set A at least to the degree is called the α leel set or α - cut. A) {x X : μ (x) A α Generalized Fuzzy Numer Any fuzzy suset of the real line R, whose memership function satisfies the following conditions, is a generalized fuzzy numer (i) μ (x) is a continuous mapping from R to the closed interal [0, ]. A (ii) μ (x) A 0, - < x a, (iii) μ (x) A L(x) is strictly increasing on [a, a ], (i) μ (x) A, a x a 3, () μ (x) A R(x) is strictly decreasing on [a 3, a ], (i) μ (x) A 0, a x <, where a, a, a 3 and a are real numers. Triangular Fuzzy Numer The fuzzy set A (a, a, a 3) where a a a 3 and defined on R, is called the triangular fuzzy numer, if the memership function of A is gien y (Q, r) Inentory Model with Fuzzy Lead Time x - a, a x a a - a a - x μ (x) A, a x a 3 a 3 - a 0, Otherwise Yagers Ranking Method If the cut of any fuzzy numer A is [A L (), A g ()]then its ranking index I( A ) is A L) + A g) dα. 0 THE DETERMINISTIC INTEGRATED VENDOR BUYER INVENTORY MODEL The assumptions and the notations of the model are as follows. Assumptions The algeraic model for the integrated two-stage endor uyer inentory model is deeloped on the asis of the following assumptions. 7

3 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) An Online International Journal Aailale at 03 Vol. 3 () January-March, pp.73-80/ritha and SSA (a) Both the production and demand rates are constant and the production rate is greater than the demand rate. () The integrated system of single-endor and single-uyer is considered. (c) The endor and the uyer hae complete knowledge of each other s information. (d) The uyer s shortage is allowed. Notations Q Buyer s lot size per deliery. n Numer of delieries from the endor to the uyer per endor s replenishment interal. nq Vendor s lot size per deliery. S Vendor s setup cost per setup. A Buyer s ordering cost per order. C Vendor s unit production cost. C Unit purchase cost paid y the uyer. π Unit ackordering cost of the uyer. r Annual inentory carrying cost per dollar inested in stocks. P Production rate per year, where P > D. D Demand rate per year. K The ackordering ratio. ( K) The non-ackordering ratio. TC(Q) The aerage integrated total cost of the endor uyer inentory model considering ackordering. The total integrated cost is TC (Q) {the endor s setup cost + carrying cost} + {the uyer s ordering cost + carrying cost}+{the uyer s ackordering cost} The total integrated cost is TC(Q) {the endor s setup cost + carrying cost} + {the uyer s ordering cost + carrying cost} + {the uyer s ackordering cost} DS rqc D P DA r( - F) QC B F Qπ + (n - ) nq P P Q TC(Q) - DS rc D D DA r( - F) QCB F Qπ + (n - ) - + Q nq P P Q TC(Q) 0 Q Q D(S + na) D D n rcr (n - ) - + r( - F) C B+F π P P (or) TC(Q) can e written as TC(Q) DS (n - )rqc D ( - n)rqc + + nq P 75

4 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) An Online International Journal Aailale at 03 Vol. 3 () January-March, pp.73-80/ritha and SSA DA r( - F) QC F Qπ + + Q TC(Q) Q Q DS +DA + D( - n)rc +P[(n-)rC +r( - F) C +F π Q nq P PD(S + na) D( - n)rc + P((n - )rc + r( - F) C + F π FUZZY INTEGRATED VENDOR-BUYER INVENTORY MODEL D, S, C, C, π, A e fuzzy trapezoidal numers and they are defined as follows. [ie. they Let are descried y the -cuts] D( D ) S( S ) C (C ) C (C ) A( A ) r( r ) TC(Q) can e rewritten as - - L D D), R D D) - - L S S), R S S) - - LC C ), R C C ) - - Lπ π ), R π π ) - - L A A), R A A) - - L r r), R r r) DS rqc D rqc TC(Q) + (n - ) + ( n) nq P DA r( - F) QC F Qπ + + Q K ( D, D ) K (, ) K 3 (, C ) K (, A ) LC - C ) + RC - C )dα. L - )dα. R - )dα L D D)dL D. L S S)dα S + L S S)dα S. R S S)dαS r r r r r r r L D D)dL D. LC r C r )dα r+ R D D )dα D. RC )dαc L D D)dL D. L A A)dα A+ R D D)dα D. R A A)dαCA

