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An experience-based system supporting inventory planning: A fuzzy approach
Expert Systems with Applications 39 (2012) 6994–7003
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An experience-based system supporting inventory planning: A fuzzy approach Suphattra Ketsarapong a, Varathorn Punyangarm b, Kongkiti Phusavat c, Binshan Lin d,⇑ a
Department of Industrial Engineering, Faculty of Engineering, Sripatum University, Bangkok 10900, Thailand Department of Industrial Engineering, Faculty of Engineering, Srinakharinwirot University, Nakhonnayok 26120, Thailand c Department of Industrial Engineering, Faculty of Engineering, Center of Advanced Studies in Industrial Technology, Kasetsart University, Bangkok 10900, Thailand d Louisiana State University in Shreveport, BE321, Business School, Shreveport, LA 71115, USA b
a r t i c l e
i n f o
Keywords: Inventory planning Qualitative information Fuzzy set theory
a b s t r a c t The study aims to extend the Uncapacitated Fuzzy Single Item Lot Sizing Problem (known as F-USILSP) model and extend it for inventory planning. The F-USILSP model is a good choice when there is no statistical data collection, but where there is verbal or qualitative information from experts with experience. Previously, the mixed integer linear programming (MILP) relied on the crisp assumption which hinders the use of the F-USILSP. In this paper, a Possibility Approach is adapted to convert the F-USILSP to a mathematically solvable equivalent crisp USILSP (EC-USILSP). The EC-USILSP model is tested with a case. The organization under study is a petrochemical company power plant with trapezoidal fuzzy demand and triangular fuzzy unit price. The overall results show that the EC-USILSP is more practical and exhibits more flexibility when there is a need to add more realistic situations. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Lot sizing problems are production planning problems of order quantity between purchasing or production lots (Brahimi, Dauzere-Peres, Najid, & Nordli, 2006). Small lot sizes lead to many orders and low inventory levels while large lot sizes lead to few orders and high inventory levels (Lee, Kramer, & Hwang, 1991). The consideration of lot sizes is, therefore, an economic problem in that the objective of inventory models is to minimize total inventory cost, which comprises unit price, ordering cost, and inventory holding cost, while satisfying demand (Brahimi et al., 2006; Hilmola & Lorentz, 2011; Lee et al., 1991). The first inventory planning model, namely Economic Order Quantity (EOQ), was proposed by Harris (1913). It was used to find an optimal order quantity in the case of an uncapacitated single stage and single item of inventory control with a well-defined demand pattern. Wagner and Whitin (1958) proposed an inventory model with time-vary demand, namely dynamic lot sizing, and used dynamic programming techniques to find an optimal order quantity. Other subsequent inventory models have been developed, based on the above models (Askin & Goldberg, 2002; Barancsi et al., 1990). The USILSP is a type of inventory model with time-vary demand. Brahimi et al. (2006) stated that there are four basic formulations of the USILSP in the form of mixed integer linear programming or MILP; i.e., aggregate formulation (AGG), formulation without ⇑ Corresponding author. E-mail addresses: [email protected] (S. Ketsarapong), Punyangarm@ hotmail.com (V. Punyangarm), [email protected] (K. Phusavat), Binshan.Lin@ lsus.edu, [email protected] (B. Lin). 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2012.01.048
inventory variables (NIF), the shortest path formulation (SHP), and facility location-based formulation without inventory variables (FAL). In addition, Brahimi et al. (2006) added that USILSP modeling is popular for inventory planning for three reasons. Firstly, some industries can aggregate products to obtain a single product, for example, products that differ only in color can be treated as a single product. Secondly, the USILSP is a basic model but can be easily extended for more complex circumstances. Lastly, many complex lot sizing problems must employ USILSP as sub-problems (see Cattrysse, Maes, & Van Wassenhove, 1990; Merle, Goffin, Trouiller, & Vial, 1999). Typically, when a classical inventory model is used, the Crisp Deterministic Assumption is required. However, often the information can be uncertain such as a situation in which only qualitative information from experienced operators or personnel. As a result, the fuzzy set theory is applied to resolve this information uncertainty (Dubois & Prade, 1980; Gumus & Guneri, 2009; Ketsarapong & Punyangarm, 2010; Zadeh, 1965; Zimmermann, 1996). In addition, Tütüncü, Akoz, Apaydin, and Petrovic (2008) suggested that some uncertainty within inventory systems should not considers the probability applications. Therefore, since 1980s, the fuzzy set theory has been widely used in modeling of inventory systems when dealing with vagueness and uncertainty (Cakir & Canbolat, 2008; Chang, Yao, & Ouyang, 2006; Chen & Chang, 2008; Dutta, Chakraborty, & Roy, 2007; Green, Inman, & Birou, 2011; Halim, Girl, & Chaudhuri, 2011; Hsieh, 2002; Mandal, Roy, & Maiti, 2005; Pai, 2003; Yao & Lee, 1999). The rest of the paper is organized as follows: Section 2 presents the research premise; Section 3 presents the research objectives; Section 4 present the four steps of research methodology including,
S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003
(1) the data collection process, (2) developing an inventory model, transforming the F-USILSP to an EC-USILSP model, developing the EC-USILSP model in the form of MILP, and an illustrative numerical example, (3) data analysis process, and (4) the decision making process. In Section 5, the case application of a petrochemical company is presented. Finally, the last Sections are the conclusions and recommendations for future studies.
2. Research problem Generally, there are two types of uncertain parameter in a typical inventory model; i.e., fuzziness and randomness, which lead to the fuzzy inventory model and the stochastic inventory model respectively. Most stochastic inventory models are described by statistical or probabilistic information, which deals with the probability distributions of inventory parameters. Since statistical information is collected from a lot of sampling data, high cost and time lost for data collection must be considered. As a result, Small and Medium Enterprises (SMEs) are often faced with the situation which no reliable and quantitative data exist, despite abundant verbal information and experiences from key personnel or operators (Martz, Neil, & Biscaccianti, 2006). Moreover, extreme events such as natural disasters, resulting in rapid changes in demand, unit price, and/or ordering cost, prohibits the use of available data. When the above circumstance take place, an organization must be able to learn and adept itself (Melton, Chen, & Lin, 2006). Therefore, this study attempts to address the following research problem. How can an inventory planning model be developed to
provide information for an inventory planner to make decisions effectively? 3. Research objectives The objective of this research is to develop an inventory planning model to deal with a single item lot sizing problem with fuzzy parameters. This inventory planning model is designed in response to circumstances where there is lack of quantitative data but abundant experiences from operators or personnel. 4. Research methodology The research methodology consists of four parts; (4.1) the data collection process (4.2) a developing inventory model (4.3) computing results and (4.4) a decision making process (see Fig. 1). The steps of each part are as follows. 4.1. Data collection process The data collection process is a very important because the verbal information obtained from this process can be used as inputs in the F-USILSP model. Verbal information must be collected from experienced decision makers. Furthermore, a computer is an essential device for this process and is used for recording verbal or qualitative information. Therefore, with a lack of verbal information and without accurate computer records the F-USILSP model cannot be operated.
4.1 Data Collection Process
~
Fuzzy demand ( d )
Input
Fuzzy unit price ( ~ p )
Verbal information from experienced decision makers
s) Fuzzy ordering cost ( ~ ~
Fuzzy holding cost ( h ) Data 4.3 Computing Results
4.2 Developing Inventory modeling Step4: An illustrative numerical example Process
Step3: Developing EC-USILSP in a form of MILP Step2: Transforming F-USILSP to EC-USILSP
Computer Basic Software was used to find the optimal solution
Step1: Developing F-USILSP Model Information 4.4 Decision – Making Process The purchasing or producing quantity (x) Output Information for Inventory Planners
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The inventory level (I) The total cost (TC)
Fig. 1. Research methodology framework.
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subject to
4.2. Developing inventory model based on fuzzy parameters 4.2.1. Developing F-USILSP model The USILSP-AGG model is selected to show how a Possibility Approach can transform the fuzzy inventory model into an equivalent crisp inventory model (see the USILSP-AGG model in Appendix A). Let the unit price,ordering cost, inventory holding cost, and demand be fuzzy variables. Thus, the F-USILSP can be modeled as follows:
ðF USILSPÞ Min TC ¼
T X
~ t It Þ ~ t xt þ h ð~st yt þ p
ð1Þ
t¼1
subject to
~t ; 8t; It1 þ xt It ¼ d ~ > xt ; 8 t; yt d kT yt 2 binary variables; xt P 0;
ð2Þ It P 0;
ð3Þ ð4Þ
8t:
The F-USILSP in Eqs. (1)–(4) is in the form of fuzzy MILP. As a result, this model cannot be solved by classical mathematical methods.
p
T X ~ t It Þ 6 H ~ t xt þ h ð~st yt þ p
! P a;
8t;
ð7Þ
t¼1
pðIt1 þ xt It ¼ d~t Þ > a; 8t;
ð8Þ
pðyt d~kT P xt Þ > a; 8t;
ð9Þ
yt 2 binary variables; xt P 0; It P 0; H P 0; 8 t:
ð10Þ
4.2.3. Developing EC-USILSP in a form of MILP From Lemma 1(see details in Appendix B), the fuzzy constraint in Eq. (7) can be converted to be equivalent crisp constraint by (Eq. (B.6)), and it becomes an objective function by Eq. (5). The fuzzy constraints in Eqs. (8) and (9) can be converted to be an equivalent crisp constraint by (Eqs. (B.8) and (B.7)), respectively. So, the ECUSILSP in the form of MILP is as follows:
ðEC USILSPÞ Min TC ¼
T X ~t ÞL It ~t ÞLa xt þ ðh ð~st ÞLa yt þ ðp a
ð11Þ
t¼1
4.2.2. Transforming F-USILSP model to EC-USILSP model In this paper, Chance-Constrained Programming (CCP) (Charnes & Cooper, 1959), which is normally used to confront stochastic linear programming (SLP), is adopted as a way to convert F-USILSP to EC-USILSP. The concept of CCP guarantees that the probability of stochastic constraints is greater than or equal to a pre-specified ^ for j = 1, . . . , n and i = 1, ^ij , and b minimum probability. Let ^cj ; a i . . . , m be continuous random variables, and xj be decision variables. The relationship between standard SLP and its probability SLP is given by:
Min Z ¼
n X
^cj xj ;
subject to constraints;
^; ^ij xj 6 b a i
xj P 0 for all xj if and only if;
j¼1
Min Z ¼ f ;
Pr
n X
! ^cj xj 6 f
Pr
ð5Þ
! ^ ^ij xj 6 b a i
P 1 ai ;
8t;
ð12Þ
~ t ÞU ; It1 þ xt It < ðd a
8t;
ð13Þ
yt
T X
~ Þ U > xt ; ðd kT a
8t;
ð14Þ
k¼t
yt 2 binary variables; xt P 0; It P 0; 8 t:
ð15Þ
~ ÞU ¼ y is yt ðd kT a t
PT
a
a
~ U k¼t ðdk Þa .
