Managing Decentralized Inventory and Transhipment

Transcription

1 Managing Decentralized Inventory and Transhipment Moshe Dror Nichalin Suakkaphong Department of MIS University of Arizona Fleet Size Mix and Inventory Routing DOMinant Workshop, Sep , 2009 (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

2 Hypothetical Consider a car dealership. If the dealership does not have a car requested by a customer, it might consider acquiring it from a competing dealer. All parties have to be appropriately compensated. It is not obvious how and if one can engineer an operational process so that the decentralized system performance matches a fully centralized setting (first-best solution). (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

3 Our results reveal that the outcome of stable solution as proposed in past work is sensitive to a number of crucial assumptions regarding Nash play by all participants, complete information, and more. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

4 We examine a single-commodity multi-player inventory procurement and storage operations in a decentralized two-stage distribution system described by Anupindi, Bassok, and Zemel (2001). (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

5 Outline 1 Model Description 2 Existence and Uniqueness of First-Best Nash Equilibrium 3 Effect of Non-Nash Strategy 4 Allocation rules and incentive compatibility 5 Relaxing the assumption on satisfying local demand first 6 Discussion (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

6 Model Description Outline 1 Model Description 2 Existence and Uniqueness of First-Best Nash Equilibrium 3 Effect of Non-Nash Strategy 4 Allocation rules and incentive compatibility 5 Relaxing the assumption on satisfying local demand first 6 Discussion (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

7 Model Description Model Description First stage: Competitive independent retailers face random demands. The cost parameters and distribution of demand are of common knowledge. Inventories are ordered individually based on anticipated demands of everybody. After the demand is realized at each retailer, she may end up with excess demand or supply. Second stage: Excess demand at one retailer can be satisfied from surplus transshipped from other retailers local inventories. Each retailer will choose to satisfy local demand first. All retailers cooperate fully to maximize the second-stage transshipment profit. The combine profit is allocated based on an allocation rule agreed ex-ante. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

8 Model Description Model Description First stage: Competitive independent retailers face random demands. The cost parameters and distribution of demand are of common knowledge. Inventories are ordered individually based on anticipated demands of everybody. After the demand is realized at each retailer, she may end up with excess demand or supply. Second stage: Excess demand at one retailer can be satisfied from surplus transshipped from other retailers local inventories. Each retailer will choose to satisfy local demand first. All retailers cooperate fully to maximize the second-stage transshipment profit. The combine profit is allocated based on an allocation rule agreed ex-ante. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

9 Model Description In contrast to decentralized system, in the centralized inventory system... the inventory and transshipment decisions maximize the expected profit (first-best profits) of the overall system. The total profit of a decentralized inventory system cannot exceed the first-best profit. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

10 Model Description In contrast to decentralized system, in the centralized inventory system... the inventory and transshipment decisions maximize the expected profit (first-best profits) of the overall system. The total profit of a decentralized inventory system cannot exceed the first-best profit. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

11 Model Description Past assumptions Past assumptions (Anupindi et al., ABZ) The conditions that result in the decentralized inventory system achieving the first-best profit are: 1 Profit allocation must be in the core of the second-stage transshipment game. The allocation rule based on dual prices for the solution of transshipment profit maximization problem is in the core (Shapley and Shubik, 1975; Samet and Zemel, 1984). 2 A unique pure strategy Nash equilibrium (PSNE) must exist in the first stage. 3 The first-stage inventory decisions must result in the same inventory levels as those that achieve the first-best profit. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

12 Model Description Past assumptions ABZ s main results An allocation based on dual price, although is in the core of the second-stage transshipment game and attains a unique PSNE, does not achieve the first-best profit. ABZ construct an allocation rule based on ex-post side payments that is claimed to satisfy all three conditions. ABZ claim that if the distribution system exhibits a unique first-best solution, then the PSNE of a system that adopts ABZ allocation rule is unique. This is not the case. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

13 Model Description Past assumptions ABZ s main results An allocation based on dual price, although is in the core of the second-stage transshipment game and attains a unique PSNE, does not achieve the first-best profit. ABZ construct an allocation rule based on ex-post side payments that is claimed to satisfy all three conditions. ABZ claim that if the distribution system exhibits a unique first-best solution, then the PSNE of a system that adopts ABZ allocation rule is unique. This is not the case. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

