Inventory policies in a fuzzy uncertain supply chain ... - Semantic Scholar
European Journal of Operational Research 197 (2009) 609–619
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
ðQ ; rÞ Inventory policies in a fuzzy uncertain supply chain environment Robert Handfield a, Don Warsing a, Xinmin Wu b,* a b
Department of Business Management, North Carolina State University, Raleigh, NC 27695-7229, USA Operations Research Graduate Program, Campus Box 7913, North Carolina State University, Raleigh, NC 27695-7913, USA
a r t i c l e
i n f o
Article history: Received 11 March 2006 Accepted 11 July 2008 Available online 29 July 2008 Keywords: ðQ ; rÞ System Inventory Fuzzy sets Optimization
a b s t r a c t Managers have begun to recognize that effectively managing risks in their business operations plays an important role in successfully managing their inventories. Accordingly, we develop a ðQ ; rÞ model based on fuzzy-set representations of various sources of uncertainty in the supply chain. Sources of risk and uncertainty in our model include demand, lead time, supplier yield, and penalty cost. The naturally imprecise nature of these risk factors in managing inventories is represented using triangular fuzzy numbers. In addition, we introduce a human risk attitude factor to quantify the decision maker’s attitude toward the risk of stocking out during the replenishment period. The total cost of the inventory system is computed using defuzzification methods built from techniques identified in the literature on fuzzy sets. Finally, we provide numerical examples to compare our fuzzy-set computations with those generated by more traditional models that assume full knowledge of the distributions of the stochastic parameters in the system. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Managers have begun to recognize that effectively managing risks in their business operations plays an important role in successfully managing their inventories. One of the most common risks is demand uncertainty, a phenomenon that is widely studied in the literature. Another risk that has attracted significant attention is supply uncertainty, especially the risk associated with direct material supplies and deliveries. This type of risk has escalated as Fortune 500 companies have sourced a great proportion of products from areas of the globe with low labor costs, such as China and India. Frequently, the hidden perils of such sourcing strategies are not fully considered (Elkins et al., 2005). There is an abundant literature that models uncertainty in demand and/or lead time using probability distributions with known parameters. However, in many cases where there is little or no historical data available to the inventory decision maker, perhaps due to recent changes in the supply chain (SC) environment, probability distributions may simply not be available, or may not be easily or accurately estimated (Xie et al., 2006). Additionally, in some cases, it may not be possible to collect data on the random variables of interest because of certain system or time constraints. Furthermore, other critical supply chain parameters, in particular the various costs that impact the system, are often ill-defined and may vary from time to time. All of these situations raise challenges for using traditional inventory models in practice. * Corresponding author. Tel.: +1 919 607 6648. E-mail address: [email protected] (X. Wu). 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.07.016
Fuzzy theory provides an alternate, flexible approach to handle such situations because it allows the model to easily incorporate various subject experts’ advice in developing critical parameter estimates (Zimmermann, 2001). Recent research has employed fuzzy logic in modeling demand or lead time uncertainty in inventory systems. For example, Pai and Hsu (2003) apply fuzzy-set theory to continuous review reorder point problems, with the assumption that uncertainties may appear not only in demand over the lead time, but also in inventory holding costs. They obtain optimal ðQ ; rÞ policies that account for these uncertainties by computing derivatives of a Type-2 fuzzy-set of total cost. Petrovic et al. (2001) develop a fuzzy model for the newsvendor problem in an uncertain environment, where inventory cost and demand are represented by fuzzy sets. Vujosevic et al. (1996) develop an alternative EOQ formula that accounts for fuzzy inventory costs and demand parameters. This solution approach is extended in Mondal and Maiti (2002) to a multi-item fuzzy EOQ system by using a genetic algorithm. Chang (1999) derives a membership function of the total cost and EOQ in a production-inventory problem where the production quantity is a triangular fuzzy number. In a broader setting, Wang and Shu (2005) develop a fuzzy decision methodology that provides an alternative framework to handle SC uncertainty and to determine inventory management strategies for a SC network with a parameter, q ð0 6 q 6 1Þ, to describe the managerial (human) attitude toward risk. Also in a SC setting, Xie et al. (2006) develop a hierarchical, two-level coordination method to compute order-up-to policies in SCs under fuzzy demand. In their model, the entire SC system is decomposed into independent sub-systems optimized at the ‘‘follower level,” while
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coordination is implemented at the ‘‘leader level.” SC simulation models have also been developed by Petrovic et al. (1998, 1999) in a fuzzy uncertain environment for demand and material supply, using base-stock control policies. Finally, Yimer and Demirli (2004) describe the performance achieved by a fuzzy inventory control simulation model in a continuous review inventory system. However, to the best knowledge of the authors, no research has simultaneously considered multiple fuzzy risk factors in ðQ ; rÞ inventory policies while also incorporating the decision maker’s risk attitude (i.e., the q factor introduced by Wang and Shu, 2005). In this paper, we present a ðQ ; rÞ system that accounts for the typical imprecision in modeling demand, lead time, supply yield, penalty cost, and inventory holding costs. Since fuzzy sets are based on human subjective opinion, however, the decision maker’s attitude towards risks driven by uncertainty affects the decision process and is, therefore, clearly worthy of study. We study the effects of three levels of risk attitude – pessimistic, neutral, and optimistic – extended from Wang and Shu (2005), and we provide corresponding managerial insights from this analysis. The rest of the paper is organized as follows: In Section 2, we present several important fuzzy set-related definitions and develop important propositions of fuzzy arithmetic to be used in our ðQ ; rÞ computations. In Section 3, we present our approach to modeling SC uncertainty using fuzzy sets, and we provide our model assumptions and notations. We derive a fuzzy total cost function based on fuzzy arithmetic rules, and use a direct search algorithm to find optimal solutions. In Section 4, we apply our fuzzy computational scheme to a numerical example adapted from a popular textbook, and we compare these results to those generated by the traditional computational approach that assumes known probability distributions with known parameters. In addition, we conduct a sensitivity analysis, accounting for different managerial risk attitudes toward stocking out. Finally, we provide some concluding remarks in Section 5.
The fuzzy-set in the above definition is called a Type-1 fuzzyset, whose membership function is determined. When an exact membership function is difficult to determine, however, we need to use the concept of Type-2 fuzzy-set, which is used to handle and measure additional uncertainty. A Type-2 fuzzy-set is actually a fuzzy-set of a fuzzy-set, defined as follows: e ¼ fðx; l Þg is a Type-2 fuzzy-set if Definition 2.2. P e P
leP ðxÞ ¼ fðui ; lui ðxÞÞjx 2 X; ui ; lui ðxÞ 2 ½0; 1g;
where li is the degree of membership of a Type-1 fuzzy-set and lui ðxÞ is its membership function (Zimmermann, 2001). Note that here we only give the definition when the element of the membership function is a fuzzy-set, but we can also define it when x is a fuzzy-set. In order to obtain a scalar from the fuzzy domain that most appropriately describes a fuzzy-set, we need to use defuzzification functions. There are many defuzzification heuristics such as extreme value (EV), center of area (COA), and center of gravity (COG) defuzzification (Zimmermann, 2001). In this paper, we adopt COG defuzzification as it has a distinct geometrical meaning and strong probability analogy. The definition is given as follows: e is defined by Definition 2.3. COG defuzzification of P
R xl ðxÞ dx e defuzzð PÞCOG ¼ R P lP ðxÞ dx
ð2Þ
(Zimmermann, 2001). Next, we give the definition for the a-level cut of a fuzzy-set, and then use it in the definition of triangular fuzzy number arithmetic. e is defined as Definition 2.4. The a-level cut of fuzzy-set P
e a ¼ fx 2 Xjl ðxÞ P ag; P e P
where 0 6 a 6 1
ð3Þ
2. Fuzzy-set definitions and propositions
(Zimmermann, 2001).
For ease of exposition in this paper, we present some important definitions and propositions that will be used in the following sections.
e be fuzzy numbers with membership Definition 2.5. Let f M and N functions l e ðxÞ and le ðyÞ, respectively, where x; y 2 R. Let } be M N a binary operation, i.e., } : R}R ! R. The operation } can be extended to the fuzzy domain by the following formula:
Definition 2.1. If X is a collection of objects denoted generically by e on X is a set of order pairs x, then a fuzzy-set P
e ¼ fðx; l Þjx 2 Xg; P e
leP ðxÞ is a membership function and leP ðxÞ P 0. If max
e is called a fuzzy number (Zadeh, 1978). ðle ðxÞÞ ¼ 1, P P
If x is continuous on X, then the fuzzy-set is continuous and the e Continuous fuzzy range of x is called the support of fuzzy-set P. sets can have various membership function shapes such as triangular and trapezoidal. In this paper, we use triangular fuzzy numbers, a special case of trapezoidal, due to its straightforward structure and computational simplicity. Throughout the paper, unless otherwise indicated, we use a bold capital letter with a wide tilde (e.g., e to represent a generic fuzzy-set and a non-bold capital letter P) e to denote a triangular fuzzy number. with a wide tilde (e.g., P) e ¼ ða; c; bÞ. Its memberFig. 1a shows a triangular fuzzy number P ship function, leðxÞ, can be written as follows: P
8 0 > > > > < x a ca leP ðxÞ ¼ ca x b > þ bc > bc > > : 0
ð4Þ
(Kaufmann and Gupta, 1988).
