Assumption 1

[10pt]

Dynamic Pricing under a Static Calendar

Will Ma

Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139, willma@mit.edu

David Simchi-Levi

Institute for Data, Systems, and Society, Department of Civil and Environmental Engineering, and Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139, dslevi@mit.edu

Jinglong Zhao

Institute for Data, Systems, and Society, Massachusetts Institute of Technology, Cambridge, MA 02139, jinglong@mit.edu

This work is motivated by our collaboration with a large Consumer Packaged Goods (CPG) company. We have found that while they appreciate the advantages of dynamic pricing, they deem it operationally much easier to plan out a static price calendar in advance.

In this paper, we investigate the efficacy of static control policies for dynamic revenue management problems. In these problems, a firm is endowed with limited inventories to sell over a finite time horizon where demand is known but stochastic. We consider both pricing and assortment controls, and derive simple static policies in the form of a price calendar or a planned sequence of assortments, respectively. We show that our policies are within 1-1/e (approximately 0.63) of the optimum under stationary (IID) demand, and 1/2 of optimum under non-stationary demand, with both guarantees approaching 1 if the starting inventories are large.

A main contribution of this work is developing a system of tools for establishing best-possible performance guarantees relative to linear programming relaxations; specifically, structural properties about static policies which provide a complete characterization of tight bounds in the stationary setting, and an adaptation of the well-known prophet inequalities from optimal stopping theory to pricing and assortment problems in the non-stationary setting. We also demonstrate on data provided by the CPG company that our simple price calendars are effective.

 

A revenue management problem can be generally described as follows. A firm has finite inventories of multiple items to sell over a finite time horizon. The starting inventories are unreplenishable and exogenously given, having been determined by supply chain constraints or a higher-level managerial decision. The firm can control its sales through sequential decisions in the form of accepting/rejecting customer requests, pricing, or adjusting the assortment of items offered. Its objective is to maximize the cumulative revenue earned before the time horizon or inventory runs out.

We consider the setting where customer demand is distributionally-known and independent over the time horizon, having been estimated from historical sales. The literature has also considered other settings, where an unknown IID demand distribution (Besbes and Zeevi 2009, Agrawal et al. 2017) or an evolving demand process correlated across time (Araman and Caldentey 2009, Ciocan and Farias 2012) must be dynamically learned, to list a few references. In our setting, the firm’s decision at one point in time has no impact on its estimate of the demand at another point in time. Instead, the time periods are linked by the inventory constraints, and the firm must trade off between revenue-centric decisions which maximize expected revenue irrespective of inventory consumption, and inventory-centric decisions which maximize the yield from the remaining inventory. Revenue-centric decisions tend to be myopic and maximize the sales volumes of the most popular items, while inventory-centric decisions tend to be conservative and charge higher prices or prioritize selling highly-stocked items.

Intuitively, the optimal control policy would make revenue-centric decisions when the overall remaining inventory is plentiful for the remaining time horizon, and inventory-centric decisions when the overall remaining inventory is scarce relative to the remaining time horizon. However, not all firms have the infrastructure to query the state of its inventory in real-time and adjust its decisions accordingly. In the case of the Consumer Packaged Goods (CPG) company we are working with, it is of great operational benefit for their brick-and-mortar stores to set a price calendar far in advance, before any sales are realized, which allows the marketing team to design flyers and advertisements accordingly.

In this paper, we derive static (or non-adaptive) policies, which must plan out all of the firm’s decisions at the start of the time horizon, for revenue management problems that are intrinsically dynamic, where the optimal policy should adapt to the realized inventory state. We demonstrate the effectiveness of our policies on data provided by the CPG company. Our policies are also structurally very simple, and have performance guarantees comparable to those of their dynamic counterparts.

We partition the time horizon into a discrete number of time periods. This does not lose generality, since a continuous time horizon can be modeled by the limiting case where the time periods are arbitrarily granular. Similarly, we model each item as having discrete “prices points” at which it could be sold. This allows us to both approximate a continuous price range and capture situations where fixed price points have been pre-determined by market standards.

Throughout this paper, we will consider two models which differ in the type of decision made by the firm.

  1. Single-item Pricing: There is a single item with a discrete starting inventory. We are given, for each time period and each feasible price , the probability of earning a sale if price is offered during time . The goal is to plan, for each time period , the price to offer during time , with no sales occurring if inventory has stocked out. We also generalize our results to the fractional-demand setting, where the demand distribution given for each time and price is over the continuous interval [0,1] (after normalizing), and the sales at a time period is the minimum of the realized demand and remaining inventory.

