,,, be any other strategy for selling items. It yields no more revenue than, based on the

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1 ONLINE SUPPLEMENT Appendix 1: Proofs for all Propositions and Corollaries Proof of Proposition 1 Proposition 1: For all 1,2,,, if, is a non-increasing function with respect to (henceforth referred to as the property of non-increasing daily marginal revenue, or NDMR property), then the greedy heuristic approach is guaranteed to find the optimal liquidation strategy. PROOF. The NDMR property states that if, then,,. Consider a liquidation problem of items. Let,,, denote the total revenue maximizing strategy for items. Then, let 1 be the strategy that yields the highest marginal revenue by selling one more item, based on strategy. Without loss of generality, let s assume that 1 achieves the maximum marginal revenue by selling an extra item on the th day, i.e., 1, 1,. Furthermore, let,,, be any other strategy for selling items. It yields no more revenue than, based on the definition of. Then, let 1, 1, is the strategy that yields the highest marginal revenue based on strategy ; here we assume that this is achieved by selling one more item on the th day, without loss of generality. Denote. as the total revenue obtained from a given strategy. Based on the above,. Then, proving the optimality of the greedy heuristic is equivalent to proving 1 1. By definition of daily marginal revenue, we have, 1 and, 1, where. Let s consider the cases and separately. If, which means there are more items sold on the th day by strategy than by strategy, then NDMR property implies:,,. Also, because selling one more item on th day gives the highest marginal revenue based on strategy, it follows that,,. Therefore,,,, and 1 1. Suppose, instead,, or equivalently 1, which means there are more items sold on the th day by strategy than by strategy. Because and, we know such that 1

2 , or 1. We can construct a strategy C as follows:, 1,, 1,. In other words, compared with strategy 1, there is 1 fewer item sold on the th day, and the same number of items sold on all other days. Based on this construction, we have, 1 and 1, 1. Because of the optimality of, we have. Because 1, NDMR implies that, 1,. Also, because selling one more item on th day gives the highest marginal revenue based on strategy A, which means:,,. Therefore,, 1,, and 1 1. Proof of Proposition 2 Proposition 2: The NDMR property is satisfied if and only if, for all 1,,2, and for all 1,,. PROOF. Given,, we can represent any customer valuation on day as,. By definition, we have,, and 1, 1,. Therefore, 1,,,,. For, to satisfy NDMR property, we need, 1,. Equivalently:,, This completes the proof., 1 1,, 1 1,, 2 1,, 2 1,,, 2

3 Proof of Corollary 2 Corollary 2. The NDMR property is satisfied if the underlying valuation distribution is uniform, exponential, or Weibull (with shape parameter larger than 1). PROOF. Suppose X,X, X are the n order statistics in descending order (i.e., X X X ). If the underlying distribution is uniform,, then, and is a constant. Therefore, 1. If the underlying distribution is exponential with parameter (i.e., 1e 0), then, and. Therefore,. If the underlying distribution is Weibull with shape parameter (we consider scale parameter to be 1 because it only changes the expected order statistics proportionally), i.e., 1 0. According to Lieblein (1955), Γ 1 where Γ. is the gamma function and 1. Since Γ is a common constant, we do not need to carry it in the computation. Therefore, we calculate /Γ !!!!!!! 1. Further, 1 1!!!!!!!!!!!!!!. Therefore,!!! 1 1!!!!!!!!. Define another function. We calculate 1 by following the same steps as we did for, and we can get 1!!!! 1. Equivalently, we write 1 1. When 1, we know 2 0,1. Denote 2,, 3

4 , then 1 1. Denote the m-th finite difference of function at point as where. is the m-th finite difference operator. By definition, 1. Equivalently, 1,,. To calculate, we need the following lemma: Lemma: Given positive integers and, suppose a function. is 1 times continuously differentiable on, and its m-th derivative exists on,, then, such that, where. represents the m-th derivative of.. Proof of Lemma: Construct a polynomial of degree, i.e.,, such that 0,1,,,. In other words, is the interpolation polynomial of at points,1,,. By this construction and the definition of finite difference,. Consider the difference..., we know that 0,1,,, 0. Equivalently, this means that. takes the same value (i.e., 0) at the end points of intervals:, 1, 1, 2,,1,. Since function. is also 1 times continuously differentiable on, and its m-th derivative exists on,, then function. satisfies the same property. According to the generalized Rolle s theorem, there, such that 0. It follows that!. On the other hand, according to the property of finite difference of polynomials, we know!. Therefore,!. Returning to the main proof, because where 0,1 is continuously differentiable on any interval to arbitrary degree, it certainly satisfies the condition specified in the above lemma. Following the lemma,, such that. Therefore, 1 1. Because,, we know 0, and because 0,1, 1 1 is positive if is odd and is negative if is even. Overall, is positive if is odd and is negative if is even. Therefore, 1 is always negative. It follows that

