Risk in Revenue Management and Dynamic Pricing


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1 OPERATIONS RESEARCH Vol. 56, No. 2, March Aril 2008, issn X eissn informs doi /ore INFORMS Risk in Revenue Management and Dynamic Pricing Yuri Levin, Jeff McGill, Mikhail Nediak School of Business, Queen s University, Kingston, Ontario, Canada K7L 3N6 We resent a new model for otimal dynamic ricing of erishable services or roducts that incororates a simle risk measure ermitting control of the robability that total revenues fall below a minimum accetable level. The formulation assumes that sales must occur within a finite time eriod, that there is a finite ossibly large set of available rices, and that demand follows a ricedeendent, nonhomogeneous Poisson rocess. This model is articularly aroriate for alications in which attainment of a revenue target is an imortant consideration for managers; for examle, in event management, in seasonal clearance of highvalue items, or for business subunits oerating under erformance targets. We formulate the model as a continuoustime otimal control roblem, obtain otimality conditions, exlore structural roerties of the solution, and reort numerical results on roblems of realistic size. Subject classifications: inventory/roduction: erishable/aging items, marketing/ricing, uncertainty; dynamic rogramming/otimal control: alications; robability: stochastic model alications. Area of review: Manufacturing, Service, and Suly Chain Oerations. History: Received December 2005; revisions received August 2006, February 2007; acceted March Published online in Articles in Advance December 3, Introduction In recent years, the widely reorted successes of revenue management in the airline and hotel industries and the exansion of online booking and retail sales systems have stimulated interest in revenue management and dynamic ricing in many new areas. Revenue management alications have been imlemented or roosed in such diverse areas as freight transortation, automobile rental services, broadcast advertising, sorts and entertainment event management, style goods inventory clearance, medical services, and real estate. Details of imlementation across these areas differ, but most share a basic set of attributes: a finite suly of some inventory item, a fixed time san over which sales can occur, and stochastically varying demand. A tyical objective is to maximize exected revenues by one or a combination of two aroaches: (1) offering multile roduct classes at different rices and varying the allocation of fixed inventory to those classes over time, or (2) offering a single roduct class and dynamically varying the rice over time. There is an extensive literature on revenue management and related ractices. For surveys, see McGill and van Ryzin (1999), Belobaba (1987), Weatherford and Bodily (1992), Bitran and Caldentey (2003), Chan et al. (2004), Elmaghraby and Keskinocak (2003), and Yano and Gilbert (2004). A broad discussion of various asects of revenue management can by found in books by Talluri and van Ryzin (2004) and Phillis (2005). Traditional revenue management otimization models in transortation and accommodation services are risk neutral the objective is to maximize exected revenues at the end of the disosal eriod without consideration for the variability of final revenues across realized roblem instances. This is often aroriate in these alications because inventory control and ricing strategies are imlemented over hundreds or thousands of roblem instances (e.g., successive flight deartures), and no single realization has a otential for severe imact on the comany revenues. In these situations, the law of large numbers ensures that longterm average revenues will be maximized as long as good riskneutral strategies are emloyed. This is not the case for other otential alications. For examle, consider the case of an event romoter who may organize a few large events er year. In this case, the romoter faces a very high fixed cost for rental of a stadium or concert hall, and her first riority will be to recover this cost. Also, roblem instances are infrequent, and a single oor realization can have a major imact on the financial condition of the business. In other businesses, a manager s rimary concern may be maintenance or exansion of market share because a slide in share can lead to negative stockmarket assessments that can far outweigh the marginal revenues involved in revenue management decisions. Both of these circumstances are encountered by consultants in ricing and revenue management (Phillis 2006). In their descrition of a otential imlementation of revenue management in the health care industry, Secomandi et al. (2002,. 7) comment: the hosital s most imortant control of its revenue stream may be its ability to negotiate and imlement contracts with ayors to meet its revenue goals while minimizing risk. Furthermore, businesses with multile divisions or rofit centers tyically set erformance targets for their subunits as a art of overall strategy. Managers of subunits cannot afford to ignore minimum targets 326