  2. Assortment (and Pricing): There are multiple items each with a discrete starting inventory. We are given, for each time period and each assortment of items which could be offered (as well as corresponding prices), the probability of selling each item in during time . We assume that these choice probabilities satisfy a substitutability condition (see Section id1), which is very mild and standard in the literature. The goal is to plan, for each time period , the assortment of items (and prices) to offer during time , with no sales occurring if the customer chooses an item that has stocked out.

These two prototypical models capture most controls used in revenue management, including the case where the firm’s decision is to accept/reject customer requests (Talluri and Van Ryzin 2006, Maglaras and Meissner 2006). The single-item pricing problem with discrete prices and {0,1}-demand is a special case of the assortment problem with pricing. Nonetheless, we consider it separately, because the optimal static policy satisfies additional structural properties in this special case, and the generalization to [0,1]-demand is what we will apply on the data provided by the CPG company.

Finally, we will consider the following two demand models separately, because the design of effective policies changes significantly when the demand is stationary (IID).

  1. Stationary: the demand distribution for a specific decision, e.g. the purchase probability associated with price , is identical for all time periods .

  2. Non-stationary: the demand distribution for any decision can vary arbitrarily over time (but the realizations are still independent across ).

Our policies are based on a deterministic linear program (see Section id1 for details), which can be formulated for a given problem instance (items, inventories, prices, and demand distributions) in advance, and hence be used to derive static policies. At a high level, the LP uses deterministic values to approximate the random execution of a policy, and we can use its optimal solution as a “guide” in designing actual policies.

Such an LP was first used for the single-item pricing problem under stationary demand (Gallego and Van Ryzin 1994). It is known111 This fact also originated from Gallego and Van Ryzin (1994), although it is not to be confused with their result in the continuous-price, regular-demand-function setting, which says that the LP suggests a single price. that in the general non-degenerate case, an optimal LP solution will suggest two prices to be offered for fractions of the time horizon, respectively. Our static policy is to simply offer the higher price for the first -fraction of the time horizon, and offer the lower price for the remaining time horizon (rounding as necessary to fit the discrete time periods). By contrast, the policy originally proposed for this situation by Gallego and Van Ryzin (1994, Sec. 4) allowed the prices to be offered in either order, but required the switching point to be dynamically determined based on the realized inventory levels. We show that if the switching point must be fixed in advance, then only the high-to-low ordering of prices is effective.

For the assortment problem under stationary demand, we propose a similar static policy which follows the choice-based deterministic LP originating from Gallego et al. (2004), in this case probabilistically. The choice-based LP can in general be solved efficiently despite having exponentially-many variables, and we defer discussion about such methods to Cheung and Simchi-Levi (2016).

Moving to non-stationary demand, we can no longer directly follow the LP solution. In fact, we may want to modify certain decisions suggested by the LP to ensure that sufficient inventory is “reserved” for higher-revenue time periods (see Example 1). To accomplish this, we introduce a bid price for each item , which can be interpreted as the opportunity cost of a unit of item ’s inventory. Our bid prices are based on the non-stationary LP, but end up being time-invariant (i.e., independent of ).

In the single-item pricing problem, letting denote the bid price of the single item, our static policy is to greedily maximize the expected “profit” under cost . That is, our policy sets the price at each time to be . Policies based on bid prices are common in revenue management, and the papers by Adelman (2007), Rusmevichientong et al. (2017) imply static policies for our problem which take the same form. The difference is in the computation of the bid prices, where their bid prices are time-varying and based on approximate dynamic programming, while our bid prices are time-invariant and based on the LP.

In the assortment problem under non-stationary demand, our static policy is to take the LP solution, remove from the suggested assortments all instances where an item is offered at a price less than , and then follow the modified solution. In doing so, we are treating as an acceptance threshold instead of a bid price. Our policy is similar to those of Wang et al. (2015), Gallego et al. (2016), in that it probabilistically imitates the LP solution and independently determines for each item when to discard it from the assortment. However, our discarding rule is static and based on a fixed , whereas their discarding rule is dynamic and based on the realized inventory levels.

We establish performance guarantees for our static policies which, in many cases, improve existing guarantees even for dynamic policies. All of our guarantees are ratios relative to the optimal LP objective value, which is an upper bound on the performance of any static or dynamic policy.

Dynamic Policies Static Policies Stationary Demand Single-item Pricing [Gallego and Van Ryzin (1994)] [Theorem 1] Assortment [Liu and Van Ryzin (2008)] [Theorem 2] General Demand [Theorem 1] Non-stationary Demand Pricing/Assignment [Wang et al. (2015)] [Rusmevichientong et al. (2017); Assortment [Gallego et al. (2016)] Theorems 34] General Demand [Theorem 3] Non-stationary Demand Single-item Pricing [Wang et al. (2015)] [Hajiaghayi et al. (2007);Theorem 5] Assortment [Gallego et al. (2016)] [Theorem 6] General Demand [Theorem 5] Note: refers to the amount of starting inventory (or the smallest starting inventory, if there are multiple items).