5 Proof of Proposition 3 Proposition 3. Under deterministic representation of stochastic demand, if NDMR property is satisfied, then the total revenue obtained by optimal liquidation strategy has the following properties: 1. Total revenue increases concavely as inventory size (N) increases; 2. Total revenue increases concavely as the length of liquidation period (D) increases, and stops increasing after a certain length; 3. Total revenue increases as number of daily arrivals (B) increases; 4. Total revenue increases convexly as daily decay rate () increases; 5. Total revenue decreases convexly as unit holding cost (h) increases; PROOF. Each of the five statements is proved as follows. 1. Inventory Size [Monotonicity] Denote as the optimal total revenue for items. Consider the heuristic assignment process for versus 1 items: the assignments of the first items based on heuristic algorithm are the same in both cases. 1 because otherwise the 1-th item would not have been assigned. [Concavity] Consider revenue gain by adding 1 more item, denote 1, and 2 1,, where, denotes the marginal revenue for assigning 1 more item on day d, with items already assigned. If, then,,, otherwise the 1-th item would have been assigned to. Instead, if and 1 (i.e., both the 1- th and the 2-th items are assigned to the same day), we also have,, because of NDMR property. Therefore, we always have 1 2 1, indicating concavity. 2. Length of Liquidation Period [Monotonicity] Denote the optimal liquidation revenue and optimal liquidation strategy for days as and,,. Suppose liquidation period is prolonged from days to 1 days. Because is still a feasible strategy for the liquidation problem of 1 days, if it is profitable to sell some number 5

6 of items on day 1, it means the seller can obtain a higher revenue than what would be obtained by the original -day strategy. Therefore, optimal revenue is monotonically increasing, until it is no longer profitable to sell any item on additional days (because of decay), i.e., 1. [Concavity] Consider,, and 1,,, where 0, we claim that 1,,,, i.e., for the first days, there are no fewer items sold under than under 1. Suppose otherwise, that there 1,, such that. Immediate from the heuristic assignment process, this means,, cannot be an optimal strategy for items, i.e.,,,. This implies that the sum of revenue from and revenue from selling items on day 1 would be higher than 1, contradiction to the optimality of 1. In other words, one can imagine the strategy change from to 1 as an action of de-allocating items in total from the first days, and re-allocate them to day 1. Consider the difference between and 1 on the first days, i.e.,,,. We know that 1,,, 0 and Therefore, we can derive the increase in optimal revenue as 1, 1,. The first term is the revenue gain from selling on day 1 whereas the second term is the cost of de- allocating from the first days. Note that there are exactly non-zero. elements in the second term. Based on heuristic assignment process, they correspond to the smallest daily marginal revenues among, where 1,, and 0,, 1. Without loss of generality, we can denote and order these elements as, then we can re-write the increase in optimal revenue as 1, 1, i.e., by pairing the marginal revenues on day 1 with in the descending order. Following the same approach, we can write 2 6

7 1, 2. Note that there are terms of, each of which is larger than even the largest among (i.e., ), again, because of the heuristic assignment process. Meanwhile, due to valuation decay, given,, 1, 2, which implies, 1, 2. Also, due to we have shown before,, and again due to valuation decay,, therefore,. Overall, it follows that, 1, 2, or 1 2 1, indicating concavity. Note that if 0 and 0, meaning that it is profitable to sell on day 1 but not any day further, then and the concavity still holds. If both and are zero, then , and optimal revenue does not change with any longer liquidation period. 3. Number of Daily Arrivals [Monotonicity] We use the following lemma from David (1997). Lemma. Let : denote the expected value of the -th largest order statistics, among order statistics in total, of an arbitrary distribution, then 1,2,,, : :. Proof. We refer to David (1997) for proof of this lemma. In other words, the largest expected order statistics increase as the total number of order statistics increases. Denote as the optimal revenue under daily arrival. According to the above lemma, the largest valuations under daily arrival 1 are larger than the corresponding largest valuations under daily arrival. As a result, the optimal strategy under daily arrival is associated with higher revenue under daily arrival 1, and therefore, 1, i.e., total revenue increases as number of daily arrivals increases. 4. Valuation Decay Rate [Monotonicity] As valuation decay rate decreases, selling on later days become less attractive, and more sells happen on 7