P
where
l Me }eN ðzÞ ¼ sup min l Me ðxÞ; leN ðyÞ z¼x}y
if x 6 a; if a < x 6 c; if c < x 6 b; if x > b:
ð1Þ
By applying the definition of an a-level cut to (4) (see Fig. 2), we have the following exact arithmetic operations for continuous fuzzy sets:
f e a ¼ ½ma ; ma þ ½na ; na ¼ ½ma þ na ; ma þ na ; Ma þ N 1 2 1 2 1 1 2 2 e a ¼ ½ma ; ma ½na ; na ¼ ½ma na ; ma na ; f Ma N 1 2 1 2 1 2 2 1 f e a ¼ ½ma ; ma ½na ; na ¼ ½ma na ; ma na ; Ma N 1 2 1 2 1 1 2 2
ð5Þ
e a ¼ ½ma ; ma ½na ; na ¼ ½ma na ; ma na : f Ma N 1 2 1 2 1 2 2 1 For triangular fuzzy numbers, fuzzy arithmetic can be simplie ¼ ða2 ; c2 ; b2 Þ. It can be easily proved e ¼ ða1 ; c1 ; b1 Þ and N fied. Let M (see, e.g., Kaufmann and Gupta, 1988) that
e þN e ¼ ða1 þ a2 ; c1 þ c2 ; b1 þ b2 Þ; M e N e ¼ ða1 b2 ; c1 c2 ; b1 a2 Þ; M e ¼ ðsa1 ; sc1 ; sb1 Þ; where s is a crisp number: sM For multiplicative, inverse, and division operations, similar triplets cannot be used, since the resulting membership function is no
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e (b) Parabolic-triangular fuzzy number with parabolic membership function P e. Fig. 1. (a) Triangular fuzzy number P.
e (b) a cut of N. e Fig. 2. (a) a cut of M.
longer linear over its left and right segments (see Fig. 1b), and this could produce erroneous results, especially when multiplication and division operands are used repeatedly (Wang and Shu, 2005; Giachetti and Young, 1997). For the purpose of differentiation, however, we add a star superscript to denote a parabolic-triangular fuzzy number with parabolic membership function. For example, e ¼ ða; c; bÞ represents a parabolic-triangular fuzzy number over P support ½a; b (See Fig. 1b). By exact methods, operations among multiple triangular fuzzy numbers can be carried out based on (5) because the arithmetic has been converted to arithmetic operations on a certain membership interval from fuzzy triangular numbers. Membership functions of the product and division of two triangular fuzzy numbers are given in the following propositions, which are obtained by interval multiplication/divison and inverse function formulas. e ¼ ða2 ; c2 ; b2 Þ. Then, the e ¼ ða1 ; c1 ; b1 Þ and N Proposition 2.6. Let M e N e is given by membership function of M
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > < e2 þe6 þ ðe2 þe6 Þ þ e1 e5 x e2 e6 2 4 l Me eN ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : e4 þe8 þ ðe4 þe8 Þ2 þ e e x e e 3 7 4 8 2 4
if a1 a2 6 x 6 c1 c2 ; if c1 c2 6 x 6 b1 b2 ; ð6Þ
( e2 e7 e1 e8 x
l Me eN ðxÞ ¼
if a1 =b2 6 x 6 c1 =c2 ;
e7 e1 x e4 e5 e3 e6 x e5 e3 x
if c1 =c2 6 x 6 b1 =a2 ;
ð7Þ
where
e1 ¼ ðc1 a1 Þ1 ; 1
e2 ¼ a1 ðc1 a1 Þ1 ; 1
e4 ¼ b1 ðb1 c1 Þ ;
e5 ¼ ðc2 a2 Þ ;
e7 ¼ ðb2 c2 Þ1 ;
e8 ¼ b2 ðb2 c2 Þ1 :
e3 ¼ ðb1 c1 Þ1 ; e6 ¼ a2 ðc2 a2 Þ1 ;
Proof. The proof is straightforward from Definition 2.5. h To measure the possibility that a fuzzy-set belongs to another fuzzy set, we need to introduce the definitions of possibility and necessity measures. The definitions are given as follows: e in the Definition 2.8. Suppose x is restricted by a fuzzy-set B universe U. Further, suppose that the possible distribution of x, px , is taken to be equal to membership function le . Then, the B e is defined as possibility of event x 2 P
e ¼ sup min l ðuÞ; l ðuÞ : q x2P e e u2U
B
P
ð8Þ
e is The dual measure of possibility, i.e., the necessity of event x 2 P, defined as
where
e1 ¼ ðc1 a1 Þ1 ;
e ¼ ða1 ; c1 ; b1 Þ and N e ¼ ða2 ; c2 ; b2 Þ. Then, the Proposition 2.7. Let M e N e is given by membership function of M
e2 ¼ a1 ðc1 a1 Þ1 ;
e3 ¼ ðb1 c1 Þ1 ;
e4 ¼ b1 ðb1 c1 Þ1 ;
e5 ¼ ðc2 a2 Þ1 ;
e6 ¼ a2 ðc2 a2 Þ1 ;
e7 ¼ ðb2 c2 Þ1 ;
e8 ¼ b2 ðb2 c2 Þ1 :
Proof. The proof is straightforward from Definition 2.5. h
e ¼ inf max 1 l ðuÞ; l ðuÞ N x2P e e u2U
B
P
ð9Þ
(Yager, 1979; Wang and Shu, 2005). e 6 RÞ represents the maxSuppose R is a crisp number, then qð B e is less than R, and Nð B e 6 RÞ imum likelihood of the event that B
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e 6 R will ocestimates the minimum likelihood of the event that B cur. By Definition 2.8, we have
e 6 R ¼ qðx 2 ð1; RÞ ¼ sup l ðuÞ ; q B eB u6R e 6 R ¼ Nðx 2 ð1; RÞ ¼ inf 1 l ðuÞ ; N B e B u>R e P R ¼ qðx 2 ½R; þ1ÞÞ ¼ sup l ðuÞ ; q B e B uPR e P R ¼ Nðx 2 ½R; þ1ÞÞ ¼ inf 1 l ðuÞ : N B e B u ðQe S=2þrjÞlA~ ðjÞ < j¼1P r
> :
l
~ ðjÞ j¼1 A
L
ðn þ rÞ if n 2 f1; 2; . . . ; maxf0; d3 l3 rgg: > : le D eL
ð17Þ
if n ¼ 0; if n > 0:
n
u is part of inventory holding cost calculated based on the lowest cycle inventory level. If it is positive, we add the safety stock. If there is a stock-out, we exclude the portion of time associated with the out-of-stock condition (Zipkin, 2000) and we assume full backlogging of demand, i.e., that no sales are lost. Other cost components in (18) include the purchasing cost for received goods, S; the handling cost for rejected items to be returned, u1 Q e e SÞ; and the penalty cost n P. u2 Q ð1 e By Definition 2.5, the parabolic-triangular membership function of e F ðr; Q ; nÞ can be written as
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > < q2 þq3 ðxhuÞ ðq2 þq3 ðxhuÞÞ 4q1 ðq4 þq5 ðxhuÞÞ if D1 6 x 6 D2 ; 2q1 leF ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : q7 þq8 ðxhuÞþ ðq7 þq8 ðxhuÞÞ2 4q6 ðq9 þq10 ðxhuÞÞ if D 6 x 6 D ; 2 3 2q6
where
D1 ¼ 0:5hQs1 þ hu þ d1 ðkQ
1
þ p1 nQ 1 þ u2 þ ðu1 u2 Þs1 Þs1 3 ;
D2 ¼ 0:5hQs2 þ hu þ d2 ðkQ
1
þ p2 nQ 1 þ u2 þ ðu1 u2 Þs2 Þs1 2 ;
D3 ¼ 0:5hQs3 þ hu þ d3 ðkQ
1
þ p3 nQ 1 þ u2 þ ðu1 u2 Þs3 Þs1 1 ;
v1 ¼ ðs2 s1 Þ ;
8 e 6 r þ ð1 qÞN D e 6 r > if n ¼ 0; q2 . Since it is always true that e 6 rÞ, it follows that l ð0Þj P l ð0Þj . Theree 6 rÞ P Nð D qð D q1 q2 e e eL eL A A e e d2 l2 , it can be easily shown from defuzzification of discrete sets that
06
e e defuzzð AÞj 1 q2 defuzzð AÞjq1 6 Prd l : 2 2 e lði þ rÞ defuzzð AÞj q1
i¼1
e ¼ ða; c; bÞ with memberRemark 3.2. Triangular fuzzy number P ship function leðxÞ, x 2 ½a; b, has the same defuzzification value P e 0 ¼ fðx; al Þj 0 < a 6 1g, and this value as triangular fuzzy-set P eP 0 aþcþb is 3 . 2675
Proof. The proof follows directly from Definitions 2.1 and 2.3. h
Avg. Annual Cost
2670
2665 2660 2655
180 2650
175
2645
170
2640
165
2635 155
150
145
160 140
r
135
130
125
155
Fig. 5. Plot of total annual cost (crisp) with neutral attitude.
