A supply chain application of fuzzy set theory to inventory control models – DRP system analysis

Expert Systems with Applications 36 (2009) 9229–9239

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A supply chain application of fuzzy set theory to inventory control models – DRP system analysis Jui-Lin Wang * Department of Engineering and Management, Hisuping Institute of Technology, No. 11, Gongye Road, Dali City, Taichung County, Taiwan, ROC Graduate Department of Business Administration, National Chung Hsing University, Taiwan, ROC

a r t i c l e

i n f o

Keywords: DRP Fuzzy set Multi-echelon distribution Inventory control Supply chain

a b s t r a c t As competition abounds, the efficient solution on inventory control of a DRP’s (Distribution Requirement Planning) supply chain management is a vital success factor for companies in today’s business world. A stochastic program of market distribution and its deterministic equivalent control program is approximated by a multi-echelon lot-sizing model based on ‘‘risk inflated effective demands”. The DRP-decomposition of this approximate model, which can be used with allocation application of Fuzzy Set Theory, is then introduced. The aim of this paper is to find methods to address traditional DRP’s weaknesses and to improve the performances of DRP systems. In this paper, the field of continuous review model will be focused in, and a new method on the model with triangular fuzzy numbers (input data) will be presented. By using the method, the maximum of order quantity under a minimum of total cost can be obtained. In many previous research, authors take a precise number approximately as the representative of a fuzzy number. But the precise number can not reflect the property of fuzzy inventory control number fully. Therefore, in a numerical example of this paper, in addition to providing a transformation for reducing a fuzzy number into a closed interval by introducing the interval mean value concept proposed by Dubios and Prude, this fuzzy system can be transformed into a more precise diagnosis system for channel members in the supply chain distribution organization. Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Research motivation DRP (Distribution Requirement Planning) is one of the important subsystems which a modernization meat supply manufacturer adopts to respond to the chain store retail environment to achieve its supply chain management objective (Vollmann, Berry, Whybark, & Jacobs, 2004). In many sectors, the network through which a given item flows takes the form of a tree and, as illustrated in Fig. 1, this tree often has more than two echelons. Typically, the intermediate nodes in the tree are warehouses, the leaves are sales points or consumption points responding to an external demand, and the time required to ship items from node to node is not negligible. Almost research works on inventory control problem from this multi-echelon distribution network are solved by converting vagunese or imprecise input data to crisp one. But, many variables in inventory control process from this supply chain distribution net-

* Address: Department of Engineering and Management, Hisuping Institute of Technology, No. 11, Gongye Road, Dali City, Taichung County, Taiwan, ROC. Tel.: +886 0933513526/886 424961100x1469; fax: +886 0422111396. E-mail address: [email protected]

work may truly be fuzzy. Some components of the setup, holding and shortage costs may be unknown with uncertainty problem (Gen, Tsujimura, & Zheng, 1996). In many previous research, authors take a precise number approximately as the representative of a fuzzy number. But the precise number can not reflect the property of fuzzy inventory control number fully. Therefore, in this paper, a transformation for reducing a fuzzy number into a closed interval by introducing the interval mean value concept proposed by Dubios and Prude will be presented. The fuzzy number can be transformed into a closed interval, and possibility theory is used here to obtain a more precise result for above interval. Overall in this paper, the field of continuous review model will be focused in, and a new method on the model with fuzzy input data will be presented. By using the method, the maximum of order quantity under a minimum of total cost can be obtained. For the reason that result should be a fuzzy number because of fuzzy input data, and the certain number about order quantity is preferred in real-world, it is necessary to transform the fuzzy result to crisp one. The application of strategic diagnosis system to supply chain management could be found in the papers of Supply Chain Management (Dahel, 2003; Huang, Uppal, & Shi, 2002; McAdam & Brown, 2001; Sadler & Hines, 2002). Sadler and Hines (2002)

0957-4174/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.12.047

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Multi-Echelon 1

2

3

Vendor 1

2

5

6

3

7

---

4

8

9

10

Fig. 1. Conceptual framework of Fuzzy CRIM model method.

proposed a conceptual model for strategic operations diagnosis without empirical results for meat business to retail business system (Sadler & Hines, 2002). However, Bhutta and Huq (2002) evaluates the problems of the strategic diagnosis system by evaluating the consistency between the result and the decisions of the strategic channel management. In other words, if the forecasting results for the manufacturing business to retail business strategic diagnosis system are highly consistent with the decisions made by the strategic experts experience, then the channel strategic diagnosis system method are considered helpful for channel strategy decision-makings. Except there must have a lot of transformation technique method talking about the fuzzy set method basis to reflect the property of fuzzy number fully, however, the research also providing a concrete bilateral channel diagnosis about whether a channel strategy is appropriate or not is in paucity. This research, based on the point of view, this fuzzy system can be transformed into a more precise diagnosis system for channel members in the supply chain distribution organization. We are trying to construct a channel strategic diagnosis system analysis by using more precious transformation technique of fuzzy set method. 1.2. Research purpose and procedure This study attempts to start filling the void of how channel strategic diagnosis system to construct a distribution requirement planning system analysis by using more precious inventory control transformation technique of fuzzy set method in the Asia-Pacific Orient supply chain distribution. Thus, the purpose of this study is to use the transformation technique method of fuzzy set method to reflect the inventory control property of fuzzy number fully, and to give the supply chain manager a concrete bilateral suggestion about whether a critical channel inventory control strategy is appropriate or not. This paper is constructed in the following way. The literature review of the DRP (Distribution Requirement Planning), the mean value of a fuzzy number for inventory control and strategic Continuous Review Inventory Model (CRIM) are analyzed in Sections 2 and 3. More specifically, Sections 4 and 5 gives the model construction process and its empirical Numerical example for the meat business to retail business data in Taiwan supply chain distribution. Finally, the Sections 6 and 7, their conclusions and future directions are discussed.