Table 1: Lower bounds on the performance of static and dynamic policies. Our new results (in Section id1) are bolded.

Our results are outlined in Table 1. The baseline performance ratio is for stationary demand and for non-stationary demand. That is, our static policies always earn at least 50% of the optimum in expectation, with the ratio improving to if the given demand distributions are stationary. Both of these ratios are tight. The ratios also increase to 100% as , the starting inventory level when demand has been normalized to lie in [0,1] (or in the assortment setting, the minimum starting inventory among the items), increases to .

In the stationary-demand pricing problem, Gallego and Van Ryzin (1994) have previously derived both the lower bound of and an asymptotic-optimality result. However, their policy is in general dynamic, unless the demand function is regular over a continuous interval. By contrast, we show that the same results can be obtained using a static policy, regardless of demand regularity, if we sort the prices from high-to-low in the static policy. Also, in our analysis, we derive the tightest possible bound for every value of and (the number of time periods), which allows us to establish asymptotic optimality in only . To our knowledge, existing analyses in the revenue management literature have scaled both and to get an asymptotic optimality result.

In the stationary-demand assortment problem, we obtain the exact same bounds which are tight in and . To our knowledge, this type of result, which inclues the baseline lower bound of , has been previously unknown222 The results in Golrezaei et al. (2014) imply performance guarantees for our problem, but their ratios are smaller than , since they are designed to hold under more general settings such as adversarial demand. for the assortment problem. Asymptotic optimality has been previously derived by Liu and Van Ryzin (2008) for their dynamic policy when both and .

Moving to non-stationary demand, the lower bound of 1/2 which improves to 1 as has been previously established using dynamic policies, in the assignment problem of Wang et al. (2015) and the more general assortment problem of Gallego et al. (2016). We establish the same bounds using static policies, with an extremely simple analysis based on prophet inequalities from optimal stopping theory. However, our convergence rate of is worse than the rate of achievable by their dynamic policies.

We should mention that the lower bound of 1/2 for static policies under non-stationary demand has also been recently established by Rusmevichientong et al. (2017). Their bound and analysis differ from ours in that they are relative to the optimal dynamic policy instead of the deterministic LP relaxation. One benefit of using the LP is it directly extends to the fractional-demand setting, since the LP does not change when demand can take any value in [0,1], which is our application of interest with the CPG company. By contrast, their framework is designed for a very general setting where resources can be reused after a random amount of time.

We now outline the new analytical techniques we used in deriving the results in Table 1.

  1. Stationary Demand (Sections id1 and id1), We establish a sequence of three lemmas about single-item pricing policies which: (i) relate the LP optimum to the expected revenue over a distribution of static price calendars; (ii) show that revenue does not decrease if we sort the prices in each calendar from high-to-low; and (iii) use a “convexity” argument to show that revenue again does not decrease if we simply prescribe the “average” of the sorted calendars. These lemmas motivate a complete characterization of tight bounds under stationary demand, which also hold for the assortment problem.

  2. Non-stationary Demand (Section id1): We adapt prophet inequalities from optimal stopping theory to pricing and assortment problems. Specifically, motivated by the clever analysis of the fixed-threshold stopping policy from Samuel-Cahn et al. (1984), we derive time-invariant thresholds based on our LP’s which allow us to balance between selling too little and selling too much in the analysis. The result is a very short and interpretable proof of the tight 1/2-optimality result for revenue management problems under non-stationary demand.

We use aggregated weekly panel data from a CPG company to validate the assumptions of our model, and test the performance of our proposed policies. We use the random forest method to build prediction models that suggest a distribution of demand under different prices. Then we take these distributions as inputs, and compare the performance of our policies to some benchmarks.

We validate our model with the following two observations. (i) Cross-SKU cannibalization is not significant. This suggests that we can do single-item calendar pricing, one SKU and another. (ii) Inter-temporal cannibalization is not significant, which can be explained by “pantry effect” Ailawadi and Neslin (1998), Bell et al. (1999). This suggests that demands can be modeled to be independent.

A representative SKU has valid weekly data over 3 years in the past. It can be tagged at several different integer prices. The planning horizon is one year weeks. We increase the total units of initial inventory from unit to units. We analyzed both stationary and non-stationary demand models. We find that under both cases, there are increases in revenue when inventory level is of moderate size.