Q
From Remark 3.1 we see that because of subjective opinion involved in estimating input parameters, different risk attitude levels will lead to different stock-out levels in the computation, which may cause the final decisions of different decision makers to diverge from one another. Remark 3.2 provides some insight to the s-fuzzification procedure, although in some cases, weighting membership functions will not change the defuzzification of the fuzzy set; rather, it will cause a different shape of the unionized membership function. Note that due to the complexity of the total cost structure, analytical comparison of optimal policies with different human risk factors could be very complicated for our model.
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Next, we would like to extend the formulation given by expression (18) from triangular fuzzy sets to two types of generic fuzzy sets: continuous and discrete. For continuous, unimodal fuzzy sets, in reality the membership function could be too complex to show in a mathematical form. Commonly encountered continuous fuzzy sets, however, are defined as fuzzy intervals, such that a-level cut arithmetic rules in (5) are always true irrespective of the shape of the membership function. Therefore, a general formula of the total cost function based on an a-level cut is given as follows. Let a a e a ¼ ½pa ; pa . Then, e a ¼ ½da ; da , e L a ¼ ½ll ; lr , e S a ¼ ½sal ; sar , and P D l r r l expression (15) can be computed directly from (5), and expression (18) can be generalized as
e F a ðr; Q ; nÞ ¼ ½fla ; fra ;
ð21Þ
where
a a dl Qsl a a þu ; a k þ npl þ c2 Q þ ðc1 c2 ÞQsl þ h 2 Qsr a a d Qs r fra ¼ ra k þ npar þ c2 Q þ ðc1 c2 ÞQsar þ h þu : 2 Qsl
leF ðr;Q ;nÞ ðf Þ ¼ sup min leSðs Þ ; leP ðp Þ ; leD ðd Þ i
fijk
j
;
ð22Þ
k
where
dk Qsi fijk ¼ k þ npj þ c2 Q þ ðc1 c2 ÞQsi þh þu : Qsi 2 Similarly, using the s-fuzzication procedure and Definition 2.3, we can defuzzificate the total annual cost for a given ðQ ; rÞ policy associated with a certain risk attitude and search for a minimum. It is worth noting that, for non-triangular fuzzy sets, the fuzzy membership function of the total cost function could be very complicated. For example, given a ðQ ; rÞ policy, for demand with 0 0 maximal possible stock-out units m ¼ maxð0; dr lr rÞ, the computational complexity of e Fðr; Q Þ, is oðm2 CÞ, where C ¼ maxðfijk Þ. 4. Numerical example
fla ¼
Membership functions generally can be obtained by taking inverse functions of a-cut functions over applicable intervals. For generic discrete fuzzy sets, it is straightforward to repeat (4) in sequence. If the three discrete fuzzy sets of demand, supplier yield, e ¼ ðp ; l and penalty cost rate are given by e S ¼ ðsi ; le Þ, P j eP ðpj Þ Þ, Sðsi Þ e and D ¼ ðdk ; le Þ, where i; j; k are positive integers, then the D ðdk Þ membership function of discrete set e Fðr; Q ; nÞ is
Table 2 Values of parameters in the fuzzy scenarios and their corresponding stochastic scenarios Parameter
Traditional model
Fuzzy-set model
Lead time Ordering cost Inventory holding cost Supplier yield Penalty cost Unit cost Short material handling cost
L ¼ 0:5 years k ¼ $50/order h ¼ $2/unit/year S ¼ 0:8 P ¼ $25/unit c1 ¼ $10/unit c2 ¼ $2/unit
Same Same Same e S ¼ ð0:7; 0:8; 0:9Þ e ¼ ð20; 25; 30Þ P Same Same
a
In this section, we illustrate the use of our fuzzy ðQ ; rÞ model through a simple example adapted from a popular textbook (Nahmias, 1997). The example problem (Harvey’s specialty shop: p. 287), as stated in the text, assumes normally distributed demand with known parameters. We also examine a skewed demand case represented by a comparable Gamma distribution, since in reality probability density functions corresponding to human activities are often skewed to the right (Law and Kelton, 2000). These more traditional demand assumptions, known distributions with known parameters, serve as baseline cases to our fuzzy models. Then we perform some sensitivity analysis for key parameters under different risk attitude levels. 4.1. Example problem The values of all parameters besides demand in the traditional stochastic formulation of our example problem (Nahmias, 1997) and the corresponding fuzzy-set formulation are given in Table 2. Regarding demand, in the example problem, the normally distributed pffiffiffidemand has a mean of 200 units and a standard deviation of 25 2 annually (see Fig. 6a). Since fuzzy-set possibility distributions are weaker representations of uncertainty than traditional probability distributions, moving from possibility to probability means a loss of information. Thus, to build a corresponding fuzzy
b
0.012
−3
8
x 10
7
0.01
6 0.008
pdf
pdf
5 0.006
4 3
0.004
2 0.002 1 0 0
50
100
150
200
250
Demand
300
350
400
450
500
0
0
50
100
150
200
250
300
350
400
450
Demand
pffiffiffi Fig. 6. (a) Normal demand Nð200; 25 2Þ in Harvey’s specialty shop example. (b) Gamma demand Cð16; 13:3Þ in Harvey’s specialty shop example.
500
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distribution, we utilize the triangular approach from Law and Kelton (2000), suggested for specifying a traditional stochastic distribution in the absence of data. Using this approach, the middle value and boundaries of our triangular fuzzy-set are determined by the mode and given quantiles of the corresponding probability distribution, respectively. For our triangular fuzzy number parallel to the normal demand distribution inp the ffiffiffi Nahmias example, with mean 200 and standard deviation 25 2, the most likely value is d2 ¼ 200, with minimal and maximal values of d1 ¼ 95 and d3 ¼ 305, obtained from the 0.15% and 99.85% quantiles, respectively, of the corresponding normal distribution. Similarly, for e ¼ comparable right skewed demand, we use fuzzy number D ð95; 200; 410Þ, obtained from Gamma distribution Cð16; 13:3Þ (see Fig. 2b). To specify the distribution of demand over the lead time for e ¼ ð95; 200; 305Þ, since L ¼ 0:5 years symmetric fuzzy number D e L ¼ ð48; 100; 153Þ. Then, given Q, r, and q (a crisp number), D (risk attitude), we can specify the fuzzy stock-out set and the fuzzy total cost set. Consider r ¼ 110, Q ¼ 140, and q ¼ 0:5, for example. By (17), the possible stock-out units are n 2 f0; 1; . . . ; 43g, and the e is membership function of the stock-out set A
leA ðnÞ ¼
0:59; if n ¼ 0; 0:0189ð120 þ nÞ þ 2:8868 if n 2 f1; 2; . . . ; 43g:
Table 3 Results of fuzzy ðQ ; rÞ model versus normal baseline case
ð23Þ By (18) and (19), we can compute the fuzzy total cost set for each stock-out possibility and unify them. Fig. 7a illustrates 44 parabolic-triangular fuzzy numbers of total cost for the 44 possible stock-out cases, and Fig. 7b and c illustrate the s-defuzzification procedure on the aggregated total cost fuzzy-set.
a
Model
Demand
Risk attitude
b r
b Q
b f
Fuzzy
e ¼ ð95; 200; 305Þ D
Traditional
pffiffiffi D Nðl ¼ 200; r ¼ 25 2Þ
q¼0 q ¼ 0:5 q¼1
141 140 140 143
139 142 139 138
2.6378e+03 2.6374e+03 2.6368e+03 2.4067e+03
8000
9000
–
b
0.9
0.6
Membership Function
0.8 0.7 0.6 0.5 0.4 0.3 0.2
0.5 0.4 0.3 0.2 0.1
0.1 0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0
1000
Annual Average Total Cost ($)
c
2000
3000
0.7
0.5 0.4 0.3 0.2 0.1 0 0
4000
5000
6000
7000
Annual Average Total Cost ($)
0.6
Membership Function
Membership Function
In Table 3, the row labeled ‘‘Fuzzy” lists optimal policies for the three risk attitude levels, found using the global DIRECT algorithm, for the symmetric-demand fuzzy scenario for our example problem. The row labeled ‘‘Traditional” displays the optimal results for the normal baseline case. The result shows that the total cost obtained from the symmetric-demand fuzzy scenario is about 20% higher than that from its baseline case. Similarly, optimal ðQ ; rÞ policies for the skewed fuzzy demand and its corresponding baseline Gamma scenario are listed in Table 4. The result indicates that the total cost obtained from fuzzy sets is around 40% higher than its baseline case. For symmetric-demand, optimal ðQ ; rÞ policies from fuzzy scenarios are closer to the stochastic cases than for skewed demand. We should be careful, however, to put these results in the proper perspective. The larger cost outcomes for the fuzzy-number models are consistent with intuition: More ‘‘fuzziness” means less information about system parameters and outcomes, which in turn means higher cost from less well-matched inventory and demand levels. The important conclusion we should draw from our examples, though, is that our fuzzy method provides a means of actually quantifying the cost of not knowing the
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Annual Average Total Cost ($) Fig. 7. Total cost structure when r ¼ 110, q ¼ 140, and q ¼ 0:5.
