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1 Elsevier Editorial System(tm) for International Journal of Production Economics Manuscript Draft Manuscript Number: IJPE-D-1-0 Title: Cooperative Grey Games and An Application on Economic Order Quantity Model Article Type: Research Paper Keywords: Grey Numbers; Cooperative Game Theory; Inventory Management; Economic Order Quantity Model Corresponding Author: Dr. Sırma Zeynep Alparslan Gök, PhD Corresponding Author's Institution: First Author: Mehmet Onur Olgun Order of Authors: Mehmet Onur Olgun; Sırma Zeynep Alparslan Gök, PhD; Gültekin Özdemir, PhD Abstract: Inventory management studies to minimize the average total cost per unit time and to determine the quantity of the stocked material to be ordered. A system on which the information is partly known and partly unknown is called the grey information. In this paper, we extend the results of Meca(00) depending on the grey information revealed by the individual firms. We introduce cooperative grey games and focus on sharing ordering cost rule (SOC-rule) to distribute the joint cost.

2 Title page including author details Click here to download Title page including author details: titlepage.pdf Cooperative Grey Games and An Application on Economic Order Quantity Model M.O. Olgun 1, S.Z. Alparslan Gök, G. Ozdemir 1 1 Department of Industrial Engineering, Faculty of Engineering, Süleyman Demirel University, 0 Isparta, Turkey, onurolgun@sdu.edu.tr, gultekinozdemir@sdu.edu.tr Department of Mathematics, Faculty of Arts and Sciences, Süleyman Demirel University, 0 Isparta, Turkey and Institute of Applied Mathematics, Middle East Technical University, 0, Ankara, Turkey, sirmagok@sdu.edu.tr 1

3 *Manuscript Click here to view linked References Cooperative Grey Games and An Application on Economic Order Quantity Model M.O. Olgun 1, S.Z. Alparslan Gök, G. Ozdemir 1 Abstract Inventory management studies to minimize the average total cost per unit time and to determine the quantity of the stocked material to be ordered. A system on which the information is partly known and partly unknown is called the grey information. In this paper, we extend the results of [1] depending on the grey information revealed by the individual rms. We introduce cooperative grey games and focus on sharing ordering cost rule (SOC-rule) to distribute the joint cost. Keywords: Grey Numbers, Cooperative Game Theory, Inventory Management, Economic Order Quantity Model. 1 Introduction Inventory management studies to minimize the average total cost per unit time and to determine the quantity of the stocked material to be ordered. In order to address these problems, [] introduced Economic Order Quantity Model (EOQ). The study of inventory management and the distribution of ordering and holding costs are very popular in the business world. If rms place their orders simultaneously, total cost can be reduced drastically. However, it is not always apparent how to distribute the total cost between these rms. Recent studies show that there has been wide interest on cooperative inventory games. The rst important study on inventory games was reported by [1]. They provide a proportional division mechanism to distribute the joint total cost and characterize the Share the Ordering Costs (SOC-rule) rule as a distribution rule for EOQ models on cooperative inventory cost games. [1] examine the wider class of n-person Economic Production Quantity (EPQ) inventory models with shortages. Further, [1] suggest a new class of inventory games called generalized holding cost games. In the study of [], the cost allocation procedure for the joint replenishment problem with rst-order interaction is considered. Finally, [] provide an excellent review of cooperative inventory games and extensions in the context of deterministic EOQ models. In real life situations, potential holding/ordering costs are not known exactly and these costs often change slightly from one period to another. For instance, ordering costs being dependent on the transportation facilities might also vary 1 Department of Industrial Engineering, Faculty of Engineering, Süleyman Demirel University, 0 Isparta, Turkey, onurolgun@sdu.edu.tr, gultekinozdemir@sdu.edu.tr Department of Mathematics, Faculty of Arts and Sciences, Süleyman Demirel University, 0 Isparta, Turkey and Institute of Applied Mathematics, Middle East Technical University, 0, Ankara, Turkey, sirmagok@sdu.edu.tr 1

