An application of fuzzy set theory to inventory control models

Computers ind. Engng Vol. 33, Nos 3---4,pp. 553-556, 1997 O 1997 Elsevier Science Ltd

Pergmon

Printed in Great Britain. All rights reserved 0360-8352/97 $17.00 + 0.00 PH: S0360-8352(97)00191-$

An Application of Fuzzy Set Theory to Inventory Control Models Mitsuo Gen

Dazhong Zheng

Yasuhiro Tsujimura

Department of Industrial and Systems Engineering Ashikaga Institute of Technology

268 Ohmae-cho, Ashikaga, 326, Japan E_mail : [email protected]

Abstract A method for solving an inventory control problem, of which input data are described by triangular fuzzy numbers will be presented here. The continuous review model of the inventory control problem with fuzzy input data will be focused in, and a new solution method will be presented. For the reason that the result should be a fuzzy number because of fuzzy input data, and the certain number about order quantity is prefered in the real-world, it is necessary to transform the fuzzy result to crisp one. The interval mean value concept is used here to help to solve this problem. Under the condition of total cost minimum, the interval order quantity maximum can be obtained. © 1997 Elsevier Science Ltd K e y words: Inventory control problem, Mean value of a fuzzy number

1

Introduction

the mean value of a fuzzy number is transformed to a crisp value by complex calculation. However, it will lose 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. Here the mean value concept of a fuzzy number proposed by Dubois and Prude [1] is introduced, i.e., "The mean value of a fuzzy number is a closed interval bounded by the expectations calculated from its upper and lower distribution functions." The fussy number is assumed as a convex set by Dubois and Prade. The concept to a triangular fuzzy number(TFN) is adoped here. Let us considering a TFN . 4 = (al,a2, a3) shown in Fig. 1.

An inventory control problem is one of the very common problems in industrial engineering. Decision regarding how much and when to order are typical of every inventory problem. Almost research works on inventory control problem are solved by converting vagunese or imprecise input data to crisp one. But, many variables in inventory control process may truly be fuzzy. Some components of the setup, carrying and shortage costs may not be known with certainty. 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 prefered in real-world, it is necessary to transform the fuzzy result to crisp one. 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 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.

2

,~(x)

0 al a2 as Figure 1: TFN ~. = (at,a~,a3)

z

The mean value of fuzzy number A is E(A) = [Eo ( i ) , E" (,4)]. The lower value is a2

E.(~) = a2 -

Z ~dz,

(i)

o,idx,

(2)

t

M e a n V a l u e of a F u z z y N u m ber

and the upper value is

E*(~i) = ~2 +

The approach to find the mean value of a fuzzy number has been proposed by Yager [4]. In his approach,

where #/i is the membership function of A.

553

Proceedings of 1996 lCC&lC

554

3

Continuous tory M o d e l

Review

Inven-

6vx s ~ , ~Ft *mix d 24s(1 - d/p) + qs ' ~s~___~d+ ~ d + ~c~[4~(I - d / p ) - , m u ] ~

In a continuous review model, the inventory level,I, is constantly monitored, and when I drops to or below a predesignated reorder level,r, then an order is placed. This order quantity may be fixed at q, irrespective of the current inventory level, It; or the order may be variable, dependent upon It. In the latter case, the order quantity, qt, is expressed as follows: qt =/max - l,

242(1 - d/p) cv2 s~ax c r 2 Smex d 2420 - d/p) + 42 ' +ss___.dd+ c d + Eta[#1(1 - d/p) Smax]2 ql 241(1 -- d/p) q2

-

CVS S2max + CF3 Smax d q---t--- ) (4)

-I-2#1(1 - d/p)

(3)

Since suppliers do not allow that orders a fuzzy quantity, 4 must be transfered into a non-fuzzy number. As stated before, we will use the interval mean value and possibility index of the fuzzy cost exceeding a total cost. According to the following analysis, we will give a detailed disscusion about how to obtain a non-fuzzy result for order quantity and total cost with intervals. Under the conditions of known number for all input data, i.e., fuzzy input data is TFN and nonfuzzy input data is positive real number, we will obtain a fuzzy order quantity 4 after the fuzzy number operations. Fuzzy quantity 4 can be viewed as a convex set because the result of fuzzy number operations, such as product operation, division operation etc. can be approximated as TFNs [3]. Applying the interval mean value concept, a closed interval E(4) -- [qL, qR] can be obtain. We also can obtain the closed interval E(~T) = [c~, ~ ] for fuzzy total cost CT by substituting 4 into eq. (4), and applying the interval mean value concept. If the decision-maker(DM) had a deterministic Total Cost Goal (gTc) predesignated, and possibility of the toatl cost exceeding gTC, can be defined as a. So we can search the result more precisely in three cases as follows: Let us illustrate the relation of interval E(6T) and value gTV :

where/max is the predetermined maximum inventory level allowed predesignated. Let us consider all input costs are fuzzy and expressed by TFNs. Here, how to find optimum inventory level with fuzzy input data will be explored in the following three subsections.