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underlying demand distribution and parameters. Indeed, this could serve as an effective means of quantifying the value of an information system that could accurately estimate demand distributions and their parameters, in addition to providing accurate estimates of various cost parameters defining the inventory system. Finally, we also observe that optimal policies and total cost do not appear to vary much for different risk attitude levels, a phenomenon we explore further in the next section.
nual costs increase as well. Interestingly, when h < 0, the reorder quantity, reorder point, and total annual cost follow similar but non-synchronized patterns for the three types of risk attitude, from optimistic to neutral to pessimistic. This pattern can be explained in two ways. On one hand, a low penalty cost leads to correspondingly less safety stock buffer against uncertainty in demand over the lead time, but higher average cycle stock instead. On the other hand, the optimistic attitude demonstrates the most pronounced shift from cycle stock to safety stock to the increasing trend (h ¼ 1 and h ¼ 0:5) due to its risk aversion, attempting to buffer most directly against the uncertainty during the lead time. Under the pessimistic case the opposite situation occurs. Also, for penalty cost levels beyond a certain point (h ¼ 0), we see that the difference in inventory policies among different risk attitudes is trivial. Fig. 9 shows plots of the optimal reorder points, reorder quantities, and total costs under optimistic, neutral, and pessimistic risk attitude levels with the following triangular fuzzy numbers:
4.2. Sensitivity analysis In this section, we use the same example problem introduced in Section 4.1 to conduct some sensitivity analysis on two key inputs to the fuzzy-set model: the demand parameters and the penalty cost. In addition, we study how the stock-out risk factor affects the optimal ðQ ; rÞ policies. Fig. 8 shows plots of the optimal reorder points, reorder quantities, and total costs under optimistic, neutral, and pessimistic risk e ¼ 10h þ ð20; 25; 30Þ. From the plots, as exattitude levels with P e increases, reorder points and total anpected, we observe that as P
e ¼ D
Demand
Risk attitude
b r
b Q
b f
Fuzzy
e ¼ ð95; 200; 410Þ D
q¼0 q ¼ 0:5 q¼1
Traditional
D Gammaða ¼ 16:0; b ¼ 13:3Þ
–
188 188 187 176
157 154 157 161
3.1585e+03 3.1579e+03 3.1573e+03 2.6311e+03
a
b
145 140
Reorder Quantity
135
125 120 115 110
260 Pessimism Neutral Optimism
220 200 180 160
105
Pessimism Neutral Optimism
100 −1.5
−1
−0.5
2
1
0
3
4
140 −0.5
−1
−1.5
−2
5
c
1
0
θ
Defuzzified Annual Cost (103)
Reorder Point
ð95 þ 16n; 200; 410Þ; if n < 0:
240
130
−2
ð95; 200; 410 þ 16nÞ; if n P 0;
With every unit change of n, the skewness of the fuzzy demand set is increased by 0:03%, as measured by the fuzziness function introduced in Yager (1979). When n > 0, the demand set is more skewed to the right with average demand increasing. While for n < 0, the situation is opposite. Fig. 9 shows very strong positive correlation between demand and reorder point, total cost, reorder quantity respectively when n > 0. When the demand set moves towards symmetry (i.e., decreasing n), we observe that reorder points are not much different, but total costs and reorder quantities decrease steadily. The phenomena can be explained by high stock-out
Table 4 Results of ðQ ; rÞ model versus Gamma baseline case Model
θ
2.64 2.62 2.6 2.58 2.56 2.54 Pessimism Neutral Optimism
2.52 −2
−1.5
−1
−0.5
0
θ
1
2
3
4
5
Fig. 8. Optimal policies with variation of penalty rate for three typical risk attitude.
2
3
4
5
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R. Handfield et al. / European Journal of Operational Research 197 (2009) 609–619
260
b 180
250
175
240
170
Reorder Quantity
Reorder Point
a
230 220 210
165 160 155 150
200
Pessimism Neutral Optimism
190 −6 −5 −4 −3 −2 −1
0
1
2
3
4
5
6
7
8
140 −6 −5 −4 −3 −2 −1
9 10
ξ
Defuzzified Annual Cost (103)
c
Pessimism Neutral Optimism
145 0
1
2
ξ
3
4
5
6
7
8
9 10
3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3
Pessimism Neutral Optimism
2.9 −6 −5 −4 −3 −2 −1
0
1
2
3
4
5
6
7
8
9 10
ξ Fig. 9. Optimal policies with variation of demand for three typical risk attitude.
penalty cost, which we believe also accounts for the close results among the three different risk attitude levels from Fig. 8. 5. Conclusion and remarks In managing the supply chain, it may not be possible to provide precise estimates of all of the critical parameters that influence inventory decision making. Such parameters include the distribution of demand, the distribution of lead time, uncertainty in supplies, and difficult to estimate costs like the penalty cost for supply shortfalls. In this paper, we develop an approach to computing ðQ ; rÞ inventory policy parameters based on fuzzy sets that account for imprecise estimates in key parameters, and for managerial risk attitudes. We present a method to optimize this model, present analytical results related to it, and demonstrate the model’s use via numerical examples and sensitivity analysis. An important advantage of our fuzzy ðQ ; rÞ model is that it can be used for virtually any empirical demand or lead time distribution, whereas non-normal demand or lead time distributions could significantly complicate the computation of policy parameters in a traditional, stochastic model of a ðQ ; rÞ system. In addition, since fuzzy modeling involves a manager’s – or some functional experts’ – subjective estimates of key parameters, accounting for attitudes toward stock-out risk is important, and fuzzy logic provides a flexible and straightforward means of doing this. We are currently utilizing our model and the computational approaches developed in this paper in a number of ways. First, we are adding a service level constraint as an alternative to an explicit penalty cost. Also, we are evaluating ðQ ; rÞ policies in multi-level
supply chains under a fuzzy risk environment to improve supply chain performance through various coordination mechanisms. Finally, we are comparing the performance of the fuzzy model and the traditional stochastic model in situations where some underlying data is available regarding demand and lead time, but where this data does not clearly fit any of the stochastic probability distributions that are traditionally used in inventory analysis. Acknowledgement We are grateful to two anonymous reviewers for their very constructive comments. References Chang, S.C., 1999. Fuzzy production inventory for fuzzy product quantity with triangular fuzzy number. Fuzzy Sets and Systems 107, 37–57. Elkins, D., Handfield, R., Blackhurst, J., Craighead, C.W., 2005. 18 Ways to guard against disruption. Supply Chain Managment Review (January/February). Finkel, D.E., 2005. Global optimization with the DIRECT algorithm, Ph.D. Thesis, NC State University, . Giachetti, R., Young, R., 1997. Analysis of the error in the standard approximation used for multiplication of triangular and trapezoidal fuzzy numbers and the development of a new approximation. Fuzzy Set and Systems 91, 1–13. Kaufmann, A., Gupta, M.M., 1988. Fuzzy Mathematical Models in Engineering and Management Science. North-Holland, Amsterdam, Netherlands. Law, A.M., Kelton, W.D., 2000. Simulation. Simulation Modeling and Analysis. McGraw-Hill. Mondal, S., Maiti, M., 2002. Multi-item fuzzy EOQ models using genetic algorithm. Computers & Indutrial Engineering 44, 105–117. Nahmias, S., 1997. Production and Operations Analysis. McGraw-Hill. Pai, P.F., Hsu, M.M., 2003. Continuous review reorder point problems in a fuzzy environment. The International Journal of Advanced Manufacturing Technology 22, 436–440.
R. Handfield et al. / European Journal of Operational Research 197 (2009) 609–619 Petrovic, D., Petrovic, R., Vujosevic, M., 2001. Fuzzy models for the newsboy problem. International Journal Production Economics 45, 435–441. Petrovic, D., Roy, R., Petrovic, R., 1998. Modeling and simulation of a supply chain in an uncertain environment. European Journal of Operational Research 109, 200– 309. Petrovic, D., Roy, R., Petrovic, R., 1999. Supply chain modeling using fuzzy sets. International Journal Production Economics 59, 443–453. Vujosevic, M., Petrovic, D., Petrovic, R., 1996. EOQ formula when inventory cost is fuzzy. International Journal Production Economics 45, 499–504. Wang, J., Shu, Y.F., 2005. Fuzzy decision modeling for supply chain management. Fuzzy Sets and Systems 150, 107–127. Xie, Y., Petrovic, D., Burnham, K., 2006. A heuristic procedure for the two-level control of serial supply chains under fuzzy customer demand. International Journal Production Economics 102, 37–50.
