Research Article

Revenue management with dynamic pricing and advertising Received (in revised form): 14th July 2009

Leo MacDonald is an assistant professor in the Department of Economics, Finance and Quantitative Analysis at the Coles College of Business, Kennesaw State University. His primary research interests are in Revenue Management and Pricing/Inventory policies. Professor MacDonald’s industry experience includes several years as a professional engineer, including consulting, project management and research/product development.

Henning Rasmussenw was formerly Professor Emeritus in the Department of Applied Mathematics, University of Western Ontario. His academic career spanned over three decades and included more than 90 refereed publications in modeling and applied numerical analysis. Correspondence: Leo MacDonald Department of Economics, Finance and Quantitative Analysis, Coles College of Business, Kennesaw State University, 1000 Chastain Road, Kennesaw, Georgia 30144, USA. E-mail: [email protected] w

Dedicated to the memory of Henning Rasmussen, mentor, colleague and friend.

ABSTRACT In this article, we analyze the temporal pricing and advertising strategy of a monopolist with a fixed inventory to sell over a finite horizon. The arrival of the customers is modeled by a Poisson process where the arrival rate is given by an increasing convex function of the advertising expenditure, and the willingness of a customer to pay is modeled by a decreasing function of the price. For specific functions for the arrival rate and willingness-to-pay, we derive and solve a system of ordinary differential equations for the optimal pricing and advertising strategy. We show that this optimal strategy is very close to a fixed optimal strategy. Journal of Revenue and Pricing Management (2010) 9, 126–136. doi:10.1057/rpm.2009.36; published online 30 October 2009 Keywords: pricing; advertising; revenue management; Poisson process; dynamic optimization

INTRODUCTION Determination of dynamic pricing strategies has become a standard approach to maximizing expected revenue in the revenue management literature. Excellent reviews of this extensive literature can be found in recent articles (McGill and van Ryzin, 1999; Bitran and Caldentey, 2003; Elmaghraby and Keskinocak, 2003) and in the text by Talluri and van Ryzin (2004). The basic premise is a service provider of a perishable asset who must sell a product with a fixed inventory level over a finite horizon with uncertain demand, often described by a Poisson

customer arrival process. The provider dynamically prices to maximize revenues over the remaining horizon and inventory (n) as sales occur. Gallego and van Ryzin (1994) establish the basic properties of dynamic pricing policies under a Poisson-type demand uncertainty. Numerous extensions of this research have been published, including that by Gallego and van Ryzin (1997), who generalize the results to a network (multi-product) setting. Zhao and Zheng (2000) extend the model to include non-homogeneous customer arrival processes, whereas Feng and Xiao (2000) consider a more

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realistic ‘menu of prices’ with multiple price changes within the price set. Somewhat separate from this stream of research has been the vast literature in marketing on advertising controls and their effect on sales. However, though numerous articles have been published evaluating price and advertising decisions separately, little has been done to combine these two key decision variables and evaluate joint effects. This is somewhat surprising given that setting price and determining the level of advertising expenditure are two of the key marketing-mix variables most businesses, regardless of sector or product, need to determine in controlling sales and overall profits. In fact, although one of the original studies that considers joint price and advertising dates to the mid-1950s (Dorfman and Steiner, 1954), the volume of literature addressing this issue is relatively small. The Dorfman and Steiner (1954) article addresses a static demand model, and determines an often cited result that the ratio of sale revenue to advertising expenditure is equal to the ratio of the price and advertising elasticities of demand. Most of the limited number of articles that consider joint pricing and advertising can generally be classified as diffusion models, extensions of the Bass (1969) new product diffusion model to include price and advertising controls for new product introductions. Of these, we can define sub-classifications based on length of horizon, finite versus infinite, and type of interactions between the controls, that is, price and advertising, on demand or sales. A short list of this literature includes studies by Thompson and Teng (1984), Teng and Thompson (1985) and Kalish (1985). The basic construct of new product diffusion models involves the purchase of durable goods, with a fixed potential market. As consumers ‘adopt’ the product, they exit the market, leading to saturation effects as the product matures. Price and/or advertising controls are then used to affect either market potential or the rate of adoption. An excellent review and analysis of numerous new product diffusion models,

