Single-Period Inventory Models with Discrete Demand Under Fuzzy Environment

J. of Mult.-Valued Logic & Soft Computing, Vol. 18, pp. 371 386 Reprints available directly from the publisher Photocopying permitted by license only 2012 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group Single-Period Inventory Models with Discrete Demand Under Fuzzy Environment Hülya Behret* and Cengi z Kahraman Industrial Engineering Department, Istanbul Technical University, Macka, 34367, Istanbul, TURKEY, E-mail: behreth@itu.edu.tr; kahramanc@itu.edu.tr Received: February 21, 2010. Accepted: November 26, 2010. This paper analysis single-period inventory models with discrete demand under fuzzy environment. In the proposed models three different cases are examined. In the first case, demand is represented by a triangular fuzzy number and a discrete membership function. In the second case, demand is a stochastic variable while inventory costs such as unit holding cost and unit shortage cost are imprecise and represented by fuzzy numbers. In the third case, both demand and inventory costs are imprecise. The objective of the models is to find the product s best order quantity that minimizes the expected total cost. The expected total cost that includes fuzzy parameters is minimized by marginal analysis and defuzzified by the centroid defuzzification method. Models are experimented with illustrative examples and supported by sensitivity analyses. Keywords: Inventory problem, fuzzy modeling, single-period, newsvendor, fuzzy demand, fuzzy inventory costs. 1 Introduction Single-period inventory problem can be defined as follows: A single order can be placed for an item before the beginning of the selling period. There is either no opportunity of placing any subsequent orders during the period, or there is a penalty cost per item for special orders placed during the period. The assumption of the single-period problem (SPP) is that if any inventory remains at the end of the period, either a discount is used to sell it or it is disposed of, and if an unsatisfied demand occurs, it results in a penalty cost. *Corresponding author: Tel: +90 212 2931300 (2670) Fax: +90 212 2407260 371

372 H. Behret et al. The objective of the single-period problem is to find product s order quantity that minimizes the expected total cost under linear purchasing, holding, and shortage costs and probabilistic demand. An extensive literature review on a variety of extensions of the singleperiod problem (or newsvendor problem) and related multi-stage, inventory control models can be found in [1] and [2]. Most of the extensions have been made in the probabilistic framework, in which the uncertainty of demand is described by probability distributions. The demand probability distribution is usually obtained from evidence recorded in the past. However, if there is not adequate evidence available, or evidence is recorded in different environments then the demand forecast will be based on subjective evaluations and linguistic expressions. When subjective evaluations are considered, the possibility theory takes the place of the probability theory [3]. The fuzzy set theory introduced by Zadeh [4], can represent linguistic data which cannot be easily modeled by other methods [5]. Furthermore, the methods based on the probability theory allow only quantitative uncertainties. In reality, most of the evaluations are imprecise and fuzzy and they cannot be quantified. Recently, interest in single-period problems under fuzzy environment has increased and many extensions to the newsvendor problem have been proposed [1]. A brief summary of the fuzzy single-period inventory models in the literature given in Table 1. Fuzzy single-period inventory model Fuzzy parameters Solution method Ishii and Konno [6] Shortage cost Fuzzy max (min) order Petrovic et al. [7] 1. Demand 2. Demand, holding cost and shortage cost Arithmetic defuzzification Li et al. [8] 1. Holding cost and shortage cost 2. Demand Fuzzy rank ordering by total integral value Kao and Hsu [9] Demand Ranking fuzzy numbers Dutta et al. [10] Demand Graded mean integration representation Dutta et al. [11] Demand Ordering of fuzzy numbers with respect to their possibilistic mean values Ji and Shau [12] Demand Hybrid intelligent algorithm based on fuzzy simulation Shau and Ji [13] Demand Hybrid intelligent algorithm based on fuzzy simulation Lu [14] Demand Centroid defuzzification Xu and Zhai [15] Demand Ordering of fuzzy numbers with respect to their possibilistic mean values TABLE 1 Summary of fuzzy single-period inventory control models.