ing horizon echelon-by-echelon approach that bases procurement decisions on time-phased projected future node requirements. The approach has several advantages (Martin, 1994). It can deal with any number of echelons, it takes the dependent nature of the demand into account, it manages lead times effectively, it can take economies of scale in transportation into account through the choice of appropriate lot-sizing algorithms, it can take any resource constraints into account indirectly through the intervention of a ‘‘master scheduler”, notwithstanding the fact that it has been implemented in several commercial software packages also supporting other needs of distribution/supply organisations (demand and order management, warehousing, transportation, personnel productivity, accounting, . . .). The main drawback of the DRP approach is that it was fundamentally designed to support deterministic time-varying demands. Several mechanisms, such as safety stocks, safety times and freeze periods were introduced to ‘‘manage” demand uncertainty, but they are often used arbitrarily. For example, fixed safety stocks are often used even if the demand follows a non-stationary stochastic process. Good approaches to compute safety buffers in DRP systems are not available currently (Bookbinder & Ng, 1986). Also when the master scheduler has to intervene to solve a resource problem, such as a warehouse inventory shortage, his or her decisions are not necessarily ‘‘optimal” (Bookbinder & Heath, 1988). Then, good DRP systems usually provide some help in the form of simulation and ‘‘pegging” facilities, for example, and they are usually flexible enough to permit the solution of such problems through ‘‘expediting” actions. In other words, if the weaknesses listed above could be corrected, DRP systems would provide an flexible and efficient environment to manage the flow of items in the complex supply/distribution trees discussed above (Bookbinder & Tan, 1988). The aim of this paper is to find methods to address these weaknesses and to improve the performances of DRP systems. In this present study, it focus that a stochastic program of market distribution and its deterministic equivalent control program is approximated by a multi-echelon lot-sizing model based on ‘‘risk inflated effective demands.” The DRP (Distribution Requirement Planning)-decomposition of this approximate model, which can be used with allocation application of fuzzy set method, is then introduced (Martel, Diaby, & Boctor, 1995).

3. Model development 2. Distribution Requirement Planning (DRP) theory 3.1. Problem description, assumptions and notation Since Whybark proposed the DRP logic under uncertainty in 1975, researchers have developed numerous models to help to solve the inventory control problems, each with their own pros and cons (Whybark & Vastag, 1993). In the same spirit as MRP logic, DRP (Distribution Requirements Planning) framework is a roll-

The multi-echelon solid distribution problem can be described as follows. Assume that a retail market separates N multi-echelon types of chain store supermarkets to satisfy the market demand for the unique modernizational meat supply manufacturer over a

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planning horizon T. The problem involves determining the most effective means of satisfying forecasted demand by adjusting optimum reorder point, reorder quantities and minimum inventory cost levels and other controllable variables by Strategic Continuous Review Inventory Model (CRIM). Based on the above characteristics of the considered Strategic Continuous Review Inventory Model (CRIM) problem, this CRIM mathematical model herein is developed on the following assumptions. 1. The piecewise linear membership functions are specified for all fuzzy sets involved. 2. The minimum operator is used to aggregate fuzzy sets for the mean value of fuzzy total cost interval. 3. The values of all parameters are certain over the next T planning horizon. 4. The escalating factors in each of the costs categories are certain over the next T planning horizon. 5. Actual reorder levels, inventory level capacity and warehouse space in each period cannot exceed their respective maximum levels. 6. The forecasted demand over a particular period can be either satisfied or backordered, but the backorder must be fulfilled in the next period. Assumptions 1 and 2 are made to convert the original Fuzzy CRIM problem into an equivalent LP (linear programming) problem that can be solved efficiently by the standard simplex method (Zimmermann, 1997). Assumption 1 follows from the fact that the DM (data model) could specify the degree of membership on distinct values for each of the objective functions, so piecewise linear functions may used to represent the fuzzy sets. Assumption 2 implies that the minimum operator suffices for aggregating the fuzzy mean value of fuzzy total cost interval. Assumptions 3 and 4 imply that the certainty property must be technically satisfied to represent an optimization problem as a LP problem. Assumption 5 represents the limits on the optimum available order quantity, reorder level, inventory capacity and warehouse capacity in a normal business operation. Assumption 6 concerns the portion of market demand that must be satisfied during any period, whereas the rest of the market demand can be backordered. However, backorders should not be carried over for more than one period in a practical situation. The following notation is used. Dnt forecasted demand for nth multi-echelon retail type in period t(U) continuous review procurement cost per unit for nth mulant ti-echelon retail type in period t(/U) overtime procurement cost per unit for nth multi-echelon bnt retail type in period t ($/U) subcontracting cost per unit of nth product in period t ($/ cnt U) inventory holding cost per unit of nth multi-echelon retail dnt type in period t($/U) backorder cost per unit of nth multi-echelon retail type in ent period t($/U) reorder level per unit of nth multi-echelon retail type in rnt period t (/U) continuous review order quantities for nth multi-echelon Qnt retail type in period t(U) inventory level in period tfor nth multi-echelon retail type Int (U) escalating factor for regular time procurement cost (%) ia escalating factor for overtime procurement cost (%) ib escalating factor for subcontract cost (%) ic escalating factor for inventory holding cost (%) id escalating factor for backorder cost (%) ie

Ont

overtime procurement for nth multi-echelon retail type in period t(U) subcontracting volume for nth multi-echelon retail type in period t(U) backorder level for nth multi-echelon retail type in period t(U) warehouse spaces per unit of nth multi-echelon retail type in period t (ft2/U) maximum inventory capacity available in period t maximum warehouse space available in period t (ft2) maximum subcontracted volume available for nth multiechelon retail type in period t (U) minimum inventory level available of nth multi-echelon retail type in period t (U) maximum backorder level available of nth product in period t (U)

Snt Bnt

vnt Mt max Vt max Snt max Int min Bnt max

3.2. Problem formulation 3.2.1. Objective function This study selected the closed interval mean value functions for solving the Fuzzy CRIM problem by reviewing the literature and considering practical situations. Since Yager proposed the approach to find the mean value of a fuzzy number (Yager, 1978). The mean value of a fuzzy number is transformed to a crisp value by complex calculation. However, it always loses the fuzziness of data in real-world. From this point of view, the mean value of a fuzzy number should be treated as an interval, and also it is very natural (Gen, Tsujimura, & Zheng, 1997). Here, the mean value concept of a fuzzy number proposed by Dubois and Prade (1987) is introduced, i.e., ‘‘The mean value of a fuzzy number is a closed interval (Yager, 1978) bounded by the expectations calculated from its upper and lower distribution functions.” The fuzzy number is assumed as a convex set by Dubois and Prade. The concept to a triangular fuzzy number (TFN) is adopted here (Kaufmann & Gupta, 1988). Let us considering a TFN A = (al, a2, a3) shown in Fig. 2. Let A be a triangular fuzzy number (TFN), denoted by (a1, a2, a3), its membership function u A is given by

