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... on R is called a level αsfuzzy interval if the membership function of [p, q; α] is given bys[ , ; ] , ; ( ) 0, otherwise.p q p x q xα αμ ≤ ≤⎧= ⎨⎩s(1)sDecomposition Theorem (see, Kaufmannan and Gupta =-=[29]-=-)sLet Dsbe a fuzzy set on R and Ds∈ Fs, 0 ≤ α ≤ 1. The α-cut of Dsis D(α) = [DL(α),sLIANG-YUH OUYANG, JINN-TSAIR TENG AND MEI-CHUAN CHENGs234sDU(α)]. Then, we haves0 1 ( ),D D α α α ≤ ≤ = ∪s(2)sor...

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...aid within the permissible delay period. Therefore,sthe retailer can earn interest on the accumulated revenue received by deferring the payment until the last moment of this permissible period. Goyal =-=[20]-=- developed an EOQsmodel under conditions of permissible delay in payments, in which he calculated interestsincome based on the purchasing cost of goods sold within the permissible delay period.sBeyond...

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...ion rate is not known withscertainty. Amongst extended studies on this topic, only Chen and Ouyang [27] havestreated interest rates as variable, wherein the authors extended the model of Jamal et al.s=-=[28]-=- by fuzzifying the carrying cost rate, interest paid rate and interest earned rate simultaneously. Summarizing the above, we observe that the inventory problem associatedswith trade credit has yet to ...

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.... ka kb k k a b kb ka k >⎧⋅ = ⎨ <⎩s(6)sIf 0 ≤ a < b and 0 ≤ c < d, thens(iv) [a, b](⋅)[c, d] = [ac, bd].sIf 0 ≤ a < b and 0 < c < d, thens(v) [a, b](÷)[c, d] = [a/d, b/c].sNext, similar as Yao and Wu =-=[30]-=-, we introduce the concept of the signed distanceswhich will be needed later. We first consider the signed distance on R.sA FUZZY INVENTORY SYSTEM UNDER SUPPLIER CREDITSs235sDefinition 2sFor any a and...

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... on the purchasing cost of goods sold within the permissible delay period.sBeyond the permissible period, interest payments are calculated based on the purchasingscost of the goods not yet sold. Teng =-=[21]-=- amended Goyal’s [20] model by calculatingsinterest earned based on the selling price of goods sold. In Chang and Teng [22], thessuppliers offer cash discounts or delay payment to retailers. Within th...

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...end points of the α-cut of ,θs0 ≤ α ≤ 1, aresθL(α) = θ − (1 − α)Δ1 > 0, and θU(α) = θ + (1 − α)Δ2 > 0,s(20)srespectively.sLet y = g(x) = exT, and by extension principle in fuzzy set (see, Zimmermann =-=[31]-=-),swe get the membership function of fuzzy set ( ) Tg eθθ = sas follows:s1 2 ( )1 1 ( )2( ) 2 ln , ; ( ) sup ( ) (ln / ) ln , ; 0, otherwise, x T T T TTg y e y T T e y e T y x y T T T y e y e T θ θ ...

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...ecise) under inflationsand incorporating the time value of money. Other related articles on this topic can besfound in work by Chen and Wang [9], Vujosevic et al. [10], Gen et al. [11], Roy andsMaiti =-=[12]-=-, Ishii and Konno [13], Lee and Yao [14], Yao and Lee [15], Chang et al. [16],sChang et al. [17], Ouyang et al.[18], Yao et al. [19].sIn today’s business environment, trade credit plays an important r...

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...roblem may be fuzzy. In this paper,swe deal with these three imprecise parameters in inventory modeling by utilizing thesfuzzy set theory. We develop the fuzzy inventory model based on Chang et al.’s =-=[1]-=-smodel by fuzzifying the rate of interest charges, the rate of interest earned, and the deterioration rate into the triangular fuzzy number. Subsequently, we discuss how to determine the optimal order...

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...ount and storehouse capacity are imprecise) under inflationsand incorporating the time value of money. Other related articles on this topic can besfound in work by Chen and Wang [9], Vujosevic et al. =-=[10]-=-, Gen et al. [11], Roy andsMaiti [12], Ishii and Konno [13], Lee and Yao [14], Yao and Lee [15], Chang et al. [16],sChang et al. [17], Ouyang et al.[18], Yao et al. [19].sIn today’s business environme...

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...,sinterest payments accrue for the goods not yet sold. There are many interesting and relevant articles related to trade credit, such as Davis and Gaither [23], Ouyang et al. [24, 25]sand Teng et al. =-=[26]-=-.sWhen discussing trade credit, interest income and interest payments are importantsissues. The above mentioned studies on trade credit assumed that the rate of interestscharges, the rate of interest ...