4 from time to time. Changes in price of oils, mailing and telephonic charges may also vary the ordering costs. It is obvious that in nature rainy times vary for di erent seasons. In the sequel di erent holding costs occur that we can only estimate the values of the parameters. Usually, researchers consider these parameters either as constant or dependent on time or probabilistic [, ]. Generally, uncertainties are considered as randomness and are handled by probability theory in stochastic inventory games. In stochastic approach, the parameters are assumed to be random variables with known probability distributions [1]. However, we cannot estimate the exact probability distributions due to lack of previous data. To solve the problem with such imprecise numbers, fuzzy and fuzzy-interval approaches may be used [1]. However, in reality, it is not always easy to specify the membership in an exact environment with fuzzy numbers. In this study instead of fuzzy and stochastic parameters grey parameters are used. Grey system theory is an approach that can integrate uncertainty and ambiguity into the evaluation process. It is an e ective approach for theoretical analysis of systems with imprecise information and incomplete samples [, ]. Therefore, grey system theory, rather than the traditional probability theory and fuzzy set theory, is better suited to model the inventory problems by using cooperative game theory. To the best of our knowledge no study exists modeling inventory situations by using cooperative grey games. From this point of view this study is a pioneering work on a promising topic. In this study we consider the model of cooperative grey games as distinct one within cooperative games. To perform such a game, rms observe a lower and an upper bound of the estimated holding/ordering costs except the demand rate. At rst we focus on cooperative grey games arising from inventory situations. Further, inspiring from the results of [1] we perform grey numbers instead of real numbers for cost allocation. Finally, we brie y discuss a real life situation which is about the production of three shotgun rms in Turkey. The paper is organized as follows. In section, some basic de nitions and notions on grey numbers are given. Classical cooperative inventory games and related allocation rules are presented in Section. Section introduces cooperative grey inventory games. An application regarding the three shotgun rms are presented in Section. Preliminaries.1 Grey Numbers and Their Operators In this study tools of grey number calculations play an important role. In the sequel some grey number notions and operators used in this paper are presented []. A grey number is such a number whose exact value is unknown, but the range in which the value lies is known. To be precise, a grey number in general is an element of a closed and bounded interval, i.e., each grey number is an

5 element of a closed and bounded interval where, a; a R: We denote by I (R) as the set of all closed and bounded intervals in R. A grey number is denoted by [a; a] ; (a; a R): For example; consider the weight of a bag of soil which is between and 0 kilograms. A speci c person s height is between 1. and 1. meters. These two grey numbers can be written as: 1 [; 0] ; [1:; 1:] ; respectively. Let 1 and be two grey numbers where, 1 [a; b] ; a < b and [c; d] ; c < d. Then, i) the sum of 1 and is de ned by 1 + [a + c; b + d] ; ii) the multiplication of 1 with a positive scalar k is de ned by k 1 [ka; kb] : By i) and ii) we note that grey numbers have a cone structure. The whitenization value of a grey number [a; b] is denoted by e and is de ned by e = a + (1 ) b; [0; 1] : Commonly, the whitenization value for a grey number is obtained by taking the value of as 1. This value is named as the equal weight mean whitenization value [, ].. Classical Cooperative Inventroy Games An inventory situation is a triplet (N; a; m) ; where N = f1; :::; ng is the group of agents agreeing to make jointly the orders of a certain good, a is the xed ordering cost and m = fm 1 ; : : : ; m n g is the optimal number of orders of the rms. The inventory game arising from the inventory situation (N; a; m) is denoted by < N; c >. Here, c(s) is the average inventory cost per unit time for the agents in S if they place their orders jointly, and de ned by c(s) = P aq is m i for each S. A classical inventory game is a cooperative game arising from an inventory situation [1]. An n person Economic Order Quantity (EOQ) situation is a well-known and simple Operational Research model constructed by an inventory situation. It considers an agent who makes orders of a certain good that he sells. The demand of agent i that must be ful lled is deterministic and equals to d i units per time (d i 0). The cost of keeping one unit of this good per unit time in stock is h i (h i 0). If the xed ordering cost and the leading time, the time between the placement of an order and the delivery of that order, are deterministic 1. 1 In this study the leading time is assumed to be zero.

6 Recall that the average inventory cost per unit time is a function of the order size Q i ; which is given by c (Q i ) = a d i Q i + h i Q i (..1) and the optimal order size ^Q i is p ad=h i : Thus, the optimal avarage inventory cost per unit time is r c( ^Q adi i ) = = a ^m i : (..) h i where; ^m i = d i is the optimal number of orders per unit time. Q i If the coalition S decide to cooperate in order to save their inventory costs, then optimal order sizes for i and j which is denoted by Q i and Q j. Here, s Q i = P ad i js d jh j ; for all i S: (..) On the other hand the minimal sum of the inventory cost per unit time is given by ad i Q + X h j Q s j X = a ^m j i : (..) js js Consequently, cooperative inventory games arising from inventory situations can be used as a natural tool to analyze cooperation for EOQ models. Additionally, the share the ordering cost rule (SOC-rule) introduced by [1] can be used for cost allocation for this model. Formally, SOC-rule is de ned by SOC i (N; c) = Cooperative Grey Games c (i) c (N) : (..) PjN c (j) A grey cooperative game is an ordered pair of < N; w 0 > where N = f1; ; :::; ng is the set of players, and w 0 : N! R is the characteristic function which assigns to each coalition S N a whitenized grey number. The value of each coalition S is de ned by w 0 (S) = e S a S ; a S ; where as ; a S represent the lower and upper bound of the grey number in I (R) such that w 0 (;) = 0: Example 1 Consider Example in []. Here, players 1 and have each one container which they want to store, and player is the owner of a holding house which has capacity for one container. If player 1 is allowed to store his container then the bene t belongs to 1 [; 0] and if player is allowed to store his containter then the bene t belongs to [0; 0] : This situation can be described as a cooperative grey game < N; w 0 > with N = f1; ; g, and w 0 (S) = e S [0; 0] if = S; w 0 (;) = w 0 () = e = e 0 = 0; w 0 (1; ) = e 1; = 0; and w 0 (N) = w 0 (; ) = e ; = 0:

7 Cooperative Grey Inventory Games The most important critic made for deterministic EOQ model is the notion that parameter values are accurately known. However, in real life situations neither the cost parameters nor demand rates are known exactly. In this study if the parameters of an EOQ model are not known accurately except the demand rate, then we use the grey numbers. Parameters used in a grey EOQ model are de ned as follows. a [a 1 ; a ] : grey number for ordering cost with lower bound a 1 and upper bound a h [h 1 ; h ] : grey number for holding cost with lower bound h 1 and upper bound h D : Deterministic demand rate dac : Average Grey total inventory cost Q : Order quantity for each period The whitenized total cost function d AC (Q) when ordering the quantity Q per order is dac (Q) = (a 1 + a ) ^D Q + (h 1 + h ) Q : (.1.1) When minimizing the average costs over all Q 0; the optimum order size ^Q is s (a 1 + a ) ^Q = ^D : (.1.) h 1 + h. Grey Ordering Situation We de ne a grey inventory ordering situation by a triple < N; a ; fm i g in >; where N = f1; :::; ng is the group of agents agree to make jointly the orders of a certain good a is the xed grey ordering cost and m = fm 1 ; :::; m n g is the optimal number of orders of the rms. If a coalition S of rms cooperate then their optimal ordering cost equals to a s X is m i : Then, the corresponding grey-based ordering game < N; c 0 0 > is de ned as fol- q P lows: For all coalitions S N; the cost c 0 0 (S) equals to cost in a is m i and c 0 0 (;) = 0: Then the corresponding inventory games < N; c w 0 > is the cost of the coalition S which equals to the minimal cost which it can obtain on its own, that is, c w 0 (S) = a s X is m i and c w 0 (;) = 0: Thus, c w 0 = c0 0: The demand rate is taken from the historical data ([])

8 If the rms place their orders simultaneously, the cycle length of the rm i N equals to Q i =d i : Thus, in optimum we have The average costs for the rms in N is Q i = d i d 1 Q 1 : (..1) d 1 AC (Q 1 ) = a + Q X 1 Q 1 d 1 in hi : (..) Minimizing this with respect to Q 1 yields an optimal ordering level s An optimal cycle length is Optimal number of order is m n = d i Q i Q i = Q i d 1 = By using the SOC-rule we have = s a m i q P jn m j. Grey holding situation P a d i jn d j hj : (..) a PjN d j hj : (..) P jn d s j hj X = m j : (..) a jn for all i N: (..) A grey holding situation is described by the tuple N; a ; f hi ; d i g in ; where N = f1; :::; ng is the group of agents agree to make jointly the orders of a certain good, a is the xed grey ordering cost h is the grey holding cost and d i is the demand rate of the i th rm. Given a grey holding situation we de ne corresponding grey holding game < N; c 0 h > as the game that assings to coalition S N a minimal cost as in p a hs PiS d i and c h 0 (;) = 0: Putting things together we see that the average cost per unit time for the rms in S equals to d 1 a + X Q i hs Q 1 : (..1) is Here, as stated in ([]) AC (Q 1 ) = AC (Q 1 ; :::; Q n) : Because it is supposed that the cycle of the rms grow hyeharcigcally and rm 1 is shorter than rm and so on.

9 If we substitute Q i = d iq 1 of Q 1 and we obtain The cost is minimized d 1 for all i S: Then we express the cost as a function d 1 a + X d i Q 1 hs : (..) Q 1 d 1 is Q 1 = s a d 1 hs PjS d j Finally, the minimal cost per unit time for coalition S equals to An Application (..) q a hs P js d i: In this section, three shotgun companies Firm-1, Firm- and Firm-, which are located in the same city (Beysehir in Turkey) are considered. All rms need same type of barrel to produce a shotgun. Each shotgun company owns a warehouse in which they store all the demands they may need. Each rm has learned how much barrel are used on the average in a year. In the shotgun sector, holding/ordering costs change slightly from one production period to another. For example, holding costs may be di erent in hunting seasons and ordering costs may be dependent on the transportation facilities changeable according to the seasons. Further, the ordering cost variables depend on the change in price of gas, mailing and telephonic charges. Therefore, it is interesting to use grey holding/ordering parameters instead of real parameters. Table 1 illustrates the ordering/holding costs of the shotgun rms with grey numbers. Table.1. The ordering/holding costs of the shotgun rms with grey numbers Firm-1 Firm- Firm- Demand Rates (items/per year) Ordering Costs (TL/year) [0; 00] [0; 00] [0; 00] Holding Costs (TL/year) [0:0; 0:0] [0:0; 0:0] [1; 1:1] * TL: Turkish Liras If the rms do not cooperate with other rms, this situation can be modeled as a grey inventory situation. The results, as shown in Table., indicate that Firm- has the lowest average total cost.