8 > < ðx  a1Þ=ða2  a1Þ; uA ðxÞ ¼ ðx  a3Þ=ða2  a3Þ; > : 0; otherwise;

a1 6 x 6 a2; a2 6 x 6 a3 ;

where a1, a2, and a3 are real numbers. Let the left and right membership functions as uA(x)L = (x  a1)/(a2  a1);uA (x)R = (x  a3)/ L and (a2  a3); the corresponding inverse functions of u A ðxÞ R ðxÞ can be expressed as follows: u A

uA ðyÞL ¼ a1 þ ða2  a1Þy and uA ðyÞR ¼ a3 þ ða2  a3Þy Here, how to find optimum inventory level with fuzzy input data will be explored in the following three subsections. Therefore, three objective functions are simultaneously considered during the development of the proposed Fuzzy CRIM model as follows.  Minimize Total inventory Costs

Min z1 ffi

N X T X  ant Q nt ð1 þ ia Þt þ bnt Ont ð1 þ ib Þt þ cnt Snt ð1 þ ic Þt n¼1 t¼1

þdnt Int ð1 þ id Þt þ ent Bnt ð1 þ ie Þt



Min z1 ffi Fðnn ðQ n ÞÞ 

N X T X  ant Q nt ð1 þ ia Þt þ bnt Ont ð1 þ ib Þt þ cnt Snt ð1 þ ic Þt n¼1 t¼1

þdnt Int ð1 þ id Þt þ ent Bnt ð1 þ ie Þt



ð1Þ

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Three relation of interval E(cTC) and gTC gTC Case for cTCR gTC Case for cTCL gTC

Case for cTCR

Strategic Variables Va lue added per item Fixed procurement cost Unit inventory holding cost Unit shorting cost Lead time (in weeks)

Fuzzy-Reorder

Channel

Level analysis

Performance ( The total Cost of

CRIM Numerical

inventory control)

Example

The fuzzy Components Fig. 2. Conceptual framework of Fuzzy CRIM model method.

where the terms of N X T X ½ant Q nt ð1 þ ia Þt þ bnt Ont ð1 þ ib Þt þ cnt Snt ð1 þ ic Þt

3.3. Fuzzy Continuous Review Inventory Model (CRIM) model

n¼1 t¼1

The original Continuous Review Inventory Model (CRIM) model for previous problems can be converted to the CRIM model using the piecewise linear membership function of Hannan to represent the fuzzy goals of the DM in the CRIM model, together with the fuzzy decision-making of Bellman and Zadeh (1970) and Hannan (1981). Yao et al. use triangular fuzzy numbers to represent the order quantity and demand in the economic order quantity (EOQ) model without backordering (Yao & Lee, 1996). The fuzzy minimal total cost is obtained through the order quantity. Trapezoid fuzzy numbers are specified by Yao and Lee to deal with EOQ problems for cases with or without backorders (Yao & Lee, 1999). A triangular fuzzy number is introduced by Chang to deal with economic production quantity (EPQ) problems when the production quantity is imprecise (Chang, 1999). The proposed procedure is able to derive a minimal total cost mathematically. Lee and Yao use a fuzzy concept to deal with the vague factors in the EOQ and EPQ problems (Lee & Yao, 1990). The approximate critical point approach is employed to achieve an optimal solution in the fuzzy EOQ model. In EPQ problems, the total cost in a fuzzy situation is slightly higher than in the crisp case model. However, it permits better use of the EPQ model in a practical situation. Chen and Hsieh use the second function principle, regraded mean integration, and an extension of the LaGrangean method to obtain optimal solutions of fuzzy backorder inventory models (Chen and Hsieh, 1999). An optimal ordering quantity is obtained to achieve a maximum profit. Chang et al. modify the economic reorder point problem by introducing a fuzzy backorder quantity (Chang, 1999). The results offer a better way of using economic fuzzy quantities. Gen et al. (1997) apply the interval mean value concept to the continuous review model where the input data is vague and is given by triangular fuzzy numbers. The maximum interval order quantity is obtained to achieve a minimum total cost. Chen et al. propose a functional principle to deal with the EOQ problem in a fuzzy environment (Chen and Hsieh, 1999). A median rule is employed to obtain the optimal EOQ. Vujosevic et al. propose three methods to calculate the EOQ problem with fuzzy costs. Yao and Lee express the fuzzy order quantity as a normal triangular fuzzy number and obtain the optimising stock quantity (Yao & Lee, 1999). Pappis and Karacapilidis (1995) derive an appropriate number of production runs and lot sizes for multi-products with fuzzy demands. Lee et al. incorporate a fuzzy demand into a part-period balancing

þ ent Bnt ð1 þ ie Þt  which are used to calculate total inventory costs. The total inventory costs include five components - regular time, overtime procurement cost per unit for nth multi-echelon retail type in period t, subcontracts, holding inventory and backordering cost per unit for nth multi-echelon retail type in period t. Escalating factors were also included for each of the cost categories.  Minimize holding and backordering costs  Optimize reorder point r for holding cost h, fix procurement cost k, and backordering costsv

Fðnn ðh; v ; kÞÞ ffi Min

N X T X 

dnt Int ð1 þ id Þt

n¼1 t¼1

 þent Bnt ð1 þ ie Þt z2 ffi r  Fðnn ðQ n ; r n ÞÞ ¼ nn ½ðant Dnt þ kDnt =Q nt 

þh ðQ nt =2 þ r  r nt Þ þ v  Dnt =Ont Þ

ð2Þ

 Optimize order quantity of change in optimum reorder levels

z3 ffi Q  Fðnn ðQ n ; r n ÞÞ 

¼ nn ½ðant Dnt þ kDnt =Q nt þ h ðQ nt =2 þ r  r nt Þ þ v  Dnt =Ont Þ

ð3Þ

s:t: r  fr 1 ða1 Þ=a1 ; r2 ða2 Þ=a2 . . . ; rn ðan Þ=an g N X

v nt Int 6 V t max 8t

n¼1

where the symbol ffi is the fuzzified version of  and refers to the fuzzification of the aspiration levels. In real-world total inventory control decision problems, the environmental coefficients and operation parameters are typically uncertain because some information is incomplete or unobtainable in a medium time horizon. Accordingly, Eqs. (1)–(3) are fuzzy with imprecise aspiration levels, and incorporate the variations in the DM’s judgments concerning the solutions of fuzzy multi-objective optimization problems. For each of the objective functions of the proposed CRIM model, this study assumes that the DM has such imprecise goals as, ‘‘the objective functions should be essentially equal to some value”. These conflicting goals are required to be simultaneously optimized by the DM in the framework of fuzzy aspiration levels.