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involving price and/or advertising controls is contained in the study by Bass et al (1994), as well as in that by Mesak and Clark (1998). Mesak and Clark (1998) describe a total of 18 different variants of the basic diffusion model involving different controls and interactions. The majority of this literature restricts itself to monopolist markets, with only a few articles involving duopolies/oligopolies. Further, the demand models are all taken to be deterministic, with the exception of that by Kalish (1985). One exception to the above diffusion models is that by Feichtinger et al (1988), which addresses price and promotion decisions for a convenience (grocery store) goods retailer. This differs from the standard diffusion models in many ways, not the least of which is the repeat purchases of non-durable goods. Reference price effects are assumed to determine the impact of price on demand, while a power model is used to incorporate advertising effects. The model is deterministic, and the steadystate solutions over an infinite horizon are established. Recently, there has been an increase in interest in combined pricing and advertising. Sethi et al evaluate pricing and advertising policies in a new product diffusion model of market share under linear and isoelastic demand functions over an infinite horizon. In an extension to this study, Krishnamoorthy et al (2010) analyze advertising and pricing policies over an infinite horizon in a differential duopoly game for both symmetric and asymmetric competitors. Results indicate that optimal prices are constant, and do not vary with cumulative sales, whereas the optimal advertising decreases with cumulative sales. In this article, we focus on a traditional revenue management problem, for example the airline or hotel sector, in which a service provider is given a fixed number of units of a perishable asset that must be sold over a finite horizon. We incorporate uncertainty in demand and evaluate the trade-off between pricing and advertising expenditure in controlling sales and

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maximizing expected revenues. To focus on the price and advertising policies, we consider a single service provider in the mode of the dynamic newsvendor, similar to the approach adopted by Gallego and van Ryzin (1994), Monahan et al (2004) and McAfee and te Velde (2008).

DYNAMIC MODEL OF PRICE AND ADVERTISING Consider a service provider with a fixed inventory to sell over a finite horizon. Demand is stochastic and follows a Poisson process, in which to incorporate the effects of both price and advertising on sales, we decompose demand into two components. Arriving customers are assumed to have a reservation price (r) or maximum willingness-to-pay (w.t.p.), where f(r) is the probability density function and F(r) is the cumulative distribution of reservation prices. Thus, for a posted price p, an arriving customer will make a purchase with probability 1F(p), that is, the probability that the reservation price is greater than the posted price. It is assumed that customers arrive according to a Poisson distribution with intensity l, and following standard convention we divide our time horizon into sufficiently small time steps of size D, so that the probability of no customer arrivals is approximately 1Dl, of a single customer arrival is Dl and of two or more arrivals is o(D). Arriving customers are assumed to demand a single unit. The intensity of the arrival process is controlled by our advertising rate (a) in each period, thus l ¼ l(a). This has an appealing/intuitive logic: the more we advertise, the more aware consumers are of our product/service and the higher the probability of an arrival, that is, of generating increased customer traffic. We assume that the effect of advertising has diminishing returns with increasing expenditure, a typical assumption in the advertising literature. Throughout the article, we will limit the discussion to a standard class of constant elasticity advertising

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functions, namely, power models of the form l(a) ¼ l0ab, where l0>0 is a constant and 0obo1, which are commonly employed in advertising models. Let t be the time variable, with 0ptpT, and Vn(t) be the expected value of having n units for sale at t. The firm’s objective is to maximize expected revenue (net of advertising cost) over the remaining horizon so