8 earlier days. Consider two decay rates, 0 1, denote as the optimal revenue under decay rate, and,, as the daily marginal revenue under decay rate. For any and, we know,,,,. Therefore,, because otherwise the liquidation strategy under would generate a higher total revenue under than. [Convexity] Regarding convexity, write total revenue as,1where,1 is the daily revenue of selling items on the first day. First, consider the case where a small increment in decay rate does not change the optimal liquidation strategy, i.e., do not change as a result of change in. Note that 1 2,10, suggesting that optimal revenue increases convexly with increasing (i.e., less rapid decay). Second, consider the case where a small increment in decay rate does change the optimal liquidation strategy. Note that such strategy change is possible to take place because of the NDMR property, otherwise it would always be more profitable to sell as many items as possible on earlier days regardless of decay rate value. Suppose and are two decay rates, such that and that optimal liquidation strategy changes exactly once as decay rate increases from to. Formally, denote as the optimal liquidation strategy under decay rate, then but,, is either the same as or the same as. Write,, and,,, and examine their difference,,. Since both and sell items in total, we know 0. Denote two sets of subscripts (indices of days): 0, 0. Clearly and are non-empty, as. Also, because, valuations under are higher than under, and sells more items in later days than. One can imagine that the liquidation strategy change from to consists of at least one re-allocation action, where an item originally assigned to an earlier day (under ) is re-allocated to a later day (under ). More precisely, denote the action of re-allocating 1 item from day to day by a pair of subscripts,, where,, and, i.e., day is later than day. Accordingly, the total number of items assigned to 8

9 these two days is, before re-allocation and becomes 1, 1 after re-allocation. Now, consider the following function of decay rate: 1,,,,. By definition of the reallocation, 0 and 0. Also,,,,,1 is a continuous function with respect to. Because is the difference between two continuous function, itself is a continuous function. Therefore, there, such that 0. Furthermore, if the strategy changes from to involves more than one re-allocation actions, then each of them must correspond to the same value where the allocations become revenue-indifferent. This is because, otherwise, there would be more than one strategy changes as decay rate increases from to, which contradicts our setup. Overall, this implies that there exists a decay rate between and, under which it is revenue-indifferent to use or. In other words, the optimal revenue is continuous with respect to decay rate. Next, we consider the left and right derivatives of optimal revenue with respect to decay rate, at point. Again, we write optimal revenue as,1, and derive the left derivative as 1,1and the right derivative as 1,1. As discussed before, the liquidation strategy change that takes place at point involves at least one re-allocation action, denoted by pair, with, where one item previously sold on day is re-allocated to be sold on day. Without loss of generality, consider one particular reallocation, and rewrite the left/right derivatives as:,, 1,1 1,11,1 and,, 1,1 1,1 1,1. Note that 1,1,,, 1,1, because no re-allocation happens on these days. Also,,1,1,1,1, because under the re-allocation is revenue-indifferent. In addition, directly based on the 9

10 re-allocation process, we know and, and therefore,1,1 and,1,1. Denote quantities,1,,1,,1, and,1, then and,. Therefore Since and, we know that. In other words, at the strategy change point, the right derivative is larger than left derivative, indicating convexity on the entire interval,. Therefore, the optimal revenue is convex with respect decay rate on the entire interval 0,1. 5. Incorporating Inventory Holding Cost [Monotonicity] Denote,, as the marginal revenue under holding cost, we know,,, 1. Consider two holding costs, then for any and,,,,,. Denote as the optimal revenue under holding cost, then, because otherwise the same strategy under would create higher revenue under than. Therefore, optimal revenue monotonically decrease as holding cost increases. [Convexity] Regarding convexity, write optimal revenue as 0, where 0 is the optimal revenue without holding cost, and 1, which represents the total number of unit holding cost incurred throughout the liquidation period. As holding cost increases, if liquidation strategy does not change, then optimal revenue decreases linearly with slope of. Instead, if liquidation strategy does change, more items would be sold to earlier days, and would decrease accordingly. As a result, optimal revenue decreases at a slower rate, indicating convexity. Reference David, H. A. (1997). Augmented order statistics and the biasing effect of outliers. Statistics & probability letters, 36(2),