A direct comparison of our contributions in relation to the existing literature has already been presented in Section id1, so here we briefly mention a few previously unmentioned papers. Inventory-constrained assortment optimization was pioneered by Talluri and Van Ryzin (2004) and has also been studied in Zhang and Adelman (2009) using approximate dynamic programming. Regarding choice of optima, in this paper we use the DLP (and CDLP), but there are other notions of optima (hindsight optimum, optimal dynamic program) used in the revenue management literature—for a discussion, see Bumpensanti and Wang (2018), Ma et al. (2018a).

Let denote the positive integers. For any positive integer , let denote the set .

A firm has items to sell over a finite time horizon of time periods. Each item is endowed with units of starting inventory, which is unreplenishable. The firm can offer items at one of prices , which are positive real numbers. The assumption that items share a common set of prices is without loss of generality, because we can always take the union of feasible prices and assign zero demand for infeasible item-price pairs.

In the single-item pricing problem, we have , and we will omit index . For each and , we are given , the probability of earning a sale if price is offered during time . If demand is stationary, then has a common value over , which we will denote using .

At the start of the time horizon, the firm sets a price for each time period, where is a price index in . Then, the demand is sequentially realized over time periods , taking values in according to probability . If and there is remaining inventory at time , then revenue is earned and the inventory is decremented by one for the future.

In the more general fractional-demand setting, the starting inventory can be any positive real number. If a demand of is realized while the remaining inventory is , then sales are made during time period , at the price of . In this setting, refers to the expected demand from offering price during time , and refers to the random demand from offering price during time , where we have normalized the starting inventory and demand so that every can only take values in [0,1]. Let denote the cumulative distribution function (CDF) of the random demand if we offer price during time period . When demands are stationary, we drop the subscript.

In the basic setting where demand conforms to the Bernoulli distribution ({0,1} demand), we will require no assumptions. But we will require two natural assumptions on the demand distributions over [0,1]. One is analogous to vertical differentiation, and the other involves the expectation of truncated variables.

The generalization to [0,1] demand gives us a lot of modeling power. From a theoretical point of view, if we only allow for {0,1}-demand, we are restricting ourselves to only Bernoulli random demands. As a limiting case when time goes to infinity, we were only dealing with Poisson random demands. This is very natural, as in many revenue management literature Gallego and Van Ryzin (1994), Maglaras and Meissner (2006), Lin et al. (2008). Yet we can generalize from it to arbitrary random demands, which makes more pratical sense. Also from a practical point of view, since we are motivated by retailing with a CPG company, demand in each week is a large number that varies on the order of hundreds or thousands. We cannot model it as either or integer values only.

In the assortment (and pricing) problem, we will refer to each item-price combination as a product. We let denote the family of feasible assortments, or subsets of products, which could be offered by the firm. can capture constraints that prevent the an item from being offered multiple times (at different prices) in an assortment, as well as operational logistics such as shelf-size constraints. For each , , and , we are given , the probability of product being chosen should assortment be offered during time . If demand is stationary, then has a common value over , which we will denote using .

At the start of the time horizon, the firm plans an assortment for each time period. Sequentially over time , up to one product is chosen from , according to choice probabilities . If item has remaining inventory, then one unit is sold for revenue . Otherwise, the sale is lost.333 This convention, motivated by e.g. parking systems where customers frequently choose “phantom” parking spots that are actually occupied, is also adopted in Rusmevichientong et al. (2017), Owen and Simchi-Levi (2017).

Our results will require the following assumption, which allows products to be judiciously withheld from assortment offerings without harming the demand for other products in the assortment.

Assumption 1

is a downward-closed family of subsets of . That is, if and , then . Furthermore, for all , , subsets , and products , .

Assumption 1 is very mild, originating from Golrezaei et al. (2014) and being nearly omnipresent in subsequent literature on inventory-constrained assortment optimization (Gallego et al. 2016, Chen et al. 2016, Ma and Simchi-Levi 2017, Rusmevichientong et al. 2017, Ma et al. 2018b, Cheung et al. 2018). The condition on the choice probabilities is often called substitutability. It is implied by any random-utility choice model, which would treat the products as substitutes.

A policy for the single-item pricing problem can be captured by the following LP:

(1)
(2)
(3)
(4)

represents the unconditional probability of price being offered during time . Constraint (\the@equationgroup@ID) ensures that at most sales are made in expectation, while constraints (\the@equationgroup@ID) ensure that only one price can be chosen for each time period. Objective (\the@equationgroup@ID) represents the expected revenue.