Vn ðtÞ ¼ max½ð1  lDÞVn ðt þ DÞ a;p

þ lDFðpÞVn ðt þ DÞ þ lDð1  FðpÞÞðp þ Vn1 ðt þ DÞÞ  aD

The first three terms represent, respectively, no customer arrival, with probability 1lD, a customer arrival but no purchase, and a customer arrival with purchase generating revenue of ‘p’. The last term is simply the cost of advertising over the increment D. Rearranging and dividing through by D, and taking the limit as D-0 results in the Hamilton–Jacobi–Bellman equation: Vn 0 ðtÞ ¼ max½lð1  FðpÞÞðp  ðVn ðtÞ a;p

 Vn1 ðtÞÞ  a

ð1Þ

Thus, given a sale, we receive revenue ‘p’ minus the cost of advertising, where (Vn(t)Vn1(t)) represents the expected marginal value of selling a unit of inventory at t. From first-order conditions, we can determine optimal solutions for price and advertising, resulting in p ðtÞ ¼

1  FðpÞ þ Vn ðtÞ  Vn1 ðtÞ f ðpÞ "

ð1  FðpÞÞ2 a ðtÞ ¼ bl0 f ðpÞ

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Revenue management with dynamic pricing and advertising

For price, it is easy to show that optimal price is decreasing in both inventory (n) and time t (increasing in time-to-go), the proof for which is identical to that given in the study by Gallego and van Ryzin (1994). To determine the effects of inventory and time on advertising, we are required to make a mild assumption as to the nature of the w.t.p. distribution. We restrict ourselves to distributions with a constant or increasing hazard function, where the hazard function is defined as f(x)/(1F(x)), whereby we can easily demonstrate that the rate of advertising expenditure is increasing in both n and t (decreasing in time-to-go). This follows directly from the assumption of an increasing or constant hazard rate distribution, thus ensuring that the optimal advertising function above is strictly decreasing in price. This assumption is not particularly restrictive, as distributions of this type describe a large number of the possible distributions of practical interest. Further, the hazard rate assumption allows us to establish that the firstorder conditions for the optimal price are sufficient. The proof is identical to that given in the study by Bitran and Mondschein (1993), in which they prove sufficiency conditions for p (t) given (1F(p))2/f(p) is a decreasing function of p(t), which holds for all distributions with constant or increasing hazard functions. The sufficiency conditions for a (t) follow directly from the assumption of l(a) ¼ l0ab, where l0>0 is a constant and 0obo1 above. In the limit, as t-T and/or n-N, p ðtÞ ¼ pmin

MODEL-SPECIFIC SOLUTION We begin by considering specific forms of F(p) and l. Let 1  FðpÞ ¼ eap and

1  FðpÞ ¼ f ðpÞ

and 

a ðtÞ ¼ amax "

a constant or increasing hazard rate, results in a maximum advertising rate. Given the divergent nature of the response of price and advertising to t and n, the ratio of price to advertising thus decreases as time decreases (time-to-go decreases) and inventory increases. Thus, we have two alternate mechanisms to control our expected sales and generate net revenue. To increase sales, we can decrease prices, which also results in lower margins, or increase advertising, maintaining our margin but increasing expenses, thus decreasing revenue net of advertising costs. However, though the effect is the same, the mechanisms through which both accomplish this goal are very different; lower prices increase the purchase incidence of potential customers, whereas increased advertising creates an increase in customer traffic. This then allows for the possibility that one may be more effective in maximizing net expected revenue under different circumstances or market situations. To further evaluate the structural properties of the pricing and advertising policies, we turn to specific model formulations, using common specifications for willingness-to-pay and advertising effects.