14 Model Description Past assumptions ABZ s main results An allocation based on dual price, although is in the core of the second-stage transshipment game and attains a unique PSNE, does not achieve the first-best profit. ABZ construct an allocation rule based on ex-post side payments that is claimed to satisfy all three conditions. ABZ claim that if the distribution system exhibits a unique first-best solution, then the PSNE of a system that adopts ABZ allocation rule is unique. This is not the case. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

15 Model Description Past assumptions ABZ s main results An allocation based on dual price, although is in the core of the second-stage transshipment game and attains a unique PSNE, does not achieve the first-best profit. ABZ construct an allocation rule based on ex-post side payments that is claimed to satisfy all three conditions. ABZ claim that if the distribution system exhibits a unique first-best solution, then the PSNE of a system that adopts ABZ allocation rule is unique. This is not the case. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

16 Model Description Details of the Decentralized Distribution Systems Notations and Definitions Let α i ( X, D) be the profit allocated to retailer i as a result of transshipment game base on some allocation rule α. For a given demand D, the profit function of retailer i is: P i ( X, D) = [r i B i + v i H i c i X i ] + α i ( X, D). The expected profit (payoff) function is: J i ( X) = E D (P i ( X, D)) For a two-retailer case, Transshipment profit = (r 1 v 2 t 2,1 ) min {E 1, H 2 } +(r 2 v 1 t 1,2 ) min {E 2, H 1 }. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

17 Model Description Details of the Decentralized Distribution Systems First-best profit and first-best solution In a case of two retailers, the combined profit is represented by: PN c ( X, D) = [r 1 B 1 + v 1 H 1 c 1 X 1 ] + [r 2 B 2 + v 2 H 2 c 2 X 2 ] +(r 1 v 2 t 2,1 ) min {E 1, H 2 } +(r 2 v 1 t 1,2 ) min {E 2, H 1 }. The expected combined profit is JN c ( X) = E D (PN c ( X, D)). The first-best (centralized) solution is X c = argmax X JN c ( X). The first-best profit is JN c ( X c ). Note: We assume JN( c X c ) 0. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

18 Model Description Details of the Decentralized Distribution Systems Fractional allocation The fractional allocation α f i ( X, D) is defined as: α f i ( X, D) = γ i P c N ( X, D) [r i B i + v i H i c i X i ] where γ i is a fraction agreed by all retailers such that i N γ i = 1 and for all i, γ i [0, 1]. Note: α f i ( X, D) can be negative. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

19 Model Description Details of the Decentralized Distribution Systems Dual allocation The dual allocation αi d ( X, D) is defined as the allocation based on dual price of transshipment problem. That is: αi d ( X, D) = λ i H i + δ i E i and αi d ( X, D) = W N ( X, D) Transshipment problem W N ( X, D) = max y (r j v i t i,j )y i,j (1) s. t. i N j N,j i j N,j i i N,i j i N y i,j H i for all i N y i,j E j for all j N for all y i,j 0. The dual prices λ i and δ j are associated with the surplus H i and shortage E j. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

20 Model Description Details of the Decentralized Distribution Systems ABZ allocation Claim 1 (Corollary 5.1 in ABZ) Consider a modified fractional allocation rule that allocates the residual profits to player i N as follows: α m i ( X, D) = α f i ( X, D) + α d i ( X c, D) α f i ( X c, D) where X c is the first-best solution. Then the PSNE using α m i ( X, D) is first-best and the α m i ( X c, D) allocation values are in the core of the transshipment game. View proof J i ( X) = γ i E D (P c N ( X, D)) + E D (α d i ( X c, D)) E D (α f i ( X c, D)). (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

21 Existence and Uniqueness of First-Best Nash Equilibrium Outline 1 Model Description 2 Existence and Uniqueness of First-Best Nash Equilibrium 3 Effect of Non-Nash Strategy 4 Allocation rules and incentive compatibility 5 Relaxing the assumption on satisfying local demand first 6 Discussion (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

22 Existence and Uniqueness of First-Best Nash Equilibrium Conditions for the Existence of PSNE In ABZ s work, they assumed the following: 1 the expected profit function J i ( X) for retailer i is simultaneously continuous in X, 2 J i ( X) is unimodal in X i for every X N\i, and 3 the distribution system exhibits a unique first-best Subsequently, they claim that there exists a unique PSNE. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

23 Existence and Uniqueness of First-Best Nash Equilibrium Conditions for the Existence of PSNE We prove: Proposition 1: There exists a PSNE for a decentralized distribution system that adopts ABZ allocation rule. View proof If there exists a first-best solution, then there must exist a PSNE. If a function f : C R is continuous and its domain is a compact subset C of R n, then there are vectors in C that maximize the function f, i.e. the first-best solution (Weierstrass s theorem). (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