2 Oerations Research 56(2), , 2008 INFORMS 327 in their search for higher revenues. In these situations and others, managers may be willing to ursue ricing olicies that sacrifice some exected revenues in return for a higher robability of attaining a minimum revenue goal. In this aer, we consider the roblem of dynamically ricing a stock of items over a finite time horizon so that both the exected revenues and the risk of oor erformance are taken into account. We assume that demand for the roduct follows a nonhomogeneous Poisson rocess in which the demand rate is rice sensitive, where rices are drawn from an arbitrary but finite redetermined set. We formulate a dynamic ricing model incororating risk, which we analyze as a roblem of otimal control. Risk is introduced to the model by augmenting the exected revenue objective with a enalty term for the robability that total revenues fall below a desired level of revenue a lossrobability risk measure. This risk measure has significant advantages: First, it is much more readily interreted by managers than the conventional variance measure; second, it is aroriate for both symmetric and skewed total revenue distributions; and third, it leads in the resent case to a tractable formulation that can be imlemented on roblems of realistic size. The introduction of risk in this form does, however, have a significant comutational drawback. In conventional dynamic inventory control roblems of this tye, the state of the system at any oint in time can be summarized by a single state variable remaining inventory whereas in the resent case, it is necessary to augment the state descrition with a second variable current revenues. Nonetheless, we show in this aer that a comutationally tractable formulation is ossible even with a twodimensional state sace. Prior studies most relevant to this aer are those on dynamic ricing by Gallego and van Ryzin (1994), Feng and Xiao (2000a, b), Zhao and Zheng (2000), and Chatwin (2000). In articular, Feng and Xiao (1999) introduced a risk factor in a dynamic ricing framework. They considered a model with two redetermined rices, and instead of looking at the exected revenue alone, used an objective function that reflects changes in the revenue variance as a result of rice changes. Chen et al. (2006) used utility theory to develo a general framework for incororating risk aversion in multieriod inventory models as well as multieriod models that coordinate inventory and ricing strategies. Other references on riskaverse inventory models include Agrawal and Seshadri (2000), Chen and Federgruen (2000), Eeckhou et al. (1995), and Martínezde Albéniz and SimchiLevi (2006). Integration of risk attitudes into dynamic ricing has also been recently addressed with different aroaches by Lim and Shanthikumar (2007) and Feng and Xiao (2004). The main contributions of this aer are: introduction of a ractical risk factor into a general roblem of otimal dynamic ricing of erishable items with stochastic demand over finite horizons, and exlicit inclusion of the revenue rocess in the descrition of the system s state to formulate a class of simle Markovian models incororating risk, which are imlementable in roblems of realistic size. This aer is organized as follows. We resent the notation and general model in 2, and derive the otimality conditions in 3. We exlore some of the structural roerties of the otimal olicies in 4, resent a few ossible generalizations of the model in 5, and discuss aroximate solution methods in 6. We rovide the results of extensive numerical exeriments in 7. Finally, 8 contains concluding remarks. 2. The Model 2.1. Statement We resent the model as a stochastic otimal control roblem in continuous time over the fixed lanning horizon T 0 (i.e., the negative value T is the beginning, and 0 is the end of the lanning interval). The notation T 0 for the lanning horizon is consistent with Zhao and Zheng (2000). Suose that the demand rocess N t is a nonhomogeneous Poisson rocess. The rate t of this rocess is nonincreasing in the current rice, and is a bounded, continuously differentiable function of time t for each. Such a model for the demand rocess is standard in the dynamic ricing literature (see, for examle, Gallego and van Ryzin 1994, Zhao and Zheng 2000). The sales rocess N t has its value limited by the initial inventory Y T ; thus, N t = min N t Y T. If the rice rocess is t, then the revenue rocess R t can be defined as a stochastic integral of the form R t = t T dn The riskneutral objective is to simly max E R 0 (1) without constraints, excet ossibly ones on the set of allowed rices. To incororate risk, we add a constraint of the form P R 0 z 0 (2) where z is a desired minimum level of revenues and 0 is the minimum accetable robability with which we want this level to be reached. Maximizing E R 0 (a rimary objective) is aroriate for a decision maker who can sell the roduct reeatedly in consecutive lanning horizons and is fully rational in the long term. On the other hand, reaching the level z reresents a shortterm (secondary) objective of the satisficing tye. The term satisfice as an alternative to otimize was used by Simon (1956,. 131) in a model of a decision maker with a definite, fixed asiration level. Formulation (1) (2) combines both of these objectives in a single model. If 0 is varied, the roblem