Similarly, a policy for the assortment problem can be captured by the following LP:

(5)
(6)
(7)
(8)

Under stationary demand, both LP’s can be simplified. In the pricing problem, since for all , we can let for each , which represents the number of time periods to offer price ; constraints (\the@equationgroup@ID) are then equivalent to the single constraint (9)

(9)

Analogously, in the assortment problem, since for all , we can let for each , which represents the number of time periods to offer assortment ; constraints (\the@equationgroup@IDa) are then equivalent to the single constraint (10).

(10)

The LP for pricing under stationary demand also has the following structure.

Lemma 1 (Gallego and Van Ryzin (1994))

The DLP-S defined by (\the@equationgroup@ID), (\the@equationgroup@ID), (9), and (\the@equationgroup@ID) has a basic optimal solution with at most two non-zeros in its support, which we will denote using (“Higher price”) and (“Lower price”), with . Furthermore, the optimal solution satisfies one of the following:

  1. Constraints (\the@equationgroup@ID) and (9) are both binding;

  2. Constraint (\the@equationgroup@ID) is binding but (9) is not, in which case , , and ;

  3. Constraint (9) is binding but (\the@equationgroup@ID) is not, in which case , , and .

Generally, these LP’s are useful because they portray a relaxation of the optimal policy, and hence an optimal LP solution can be used as a “guide” in designing a policy for the corresponding problem. In this paper, we will focus on converting the LP solution into a static policy. However, as we will see in Section id1, such a procedure allow us to compare our static policy’s revenue against even the best dynamic policies.

Definition 1

Single-item pricing policy (for both - and -demand) when demand is stationary:

  1. Solve DLP-S, and let correspond to an optimal solution as described in Lemma 1;

  2. Set the price to be for and for , where the duration for which the lower price is offered, equals to , is either or . 444 We define based on instead of , becaues can span the full range of values from 0 to , unlike (see Lemma 1).

Our policy offers the prices in high-to-low order, with a static switching point. Intuitively, the high-to-low ordering is desirable, because should we stock out early from higher-than-expected demand realizations, we would rather lose low-priced sales at the end.

Definition 2

Assortment (and pricing) policy when demand is stationary:

  1. Solve (CDLP-S), and let denote an optimal solution;

  2. Independently for each time , set the assortment to be with probability proportional to , for all . 555 If then this is already a probability; if then we divide each by to normalize their sum to one.

Our assortment policy simply probabilistically follows the LP solution, without specifically re-ordering the decisions portrayed in the LP. Here we do not have a high-to-low order, because it is hard to define a “higher” assortment when there are multiple items.

However, under the more general setting of non-stationary demand, following the LP solution may be undesirable, because it may be beneficial to “reserve” inventory for the highest-revenue time periods. The following example demonstrates this.

Example 1

Let there be periods and unit of initial inventory. Let be some small positive number. Let there be two prices: . During day , the purchase probability of offering the higher price is ; and the purchase probability of offering the lower price is . During day , the purchase probabilities of offering both prices are .

Probability Period 1 Period 2

DLP-N suggests that we offer in the first period, then in the second period. The objective value of DLP-N is . By simply using the DLP-N solution as a calendar, the expected revenue is . We can pick to be arbitrarily small so directly using LP can be arbitrarily bad.

Nonetheless, we can still use the LP as a guide for our reservation policies.

Definition 3

Single-item pricing policy (for both - and -demand) when demand is non-stationary:

  1. Solve DLP-N, and let denote the optimal objective value;

  2. For each time , set the price to be , where

    (11)

In (11), can be interpreted as the per-inventory revenue of the LP. The policy from Definition 3 is guaranteed to sell inventory for at least half of this value, since at each time , it maximizes the expected profit with a bid price (opportunity cost) of .

Definition 4

Assortment (and pricing) policy when demand is non-stationary, but under Assumption 1:

  1. Solve CDLP-N, and let denote an optimal solution;

  2. For each item , let be the contribution from item to the optimal objective value (note that );

  3. Independently for each time , first randomly select a , to be equal to each with probability , which is a proper probability distribution by constraint (\the@equationgroup@IDa). If , then select to be the empty set with the remaining probability, where is guaranteed by the downward-closed statement in Assumption 1.

    After has been selected, set the final assortment to offer at time to be

    (12)

    which is a feasible assortment to offer since is downward-closed.

Our assortment policy under non-stationary demand differs from our pricing policy in that the cost is used as an acceptance threshold instead of a bid price. That is, we remove from the planned assortments all instances of items being offered at prices below their thresholds.

Finally, we present alternative policies for non-stationary demand which conduct “reservation” to a lesser degree than in Definitions 34. Our policies have better performance if starting inventory is large, where the law of large numbers reduces the necessity of reservation, even under non-stationary demand.