11 Appendix 2: Performance Benchmarking when NDMR Property is violated We conduct an additional set of simulations, where customer valuations are randomly drawn from a Weibull distribution with a shape parameter less than 1, which is a distribution not typically used to model customer valuations in the literature. Because expected order statistics from such a Weibull distribution do not satisfy the NDMR property, our proposed heuristic approach is not guaranteed to find the optimal liquidation strategy under the deterministic demand representation. For the sake of experimental completeness, we use this set of simulations to illustrate the revenue performance of greedy heuristic under not only demand stochasticity, but also NDMR violation. For benchmarking purposes, we also incorporate the proposed dynamic programming (DProg in short) approach discussed in Section 3.3. Under deterministic demand representation, our proposed heuristic can find the optimal liquidation strategy when NDMR property is satisfied; in contrast, our proposed dynamic programming approach can find the optimal liquidation strategy regardless of whether NDMR property is satisfied or violated. In addition, similarly to SDP, the DProg approach is also naturally suitable to adjustable daily prices. We compare the revenue performance of both PDS and DProg with SDP. Specifically, we use the same basic small problem configuration as in Section 6.2 (i.e., arrival ~ 100, 100, 10, and 0.99 ), and simulate customer valuations from Weibull distributions with shape parameters of 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. Simulation results are reported in the following table. Table A2. Revenue and Running Time Comparisons when NDMR is violated Valuation Total Revenue Running Time Distribution SDP PDS DProg SDP PDS DProg (-7.35%) (-2.00%) (7209x) 0.05 (317x) (-0.48%) (-0.48%) (6109x) 0.05 (269x) (-0.15%) (-0.15%) (6729x) 0.05 (282x) (-0.11%) (-0.11%) (6057x) 0.05 (254x) (-0.09%) (-0.09%) (5090x) 0.05 (214x) (-0.11%) (-0.11%) (5919x) 0.05 (249x) Note. Values in parentheses represent the difference between PDS or DProg with respect to SDP in terms of percentage revenue loss and running speed. We can see that, even when NDMR property is violated, PDS strategy is able to obtain a major percentage 11

12 of the optimal expected revenue under stochastic demand, while continuing to be several orders of magnitude faster than SDP. In particular, when shape parameter is larger than or equal to 0.5, PDS is associated with less than 0.5% revenue loss, which is comparable to the magnitude of revenue loss when NDMR property is satisfied. Furthermore, despite being computationally less efficient, the DProg strategy can produce revenue improvement upon PDS, especially when the shape parameter is 0.4. Therefore, DProg can be a viable alternative to our proposed heuristic in the less common scenarios where NDMR property for deterministic demand representation is violated. Again, the aforementioned results are included to provide a comprehensive picture of the performance of the proposed approach, i.e., even in highly atypical scenarios. However, overall, we argue that the NDMR property is an important characteristic of practical inventory liquidation problems for several reasons. First, we have analytically proved that uniform, exponential, and Weibull (with shape parameter larger than 1) satisfy the NDMR property. These distributions are very commonly used in the inventory management literature to model or simulate customer valuations. For example, uniform distribution was used in Araman and Caldentey (2009) and Aviv and Pazgal (2002); exponential distribution was used in Gallego and van Ryzin (1994), Smith and Achabal (1998), Araman and Caldentey (2009), and Farias and Van Roy (2010); Weibull distribution was used in Bitran and Mondschein (1997), Bitran et al. (1998), and Bitran and Caldentey (2003). We believe that these distributions are so widely used in the literature because they represent canonical and realistic types of customer valuations. In particular, uniform distribution represents the case where all possible valuations (within some specified range) are equally likely. Exponential distribution is a natural choice to model the long-tail, where many customers have relatively low valuations and a few have relatively high valuations. Weibull distribution (with shape parameter larger than 1) is a natural choice to model unimodal valuations, where the mass of customers have valuations within certain range and few customers have extremely small or large valuations. Second, we numerically checked the NDMR condition for several other common distributions in the exponential family that are continuous and are defined within 0, (i.e., we do not consider negative valuations), including Gamma distribution, Chi-Squared distribution, and Erlang distribution. We found 12

13 that Gamma distribution with shape parameter and scale parameter (denoted as, ) satisfies the NDMR property when 1. The Chi-Squared distribution,, is equivalent to,2, and therefore satisfies the NDMR property when 2. The Erlang distribution is equivalent to, with, and therefore always satisfies the NDMR property. Third, expected order statistics of some typical NDMR-incompliant distributions do not represent reasonable customer valuations. For example, in Weibull distributions with small shape parameter (e.g., shape parameter equals 0.1, 0.2, or 0.3), the largest expected order statistics is even larger than twice the second-largest expected order statistics. In other words, under such valuation distributions, it would not even be profitable to sell more than 1 item per day, which is not representative of the vast majority of realistic inventory liquidation scenarios. Finally, NDMR-incompliant valuation distributions do not necessarily result in substantial revenue loss of our proposed approach. The actual performance depends on the specific liquidation problem configurations. In the above simulations, if 0.5, PDS is associated with comparable revenue loss when NDMR property is satisfied. This is because, under these simulated Weibull distributions, the NDMR condition is not violated for every expected order statistics the largest few expected order statistics satisfy the NDMR condition, whereas the subsequent smaller expected order statistics violate the NDMR condition. The same is true for Gamma distribution with shape parameter less than 1. For example, even for a Gamma distribution with shape parameter as small as 0.05, the largest 6 expected order statistics (out of 100 in total) still satisfy the NDMR property. In summary, the NDMR property reflects a broad problem context, which has applicability to a very broad set of distributions representative of a variety of real-world phenomena and under which our approach has strong performance guarantees. 13