3 328 Oerations Research 56(2), , 2008 INFORMS will have different otimal solutions resulting in an efficient frontier in the lane of otimal P R 0 z and E R 0 ; that is, the efficient frontier is the set of attainable exected revenue/robability airs that are not dominated by any other attainable air. An alternative way to obtain this frontier is by solving the roblem max E R 0 CP R 0 <z (3) for a range of values of the coefficient C 0 +. The enalty arameter C can be interreted as the maximum cost associated with not meeting the desired level of revenue z. In some cases, z may corresond to sunk costs that must be recovered to avoid negative consequences of a severity reresented by the value C. In this case, z reresents a reference oint from which losses or gains are comuted. Furthermore, the hybrid objective in (3), adjusted by subtracting the reference oint z, is equal to E u R 0 z for a value function of the form u x = x CI x < 0 alied to the gain/loss x = R 0 z. (Here, I A is an indicator function of event A.) Note that u x is a somewhat extreme case of the value function roosed by the rosect theory of Kahneman and Tversky (1979): concave (linear in our case) for gains, convex for losses, and steeer for losses than for gains. We note that for many decision makers the values of the arameters z, 0, and C may be unknown or available only as rough estimates; however, by otimizing the system over ranges of values of these arameters, managers can gain insight into their ersonal assessments of the risk/return tradeoffs. For examle, selecting an aroriate value for C reduces to selecting a oint on the efficient frontier, which is quite similar to choosing an aroriate tradeoff between the return and risk for a ortfolio of securities. Furthermore, different values of z will result in different ranges of robabilities P R 0 z that can be attained on the same curve. Tyically, a decision maker will have some idea of the range of robabilities he/she is interested in. Due to the equivalence of the families of roblems reresented by (1) (2) and (3), and our articular interest in the efficient frontiers, we will focus on (3). For future reference, we will denote the otimal value of (3) as J Y T T C z, and we will refer to this roblem as the riskadjusted maximization roblem with initial inventory Y T, time horizon T, cost of risk C, and desired level of revenue z. In the terminology of Gīhman and Skorohod (1979), the system can be described as a jum Markovian controlled rocess. Indeed, for any fixedrice trajectory t, t T 0, a air Y t R t consisting of the remaining inventory Y t = Y T N t and the revenues R t obtained to date, is, by construction, a Markov jum rocess. In standard continuoustime dynamic ricing models, it is usually enough to only include remaining inventory Y t in the state descrition. In our case, due to the resence of revenue targets, the state needs to be described as Y t R t ; that is, we need to kee track of current inventory and the value of revenues to date. In this aer, we will consider admissible rice controls in a class of statefeedback (Markovian) controls, which are functions of time t, remaining inventory Y t, and current value of revenue R t ; that is, rice control: = t Y t R t. Given the time oint t and the value of Y t, it is imractical to comute the otimal rice for each value of R t if it is allowed to take a continuum of values. Therefore, it is desirable to have a discretevalued revenue rocess R t. In fact, the rice of items is always discrete in ractice. Thus, we assume that there is a smallest rice unit (e.g., one cent), and rices belong to a finite discrete set with the maximum rice max. This results in admissible controls taking values in the set = 0 max. Discrete sets of allowed rices were considered, for examle, in Feng and Xiao (2000a, b), and one can further restrict the set of allowed rices to some roer subset of. Because the rice at each time t belongs to a fixed discrete set, the revenue rocess R t is also discrete, taking values only in the set 0 Y T max. Thus, the state of the system at time t can be secified by a air in the finite set = n r n 0 1 Y T r 0 Y T n max. The states with n = 0 are absorbing. We will denote the set of all nonabsorbing states by. The secial case of our model with C = 0orz = 0is the discreterice version of the model of Gallego and van Ryzin (1994), studied extensively by Feng and Xiao (2000a, b) and by Zhao and Zheng (2000) for the more general case of a nonstationary distribution of reservation rice. In this aer, we will often refer to that model as the riskneutral model. We see that under the finite discreterice assumtion, the rocess Y t R t is an intensity controlled, nonhomogeneous, continuoustime, finitestate Markov chain. In 2.2, we will see that roblem (3) over the class of statefeedback controls can be formulated as a deterministic control roblem for the state robability distribution vector of the rocess Y t R t and analyzed using deterministic continuoustime otimal control techniques via the Pontryagin maximum rincile (see, for examle, Vinter 2000) Solution by Deterministic Otimal Control Methods Recall that the demand rate at time t and rice level is t. Suose that a comany sets the rice t n r in the state n r at time t. Then, our continuoustime Markov chain makes transitions from the state n r to the state n 1 r+ with a rate t t n r. Let P n r t be the robability that the system is in the state n r at time t. Then, P n r t is governed by the following system of ordinary differential equations, which follow from the Kolmogorov forward equations for nonhomogeneous