Single-Period Inventory Models 373 Among the examined single-period inventory control models, the model proposed by Petrovic et al. [7] considers both imprecise demand and imprecise inventory costs. In the other models either demand or inventory costs are fuzzy. The solution methods of single-period inventory control models generally need either a defuzzification method or a ranking method for fuzzy numbers. In this study, single-period inventory control models are analyzed under three different fuzzy environments and the solutions of the models are compared with each other. The objective of these models is to find the best order quantity that minimizes the expected total cost. The expected total cost including fuzzy parameters is minimized by marginal analysis and defuzzified by centroid defuzzification method. The models are experimented with illustrative examples and supported by sensitivity analyses. The remainder of the paper is organized as follows. The single-period (newsvendor) problem is subjected in Section 2. In Section 3, firstly preliminary definitions about fuzzy modeling are presented and then three different fuzzy single-period inventory control models are described and the results of these models are compared with each other and sensitivity analyses are performed. Finally the paper is concluded in Section 4. 2 Single-period (newsvendor) problem In the literature, single-period inventory problems are known as newsvendor or newsboy problems. Such problems are associated with the inventory of items having one or more of the following characteristics [16]; They become obsolete quickly, e.g. newspapers, fashion goods etc. They spoil quickly, e.g. fruit, vegetable etc. They are seasonal goods where a second order during the season is difficult. They are stocked only once, e.g. spare parts for a single production run of products. They have a future that is uncertain beyond the planning horizon. The objective of the stochastic single-period (newsvendor) model is to determine the order quantity Q* for a fixed time period that will minimize the expected total cost. The expected total cost function is the combination of unit production cost, unit overage and unit underage costs. Items are purchased (or produced) for a single-period at the cost of c p. The holding cost which is the cost of storing excess products minus their salvage value is c h and the shortage cost which is the cost of lost sales due to the inability to supply the demand is c s. It is assumed that there is no initial inventory on hand. As we know in the stochastic single-period problem, demand is a random variable and represented by probability distributions. The total cost function

374 H. Behret et al. [TC (Q; X)] will be as follows, where Q represents order quantity and X stands for the demand; TC( QX ; )= cq+ c {( Q- X), 0}+ c ( X-Q), 0 max max{ } p h s Production cost Overage cost (OC) Underage cost (UC) (1) The expected total cost in the discrete case is; ETCQX [ ( ; )] = c Q+ c ( Q- x ) p ( x ) p Q-1 x0 = 0 + c ( x -Qp ) ( x ) x = Q s 0 X 0 0 h 0 X 0 (2) where p x (x 0 ) is the probability that the demand X is equal to the value x 0. Let ETCQX [ ( ; )] = E[ TC( Q+ 1 ; X)] -ETCQX [ ( ; )] (3) Then, E[ TCQX ( ; )] is the change in expected total cost when we switch from Q to Q + 1. For a convex cost function, the best Q will be the lowest Q where E[ TCQX ( ; )] is greater than zero. Therefore, we select the smallest Q for which, ETCQX [ ( ; )] 0 (4) The equation above holds if, ETCQ [ ( + 1; X)] -E[ TC( QX ; )] 0 (5) Substituting Equation (2) into Equation (5) leads to; c ( Q+ 1) + c ( Q+ 1-x ) p ( x ) c p p p + X x Q x0 = 0 h 0 X 0 + c ( x -Q-1) p ( x )-[ c Q x0 = Q+ 1 s 0 X 0 p Q-1 + c ( Q-x ) p ( x ) Q c 0 = 0 h 0 X 0 c x Q s ( x 0 = 0 Qp ) X ( x0) ] 0 + - p ( x )- cp ( x ) 0 x = h 0 0 X 0 x 1 s 0 0 = Q+ X cs c ( Q) - c + c h p s (6) where px ( Q) is the probability that the demand X is smaller or equal to the order quantity Q. The expected total cost, E[TC(Q;X)] will be minimized by the smallest value of Q (call it Q*) satisfying the equation above.