P 1 a;
j¼1
n X
~ t ÞL ; It1 þ xt It P ðd a
~ ~ ¼ PT d ~ Note that focusing on Eq. (14), since d kT k¼t k and dk P 0 then ~ the summation of dk for t = 1, . . . , T can be computed based on an PT ~ L PT ~ U ~ ¼ d d ; extension principle as follows, d . That kT k k k¼t k¼t
j¼1 n X
subject to
f and xj P 0 for all xj ;
j¼1
where f is an artificial variable, which should be the minimum value when it is not greater than the objective function of standard SLP, Pr means probability, 1 a and 1 ai are pre-specified minimum probabilities. Researchers who are interested in the field of SLP can refer to Birge and Louveaux (1997) for further details. In the same way, let H be an artificial variable. The objective function of F-USILSP in Eq. (5) can be separated to be a crisp objective function and fuzzy constraint of possibility USILSP. All fuzzy constraints can be guaranteed to be greater than or equal to a pre-specified minimum possibility as follows:
ðPoss USILSPÞ Min TC ¼ H
ð6Þ
4.2.4. An illustrative numerical example The numerical example of F-USILSP with fuzzy parameters is illustrated to show how to use the EC-USILSP model. The details are as follows: Assume that the beginning and ending stock of the planning horizon are both zero. Let the unit price be classified as three ver~c Þ, normal ðp ~n Þ and expensive ðp ~e Þ. The debal levels, i.e., cheap ðp ~ Þ, medium mand is classified as three verbal levels, i.e., low ðd l ~m Þ, and high ðd ~ Þ. The ordering cost is classified as three verbal ðd h levels, i.e., low ð~sl Þ, medium ð~sm Þ, and high ð~sh Þ. The inventory hold~ Þ, medium ing cost is classified as three verbal levels, i.e., low ðh l ~m Þ, and high ðh ~ Þ. Based on the experience of the decision maker, ðh h the parameter levels of the F-USILSP in terms of verbal classification for five periods are shown in Table 1. From Table 1, the EC-USILSP of this numerical example is modeled in the form of MILP as follows:
Table 1 Verbal estimation of F-USILSP parameters for five periods. t
~Þ Unit price ðp
~ Demand ðdÞ
Ordering cost ð~sÞ
~ Holding cost ðhÞ
1
~m ¼ ð100; 120; 140; 160Þ d ~ ¼ ð140; 160; 180; 200Þ d h ~ ¼ ð60; 80; 100; 120Þ d
~sm ¼ ð300; 400; 500Þ ~sh ¼ ð400; 500; 600Þ
3
~c ¼ ð10; 15; 20; 25Þ p ~e ¼ ð30; 35; 40; 45Þ p ~c ¼ ð10; 15; 20; 25Þ p
~ ¼ ð3; 4; 5Þ h h ~ m ¼ ð2; 3; 4Þ h ~ ¼ ð3; 4; 5Þ h
4
~n ¼ ð20; 25; 30; 35Þ p
~sl ¼ ð200; 300; 400Þ
5
~e ¼ ð30; 35; 40; 45Þ p
~m ¼ ð100; 120; 140; 160Þ d ~ ¼ ð140; 160; 180; 200Þ d
2
l
h
~sl ¼ ð200; 300; 400Þ ~sh ¼ ð400; 500; 600Þ
h
~ m ¼ ð2; 3; 4Þ h ~ ¼ ð1; 2; 3Þ h l
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ðEC USILSPÞ Min TC ¼ ð~sm ÞLa y1 þ ð~sh ÞLa y2 þ ð~sl ÞLa y3 þ ð~sl ÞLa y4 L
L
L
~ c Þ a x1 þ ð p ~ e Þ a x2 þ ð~sh Þa y5 þ ðp ~c ÞLa x3 þ ðp ~n ÞLa x4 þ ðp ~e ÞLa x5 þ ðp ~ ÞL I1 þ ðh ~m ÞL I2 þ ðh ~ ÞL I3 þ ðh h a h a a ~ m ÞL I 4 þ ðh ~ ÞL I 5 þ ðh l a a
ð16Þ
Subject to
~ m ÞL ; x1 I 1 > ð d a ~ ÞU ; x I < ðd
ð17Þ
~ ÞL ; I1 þ x2 I2 > ðd h a ~ ÞU ; I1 þ x2 I2 < ðd
ð19Þ
~ ÞL ; I2 þ x3 I3 > ðd l a ~ I2 þ x3 I3 < ðd ÞU ;
ð21Þ
~ m ÞL ; I3 þ x4 I4 > ðd a ~ m ÞU ; I3 þ x4 I4 < ðd
ð23Þ
1
1
ð20Þ
ð22Þ
l a
ð24Þ
a
~ ÞL ; I 4 þ x5 > ð d h a ~ ÞU ; I 4 þ x5 < ð d
ð25Þ ð26Þ
h a
5 X
~ Þ U > x1 ; ðd k a
ð27Þ
5 X ~ Þ U > x2 ; ðd k a
ð28Þ
k¼1
y2
r
ð18Þ
m a
h a
y1
used (Liang & Cheng, 2009; Ordoobadi, 2008). Trapezoidal and triangular fuzzy numbers, which are basically fuzzy numbers because its membership function is a straight line, are focused in this sube be the trapezoidal fuzzy number with a lower and section. Let A upper crisp value at a = 0 and 1 at each corner points of a trapezoie L ; ð AÞ e L ; ð AÞ e U, dal membership function, which are denoted by ð AÞ 0 1 1 U e and ð AÞ0 , respectively (see Fig. 2a). Let le ðrÞ 2 ½0; 1, and r 2 R. Thus, the lower and upper crisp vale Aat each a-cut are respectively defined by ues of A
k¼2 5 X ~ Þ U > x3 ; y3 ðd k a
Lower=e A
e L a þ ð AÞ e L ð1 aÞ; and r ¼ ð AÞ 1 0
Upper=e A
e U a þ ð AÞ e U ð1 aÞ: ¼ ð AÞ 1 0 ð33Þ
e be a trapezoidal fuzzy number with ð BÞ e L ¼ ð BÞ e U ¼ ð BÞ e , then B e Let B 1 1 1 is called a triangular fuzzy number (see Fig. 2b). The lower and e at each a-cut are respectively defined by upper crisp value of B
r
Lower=e B
e a þ ð BÞ e L ð1 aÞ; ¼ ð BÞ 0 1
and r
Upper=e B
e a þ ð BÞ e U ð1 aÞ: ¼ ð BÞ 0 1 ð34Þ
~t ; ~st and h ~ t are trapezoi~t ; d If the fuzzy parameters of F-USILSP, i.e., p dal fuzzy numbers, then the general model of EC-USILSP for trapezoidal parameters is the following MILP.
ðEC USILSP TrapezoidalÞ Min TC T X að~st ÞL1 þ ð1 aÞð~st ÞL0 yt þ aðp~t ÞL1 þ ð1 aÞðp~t ÞL0 xt ¼ t¼1
~ t ÞL þ ð1 aÞðh ~t ÞL It þ aðh 1 0
ð35Þ
ð29Þ
k¼3
subject to
5 X ~ Þ U > x4 ; y4 ðd k a
ð30Þ
k¼4
~ 5 Þ > x5 ; y15 ðd a
ð31Þ
yt 2 binary variables; xt P 0; It P 0 for t ¼ 1; 2; . . . ; 5:
ð32Þ
U
The membership functions of fuzzy parameters can take several shapes, for example; S-curve representations, ‘bell’ curve representations, and triangular and trapezoidal representations (Cox, 1994). Two cases of the calculation procedure of EC-USILSP, which are shown in this paper, are (i) in cases where the membership function of fuzzy parameters are linear membership functions (see details in Section 4.2.4.1) and (ii) a cases where the membership function of fuzzy parameters are non-linear membership functions (see details in Section 4.2.4.2). 4.2.4.1. The first case: fuzzy parameters with linear membership function. The trapezoidal and triangular are the most frequently
~t ÞL þ ð1 aÞðd ~t ÞL ; 8t; It1 þ xt It P aðd 1 0 ~t ÞU þ ð1 aÞðd ~t ÞU ; 8t; It1 þ xt It 6 aðd 1 0 ~ ÞU þ ð1 aÞðd ~ ÞU P xt ; 8 t; yt aðd kT 1 kT 0
ð37Þ
yt 2 binary variables; xt and It 2 integer variables; 8t;
ð39Þ
ð36Þ
ð38Þ
Note: a triangular fuzzy number is a trapezoidal fuzzy number with ðÞ ~ L1 ¼ ðÞ ~ U1 . Thus, the EC-USILSP with triangular fuzzy parameters can be modeled in the same way as the EC-USILSP-trapezoidal. 4.2.4.2. The second case: fuzzy number with non-linear membership function. In this sub-section, F-USILSP with non-linear membership function of fuzzy parameters will be shown. Let the fuzzy parameters of F-USILSP be fuzzy numbers with S and Z membership functions at lower and upper spreads, respectively. To make it easier to refer to these fuzzy numbers, the big bell and small bell
Fig. 2. Membership function of (a) trapezoidal fuzzy number, and (b) triangular fuzzy number.
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S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003
Fig. 3. Membership function of (a) big bell fuzzy number, and (b) small bell fuzzy number.
will be called for fuzzy numbers with S and Z membership functions at lower and upper spreads respectively and the crisp value is an interval number and point number at a = 1 (see Fig. 3). Let e be the big bell fuzzy number with a lower and upper crisp value A at a = 0 and 1 at each corner point of S and Z membership funce L ; ð AÞ e L ; ð AÞ e U , and ð AÞ e U , respectively tions, which are denoted by ð AÞ 0 1 1 0 (see Fig. 3a). Let le ðrÞ 2 ½0; 1; and r 2 R. Thus, the lower and upper crisp vae Aat each a-cut are respectively defined by lue of A
subject to
8 pffiffia e L pffiffia e L > for a ¼ ½0; 0:5; < ð AÞ0 1 2 þ ð AÞ1 2 r ¼ ffiffiffiffiffiffi ffi q ffiffiffiffiffiffi ffi q e Lower= A > e L 1a þ ð AÞ e L 1 1a : ð AÞ for a ¼ ½0:5; 1; 0 1 2 2
yt 2 binary variables; xt and It 2 integer variables; 8t; ð40Þ
8 pffiffia e U pffiffia e U > for a ¼ ½0; 0:5; < ð AÞ0 1 2 þ ð AÞ1 2 ¼ r ffiffiffiffiffiffi ffi q ffiffiffiffiffiffi ffi q e Upper= A > e U 1a þ ð AÞ e U 1 1a : ð AÞ for a ¼ ½0:5; 1: 0 1 2 2
ð41Þ
e be a big bell fuzzy number with ð BÞ e L ¼ ð BÞ e U ¼ ð BÞ e , then B e is Let B 1 1 1 called a small bell fuzzy number (see Fig. 3b). The lower and upper e at each a-cut are respectively defined by crisp values of B
8 pffiffia e L e pffiffia > for a ¼ ½0; 0:5; < ð BÞ0 1 2 þ ð BÞ1 2 r ¼ ffiffiffiffiffiffi ffi q ffiffiffiffiffiffi ffi q Lower=e A > e L 1a þ ð BÞ e 1 1a : ð BÞ for a ¼ ½0:5; 1; 0 1 2 2 8 pffiffia e U e pffiffia > for a ¼ ½0; 0:5; < ð BÞ0 1 2 þ ð BÞ1 2 ¼ r ffiffiffiffiffiffi ffi q ffiffiffiffiffiffi ffi q e Upper= A > e U 1a þ ð BÞ : ð BÞ e 1 1a for a ¼ ½0:5; 1: 0 1 2 2
ð42Þ
ð43Þ
ðEC USILSP Big Bell;a ¼ ½0; 0:5ÞMin TC ! rffiffiffi rffiffiffi! T X a a L L ð~st Þ1 ð~st Þ0 yt ¼ þ 1 2 2 t¼1 ~t ÞL1 þ ðp
~ ÞL þ ðh t 1
rffiffiffi!
a 2
þ 1
a 2
2
þ 1
! !
a ~ L ðht Þ0 It 2
2 rffiffiffi
þ 1
a
a ~ L ðdt Þ0 ; 8t;
2 rffiffiffi!