24 Existence and Uniqueness of First-Best Nash Equilibrium Uniqueness of PSNE Uniqueness of PSNE When using ABZ allocation rule, it is important to show that the PSNE/first-best inventory level of the decentralized distribution system is unique. The side payment calculation is based on the value of the unique PSNE/first-best inventory level. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

25 Existence and Uniqueness of First-Best Nash Equilibrium Uniqueness of PSNE Lemma 1: (New result) Suppose that JN c ( X) is strictly quasi-concave in X i for each i N. If there is a unique point X where JN c ( X) is strictly increasing in X i for X i < Xi, and JN c ( X) is strictly decreasing in X i for X i > Xi for all i N, then there is a unique PSNE that corresponds to the first-best solution. View proof A pertinent question is what are the demand distributions and cost parameters that would satisfy the conditions in Lemma (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

26 Existence and Uniqueness of First-Best Nash Equilibrium Uniqueness of PSNE Uniqueness of PSNE In summary, there is a unique PSNE if... (i) there is no ridge present for JN c ( X), (ii) JN c ( X) is strictly quasi-concave in each X i and (iii) weakly quasi-concave in X. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

27 Existence and Uniqueness of First-Best Nash Equilibrium Uniqueness of PSNE Roughly speaking, the following are the requirements: The best response functions of any two retailers do not describe the same graph in any neighborhood of X. Each demand density function is strictly log-concave in D. e.g. normal distribution and exponential distribution The marginal profits from own selling are more than the marginal profits from buying and selling through transshipment. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

28 Existence and Uniqueness of First-Best Nash Equilibrium Uniqueness of PSNE Example 4: Two retailers, Uniform[49,51] When the conditions are not met, multiple PSNE may exist as shown in the following example. Cost parameters: r 1 = 5.4, r 2 = 5.6, c 1 = 3.2, c 2 = 1.2, v 1 = 4, v 2 = 1, t 1,2 = 0, t 2,1 = 2, and γ i = 0.5. Demands are independent and uniform[49,51]. Strategy space is bounded in [48,52]. The unique first-best solution is at (52,50) with expected profit of $ Infinite number of PSNE, e.g. (48,52), (48.5,51.5), etc., with the combined expected profit of $ Expected Centralized Profit Player Player (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

29 Existence and Uniqueness of First-Best Nash Equilibrium Uniqueness of PSNE Importance of Unique first-best/psne The side payment calculation is based on the value of the unique first-best/psne inventory level. For instance, consider an n-retailers decentralized distribution system that has two different first-best/psne inventory levels: X A and X B where { } X A = X1 A, X2 A,..., Xn A (2) { } X B = X1 B, X2 B,..., Xn B. Retailer 1 may hope to calculate the side payment A because it benefit her more and choose inventory level X A 1. On the other hand, Retailer 2 may be better off with B and choose inventory level X B 2. The resulting inventory levels are not a member of PSNE, hence, the first-best expect profit is not achieved. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

30 Effect of Non-Nash Strategy Outline 1 Model Description 2 Existence and Uniqueness of First-Best Nash Equilibrium 3 Effect of Non-Nash Strategy 4 Allocation rules and incentive compatibility 5 Relaxing the assumption on satisfying local demand first 6 Discussion (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

31 Effect of Non-Nash Strategy Do players always play Nash equilibrium? Polls and laboratory experiments indicate that people often fail to conform to some of the basic assumptions of rational decision theory. Aumann (1997) (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

32 Effect of Non-Nash Strategy Nash equilibrium and core allocation In centralized inventory game literature, it is proven that some cost games only have a nonempty core for the expected cost game (Hartman and Dror, 2005). For any specific demand realization, the core of the game is likely to be empty. Our decentralized distribution systems do not have that pitfall because the core is always non-empty regardless of demand realization. Allocation based on dual prices is always a member of the core. Once the demand is realized, if the chosen inventory levels are at unique PSNE, ABZ allocation is guaranteed to be in the core. Otherwise, ABZ allocation is not guaranteed to be in the core. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