5 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) An Online International Journal Aailale at 03 Vol. 3 () January-March, pp.73-80/ritha and SSA K 5 ( r, C ) K 6 ( ) TC(Q) LC C ) + RC C ) dαc. L r r)dα r. R r r)dαr Lπ π ) + Rπ π ) dα - - π K α D, α S (n - )rcrk C r) rq( - n)k 3 D, αc ) + + nq P K D, α A) r( - F) QK 5C ) F K 6π)C r Cr To determine the optimal order quantity TC(Q) K α D, αs K α D, α A (n - )rk C ) + Q Q r( - n)k, αc ) r( - F) K C ) K π)f P TC(Q) 0 Q 3 D 5 6 Q D S D A P K α, α K α, α P((n - )K α(r) + r( - n)k3 α D,α() P r( - F) K5 αc h + K5 απ F NUMERICAL EXAMPLE To alidate the proposed model consider the data D (560, 580, 60, 60) r (8, 9,, ) S (80, 90, 0, 0) n. C (3,, 6, 7) F 0% C (, 3, 5, 6) (, 3, 5, 6) A (90, 00, 0, 30) P 700 D( D ) ( , 60-0) S( S ) (80 + 0, 0-0) C ( C ) (3 +, 7 - ) C ( C ) ( +, 6 - ) ( ) ( +, 6 - ) A( A ) (90 + 0, 30-0) P( r ) (8 +, - ) P

6 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) An Online International Journal Aailale at 03 Vol. 3 () January-March, pp.73-80/ritha and SSA K (, s ) 560+0α dα. 80 0α dα α dα. 0-0α dα α 0α 0α 0α 560α+. 80α+ + 60α-. 0α K (, s ) 305 K ( r, C ) (8+α) dα. (3 α) dα + (-α) dα. (7-α) dα K 3 (, r, C ) (570)(3.5) + (630)(6.5) K (, A )

7 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) An Online International Journal Aailale at 03 Vol. 3 () January-March, pp.73-80/ritha and SSA (570)(95) + (630)(5) 335 K 5 ( r, C5 ) (8.5)(.5) + (.5)(5.5).5 K 6 ( ) Q 700[ ] [(0.8)(.5) ] Q 66.8 CONCLUSION The purpose of this paper is to study the integrated models under fuzzy enironment. This fuzzy model assists in determining the optimal order quantity amidst the existing fluctuations. This model enefits the enterprise manager in decision making. In this paper the yager ranking method is employed as this does not require the explicit form of the memership function. REFERENCES Ben-Daya M and Raouf A (99). Inentory models inoling lead time as a decision ariale. Journal of the Operational Research Society 5(5) Chang HC (00). An application of fuzzy theory to the EOQ model imperfect quality items. Computers and Operations Research Chang, SC and Yao JS (998). Economic reorder point for fuzzy ackorder quantity. European Journal of Operational Research Hayya JC, Bagchi U, Kim JG and Sun D (008). On Static Stochastic Order Crossoer. International Journal of Production Economics () 0-3. IShii H and Konno T (998). A Stochastic inentory prolem with shortage cost European Journal of Operational Research Jack C Hayya, Terry P Harrison and James X He (0). The impact of stochastic lead time reduction on inentory cost under cross oer. European Journal of Operational Research 7-8. Kao C and Hsu WK (00). A single-period inentory model with fuzzy demand. Computer and Mathematic with Application

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