4 Oerations Research 56(2), , 2008 INFORMS 329 rocesses (see, for examle, Gnedenko 1967,. 37): dp n r dp 0 r dp YT 0 = t t n r P n r t + t t n+1 r P n+1 r t = r r =r t n+1 r r r =r t 1 r for 0 <n<y T 0 r Y T n max (4) t t 1 r P 1 r t for 0 r Y T max (5) = t t Y T 0 P YT 0 t (6) with the initial conditions P n r T = 0 for n r \ Y T 0 (7) P YT 0 T = 1 (8) To be able to emloy standard results from control theory (standard existence theorem and otimality conditions), we convexify the controls by embedding them into a larger olicy sace of randomized olicies formed by timedeendent measures t n r on the set of allowable rices in all ossible states n r. This extended olicy sace (a set of randomized statefeedback controls) can be interreted as follows. Suose that at time t, the comany has inventory n and revenue r. When n>0, the rice is offered to a customer with robability t n r. When n = 0, no rice is offered the zero inventory state is absorbing. Both situations can be catured using normalizations t n r = 1 for n r (9) t 0 r = 0 (10) where, for brevity, we droed the summation range condition. (We also dro it in all subsequent summations over the index for the same reason.) While this extended form of controls hels us establish existence, we emhasize that the extension is merely an analytical device, and the original sace of discrete rice controls remains our rimary interest. The corresonding rate of transition out of state n r is t t n r ; thus, for each, t t n r transitions er unit of time are directed into the state n 1 r +, and the system of equations governing P n r t can be rewritten succinctly as dp n r ( ) = t t n r P n r t + n+1 r t t n+ 1 r P n+1 r t for n r (11) Note that t n r is reresented by a nonnegative vector in the max dimensional real sace satisfying (9) (10). System (11) immediately generalizes Equation (4). We also note that it holds for all n r with n = 0 because (10) imlies that the first term is zero, and for n r = Y T 0 because does not contain a state with inventory level Y T + 1, and consequently the second term is trivially zero. Thus, (11) incororates (4) and (6) as well. Formulation (3) involves maximizing the objective E R 0 CI R 0 < z = rp n r 0 C n r n r r<z P n r 0 (12) subject to evolution system (11) with the initial conditions (7) (8). We will refer to this otimal control roblem as PH. Note that PH is a standard otimal control roblem; see Vinter (2000). In fact, because the evolution system (11) is bilinear in P n r and t n r (i.e., linear in the state and control variables searately, but jointly quadratic with constant coefficients), and the objective is linear in P n r, this formulation belongs to a secial class of bilinear otimal control roblems, long used in alied models in engineering (see Mohler 1973). 3. Existence of a Solution and Otimality Conditions In this section, we rove the existence of an otimal solution to the riskadjusted roblem (in its otimal control form PH) and derive its otimality conditions. First, we note that roblem PH satisfies the conditions of the existence theorem in Lee and Markus (1967, Theorem 4,. 259), which allows us to assert the existence of the otimal solution in the aroriate class of admissible controls. One of the theorem s requirements is that the image of the admissible control set at time t under the maing given by the righthand side of system (11) is a convex set. Note that the discreterice controls t n r do not satisfy this requirement (image would be a discrete set), whereas the extended controls t n r do. This is our rimary reason for control embedding. Proosition 1. Problem PH has an otimal solution: There exists a control that maximizes (12) for the system governed by (11) with initial conditions (7) (8) in the class of measurable functions t = t n r n r subject to constraints (9) (10). Corollary 1. The solution of the stochastic control roblem (3), under the finite discreterice assumtion in the class of randomized statefeedback controls, exists. A standard analysis technique for otimal control is the necessary otimality condition in the form of the Pontryagin maximum rincile; see Vinter (2000). To aly the maximum rincile to PH, we introduce the adjoint
5 330 Oerations Research 56(2), , 2008 INFORMS variables n r t for each of the evolution equations in (11). As often occurs in these tyes of roblems, n r t can be interreted as a conditional exectation of the objective term R 0 CI R 0 <z (see Lemma 1 below). We will denote the vector of all adjoint variables by, and the vectors of all state robabilities and olicies by P and, resectively (as functions of time). The following roosition is roved in the online aendix. An electronic comanion to this aer is available as art of the online version that can be found at htt://or.journal.informs.org/. Proosition 2. For an otimal state trajectory P and control in roblem PH, there exists satisfying the adjoint system of differential equations d n r d n r = t t n r n r t n 1 r+ t n r (13) = 0 n r \ (14) with the terminal condition n r 0 = r CI r < z n r (15) such that t n r delivers the maximum in the auxiliary roblem max P n r t t n 1 r+ t n r t t n r (16) s.t. t n r = 1 (17) for all