Single-Period Inventory Models 375 Demand Probability 1,000 0 2,000 0.0625 3,000 0.125 4,000 0.1875 5,000 0.25 6,000 0.1875 7,000 0.125 8,000 0.0625 9,000 0 TABLE 2 Probability distribution of demand. Let the demand of a product has the probability distribution represented in Table 2. Items are produced for a single-period at the cost of c p = $4. The holding cost is c h = $3 and the shortage cost is c s = $6. It is assumed that there is no initial inventory on hand. For the given parameters optimum order quantity is found as 4,000 from Equation (6) and the expected value of total cost for the best order quantity is found as $24,250. 3 Single-period inventory models under fuzzy environment The fuzzy set theory provides a proper framework for description of uncertainty related to vagueness of natural language expressions and judgments. In this section, firstly preliminary definitions about fuzzy modeling are presented and then three different fuzzy single-period inventory control models are developed and the solutions of the models are compared with each other. 3.1 Preliminaries In this section, some introductory definitions of the fuzzy set theory are presented. Our models are based on these definitions. Definition 1: Fuzzy Sets [4] Let X be a classical set of objects, called the universe, whose generic elements are denoted by x. Membership in a classical subset A of X is often viewed as a characteristic function, µ A from X to {0,1} such that

376 H. Behret et al. iff x A µ A ( x)= 1 0 iff x A (7) If the valuation set ({0,1}) is allowed to be the real interval [0,1], A is called a fuzzy set, µ A (x) is the grade of membership of x in A. The closer the value of µ A (x) is to 1, the more x belongs to A. A is completely characterized by the set of pairs. A= {( x, µ ( x) ), x A} (8) A Definition 2: Fuzzy Numbers [5] Fuzzy numbers are a particular kind of fuzzy sets. A fuzzy number is a fuzzy set R of the real numbers set with a continuous, compactly supported, and convex membership function. Let U be a universal set; a fuzzy subset à of X is defined by a function µ à (x): X [0,1] is called membership function. Here, X is assumed to be the set of real numbers R and F the space of fuzzy sets. The fuzzy set à F is a fuzzy number iff: α α [ 01, ] the set A = { x R: µ ( x) α}, which is called α-cut of Ã, à is a convex set. I. µ Ã(x) is a continuous function. II. sup(ã) = {x R: µ à (x) 0} is a bounded set in R. III. height à = max x X µ à (x) = h 0. By conditions (I) and (II), each a-cut is a compact and convex subset of R hence it is a closed interval in R, A a = [A L (a); A R (a)]. If h = 1 we say that the fuzzy number is normal. For example, the fuzzy number à is a triangular fuzzy number à = (a 1 ; a 2 ; a 3 ), a 1 a 2 a 3 if its membership function µ à (x): R [0,1] is equal to as follows; µ à ( x) 0 x a1 x-a 1 a1 < x a2 a -a 2 1 = a - x 3 a < x a a3- a 2 0 x> a3 2 3 (9) The graphical representations of symmetrical and non-symmetrical triangular membership functions are shown in Figures 1.(a) and 1.(b). Definition 3: Possibility measure [3] A possibility measure is a function from P(X) to [0,1] such that

Single-Period Inventory Models 377 FIGURE 1. (a) symmetrical and (b) non-symmetrical triangular membership functions. I. Π( ) = 0; Π( X) = 1; II. For any collection { A i } of subsets of X, ( i A i ) = sup i ( (A i )). A possibility measure can be built from a possibility distribution, i.e., a function from X to [0,1] such that sup x X ( (x)) = 1 (normalization condition). More specifically, we have A, Π( A) = sup x AΠ( x) (10) Definition 4: Level-k fuzzy set [17] The term level-2 fuzzy set indicates fuzzy sets whose elements are fuzzy sets (See Figures 2.(a) and 2.(b)). The term level-1 fuzzy set is applicable to fuzzy sets whose elements are (no fuzzy sets) ordinary elements. In the same way, we can derive up to level-k fuzzy set. Definition 5: Defuzzification [18] Defuzzification is the conversion of a fuzzy quantity to a precise quantity; in contrast fuzzification is the conversion of a precise quantity to a fuzzy quantity. Usually, a fuzzy system will have a number of rules that transform a number of variables into a fuzzy result. Defuzzification would transform this result into a single number. Centroid method (also called center of area or center of gravity method) is the most common of all the defuzzification methods. It is given by the algebraic expression as follows; z * z = * µ. zdz. C( Z ) µ =. dz C( Z ) µ. z C( Z ) µ C( Z ), for continuous functions (11), for discrete functions (12)

