order quantity, total cost function, which is also an interval grey numer ... Keywords: Grey interval number; EOQ models; Fuzzy mathematical programming.
The Journal of Grey System 1 (2011) 71-82
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Grey System Approach for Economic Order Quantity Models under Uncertainty Wekan K Ö S E ' * , Izzettin TEMIZ^ Serpil 1.
Turkish Military Academy, Institute of Defense Science, Bakanliklar 06654, Ankara, Turkey
2.
Gazi University, Faculty of Engineering, Department of Industrial Engineering, Maltepe 06570, Ankara, Turkey
Received September 2010
Abstract — In this study, contrary to the classical Economic Order Quantity (EOQ) models unit holding and order cost parameters are included in the model as interval grey numbers and the grey prediction model has been used to forecast annual demand rate. To determine optimal order quantity, total cost function, which is also an interval grey numer mathematical operations. The total cost function has been whitenized first by using equal qeight mean whitenization then fuzzy mathematical programming methods and obtained results have been compared for different cost parameters. The analysis of the results show that for whetinization of interval valued functions equal weight mean whitenization method is better than fuzzy mathematical programming method which needs very complicated mathematical operations. Keywords: Grey interval number; EOQ models; Fuzzy mathematical programming.
Introduction The aim in inventory management is to determine when and how much the stock material that minimizes the total cost to be ordered. These two questions can be answered through analytic models established under various assumptions. Many mathematical models have been developed in order to solve inventory problems. At first. Ford Harris formulized deterministic EOQ model mathematically in 1915. Until the end of 1940s, many researchers worked on deterministic EOQ models and the derivatives of these models. Although real life problems entail various types of uncertainties, deterministic EOQ models have been emerged with the tendency of
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ignoring these uncertainties because of simplicity. In deterministic EOQ models it is assumed that all parameters are accurately known. In order to extinguish these assumptions, not possible to be faced in real life, stochastic and fuzzy logic based EOQ models have been suggested later. But it is not quite possible to explain the uncertainties faced in inventory planning duration through fuzzy mathematics, probability or statistics. Fuzzy mathematics tries to solve theoretically uncertainty through the help of past experiences by using membership functions. Probability and statistics feel the need of special distributions and sampling in plausible size in order to make accurate inferences. However, it is often encountered with situations with no past experience, that no required distributions have been established, and only small sized samplings are reached. Grey systeih approach, which has the power to make satisfactory application over these kinds of uncertain systems, have been put forward in 1982 by Professor Deng. Although grey system theory has been applied in many fields since it has been emerged in 1982, it has not been encountered with any work granting the parameters grey in EOQ model. Therefore, this is the first attempt that grey system theory is applied to EOQ model. In this study an application has been made aimed at determining the optimal order quantity for a company active for four years in the sector that has been working. Company authorities have only four years data and as the duration is only too new, it is not quite possible to determine unit holding cost, and order cost parameters exactly. When the papers in literature are examined, it can be seen that during these kinds of situations, the uncertainties related with demand rate are mostly handled with stochastic approachesyand the uncertainties related with cost parameters are handled with fuzzy logic based approaches. In this study, different from the studies in the literature, the uncertainties encountered in EOQ models are handled with grey system approach, and a new model named Grey EOQ model has been suggested. Grey EOQ model is an inventory model in which holding and ordering cost parameter are to be used as interval grey numbers [1]. Annual demand rate in the model has been calculated by using grey prediction method which supplies the best solutions for short time estimations in the situations where small amount of observations can be reached. Demand rate estimation obtained by grey prediction method has been attached to the model as a unique number and by using grey niiniber mathematical operations, interval total cost function has been calculated. In order to clarify grey total cost function two