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Yager, R.R., 1979. On the measure of fuzziness and negation part I: Membership in the unit interval. International Journal of General Systems 5, 221–229. Yimer, A.D., Demirli, K., 2004. Fuzzy modeling and simulation of a single item inventory system with variable demand. In: Annual Conference of the North American Fuzzy Information Processing Society – NAFIPS 2004, vol. 2, pp. 985– 989. Zadeh, L.A., 1978. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1, 3–28. Zimmermann, H.J., 2001. Fuzzy Set Theory and Its Application. International Thomson Publishing, Norwell, Massachusetts. Zipkin, P.H., 2000. Foundations of Inventory Management. McGraw-Hill.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
ðQ ; rÞ Inventory policies in a fuzzy uncertain supply chain environment Robert Handfield a, Don Warsing a, Xinmin Wu b,* a b
Department of Business Management, North Carolina State University, Raleigh, NC 27695-7229, USA Operations Research Graduate Program, Campus Box 7913, North Carolina State University, Raleigh, NC 27695-7913, USA
a r t i c l e
i n f o
Article history: Received 11 March 2006 Accepted 11 July 2008 Available online 29 July 2008 Keywords: ðQ ; rÞ System Inventory Fuzzy sets Optimization
a b s t r a c t Managers have begun to recognize that effectively managing risks in their business operations plays an important role in successfully managing their inventories. Accordingly, we develop a ðQ ; rÞ model based on fuzzy-set representations of various sources of uncertainty in the supply chain. Sources of risk and uncertainty in our model include demand, lead time, supplier yield, and penalty cost. The naturally imprecise nature of these risk factors in managing inventories is represented using triangular fuzzy numbers. In addition, we introduce a human risk attitude factor to quantify the decision maker’s attitude toward the risk of stocking out during the replenishment period. The total cost of the inventory system is computed using defuzzification methods built from techniques identified in the literature on fuzzy sets. Finally, we provide numerical examples to compare our fuzzy-set computations with those generated by more traditional models that assume full knowledge of the distributions of the stochastic parameters in the system. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Managers have begun to recognize that effectively managing risks in their business operations plays an important role in successfully managing their inventories. One of the most common risks is demand uncertainty, a phenomenon that is widely studied in the literature. Another risk that has attracted significant attention is supply uncertainty, especially the risk associated with direct material supplies and deliveries. This type of risk has escalated as Fortune 500 companies have sourced a great proportion of products from areas of the globe with low labor costs, such as China and India. Frequently, the hidden perils of such sourcing strategies are not fully considered (Elkins et al., 2005). There is an abundant literature that models uncertainty in demand and/or lead time using probability distributions with known parameters. However, in many cases where there is little or no historical data available to the inventory decision maker, perhaps due to recent changes in the supply chain (SC) environment, probability distributions may simply not be available, or may not be easily or accurately estimated (Xie et al., 2006). Additionally, in some cases, it may not be possible to collect data on the random variables of interest because of certain system or time constraints. Furthermore, other critical supply chain parameters, in particular the various costs that impact the system, are often ill-defined and may vary from time to time. All of these situations raise challenges for using traditional inventory models in practice. * Corresponding author. Tel.: +1 919 607 6648. E-mail address: [email protected] (X. Wu). 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.07.016
Fuzzy theory provides an alternate, flexible approach to handle such situations because it allows the model to easily incorporate various subject experts’ advice in developing critical parameter estimates (Zimmermann, 2001). Recent research has employed fuzzy logic in modeling demand or lead time uncertainty in inventory systems. For example, Pai and Hsu (2003) apply fuzzy-set theory to continuous review reorder point problems, with the assumption that uncertainties may appear not only in demand over the lead time, but also in inventory holding costs. They obtain optimal ðQ ; rÞ policies that account for these uncertainties by computing derivatives of a Type-2 fuzzy-set of total cost. Petrovic et al. (2001) develop a fuzzy model for the newsvendor problem in an uncertain environment, where inventory cost and demand are represented by fuzzy sets. Vujosevic et al. (1996) develop an alternative EOQ formula that accounts for fuzzy inventory costs and demand parameters. This solution approach is extended in Mondal and Maiti (2002) to a multi-item fuzzy EOQ system by using a genetic algorithm. Chang (1999) derives a membership function of the total cost and EOQ in a production-inventory problem where the production quantity is a triangular fuzzy number. In a broader setting, Wang and Shu (2005) develop a fuzzy decision methodology that provides an alternative framework to handle SC uncertainty and to determine inventory management strategies for a SC network with a parameter, q ð0 6 q 6 1Þ, to describe the managerial (human) attitude toward risk. Also in a SC setting, Xie et al. (2006) develop a hierarchical, two-level coordination method to compute order-up-to policies in SCs under fuzzy demand. In their model, the entire SC system is decomposed into independent sub-systems optimized at the ‘‘follower level,” while
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coordination is implemented at the ‘‘leader level.” SC simulation models have also been developed by Petrovic et al. (1998, 1999) in a fuzzy uncertain environment for demand and material supply, using base-stock control policies. Finally, Yimer and Demirli (2004) describe the performance achieved by a fuzzy inventory control simulation model in a continuous review inventory system. However, to the best knowledge of the authors, no research has simultaneously considered multiple fuzzy risk factors in ðQ ; rÞ inventory policies while also incorporating the decision maker’s risk attitude (i.e., the q factor introduced by Wang and Shu, 2005). In this paper, we present a ðQ ; rÞ system that accounts for the typical imprecision in modeling demand, lead time, supply yield, penalty cost, and inventory holding costs. Since fuzzy sets are based on human subjective opinion, however, the decision maker’s attitude towards risks driven by uncertainty affects the decision process and is, therefore, clearly worthy of study. We study the effects of three levels of risk attitude – pessimistic, neutral, and optimistic – extended from Wang and Shu (2005), and we provide corresponding managerial insights from this analysis. The rest of the paper is organized as follows: In Section 2, we present several important fuzzy set-related definitions and develop important propositions of fuzzy arithmetic to be used in our ðQ ; rÞ computations. In Section 3, we present our approach to modeling SC uncertainty using fuzzy sets, and we provide our model assumptions and notations. We derive a fuzzy total cost function based on fuzzy arithmetic rules, and use a direct search algorithm to find optimal solutions. In Section 4, we apply our fuzzy computational scheme to a numerical example adapted from a popular textbook, and we compare these results to those generated by the traditional computational approach that assumes known probability distributions with known parameters. In addition, we conduct a sensitivity analysis, accounting for different managerial risk attitudes toward stocking out. Finally, we provide some concluding remarks in Section 5.
The fuzzy-set in the above definition is called a Type-1 fuzzyset, whose membership function is determined. When an exact membership function is difficult to determine, however, we need to use the concept of Type-2 fuzzy-set, which is used to handle and measure additional uncertainty. A Type-2 fuzzy-set is actually a fuzzy-set of a fuzzy-set, defined as follows: e ¼ fðx; l Þg is a Type-2 fuzzy-set if Definition 2.2. P e P
leP ðxÞ ¼ fðui ; lui ðxÞÞjx 2 X; ui ; lui ðxÞ 2 ½0; 1g;
where li is the degree of membership of a Type-1 fuzzy-set and lui ðxÞ is its membership function (Zimmermann, 2001). Note that here we only give the definition when the element of the membership function is a fuzzy-set, but we can also define it when x is a fuzzy-set. In order to obtain a scalar from the fuzzy domain that most appropriately describes a fuzzy-set, we need to use defuzzification functions. There are many defuzzification heuristics such as extreme value (EV), center of area (COA), and center of gravity (COG) defuzzification (Zimmermann, 2001). In this paper, we adopt COG defuzzification as it has a distinct geometrical meaning and strong probability analogy. The definition is given as follows: e is defined by Definition 2.3. COG defuzzification of P
R xl ðxÞ dx e defuzzð PÞCOG ¼ R P lP ðxÞ dx
ð2Þ
(Zimmermann, 2001). Next, we give the definition for the a-level cut of a fuzzy-set, and then use it in the definition of triangular fuzzy number arithmetic. e is defined as Definition 2.4. The a-level cut of fuzzy-set P
e a ¼ fx 2 Xjl ðxÞ P ag; P e P
where 0 6 a 6 1
ð3Þ
2. Fuzzy-set definitions and propositions
(Zimmermann, 2001).
For ease of exposition in this paper, we present some important definitions and propositions that will be used in the following sections.
e be fuzzy numbers with membership Definition 2.5. Let f M and N functions l e ðxÞ and le ðyÞ, respectively, where x; y 2 R. Let } be M N a binary operation, i.e., } : R}R ! R. The operation } can be extended to the fuzzy domain by the following formula:
Definition 2.1. If X is a collection of objects denoted generically by e on X is a set of order pairs x, then a fuzzy-set P
e ¼ fðx; l Þjx 2 Xg; P e
leP ðxÞ is a membership function and leP ðxÞ P 0. If max
e is called a fuzzy number (Zadeh, 1978). ðle ðxÞÞ ¼ 1, P P
If x is continuous on X, then the fuzzy-set is continuous and the e Continuous fuzzy range of x is called the support of fuzzy-set P. sets can have various membership function shapes such as triangular and trapezoidal. In this paper, we use triangular fuzzy numbers, a special case of trapezoidal, due to its straightforward structure and computational simplicity. Throughout the paper, unless otherwise indicated, we use a bold capital letter with a wide tilde (e.g., e to represent a generic fuzzy-set and a non-bold capital letter P) e to denote a triangular fuzzy number. with a wide tilde (e.g., P) e ¼ ða; c; bÞ. Its memberFig. 1a shows a triangular fuzzy number P ship function, leðxÞ, can be written as follows: P
8 0 > > > > < x a ca leP ðxÞ ¼ ca x b > þ bc > bc > > : 0
ð4Þ
(Kaufmann and Gupta, 1988).