378 H. Behret et al. FIGURE 2. (a) level-2 fuzzy set, (b) elements of level-2 fuzzy set A1, A2 and A3 where k C C k = i =1 i and C i is one of the membership functions those figure the fuzzy output. This method is represented in Figure 3. Definition 6: S-fuzzification [19] Let à be a level-2 fuzzy set and let à takes fuzzy values C ( x ), x i X, i = 12,,, n with the possibility µ Ã(i). à can be transformed into ordinary set s - fuzz(ã) using the s-fuzzification; µ ( x ) = supp µ ()* i µ ( x ), x X 123 (13) s- fuzz( Ã) i=,,,, n à ci 3.2 Fuzzy Single-Period Inventory Control Model with Imprecise Demand Consider a single-period inventory problem.the demand is a trğinagular fuzzy number X (see Section 3.1) given by domain X = { x 0; x 1; x 2 ; ; xn } with membership function µ x ( x i ), i= 012,,,, n. Unit production cost, (c p ), unit holding cost (c h ) and unit shortage cost (c s ), are precise in this model. The uncertain demand causes uncertain overage and underage costs. For a given Q and x i X, the fuzzy total cost is as follows;

Single-Period Inventory Models 379 FIGURE 3. Centroid defuzzification method TC( QX ; ) = cq p + chmax{( Q- X ), 0} + csmax{( X -Q), 0 } (14) Production cost verage cost OC Underage cost UC The membership functions of OC and UC are the same as the membership function of demand and according to the properties of possibility measure (see Section 3.1), µ OC ~ + UC ~ (x i ) is obtained as follows; µ OC ~(x i ) = µ UC ~(x i ) = µ X ~(x i ) µ OC ~ +UC ~(x i ) = max xi X ~{µ X ~(x i )}, i = 0,1,2,, n The expected value of fuzzy total cost in the discrete case is; E[ TCQX ( ; )] = cpq+ defuzz( OC+ UC) n [( O xi = 0 Cx ( i) + UC( xi))* µ ( x OC UC i )] + E[ TCQX ( ; )] = cpq + n [ µ ( x )] x OC UC i i = 0 + (15) (16) Here, the operator defuzz denotes the centroid method for defuzzification, (see Section 3.1). Best order quantity (Q*) which minimizes the fuzzy total cost is found by marginal analysis. The best (Q*) will be the lowest Q where E[ TCQX ( ; )] is greater than zero. Therefore, we select the smallest Q from the set {x 0 ; x 1 ; x 2 ; ; x n } for which, ETCQX [ ( ; )] > 0 (17) Let the unit inventory costs are considered as precise, c p = $4, c h = $3 and c s = $6. The demand is a triangular fuzzy number X given by domain X = {1,000; 2,000; 3,000;,; 9,000}and have a discrete membership function µ X ~ (x i ) as follows;