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... the time value of money. Other related articles on this topic can besfound in work by Chen and Wang [9], Vujosevic et al. [10], Gen et al. [11], Roy andsMaiti [12], Ishii and Konno [13], Lee and Yao =-=[14]-=-, Yao and Lee [15], Chang et al. [16],sChang et al. [17], Ouyang et al.[18], Yao et al. [19].sIn today’s business environment, trade credit plays an important role and the inventory problem associated...

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...zzy random variable and the total demand to be thestriangular fuzzy number. Based on the centroid method of defuzzification, they derivedsan estimate of the total cost in the fuzzy sense. Chen et al. =-=[7]-=- introduced a fuzzy economic production quantity model with defective products in which they considered asfuzzy opportunity cost, trapezoidal fuzzy cost and quantities in the context of the traditiona...

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...se cost, investment amount and storehouse capacity are imprecise) under inflationsand incorporating the time value of money. Other related articles on this topic can besfound in work by Chen and Wang =-=[9]-=-, Vujosevic et al. [10], Gen et al. [11], Roy andsMaiti [12], Ishii and Konno [13], Lee and Yao [14], Yao and Lee [15], Chang et al. [16],sChang et al. [17], Ouyang et al.[18], Yao et al. [19].sIn tod...

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... money. Other related articles on this topic can besfound in work by Chen and Wang [9], Vujosevic et al. [10], Gen et al. [11], Roy andsMaiti [12], Ishii and Konno [13], Lee and Yao [14], Yao and Lee =-=[15]-=-, Chang et al. [16],sChang et al. [17], Ouyang et al.[18], Yao et al. [19].sIn today’s business environment, trade credit plays an important role and the inventory problem associated with trade credit...

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...ELSP problem. Yao etsal. [4] presented a fuzzy inventory system without the backorder model in which bothsthe order quantity and the total demand were fuzzified as the triangular fuzzy numbers.sChang =-=[5]-=- discussed the Economic Order Quantity (EOQ) model with imperfect qualitysReceived December 25, 2007; revised May 8 & August 20 & October 27, 2008; accepted November 20, 2008.sCommunicated by Chin-Ten...

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...nd the permissible period,sinterest payments accrue for the goods not yet sold. There are many interesting and relevant articles related to trade credit, such as Davis and Gaither [23], Ouyang et al. =-=[24, 25]-=-sand Teng et al. [26].sWhen discussing trade credit, interest income and interest payments are importantsissues. The above mentioned studies on trade credit assumed that the rate of interestscharges, ...

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...nd the permissible period,sinterest payments accrue for the goods not yet sold. There are many interesting and relevant articles related to trade credit, such as Davis and Gaither [23], Ouyang et al. =-=[24, 25]-=-sand Teng et al. [26].sWhen discussing trade credit, interest income and interest payments are importantsissues. The above mentioned studies on trade credit assumed that the rate of interestscharges, ...

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...e delay period. Beyond the permissible period,sinterest payments accrue for the goods not yet sold. There are many interesting and relevant articles related to trade credit, such as Davis and Gaither =-=[23]-=-, Ouyang et al. [24, 25]sand Teng et al. [26].sWhen discussing trade credit, interest income and interest payments are importantsissues. The above mentioned studies on trade credit assumed that the ra...

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...ion quantity model with defective products in which they considered asfuzzy opportunity cost, trapezoidal fuzzy cost and quantities in the context of the traditional production inventory model. Maiti =-=[8]-=- developed a multi-item inventory modelswith stock-dependent demand and two-storage facilities in a fuzzy environment (wherespurchase cost, investment amount and storehouse capacity are imprecise) und...

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...[3] investigated the EconomicsLot Scheduling Problem (ELSP) with fuzzy demands. They used the ‘Independent Solution’ as well as the ‘Common Cycle’ approach to solve the fuzzy ELSP problem. Yao etsal. =-=[4]-=- presented a fuzzy inventory system without the backorder model in which bothsthe order quantity and the total demand were fuzzified as the triangular fuzzy numbers.sChang [5] discussed the Economic O...

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...se capacity are imprecise) under inflationsand incorporating the time value of money. Other related articles on this topic can besfound in work by Chen and Wang [9], Vujosevic et al. [10], Gen et al. =-=[11]-=-, Roy andsMaiti [12], Ishii and Konno [13], Lee and Yao [14], Yao and Lee [15], Chang et al. [16],sChang et al. [17], Ouyang et al.[18], Yao et al. [19].sIn today’s business environment, trade credit ...