14 Appendix 3: Estimation of Valuation Decay Rate based on Valuation Distribution Suppose the CDFs of customer valuation distributions for times and 1 are and, respectively. Recall that the valuation decay rate,, is defined as a quantity that satisfies,. Suppose we have estimated the valuation distributions for times and 1; then, the valuation decay from time to 1 can be estimated based on parameters of the specific distribution. Below we describe the estimation approach for three common valuation distributions: exponential, Weibull, and uniform. Exponential valuation distribution. Suppose 1 and 1 ; then 1. The definition of decay rate implies: /. Weibull valuation distribution. Suppose 1exp and 1 exp, where is the scale parameter and is the shape parameter. Note that only the scale parameter changes over time, as it characterizes the magnitude of valuations. It follows that 1exp, and the definition of decay rate implies /. Uniform valuation distribution. Suppose and. In the special case where 0, i.e., the lower bound of valuation distribution is 0, we can obtain a unique decay rate estimation, /. In the general case where,,, 0, there can be different decay rate estimations for different choices of valuations (i.e., this means that customer valuations of different magnitudes may decay at different rates). Nonetheless, our proposed heuristic would still be able to find the optimal liquidation strategy under deterministic demand representation, because the uniform distribution on each day satisfies the NDMR property. We summarize the above results in the following table. Table A3. Estimation of Daily Decay Rate under Common Valuation Distributions Valuation Distribution at Time Valuation Distribution at Time 1 Estimated 0, 0, / /,, / 14

15 Appendix 4: Performance Benchmarking with Other Scalable Liquidation Strategies (Poisson Arrival) Table A4.1. Performance Benchmarking under Poisson Arrival and Uniform Valuations N D Without adjustable daily prices With adjustable daily prices PAS FP FQ PDS DP DQ (26.57) (14.42) (30.89) (11.96) (8.33) (15.55) (24.86) (16.78) (27.40) (11.66) (7.91) (14.00) (21.46) (15.69) (26.95) (9.92) (6.64) (13.77) (25.37) (18.19) (27.21) (9.62) (6.61) (13.15) (23.76) (18.07) (23.32) (8.81) (6.01) (10.64) (25.12) (20.03) (21.91) (7.75) (5.63) (10.19) (33.39) (19.73) (41.27) (15.40) (10.23) (19.06) Valuations ~ (26.22) (21.75) (28.53) (11.69) (9.05) (12.76) 0, (32.46) (22.34) (36.16) (15.77) (10.00) (18.88) (31.70) (20.15) (33.98) (17.22) (9.95) (19.84) (23.22) (16.86) (24.68) (17.40) (12.07) (19.02) (23.64) (14.24) (27.16) (14.86) (9.29) (18.66) (19.76) (7.50) (22.29) (15.90) (9.02) (18.21) (15.68) (6.62) (16.38) (13.76) (7.93) (14.72) (15.77) (4.69) (15.93) (15.02) (7.18) (14.95) (13.64) (3.13) (13.85) (13.31) (5.68) (13.37) Note. For each strategy, the first number represents average revenue, the second number in parentheses represents standard deviation of revenue. 15

16 Table A4.2. Performance Benchmarking under Poisson Arrival and Exponential Valuations N D Without adjustable daily prices With adjustable daily prices PAS FP FQ PDS DP DQ (64.94) (41.24) (66.20) (48.11) (39.41) (51.23) (74.57) (48.55) (71.85) (49.01) (38.37) (50.71) (76.94) (51.89) (77.95) (48.68) (39.02) (55.27) (73.04) (49.44) (82.15) (42.69) (33.38) (55.46) (67.89) (50.88) (73.44) (36.94) (29.02) (47.96) (73.04) (49.84) (76.46) (35.65) (28.41) (47.01) (72.04) (56.87) (80.14) (48.95) (41.42) (55.79) Valuations ~ (64.25) (46.24) (74.06) (53.63) (42.07) (57.93) (53.69) (40.45) (63.21) (46.61) (39.60) (50.29) (56.48) (44.94) (62.44) (58.28) (51.02) (61.35) (45.28) (35.28) (45.47) (53.16) (46.48) (54.48) (51.70) (32.60) (54.42) (38.38) (27.28) (44.19) (41.30) (18.13) (46.88) (36.05) (20.82) (42.33) (32.11) (10.46) (32.58) (30.66) (15.85) (30.69) (25.74) (8.88) (32.41) (25.50) (13.21) (31.24) (23.64) (5.13) (28.84) (23.50) (9.54) (28.12) Note. For each strategy, the first number represents average revenue, the second number in parentheses represents standard deviation of revenue. 16