33 Effect of Non-Nash Strategy Example: Non-core allocation Recall that the first-best/psne solution (X c 1, X c 2 ) = (76.81, 62.35). Assume that retailer 1 does not order any units of inventory. she does not have enough funding i.e. cannot get credit. Assume that retailer 2 orders units of inventory. Realized demand: D 1 = 75, D 2 = 70 Thus, no transshipment since D 1 > X 1, D 2 > X 2. Local profits = ($0, $548.68) (α f 1( X, D), α f 2( X, D)) = ($274.34, -$274.34) Side payment (2-to-1) = α d 1( X c, D) α f 1( X c, D) = $62.72 (α m 1 ( X, D), α m 2 ( X, D)) = ($337.06, -$337.06) Retailer 2 would want to break from cooperation as she can perform better on her own. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

34 Allocation rules and incentive compatibility Outline 1 Model Description 2 Existence and Uniqueness of First-Best Nash Equilibrium 3 Effect of Non-Nash Strategy 4 Allocation rules and incentive compatibility 5 Relaxing the assumption on satisfying local demand first 6 Discussion (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

35 Allocation rules and incentive compatibility Allocation rules and incentive compatibility So far, we assumed complete information. However, (an observation) if the demand distribution is not of common knowledge, the decentralized distribution system that adopts ABZ allocation rules is not necessarily incentive compatible. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

36 Allocation rules and incentive compatibility Example: We revisit the previous example. Let the distribution of demand assumed by retailer 2 be [0,100] for both players, while retailer 1 knows that the true distribution of demand is [0,110] for herself and [0,100] for retailer 2. Cost parameters: r i = 10, c i = 1.2, v i = 1, t 1,2 = 1, and t 2,1 = 2. for i = 1, 2 They agree to use ABZ allocation with γ i = 0.5. If retailer 1 communicates the true distribution of demand to retailer 2 First-best/PSNE solution (X1 c, X 2 c ) = (84.20, 62.00). Expected profit (J 1 ( X c ), J 2 ( X c )) = ($406.99, $375.16). If retailer 1 does not communicate the true distribution of demand to retailer 2 First-best/PSNE solution (X1 c, X 2 c ) = (79.31, 62.35) was the first-best solution for retailer 2 for [0,100]. Expected profit (J 1 ( X c ), J 2 ( X c )) = ($407.23, $374.15). Retailer 2 would not know that her expected profit is $ Retailer 1 would choose to not communicate the true demand distribution to retailer 2. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

37 Relaxing the assumption on satisfying local demand first Outline 1 Model Description 2 Existence and Uniqueness of First-Best Nash Equilibrium 3 Effect of Non-Nash Strategy 4 Allocation rules and incentive compatibility 5 Relaxing the assumption on satisfying local demand first 6 Discussion (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

38 Relaxing the assumption on satisfying local demand first What if we relax the assumption on satisfying local demand first? For a two-retailer case, if retailer 2 can order inventory at a cost c 2 that is higher than cost c 1 + t 1,2 of obtaining transshipment from retailer 1, one might misinterpret that retailer 2 would not order at all and let retailer 1 order for her. This is not correct because the second-stage game is a cooperative game. The profit made from sales at retailer 2 must be shared with retailer 1 according to an agreed allocation rule. Retailer 2 s share of profit per unit from transshipment might be lower than the profit per unit when retailer 2 sells from her own local inventory. The behavior of retailers depends on the allocation rules that they agree among themselves. We have the same results as before without this assumption. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

39 Discussion Outline 1 Model Description 2 Existence and Uniqueness of First-Best Nash Equilibrium 3 Effect of Non-Nash Strategy 4 Allocation rules and incentive compatibility 5 Relaxing the assumption on satisfying local demand first 6 Discussion (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

40 Discussion Discussion Decentralized inventory system is applicable to many industries and supply chain settings. Retailers might find decentralized distribution systems with cooperative transshipment attractive because... it improves customer satisfaction, reduces excess inventory, and may potentially generate higher profit than traditional decentralized distribution systems without transshipment. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

41 Discussion Thank you! (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

42 References References I Anupindi, R., Y. Bassok, E. Zemel A general framework for the study of decentralized distribution systems. Manufacturing Service Oper. Management 4(3) Aumann, R. J Rationality and bounded rationality. Games Econom. Behavior Aumann, R. J., M. B. Maschler Repeated Games with Incomplete Information. MIT Press, Cambridge, MA. Bagnoli, M., T. Bergstrom Log-concave probability and its applications. Econom. Theory 26(2) Dror, M., L. A. Guardiola, A. Meca, J. Puerto Dynamic realization games in newsvendor inventory centralization. Internat. J. Game Theory 37(1) Fudenberg, D., J. Tirole Game Theory. MIT Press, Cambridge, MA. Granot, D., G. Sošić A three-stage model for a decentralized distribution system of retailers. Oper. Res. 51(5) Hartman, B. C., M. Dror Allocation of gains from inventory centralization in newsvendor environments. IIE Trans Karaesmen, I., G. J. van Ryzin Overbooking with Substitutable Inventory Classes. Oper. Res. 52(1) Kolmogorov, A. N., S. V. Fomin Introductory Real Analysis. (Translated by R. A. Silverman), Prentice Hall, Englewood Cliffs, NJ. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