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lot-sizing algorithm (Lee & Yao, 1990). Three lot size algorithms with fuzzy demand are compared in the papers. Generally, the piecewise linear membership function and the fuzzy decision-making of Bellman and Zadeh (1970) can be adopted to convert the problem to be solved into an ordinary LP problem. The algorithm includes the following steps.

1.0 f 1 ( z1 )

0.8 0.6 0.4 0.2 0.0 59

0 33 59 86 60 39 60 93 61 55 61 66 62 40 62 56 63 11 63 66

Algorithm Step 1: Specify the degree of membership for several values of each objective function zi(i = 1,2, . . . ,k). Step 2: Draw the piecewise linear membership function. Step 3: Formulate the linear equations for each of the piecewise linear membership functions fi(zi)(i = 1,2, . . ., k). Step 2: Introduce the auxiliary variable L; the problem can be transformed into the equivalent conventional LP problem. The variable L can be interpreted as representing an overall degree of satisfaction with the DM’s multiple fuzzy goals. Step 5: Execute and modify the interactive decision process. If the DM is not satisfied with the initial solution, the model must be changed until a satisfactory solution is found. Fig. 3 present the conceptual framework of Fuzzy CRIM model method. The Appendix details the derivation. Fig. 4 present the block diagram of the interactive CRIM model development. 4. Numerical example 4.1. Basic data for numerical example There are about 802 modernization supermarkets (27 chainstore company) in Taiwan. Modernization supermarket channel has the common developing trend from Metropolitan area to the

TC (Total inventory Costs) Fig. 4. Shape of membership function (z1, f1(z1)).

other area in a New Developing Industrial Country. Fresh food production is focusing in central-south region. The Modernization Fresh food supply chain Manufacture process is limited and franchised by the government during to the food safeguarding. The monopoly supplier supplying chilly (frozen) pork (i.e.,Y&L corporation) are the major (above 80%) modernization meat supplier in Taipei’s metropolitan area. Of the 622 Supermarkets in Taipei’s metropolitan area chosen for the study, a representative store rate of 77.6% was responsed. Of the 12 chain-store cooperation in Taipei’s metropolitan, the study was conducted with three multi-echelon retail types of chain-store supermarkets (2 oversea large, 5 domestic large, and 10 domestic small). A database containing the items’ construct of Inventory Control models was efficiently administered to a statistic population of all volume-retail supermarkets in Taipei, reporting their relationship with their monopoly supplier supplying chilly (frozen) pork(i.e.,Y&L corporation). We collected these meat product supply data for two years and 17 channel distribution series of 104 weeks (yielding 1768 multi-echelon procurement planning problems for each of the inventory policy-factor set).

Start Formulate the CRIM model Specify the degree of membership for each zi (i 1, 2, Λ , k ) Draw the piecewise linear membership function for each ( zi , f i ( zi ))

(i 1, 2, Λ , k ) Formulate the linear equations for each f i ( zi ) (i 1, 2, Λ , k ) Transformed into the equivalent conventional Min TC problem Solve the CRIM model Modify the model

No Is the solution acceptable?

Ye s

Fig. 3. The block diagram of the interactive CRIM model development.

Stop

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Table 1 Summarized data in the numerical example (in units of TW dollars). Retail type

Multi-echelon

Dnt (U)

ant ($/U)

bnt ($/U)

cnt ($/U)

dnt ($/U)

ent ($/U)

nnt (h/U)

rnt (h/U)

vnt (ft2/U)

Mt max (machine-hours)

1

1 2

1000 1000

20 10

32 16

30 15

0.30 0.15

40 20

0.05 0.07

0.10 0.08

2 3

400

2

1 2

3000 500

20 10

32 16

30 15

0.30 0.15

40 20

0.05 0.07

0.10 0.08

2 3

500

3

1 2

5000 3000

20 10

32 16

30 15

0.30 0.15

40 20

0.05 0.07

0.10 0.08

2 3

600

Table 2 Membership functions.

4.2. Formulate the Fuzzy CRIM model Table 1 summarizes the basic data of the numerical example (Wang, Ho, Lin, & Chou, 2007). Other relevant data are as follows (in units of TW dollars). The expected escalating factor in each of the costs categories is 1% (Susan, 1989). The Reorder point level demand over the lead time is estimated to be ‘about 20 products’ and its membership function is represented as: r = {16/0, 17/0.232, 18/0.4, 19/0.9, 20/1, 21/0.9, 22/ 0.4, 23/0.23, 24/0}. Suppose the holding cost is imprecise and is linguistically expressed as ‘the holding cost is about 0.3’. A triangular fuzzy set (0.2, 0.3, 0.4) is used to express the holding cost. Let the costs per product be precise: The demand during the planning period (D) is 5000. The reorder point is selected from the set {16,17, . . ., 24} to minimise the possible total cost. The complete Fuzzy CRIM model of numerical example is as follows. Min TC (total inventory costs)