17 Table A4.3. Performance Benchmarking under Poisson Arrival and Weibull Valuations N D Without adjustable daily prices With adjustable daily prices PAS FP FQ PDS DP DQ (46.13) (33.46) (49.74) (15.11) (10.37) (22.59) (50.97) (35.51) (51.48) (13.55) (9.70) (21.67) (37.73) (25.96) (37.38) (11.78) (8.91) (16.41) (35.11) (22.67) (36.24) (9.35) (6.60) (15.74) (43.52) (30.49) (44.75) (10.38) (7.79) (18.83) (39.89) (25.64) (36.74) (8.71) (6.99) (14.63) (50.11) (33.06) (57.79) (15.01) (11.24) (20.40) Valuations ~ (46.71) (30.42) (49.94) (15.27) (10.26) (18.81) (43.40) (30.40) (51.02) (14.39) (10.51) (18.23) (34.79) (26.05) (43.86) (13.19) (9.66) (16.94) (41.62) (32.94) (45.78) (16.44) (10.80) (20.82) (38.28) (21.54) (33.33) (16.35) (11.46) (20.31) (31.35) (18.89) (31.61) (20.55) (12.70) (25.13) (23.99) (13.46) (23.43) (19.31) (10.73) (20.99) (28.07) (11.56) (20.29) (24.11) (8.99) (18.94) (22.44) (8.11) (18.01) (20.39) (6.98) (17.09) Note. For each strategy, the first number represents average revenue, the second number in parentheses represents standard deviation of revenue. 17

18 Figure A4. Performance Benchmarking under Poisson Arrival Note. Daily arrival drawn from 300. The first, second, and third rows correspond to customer valuations of 0,1, 1, and 4, respectively. The first, second, and third columns correspond to simulations varying inventory size, liquidation period length, and decay rate, respectively. 18

19 Appendix 5: Performance Benchmarking with Other Scalable Liquidation Strategies (Uniform Arrival) Table A5.1. Performance Benchmarking under Uniform Arrival and Uniform Valuations N D Without adjustable daily prices With adjustable daily prices PAS FP FQ PDS DP DQ (76.55) (67.75) (66.63) (35.84) (27.40) (36.91) (72.23) (66.81) (62.49) (33.81) (27.21) (33.69) (61.58) (51.97) (53.54) (27.78) (21.53) (27.84) (62.94) (60.81) (55.36) (26.35) (21.55) (26.68) (58.66) (49.11) (49.14) (24.26) (18.82) (24.12) (47.63) (38.80) (43.92) (17.59) (13.59) (19.27) (74.58) (64.69) (75.43) (36.76) (29.01) (38.41) Valuations ~ (78.75) (66.70) (77.00) (40.35) (30.39) (41.26) 0, (72.17) (65.07) (76.44) (38.67) (27.46) (43.73) (81.78) (72.54) (84.81) (54.81) (36.31) (60.38) (76.53) (68.75) (76.66) (63.20) (48.00) (65.88) (62.30) (48.14) (50.84) (43.20) (35.25) (34.21) (62.24) (34.60) (46.31) (54.20) (38.60) (36.95) (60.15) (30.12) (40.25) (56.30) (34.45) (35.06) (53.40) (27.16) (33.19) (51.74) (19.70) (29.86) (51.73) (19.23) (32.20) (51.09) (14.38) (29.89) Note. For each strategy, the first number represents average revenue, the second number in parentheses represents standard deviation of revenue. 19

20 Table A5.2. Performance Benchmarking under Uniform Arrival and Exponential Valuations N D Without adjustable daily prices With adjustable daily prices PAS FP FQ PDS DP DQ (147.00) (123.79) (141.42) (113.13) (97.55) (117.48) (135.27) (108.62) (142.43) (102.31) (86.49) (116.20) (153.32) (124.85) (152.30) (105.96) (92.36) (112.00) (141.99) (109.72) (139.47) (96.91) (83.39) (103.39) (124.96) (97.42) (120.29) (75.43) (66.79) (82.42) (133.49) (102.07) (124.08) (72.74) (61.01) (81.00) (160.70) (129.36) (174.89) (134.81) (112.80) (141.90) Valuations ~ (125.85) (103.35) (139.48) (114.32) (101.40) (120.26) (123.09) (100.20) (131.51) (125.17) (110.77) (129.50) (154.11) (133.31) (160.96) (166.18) (143.68) (170.35) (127.63) (101.39) (128.47) (175.60) (156.16) (176.03) (126.47) (91.48) (122.57) (105.26) (85.61) (103.33) (115.20) (61.03) (108.50) (107.54) (85.83) (96.79) (96.93) (39.16) (91.99) (94.97) (65.67) (84.62) (88.78) (35.63) (77.93) (87.56) (51.57) (73.89) (70.99) (34.29) (59.91) (70.89) (18.97) (57.80) Note. For each strategy, the first number represents average revenue, the second number in parentheses represents standard deviation of revenue. 20