ð1  Fðpmin ÞÞ2 ¼ bl0 f ðpmin Þ

1 #1b

l ¼ l0 ab Here the distribution of reservation prices is an exponential distribution, with the power model described above modeling advertising effects, and a and l0 positive parameters. Substituting these functions into (1) we get Vn 0 ðtÞ

The existence of pmin is easy to establish, with the advertising result (amax) a direct extension of the existence of pmin, which given

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¼  max½l0 ab eap ðp þ Vn1 ðtÞ a;p

 Vn ðtÞÞ  a

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Now let

where ap

gða; pÞ ¼ l0 ab e

ðp þ Vn1 ðtÞ  Vn ðtÞÞ  a

and we find the optimal values of a and p by maximizing g(a, p). Solving qg qg ¼ ¼0 qa qp we find the optimal values of p and a from these first-order conditions as 1 p ðtÞ ¼ þ Vn ðtÞ  Vn1 ðtÞ a

Bn ðtÞ ¼

Substituting the solution for Vn(t) Vn1(t) into equations (5) and (6) for p a we get for our optimal dynamic pricing advertising policies, respectively,   1 1b Bn ðtÞ  log pn ðtÞ ¼ þ a a Bn1 ðtÞ and

ð5Þ an ðtÞ

and 

bl0 aða1þVn Vn1 Þ a ðtÞ ¼ e a 

1 1b

ð6Þ

Following an approach similar to that given in the study by McAfee and te Velde (2006), we collect terms and define b  1b 1 b 1 l01b e1b b¼ a

which is the maximum per-period arrival intensity, that is, the intensity generated at pmin and amax. We can then write the equation Vn0 (t) as

Vn 0 ðtÞ ¼

a ð1  bÞb ð1bÞ e ðVn ðtÞVn1 ðtÞÞ a

ð7Þ

Given initial and boundary conditions Vn(T) ¼ V0(t) ¼ 0, 8n,t, we can find the expected revenue for ‘n’ and ‘t’ by solving the system of initial value problems given by (7), starting with n ¼ 1, resulting in Vn ðtÞ ¼

130

1b logðBn ðtÞÞ a

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ð8Þ

n X ½bðT  tÞj j! j¼0

  bb Bn1 ðtÞ ¼ a Bn ðtÞ

and and and

ð9Þ

ð10Þ

A sample optimal price and advertising path for given simulation of sales (inventory) is presented below. From Figure 1, we observe that the changes in optimal price and advertising, as both time or inventory decrease, are as expected from the above analysis. As time increases (time-to-go decreases) without a sale, advertising starts to rise and price falls; however, once a sale has occurred, in the next period price increases whereas advertising decreases. For the specific form of the model, pmin and amax can now be specified as b  1b 1 bl0 and amax ¼ pmin ¼ a ae1 Further, it is easy to demonstrate that pn (t)/ and thus the ratio of price to advertising decreases as both remaining time (Tt) and n increase, that is, with increasing time-to-go or with higher inventory, the optimal advertising and pricing policies begin to converge, approaching amax and pmin, respectively. More interestingly, the relative rate of change in advertising is always at least as great as that of price, and thus the optimal advertising response to decreasing time-to-go and higher remaining inventory is more pronounced than that of the price response. an (t)X1,

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Figure 1: Simulated sample paths for price, advertising and inventory over the sales horizon (generated at a ¼ 0.02, l0 ¼ 0.3, b ¼ 0.5, initial inventory of 25 and T ¼ 100).

We can also characterize the expected sales for the optimal dynamic policies over the sales horizon T. Given the optimal policies for a and p in any period t with n units remaining, we can determine the probability of a sale as  b     b Bn1 ðtÞ Bn1 ðtÞ ¼b l0 e 2a Bn ðtÞ Bn ðtÞ 1

Then, for a remaining inventory of n at any time t with a sales horizon T and optimal price and advertising strategies, we have over the remaining time-to-go E½Sales ¼ b

Bn1 ðtÞ ðBn ðtÞÞðT  tÞ

or, specifically, at the start of a sales horizon with our starting inventory N, we have E½Sales ¼ b

BN 1 ðtÞ T BN ðtÞ

Identical results are obtained by integrating the intensity function over the sales horizon for

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the optimal controls. Thus, for an exponential willingness-to-pay reservation price distribution and an advertising power model, we can specify the dynamic optimal price and advertising problem.