ðEC USILSP Big Bell;a ¼ ½0:5; 1ÞMin TC rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffi! T X 1a 1a L L ð~st Þ1 þ ð~st Þ0 yt ¼ 1 2 2 t¼1 ! ! rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 1a 1a ~t ÞL0 ~t ÞL1 þ ðp þ 1 ðp xt 2 2 rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffi! ! 1a ~ L ~t ÞL 1 a It þ 1 ðht Þ1 þ ðh 0 2 2
ð45Þ ð46Þ ð47Þ ð48Þ
ð49Þ
subject to
rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi! 1a 1a ~ L þ 1 ðdt Þ0 ; 8t; 2 2 rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffi 1a ~ U ~t ÞU 1 a; 8t; ðdt Þ1 þ ðd It1 þ xt It 6 1 0 2 2 rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffi! 1a ~ U ~ ÞU 1 a P xt ; 8t; ðdkT Þ1 þ ðd yt 1 kT 0 2 2
yt 2 binary variables; xt
and It 2 integer variables; 8t;
ð50Þ ð51Þ ð52Þ ð53Þ
Note that a small bell fuzzy number is a big bell fuzzy number ~ L1 ¼ ðÞ ~ U1 . Thus, the EC-USILSP with small bell fuzzy paramwith ðÞ eters can be modeled in the same way as the EC-USILSP-Big Bell. After using the Possibility Approach to transform the F-USILSP to the EC-USILSP,fuzzy inventory parameters can be formed in terms of corner points for all membership functions (see Eqs. (35)–(39), (44)–(48) and (49)–(53)).
4.3. Computing results
!
a ~ L ðpt Þ0 xt
rffiffiffi!
rffiffiffi
a
a ~ U þ 1 ðdt Þ0 ; 8t; 2 2 ! ! rffiffiffi rffiffiffi a a ~ U aðd~kT ÞU1 þ 1 ðdkT Þ0 P xt ; 8t; 2 2
~ t ÞU It1 þ xt It 6 ðd 1 yt
rffiffiffi!
rffiffiffi
~ ÞL It1 þ xt It P ðd t 1
~t ; ~st and h ~ t are big ~t ; d When the fuzzy parameters of F-USILSP, i.e., p bell fuzzy numbers, then the general model of EC-USILSP for big bell fuzzy parameters is the following MILP.
rffiffiffi
~ ÞL It1 þ xt It P ðd t 1
ð44Þ
After using the Possibility Approach to transform the F-USILSP model to an EC-USILSP model in the form of MILP,this model can be solved with basic software. In this paper, Excel solver was used to find optimal solutions. The results of numerical examples are shown in Table 2. The results of this process provide useful information including the total cost, ordering variables in each period,
S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003
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Table 2 Results of numerical examples.
a
Case 1 Plan
Total cost
Plan
Total cost
0
Order 240 units at period 1 for period 1, and 2 Order 300 units at period 3 for period 3, 4, and 5
7320.00
Order 240 units at period 1 for period 1, and 2 Order 300 units at period 3 for period 3, 4, and 5
7320.00
0.1
Order 244 units at period 1 for period 1, and 2 Order 306 units at period 3 for period 3, 4, and 5
7789.80
Order 250 units at period 1 for period 1, and 2 Order 315 units at period 3 for period 3, 4, and 5
8386.54
0.2
Order 248 units at period 1 for period 1, and 2 Order 312 units at period 3 for period 3, 4, and 5
8271.20
Order 254 units at period 1 for period 1, and 2 Order 321 units at period 3 for period 3, 4, and 5
8892.69
0.3
Order 252 units at period 1 for period 1, and 2 Order 318 units at period 3 for period 3, 4, and 5
8764.20
Order 256 units at period 1 for period 1, and 2 Order 324 units at period 3 for period 3, 4, and 5
9222.41
0.4
Order 256 units at period 1 for period 1, and 2 Order 324 units at period 3 for period 3, 4, and 5
9268.80
Order 258 units at period 1 for period 1, and 2 Order 327 units at period 3 for period 3, 4, and 5
9515.19
0.5
Order 260 units at period 1 for period 1, and 2 Order 330 units at period 3 for period 3, 4, and 5
9785.00
Order 260 units at period 1 for period 1, and 2⁄ Order 330 units at period 3 for period 3, 4, and 5⁄
9785.00⁄
0.6
Order 264 units at period 1 for period 1, and 2 Order 336 units at period 3 for period 3, 4, and 5
10,312.80
Order 264 units at period 1 for period 1, and 2 Order 336 units at period 3 for period 3, 4, and 5
10,134.90
0.7
Order 268 units at period 1 for period 1, and 2 Order 342 units at period 3 for period 3, 4, and 5
10,852.20
Order 266 units at period 1 for period 1, and 2 Order 339 units at period 3 for period 3, 4, and 5
10,439.43
0.8
Order 272 units at period 1 for period 1, and 2 Order 348 units at period 3 for period 3, 4, and 5
11,403.20
Order 268 units at period 1 for period 1, and 2 Order 342 units at period 3 for period 3, 4, and 5
10,790.11
0.9
Order 276 units at period 1 for period 1, and 2 Order 354 units at period 3 for period 3, 4, and 5
11,965.80
Order 272 units at period 1 for period 1, and 2 Order 348 units at period 3 for period 3, 4, and 5
11,311.51
1
Order 280 units at period 1 for period 1, and 2 Order 360 units at period 3 for period 3, 4, and 5
12,540.00
Order 280 units at period 1 for period 1, and 2 Order 360 units at period 3 for period 3, 4, and 5
12,540.00
Case 2
Note: ⁄ refers to the equal results from Eqs. (44)–(48) and (49)–(53).
purchasing quantity in each period, and the inventory level at the end of each period,and this information can be used in the decision making process. From Table 2, the trend of total cost from EC-USILSP, solved with a = 0, 0.1, 0.2, . . . , 1, is an increasing trend, which is illustrated by plotting TC and a in x and y axis, respectively. It means that when higher possible levels are required, the total cost will increase. In addition, the differences of membership functions result in differences in the total cost at each a-level. The total cost of both cases will be equal at a = 0, then the rate of increase in the initial phase of case 2 increases faster than in case 1 until at a = 0.5 (the change curve of case 2), the total cost of two cases are equal (total cost = 9785.00) and the rate of increase in the total cost of case 2 is lower than incase 1 until at a = 1, where the total cost of both cases are equal. 4.4. Decision making process Inventory planners can make decisions under the acceptable possible levels (a). The effectiveness of the decision making process in inventory planning depends on verbal information from experienced decision makers and the F-USILSP model. For example, if the experienced personnel give erroneous information, inventory planning errors can be made. 5. The case study of a petrochemical company The case study of this paper focuses on the inventory planning of bituminous coal in a power plant of a petrochemical company. The main purpose of this power plant is to produce electrical power for the company’s petrochemical plants, which is the core business of this company. Bituminous coal is the main raw material for the generation of electrical power. Demand for bituminous coal,measured in metric tons, is dependent on the demand for electrical power, which is required for petrochemical production. In this study, the demand for bituminous coal is classified as 5
trapezoidal fuzzy numbers, i.e., lowest, low, medium, high, and highest (see the membership function of fuzzy demand in Fig. 4). Based on the experience of the production manager and the fuzzy information in Fig. 3, the demand for bituminous coal in the planning horizon (12 periods) is predicted in the form of verbal demand, which is shown in Table 3. The price of bituminous coal per metric ton is classified as 7 triangular fuzzy numbers, i.e., the cheapest, very cheap, cheap, normal, expensive, very expensive, the most expensive (see the membership function of fuzzy unit prices in Fig. 5). Based on the experience of the purchasing manager and the fuzzy information in Fig. 5, the unit price of bituminous coal in the planning horizon is predicted in the form of verbal unit prices, which are shown in Table 3. The company has two outsourcers, companies A and B, which transport bituminous coal from the distributor to the inventory area. There is a large area for the storage of 80,000 metric tons of bituminous coal. According to company policies, 20,000 metric tons of bituminous coal must be stored as a safety stock. It cannot be used up. Thus, the net inventory space is equal to 60,000 metric tons. The inventory level at the end of period 0, or the beginning stock of period 1, is equal to 15,000 tons. So the net demand of period 1 is equal to (20,000, 25,000, 30,000, 35,000) metric tons minus 15,000 metric tons, that is (5000, 10,000, 15,000, 20,000) metric tons. Company A charges1 million Thai Baht (THB) per order, and the maximum transportation capacity of company A is 50,000 metric tons per period. Company B charges 1.5 million THB per order, and the maximum transportation capacity of company B is equal to that of company A. Note that although the quantity of each order is less than the maximum transportation capacity of each company, companies A and B cannot discount the charge rate per order. Based on the policies of the company, the holding cost is fixed as a crisp value. It is equal to 200 THB per metric ton per period. In addition, 30.5 THB is equivalent to $1US (as of May 2011). This case study is different from the numerical example. Since there are two transporters, and transportation capacity of them is limited, sometimes only company A can be selected (the service
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Fig. 4. Membership function of fuzzy demand.
Table 3 Demand (1000 metric tons) and unit price (100 THB per metric tons). Period t
1 2 3 4 5 6 7 8 9 10 11 12
Demand
Unit Price
Verbal prediction
Trapezoidal fuzzy number
Verbal prediction
Triangular fuzzy number
Low Low Medium Medium Medium Medium High High Medium Low Lowest Lowest
(20, 25, 30, 35) (20, 25, 30, 35) (30, 35, 40, 45) (30, 35, 40, 45) (30, 35, 40, 45) (30, 35, 40, 45) (40, 45, 50, 55) (40, 45, 50, 55) (30, 35, 40, 45) (20, 25, 30, 35) (15, 15, 20, 25) (15, 15, 20, 25)
Normal Cheap Cheap Very Cheap Cheap Cheap Normal Expensive Normal Expensive Normal Expensive
(20, 25, 30) (15, 20, 25) (15, 20, 25) (10, 15, 20) (15, 20, 25) (15, 20, 25) (20, 25, 30) (25, 30, 35) (20, 25, 30) (25, 30, 35) (20, 25, 30) (25, 30, 35)
Fig. 5. Membership function of fuzzy unit prices.
charge of company A is less than that of company B), but sometimes both companies must be selected (when the buying quantity exceeds the transportation capacity of company A). Therefore, the decision variables are as follows: t is the tth period in the planning horizon t = 1, 2, . . ., 12, xtA and xtBare the buying quantities which are transported by companies A and B in period t, respectively. Binary variable ytA and ytB are ordering variable by companies A and B in period t, and It be the inventory level at the end of period t. Based on transportation capacity limitations, the maximum transportation capacity of companies A and B is 50,000 metric tons per period. The constraint for the control ordering variable can be formulated as follows:
50; 000ytA > xtA
and 50; 000ytB P xtB
for t ¼ 1; . . . ; 12:
ð54Þ
Focusing on inventory space limitations,the bituminous coal cannot be stored at a quantity greater than 60,000 metric tons. The warehouse capacity constraints are thus added as follows:
It1 þ xtA þ xtB 6 60; 000 for t ¼ 1; . . . ; 12:
ð55Þ
Substitute inventory parameters, decision variables, and add Eqs. (54) and (55) in the EC-USILSP model. The results at a = 0 and 1 are shown in Table 4. If a decision maker wants to plan for the lowest total cost, the lowest possible level (a = 0) is chosen. The inventory plan orders 5000, 20,000, and 30,000 metric tons of bituminous coal, which is transported by company A, in periods one, two, and three, respectively. To serve the demand of periods four and five, 60,000 metric tons must be refilled. Since the maximum transportation capacity of company A is less than 60,000 metric tons, an
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S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003 Table 4 Inventory plan at a = 0 and 1. Company
a = 0 and TC = 540,000,000 y
X
I
Y
x
I
1
A B
1 0
5000 0
0
1 0
10,000 0
0
2
A B
1 0
20,000 0
0
1 0
25,000 0
0
3
A B
1 0
30,000 0
0
1 0
35,000 0
0
4
A B
1 1
50,000 10,000
30,000
1 1
50,000 10,000
25,000
5
A B
0 0
0 0
0
1 0
10,000 0
0
6
A B
1 1
50,000 10,000
30,000
1 1
50,000 10,000
25,000
7
A B
1 0
30,000 0
20,000
1 0
35,000 0
15,000
8
A B
1 0
20,000 0
0
1 0
30,000 0
0
9
A B
1 0
50,000 0
20,000
1 1
50,000 10,000
25,000
10
A B
0 0
0 0
0
0 0
0 0
0
11
A B
1 0
30,000 0
15,000
1 0
30,000 0
15,000
12
A B
0 0
0 0
0
0 0
0 0
0
Period t
excess demand (10,000 metric tons) must be transported by company B. In period six, 60,000 metric tons of bituminous coal is ordered. An aggregate total of 30,000 metric tons, which is ordered in period seven, and 30,000 metric tons, which was transferred from period six, must be used to serve the demand of periods seven and eight. In period nine, 50,000 metric tons is ordered to cover periods nine and ten, and 30,000 metric tons is ordered in period eleven to cover the demand of the last two periods. The total cost is 540 million THB. On the other hand, when the highest possible level (a = 1) is required, the total cost increases to 813 million THB. The inventory plan of this scenario orders 10,000, 25,000, and 35,000 metric tons in periods one, two, and three, respectively. To serve the demand of periods six, seven, and eight, 60,000 and 35,000 metric tons is ordered respectively, in periods six and seven. In period nine, 60,000 metric tons is ordered to cover periods nine and ten, and 30,000 metric tons is ordered in period eleven to cover the last two periods.
