Niagara Falls, Ontario University of Manitoba ON FUZZY ECONOMIC ORDER QUANTITY USING POSSIBILISTIC APPROACH

Transcription

1 ASAC 2009 Niagara Falls, Ontario S. S. Appadoo A. Dua University of Manitoba V. N. Sharma Department of Civil Engineering Technology Red River College ON FUZZY ECONOMIC ORDER QUANTITY USING POSSIBILISTIC APPROACH Recently, fuzzy sets theory has been used to generalize, under fuzzy environment, the economic order quantity (EOQ) inventory model. To widen the application of this approach, this paper considers the parameters in the EOQ model as trapezoidal fuzzy numbers (Tr.F.N. s) and derives certain results using possibilistic approach. Finally certain comments are suggested for further studies in supply chain management. Introduction The well-known EOQ inventory model has been extended to fuzzy versions by Park (1987), Chen and Wang (1996), Roy and Maiti (1997), Yao et al. (2000), Yao et al. (2003) and Chang (2004) among others. In this paper, we develop a fuzzy possibilistic version of the EOQ model under the assumptions that the parameters involved are trapezoidal fuzzy numbers.

2 Figure 1 Inventory Model with Constant Demand and Instantaneous Supply Source: Sharma (1997)

3 Figure 2 Relationship Between EOQ and Total Cost Source Sharma (1997) This paper is organized as follows. We summarize the preliminaries in Section 1. In Section 2, we discuss the fuzzy possibilistic EOQ inventory model along with some important results. The approach used to derived the optimal economic order quantity is somewhat close to the method adopted by Hsieh (2002) for deriving the optimal order quantity. It is of interest to note here that Hsieh (2002) used an approximation to deffuzify the fuzzy total cost, whereas, our approach does not use any kind of approximation. Concluding remarks are made in Section 4. Preliminaries and Notation Before discussing the fuzzy possibilistic EOQ inventory model, we introduce some definitions and with relevant operations. Most of these related definitions and properties may be found in Bector and Chandra (2005), Kaufmann and Gupta (1985), Zimmermann (2001). Let be the set of real numbers, be the set of positive real numbers, and. Definition 1.1

4 Definition 1.2 Definition 1.3 Definition 1.4 Definition 1.5 Definition 1.6 A Trapezoidal Fuzzy Number (Tr.F.N.) is represented completely by a quadruplet, where with membership function given by (1.1) Alternatively Kaufmann and Gupta (1985), p.26,27, defining the interval of confidence at level as,and setting, and, we get and. Thus, we get the -cut representation of the Tr.F.N., where as (1.2)

5 Possibilistic Mean, Variance and Covariance of Fuzzy Numbers As in Appadoo et al. (2008) and Carlsson and Fuller (2001) we use the following equalities given in (1.3) and (1.4) in deriving some of the results in the lot size inventory model. (1.3) (1.4) For fuzzy number, and, Carlsson and Fuller (2001) define crisp lower possibilistic mean value, crisp upper possibilistic mean value and crisp possibilistic mean value. Consider a fuzzy number whose -cut is written as and. Carlsson and Fuller (2001), define the lower and upper possibilistic mean and respectively, and the possibilistic mean of a fuzzy number as follows (1.5) and (1.6) In view of expression (1.5) and (1.6), the interval valued possibilistic mean values of fuzzy number, Carlsson and Fuller (2003) can be written as (1.7) (1.8) The possibilistic variance of is given by The possibilistic covariance between two fuzzy numbers, and, is

6 Deterministic Economic Order Quantity (EOQ) Model We now consider the classical inventory model in which we assume the parameters as crisp. The only variable in the classical EOQ model is ;, and are constant parameters. In other words, demand is known with certainty. Demand is constant and replenishment is assumed to be instantaneous. Thus, Harris (1990) Therefore, (2.1) (2.2) where is the carrying cost per unit per year, is the cost per order, is the order size and is the annual demand. Therefore, in this case the optimal order quantity is given by. (2.3) Fuzzy Economic Order Quantity Model In the EOQ model we now take the parameters, and as fuzzy numbers, and denote the fuzzy ordering cost by, the fuzzy annual demand and the fuzzy carrying cost by. Let, and denote the -cuts of, and respectively. Thus, for we have Analogous to (2.3), we have (2.4) (2.5) (2.6) (2.7) Let and for denote the lower bound and the upper bound, respectively, of the cuts for fuzzy total cost. Then, (2.8) (2.9)

7 The lower possibilistic mean and upper possibilistic mean for fuzzy total cost are as follows. (2.10) (2.11) Now, the possibilistic mean for fuzzy total cost is (2.12) The interval valued possibilistic mean is (2.13) (a) In order to find the lower possibilistic optimal economic order quantity, we set (2.14) and then solve for. That value of that minimize (2.14) is the lower possibilistic order quantity ( = ). (b) In order to find the upper possibilistic optimal economic order quantity, we set (2.15) and then solve for. That value of quantity ( = ). that minimize(2.15) is the upper possibilistic order (c) In order to find the possibilistic optimal economic order quantity, we set (2.16) and then solve for. That value of that minimize (2.16) is the possibilistic order quantity ( = ).