FIXED-POLICY HEURISTICS One limitation of the optimal dynamic polices determined above is in the implementation. One solution to this is the determination of fixedprice advertising policies that are near-optimal and more readily implemented. We begin by evaluating the deterministic dynamic solution to our problem. Under a deterministic scenario, we take demand to be the expected value of the Poisson arrival process. Solving for the optimal deterministic price (pD) and advertising (aD), where subscript ‘D’ denotes a deterministic dynamic optimal policy, we arrive at 1 pD ¼ þ m and a  b   bl0 apD 1b b aD ¼ ¼ b e a a

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where m is the adjoint variable of the deterministic dynamic optimization and is similarly interpreted as the expected marginal value of the stochastic solution. For the unconstrained inventory situation, the deterministic price and advertising equal the minimum and maximum optimal price and advertising respectively, that is, pD, u ¼ pmin and aD, u ¼ amax, with m ¼ 0, where the subscript ‘u’ indicates an unconstrained optima. Thus, it is obvious that with high inventories and/or long sales horizons, resulting in an unconstrained demand situation, the deterministic optimal solution converges to a fixed-policy solution that is also equivalent to the dynamic stochastic polices. For the constrained inventory case, m will be a constant strictly greater than 0, still indicative of a constant price and advertising policies. The deterministic policies under inventory constraint are often referred to as the run-out policies, in that the controls are set so that demand matches the available inventory, resulting in an isoperimetric inventory constraint, that is, n ¼ 0 @ t ¼ T. This result allows us to calculate the fixed deterministic policies, as the following relationship must hold: l0 ab eap ðT  tÞ ¼ DðT  tÞ ¼ n

ð11Þ

where D ¼ l0abeap, that is, the instantaneous per-period demand for a given price and advertising rate. That is, given the remaining time-to-go (Tt) and current inventory position (n), the appropriate advertising rate and price will sell exactly the remaining inventory, as defined by a run-out policy. Rearranging (11) to solve for ‘p’ and substituting this relationship into the original optimization problem results in aD; c ¼

bn bD ¼ aðT  tÞ a

ð12Þ

and pD; c

1 D ¼ ln þ a l0 ðaD; c Þb

! ð13Þ

as the optimal advertising rate and pricing policy,

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respectively, of the inventory constrained deterministic problem. To evaluate the effectiveness of any fixedpolicy solution in maximizing net revenue under stochastic demand, we can calculate the expected value function for any level of inventory (n) at time t by integrating (4) for arbitrary fixed ‘a’ and ‘p’, which after some manipulation results in the following equation:  a E½VnFP ðtÞ ¼ p  D   DðT  tÞGðn; DðT  tÞÞ  Gðn þ 1; DðT  tÞÞ ð14Þ nþ GðnÞ

where G(x) and G(x, y) are the Gamma and incomplete Gamma functions, respectively. The second bracketed term represents the expected sales for a given inventory position (n) and time-to-go (Tt). We can also define the deterministic value function under both unconstrained and constrained inventory positions, by solving using the appropriate price and advertising policies generated above for the two situations. Specifically for the unconstrained problem, we have VnD ðtÞ ¼ pD; u D ðT  tÞ  aD; u ðT  tÞ   b  ¼ pD; u  ðT  tÞb a   1 b  ðT  tÞb ¼ a a

ð15Þ

while for the constrained problem VnD ðtÞ ¼ pD; c D ðT  tÞ  aD; c ðT  tÞ   b ¼ pD; c D  n a

ð16Þ

Using the above, we can establish that VnD(t)XE[Vn (t)], as E[Vn (t)] converges to the deterministic value function in the limit as n-N. Further, by definition, [E[V ]n (t)]X [E[V ]FP n (t)], as the optimal dynamic policy must be greater than or equal to any arbitrary price and advertising policies. Thus, by establishing the relationship between VnD(t) and [E[V ]FP n (t)], we can subsequently define the