10 Table.. Grey inventory situation results Firm-1 Firm- Firm- Optimal Order Quantity (Q i ) : :0 : Items Per Cycle Length (t i ) 0: 0: 0: Orders Per Year (m i ) 1:1 1: 1: Average Total Cost ( AC) d 1: : : The cost of various coalitions in the cooperative grey inventory games are presented in Table.. Table.. The cost of various coalitions in the cooperative grey inventory games S f1g fg fg f1; g f1; g f; g f1; ; g c w 0 (S) 1: : : 1:0 1: 1: 1:1 If the rms work together, the cycle length of Firm-1 and Firm- equals to 0. which is shorter than the cycle length of Firm-. The SOC-rule assigns total cost c w 0 (N) proportionally to square of the individual costs, therefore it assigns the cost (1.0,.,1.) to the rms. All calculations are based on the individual optimal number of times to place an order, m i, for all rms i in N. These m i s depends r on the demand and grey holding cost of the di hi corresponding rm since m i =. As shown in Table.1, Firm-1 has a a very attractive house, since its holding cost is the lowest one. The grey holding cost game coalition results are given in Table.. Table.. The cost of various coalitions in the cooperative grey holding games S f1g fg fg f1; g f1; g f; g f1; ; g c h 0 (S) 1: : : 1: : : : The SOC-rule which assigns the demands proportionally the cost (00.0,.0,.) to the rms. Firm- pays the smallest amount since its demand is the smallest one. Notice that although all rms store their items in the warehouse of Firm-1, this rm has to pay the greatest part of the total cost. Conclusion In this study, we introduce a model, where inventory costs are assumed as grey numbers instead of crisp or stochastic ones studied in literature. At rst, grey numbers and classical cooperative inventory games are recalled. Then, cooperative grey games are introduced and related results are given. Finally, an application is performed for three shotgun companies in Turkey. For future

11 research, some possible model extensions may be considered, such as grey purchasing cost, grey allowing for stock outs, grey defective goods and grey quantity discounts situations. References [1] Alparslan, Gök, S.Z., Branzei, R. and Tijs, S., 00, Convex interval games, Journal of Applied Mathematics and Decision Sciences, Vol. 00, Article ID 0, 1. pages. [] Alparslan, Gök, S.Z., Branzei R. and Tijs, S., 0, Big boss interval games, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 1, -1. [] Anily, S. and Haviv, M., 00, The cost allocation problem for the rst order interaction joint replenishment model, Operations Research,, - 0. [] Bai C., Sarkis J., 01, A grey-based DEMATEL model for evaluating business process management critical success factors, International Journal of Production Economics, 1, 1-. [] Borm P., Hamers H., Hendrickx R., 001, Operations research games: A Survey, TOP, (), 1-1. [] Chakrabotty, S., Madhumangal, P., Kumar, P. and Nayak, K., 01, An algorithm for solution of an interval valued EOQ model, An International Journal of Optimization and Control: Theories & Applications, (1), -. [] Deng, J.L.,1, Introduction to grey system theory, The Journal of Grey System 1 (1), 1-. [] Dror, M. and Hartman, B.C., 0, Survey of cooperative inventory games and extensions, Journal of the Operational Research Society,, -0. [] Harris, F.W.,, How many parts to make at once, Factory, The Magazine of Management, (), 1-1. [] Kose, E., Temiz, I. and Erol, S., 0, Grey system approach for economic order quantity models under uncertainty, The Journal of Grey System, 1, 1-. [] Liu, S., Lin, Y., 00. Grey Information: Theory and Practical Applications, Springer, Germany. [1] Mallozzi L., Scalzo V., Tijs S., 0. Fuzzy interval cooperative games, Fuzzy Sets and Systems, 1(1), -. [1] Meca, A., 00, A core-allocation family for generalized holding cost games, Mathematical Methods of Operations Research,, -1.

12 [1] Meca, A., Timmer, J., Garc a-jurado, I. and Borm, P., 00, Inventory games, European Journal of Operational Research, 1, 1-1. [1] Mosquera, M.A., Garcia-Jurado, I. and Borm, P., 00, A note on coalitional manipulation and centralized inventory management, Annals of Operations Research, 1, 1-1. [1] Suijs, J., 0. Cooperative Decision-making Under Risk. Kluwer: Boston.