21 Table A5.3. Performance Benchmarking under Uniform Arrival and Weibull Valuations N D Without adjustable daily prices With adjustable daily prices PAS FP FQ PDS DP DQ (113.05) (93.62) (96.34) (42.05) (34.55) (44.00) (105.94) (100.21) (86.37) (36.58) (30.42) (39.80) (120.91) (116.09) (86.31) (37.12) (31.64) (37.25) (109.55) (97.75) (83.31) (33.11) (27.86) (35.78) (88.28) (78.39) (64.77) (27.46) (22.37) (29.53) (75.91) (66.61) (61.76) (21.71) (18.04) (26.32) (132.89) (122.88) (110.91) (50.40) (41.48) (48.09) Valuations ~ (117.44) (106.51) (104.97) (45.66) (36.94) (48.34) (121.76) (118.03) (113.57) (46.89) (39.53) (52.38) (100.20) (91.61) (105.70) (43.38) (35.18) (51.84) (118.40) (109.88) (120.09) (50.09) (39.98) (57.07) (114.23) (95.02) (88.00) (64.28) (45.80) (54.08) (108.73) (96.01) (63.25) (78.67) (50.08) (49.43) (99.32) (67.08) (48.02) (79.90) (43.05) (41.45) (105.91) (47.24) (47.17) (92.99) (38.08) (42.60) (96.94) (34.97) (45.28) (91.68) (34.95) (41.75) Note. For each strategy, the first number represents average revenue, the second number in parentheses represents standard deviation of revenue. 21

22 Figure A5. Performance Benchmarking under Uniform Arrival Note. Daily arrival drawn from 150,450. The first, second, and third rows correspond to customer valuations of 0,1, 1, and 4, respectively. The first, second, and third columns correspond to simulations varying inventory size, liquidation period length, and decay rate, respectively. 22

23 Appendix 6: Total Revenue and Running Time Comparisons among PDS, SDP, and ADP Table A6.1. Total Revenue and Running Time Comparisons under Uniform Valuations N D Total Revenue Running Time (Seconds) PDS SDP ADP PDS SDP ADP (+0.69%) (-0.12%) (4642x) (31809x) (+0.65%) (-0.21%) (4538x) (30888x) (+0.61%) (-0.34%) (4234x) (29287x) (+0.54%) (-0.57%) (4164x) (29237x) (+0.40%) (-0.91%) (4133x) (28895x) (+0.27%) (-0.90%) (4210x) (29717x) (+0.73%) (-0.17%) (4310x) (28047x) Valuations ~ (+0.78%) (-0.09%) (3929x) (24911x) 0, (+0.90%) (-0.02%) (3560x) (21726x) (+0.95%) (0.00%) (3425x) (20325x) (+1.04%) (+0.02%) (3040x) (17300x) (+0.66%) (0.00%) (4785x) (31071x) (+0.40%) (-0.26%) (5733x) (36361x) (+0.14%) (-0.35%) (6900x) (43560x) (+0.03%) (-0.55%) (7471x) (46635x) (+0.03%) (-0.38%) (8223x) (50292x) Note. The numbers in parentheses represent the differences in revenue or running time between SDP/ADP and PDS. 23

24 Table A6.2. Total Revenue and Running Time Comparisons under Exponential Valuations N D Total Revenue Running Time (Seconds) PDS SDP ADP PDS SDP ADP (+0.14%) (-0.67%) (8104x) (28738x) (+0.20%) (-0.76%) (7511x) (27029x) (+0.27%) (-0.80%) (7324x) (26112x) (+0.31%) (-0.85%) (6892x) (24945x) (+0.35%) (-0.89%) (6569x) (23122x) (+0.35%) (-0.81%) (6325x) (22575x) (+0.19%) (-0.63%) (7315x) (25773x) Valuations ~ (+0.25%) (-0.67%) (6311x) (21611x) (+0.29%) (-0.61%) (6046x) (20080x) (+0.46%) (-0.56%) (5292x) (17253x) (+0.61%) (-0.47%) (4545x) (14499x) (+0.18%) (-0.57%) (7540x) (27240x) (+0.19%) (-0.56%) (7957x) (28614x) (+0.18%) (-0.66%) (8440x) (30210x) (+0.23%) (-0.55%) (8768x) (31684x) (+0.20%) (-0.58%) (9005x) (31689x) Note. The numbers in parentheses represent the differences in revenue or running time between SDP/ADP and PDS. 24