380 H. Behret et al. 0 xi 1, 000 xi -1, 000 1, 000 < xi 5, 000 4, 000 ( )= 90, 00- x i 5, 000 < x 9 000 4 000 i,, 0 xi > 9, 000 µ X xi (18) For example, let us order a quantity of 4,000 units (Q = 4,000). For x 2 = 2,000, OC = $3 * 2,000 = $6,000 and UC = $6 * 0 = $0 with possibility of 0.25, for x 6 = 6,000, OC = $3 * 0 = $0 and UC = $6 * 2,000 = $12,000 with possibility of 0.75 and so on. The possibility distribution of (OC + UC ) is represented in Table 3. For example, (OC + UC ) is 6,000 for both x2 and x 5 with possibility 0.25 and 1 respectively. From the properties of possibility measure (see Section 3.1), the possibility of (OC + UC ) = 6,000 will be µ OC ~ +UC ~(6,000) = 1. The defuzzified value of fuzzy overage and underage costs is; defuzz (OC + UC ) = $8,400 (from Equation (16)) and the expected value of fuzzy total cost is, E[ TC (4,000; X )] = $24,400. The same procedure is applied for other order quantities, best order quantity (Q*) which minimizes the expected value of fuzzy total cost is found by marginal analysis. 3.3 Fuzzy Single-Period Inventory Control Model with Stochastic Demand and Imprecise Inventory Costs In this model demand is a stochastic variable with probability function p X (x i ), while inventory costs such as holding and shortage cost are imprecise and represented by fuzzy numbers (see Section 3.1). The membership functions of OC and UC are the same as the membership function of holding and shortage cost respectively. µ ( x ) = µ µ ( x i ) = µ c (19) OC i ch UC s The expected value of fuzzy total cost is; n E[ TCQX ( ; )] = cpq+ ( defuzz( OCx ( i) * px( xi) ) x i = 0 + defuzz( UC( x )* p ( x )) i X i (20) OC + UC 0 3,000 6,000 9,000 12,000 18,000 24,000 30,000 µ OC ~ + UC ~ 0.75 0.5 1 0 0.75 0.5 0.25 0 TABLE 3 Possibility distribution of ( OC + UC ).

Single-Period Inventory Models 381 Here, as in Section 3.2 best order quantity (Q*) which minimizes the fuzzy total cost is found by marginal analysis. Consider that the demand of a product has the probability distribution represented in Table 2 and the inventory costs such as holding and shortage cost are considered as imprecise and represented by triangular fuzzy numbers, c ~ h = $ (2; 3; 4) and c ~ s= $ (2; 3; 4), respectively. Items are produced for a single-period at the cost of c p = $4. As an example, let us order a quantity of 4,000 units (Q = 4,000). For x 2 = 2,000, OC = $(2; 3; 4) * 2,000 = $ (4000; 6000; 8000) and UC = $ (5; 6; 7) * 0 = $0. The probability of the demand X is equal to the value 2,000 is p x (2,000) = 0.0625. The addition of the defuzzified values is; defuzz (OC (2,000) * p x (2,000)) + defuzz (UC (2,000) * p x (2,000)) = 375. The defuzzified values for other demand parameters are found by the same way. From Equation (20), the expected value of fuzzy total cost is obtained as E[TC (4,000; X )] = $24,250. The same procedure is applied for all order quantities to find the best order quantity (Q*) which minimizes the expected value of fuzzy total cost. 3.4 Fuzzy Single-Period Inventory Control Model with Imprecise Demand and Imprecise Inventory Costs This model considers both imprecise demand and imprecise inventory costs. As in the first case the demand is a fuzzy number X given by domain X = {x 0 ; x 1 ; x 2 ; ; x n } which has a membership function µ X ( x i ), i = 012,,,, n. Additionally, holding and shortage costs are imprecise and represented by fuzzy numbers. The uncertain demand and uncertain inventory costs cause uncertain overage and underage costs. The unit penalty cost (PC ) is the sum of unit overage cost and unit underage cost with the membership function µ PC ~. PC = OC + UC (21) The unit penalty cost (PC ) is a level-2 fuzzy set (see Section 3.1) which means that it contains two fuzzy values and there are corresponding membership degrees of these fuzzy values. A level-2 fuzzy set can be reduced to an ordinary fuzzy set by s-fuzzification process (see Section 3.1). The membership function of an ordinary fuzzy set is maintained via s-fuzzification as follows; µ ( x) = supp µ () i * µ 123 ( x), x X (22) s- fuzz( PC ) i=,,,, n PC ci where c ~ i(x) is the i th possible fuzzy cost of PC and µ PC ~(i) is the possibility of that cost. According to the properties of possibility measures, µ PC ~(i) is obtained as, µ () i = maxx Xµ X ( xi ), i= 123,,,, n (23) PC i The expected value of fuzzy total cost in the discrete case is;