a = 1 and TC = 813,000,000
The Possibility Approach was used to transform the F-USILSP model to the EC-USILSP model. As a result, this model can be solved by using basic software. Excel solver can be used to find an optimal solution effectively and provide useful information including total cost, ordering variables, purchasing and production quantity, and the inventory level. Furthermore, the result of Excel solver processing also showed that when the higher possible levels (a) are required the total cost will increase accordingly. Planners can therefore make decisions under the possible levels (a) that can be accepted. This article has several important implications. The EC-USILSP model is very flexible, so other conditions can be added to this model tobe consistent with events occurring in each business. For example, in the case study of a petrochemical company, restrictions on two issues were added: transportation capacity limitations and inventory space limitations in the EC-USILSP model to better represent the actual situation. 7. Future studies
6. Conclusions The F-USILSP model was developed mainly for inventory planning. This model is suitable for inventory planning in cases where there is no information from statistical data collection, but there is verbal information from experienced operators or personnel. Such situations frequently occur for SMEs or in cases where there is information from statistical data collection which cannot be used for inventory planning due to unexpected circumstances such as natural disasters resulting in rapid changes in demand, unit price, or ordering cost.
In this paper, SILSP with fuzzy parameters, which is a small problem, was selected as the sample to study. Future work could focus on, firstly, using other more sophisticated inventory models, such as a multiple item lot sizing problem in a single-stage and multiple-stage production; and secondly, a study of SILSP over several periods, where these problems would be more complex. Consequently, solving these problems requires a lot of time. For future work, comparisons should therefore be made between ‘optimization methods and Meta Heuristics’and‘optimization methods or Meta Heuristics’ to test which is more appropriate in these cases.
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Lastly, this article focuses only on inventory models with fuzzy parameters, but some parameters of inventory models may have variations in other characteristics, such as random variables or random fuzzy variables. In future studies,models should therefore be developed to support other parameters. Appendix A. USILSP-AGG model In this research, the AGG of USILSP in a form of MILP is selected for developing F-USILSP. The AGG of USILSP is shown as follow: A.1. Notations The following notations are used in the paper: t yt St xt It pt ht dt dkT
the tth period in the planning horizon t = 1, 2, . . ., T the ordering (or setup) variable, which is 1 when order or setup occurs in period t, and 0 otherwise the ordering (or setup) cost in period t the purchasing (or producing) quantity in period t the inventory level at the end of period t the unit price (or production cost) in period t the inventory holding cost in period t the demand in period t P the sufficiently large number, where dkT ¼ Tt¼1 dt ; 8t
classification shows that when an event is classified by the judgment of decision maker(s), an exact set boundary is inappropriate. On the other hand, the fuzzy set boundary is not an exact demarcation but an area of boundary. Members in a fuzzy set must be linked with a degree of possibility in the range of [0, 1], which is called a membership function. Fuzzy variables have membership functions as indicators of degrees of truth. Therefore, in vague situations, classical mathematical operations or arithmetic cannot be employed. One of the basic methods of fuzzy set theory which relies on a crisp mathematical concept applied to the fuzzy set is ‘‘the extension principle’’, as follows: Definition 1. Let X be a Cartesian product of universes X = e1; . . . ; A e r be r fuzzy set in X1, . . . , Xr, respectively. X1 x, . . . , x Xr and A f is a mapping from X to a universe Y, y = f(x1, . . . , xr). Then the e in Y by extension principle allows us to define a fuzzy set B
e ¼ fðy; l ðyÞÞ=y ¼ f ðx1 ; . . . ; xr Þ; ðx1 ; . . . ; xr Þ 2 Xg; B e
ðB:1Þ
B
where
leB ðyÞ ¼
8 < :
sup ðx1 ;...xr Þ2f 1 ðyÞ
0;
fminðle ðx1 Þ; . . . ; le ðxr ÞÞg; A1
Ar
f 1 ðyÞ – h; ;
Otherwise; ðB:2Þ
1
Focusing on the AGG model in USILSP, it is assumed that there are no beginning and ending stocks of the planning horizon (I0 = IT = 0), and inventory parameters i.e.,st, pt, ht, and dt are deterministic. The AGG of USILSP in a form of MILP is as follows:
ðUSILSP AGGÞ Min TC ¼
T X
ðst yt þ pt xt þ ht It Þ
ðA:1Þ
t¼1
subject to
It1 þ xt It ¼ dt ; yt dkT P xt ;
8t;
8t;
yt 2 binary variables; xt P 0; It P 0; 8 t:
ðA:2Þ ðA:3Þ ðA:4Þ
The objective function of USILSP-AGG in equation (A.1) is to minimize the total cost over the t period horizon. Constraints in equation (A.2) are inventory balance equations. Demand in period t(dt) can be satisfied when it is equal to the beginning inventory of period t(It1) plus the purchasing or producing quantities in period t(xt) minus the available on-hand inventory at the end of period t(It). Constraints in equation (A.3) are used to control ordering variables. Since dkT is a large number, then yt is treated as 1 when inventory is ordered in period t(xt > 0), and yt is treated as 0 when inventory is not ordered in period t(xt = 0) Appendix B. Fuzzy set background Fuzzy set theory was first proposed by Zadeh (1965). It is a mathematical tool to describe imprecision in a fuzzy environment. Imprecision refers to the sense of vagueness rather than a lack of knowledge about the value of parameters. The vagueness is due to the unique experiences and judgments of decision makers. For example, demand is classified as two verbal sets: medium and high. When a classical set is chosen to classify this demand as verbal classes, the range of 10,000–15,000 units of demand is defined as medium, and 15,001–20,000 units of demand is defined as high. The exact boundary of medium and high is 15,000 units of demand. It means that 15,000 units of demand is assigned to the medium set, but 15,001 units of demand is assigned to the high set. This
where f is the inverse off. Possibility theory in the context of fuzzy set theory was introduced by Zadeh (1978) to deal with non-stochastic imprecision and vagueness. Possibility theory and probability theory are similar because both theories are based on set functions. However, possibility theory differs from probability theory in the use of dual set functions, called possibility and necessity measures. Both measures can be defined as the basis of fully-fledged representation graded semantics for language statements on possibility distributions (Dubois & Prade, 1980). Furthermore, Nguyen and BouchonMeunier (2003) stated the theory of large deviations in probability theory also handles set functions that have similar possibility measures. This is consistent with Dubois (2006) who stated that possibility theory refers to possibility and necessity measures in the study of ‘‘maxitive and minitive’’ set functions, respectively. The possibility degree of a disjunction of events defines the maximum possibility degree of events in the disjunction. On the other hand,the necessity degree of a conjunction of events defines the minimum necessity degree of events in the conjunction. These basic concepts are simple and suitable to be applied to model various kinds of information items with linguistic information, and uncertain formulae in logical settings (Dubois, 2006; Dubois & Prade, 1988). The possibility theory is as follow. ~ be fuzzy variables, a ~0 be the complements of a ~ and b ~0 and b ~ Let a ~ respectively. Operation ⁄ means any one of operators >, P, =, and b, ~ are ~b 6, <. The possibility and necessity measure of fuzzy event a respectively defined by Zimmermann, 1996.
~ ¼ supfminðl ðxÞ; l~ ðyÞÞ=x y; x; y 2 Rg; pða~ bÞ ~ a b ~0 Þ; ~ ¼ 1 pða ~0 b ~ bÞ Nessða
ðB:3Þ ðB:4Þ
where pðÞ ~ and NessðÞ ~ means possibility and necessity of fuzzy ~ variable . ~ be~i for i = 1, . . ., n be n fuzzy variables, the right hand side b Let a n comes to a crisp variable, and f : R ! R be a real value function. ~i Þ b is defined by (Zimmermann, The possibility of fuzzy event f ða 1996).
pðf ða~i Þ bÞ ¼ supxi 2R fmin16i6n fla~i ðxi Þg=f ðxi Þ b; xi ; b 2 R; 8ig ðB:5Þ
S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003
From equation (B.5), the fuzzy ranking condition can be proved to be Lemma 1 based on possibility measure, as follows: ~i for i = 1, . . . , n be fuzzy variables with normal and Lemma 1. Let a convex membership functions and b be a crisp variable. The lower and ~i are denoted by ða ~i ÞLa and ða ~i ÞUa , upper bounds of the a-level set of a respectively. Then, for any given possibility levels a1, a2 and a3 with 0 6 a1, a2, a3 6 1, (i)
(ii)
pða~1 þ þ a~n 6 bÞ P a1 iff ða~1 ÞLa1 þ þ ða~n ÞLa1 6 b;
ðB:6Þ
pða~1 þ þ a~n P bÞ P a2 iff ða~1 ÞUa2 þ þ ða~n ÞUa2 P b; ðB:7Þ
(iii)
pða~1 þ þ a~n ¼ bÞ P a3 iff ða~1 ÞLa3 þ þ ða~n ÞLa3 6 b ~1 ÞUa þ þ ða ~n ÞUa P b: ðB:8Þ and ða 3 3
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Dubois, D., & Prade, H. (1988). Possibility theory. New York: Plenum Press. Dutta, P., Chakraborty, D., & Roy, A. R. (2007). Continuous review inventory model in mixed fuzzy and stochastic environment. Applied Mathematics and Computation, 188(1), 970–980. Green, K. W., Jr., Inman, A., & Birou, L. M. (2011). Impact of JIT-selling strategy on organizational structure. Industrial Management & Data Systems, 111(1), 63–83. Gumus, A. T., & Guneri, A. F. (2009). A multi-echelon inventory management framework for stochastic and fuzzy supply chain. Expert Systems with Applications, 36, 5565–5575. Halim, K. A., Girl, B. C., & Chaudhuri, K. S. (2011). Fuzzy production planning models for an unreliable production system with fuzzy production rate and stochastic/ fuzzy demand rate. International Journal of Industrial Engineering Computations, 2(1), 179–192. Harris, F. W. (1913). How many parts to make at once. The Magazine of Management, 10(2), 135–136. Hilmola, O-P., & Lorentz, H. (2011). Warehousing in Northern Europe: longitudinal survey findings. Industrial Management & Data Systems, 110(3), 320–340. Hsieh, C. H. (2002). Optimization of fuzzy production inventory models. Information Sciences, 146, 29–40. Ketsarapong, S., & Punyangarm, V. (2010). An application of fuzzy data envelopment analytical hierarchy process for reducing defects in the production of liquid medicine. Industrial Engineering & Management System, 9(3), 251–261. Lee, Y. Y., Kramer, B. A., & Hwang, C. L. (1991). A comparative study of three lotsizing methods for the case of fuzzy demand. International Journal of Operations & Production Management, 11(7), 72–80. Lertworasirikul, S., Fang, S. C., Joines, J. A., & Nuttle, H. L. W. (2003). Fuzzy data envelopment analysis (DEA): A possibility approach. Fuzzy Sets and Systems, 139(2), 379–394. Liang, T. F., & Cheng, H. W. (2009). Application of fuzzy sets to manufacturing/ distribution planning decisions with multi-product and multi-time period in supply chains. Expert Systems with Applications, 36, 3367–3377. Mandal, N. K., Roy, T. K., & Maiti, M. (2005). Multi-objective fuzzy inventory model with three constraints: a geometric programming approach. Fuzzy Sets and Systems, 150(1), 87–106. Martz, W., Neil, T., & Biscaccianti, A. (2006). Exploring entrepreneurial decisionmaking strategies. International Journal of Innovation and Learning, 3(6), 658–672. Melton, C., Chen, J., & Lin, B. (2006). Organisational knowledge and learning: leveraging it to accelerate the creation of competitive advantages. International Journal of Innovation and Learning, 3(3), 254–266. Merle, O. D., Goffin, J. L., Trouiller, C., & Vial, J. P. (1999). A Lagrangian relaxation of the capacitated multi-item lot sizing problem solved with an interior point cutting plane algorithm. In P. M. Pardalos (Ed.), Approximation and complexity in numerical optimization: Continuous and discrete problem. Kluwer Academic Publishers. Nguyen, H. T., & Bouchon-Meunier, B. (2003). Random sets and large deviations principle as a foundation for possibility measures. Soft Computing, 8, 61–70. Ordoobadi, S. M. (2008). Fuzzy logic and evaluation of advanced technologies. Industrial Management & Data Systems, 108(7), 928–946. Pai, P. F. (2003). Lot size problems with fuzzy capacity. Mathematical and Computer Modeling, 38(5–6), 661–669. Tütüncü, C. Y., Akoz, O., Apaydin, A., & Petrovic, D. (2008). Continuous review inventory control in the presence of fuzzy costs. International Journal of Production Economics, 113(2), 775–784. Wagner, H. M., & Whitin, T. M. (1958). Version of the economic lot sizing model. Management Science, 5(1), 89–96. Yao, J. S., & Lee, H. M. (1999). inventory with or without backorder for fuzzy order quantity with trapezoid fuzzy number. Fuzzy Sets and Systems, 105(3), 311–337. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1(1), 3–28. Zimmermann, H. J. (1996). Fuzzy set theory and its application. London: Kluwer Academic Publishers.
Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
An experience-based system supporting inventory planning: A fuzzy approach Suphattra Ketsarapong a, Varathorn Punyangarm b, Kongkiti Phusavat c, Binshan Lin d,⇑ a
Department of Industrial Engineering, Faculty of Engineering, Sripatum University, Bangkok 10900, Thailand Department of Industrial Engineering, Faculty of Engineering, Srinakharinwirot University, Nakhonnayok 26120, Thailand c Department of Industrial Engineering, Faculty of Engineering, Center of Advanced Studies in Industrial Technology, Kasetsart University, Bangkok 10900, Thailand d Louisiana State University in Shreveport, BE321, Business School, Shreveport, LA 71115, USA b
a r t i c l e
i n f o
Keywords: Inventory planning Qualitative information Fuzzy set theory
a b s t r a c t The study aims to extend the Uncapacitated Fuzzy Single Item Lot Sizing Problem (known as F-USILSP) model and extend it for inventory planning. The F-USILSP model is a good choice when there is no statistical data collection, but where there is verbal or qualitative information from experts with experience. Previously, the mixed integer linear programming (MILP) relied on the crisp assumption which hinders the use of the F-USILSP. In this paper, a Possibility Approach is adapted to convert the F-USILSP to a mathematically solvable equivalent crisp USILSP (EC-USILSP). The EC-USILSP model is tested with a case. The organization under study is a petrochemical company power plant with trapezoidal fuzzy demand and triangular fuzzy unit price. The overall results show that the EC-USILSP is more practical and exhibits more flexibility when there is a need to add more realistic situations. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Lot sizing problems are production planning problems of order quantity between purchasing or production lots (Brahimi, Dauzere-Peres, Najid, & Nordli, 2006). Small lot sizes lead to many orders and low inventory levels while large lot sizes lead to few orders and high inventory levels (Lee, Kramer, & Hwang, 1991). The consideration of lot sizes is, therefore, an economic problem in that the objective of inventory models is to minimize total inventory cost, which comprises unit price, ordering cost, and inventory holding cost, while satisfying demand (Brahimi et al., 2006; Hilmola & Lorentz, 2011; Lee et al., 1991). The first inventory planning model, namely Economic Order Quantity (EOQ), was proposed by Harris (1913). It was used to find an optimal order quantity in the case of an uncapacitated single stage and single item of inventory control with a well-defined demand pattern. Wagner and Whitin (1958) proposed an inventory model with time-vary demand, namely dynamic lot sizing, and used dynamic programming techniques to find an optimal order quantity. Other subsequent inventory models have been developed, based on the above models (Askin & Goldberg, 2002; Barancsi et al., 1990). The USILSP is a type of inventory model with time-vary demand. Brahimi et al. (2006) stated that there are four basic formulations of the USILSP in the form of mixed integer linear programming or MILP; i.e., aggregate formulation (AGG), formulation without ⇑ Corresponding author. E-mail addresses: [email protected] (S. Ketsarapong), Punyangarm@ hotmail.com (V. Punyangarm), [email protected] (K. Phusavat), Binshan.Lin@ lsus.edu, [email protected] (B. Lin). 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2012.01.048
inventory variables (NIF), the shortest path formulation (SHP), and facility location-based formulation without inventory variables (FAL). In addition, Brahimi et al. (2006) added that USILSP modeling is popular for inventory planning for three reasons. Firstly, some industries can aggregate products to obtain a single product, for example, products that differ only in color can be treated as a single product. Secondly, the USILSP is a basic model but can be easily extended for more complex circumstances. Lastly, many complex lot sizing problems must employ USILSP as sub-problems (see Cattrysse, Maes, & Van Wassenhove, 1990; Merle, Goffin, Trouiller, & Vial, 1999). Typically, when a classical inventory model is used, the Crisp Deterministic Assumption is required. However, often the information can be uncertain such as a situation in which only qualitative information from experienced operators or personnel. As a result, the fuzzy set theory is applied to resolve this information uncertainty (Dubois & Prade, 1980; Gumus & Guneri, 2009; Ketsarapong & Punyangarm, 2010; Zadeh, 1965; Zimmermann, 1996). In addition, Tütüncü, Akoz, Apaydin, and Petrovic (2008) suggested that some uncertainty within inventory systems should not considers the probability applications. Therefore, since 1980s, the fuzzy set theory has been widely used in modeling of inventory systems when dealing with vagueness and uncertainty (Cakir & Canbolat, 2008; Chang, Yao, & Ouyang, 2006; Chen & Chang, 2008; Dutta, Chakraborty, & Roy, 2007; Green, Inman, & Birou, 2011; Halim, Girl, & Chaudhuri, 2011; Hsieh, 2002; Mandal, Roy, & Maiti, 2005; Pai, 2003; Yao & Lee, 1999). The rest of the paper is organized as follows: Section 2 presents the research premise; Section 3 presents the research objectives; Section 4 present the four steps of research methodology including,
S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003
(1) the data collection process, (2) developing an inventory model, transforming the F-USILSP to an EC-USILSP model, developing the EC-USILSP model in the form of MILP, and an illustrative numerical example, (3) data analysis process, and (4) the decision making process. In Section 5, the case application of a petrochemical company is presented. Finally, the last Sections are the conclusions and recommendations for future studies.
2. Research problem Generally, there are two types of uncertain parameter in a typical inventory model; i.e., fuzziness and randomness, which lead to the fuzzy inventory model and the stochastic inventory model respectively. Most stochastic inventory models are described by statistical or probabilistic information, which deals with the probability distributions of inventory parameters. Since statistical information is collected from a lot of sampling data, high cost and time lost for data collection must be considered. As a result, Small and Medium Enterprises (SMEs) are often faced with the situation which no reliable and quantitative data exist, despite abundant verbal information and experiences from key personnel or operators (Martz, Neil, & Biscaccianti, 2006). Moreover, extreme events such as natural disasters, resulting in rapid changes in demand, unit price, and/or ordering cost, prohibits the use of available data. When the above circumstance take place, an organization must be able to learn and adept itself (Melton, Chen, & Lin, 2006). Therefore, this study attempts to address the following research problem. How can an inventory planning model be developed to
provide information for an inventory planner to make decisions effectively? 3. Research objectives The objective of this research is to develop an inventory planning model to deal with a single item lot sizing problem with fuzzy parameters. This inventory planning model is designed in response to circumstances where there is lack of quantitative data but abundant experiences from operators or personnel. 4. Research methodology The research methodology consists of four parts; (4.1) the data collection process (4.2) a developing inventory model (4.3) computing results and (4.4) a decision making process (see Fig. 1). The steps of each part are as follows. 4.1. Data collection process The data collection process is a very important because the verbal information obtained from this process can be used as inputs in the F-USILSP model. Verbal information must be collected from experienced decision makers. Furthermore, a computer is an essential device for this process and is used for recording verbal or qualitative information. Therefore, with a lack of verbal information and without accurate computer records the F-USILSP model cannot be operated.
4.1 Data Collection Process
~
Fuzzy demand ( d )
Input
Fuzzy unit price ( ~ p )
Verbal information from experienced decision makers
s) Fuzzy ordering cost ( ~ ~
Fuzzy holding cost ( h ) Data 4.3 Computing Results
4.2 Developing Inventory modeling Step4: An illustrative numerical example Process
Step3: Developing EC-USILSP in a form of MILP Step2: Transforming F-USILSP to EC-USILSP
Computer Basic Software was used to find the optimal solution
Step1: Developing F-USILSP Model Information 4.4 Decision – Making Process The purchasing or producing quantity (x) Output Information for Inventory Planners
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The inventory level (I) The total cost (TC)
Fig. 1. Research methodology framework.
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subject to
4.2. Developing inventory model based on fuzzy parameters 4.2.1. Developing F-USILSP model The USILSP-AGG model is selected to show how a Possibility Approach can transform the fuzzy inventory model into an equivalent crisp inventory model (see the USILSP-AGG model in Appendix A). Let the unit price,ordering cost, inventory holding cost, and demand be fuzzy variables. Thus, the F-USILSP can be modeled as follows:
ðF USILSPÞ Min TC ¼
T X
~ t It Þ ~ t xt þ h ð~st yt þ p
ð1Þ
t¼1
subject to
~t ; 8t; It1 þ xt It ¼ d ~ > xt ; 8 t; yt d kT yt 2 binary variables; xt P 0;
ð2Þ It P 0;
ð3Þ ð4Þ
8t:
The F-USILSP in Eqs. (1)–(4) is in the form of fuzzy MILP. As a result, this model cannot be solved by classical mathematical methods.
p
T X ~ t It Þ 6 H ~ t xt þ h ð~st yt þ p
! P a;
8t;
ð7Þ
t¼1
pðIt1 þ xt It ¼ d~t Þ > a; 8t;
ð8Þ
pðyt d~kT P xt Þ > a; 8t;
ð9Þ
yt 2 binary variables; xt P 0; It P 0; H P 0; 8 t:
ð10Þ
4.2.3. Developing EC-USILSP in a form of MILP From Lemma 1(see details in Appendix B), the fuzzy constraint in Eq. (7) can be converted to be equivalent crisp constraint by (Eq. (B.6)), and it becomes an objective function by Eq. (5). The fuzzy constraints in Eqs. (8) and (9) can be converted to be an equivalent crisp constraint by (Eqs. (B.8) and (B.7)), respectively. So, the ECUSILSP in the form of MILP is as follows:
ðEC USILSPÞ Min TC ¼
T X ~t ÞL It ~t ÞLa xt þ ðh ð~st ÞLa yt þ ðp a
ð11Þ
t¼1
4.2.2. Transforming F-USILSP model to EC-USILSP model In this paper, Chance-Constrained Programming (CCP) (Charnes & Cooper, 1959), which is normally used to confront stochastic linear programming (SLP), is adopted as a way to convert F-USILSP to EC-USILSP. The concept of CCP guarantees that the probability of stochastic constraints is greater than or equal to a pre-specified ^ for j = 1, . . . , n and i = 1, ^ij , and b minimum probability. Let ^cj ; a i . . . , m be continuous random variables, and xj be decision variables. The relationship between standard SLP and its probability SLP is given by:
Min Z ¼
n X
^cj xj ;
subject to constraints;
^; ^ij xj 6 b a i
xj P 0 for all xj if and only if;
j¼1
Min Z ¼ f ;
Pr
n X
! ^cj xj 6 f
Pr
ð5Þ
! ^ ^ij xj 6 b a i
P 1 ai ;
8t;
ð12Þ
~ t ÞU ; It1 þ xt It < ðd a
8t;
ð13Þ
yt
T X
~ Þ U > xt ; ðd kT a
8t;
ð14Þ
k¼t
yt 2 binary variables; xt P 0; It P 0; 8 t:
ð15Þ
~ ÞU ¼ y is yt ðd kT a t
PT
a
a
~ U k¼t ðdk Þa .