8 Remark 2.1 Theorem 2.1 For, let the fuzzy ordering cost, fuzzy Demand and the fuzzy holding cost be given by (2.17) (2.18) (2.19) respectively. Then, the fuzzy total cost sets is given by in terms of the model parameters and the -level where, and (2.20) (2.21) and (2.22) In what follows, we will always refer to, and as fuzzy possibilistic costs (total inventory cost). The proof for and follows from definitions given in Section 1.2 and therefore is omitted. In the Theorem below, we will discuss a possibilistic approach to find the economic order quantity in a possibilistic setup. Note that is the fuzzy possibilistic economic order quantity using expression (2.20) is the fuzzy possibilistic economic order quantity using expression (2.21) and is the fuzzy possibilistic economic order quantity using expression (2.22) respectively.

9 Theorem 2.2 Using expression for, and from Theorem 2.1, we have the following values for lower, and respectively. (2.23) (2.24) (2.25) and total cost at is (2.26) The total cost at is given by (2.27) and the total cost at is (2.28) We will established the proof for (2.23) and (2.26) only. The proof for (2.24), (2.25), (2.27) and (2.28) follow along similar lines. Proof: Since,

10 To find the optimal we solve, thus Solving for, we have (2.29) It is important to note here that in (2.29) we only consider the positive value of, since cannot be negative. The value of that optimizes will be referred to as. Since, Thus, (2.23) is proved., thus we have minimum value at. The result for expression (2.26) follows directly by evaluating at and is as follows, This proves the results (2.26). Conclusion In this paper we have developed possibilistic approach to the economic order quantity inventory model. The methodology proposed in this paper may also be applicable to other inventory models within a supply chain environment.

11 References Appadoo, S. S., Bhatt, S. K., Bector, C. R., Application of possibility theory to investment decisions, Fuzzy Optimization and Decision Making, 7, 2008, Bector, C. R. and Chandra, S., Fuzzy Mathematical Programming and Fuzzy Matrix Games, Studies in Fuzziness and Soft Computing, Springer Verlag 169, Carlsson, C and Fuller, R., On Possibilistic Mean Value and Variance of Fuzzy Numbers", Fuzzy Sets and Systems 122, 2001, Chang, H.C., An Application of Fuzzy Sets Theory to the EOQ Model With Imperfect Quality Items", Comput. Oper. Res., 2004, 31, Chen, S.H. and Wang, C. C., Backorder Fuzzy Inventory Model Under Functional Principle", Inf. Sci., 1996, 95, Harris F. How many parts to make at once. The Magazine of Management, 1913, 10(2): , 152; reprinted in Operations Research, 1990, 38(6): Hsieh. C. H., Optimization of fuzzy production inventory models", Information Sciences 146 (2002) Kaufmann, A. and Gupta, M. M., Introduction to Fuzzy Arithmetic Theory and Applications, Von Nostrand Reinhold Company, New York, U.S.A Park, K. S., Fuzzy-Set Theoretic Interpretation of Economic Order Quantity", IEEE Trans.Syst. Man Cybern, 17, 1987, Roy, T. K. and Maiti, M., A Fuzzy EOQ model With Demand Dependent Unit Cost Under Limited Storage Capacity", Eur. J. Oper. Res, 99, 1997, Sharma J. K., Operations Research Theory and Applications, 1st Edition, Macmillan India Limited, Sriniwaspuri, New Delhi, India Yao, J. S., Chang, S.C. and Su, J. S., Fuzzy Inventory Without Backorder For Fuzzy Order Quantity and Fuzzy Total Demand Quantity", Comput. Oper. Res., 27, 2000, Yao, J. S., Ouyang, L.Y. and Chang, H. C., Models for a Fuzzy Inventory of two Replaceable Merchandises Without Backorder Based on the Signed Distance of Fuzzy Sets", Eur. J. Oper. Res., 150, 2003, Zimmermann H. J., Fuzzy Sets Theory and Its Applications, 4th Edition, Kluwer Academic Publishers, Nowell, MA, U.S.A