382 H. Behret et al. E[ TC( Q; X )] = c * Q+ defuzz( s- fuzz( PC)) (24) p Here, s-fuzzified penalty cost is defuzzified via centroid method. Furthermore best order quantity (Q*) is found by the marginal analysis given in Section 3.2. Let the demand ( X ), given by domain X = {1,000; 2,000; 3,000; ; 9,000} represented by a triangular membership function µ X ~(x i ) given by Equation (18). The holding and shortage costs are imprecise and represented by triangular fuzzy numbers, c h ~ = $ (2;3;4) and cs ~ = $ (5;6;7), respectively. Items are produced for a single-period at the cost of c p = $4. For example, let us order a quantity of 4,000 units. The unit penalty cost (PC ) is a level-2 fuzzy set including imprecise demand and costs. For x 2 = 2,000, the fuzzy penalty cost will be PC = $(2;3;4) * 2,000 = $(4,000; 6,000; 8,000) with possibility of 0.25. For x 6 = 6,000, the fuzzy penalty cost will be PC = $(5;6;7) * 2,000 = $(1,000; 12,000; 14,000) with the possibility 0.75 and so on. Fuzzy unit penalty cost values for Q = 2,000 is given in Table 4. The graphical representations of level-2 fuzzy sets of PC when Q = 4,000 and the corresponding s-fuzzified set (s - fuzz(pc )) are shown in Figure 4.(a) and Figure 4.(b), respectively. According to the s-fuzzified value of the penalty cost, the expected value of fuzzy total cost when Q = 4,000 is calculated by using Equation (24). Centroid defuzzification values have been obtained by using MATLAB R2008a Fuzzy Logic Toolbox as in Figure 5. E[ TC (4,000; X )] = $4 * 4,000 + $13,498 = $29,498 (25) The same procedure is applied for all order quantities and best order quantity (Q*) which minimizes the expected value of fuzzy total cost. x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 X ~ 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 µ X~ (x i ) 0 0.25 0.5 0.75 1 0.75 0.5 0.25 0 OC (6,000; 9,000; 12,000) (4,000; 6,000; 8,000) (2,000; 3,000; 4,000) UC (0; 0; 0) (0; 0; 0) (0; 0; 0) (0; 0; 0) PC (6,000; 9,000; 12,000) (4,000; 6,000; 8,000) (2,000; 3,000; 4,000) TABLE 4 Unit penalty cost values for Q = 2,000. (0; 0; 0) (0; 0; 0) (0; 0; 0) (0; 0; 0) (0; 0; 0) (0; 0; 0) (0; 0; 0) (5,000; 6,000; 7,000) (5,000; 6,000; 7,000) (10,000; 12,000; 14,000) (10,000; 12,000; 14,000) (15,000; 18,000; 21,000) (15,000; 18,000; 21,000) (20,000; 24,000; 28,000) (20,000; 24,000; 28,000) (25,000; 30,000; 35,000) (25,000; 30,000; 35,000)

Single-Period Inventory Models 383 FIGURE 4 (a) level-2 fuzzy set (PC ), (b) s - fuzz(pc ) FIGURE 5 Centroid defuzzification of s - fuzz(pc). 3.5 Comparison of the models In the previous sections four different single-period models have been analyzed. We call them as follows;