Due to the law of large numbers, one may have the misperception that directly using the optimal solutions from the LP, without any “reservation”, is asymptotically optimal. But we will show in Example EC.1 that even both inventory and horizon scales up asymptotically, directly following LP can be arbitrarily bad. This motivates the following two static policies.

Definition 5

Single-item pricing policy (for both - and -demand) when demand is non-stationary and inventory is large:

  1. Solve DLP-N, and let denote an optimal solution;

  2. For each time , set the price to be with probability , and (the highest price) with probability , where

In our static policy, can be interpreted as the “reservation” probability, which decreases to zero as initial inventory increases. We reserve inventory by offering the highest price in each time period. Intuitively, the probability of us offering is the sum of the reservation probability, plus the probability that LP suggested us to offer .

Definition 6

Assortment (and pricing) policy when demand is non-stationary and inventory is large, but under Assumption 1:

  1. Solve CDLP-N, and let denote an optimal solution;

  2. For each time , offer the asortment with probability , and offer with probability , where

Again, can be interpreted as the “reservation” probability, which decreases to zero as initial inventory increases. Here we reserve inventory by offering the empty set, which is always available.

We derive performance guarantees for our static policies, which are based on the deterministic LP’s from Section id1, relative the optimal objective values of those LP’s. This also provides a performance guarantee relative to the revenue of any dynamic policy, which is upper-bounded by the LP objective value—this is a well-known type of result in revenue management.

Lemma 2 (Gallego and Van Ryzin (1994), Gallego et al. (2004))

The expected revenue of any (static or dynamic) policy for the single-item pricing problem is upper-bounded by the optimal objective value of DLP-N. Analogously, the expected revenue of any policy for the assortment problem is upper-bounded by the optimal objective value of CDLP-N.

Hereinafter, we will always use the LP objective value as our optimum and denote it using , where the distinction between the LP’s will be clear from context.

Theorem 1

For the single-item pricing problem (both - and -demand) under stationary demand, if there are periods to sell units of inventory, then the static policy from Definition 1 earns expected revenue at least

(13)

where and denotes a Binomial random variable consisting of trials of probability (note that if , then the performance guarantee is 100%).

Expression (13) is in turn at least

(14)

where the factor in parentheses has an order of , and increasing from to 1 as .

We prove Theorem 1 in Section id1, by establishing a sequence of three properties about the expected revenue of static policies. We also show that the bound (13) is tight for every and , in that it is the best-possible LP-relative bound achievable by any (static or dynamic) policy when there are time periods and starting inventory, in Lemma 5. Bound (13) is achieved by our static policy from Definition 1 which sorts and rounds the LP solution, and we furthermore show that sorting in the opposite order (low-to-high) is infeasible for static policies, in Example 2.

We focus on the case of -demand in Section id1, for clarity. The details of the generalization to fractional demand is deferred to Section id1, where we will require some additional assumptions on the demand distributions over [0,1].

Theorem 2

For the assortment (and pricing) problem under stationary demand, if there are time periods and denotes the minimum starting inventory among , then the static policy from Definition 2 earns expected revenue at least the expression given in (13).

Theorem 2 is motivated by the first property from the proof of Theorem 1, and we prove it in Section id1.

Theorem 3

For the single-item pricing problem (both - and -demand) where demand may be non-stationary, the static policy from Definition 3 earns expected revenue at least .

Theorem 4

For the assortment (and pricing) problem where demand may be non-stationary, the static policy from Definition 4 earns expected revenue at least .

We prove Theorem 3 (for general [0,1]-demand) and Theorem 4 in Section id1. Although the pricing policy employs a bid price while the assortment policy employs acceptance thresholds, the analyses are similar, using the technique of prophet inequalities.

To start with asymptotic analyses, we distinguish the following two cases. , and . In the first case where , we can think of it having way more inventory than we could sell. As a result, we would always offer the price that yields the highest expected revenue, as suggested by the degenerate DLP-N (degenerate in inventory constraint). In this case, calendar pricing achieves as much as DLP-N suggests. So this first case is trivially resolved.

Theorem 5

For the single-item pricing problem (both - and -demand) where demand may be non-stationary, when , the static policy from Definition 5 earns expected revenue at least , i.e. it earns as much as the optimal dynamic pricing policy when initial inventory goes to infinity.

Theorem 6

For the assortment (and pricing) policy where demand may be non-stationary, when , the static policy from Definition 6 earns revenue that is at least in expectation, i.e. it earns as much as the optimal dynamic pricing policy when initial inventory of any item goes to infinity.

We prove Theorems 5-6 for large starting inventories in Section id1.

Our main goal in this section is to prove Theorem 1, our result for single-item pricing under stationary demand. We prove Theorem 2, our result for assortment, in Section id1.