3.1

Fixed Order Quantity

In this subsection, it is assumed that the reorder point, r, is deterministic. The task is to determine a fixed, deterministic order quantity, when all of input costs are fuzzy numbers, such as set up, carrying and shortage costs. The following notations are used when determining the fuzzy order quantity:

p : production rate (if p is not given, assume p is oo ) d : constant demand rate Smax: maximum shortage level c : cost of item r : reorder level # : fuzzy order quantity cs : fuzzy setup cost for order or productiuon 6c : fuzzy inventory carrying cost 6v : fuzzy variable shortage cost 6F : fuzzy fixed shortage cost CT : fuzzy total annum cost

(1) Case for c~ < gTC

I

Fuzzy order quantity 4 is given as follows approximately [2,5]:

4

(2) Case for CT R > grc

2d(c$2 + CF2 Smax)(1 -- d/p) + S2m.(~C2 + eV2) CC2(1 diP) 2

~2d(cs3+cr3

~CT(q)

E(4) = [qL, q.].

c'c3(1 - d/p) ~

~

[

We can decide the E(~r) = [C~T,e~] as ultimate result. The DM can select a order quantity from interval

( /2d(~s~ + e~ ~m.~)(l- d/p) + S2m..~(~CX+ ~vl)

V

I

- -

Smax)(1 - d/p) + s~ax(c~3 + c v 3 ) ~ex(1 - d/p) ~

and the fuzzy total cost is expressed as:

~T('~) ~ ( ~s~ d + c d + Cell#X(1 - d/p) - Smax]2 2#3(1 - d/p) q3

d

cL

9~o

~f

) W h e n ( c~T- gTc ) / gTC <_ a, same as case (I). Otherwise, the right side of interval ~ can't satisfy the DM's demand. Therefore, the number CTX E [gTC,c~) which satisfy the demand can be found.

Proceedin8s of 1996 lCC&lC

So let us consider following equation: CTX gTC < Ct gTC

(5)

- -

We can find o r x definitely in the interval Lcrc, ~ ) s ~ t t i ~ a g (¢TX -- g T c ) / g T c < a, the ultimate result is

(3) Case for 4 > gTc

d

I

gTC

4

[

cf

I

When ( ~ --gTC ) / g'rc _< a, same as case (1). Otherwise, there is not a satisfatory solution. However, it is need to note that a satisfactory solution eTX can be found in closed interval [4, cTR]• W e can consider ( orx - gTc ) / gTC <--a . The solution CTX will be given up when CTX < 4 , and we can obtain a new interval [ 4 , cTX] as the ultimate result when cTX >_ 4 . Except for the first case, we can obtain a new closed interval about ~r(q). Substituting the left and right sides of the closed interval E(~T(q)) into eq. (4), respectively. The ultimate result about order quantity can be found in &closed interval E ( ~ ) -= [qL', qR ] under the new minimum total cost conditions. The DM can select a order quantity from the closed interval

order must be placed which the inventory reaches the order quantity. Up until now, it has been assumed ~hat the r was fixed and deterministic previously. The reorder level is a level of inventory designed to service demand over the order delivery lead time (time from placing to receiving an order). It is, therefore, dependent upon the demand d and lead time m. If the safety stock ss, demand and lead time axe deterministic, then the reorder level is the sum of safety stock and demand over the lead time, i.e., r = ss + dD m, where dD is daily demand and m is expressed in days. However, the demand, lead time and/or safety stock level may not be known with certainty, so it should be represented by fuzzy numbers. Here we discuss that all input data for r axe fuzzy, i.e.,

=

Variable

Order

q

CV S2ax CF Srnax d A 2 q ( 1 - d/p) "4" - - , q

cs d q +cd+cc[2+r2-dL]

As stated before, if the order quantity is variable, it depends on the inventory level at the orde r is placed, It. The most common variable order quantity is defined as eq. (3): We can consider It is a crisp number because it is monitored often. On the contray, the lraffixshould be a fuzzy number since it is alfected by some uncertainty conditions. It is assumed that the reorder level, r, has already been specified and is deterministic, and all input costs axe deterministic. By using fuzzy substraction, the order quantity ~ will be obtained as follows: (fi)

Using the fuzzy mean value concept, we will obtain a closed interval E(qt). Substituting the left and right sides of E(~,) into eq. (4) and using the interval mean value concept, we can get a closed interval E(6T) for cT which corresponding to interval E(~t). The DM can select a satisfactory value in the interval E ( ~ ) , and the corresponding CT can be obtained. 3.3

Fuzzy

Reorder

(7)

CT ~- ( CS d ")-c d't-cc[~ -i-r1 -dL]

Quantity

~, = & ~ - t,

+

and input data along with any fuzzy components combinations will be ommitted. The reorder point r Mso is represented as a T F N because the input data are given as TFNs. The total cost of a continuous review system taking into account the reorder point is [2]:

E(¢'). 3.2

555

Level

In continuous review model, a reorder point r is usually specified by the inventory level at which a new

cv s~ax CF Sm~ d +2q(1 - d/p) + - cs~d + c d + cc[ q + r3 - d L ] q CV $2max CF d )

+2q(1-

d/p) +

q

(8)

where dL is demand over the lead time (dD m) and the remaining notations axe the same as before. The DM can obtain q and CT as previously shown in Section 3.1.