43 References References II Prékopa, A On logarithmic concave measures and functions. Acta Scientiarum Mathematicarum Samet, D., E. Zemel On the core and dual set of linear production games. Math. Oper. Res. 9(2) Sánchez-Soriano, J., M. A. López, I. García-Jurado On the core of transportation games. Math. Soc. Sci Sandsmark, M Spatial oligopolies with cooperative distribution. Forthcoming in International Game Theory Review. Shapley, L., M. Shubik Competitive outcomes in the cores of market games. Internat. J. Game Theory 4(4) Topkis, D. M Supermodularity and Complementarity. Princeton University Press, Princeton, NJ. Wong, H., G. J. Van Houtum, D. Cattrysse, D. Van Oudheusden Multi-item spare parts systems with lateral transshipments and waiting time constraints. Eur. J. Oper. Res. 171(3) (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

44 References ABZ allocation Proof of claim 1. When X = X c, Back α m i ( X c, D) = α f i ( X c, D) + α d i ( X c, D) α f i ( X c, D) = α d i ( X c, D) Hence, the allocation α m i ( X c, D) will also be in the core of the transshipment game when the inventory levels X c are ordered by all retailers. P i ( X, D) = [r i B i + v i H i c i X i ] + α m i ( X, D) = [r i B i + v i H i c i X i ] + γ i P c N( X, D) [r i B i + v i H i c i X i ] +α d i ( X c, D) α f i ( X c, D) = γ i P c N( X, D) + α d i ( X c, D) α f i ( X c, D). J i ( X) = γ i E D (P c N( X, D)) + E D (α d i ( X c, D)) E D (α f i ( X c, D)). The value E D (αi d ( X c, D)) and E D (αi f ( X c, D)) are essentially constants as X c only depends on D. The vector X that maximizes J i ( X) is the same as X that maximizes the expected centralized profit JN( c X). (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

45 References Conditions for the Existence of PSNE Proof of proposition 1. Back We know that if a vector X is a first-best solution, then the vector X is also a member of a set of PSNE. This is because the best-response to other retailers playing first-best strategies is to play first-best strategy. If there exists a first-best solution, then there must exist a PSNE. If a function f : C R is continuous and its domain is a compact subset C of R n, then there are vectors in C that maximize the function f, i.e. the first-best solution (Weierstrass s theorem). In our case, each retailer s inventory level falls in a closed and bounded interval of R, hence the domain is a compact subset of R n. We proceed to check whether the expected centralized profit J c N ( X) is continuous in X. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

46 References Conditions for the Existence of PSNE Proof of proposition 1. (Cont d) Recall that Back PN c ( X, D) = r i B i + v i H i c i X i + W N ( X, D) i N For any given D, the centralized profit P c N ( X, D) is continuous in X because there is no fixed cost related to transshipment profits and local profits. According to Kolmogorov and Fomin (1970, p.109), a real function continuous on a compact metric space R is uniformly continuous on R. In our case, for all i, X i are defined on nonempty compact convex subsets of R and P c N ( X, D) is continuous in X. Hence, PN c ( X, D) is uniformly continuous in X, and it follows that JN c ( X) is continuous in X. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42

47 References Uniqueness of PSNE Proof of Lemma 1. Back Recall that the expected profit function for player i is: J i ( X) = γ i J c N( X) + E D (α d i ( X c, D)) E D (α f i ( X c, D)). A player i s strategy X i that maximizes her expected profit function, also maximizes the expected centralized profit J c N( X). Because there is a unique point X where JN( c X) is strictly increasing in X i for X i < Xi, and JN( c X) is strictly decreasing in X i for X i > Xi for all i N, no other points are local maxima. Therefore, the point X is a global maximum, i.e., a unique first-best solution. The point X is also a unique PSNE because (i) no player has an incentive to deviate from it, and (ii) for any other point, says at X X, each player i would be better off not playing Xi, given that other players play X N\i. (Eller College of Management, U of Arizona) DOMinant2009 September 21, / 42