P 1 a;
j¼1
n X
~ t ÞL ; It1 þ xt It P ðd a
~ ~ ¼ PT d ~ Note that focusing on Eq. (14), since d kT k¼t k and dk P 0 then ~ the summation of dk for t = 1, . . . , T can be computed based on an PT ~ L PT ~ U ~ ¼ d d ; extension principle as follows, d . That kT k k k¼t k¼t
j¼1 n X
subject to
f and xj P 0 for all xj ;
j¼1
where f is an artificial variable, which should be the minimum value when it is not greater than the objective function of standard SLP, Pr means probability, 1 a and 1 ai are pre-specified minimum probabilities. Researchers who are interested in the field of SLP can refer to Birge and Louveaux (1997) for further details. In the same way, let H be an artificial variable. The objective function of F-USILSP in Eq. (5) can be separated to be a crisp objective function and fuzzy constraint of possibility USILSP. All fuzzy constraints can be guaranteed to be greater than or equal to a pre-specified minimum possibility as follows:
ðPoss USILSPÞ Min TC ¼ H
ð6Þ
4.2.4. An illustrative numerical example The numerical example of F-USILSP with fuzzy parameters is illustrated to show how to use the EC-USILSP model. The details are as follows: Assume that the beginning and ending stock of the planning horizon are both zero. Let the unit price be classified as three ver~c Þ, normal ðp ~n Þ and expensive ðp ~e Þ. The debal levels, i.e., cheap ðp ~ Þ, medium mand is classified as three verbal levels, i.e., low ðd l ~m Þ, and high ðd ~ Þ. The ordering cost is classified as three verbal ðd h levels, i.e., low ð~sl Þ, medium ð~sm Þ, and high ð~sh Þ. The inventory hold~ Þ, medium ing cost is classified as three verbal levels, i.e., low ðh l ~m Þ, and high ðh ~ Þ. Based on the experience of the decision maker, ðh h the parameter levels of the F-USILSP in terms of verbal classification for five periods are shown in Table 1. From Table 1, the EC-USILSP of this numerical example is modeled in the form of MILP as follows:
Table 1 Verbal estimation of F-USILSP parameters for five periods. t
~Þ Unit price ðp
~ Demand ðdÞ
Ordering cost ð~sÞ
~ Holding cost ðhÞ
1
~m ¼ ð100; 120; 140; 160Þ d ~ ¼ ð140; 160; 180; 200Þ d h ~ ¼ ð60; 80; 100; 120Þ d
~sm ¼ ð300; 400; 500Þ ~sh ¼ ð400; 500; 600Þ
3
~c ¼ ð10; 15; 20; 25Þ p ~e ¼ ð30; 35; 40; 45Þ p ~c ¼ ð10; 15; 20; 25Þ p
~ ¼ ð3; 4; 5Þ h h ~ m ¼ ð2; 3; 4Þ h ~ ¼ ð3; 4; 5Þ h
4
~n ¼ ð20; 25; 30; 35Þ p
~sl ¼ ð200; 300; 400Þ
5
~e ¼ ð30; 35; 40; 45Þ p
~m ¼ ð100; 120; 140; 160Þ d ~ ¼ ð140; 160; 180; 200Þ d
2
l
h
~sl ¼ ð200; 300; 400Þ ~sh ¼ ð400; 500; 600Þ
h
~ m ¼ ð2; 3; 4Þ h ~ ¼ ð1; 2; 3Þ h l
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ðEC USILSPÞ Min TC ¼ ð~sm ÞLa y1 þ ð~sh ÞLa y2 þ ð~sl ÞLa y3 þ ð~sl ÞLa y4 L
L
L
~ c Þ a x1 þ ð p ~ e Þ a x2 þ ð~sh Þa y5 þ ðp ~c ÞLa x3 þ ðp ~n ÞLa x4 þ ðp ~e ÞLa x5 þ ðp ~ ÞL I1 þ ðh ~m ÞL I2 þ ðh ~ ÞL I3 þ ðh h a h a a ~ m ÞL I 4 þ ðh ~ ÞL I 5 þ ðh l a a
ð16Þ
Subject to
~ m ÞL ; x1 I 1 > ð d a ~ ÞU ; x I < ðd
ð17Þ
~ ÞL ; I1 þ x2 I2 > ðd h a ~ ÞU ; I1 þ x2 I2 < ðd
ð19Þ
~ ÞL ; I2 þ x3 I3 > ðd l a ~ I2 þ x3 I3 < ðd ÞU ;
ð21Þ
~ m ÞL ; I3 þ x4 I4 > ðd a ~ m ÞU ; I3 þ x4 I4 < ðd
ð23Þ
1
1
ð20Þ
ð22Þ
l a
ð24Þ
a
~ ÞL ; I 4 þ x5 > ð d h a ~ ÞU ; I 4 þ x5 < ð d
ð25Þ ð26Þ
h a
5 X
~ Þ U > x1 ; ðd k a
ð27Þ
5 X ~ Þ U > x2 ; ðd k a
ð28Þ
k¼1
y2
r
ð18Þ
m a
h a
y1
used (Liang & Cheng, 2009; Ordoobadi, 2008). Trapezoidal and triangular fuzzy numbers, which are basically fuzzy numbers because its membership function is a straight line, are focused in this sube be the trapezoidal fuzzy number with a lower and section. Let A upper crisp value at a = 0 and 1 at each corner points of a trapezoie L ; ð AÞ e L ; ð AÞ e U, dal membership function, which are denoted by ð AÞ 0 1 1 U e and ð AÞ0 , respectively (see Fig. 2a). Let le ðrÞ 2 ½0; 1, and r 2 R. Thus, the lower and upper crisp vale Aat each a-cut are respectively defined by ues of A
k¼2 5 X ~ Þ U > x3 ; y3 ðd k a
Lower=e A
e L a þ ð AÞ e L ð1 aÞ; and r ¼ ð AÞ 1 0
Upper=e A
e U a þ ð AÞ e U ð1 aÞ: ¼ ð AÞ 1 0 ð33Þ
e be a trapezoidal fuzzy number with ð BÞ e L ¼ ð BÞ e U ¼ ð BÞ e , then B e Let B 1 1 1 is called a triangular fuzzy number (see Fig. 2b). The lower and e at each a-cut are respectively defined by upper crisp value of B
r
Lower=e B
e a þ ð BÞ e L ð1 aÞ; ¼ ð BÞ 0 1
and r
Upper=e B
e a þ ð BÞ e U ð1 aÞ: ¼ ð BÞ 0 1 ð34Þ
~t ; ~st and h ~ t are trapezoi~t ; d If the fuzzy parameters of F-USILSP, i.e., p dal fuzzy numbers, then the general model of EC-USILSP for trapezoidal parameters is the following MILP.
ðEC USILSP TrapezoidalÞ Min TC T X að~st ÞL1 þ ð1 aÞð~st ÞL0 yt þ aðp~t ÞL1 þ ð1 aÞðp~t ÞL0 xt ¼ t¼1
~ t ÞL þ ð1 aÞðh ~t ÞL It þ aðh 1 0
ð35Þ
ð29Þ
k¼3
subject to
5 X ~ Þ U > x4 ; y4 ðd k a
ð30Þ
k¼4
~ 5 Þ > x5 ; y15 ðd a
ð31Þ
yt 2 binary variables; xt P 0; It P 0 for t ¼ 1; 2; . . . ; 5:
ð32Þ
U
The membership functions of fuzzy parameters can take several shapes, for example; S-curve representations, ‘bell’ curve representations, and triangular and trapezoidal representations (Cox, 1994). Two cases of the calculation procedure of EC-USILSP, which are shown in this paper, are (i) in cases where the membership function of fuzzy parameters are linear membership functions (see details in Section 4.2.4.1) and (ii) a cases where the membership function of fuzzy parameters are non-linear membership functions (see details in Section 4.2.4.2). 4.2.4.1. The first case: fuzzy parameters with linear membership function. The trapezoidal and triangular are the most frequently
~t ÞL þ ð1 aÞðd ~t ÞL ; 8t; It1 þ xt It P aðd 1 0 ~t ÞU þ ð1 aÞðd ~t ÞU ; 8t; It1 þ xt It 6 aðd 1 0 ~ ÞU þ ð1 aÞðd ~ ÞU P xt ; 8 t; yt aðd kT 1 kT 0
ð37Þ
yt 2 binary variables; xt and It 2 integer variables; 8t;
ð39Þ
ð36Þ
ð38Þ
Note: a triangular fuzzy number is a trapezoidal fuzzy number with ðÞ ~ L1 ¼ ðÞ ~ U1 . Thus, the EC-USILSP with triangular fuzzy parameters can be modeled in the same way as the EC-USILSP-trapezoidal. 4.2.4.2. The second case: fuzzy number with non-linear membership function. In this sub-section, F-USILSP with non-linear membership function of fuzzy parameters will be shown. Let the fuzzy parameters of F-USILSP be fuzzy numbers with S and Z membership functions at lower and upper spreads, respectively. To make it easier to refer to these fuzzy numbers, the big bell and small bell
Fig. 2. Membership function of (a) trapezoidal fuzzy number, and (b) triangular fuzzy number.
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S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003
Fig. 3. Membership function of (a) big bell fuzzy number, and (b) small bell fuzzy number.
will be called for fuzzy numbers with S and Z membership functions at lower and upper spreads respectively and the crisp value is an interval number and point number at a = 1 (see Fig. 3). Let e be the big bell fuzzy number with a lower and upper crisp value A at a = 0 and 1 at each corner point of S and Z membership funce L ; ð AÞ e L ; ð AÞ e U , and ð AÞ e U , respectively tions, which are denoted by ð AÞ 0 1 1 0 (see Fig. 3a). Let le ðrÞ 2 ½0; 1; and r 2 R. Thus, the lower and upper crisp vae Aat each a-cut are respectively defined by lue of A
subject to
8 pffiffia e L pffiffia e L > for a ¼ ½0; 0:5; < ð AÞ0 1 2 þ ð AÞ1 2 r ¼ ffiffiffiffiffiffi ffi q ffiffiffiffiffiffi ffi q e Lower= A > e L 1a þ ð AÞ e L 1 1a : ð AÞ for a ¼ ½0:5; 1; 0 1 2 2
yt 2 binary variables; xt and It 2 integer variables; 8t; ð40Þ
8 pffiffia e U pffiffia e U > for a ¼ ½0; 0:5; < ð AÞ0 1 2 þ ð AÞ1 2 ¼ r ffiffiffiffiffiffi ffi q ffiffiffiffiffiffi ffi q e Upper= A > e U 1a þ ð AÞ e U 1 1a : ð AÞ for a ¼ ½0:5; 1: 0 1 2 2
ð41Þ
e be a big bell fuzzy number with ð BÞ e L ¼ ð BÞ e U ¼ ð BÞ e , then B e is Let B 1 1 1 called a small bell fuzzy number (see Fig. 3b). The lower and upper e at each a-cut are respectively defined by crisp values of B
8 pffiffia e L e pffiffia > for a ¼ ½0; 0:5; < ð BÞ0 1 2 þ ð BÞ1 2 r ¼ ffiffiffiffiffiffi ffi q ffiffiffiffiffiffi ffi q Lower=e A > e L 1a þ ð BÞ e 1 1a : ð BÞ for a ¼ ½0:5; 1; 0 1 2 2 8 pffiffia e U e pffiffia > for a ¼ ½0; 0:5; < ð BÞ0 1 2 þ ð BÞ1 2 ¼ r ffiffiffiffiffiffi ffi q ffiffiffiffiffiffi ffi q e Upper= A > e U 1a þ ð BÞ : ð BÞ e 1 1a for a ¼ ½0:5; 1: 0 1 2 2
ð42Þ
ð43Þ
ðEC USILSP Big Bell;a ¼ ½0; 0:5ÞMin TC ! rffiffiffi rffiffiffi! T X a a L L ð~st Þ1 ð~st Þ0 yt ¼ þ 1 2 2 t¼1 ~t ÞL1 þ ðp
~ ÞL þ ðh t 1
rffiffiffi!
a 2
þ 1
a 2
2
þ 1
! !
a ~ L ðht Þ0 It 2
2 rffiffiffi
þ 1
a
a ~ L ðdt Þ0 ; 8t;
2 rffiffiffi!