Throughout this section, we use to denote the price index in a calendar, and to denote the price index in a revenue-maximizing calendar. We use to describe the calendar, a vector of many price indices.

We quickly establish a structural property, Lemma 3, as a warm-up to the proof of Lemma 4, although Lemma 4 is going to be proved in a self-contained fashion. The structural property says:

Lemma 3

In any calendar , if two consecutive prices , are in increasing order, this calendar cannot be optimal. More specifically, we can have prices , exchanged to achieve a greater revenue.

The proof is deferred to Section id1

Corollary 1

The optimal prices in a static calendar is non-increasing over time, i.e. .

\@trivlist

Directly follows from Lemma 3. We can start from any calendar and use a finite number (no more than ) of exchange operations to achieve the optimal non-decreasing structure.  \@endparenv

The proof can be divided into three steps, which we will illustrate using the following example. Consider a problem instance with time periods and starting inventory . Suppose we have two prices. The higher price earns units of revenue with probability ; the lower price earns units of revenue with probability , i.e. deterministically. The optimal LP solution (according to Lemma 1) suggests to offer a higher price with index for 1.5 time periods, and a lower price with index for 1.5 time periods.

We let denote the expected revenue of a static policy which offers for the first time period, then flips a coin with half probability to offer and half probability to offer for the second time period, and finally offers for the third time period (and define analogously).

We then establish the following sequence of three inequalities:

(15)
(16)
(17)

Inequality (15) relates the LP optimum to the expected revenue of a randomized policy which independently selects either or (with probability 1/2 each) for each time period. The interpretation of the ratio on the LHS is the following: the LP sells exactly units of inventory at some average price (which is a convex combination of and ), while the randomized policy sells at the same price a number of units equal to a truncated Binomial (which had expectation pre-truncation). The ratio is less than 1 unless is greater than the number of time periods (i.e. there is zero possibility of stockout), because the LP is consistent, while the randomized policy is vulnerable to variance.

Inequality (16) is an argument that augments the monotonicity property, which says that if there is a positive probability that the static policy offers a lower price before a higher price, then this policy can be improved. That is, there is probability that the policy offers both in period and in period (regardless of what it offers in period ). But the policy only suggests two possible policies: and . Either way it has zero probability to offer a lower price before a higher price. So the first policy can be improved to the second one, while keeping the total probabilities to offer both a higher price for 1.5 units unchanged, and a lower price for 1.5 units unchanged.

Finally, Inequality (17) is true and we pick the best of the two static policies in Definition (1).

We now formalize Inequality (16) in the following lemma. We will incorporate Inequalities (15) and  (17) in the proof of Theorem 1. Let be the fractional part of a real number .

Lemma 4

Suppose there are two calendars:

  1. A calendar that offers in each period the same probabilistic mixture of offering two prices, i.e. with probability offering the higher price and offering the lower price;

  2. A calendar that starts with deterministically offering the higher price for periods, then in the next period offers the higher price with probability and the lower price with probability , and finally switchs to offering the lower price in the last periods.

The first calendar can be improved to the second one, whose expected revenue is not decreased.

The proof of Lemma 4 is deferred to Section id1. Now we formalize the proof of Theorem 1.

\@trivlist

Notice that the optimal solution of DLP-S has only no more than two non-zero variables. If our algorithm only selects one single price, we enumerate all the possibilities as follows:

DLP-S has either one tight constraint or two. If it has two tight constraints and has only one non-zero variable, this suggests that . The static policy from Definition 1 yields exactly that of the upper bound.

If there is only one tight constraint, the static policy from Definition 1 would offer the corresponding price for all time periods. In one case, the tight constraint is the inventory constraint (the optimal solution has satisfying and all other variables being zero), which suggests . Therefore, the static policy from Definition 1 gets as much expected revenue as

where the last equality follows from .

In the other case, the tight constraint is the legitimate constraint (the optimal solution has satisfying and all other variables being zero), which suggests . Therefore, the static policy from Definition 1 gets as much expected revenue as

where the last equality follows from . The proof of the second line inequality is deferred to Proposition 2, where it serves as an elegant example of Assumption 3.

If our algorithm selects two prices, we can prove the theorem in three steps. We will first construct a two-price randomized calendar that achieves exactly fraction of the DLP-S upper bound. Since , we can divide and by , and naturally obtain two probabilities. This randomized calendar offers price with probability , and price with probability , in each period.