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...a [29])sLet Dsbe a fuzzy set on R and Ds∈ Fs, 0 ≤ α ≤ 1. The α-cut of Dsis D(α) = [DL(α),sLIANG-YUH OUYANG, JINN-TSAIR TENG AND MEI-CHUAN CHENGs234sDU(α)]. Then, we haves0 1 ( ),D D α α α ≤ ≤ = ∪s=-=(2)-=-sors( ) 0 1 ( ) ( ),DD x C xαα μ α ≤ ≤ = ⋅ Vs(3)swheres(i) αD(α) is a fuzzy set with membership functions( ) , ( ); ( ) 0, otherwise.D x D xα α α αμ ∈⎧= ⎨⎩s(ii) CD(α)(x) is a characteristic function ...

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...Lin.sLIANG-YUH OUYANG, JINN-TSAIR TENG AND MEI-CHUAN CHENGs232sitems by applying the fuzzy sets theory, and proposed the model with both a fuzzy defective rate and a fuzzy annual demand. Chang et al. =-=[6]-=- considered the mixture inventorysmodel involving variable lead time with backorders and lost sales. They fuzzified thesrandom lead-time demand to be a fuzzy random variable and the total demand to be...

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...ed articles on this topic can besfound in work by Chen and Wang [9], Vujosevic et al. [10], Gen et al. [11], Roy andsMaiti [12], Ishii and Konno [13], Lee and Yao [14], Yao and Lee [15], Chang et al. =-=[16]-=-,sChang et al. [17], Ouyang et al.[18], Yao et al. [19].sIn today’s business environment, trade credit plays an important role and the inventory problem associated with trade credit has become a popul...

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... temperature, humidity, and storage equipment. Therefore, in the real world, the inventory deterioration rate is not known withscertainty. Amongst extended studies on this topic, only Chen and Ouyang =-=[27]-=- havestreated interest rates as variable, wherein the authors extended the model of Jamal et al.s[28] by fuzzifying the carrying cost rate, interest paid rate and interest earned rate simultaneously. ...

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... are calculated based on the purchasingscost of the goods not yet sold. Teng [21] amended Goyal’s [20] model by calculatingsinterest earned based on the selling price of goods sold. In Chang and Teng =-=[22]-=-, thessuppliers offer cash discounts or delay payment to retailers. Within the permissible delaysperiod, retailers earn interest on sales revenue. Beyond the permissible period, interest isscharged fo...

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...sMultiple-occurrence of same fuzzy parameters in a fuzzy function will cause a fuzzysnumber been overestimated or illegal when the standard fuzzy arithmetic is applied tossolve the problem (see Chang =-=[32]-=-). In this paper, instead of the standard fuzzy arithmetic operations, we applied the interval operations to deal with our model. The intervalsoperations are adopted by many researchers to handle comp...

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... fuzzified the demand, lead time and inventory level into triangular fuzzy numbers in an inventory control model, and then determined the ordersquantity with the fuzzy propositions method. Yao et al. =-=[3]-=- investigated the EconomicsLot Scheduling Problem (ELSP) with fuzzy demands. They used the ‘Independent Solution’ as well as the ‘Common Cycle’ approach to solve the fuzzy ELSP problem. Yao etsal. [4]...

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... topic can besfound in work by Chen and Wang [9], Vujosevic et al. [10], Gen et al. [11], Roy andsMaiti [12], Ishii and Konno [13], Lee and Yao [14], Yao and Lee [15], Chang et al. [16],sChang et al. =-=[17]-=-, Ouyang et al.[18], Yao et al. [19].sIn today’s business environment, trade credit plays an important role and the inventory problem associated with trade credit has become a popular topic in the inv...

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... in work by Chen and Wang [9], Vujosevic et al. [10], Gen et al. [11], Roy andsMaiti [12], Ishii and Konno [13], Lee and Yao [14], Yao and Lee [15], Chang et al. [16],sChang et al. [17], Ouyang et al.=-=[18]-=-, Yao et al. [19].sIn today’s business environment, trade credit plays an important role and the inventory problem associated with trade credit has become a popular topic in the inventorysfield. A sup...

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...sand incorporating the time value of money. Other related articles on this topic can besfound in work by Chen and Wang [9], Vujosevic et al. [10], Gen et al. [11], Roy andsMaiti [12], Ishii and Konno =-=[13]-=-, Lee and Yao [14], Yao and Lee [15], Chang et al. [16],sChang et al. [17], Ouyang et al.[18], Yao et al. [19].sIn today’s business environment, trade credit plays an important role and the inventory ...

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...and Wang [9], Vujosevic et al. [10], Gen et al. [11], Roy andsMaiti [12], Ishii and Konno [13], Lee and Yao [14], Yao and Lee [15], Chang et al. [16],sChang et al. [17], Ouyang et al.[18], Yao et al. =-=[19]-=-.sIn today’s business environment, trade credit plays an important role and the inventory problem associated with trade credit has become a popular topic in the inventorysfield. A supplier usually per...