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lower bounds of the expected value for the deterministic heuristic relative to the optimal dynamic solution as VnD(t)XE[Vn (t)]X [E[V ]FP n (t)]. In general, it is easy to establish that in the limit as n-N, under both constrained and unconstrained scenarios, all  three results are equivalent (VD n (t) ¼ E[Vn (t)] ¼ FP [E[V]n (t)]), with pricing and advertising policies converging to pmin and amax, respectively. To establish the bounds for the deterministic heuristic with respect to available inventory and timeto-go, we separately evaluate the constrained and unconstrained deterministic problems. Starting with the constrained inventory position, our optimal deterministic policies are as given in (12) and (13) above. Substituting these policies into the fixed-policy value function given in (14) and comparing to the deterministic value function (16), we have, after some simplification VnFP ðtÞ nn1 ¼ 1  VnD ðtÞ ðn  1Þ!en

ð17Þ

It is relatively easy to show that the ratio of the value functions is a strictly increasing function of n, and as described above, as the inventory position (n) increases VnFP(t)-VD n (t). Following an equivalent approach when the inventory position is not constrained results in a slightly more complicated functional relationship, specifically VnFP ðtÞ n ¼ VnD ðtÞ DðT  tÞ DðT  tÞGðn; DðT  tÞÞ  Gðn þ 1; DðT  tÞÞ ð18Þ þ DðT  tÞGðnÞ

where, for unconstrained situations, b(Tt)on. The expression is not as straightforward to analyze as that generated for the constrained situation; however, there are a few basic properties that can be determined. For b(Tt)-n, that is, as expected demand approaches available inventory, the above expression (18) reduces to that given for the constrained problem (17), with equivalent results. Further, similar to above, it is relatively

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easy to show that (18) is strictly decreasing in the demand rate b(Tt). Thus, for b(Tt) less than n (unconstrained situation), as the ratio n/ b(Tt) increases, that is, as the inventory position greatly exceeds demand, (18) goes 1 D and VFP n (t)-Vn (t), as expected sales converge to expected demand b(Tt). Of course, the results above provide the lower bound on the heuristic approximation of the optimal dynamic controls as described above. Various methods of improving the fixed-policy heuristic exist. One approach would be to simply update the fixed price and advertising policy periodically, say on a weekly or bi-weekly basis, or more frequently towards the end of the sales horizon. Another improvement would be to select the optimal fixed price and advertising policies, which may differ from those determined via the deterministic heuristic. Optimal fixed values of price (pOFP) and advertising (aOFP) of expression (12) cannot be found analytically; however, numeric approaches are readily available to estimate these policies for given parameter values. A graph of the expected revenue function over fixed-price advertising policies is given in Figure 2; here negative values of the expected revenue have been set to zero in order to improve the layout of the figure. From Figure 2 we observe that there is a single maximum for the expected revenue. For comparison, results for the deterministic heuristic and optimal fixed policies approaches are compared in Tables 1 and 2. Table 1 shows the policies generated over a range of initial inventories, while Table 2 shows the resultant expected revenues as compared to the optimal dynamic solution. Results in both tables were generated at a ¼ 0.02, l0 ¼ 0.3, b ¼ 0.5 and T ¼ 100, for a maximum expected demand at pmin and amax of bTD30 units. From Tables 1 and 2, we observe, as expected, that as initial inventory (N) increases, the optimal dynamic policy and the different heuristic approaches converge to the same values, specifically pmin and amax, and thus equivalent expected revenues. However, at lower initial levels of stock as

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Figure 2: Expected revenues for fixed price and advertising policies (generated at a ¼ 0.02, l0 ¼ 0.3, b ¼ 0.5, initial inventory of 25 and T ¼ 100). Table 1: Comparison of price and advertising policies

Optimal policies

pOFP aOFP a 1aOFP pD 1aD a Total aD a

Starting inventory N=5

N=10

N=15

N=20

N=25

N=30

N=35

N=40

N=45

N=50

90.6 1.50 115 95.2 1.25 108

77.0 2.59 228 77.8 2.50 224

68.6 3.61 338 67.7 3.75 345

62.8 4.57 442 60.5 5.00 467

58.3 5.45 538 54.9 6.25 590

55.0 6.24 622 50.4 7.50 713

52.5 6.90 690 50.0 7.61 758

50.9 7.35 736 50.0 7.61 762

50.2 7.56 756 50.0 7.61 762

50.0 7.61 761 50.0 7.61 761

Advertising as total over the sales horizon.