Min z1 ffi Fðnn ðQ n ÞÞ 

N X T X  ant Q nt ð1 þ ia Þt þ bnt Ont ð1 þ ib Þt n¼1 t¼1

þcnt Snt ð1 þ ic Þt þ dnt Int ð1 þ id Þt þ ent Bnt ð1 þ ie Þt



z2 ffi r  Fðnn ðQ n ; r n ÞÞ ¼ nn ½ðant Dnt þ kDnt =Q nt þ h  ðQ nt =2 þ r  r nt Þ þv  Dnt =Ont Þ 

z3 ffi Q  Fðnn ðQ n ; r n ÞÞ ¼ nn ½ðant Dnt þ kDnt =Q nt þ h ðQ nt =2 þ r  r nt Þ þv  Dnt =Ont Þ

v nt Int 6 V t max 8t

n¼1 

þ



þ



þ



þ

L;d11 ;d11 ;d12 ;d12 ;d21 ;d21 ;d31 ;d31 ;P nt ;Ont ;Int ;Bnt ;Snt ;Ht ;F t P 0

>6256 0 >24 0 >77.5 0

6240 0.23 23 0.23 67.1 0.23

6166 0.4 22 0.4 54.8 0.4

6155 0.9 21 0.9 44.7 0.9

6093 1.0 20 1.0 38.7 1.0

<6093 1.0 19 0.9 <38.7 1.0

Fh ((Qn)) = [(Qnt/D)(h*(Qnt/2 + r  An))]. For example, let the reorder point be set to 22. If the demand over lead time is 16 (a1) then the possible total cost is equal to a triangular fuzzy set (6105.5, 6152.3, 6211.1). The possibility of possible total cost is 0. If the demand over lead time is 23 (a8) then the possible total cost is a triangular set, (6112.5, 6161.3, 6255.3), with a possibility of 0.23. LINDO computer package was used to run this Fuzzy CRIM model which yielded the following optimum results. z1 = $6166.2, z2 = 50.3 and z3 = 22 order quantities and the overall degree of channel performance with the DM’s multiple fuzzy goals was 0.9598. Table 4 summarizes the results of implementing the previous CRIM model solutions. Table 5 summarizes the results of implementing the Fuzzy CRIM model solutions. 5. Results analysis 5.1. Analysis of results Several significant management implications that emerged when practically applying the proposed model are as follows.

s:t: r  fr1 ða1 Þ=a1 ;r2 ða2 Þ=a2 ...;rn ðan Þ=an g N X

z1 f1(z1) z2 f2(z2) z3 f3(z3)

8n; 8t

4.3. Output solutions The following procedure is proposed to determine the possible total cost E(cTC). First, determine the initial solutions for each of the objective functions using the conventional LP model. Then, formulate the Fuzzy CRIM model using the initial solutions and the CRIM model presented in the previous section. Table 2 gives the piecewise linear membership functions of the proposed model. Table 3 shows the related values when the reorder point is 22. After defuzzification of the possible total cost, the total cost is 6166.2. In summary, the possible total cost E(cTC) is combined with the fuzzy sets of the total procurement costs Fo((Qn)) and the total holding costs Fh((Qn)). Let the total procurement costs is represented by a level fuzzy set of Fo((Qn)) = [(Qnt/D)(CD + kntD/Qnt)]. Let the total holding costs is represented by a level fuzzy set of

1. Comparing previous Fuzzy CRIM model solutions with the multi-echelon type retail DRP of numerical example (Run Taiwan Retail channel distribution Scenario), reveals the tradeoffs and conflicts among dependent objective functions such as minimum total inventory cost, optimum reorder level and order quantities for the channel manager’s decision solution. Consequently, the proposed model can satisfy the requirement for the practical application because it aims to minimize total inventory costs, holding and backordering costs, and the rates of change in optimum reorder level and order quantities levels. 2. The results of Fuzzy CRIM model solutions indicate that the escalating factor for each cost category affects the objective and L values. In particular, increasing the escalating factor increases objective values including the total inventory costs and the holding and the backordering costs, while sharply decreasing the Lvalue. This finding implies that the DM must consider the time value of money in a practical CRIM problem. 3. Additionally, the DM must also increase the efficiency of internal management and seek to reduce the cost of capital to reduce the escalating factor. The optimum sensitivity analysis of inventory cost per unit, reorder poiny level and order quantities per unit reveals that the

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J.-L. Wang / Expert Systems with Applications 36 (2009) 9229–9239 Table 3 The related defuzzicative values when the reorder point is 22. a1 16

r r – interval mean value E(q) E(cTC)

a2 17

[47.7, 51.3, 67.3] [6105.5, 6152.3, 6211.1]

a3 18

[46.5, 49.2, 65.1] [6103.5, 6149.3, 6207.1]

a4 19

[44.9, 48.5, 62.5] [6101.5, 6146.3, 6203]

a5 20

[43.2, 47.3, 59.1] [6099.5, 6143.3, 6199]

a6 21

[41.1, 46.3, 57.7] [6097.5, 6140.3, 6195]

a7 22

[39.2, 45.1, 55.6] [6095.5, 6137.3,6191]

E(cTC)

Table 4 CRIM model solutions – the total costs for all reorder points.

a9 24

[47.4, 54.8, 67.1] [6112.5, 6161.3, 6255.3]

[54.8, 63.2, 77.5] [6128.2, 6183.8, 6256]

1.0

Order quantities

Total cost

16 17 18 19 20 21 22 23 24

81.7 74.2 68.8 63.5 59.4 55.1 50.3 46.4 46.4

6308.6 6292 6275.5 6257.2 6234 6206.3 6166.2* 6171.5 6171.7

0.8

f 2 ( z2 )

Reorder point

*

[15.5, 21.7, 31] [38.7, 44.7, 54.8] [6093.5, 6134.3, 6187] 6166.2

a8 23

0.6 0.4 0.2 0.0 16

17

18

19

20

21

22

23

24

Reorder point level Fig. 5. Shape of membership function (z2, f2(z2)).

means the optimum result of the CRIM model.

change in each cost category influences the objective functions, L values and other output solutions, implying that the DM should improve the multi-echelon type retail DRP resources to reduce effectively the inventory cost, holding cost, backordering, reorder poiny level and order quantities. Figs. 4–6 presents the changes of the object and the shape of membership function is (z1,f1(z1)), (z2,f2 (z2)) and (z3,f3(z3)).

1.0

f 3 ( z3 )

0.8 0.6 0.4 0.2 0.0

5.2. Models comparisons

0 Table 6 compares the CRIM model presented in this study to the LP-CRIM (Charnes & Cooper, 1961; Singhal & Adlakha, 1989) and Stochastic-CRIM (Lee & Yao, 1990) models. Table 7. Comparisons of major fuzzy inventory programming models. Several significant characteristics distinguish the proposed model from the other models.