ðEC USILSP Big Bell;a ¼ ½0:5; 1ÞMin TC rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffi! T X 1a 1a L L ð~st Þ1 þ ð~st Þ0 yt ¼ 1 2 2 t¼1 ! ! rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 1a 1a ~t ÞL0 ~t ÞL1 þ ðp þ 1 ðp xt 2 2 rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffi! ! 1a ~ L ~t ÞL 1 a It þ 1 ðht Þ1 þ ðh 0 2 2
ð45Þ ð46Þ ð47Þ ð48Þ
ð49Þ
subject to
rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi! 1a 1a ~ L þ 1 ðdt Þ0 ; 8t; 2 2 rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffi 1a ~ U ~t ÞU 1 a; 8t; ðdt Þ1 þ ðd It1 þ xt It 6 1 0 2 2 rffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffi! 1a ~ U ~ ÞU 1 a P xt ; 8t; ðdkT Þ1 þ ðd yt 1 kT 0 2 2
yt 2 binary variables; xt
and It 2 integer variables; 8t;
ð50Þ ð51Þ ð52Þ ð53Þ
Note that a small bell fuzzy number is a big bell fuzzy number ~ L1 ¼ ðÞ ~ U1 . Thus, the EC-USILSP with small bell fuzzy paramwith ðÞ eters can be modeled in the same way as the EC-USILSP-Big Bell. After using the Possibility Approach to transform the F-USILSP to the EC-USILSP,fuzzy inventory parameters can be formed in terms of corner points for all membership functions (see Eqs. (35)–(39), (44)–(48) and (49)–(53)).
4.3. Computing results
!
a ~ L ðpt Þ0 xt
rffiffiffi!
rffiffiffi
a
a ~ U þ 1 ðdt Þ0 ; 8t; 2 2 ! ! rffiffiffi rffiffiffi a a ~ U aðd~kT ÞU1 þ 1 ðdkT Þ0 P xt ; 8t; 2 2
~ t ÞU It1 þ xt It 6 ðd 1 yt
rffiffiffi!
rffiffiffi
~ ÞL It1 þ xt It P ðd t 1
~t ; ~st and h ~ t are big ~t ; d When the fuzzy parameters of F-USILSP, i.e., p bell fuzzy numbers, then the general model of EC-USILSP for big bell fuzzy parameters is the following MILP.
rffiffiffi
~ ÞL It1 þ xt It P ðd t 1
ð44Þ
After using the Possibility Approach to transform the F-USILSP model to an EC-USILSP model in the form of MILP,this model can be solved with basic software. In this paper, Excel solver was used to find optimal solutions. The results of numerical examples are shown in Table 2. The results of this process provide useful information including the total cost, ordering variables in each period,
S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003
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Table 2 Results of numerical examples.
a
Case 1 Plan
Total cost
Plan
Total cost
0
Order 240 units at period 1 for period 1, and 2 Order 300 units at period 3 for period 3, 4, and 5
7320.00
Order 240 units at period 1 for period 1, and 2 Order 300 units at period 3 for period 3, 4, and 5
7320.00
0.1
Order 244 units at period 1 for period 1, and 2 Order 306 units at period 3 for period 3, 4, and 5
7789.80
Order 250 units at period 1 for period 1, and 2 Order 315 units at period 3 for period 3, 4, and 5
8386.54
0.2
Order 248 units at period 1 for period 1, and 2 Order 312 units at period 3 for period 3, 4, and 5
8271.20
Order 254 units at period 1 for period 1, and 2 Order 321 units at period 3 for period 3, 4, and 5
8892.69
0.3
Order 252 units at period 1 for period 1, and 2 Order 318 units at period 3 for period 3, 4, and 5
8764.20
Order 256 units at period 1 for period 1, and 2 Order 324 units at period 3 for period 3, 4, and 5
9222.41
0.4
Order 256 units at period 1 for period 1, and 2 Order 324 units at period 3 for period 3, 4, and 5
9268.80
Order 258 units at period 1 for period 1, and 2 Order 327 units at period 3 for period 3, 4, and 5
9515.19
0.5
Order 260 units at period 1 for period 1, and 2 Order 330 units at period 3 for period 3, 4, and 5
9785.00
Order 260 units at period 1 for period 1, and 2⁄ Order 330 units at period 3 for period 3, 4, and 5⁄
9785.00⁄
0.6
Order 264 units at period 1 for period 1, and 2 Order 336 units at period 3 for period 3, 4, and 5
10,312.80
Order 264 units at period 1 for period 1, and 2 Order 336 units at period 3 for period 3, 4, and 5
10,134.90
0.7
Order 268 units at period 1 for period 1, and 2 Order 342 units at period 3 for period 3, 4, and 5
10,852.20
Order 266 units at period 1 for period 1, and 2 Order 339 units at period 3 for period 3, 4, and 5
10,439.43
0.8
Order 272 units at period 1 for period 1, and 2 Order 348 units at period 3 for period 3, 4, and 5
11,403.20
Order 268 units at period 1 for period 1, and 2 Order 342 units at period 3 for period 3, 4, and 5
10,790.11
0.9
Order 276 units at period 1 for period 1, and 2 Order 354 units at period 3 for period 3, 4, and 5
11,965.80
Order 272 units at period 1 for period 1, and 2 Order 348 units at period 3 for period 3, 4, and 5
11,311.51
1
Order 280 units at period 1 for period 1, and 2 Order 360 units at period 3 for period 3, 4, and 5
12,540.00
Order 280 units at period 1 for period 1, and 2 Order 360 units at period 3 for period 3, 4, and 5
12,540.00
Case 2
Note: ⁄ refers to the equal results from Eqs. (44)–(48) and (49)–(53).
purchasing quantity in each period, and the inventory level at the end of each period,and this information can be used in the decision making process. From Table 2, the trend of total cost from EC-USILSP, solved with a = 0, 0.1, 0.2, . . . , 1, is an increasing trend, which is illustrated by plotting TC and a in x and y axis, respectively. It means that when higher possible levels are required, the total cost will increase. In addition, the differences of membership functions result in differences in the total cost at each a-level. The total cost of both cases will be equal at a = 0, then the rate of increase in the initial phase of case 2 increases faster than in case 1 until at a = 0.5 (the change curve of case 2), the total cost of two cases are equal (total cost = 9785.00) and the rate of increase in the total cost of case 2 is lower than incase 1 until at a = 1, where the total cost of both cases are equal. 4.4. Decision making process Inventory planners can make decisions under the acceptable possible levels (a). The effectiveness of the decision making process in inventory planning depends on verbal information from experienced decision makers and the F-USILSP model. For example, if the experienced personnel give erroneous information, inventory planning errors can be made. 5. The case study of a petrochemical company The case study of this paper focuses on the inventory planning of bituminous coal in a power plant of a petrochemical company. The main purpose of this power plant is to produce electrical power for the company’s petrochemical plants, which is the core business of this company. Bituminous coal is the main raw material for the generation of electrical power. Demand for bituminous coal,measured in metric tons, is dependent on the demand for electrical power, which is required for petrochemical production. In this study, the demand for bituminous coal is classified as 5
trapezoidal fuzzy numbers, i.e., lowest, low, medium, high, and highest (see the membership function of fuzzy demand in Fig. 4). Based on the experience of the production manager and the fuzzy information in Fig. 3, the demand for bituminous coal in the planning horizon (12 periods) is predicted in the form of verbal demand, which is shown in Table 3. The price of bituminous coal per metric ton is classified as 7 triangular fuzzy numbers, i.e., the cheapest, very cheap, cheap, normal, expensive, very expensive, the most expensive (see the membership function of fuzzy unit prices in Fig. 5). Based on the experience of the purchasing manager and the fuzzy information in Fig. 5, the unit price of bituminous coal in the planning horizon is predicted in the form of verbal unit prices, which are shown in Table 3. The company has two outsourcers, companies A and B, which transport bituminous coal from the distributor to the inventory area. There is a large area for the storage of 80,000 metric tons of bituminous coal. According to company policies, 20,000 metric tons of bituminous coal must be stored as a safety stock. It cannot be used up. Thus, the net inventory space is equal to 60,000 metric tons. The inventory level at the end of period 0, or the beginning stock of period 1, is equal to 15,000 tons. So the net demand of period 1 is equal to (20,000, 25,000, 30,000, 35,000) metric tons minus 15,000 metric tons, that is (5000, 10,000, 15,000, 20,000) metric tons. Company A charges1 million Thai Baht (THB) per order, and the maximum transportation capacity of company A is 50,000 metric tons per period. Company B charges 1.5 million THB per order, and the maximum transportation capacity of company B is equal to that of company A. Note that although the quantity of each order is less than the maximum transportation capacity of each company, companies A and B cannot discount the charge rate per order. Based on the policies of the company, the holding cost is fixed as a crisp value. It is equal to 200 THB per metric ton per period. In addition, 30.5 THB is equivalent to $1US (as of May 2011). This case study is different from the numerical example. Since there are two transporters, and transportation capacity of them is limited, sometimes only company A can be selected (the service
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Fig. 4. Membership function of fuzzy demand.