P
where
l Me }eN ðzÞ ¼ sup min l Me ðxÞ; leN ðyÞ z¼x}y
if x 6 a; if a < x 6 c; if c < x 6 b; if x > b:
ð1Þ
By applying the definition of an a-level cut to (4) (see Fig. 2), we have the following exact arithmetic operations for continuous fuzzy sets:
f e a ¼ ½ma ; ma þ ½na ; na ¼ ½ma þ na ; ma þ na ; Ma þ N 1 2 1 2 1 1 2 2 e a ¼ ½ma ; ma ½na ; na ¼ ½ma na ; ma na ; f Ma N 1 2 1 2 1 2 2 1 f e a ¼ ½ma ; ma ½na ; na ¼ ½ma na ; ma na ; Ma N 1 2 1 2 1 1 2 2
ð5Þ
e a ¼ ½ma ; ma ½na ; na ¼ ½ma na ; ma na : f Ma N 1 2 1 2 1 2 2 1 For triangular fuzzy numbers, fuzzy arithmetic can be simplie ¼ ða2 ; c2 ; b2 Þ. It can be easily proved e ¼ ða1 ; c1 ; b1 Þ and N fied. Let M (see, e.g., Kaufmann and Gupta, 1988) that
e þN e ¼ ða1 þ a2 ; c1 þ c2 ; b1 þ b2 Þ; M e N e ¼ ða1 b2 ; c1 c2 ; b1 a2 Þ; M e ¼ ðsa1 ; sc1 ; sb1 Þ; where s is a crisp number: sM For multiplicative, inverse, and division operations, similar triplets cannot be used, since the resulting membership function is no
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e (b) Parabolic-triangular fuzzy number with parabolic membership function P e. Fig. 1. (a) Triangular fuzzy number P.
e (b) a cut of N. e Fig. 2. (a) a cut of M.
longer linear over its left and right segments (see Fig. 1b), and this could produce erroneous results, especially when multiplication and division operands are used repeatedly (Wang and Shu, 2005; Giachetti and Young, 1997). For the purpose of differentiation, however, we add a star superscript to denote a parabolic-triangular fuzzy number with parabolic membership function. For example, e ¼ ða; c; bÞ represents a parabolic-triangular fuzzy number over P support ½a; b (See Fig. 1b). By exact methods, operations among multiple triangular fuzzy numbers can be carried out based on (5) because the arithmetic has been converted to arithmetic operations on a certain membership interval from fuzzy triangular numbers. Membership functions of the product and division of two triangular fuzzy numbers are given in the following propositions, which are obtained by interval multiplication/divison and inverse function formulas. e ¼ ða2 ; c2 ; b2 Þ. Then, the e ¼ ða1 ; c1 ; b1 Þ and N Proposition 2.6. Let M e N e is given by membership function of M
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > < e2 þe6 þ ðe2 þe6 Þ þ e1 e5 x e2 e6 2 4 l Me eN ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : e4 þe8 þ ðe4 þe8 Þ2 þ e e x e e 3 7 4 8 2 4
if a1 a2 6 x 6 c1 c2 ; if c1 c2 6 x 6 b1 b2 ; ð6Þ
( e2 e7 e1 e8 x
l Me eN ðxÞ ¼
if a1 =b2 6 x 6 c1 =c2 ;
e7 e1 x e4 e5 e3 e6 x e5 e3 x
if c1 =c2 6 x 6 b1 =a2 ;
ð7Þ
where
e1 ¼ ðc1 a1 Þ1 ; 1
e2 ¼ a1 ðc1 a1 Þ1 ; 1
e4 ¼ b1 ðb1 c1 Þ ;
e5 ¼ ðc2 a2 Þ ;
e7 ¼ ðb2 c2 Þ1 ;
e8 ¼ b2 ðb2 c2 Þ1 :
e3 ¼ ðb1 c1 Þ1 ; e6 ¼ a2 ðc2 a2 Þ1 ;
Proof. The proof is straightforward from Definition 2.5. h To measure the possibility that a fuzzy-set belongs to another fuzzy set, we need to introduce the definitions of possibility and necessity measures. The definitions are given as follows: e in the Definition 2.8. Suppose x is restricted by a fuzzy-set B universe U. Further, suppose that the possible distribution of x, px , is taken to be equal to membership function le . Then, the B e is defined as possibility of event x 2 P
e ¼ sup min l ðuÞ; l ðuÞ : q x2P e e u2U
B
P
ð8Þ
e is The dual measure of possibility, i.e., the necessity of event x 2 P, defined as
where
e1 ¼ ðc1 a1 Þ1 ;
e ¼ ða1 ; c1 ; b1 Þ and N e ¼ ða2 ; c2 ; b2 Þ. Then, the Proposition 2.7. Let M e N e is given by membership function of M
e2 ¼ a1 ðc1 a1 Þ1 ;
e3 ¼ ðb1 c1 Þ1 ;
e4 ¼ b1 ðb1 c1 Þ1 ;
e5 ¼ ðc2 a2 Þ1 ;
e6 ¼ a2 ðc2 a2 Þ1 ;
e7 ¼ ðb2 c2 Þ1 ;
e8 ¼ b2 ðb2 c2 Þ1 :
Proof. The proof is straightforward from Definition 2.5. h
e ¼ inf max 1 l ðuÞ; l ðuÞ N x2P e e u2U
B
P
ð9Þ
(Yager, 1979; Wang and Shu, 2005). e 6 RÞ represents the maxSuppose R is a crisp number, then qð B e is less than R, and Nð B e 6 RÞ imum likelihood of the event that B
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e 6 R will ocestimates the minimum likelihood of the event that B cur. By Definition 2.8, we have
e 6 R ¼ qðx 2 ð1; RÞ ¼ sup l ðuÞ ; q B eB u6R e 6 R ¼ Nðx 2 ð1; RÞ ¼ inf 1 l ðuÞ ; N B e B u>R e P R ¼ qðx 2 ½R; þ1ÞÞ ¼ sup l ðuÞ ; q B e B uPR e P R ¼ Nðx 2 ½R; þ1ÞÞ ¼ inf 1 l ðuÞ : N B e B u ðQe S=2þrjÞlA~ ðjÞ < j¼1P r
> :
l
~ ðjÞ j¼1 A
L
ðn þ rÞ if n 2 f1; 2; . . . ; maxf0; d3 l3 rgg: > : le D eL
ð17Þ
if n ¼ 0; if n > 0:
n
u is part of inventory holding cost calculated based on the lowest cycle inventory level. If it is positive, we add the safety stock. If there is a stock-out, we exclude the portion of time associated with the out-of-stock condition (Zipkin, 2000) and we assume full backlogging of demand, i.e., that no sales are lost. Other cost components in (18) include the purchasing cost for received goods, S; the handling cost for rejected items to be returned, u1 Q e e SÞ; and the penalty cost n P. u2 Q ð1 e By Definition 2.5, the parabolic-triangular membership function of e F ðr; Q ; nÞ can be written as
8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > < q2 þq3 ðxhuÞ ðq2 þq3 ðxhuÞÞ 4q1 ðq4 þq5 ðxhuÞÞ if D1 6 x 6 D2 ; 2q1 leF ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : q7 þq8 ðxhuÞþ ðq7 þq8 ðxhuÞÞ2 4q6 ðq9 þq10 ðxhuÞÞ if D 6 x 6 D ; 2 3 2q6
where
D1 ¼ 0:5hQs1 þ hu þ d1 ðkQ
1
þ p1 nQ 1 þ u2 þ ðu1 u2 Þs1 Þs1 3 ;
D2 ¼ 0:5hQs2 þ hu þ d2 ðkQ
1
þ p2 nQ 1 þ u2 þ ðu1 u2 Þs2 Þs1 2 ;
D3 ¼ 0:5hQs3 þ hu þ d3 ðkQ
1
þ p3 nQ 1 þ u2 þ ðu1 u2 Þs3 Þs1 1 ;
v1 ¼ ðs2 s1 Þ ;
8 e 6 r þ ð1 qÞN D e 6 r > if n ¼ 0; q2 . Since it is always true that e 6 rÞ, it follows that l ð0Þj P l ð0Þj . Theree 6 rÞ P Nð D qð D q1 q2 e e eL eL A A e e d2 l2 , it can be easily shown from defuzzification of discrete sets that
06
e e defuzzð AÞj 1 q2 defuzzð AÞjq1 6 Prd l : 2 2 e lði þ rÞ defuzzð AÞj q1
i¼1
e ¼ ða; c; bÞ with memberRemark 3.2. Triangular fuzzy number P ship function leðxÞ, x 2 ½a; b, has the same defuzzification value P e 0 ¼ fðx; al Þj 0 < a 6 1g, and this value as triangular fuzzy-set P eP 0 aþcþb is 3 . 2675
Proof. The proof follows directly from Definitions 2.1 and 2.3. h
Avg. Annual Cost
2670
2665 2660 2655
180 2650
175
2645
170
2640
165
2635 155
150
145
160 140
r
135
130
125
155
Fig. 5. Plot of total annual cost (crisp) with neutral attitude.