384 H. Behret et al. Model-I: Model-II: Classical stochastic single-period (newsvendor) model, Fuzzy single-period inventory control model with imprecise demand, Model-III: Fuzzy single-period inventory control model with stochastic demand and imprecise inventory costs, Model-IV: Fuzzy single-period inventory control model with imprecise demand and imprecise inventory costs. In this section, we experiment the models for all order quantities under given parameters in the previous sections and compare the results with eachother to have a better understanding of the difference between crisp and fuzzy models. In these experiments, for all of the models, we consider that items are produced for a single-period at the cost of c p = $4. For the models I and II, the crisp values of holding and shortage costs are considered as c ~ h = $3 and c ~ s = $6. For the models I and III, the demand has the probability distribution represented in table 2. In the models III and IV, inventory costs are imprecise and represented by triangular fuzzy numbers, c ~ h = $(2;3;4) and c ~ s = $(5;6;7). Additionally, in the models II and IV, the demand is given by domain X = {1,000; 2,000; 3,000; ; 9,000} and has a discerete triangular membership function µ X ~(x i ) as in Equation (18). The comparison of the results of experimented models for the given parameters are presented in Table 5. The minimum total cost values for the models are found by marginal analysis. The order quantities corresponding to the minimum total cost values are the best order quantities (Q*) for the related model. When we analyze the results of the models, we observe that the results of model-i and model-iii are the same for all order quantities. In model-iii, we consider stochastic demand Q Model-I Model-II Model-III Model-IV TABLE 5 Comparison of the results. E[TC (Q;X)] E[TC (Q;X ~ )] E[TC (Q;X)] E[TC (Q;X ~ )] 1,000 28,000 28,000 28,000 31,098 2,000 26,000 26,000 26,000 30,547 3,000 24,563 24,563 24,563 29,982 4,000 24,250* 24,400* 24,250* 29,498 5,000 25,625 25,571 25,625 29,231* 6,000 29,250 28,615 29,250 31,166 7,000 34,563 34,600 34,563 36,928 8,000 41,000 41,000 41,000 42,998 9,000 48,000 48,000 48,000 49,391 * minimum total cost values.

Single-Period Inventory Models 385 Q 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 E[TC (Q;X)] 26,667 25,000 23,896* 23,917 25,625 29,583 35,229 42,000 49,333 * minimum total cost value TABLE 6 Results for revised Model-III. and imprecise inventory costs which are represented by triangular fuzzy numbers. The triangular fuzzy numbers which are used in model-iii have symmetrical shapes (see Figure 1.(a)). When we defuzzify these numbers by the centroid method, we obtain average values of these numbers which are equal to the values of crisp costs. Therefore, we have the same results with model-i. However, if we use non-symmetrical fuzzy numbers (see Figure 1.(b)) then the defuzzified values of these numbers will be different from the values of crisp costs and the total cost values will also be different for these models. To analyze this situation, we use the following fuzzy numbers for unit holding and unit shortage costs in model-iii; c ~ h = $(2;3;5) and cs ~ = $(4;6;7) and. The results for the revised model-iii are given below (Table 6). When we compare the results of revised model-iii with our other models, we observe that the expected values of total costs vary from one model to another. By changing the shapes of fuzzy triangular numbers in the revised model-iii, we increased the value of unit holding cost and decreased the value of unit shortage cost. Therefore, best order quantity decreased to 3,000 in the revised model. This situation shows that contrary to the crisp model, fuzzy models propose highly flexible solutions for all possible states. 4 Conclusion This paper proposes single period inventory models with discrete demand under fuzzy environment. In the proposed models inventory costs, demand and both inventory costs and demand are imprecise, respectively. The objective of the models is to find the product s best order quantity that minimizes the expected total cost. The expected total cost that includes fuzzy parameters is minimized by marginal analysis and defuzzified by centroid defuzzification method. Contrary to the crisp model, fuzzy models propose highly flexible solutions for all possible states. The proposed fuzzy models operate with both precise and imprecise data. The developed models could be modified for solving similar inventory problems such as inventory replenishment models. For further research, we suggest the examination of an imprecise continuous demand function instead of the discrete case of this paper. This will require optimization techniques for solution procedure. Furthermore, in the fuzzy models, we can also increase or decrease the fuzziness of the imprecise

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