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different methods have been used which are equal weight mean whitenization and fuzzy mathematical programming methods. Equal weight mean whitenization method is a method suggested for the usage where grey numbers does not have distribution information [2]. Fuzzy programming approach is suggested by Mahata and Goswami [3] for the clariflcation of interval functions. For the clarification with the fuzzy programming approach, interval valued total cost function has been tumed into equivalent multi objective mathematical models by using 0
(1)
For each period, optimal order quantity is calculated as shown in equation (2). (2, The most important critic made for deterministic EOQ model, is the notion that parameter values are accurately knovwi in the model. However, in real life neither the cost parameter values nor demand rate can be known accurately. In this section, uncertainties existent in EOQ model are handled with grey system approach which is the most effective method used in the cases where no data is reachable to establish related distributions or decision makers do not have enough experiences [2]. While establishing grey EOQ model, cost parameters in EOQ model have been represented by interval grey numbers other than exact values. And the demand rate is calculated by the grey prediction model (GM(1,1)) made by the usage of existent data in hand. Variable and parameter values used in grey EOQ model have been described as it is shown below. D: Estimated annual demand rate by using GM (1,1); ®i6[ci,C2]: Interval grey number for order cost with lower bound Ci and upper bound C2; ®2e[h\,h'2\: Interval A
grey number for holding cost with lower bound hi, and upper bound h2; TC : Grey total cost; Q: Order quantity for each period. For an annual planning period if total cost function is rewritten by the usage of grey parameter values, equation (3) is obtained. A
TC{Q) = [c^,C2\— + [h^,h2]—,
Q>0
(3)
If grey total cost value shown by equation (3), is calculated by grey number mathematical operations, equation (4) is obtained. [c:,| + A , | , c , | +A,|],
ß>0
(4)
For the whitenization of grey total cost function calculated as an interval number, two
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different approaches have been handled in this study. Total cost function is whitenized firstiy by using equal weight mean whitenization method suggested to be used in the cases where distribution data is not existent and then by fuzzy mathematical programming method. Equal weight mean whitenization method Whitenized values of interval grey numbers which are shown as ®e[a, b] are calculated as below. a)è, ae[l,O]
(5)
Definition 1. Whitenization method shown as ® = aa + ( l - a ) ô is called equal weight whitenization method for a e [1,0]. Definition 2. In equal weight whitenization method, whitenization value obtained for a=(l/2), is called as equal weight mean whitenization method. While interval grey numbers are whitenized, in the cases where there is no distribution data, generally equal weight mean whitenization method is used [2]. If grey total cost function shown in equation (4) is whitenized by equal weight mean whitenization method, equation (6) is obtained. r C ( 0 = ( c , + C 2 ) ^ + (Ä,+A2)f,
Q>0
(6)
Optimal order quantity (g*) and total cost value corresponding to g * is calculated as shovm below.
Fuzzy mathematical programming approach Total cost function in equation (4) is actually an interval grey number with its upper and lower bounds as shown below. A
7 ' C ( 0 i = c , ^ + /i,f
A
and
TC{Q)^=c^^ + h^%,
Q>0
(8)
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For minimization problems, Ishibuchi and Tanaka [27] have defined order relations which represent the decision maker's preference between interval costs. Let the uncertain costs from two alternatives be represented by interval A=[AL , AR] and B=[BL, BR] respectively. It is assumed that the cost of each alternative is known only to lie in corresponding interval. Definition 3. For two interval numbers given as A=[AL, AR] and B=[BL, BR], order relation 0
(17)
If grey total cost function given in equation (17) is arranged by using grey number mathematical operations, equation (18) is obtained. r c ( 0 = [ ^ ^ ^ ^ + 1 2 . 5 0 , ^^5900^200,
Q>0
(18)
If grey total cost function given in equation (18) is clarified by using equal weight mean whitenization method, total cost function's clarified value is calculated as follows:
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16.25Ô,
Ö>0
(19)
Optimal order quantity Q* and total cost function's value corresponding to Q* is calculated as shown below. ^ dQ
391050^,16.25=0 ^.ß'=155.1277^ 156
7Ü(Ö') =5041.653 $
(20)
(21)
According to this result, company authorities have to meet the demand by means of giving seven orders made each of 156 units. The total inventory cost corresponding optimal order quantity would be 5041.653 $. Let's clarify total cost function given in equation (18) by the fuzzy programming method and compare the results obtained above. According to 0
(22)
If fuzzy mathematical programming conversion steps are followed for the solution of multi objective deterministic model given in equation (23), equivalent model is obtained as follows; 4639 99 A0, Q