Q
From Remark 3.1 we see that because of subjective opinion involved in estimating input parameters, different risk attitude levels will lead to different stock-out levels in the computation, which may cause the final decisions of different decision makers to diverge from one another. Remark 3.2 provides some insight to the s-fuzzification procedure, although in some cases, weighting membership functions will not change the defuzzification of the fuzzy set; rather, it will cause a different shape of the unionized membership function. Note that due to the complexity of the total cost structure, analytical comparison of optimal policies with different human risk factors could be very complicated for our model.
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Next, we would like to extend the formulation given by expression (18) from triangular fuzzy sets to two types of generic fuzzy sets: continuous and discrete. For continuous, unimodal fuzzy sets, in reality the membership function could be too complex to show in a mathematical form. Commonly encountered continuous fuzzy sets, however, are defined as fuzzy intervals, such that a-level cut arithmetic rules in (5) are always true irrespective of the shape of the membership function. Therefore, a general formula of the total cost function based on an a-level cut is given as follows. Let a a e a ¼ ½pa ; pa . Then, e a ¼ ½da ; da , e L a ¼ ½ll ; lr , e S a ¼ ½sal ; sar , and P D l r r l expression (15) can be computed directly from (5), and expression (18) can be generalized as
e F a ðr; Q ; nÞ ¼ ½fla ; fra ;
ð21Þ
where
a a dl Qsl a a þu ; a k þ npl þ c2 Q þ ðc1 c2 ÞQsl þ h 2 Qsr a a d Qs r fra ¼ ra k þ npar þ c2 Q þ ðc1 c2 ÞQsar þ h þu : 2 Qsl
leF ðr;Q ;nÞ ðf Þ ¼ sup min leSðs Þ ; leP ðp Þ ; leD ðd Þ i
fijk
j
;
ð22Þ
k
where
dk Qsi fijk ¼ k þ npj þ c2 Q þ ðc1 c2 ÞQsi þh þu : Qsi 2 Similarly, using the s-fuzzication procedure and Definition 2.3, we can defuzzificate the total annual cost for a given ðQ ; rÞ policy associated with a certain risk attitude and search for a minimum. It is worth noting that, for non-triangular fuzzy sets, the fuzzy membership function of the total cost function could be very complicated. For example, given a ðQ ; rÞ policy, for demand with 0 0 maximal possible stock-out units m ¼ maxð0; dr lr rÞ, the computational complexity of e Fðr; Q Þ, is oðm2 CÞ, where C ¼ maxðfijk Þ. 4. Numerical example
fla ¼
Membership functions generally can be obtained by taking inverse functions of a-cut functions over applicable intervals. For generic discrete fuzzy sets, it is straightforward to repeat (4) in sequence. If the three discrete fuzzy sets of demand, supplier yield, e ¼ ðp ; l and penalty cost rate are given by e S ¼ ðsi ; le Þ, P j eP ðpj Þ Þ, Sðsi Þ e and D ¼ ðdk ; le Þ, where i; j; k are positive integers, then the D ðdk Þ membership function of discrete set e Fðr; Q ; nÞ is
Table 2 Values of parameters in the fuzzy scenarios and their corresponding stochastic scenarios Parameter
Traditional model
Fuzzy-set model
Lead time Ordering cost Inventory holding cost Supplier yield Penalty cost Unit cost Short material handling cost
L ¼ 0:5 years k ¼ $50/order h ¼ $2/unit/year S ¼ 0:8 P ¼ $25/unit c1 ¼ $10/unit c2 ¼ $2/unit
Same Same Same e S ¼ ð0:7; 0:8; 0:9Þ e ¼ ð20; 25; 30Þ P Same Same
a
In this section, we illustrate the use of our fuzzy ðQ ; rÞ model through a simple example adapted from a popular textbook (Nahmias, 1997). The example problem (Harvey’s specialty shop: p. 287), as stated in the text, assumes normally distributed demand with known parameters. We also examine a skewed demand case represented by a comparable Gamma distribution, since in reality probability density functions corresponding to human activities are often skewed to the right (Law and Kelton, 2000). These more traditional demand assumptions, known distributions with known parameters, serve as baseline cases to our fuzzy models. Then we perform some sensitivity analysis for key parameters under different risk attitude levels. 4.1. Example problem The values of all parameters besides demand in the traditional stochastic formulation of our example problem (Nahmias, 1997) and the corresponding fuzzy-set formulation are given in Table 2. Regarding demand, in the example problem, the normally distributed pffiffiffidemand has a mean of 200 units and a standard deviation of 25 2 annually (see Fig. 6a). Since fuzzy-set possibility distributions are weaker representations of uncertainty than traditional probability distributions, moving from possibility to probability means a loss of information. Thus, to build a corresponding fuzzy
b
0.012
−3
8
x 10
7
0.01
6 0.008
5 0.006
4 3
0.004
2 0.002 1 0 0
50
100
150
200
250
Demand
300
350
400
450
500
0
0
50
100
150
200
250
300
350
400
450
Demand
pffiffiffi Fig. 6. (a) Normal demand Nð200; 25 2Þ in Harvey’s specialty shop example. (b) Gamma demand Cð16; 13:3Þ in Harvey’s specialty shop example.
500
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distribution, we utilize the triangular approach from Law and Kelton (2000), suggested for specifying a traditional stochastic distribution in the absence of data. Using this approach, the middle value and boundaries of our triangular fuzzy-set are determined by the mode and given quantiles of the corresponding probability distribution, respectively. For our triangular fuzzy number parallel to the normal demand distribution inp the ffiffiffi Nahmias example, with mean 200 and standard deviation 25 2, the most likely value is d2 ¼ 200, with minimal and maximal values of d1 ¼ 95 and d3 ¼ 305, obtained from the 0.15% and 99.85% quantiles, respectively, of the corresponding normal distribution. Similarly, for e ¼ comparable right skewed demand, we use fuzzy number D ð95; 200; 410Þ, obtained from Gamma distribution Cð16; 13:3Þ (see Fig. 2b). To specify the distribution of demand over the lead time for e ¼ ð95; 200; 305Þ, since L ¼ 0:5 years symmetric fuzzy number D e L ¼ ð48; 100; 153Þ. Then, given Q, r, and q (a crisp number), D (risk attitude), we can specify the fuzzy stock-out set and the fuzzy total cost set. Consider r ¼ 110, Q ¼ 140, and q ¼ 0:5, for example. By (17), the possible stock-out units are n 2 f0; 1; . . . ; 43g, and the e is membership function of the stock-out set A
leA ðnÞ ¼
0:59; if n ¼ 0; 0:0189ð120 þ nÞ þ 2:8868 if n 2 f1; 2; . . . ; 43g:
Table 3 Results of fuzzy ðQ ; rÞ model versus normal baseline case
ð23Þ By (18) and (19), we can compute the fuzzy total cost set for each stock-out possibility and unify them. Fig. 7a illustrates 44 parabolic-triangular fuzzy numbers of total cost for the 44 possible stock-out cases, and Fig. 7b and c illustrate the s-defuzzification procedure on the aggregated total cost fuzzy-set.
a
Model
Demand
Risk attitude
b r
b Q
b f
Fuzzy
e ¼ ð95; 200; 305Þ D
Traditional
pffiffiffi D Nðl ¼ 200; r ¼ 25 2Þ
q¼0 q ¼ 0:5 q¼1
141 140 140 143
139 142 139 138
2.6378e+03 2.6374e+03 2.6368e+03 2.4067e+03
8000
9000
–
b
0.9
0.6
Membership Function
0.8 0.7 0.6 0.5 0.4 0.3 0.2
0.5 0.4 0.3 0.2 0.1
0.1 0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0
1000
Annual Average Total Cost ($)
c
2000
3000
0.7
0.5 0.4 0.3 0.2 0.1 0 0
4000
5000
6000
7000
Annual Average Total Cost ($)
0.6
Membership Function
Membership Function
In Table 3, the row labeled ‘‘Fuzzy” lists optimal policies for the three risk attitude levels, found using the global DIRECT algorithm, for the symmetric-demand fuzzy scenario for our example problem. The row labeled ‘‘Traditional” displays the optimal results for the normal baseline case. The result shows that the total cost obtained from the symmetric-demand fuzzy scenario is about 20% higher than that from its baseline case. Similarly, optimal ðQ ; rÞ policies for the skewed fuzzy demand and its corresponding baseline Gamma scenario are listed in Table 4. The result indicates that the total cost obtained from fuzzy sets is around 40% higher than its baseline case. For symmetric-demand, optimal ðQ ; rÞ policies from fuzzy scenarios are closer to the stochastic cases than for skewed demand. We should be careful, however, to put these results in the proper perspective. The larger cost outcomes for the fuzzy-number models are consistent with intuition: More ‘‘fuzziness” means less information about system parameters and outcomes, which in turn means higher cost from less well-matched inventory and demand levels. The important conclusion we should draw from our examples, though, is that our fuzzy method provides a means of actually quantifying the cost of not knowing the
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Annual Average Total Cost ($) Fig. 7. Total cost structure when r ¼ 110, q ¼ 140, and q ¼ 0:5.