Table 3 Demand (1000 metric tons) and unit price (100 THB per metric tons). Period t
1 2 3 4 5 6 7 8 9 10 11 12
Demand
Unit Price
Verbal prediction
Trapezoidal fuzzy number
Verbal prediction
Triangular fuzzy number
Low Low Medium Medium Medium Medium High High Medium Low Lowest Lowest
(20, 25, 30, 35) (20, 25, 30, 35) (30, 35, 40, 45) (30, 35, 40, 45) (30, 35, 40, 45) (30, 35, 40, 45) (40, 45, 50, 55) (40, 45, 50, 55) (30, 35, 40, 45) (20, 25, 30, 35) (15, 15, 20, 25) (15, 15, 20, 25)
Normal Cheap Cheap Very Cheap Cheap Cheap Normal Expensive Normal Expensive Normal Expensive
(20, 25, 30) (15, 20, 25) (15, 20, 25) (10, 15, 20) (15, 20, 25) (15, 20, 25) (20, 25, 30) (25, 30, 35) (20, 25, 30) (25, 30, 35) (20, 25, 30) (25, 30, 35)
Fig. 5. Membership function of fuzzy unit prices.
charge of company A is less than that of company B), but sometimes both companies must be selected (when the buying quantity exceeds the transportation capacity of company A). Therefore, the decision variables are as follows: t is the tth period in the planning horizon t = 1, 2, . . ., 12, xtA and xtBare the buying quantities which are transported by companies A and B in period t, respectively. Binary variable ytA and ytB are ordering variable by companies A and B in period t, and It be the inventory level at the end of period t. Based on transportation capacity limitations, the maximum transportation capacity of companies A and B is 50,000 metric tons per period. The constraint for the control ordering variable can be formulated as follows:
50; 000ytA > xtA
and 50; 000ytB P xtB
for t ¼ 1; . . . ; 12:
ð54Þ
Focusing on inventory space limitations,the bituminous coal cannot be stored at a quantity greater than 60,000 metric tons. The warehouse capacity constraints are thus added as follows:
It1 þ xtA þ xtB 6 60; 000 for t ¼ 1; . . . ; 12:
ð55Þ
Substitute inventory parameters, decision variables, and add Eqs. (54) and (55) in the EC-USILSP model. The results at a = 0 and 1 are shown in Table 4. If a decision maker wants to plan for the lowest total cost, the lowest possible level (a = 0) is chosen. The inventory plan orders 5000, 20,000, and 30,000 metric tons of bituminous coal, which is transported by company A, in periods one, two, and three, respectively. To serve the demand of periods four and five, 60,000 metric tons must be refilled. Since the maximum transportation capacity of company A is less than 60,000 metric tons, an
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S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003 Table 4 Inventory plan at a = 0 and 1. Company
a = 0 and TC = 540,000,000 y
X
I
Y
x
I
1
A B
1 0
5000 0
0
1 0
10,000 0
0
2
A B
1 0
20,000 0
0
1 0
25,000 0
0
3
A B
1 0
30,000 0
0
1 0
35,000 0
0
4
A B
1 1
50,000 10,000
30,000
1 1
50,000 10,000
25,000
5
A B
0 0
0 0
0
1 0
10,000 0
0
6
A B
1 1
50,000 10,000
30,000
1 1
50,000 10,000
25,000
7
A B
1 0
30,000 0
20,000
1 0
35,000 0
15,000
8
A B
1 0
20,000 0
0
1 0
30,000 0
0
9
A B
1 0
50,000 0
20,000
1 1
50,000 10,000
25,000
10
A B
0 0
0 0
0
0 0
0 0
0
11
A B
1 0
30,000 0
15,000
1 0
30,000 0
15,000
12
A B
0 0
0 0
0
0 0
0 0
0
Period t
excess demand (10,000 metric tons) must be transported by company B. In period six, 60,000 metric tons of bituminous coal is ordered. An aggregate total of 30,000 metric tons, which is ordered in period seven, and 30,000 metric tons, which was transferred from period six, must be used to serve the demand of periods seven and eight. In period nine, 50,000 metric tons is ordered to cover periods nine and ten, and 30,000 metric tons is ordered in period eleven to cover the demand of the last two periods. The total cost is 540 million THB. On the other hand, when the highest possible level (a = 1) is required, the total cost increases to 813 million THB. The inventory plan of this scenario orders 10,000, 25,000, and 35,000 metric tons in periods one, two, and three, respectively. To serve the demand of periods six, seven, and eight, 60,000 and 35,000 metric tons is ordered respectively, in periods six and seven. In period nine, 60,000 metric tons is ordered to cover periods nine and ten, and 30,000 metric tons is ordered in period eleven to cover the last two periods.
a = 1 and TC = 813,000,000
The Possibility Approach was used to transform the F-USILSP model to the EC-USILSP model. As a result, this model can be solved by using basic software. Excel solver can be used to find an optimal solution effectively and provide useful information including total cost, ordering variables, purchasing and production quantity, and the inventory level. Furthermore, the result of Excel solver processing also showed that when the higher possible levels (a) are required the total cost will increase accordingly. Planners can therefore make decisions under the possible levels (a) that can be accepted. This article has several important implications. The EC-USILSP model is very flexible, so other conditions can be added to this model tobe consistent with events occurring in each business. For example, in the case study of a petrochemical company, restrictions on two issues were added: transportation capacity limitations and inventory space limitations in the EC-USILSP model to better represent the actual situation. 7. Future studies
6. Conclusions The F-USILSP model was developed mainly for inventory planning. This model is suitable for inventory planning in cases where there is no information from statistical data collection, but there is verbal information from experienced operators or personnel. Such situations frequently occur for SMEs or in cases where there is information from statistical data collection which cannot be used for inventory planning due to unexpected circumstances such as natural disasters resulting in rapid changes in demand, unit price, or ordering cost.
In this paper, SILSP with fuzzy parameters, which is a small problem, was selected as the sample to study. Future work could focus on, firstly, using other more sophisticated inventory models, such as a multiple item lot sizing problem in a single-stage and multiple-stage production; and secondly, a study of SILSP over several periods, where these problems would be more complex. Consequently, solving these problems requires a lot of time. For future work, comparisons should therefore be made between ‘optimization methods and Meta Heuristics’and‘optimization methods or Meta Heuristics’ to test which is more appropriate in these cases.
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Lastly, this article focuses only on inventory models with fuzzy parameters, but some parameters of inventory models may have variations in other characteristics, such as random variables or random fuzzy variables. In future studies,models should therefore be developed to support other parameters. Appendix A. USILSP-AGG model In this research, the AGG of USILSP in a form of MILP is selected for developing F-USILSP. The AGG of USILSP is shown as follow: A.1. Notations The following notations are used in the paper: t yt St xt It pt ht dt dkT
the tth period in the planning horizon t = 1, 2, . . ., T the ordering (or setup) variable, which is 1 when order or setup occurs in period t, and 0 otherwise the ordering (or setup) cost in period t the purchasing (or producing) quantity in period t the inventory level at the end of period t the unit price (or production cost) in period t the inventory holding cost in period t the demand in period t P the sufficiently large number, where dkT ¼ Tt¼1 dt ; 8t
classification shows that when an event is classified by the judgment of decision maker(s), an exact set boundary is inappropriate. On the other hand, the fuzzy set boundary is not an exact demarcation but an area of boundary. Members in a fuzzy set must be linked with a degree of possibility in the range of [0, 1], which is called a membership function. Fuzzy variables have membership functions as indicators of degrees of truth. Therefore, in vague situations, classical mathematical operations or arithmetic cannot be employed. One of the basic methods of fuzzy set theory which relies on a crisp mathematical concept applied to the fuzzy set is ‘‘the extension principle’’, as follows: Definition 1. Let X be a Cartesian product of universes X = e1; . . . ; A e r be r fuzzy set in X1, . . . , Xr, respectively. X1 x, . . . , x Xr and A f is a mapping from X to a universe Y, y = f(x1, . . . , xr). Then the e in Y by extension principle allows us to define a fuzzy set B
e ¼ fðy; l ðyÞÞ=y ¼ f ðx1 ; . . . ; xr Þ; ðx1 ; . . . ; xr Þ 2 Xg; B e
ðB:1Þ
B
where
leB ðyÞ ¼
8 < :
sup ðx1 ;...xr Þ2f 1 ðyÞ
0;
fminðle ðx1 Þ; . . . ; le ðxr ÞÞg; A1
Ar
f 1 ðyÞ – h; ;
Otherwise; ðB:2Þ
1
Focusing on the AGG model in USILSP, it is assumed that there are no beginning and ending stocks of the planning horizon (I0 = IT = 0), and inventory parameters i.e.,st, pt, ht, and dt are deterministic. The AGG of USILSP in a form of MILP is as follows:
ðUSILSP AGGÞ Min TC ¼
T X
ðst yt þ pt xt þ ht It Þ
ðA:1Þ
t¼1
subject to
It1 þ xt It ¼ dt ; yt dkT P xt ;
8t;
8t;
yt 2 binary variables; xt P 0; It P 0; 8 t:
ðA:2Þ ðA:3Þ ðA:4Þ
The objective function of USILSP-AGG in equation (A.1) is to minimize the total cost over the t period horizon. Constraints in equation (A.2) are inventory balance equations. Demand in period t(dt) can be satisfied when it is equal to the beginning inventory of period t(It1) plus the purchasing or producing quantities in period t(xt) minus the available on-hand inventory at the end of period t(It). Constraints in equation (A.3) are used to control ordering variables. Since dkT is a large number, then yt is treated as 1 when inventory is ordered in period t(xt > 0), and yt is treated as 0 when inventory is not ordered in period t(xt = 0) Appendix B. Fuzzy set background Fuzzy set theory was first proposed by Zadeh (1965). It is a mathematical tool to describe imprecision in a fuzzy environment. Imprecision refers to the sense of vagueness rather than a lack of knowledge about the value of parameters. The vagueness is due to the unique experiences and judgments of decision makers. For example, demand is classified as two verbal sets: medium and high. When a classical set is chosen to classify this demand as verbal classes, the range of 10,000–15,000 units of demand is defined as medium, and 15,001–20,000 units of demand is defined as high. The exact boundary of medium and high is 15,000 units of demand. It means that 15,000 units of demand is assigned to the medium set, but 15,001 units of demand is assigned to the high set. This
where f is the inverse off. Possibility theory in the context of fuzzy set theory was introduced by Zadeh (1978) to deal with non-stochastic imprecision and vagueness. Possibility theory and probability theory are similar because both theories are based on set functions. However, possibility theory differs from probability theory in the use of dual set functions, called possibility and necessity measures. Both measures can be defined as the basis of fully-fledged representation graded semantics for language statements on possibility distributions (Dubois & Prade, 1980). Furthermore, Nguyen and BouchonMeunier (2003) stated the theory of large deviations in probability theory also handles set functions that have similar possibility measures. This is consistent with Dubois (2006) who stated that possibility theory refers to possibility and necessity measures in the study of ‘‘maxitive and minitive’’ set functions, respectively. The possibility degree of a disjunction of events defines the maximum possibility degree of events in the disjunction. On the other hand,the necessity degree of a conjunction of events defines the minimum necessity degree of events in the conjunction. These basic concepts are simple and suitable to be applied to model various kinds of information items with linguistic information, and uncertain formulae in logical settings (Dubois, 2006; Dubois & Prade, 1988). The possibility theory is as follow. ~ be fuzzy variables, a ~0 be the complements of a ~ and b ~0 and b ~ Let a ~ respectively. Operation ⁄ means any one of operators >, P, =, and b, ~ are ~b 6, <. The possibility and necessity measure of fuzzy event a respectively defined by Zimmermann, 1996.
~ ¼ supfminðl ðxÞ; l~ ðyÞÞ=x y; x; y 2 Rg; pða~ bÞ ~ a b ~0 Þ; ~ ¼ 1 pða ~0 b ~ bÞ Nessða
ðB:3Þ ðB:4Þ
where pðÞ ~ and NessðÞ ~ means possibility and necessity of fuzzy ~ variable . ~ be~i for i = 1, . . ., n be n fuzzy variables, the right hand side b Let a n comes to a crisp variable, and f : R ! R be a real value function. ~i Þ b is defined by (Zimmermann, The possibility of fuzzy event f ða 1996).
pðf ða~i Þ bÞ ¼ supxi 2R fmin16i6n fla~i ðxi Þg=f ðxi Þ b; xi ; b 2 R; 8ig ðB:5Þ
S. Ketsarapong et al. / Expert Systems with Applications 39 (2012) 6994–7003
From equation (B.5), the fuzzy ranking condition can be proved to be Lemma 1 based on possibility measure, as follows: ~i for i = 1, . . . , n be fuzzy variables with normal and Lemma 1. Let a convex membership functions and b be a crisp variable. The lower and ~i are denoted by ða ~i ÞLa and ða ~i ÞUa , upper bounds of the a-level set of a respectively. Then, for any given possibility levels a1, a2 and a3 with 0 6 a1, a2, a3 6 1, (i)
(ii)
pða~1 þ þ a~n 6 bÞ P a1 iff ða~1 ÞLa1 þ þ ða~n ÞLa1 6 b;
ðB:6Þ
pða~1 þ þ a~n P bÞ P a2 iff ða~1 ÞUa2 þ þ ða~n ÞUa2 P b; ðB:7Þ
(iii)
pða~1 þ þ a~n ¼ bÞ P a3 iff ða~1 ÞLa3 þ þ ða~n ÞLa3 6 b ~1 ÞUa þ þ ða ~n ÞUa P b: ðB:8Þ and ða 3 3
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