10 13.5 21.6 30.2 38.7 44.7 54.8 67.1 77.5 88.2 102 122

Rates of changes in order quantities (continue-reviews) Fig. 6. Shape of membership function (z3, f3(z3)).

2. The proposed model provides the most flexible decision-making and adjustment processes. For instance, if the DM does not accept the initial overall degree of channel performance of 0.9598 as in the numerical example, then he may try to adjust this L value by taking account of relevant information to seek a set of rational output solutions for CRIM decision-making. 3. The proposed model considers the time value of money in relation to relevant cost categories. Generally, channel performance decision performance and objective values are affected by the interest factor. The DM computes the value in each cost cate-

1. The proposed model includes fuzzy multi-objective functions. In practice, the DM usually faces a fuzzy multi-objective planning problem when making an CRIM decision. The proposed model can satisfy the requirement for practical application because it minimizes total inventory costs, holding, carrying and backordering costs, and the rates of changes in order quantities levels, and can determine the DM’s overall degree of channel performance.

Table 5 Fuzzy CRIM model solutions – the total costs for all reorder points s. Retail type

Multiechelon

Qnt (U)

Int (U)

Ont (U)

Snt (U)

Bnt (U)

Ht (quantities)

Ft (quantities)

Reorder levels (quantities)

Capacity (inventoryelevels)

Warehouse space (ft2)

1

1 2

500 2693

0 1891

0 0

0 0

0 0

0

12

213

265

5674

2

1 2

3895 1345

895 1736

0 0

0 0

0 0

76

0

289

497

10 000

3

1 2

4505 464

400 200

0 0

0 0

0 0

0

31

258

488

1400

z1 = $6093; z2 = 20; z3 = 38.7 (continue-review); L = 0.9598.

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J.-L. Wang / Expert Systems with Applications 36 (2009) 9229–9239

Table 6 Comparisons of major CRIM models. Factor

Fuzzy CRIM

CRIM–CRIM

Stochastic-CRIM

Objective function

Multiple (Total costs, reorder point levels, and order quantity levels change rates) Fuzzy Multi-echelon Presented Multi-period Fuzzy Limited Limited Deterministic

Single (Total costs)

Single (Total costs)

Deterministic Single item Presented Single -period Deterministic Limited Limited Deterministic

Stochastic Single item Presented Multi-period Stochastic Infinite Infinite Deterministic

Allowed with fuzzy cost Considered Considered

Deterministic Negligible Deterministic

Stochastic Negligible Deterministic

Objective value Distribution network Degree of satisfaction Planning horizon Market demand Inventory capacity Warehouse space Costs of procurement, holding, and shorting Costs of regular time inventory Value added per item Lead Time

Table 7 Comparisons of major fuzzy inventory programming models. Model

The approach

The mean value operator

Yager (1978)

Find the mean value of a fuzzy number

Gen et al. (1996) Yao and Lee (1996) Lee and Yao (1990)

Use triangular fuzzy number to represent reorder level and its upper and lower distribution of order quantity Use triangular fuzzy number to represent the order quantity and demand in the economic order quantity (EOQ) model Use triangular fuzzy number to represent the inventory level without backorder models

The mean value of a fuzzy number is transformed to a crisp value (it will lose the fuzziness of data in real-world) The mean value of a fuzzy number is a closed interval bounded by the expectations for demand possibility. (it will be very natural) The mean value of a fuzzy number is represented by a crisp value of probability distribution for the demand over the lead time. The mean value of a fuzzy number is represented by a part-period balancing lot-sizing value of algorithm approach

Table 8 Comparisons of major fuzzy total cost programming models. Model

Membership function

Aggregation operator

Brief description

Zimmermann (1978)

Linear

Minimum (maximize the minimum membership function value)

 The rate of increased membership satisfaction is considered to be constant  The Fuzzy CRIM problem can be converted into the equivalent crisp LP problem  Has higher computational efficiency

Leberling (1981)

Hyperbolic

Minimum (as Zimmermann, 1978)

 The resulting nonlinear programming problem can be converted into a equivalent crisp LP problem  The computational efficiency is reduced

Hannan (1981)

Piecewise linear and continuous

 Minimum (as Zimmermann, 1978)  Minimize weighted sum of goals deviations  Maximize priorities sum of goals deviations

 Enables the nonlinear membership function to approximate piecewise by linear functions  The Fuzzy CRIM problem can be converted into the equivalent crisp LP problem  Has higher computational efficiency

Sakawa (1988)

    

Minimum (as Zimmermann, 1978)

 The resulting problem with the five types of membership functions is a nonlinear programming problem  Combines the bisection and LP methods  The computational efficiency is reduced

Linear Exponential Hyperbolic Hyperbolic inverse Piecewise linear

gory by considering the time value of money in the proposed model, which is appropriate for practical application to the CRIM problem. 4. The proposed model outputs more wide-ranging decision information than other models. This CRIM model focuses on the multi-periods and multi-echelon (retail type) problem in CRIM decision-making. It can also provide information on alternative strategies for overtime, subcontracting, inventory carrying, and backorders, but not regular time inventory, in response to variations in forecasted demand. The proposed CRIM model is constructed using the piecewise linear membership function of Hannan (1981) to represent the fuz-

zy goals of the DM in the CRIM model, together with the minimum operator of the fuzzy decision-making of Bellman and Zadeh (1970). Moreover, the original fuzzy CRIM problem can be converted into an equivalent crisp LP problem and is easily solved by the standard simplex method. Table 5 compares Hannan’s approach to the representative fuzzy goal programming models, including those of Zimmermann (1978), Leberling (1981) and Sakawa (1988). The important differences among these models result from the types of membership functions and aggregation operators they apply. In general, aggregation operators can be roughly classified into three categories - intersection, union, and averaging operators (Zimmermann, 1996). Table 8. Compare the comparisons of major fuzzy inventory programming models.