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underlying demand distribution and parameters. Indeed, this could serve as an effective means of quantifying the value of an information system that could accurately estimate demand distributions and their parameters, in addition to providing accurate estimates of various cost parameters defining the inventory system. Finally, we also observe that optimal policies and total cost do not appear to vary much for different risk attitude levels, a phenomenon we explore further in the next section.
nual costs increase as well. Interestingly, when h < 0, the reorder quantity, reorder point, and total annual cost follow similar but non-synchronized patterns for the three types of risk attitude, from optimistic to neutral to pessimistic. This pattern can be explained in two ways. On one hand, a low penalty cost leads to correspondingly less safety stock buffer against uncertainty in demand over the lead time, but higher average cycle stock instead. On the other hand, the optimistic attitude demonstrates the most pronounced shift from cycle stock to safety stock to the increasing trend (h ¼ 1 and h ¼ 0:5) due to its risk aversion, attempting to buffer most directly against the uncertainty during the lead time. Under the pessimistic case the opposite situation occurs. Also, for penalty cost levels beyond a certain point (h ¼ 0), we see that the difference in inventory policies among different risk attitudes is trivial. Fig. 9 shows plots of the optimal reorder points, reorder quantities, and total costs under optimistic, neutral, and pessimistic risk attitude levels with the following triangular fuzzy numbers:
4.2. Sensitivity analysis In this section, we use the same example problem introduced in Section 4.1 to conduct some sensitivity analysis on two key inputs to the fuzzy-set model: the demand parameters and the penalty cost. In addition, we study how the stock-out risk factor affects the optimal ðQ ; rÞ policies. Fig. 8 shows plots of the optimal reorder points, reorder quantities, and total costs under optimistic, neutral, and pessimistic risk e ¼ 10h þ ð20; 25; 30Þ. From the plots, as exattitude levels with P e increases, reorder points and total anpected, we observe that as P
e ¼ D
Demand
Risk attitude
b r
b Q
b f
Fuzzy
e ¼ ð95; 200; 410Þ D
q¼0 q ¼ 0:5 q¼1
Traditional
D Gammaða ¼ 16:0; b ¼ 13:3Þ
–
188 188 187 176
157 154 157 161
3.1585e+03 3.1579e+03 3.1573e+03 2.6311e+03
a
b
145 140
Reorder Quantity
135
125 120 115 110
260 Pessimism Neutral Optimism
220 200 180 160
105
Pessimism Neutral Optimism
100 −1.5
−1
−0.5
2
1
0
3
4
140 −0.5
−1
−1.5
−2
5
c
1
0
θ
Defuzzified Annual Cost (103)
Reorder Point
ð95 þ 16n; 200; 410Þ; if n < 0:
240
130
−2
ð95; 200; 410 þ 16nÞ; if n P 0;
With every unit change of n, the skewness of the fuzzy demand set is increased by 0:03%, as measured by the fuzziness function introduced in Yager (1979). When n > 0, the demand set is more skewed to the right with average demand increasing. While for n < 0, the situation is opposite. Fig. 9 shows very strong positive correlation between demand and reorder point, total cost, reorder quantity respectively when n > 0. When the demand set moves towards symmetry (i.e., decreasing n), we observe that reorder points are not much different, but total costs and reorder quantities decrease steadily. The phenomena can be explained by high stock-out
Table 4 Results of ðQ ; rÞ model versus Gamma baseline case Model
θ
2.64 2.62 2.6 2.58 2.56 2.54 Pessimism Neutral Optimism
2.52 −2
−1.5
−1
−0.5
0
θ
1
2
3
4
5
Fig. 8. Optimal policies with variation of penalty rate for three typical risk attitude.
2
3
4
5
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260
b 180
250
175
240
170
Reorder Quantity
Reorder Point
a
230 220 210
165 160 155 150
200
Pessimism Neutral Optimism
190 −6 −5 −4 −3 −2 −1
0
1
2
3
4
5
6
7
8
140 −6 −5 −4 −3 −2 −1
9 10
ξ
Defuzzified Annual Cost (103)
c
Pessimism Neutral Optimism
145 0
1
2
ξ
3
4
5
6
7
8
9 10
3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3
Pessimism Neutral Optimism
2.9 −6 −5 −4 −3 −2 −1
0
1
2
3
4
5
6
7
8
9 10
ξ Fig. 9. Optimal policies with variation of demand for three typical risk attitude.
penalty cost, which we believe also accounts for the close results among the three different risk attitude levels from Fig. 8. 5. Conclusion and remarks In managing the supply chain, it may not be possible to provide precise estimates of all of the critical parameters that influence inventory decision making. Such parameters include the distribution of demand, the distribution of lead time, uncertainty in supplies, and difficult to estimate costs like the penalty cost for supply shortfalls. In this paper, we develop an approach to computing ðQ ; rÞ inventory policy parameters based on fuzzy sets that account for imprecise estimates in key parameters, and for managerial risk attitudes. We present a method to optimize this model, present analytical results related to it, and demonstrate the model’s use via numerical examples and sensitivity analysis. An important advantage of our fuzzy ðQ ; rÞ model is that it can be used for virtually any empirical demand or lead time distribution, whereas non-normal demand or lead time distributions could significantly complicate the computation of policy parameters in a traditional, stochastic model of a ðQ ; rÞ system. In addition, since fuzzy modeling involves a manager’s – or some functional experts’ – subjective estimates of key parameters, accounting for attitudes toward stock-out risk is important, and fuzzy logic provides a flexible and straightforward means of doing this. We are currently utilizing our model and the computational approaches developed in this paper in a number of ways. First, we are adding a service level constraint as an alternative to an explicit penalty cost. Also, we are evaluating ðQ ; rÞ policies in multi-level
supply chains under a fuzzy risk environment to improve supply chain performance through various coordination mechanisms. Finally, we are comparing the performance of the fuzzy model and the traditional stochastic model in situations where some underlying data is available regarding demand and lead time, but where this data does not clearly fit any of the stochastic probability distributions that are traditionally used in inventory analysis. Acknowledgement We are grateful to two anonymous reviewers for their very constructive comments. References Chang, S.C., 1999. Fuzzy production inventory for fuzzy product quantity with triangular fuzzy number. Fuzzy Sets and Systems 107, 37–57. Elkins, D., Handfield, R., Blackhurst, J., Craighead, C.W., 2005. 18 Ways to guard against disruption. Supply Chain Managment Review (January/February). Finkel, D.E., 2005. Global optimization with the DIRECT algorithm, Ph.D. Thesis, NC State University, . Giachetti, R., Young, R., 1997. Analysis of the error in the standard approximation used for multiplication of triangular and trapezoidal fuzzy numbers and the development of a new approximation. Fuzzy Set and Systems 91, 1–13. Kaufmann, A., Gupta, M.M., 1988. Fuzzy Mathematical Models in Engineering and Management Science. North-Holland, Amsterdam, Netherlands. Law, A.M., Kelton, W.D., 2000. Simulation. Simulation Modeling and Analysis. McGraw-Hill. Mondal, S., Maiti, M., 2002. Multi-item fuzzy EOQ models using genetic algorithm. Computers & Indutrial Engineering 44, 105–117. Nahmias, S., 1997. Production and Operations Analysis. McGraw-Hill. Pai, P.F., Hsu, M.M., 2003. Continuous review reorder point problems in a fuzzy environment. The International Journal of Advanced Manufacturing Technology 22, 436–440.
R. Handfield et al. / European Journal of Operational Research 197 (2009) 609–619 Petrovic, D., Petrovic, R., Vujosevic, M., 2001. Fuzzy models for the newsboy problem. International Journal Production Economics 45, 435–441. Petrovic, D., Roy, R., Petrovic, R., 1998. Modeling and simulation of a supply chain in an uncertain environment. European Journal of Operational Research 109, 200– 309. Petrovic, D., Roy, R., Petrovic, R., 1999. Supply chain modeling using fuzzy sets. International Journal Production Economics 59, 443–453. Vujosevic, M., Petrovic, D., Petrovic, R., 1996. EOQ formula when inventory cost is fuzzy. International Journal Production Economics 45, 499–504. Wang, J., Shu, Y.F., 2005. Fuzzy decision modeling for supply chain management. Fuzzy Sets and Systems 150, 107–127. Xie, Y., Petrovic, D., Burnham, K., 2006. A heuristic procedure for the two-level control of serial supply chains under fuzzy customer demand. International Journal Production Economics 102, 37–50.
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Yager, R.R., 1979. On the measure of fuzziness and negation part I: Membership in the unit interval. International Journal of General Systems 5, 221–229. Yimer, A.D., Demirli, K., 2004. Fuzzy modeling and simulation of a single item inventory system with variable demand. In: Annual Conference of the North American Fuzzy Information Processing Society – NAFIPS 2004, vol. 2, pp. 985– 989. Zadeh, L.A., 1978. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1, 3–28. Zimmermann, H.J., 2001. Fuzzy Set Theory and Its Application. International Thomson Publishing, Norwell, Massachusetts. Zipkin, P.H., 2000. Foundations of Inventory Management. McGraw-Hill.