well as inventory positions approaching the unconstrained sales level of approximately 30 units, this approach yields smaller expected revenues than the optimal stochastic dynamic solution. The deterministic heuristic and the optimal fixed policies are both least effective at the

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lowest inventory level (N ¼ 5), generating expected revenues that are 92.2 per cent and 92.8 per cent of that of the expected revenue for the optimal dynamic solution. In addition, at a starting inventory position of 30 units, where b(Tt)Dn, the deterministic price and advertising policies yield an expected revenue

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Table 2: Comparison of expected revenue under different policies

Expected revenue

E[Vn(T); p, a ] E[Vn(T); pOFP, aOFP] E[Vn(T); pD, aD]

Starting inventory N=5

N=10

N=15

N=20

N=25

N=30

N=35

N=40

N=45

N=50

312 294 289

486 463 462

599 576 575

673 652 647

719 703 689

745 735 706

756 752 745

760 759 758

761 761 761

761 761 761

that is 5.2 per cent less than the optimal dynamic stochastic approach, while the optimal fixed policy results in expected revenues that are only 1.3 per cent lower. Overall, as would be expected, the optimal fixed policies (pOFP, aOFP) outperform the deterministic policy heuristic, though the differences in most scenarios are not substantial, with the differences in expected revenues typically in the range of 2 per cent. The numeric results are also consistent with the properties established based on (17) and (18) above. Specifically, the deterministic heuristic improves with increasing levels of inventory, and when the inventory position is much higher than demand. Thus, although the deterministic solution may be asymptotically optimal as inventory increases under both constrained and unconstrained scenarios, for small inventories or in highly constrained situations a deterministic fixed-price heuristic may not necessarily perform as well.

DISCUSSION AND CONCLUSIONS In this article, we extended the standard dynamic pricing revenue management problem to include both dynamic pricing and advertising controls under stochastic demand. General properties of the price and advertising policies are established, which are consistent with respect to those determined in previous research involving dynamic price, while extending the results to dynamic advertising. Further, under specific model formulation, namely, exponential reservation price distribution and a power function model of advertising, we can find an analytic solution, which is

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similar to that derived by Gallego and van Ryzin (1994) with dynamic price only. In addition, we also evaluate a deterministic heuristic pricing and advertising policy, and show that it gives expected revenues that are often close to those obtained using the dynamic optimal policy, particularly with higher inventory levels or if the problem involves inventory positions that greatly exceed demand. The heuristic, however, may not perform as well with low inventory levels and highly inventory-constrained problems. Several extensions to the analysis presented in this article are currently under development. First, thus far we have only considered the case in which the inventory position is fixed at the start of the sales horizon, with no reordering possible. Although this is a standard assumption in traditional revenue management scenarios, it may not be appropriate for non-traditional areas such as retail environments. Thus, inclusion of reordering would broaden the scope of the results beyond traditional industries. It is also reasonable to consider other models for the pricing function in addition to the exponential function, for example a constant demand elasticity function as applied in the dynamic pricing optimization of McAfee and te Velde (2008). In addition, we have only considered a relatively simple model on the effect of advertising expenditure, that is, involving a constant times an advertising power function. Extensions to this model could include different functional forms, as well as addition of a constant term. It will need to be determined whether analytic solutions could be obtained for the expected net revenue on a case-by-case basis, or whether a numerical

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procedure is required. Finally, the impact of competition with respect to both advertising and price should be considered in future extensions to this research.

ACKNOWLEDGEMENT Henning Rasmussen wanted to acknowledge support from the Natural Sciences and Engineering Research Council (NSERC), Canada.

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