J.-L. Wang / Expert Systems with Applications 36 (2009) 9229–9239

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Table 9 Comparisons of common aggregation operators for the mean value of fuzzy total cost interval. Operator

Example

Intersection (t-norms)

   

Minimum Algebraic product Bounded sum Drastic intersection

 An aggregation scheme is implemented where fuzzy sets are connected by a logical ‘and’  The result of combination is high if and only if all values are high  The minimum operator is a greatest t-norm

Union (t-conorms)

   

Maximum Algebraic sum Bounded difference Drastic union

 An aggregation scheme is implemented where fuzzy sets are connected by a logical ‘or’  The result of combination is high if some values are high  The minimum operator is a smallest t-conorm

Averaging (compensative)

   

Mean Weighted

   

c OWA (the ordered weighted averaging)

Brief description

Have the compensative property so that the result of combination will be medium Consider the relative importance of the fuzzy sets The c-operator is the convex combination of the min-operator and the max-operator OWA enables a DM to specify linguistically his agenda for aggregating a collection of fuzzy sets

Table 9 compares the common aggregation operators. The minimum operator used in this study is preferable when the DM wishes to make the optimal membership function values approximately equal or when the DM feels that the minimum operator is an approximate representation. However, for some practical situations, the application of the aggregation operator to draws maps above the maximum operator and below the minimum operator is important. Alternatively, as shown in Table A1, averaging operators consider the relative importance of fuzzy sets and have the compensative property so that the result of combination will be medium. The c-operator (Zimmermann & Zysno, 1980), which yields an acceptable compromise between empirical fit and computational efficiency, seems to be the convex combination of the minimum and maximum operators (Zimmermann, 1997). Zimmermann (1996) pointed out that the following eight important criteria must be applied selecting an adequate aggregation operator – axiomatic strength, empirical fit, adaptability, numerical efficiency, compensation, range of compensation, aggregating behavior, and required scale level of membership function. 6. Conclusions Fuzzy Continuous Review Inventory Model (CRIM) deals with matching optimum inventory level control for supply and forecasted possibility of fuzzy set theory for demand, varying channel multi-echelon retail type orders over the medium term of lead time etc. The aim of Fuzzy CRIM decision-making is to set overall Inventory levels for each channel multi-echelon retail type orders category to meet fluctuating or uncertain demands in the near future, such that Fuzzy CRIM also determines the appropriate reorder point and order quantities to be used. This study develops a Fuzzy CRIM programming model in a fuzzy environment. The proposed model aims to minimize total inventory costs with reference to optimum reorder point, order quantity level, channel multi-echelon retail type capacity and warehouse space. The purpose of channel strategic diagnosis is to find the most efficiency method of inventory control for supply chain management, which will contribute to the channel organization in the future. Therefore, an effective model to select the efficiency and efficacy method is important in channel supply chain management practical sense. In this paper, a new method which use interval mean value concept for inventory control problem was presented here to manage material flows in multi-echelon supply-distribution networks. The method combined the interval mean value concept with possibility fuzzy set theory to obtain a crisp solution for solving the practical demand. Using a closed interval as a solution for order quantity and total cost in inventory control, the DM can make a decision more flexibly and adaptably in the real-world. Therefore, This research can provide a concrete bilateral channel

diagnosis about whether a channel strategy is appropriate or not is in paucity. After all, the most of decisions can be made based on a synthetical consideration rather than according to calculation result only. Furthermore, we made a few exploration on the point which input data is fuzzy number rather than one only coefficients is fuzzy like the work done before. We could try to construct a channel strategic diagnosis system analysis by using more precious transformation technique of Fuzzy Set Theory. Fuzzy logic is used to formulate the relationship between the input and output variables. The learning ability of Fuzzy logic network is used to refine the knowledge base. The distributions of this paper are as follows. First it gives an empirical testing for the interval mean value concept of critical strategic diagnosis. Second, it gives an alternative explanation for the interval mean value. Third, it provides the channel practitioners a more concrete bilateral and scientific model to diagnosis the critical interval mean value. Fourth, it is an integration between artificial intelligence and Distribution Requirement Planning (DRP), providing a successful application example of Fuzzy logic. Five, this model can also be used as the basis for the channel supply chain training and development. 7. Future research direction The major limitations of the proposed CRIM model concern the assumptions made for each of the decision parameters with reference to Channel Value added per item, Fixed procurement cost, Unit inventory holding cost, Unit shorting cost and Lead time (in weeks). Hence, the proposed model must be modified make it better suited to the practical application. Furthermore, future researchers may explore the other fuzzy properties of inventory control decision variable, coefficients, and relevant decision parameters in DRP problems. The proposed CRIM model is based on Mitsuo Gen’s approach (1997), which implicitly assumes that the fuzzy set theory is the proper representation of the fuzzy logic and fuzzy optimization. Therefore, future research may also apply the evolutionary optimization of Genetic Algorithms to solve inventory control problems in a fuzzy Supply Chain Management environment. Appendix 1 The Fuzzy CRIM model is derived as follows. Step 1: Specify the degree of membership fi(zi)(i = 1, 2, 3) for several values for each of the objective functions zi(i = 1, 2, 3). Table A1 presents the piecewise linear membership functions, f1 (z1), f2(z2) and f3(z3). (z1,f1(z1)), TC (total inventory costs);

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J.-L. Wang / Expert Systems with Applications 36 (2009) 9229–9239 N X T X ½ant Q nt ð1 þ ia Þt þ bnt Ont ð1 þ ib Þt þ cnt Snt ð1 þ ic Þt þ dnt Int ð1 þ id Þt

Table A1 Membership function fi(zi). z1

>X10

X10

X11

X12

...