By a coupling argument we can show that this calendar achieves exactly fraction of the DLP-S upper bound. Imagine there was a virtual calendar which in each day offers a virtual price Under this price the purchase probability is . So the unit-earned revenue from this virtual calendar is

The units of inventory sold, under the randomized calendar , conforms a Bernoulli variable with a success probability of , which is exactly the same as the virtual calendar . And once one unit is sold, it has a probability sold at price , and a probability sold at price , which means that the unit-earned revenue is exactly the same as the virtual calendar , as well. By a coupling argument we know the randomized calendar earns exactly the same as the virtual calendar , which is

(18)

in expectation.

From Lemma 4, the static policy from Definition 1 achieves greater revenue than the constructed randomized calendar , which proves the bound.  \@endparenv

We now show that the ratio produced in Lemma 13, which is dependent on and , is tight.

Lemma 5

For any positive integers and , there exists an instance of the stationary-demand single-item pricing problem with time periods and starting inventory, for which the expected revenue of any policy is upper-bounded by expression (13).

\@trivlist

Just consider such an instance. We only have one price option, i.e. . So we only have one calendar to sell at this single price everyday. For any given and , the only price option has a purchase probability of , and earns unit revenue.

Then DLP-S would yield us a total of units of revenue. And the only calendar would yield units of revenue. So the expected revenue of the only policy is exactly expression (13), which finishes the tightness proof.  \@endparenv

Also, we show that switching from a higher price to lower price is necessary, in the sense that if we switch from a lower price to a higher price, then we cannot achieve the bound by expression (13).

Example 2

Let there be periods and unit of initial inventory. Let there be two prices: . The corresponding purchase probabilities are: . The LP suggests us to offer both and for exactly one period. And the LP objective is

We calculate the bound in expression (13): it gives us a guarantee.

If we offer in period and then in period , this earns an expected revenue of , which is of the LP upper bound.

If we offer in period and then in period , this earns an expected revenue of , which is of the LP upper bound.

This example depicts that switching from a lower price to a higher price is worse than the bound by expression (13), and it is even worse than the ratio. On the other hand, switching from a higher price to a lower price performs much better than that.

Since in the assortment (and pricing) setting, it is hard to define which assortment is prioritized (like the higher price in the single-item pricing problem). We do not have a nice structural property. But we can still offer a probabilistic mixture of assortments to achieve the desired bound in Theorem 2. This probabilistic mixture resembles the one in expression (18).

\@trivlist

Consider the static policy from Definition 2 and let . Regardless of inventory availability at time , the planned assortment will be with probability for all , after which the customer will attempt to purchase a product in according to probabilities . Therefore, the total probability of the customer attempting to consume inventory is

(19)

Note that does not depend on . By the independence of both policy decisions and customer decisions across time, the total sales of inventory is Binomially distributed with trials of probability , truncated by the starting inventory . To summarize, the expected sales of inventory is , with defined as in (19), for all .

Now, conditional on the policy successfully selling inventory during a time period, what is the revenue earned in expectation? The unconditional probability of the customer purchasing at any price is , hence the total conditional expectation is

Summing over all , the expected revenue of the policy is

Now, for all , since the LP solution satisfies constraints (\the@equationgroup@IDa). Invoking Proposition 2, the expected revenue of our policy is lower-bounded by

(20)
(21)

We explain both inequalities. The first inequality requires the fact that , which is implied by LP constraint (\the@equationgroup@IDa). Analyzing the two cases separately: if , then the large fraction in (20) equals while the large fraction in (21) cannot be greater than 1; otherwise, the denominator doesn’t change from (20) to (21) while probability in the Binomial decreases. The second inequality follows by the following argument: by rearranging the fraction in (21) and letting we have

where the inequality is due to the technical Proposition 10. Finally, this completes the proof of the theorem, because is an optimal LP solution.  \@endparenv

Our main goal in this section is to prove Theorems 36. We first prove Theorems 34 for non-stationary demand and small starting inventories. We prove Theorems 56 for large starting inventories in Section id1.

By finding the price suggested by expression (11) (or expression (12) under assortment setting), we makes sure that each unit sold gives us at least one half of the per-inventory revenue of the LP. If we run out inventory during the horizon, then every unit sold earns as much as one half of the per-inventory revenue, so we earns at least one half of the LP upper bound. If we never run out inventory, we use prophet inequality (Krengel and Sucheston (1977), Samuel-Cahn et al. (1984)) to show that it happens to earn at least one half of the LP upper bound, as well. So setting one half of the per-inventory revenue as a bid price is a “win-win” selection.

Here we prove Theorem 3 in the general [0,1]-demand setting. The {0,1}-demand setting is a special case of it.

\@trivlist

Let denote the prices selected from expression (11). Denote the following random variables, which depict a run of our static assortment policy from Definition 3.

  • : remaining inventory at the end of time , with ;

  • : the inventory at time that customer would have demanded if price