25 Table A6.3. Total Revenue and Running Time Comparisons under Weibull Valuations N D Total Revenue Running Time (Seconds) PDS SDP ADP PDS SDP ADP (+0.85%) (-0.02%) (47200x) (30149x) (+0.73%) (-0.20%) (46140x) (29151x) (+0.57%) (-0.33%) (46963x) (28843x) (+0.34%) (-0.40%) (46172x) (27561x) (+0.13%) (-0.46%) (48478x) (27599x) (+0.12%) (-0.57%) (47231x) (27089x) (+0.88%) (-0.09%) (42866x) (27388x) Valuations ~ (+0.96%) (-0.07%) (39543x) (24581x) (+1.07%) (-0.15%) (36550x) (21850x) (+1.07%) (-0.34%) (33266x) (19025x) (+1.12%) (-0.46%) (25509x) (14563x) (+0.54%) (-0.17%) (51538x) (34977x) (+0.52%) (-0.45%) (62060x) (41113x) (+0.28%) (-0.54%) (72607x) (46930x) (+0.13%) (-0.65%) (79925x) (50100x) (+0.13%) (-0.57%) (86145x) (54545x) Valuations ~ (+0.03%) (-0.49%) (44861x) (28647x) Valuations ~ (+0.64%) (-0.36%) (44976x) (28828x) Valuations ~ (+0.73%) (-0.22%) (44409x) (28642x) Valuations ~ (+0.83%) (-0.09%) (45414x) (28638x) Valuations ~ (+0.84%) (-0.04%) (47945x) (30105x) Valuations ~ (+0.86%) (+0.01%) (48010x) (30250x) Note. The numbers in parentheses represent the differences in revenue or running time between SDP/ADP and PDS. 25

26 Figure A6.1. Total Revenue Comparisons among PDS, SDP, and ADP Note. Daily arrival drawn from 100. The first three rows correspond to customer valuations of 0,1, 1, and 4, respectively. The first, second, and third columns correspond to simulations varying inventory size, liquidation period length, and decay rate, respectively. The bottom left figure represents the simulation varying Weibull shape parameter under the basic configuration. 26

27 Figure A6.2. Running Time (Seconds) Comparisons among PDS, SDP, and ADP Note. Daily arrival drawn from 100. The first three rows correspond to customer valuations of 0,1, 1, and 4, respectively. The first, second, and third columns correspond to simulations varying inventory size, liquidation period length, and decay rate, respectively. The bottom left figure represents the simulation varying Weibull shape parameter under the basic configuration. 27

28 Appendix 7. Additional Simulation Results for Variations of Price-Based and Quantity-Based Strategies While we discussed the most natural constructions of price-based (FP and DP) and quantity-based (FQ and DQ) strategies in Section 6.1, other variations of these strategies are possible. In particular, we consider the following variations. For price-based strategies, we define to be a fixed-price strategy where the seller sets the fixed price to be the -th expected order statistic across the entire liquidation horizon. Clearly, is equivalent to the FP strategy we considered in Section 6.1. Similarly, we define to be a dynamic-price strategy where the seller sets price on day d to be the -th expected order statistic across the remaining liquidation horizon, where is the amount of remaining inventory at the beginning of day d. Again, is equivalent to the DP strategy we considered in Section 6.1. For quantity-based strategies, we define to be a fixed-quantity strategy where the seller sets the price on day d to be the -th expected order statistics on that day. Clearly, is equivalent to the FQ strategy we considered in Section 6.1. Similarly, we define to be a dynamic-quantity strategy where the seller sets price on day d to be the -th expected order statistic on day d, where / 1 is the average daily quantity that needs to be sold, i.e., the total remaining quantity divided by the number of remaining days. Again, is equivalent to the DQ strategy we considered in Section 6.1. We repeat the same simulation experiments conducted in Section 6.1 under Poisson arrival. For each price-/quantity-based strategy, we simulate 7 different variations with 10, 5, 1,0,1,5,10. For instance, for fixed-price strategy, we simulate,,,,,,. We chose to simulate these values of as they represent different magnitudes of adjustments with respect to the original price-/quantity-based strategies. Given the liquidation problems we have simulated, 10,10 represent relatively large adjustments, 5,5 represent medium-sized adjustments, and 1,1 represent small adjustments. Below we report the average liquidation revenue comparisons for (1) PAS vs. 28

29 ; (2) vs. ; (3) PDS vs. ; and (4) PDS vs., respectively in four tables. Table A7.1. Performance Comparison between PAS and N D PAS Valuations ~ 0,1 Valuations ~ 1 Valuations ~

30 Table A7.2. Performance Comparison between PAS and N D PAS Valuations ~ 0,1 Valuations ~ 1 Valuations ~