4

Numerical Example

Let us consider numerical examples with the following input data: p=oo d = 500 Sm~x = 10 c = 2.80

~s ~c ~v ~r

=(4,5,7) = (0.5,1.0,2.0) =(0.1,0.2,0.3) = ( 0.I, 0,3, 0,5 )

Proceedings of 1996 ICC&IC

556 4.1

Example Model

of Fixed

Order

Quantity

Substituting these input data into eq. (4), we will obtain p~(z) is expressed as follows:

{

(2z 2 - 5060)/(3060+z 2)

; 50.3 < z ~ 90.1

(12230-0.5z2)/(4110+0.5z 2) ; 90.1 < z < 156.4 0 ; otherwise

4.3

CTX - 1480 _< 0.02 1480

(9)

We can find CTX = 1509.80, and the total cost is limited to the interval [1466.44, 1509.80]. By substituting this new total cost condition into eq. (4), a new ~nterval for order quantity is [ 68, 111].

of Fuzzy

Reorder

Level

Assume the safety stock ss is "about 5", and is described by the TFN (2,5,7). Recall from the input data in the beginning of this section, that the demand is "about 500" per year, expressed as (450, 500, 600) and one year is equal to 300 working days. Lead time is also fuzzy at "about 10 working days", expressed as (9,19,12). By eq. (7), we can obtain the reorder level as follows: = (2, 5, 7) + [(450, 500, 600)/300] (9, I0, 12) = (15.5, 22, 31)

According to the introduction in Section 2, we will obtain a close interval E(~) = [68,118] as the mean value for order quantity ~ and mean value E(~T) = [1466.44, 1516.57] for total cost. The DM can select a order quantity in E(~) and obtain the corresponding cost. If the DM had a deterimistic gTV = 1480, and wishes to limit the possibility of the cr(~) exceeding 1480 to a = 0.02. Then we can decide E(~) and E(~T) more precisely as the debation for three cases above mentioned. We can see the currently data is corresponding with Case 2. Because (1516.571480)/1480 = 0.025 > a, the right-side value 1516.57 of E(br) can't satisfy the DM's demand, so we need to search a new right-side CTX for interval E(~T) to replace its old one. Let us consider following equation.

Example Model

Substituting ~ into eq. (8), and the remaining input data is same as previous subsection. As shown in subsection 3.1, we have the following results: E(~) = [42, 63], E(~) = [1476.85, 1489.40]

5

Conclusion

A new method which use interval mean value concept for inventory control problem was presented here. The method combined the interval mean value concept with possibility theory to obtain a crisp solution for solving the practical demand. Using a closed invertal 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. After all, the most of decisions can be made based on a synthetical consideration rather than according to calculation result only. Futhermore, 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.

References 4.2

Example of Variable tity Model

Order

Quan-

It is assumed that r has already been specified and is deterministic, and all input costs are deterministic. Predetermined maximum inventory level allowed is "about 100" and modeled as (75,100,115). Current inventory level is 40. For fuzzy input data, the mode value will be selected. According to eq. (6), the following result can be obtained:

[1] Dubois, D. and H. Prade, The Mean Value of a Fuzzy Number, Fuzzy Sets and Systems., 24, pp.279-300, 1987. [2] Susan, M. L., Fuzzy Set Theory Applied to Production and Inventory Control, P h . D . Dissertation, Kanasas State University, 1989. [3] Kaufmann, A. and M. M. Gupta, Fuzzy Mathematical Models in Engineering and Ma,agement Science, Elsevier Science Publishers B.V., NorthHolland, 1988.

(10)

[4] Yager, R. R., Ranking Fuzzy Subsets Over the Unit Interval, Proceedings of the 1978 CDC, pp.1435-1437, 1978.

Using fuzzy mean value concept, a closed interval E(~t) -- [47, 67] can be obtained as the mean value of ~. Substituting the left and right side of E(~t) into eq. (4), the corresponding E(~t) -- [1483.90, 1499.22] can be obtained.

[5] Gun, M., Y. Tsujimura and D. Z. Zheng, An Application of Fuzzy Set Theory to Inventory Control Models, l~th Fuzzy System Symposium, pp.855-856, 1996.

qt

= (75,100,115)- 40

= (35, 80, 75)