X1P

X1,P+1


f1(z1) z2 f2(z2) z3 f3(z3)

0 >X20 0 >X30 0

0 X20 0 X30 0

q11 X21 q21 X31 q31

q12 X22 q22 X32 q32

... ... ... ... ...

q1P X2P q2P X3P q3P

1.0 X2,P+1 1.0 X3,P+1 1.0

1.0
Note: 0 6 qij 6 1.0; qij 6 qi;

n¼1 t¼1

þ ent Bnt ð1 þ ie Þt  þ

n¼1 t¼1

ðA6Þ

aij j zi  X ij j þbi zi þ ci 8i t

t

    t 12  t 11 t13  t 12  þ  þ ðd11  d11 Þ þ ðd12  d12 Þ 2 2   t 1;Pþ1  t 1P  þ ðd1P  d1P Þ þ þ 2  (X N X T t 1;Pþ1 þ t11 ½ant Q nt ð1 þ ia Þt þ bnt Ont ð1 þ ib Þt þ 2 n¼1 t¼1

f1 ðz1 Þ ¼

þS

where aij ¼ i;jþ12 ij ; bi ¼ i;Pþ12 i1 ; ci ¼ i;Pþ12 i1 Assumed that fi(zi) = tirzi + Sir for each segment Xi,r1 6 zi 6 Xir, where tir is the slope and Sir is the y-intercept for the section of the line that begins at Xi,r1 and ends at Xir. Hence,

f1 ðz1 Þ ¼

    t12  t 11 t 13  t 12 j z1  X 11 j þ j z1  X 12 j 2 2     t1;Pþ1  t 1P t1;Pþ1 þ t 21 þ þ j z1  X 1P j þ z1 2 2   S1;Pþ1 þ S11 t 1;jþ1  t 1j þ –0 j ¼ 1; 2; ; P 2 2

where t 11 ¼



q11 0 X 11 X 10

 ;

t 12 ¼



q12 q11 X 12 X 11

þcnt Snt ð1 þ ic Þt þ dnt Int ð1 þ id Þt T  X S1;Pþ1 þ S11 þent Bnt ð1 þ ie Þt  þ ðA7Þ 2 t¼1     t 22  t 21 t 23  t22  þ  þ ðd21  d21 Þ þ ðd22  d22 Þ f2 ðz2 Þ ¼ 2 2     t 2;Pþ1  t2P t 2;Pþ1 þ t 21  þ ðd2P  d2P Þ þ þ þ 2 2 ( ) N X T X ½dnt Int ð1 þ id Þt þ ent Bnt ð1 þ ie Þt 

ðA2Þ

n¼1 t¼1

  S2;Pþ1 þ S21 þ 2     t 32  t 31 t 33  t32  þ  þ ðd31  d31 Þ þ ðd32  d32 Þ f3 ðz3 Þ ¼ 2 2   t 3;Pþ1  t3P  þ ðd3P  d3P Þ þ þ 2     t3;Pþ1 þ t31 S3;Pþ1 þ S31 þ þ 2 2

   1:0q1P ; ; t1;Pþ1 ¼ X 1;Pþ1 , and X 1P

S1, P+1 is the y-intercept of the section of the line that begins at X1P and ends at X1,P+1, and can be derived using f1 (z1) = t1rz1 + S1r.

f2 ðz2 Þ ¼

where

    t22  t 21 t 23  t 22 j z2  X 21 j þ j z2  X 22 j 2 2     t2;Pþ1  t 2P t2;Pþ1 þ t 21 þ þ j z2  X 2P j þ z2 2 2   t 2;jþ1  t 2j S2;Pþ1 þ S21 –0 j ¼ 1; 2; . . . ; P þ 2 2 t 21 ¼



q21 0 X 21 X 20

     q21 1:0q2P ; t 22 ¼ Xq22 ; ; t 2;Pþ1 ¼ X2;Pþ1 , X 2P 22 X 21

ðA3Þ and

S2, P+1 is the y-intercept of the section of the line that begins at X2P and ends at X2,P+1, and can be derived using f2 (z2) = t2rz2 + S2r.

    t32  t 31 t 33  t 32 j z3  X 31 j þ j z3  X 32 j f3 ðz3 Þ ¼ 2 2     t3;Pþ1  t 3P t3;Pþ1 þ t 31 j z3  X 3P j þ z2 þ þ 2 2   S3;Pþ1 þ S31 t 3;jþ1  t 3j –0 j ¼ 1; 2; . . . ; P þ 2 2 where t 31



   0 q32 q31 ; t ; ; ¼ Xq3131X 32 ¼ X X 30 32 31

t3;Pþ1

 dij

where and denote the deviational variables at the jth point and Xij represent the values zi(i = 1,2,3) of theith objective function at the jth point. Step 3-3: Substituting Eqs. (A5) and (A6) into Eqs. (A2)–(A4), respectively, yields (A7)–(A9),

ðA1Þ S

þt

ðA5Þ

N X T X  þ ½dnt Int ð1 þ id Þt þ ent Bnt ð1 þ ie Þt  þ d2j  d2j ¼ X 2j j ¼ 1; 2; . . . ; P

j¼1 t

þ

i = 1, 2, 3 and j = 1,2 . . ., P.

j+1

(z2,f2(z2)), reorder point level; (z3,f3(z3)), rates of changes in order quantities (continuereviews). Step 2: Draw the piecewise linear membership function. Step 3-1: Convert the membership function fi(zi)(i = 1,2,3) into the form P X



d1j  d1j ¼ X 1j j ¼ 1; 2; . . . ; P

j¼1

þ dij

fi ðzi Þ ¼

p X



þ v  Db=Ont Þ ln ¼ minfFðnn ðQ n ; rn ÞÞg qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q n ¼ ð2D½k þ v b=hÞ



 1:0q3P , and ¼ X 3;Pþ1 X 3P

S3,P+1 is the y-intercept for the section of the line that begins at X3P and ended at X3,P+1, and can be derived using f3 (z3) = t3rz3 + S3r. Step 3-2: Introduce the following nonnegative deviational variþ  ables dij and dij

ðA9Þ

Step 4: Introduce the auxiliary variable L, the problem can be transformed into the equivalent conventional LP problem. The variable L can be interpreted as representing an overall degree of satisfaction with the DM’s multiple fuzzy goals. The complete Fuzzy CRIM model is as follows. Min TC (total inventory costs)

Fðnn ðQ n ; r n ÞÞ ¼ nn ½ðCD þ kD=Q nt þ h ðQ nt =2 þ r  AnÞ

ðA4Þ

ðA8Þ

s:t: r  fr 1 ða1 Þ=a1 ; r 2 ða2 Þ=a2 . . . ; r n ðan Þ=an g b  Maxfr  r n ðan Þ=an ; 0g qn ¼ lmax  ln N X

v

8

t nt Int 6 V t max n¼1  þ  þ  þ L; d1j ; d1j ; d2j ; d2j ; d3j ; d3j ; Pnt ; Ont ; Int ; Bnt ; Snt

P 0 8j; 8n; 8t

J.-L. Wang / Expert Systems with Applications 36 (2009) 9229–9239

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