n r and for almost all t T 0. Remark 1. In (16), P n r t is merely a coefficient in a linear exression to be maximized over t n r. Then, if t n r delivers the maximum in (16) for some strictly ositive value of P n r t, then that same t n r would also deliver the maximum for any other P n r t, including P n r t = 0. We see that the initial value roblem for state system (11) with initial conditions (7) (8) and the terminal value roblem for adjoint Equations (13) (14) with terminal condition (15) searate in the following sense. If one wants to find some P,, and satisfying the otimality conditions, it is ossible to do so by first finding a solution to the terminal value roblem for the adjoint variables, where delivers the maximum in (16) indeendently of P. Second, one solves the initial value roblem for P using found in the first ste. Because this rocedure is very efficient, control roblems where the state and adjoint equations searate have a reutation for being quite easy to solve in ractice. The following lemma (roved in the online aendix) rovides an interretation for n r t : Lemma 1. Under any control, the solution of the adjoint system (13) (14) with the terminal condition (15) is such that n r t = E R 0 CI R 0 <z Y t = n R t = r Remark 2. Lemma 1 imlies that roblem (16) (17) has a very natural interretation: For any inventory n and revenue r at any time t, the otimal decision is to maximize the exected rate of increment in our objective er unit of time. Remark 3. The existence results imly that the value function J Y T T C z of (3) is defined for all Y T, T<0, C, and z. We may also extend it to Y T = 0 and T = 0by J 0 T C z = J Y T 0 C z = CI 0 <z. For any solution of the adjoint system (13) (14), Lemma 1 imlies that n r t r corresonds to the objective function of the roblem of the form (3) with initial inventory n, time horizon t 0, and the revenue threshold z r. The otimal value of such a roblem is equal to J n t C z r. If under the otimal controls the state n r has a nonzero robability at time t that is P n r t > 0 then the adjoint variables satisfy n r t = r + J n t C z r. For an arbitrary trile P,, satisfying the conditions of Proosition 2, this equality relation need not be satisfied in states of robability zero at time t. The reason is that the olicies in these states do not affect the otimal value of the roblem for a system whose initial state is Y T 0. This can also be seen from the fact that, when P n r t = 0, the otimal value in (16) is delivered by any t n r satisfying (17). However, for any trile P,, that satisfies the conditions of Proosition 2, and for any state n r at time t including those of robability zero, Lemma 1 imlies that n r t r +J n t C z r. Otherwise, we have a contradiction with the otimality of J n t C z r. The following corollary gives a generalization of Lemma 1 in case of arbitrary terminal conditions (such as situations with salvage value or disosal costs, to be discussed later). Corollary 2. Under any control, the solution of the adjoint system (13) (14) with arbitrary terminal conditions n r 0 = F n r is such that n r t = E F Y 0 R 0 Y t = n R t = r
6 Oerations Research 56(2), , 2008 INFORMS 331 From the maximum rincile, we know that the otimal control satisfies (16) (17) for almost all t. IfP n r t > 0, then by dividing the objective term (16) by P n r t, we get the negative of the righthand side of the adjoint equation (13) at time t. Thus, when P n r t > 0, the value of the righthand side in (13) is the same for all controls delivering the maximum in (16) (17). We can now combine (16) (17) with (13) to get the HamiltonJacobi Bellman equation d n r = min t n r { t t n r } n r t n 1 r+ t (18) For given model arameters Y T T C z, a system of equations consisting of (18) for all n r and d n r / = 0 for n r \ with the boundary condition (15) rovides a sufficient otimality condition. A solution n r t to (18) will satisfy n r t = J n t C z r + r due to Remark 3. Remark 4. We now show that the otimal controls in the original control sace (i.e., the discrete rice set) exist as well. In doing so, we also establish that the righthand side of (18) coincides with min t n r t n 1 r+ t (19) Note that the righthand side of (18) is linear in controls t n r, and the set of feasible controls for given t n r has a olyhedral structure: t n r are nonnegative and satisfy (9). Recall that the otimum in linear otimization roblems is attained at a vertex that in our case, corresonds to t n r = 1 for some. Thus, if the term attaining the minimum in (19) is unique (e.g., if the terms t n r t n 1 r+ t are all distinct for different ), then the minimum in (18) is uniquely attained by t n r = 1 for some. Then, the righthand side of (18) reduces to (19), and the olicy in the state n r at time t is not randomized. If several terms attain the minimum in (19) with resect to, then the minimum in (18) is not unique and may be attained by t n r with several ositive comonents (i.e., several vertices and their linear combinations attain the minimum). In articular, such a henomenon occurs at switch oints between different rices. However, we can modify any such t n r by concentrating all robability on an aroriately chosen singleton set of rices without modifying the value of t t n r n r t n 1 r+ t (and thus showing that this value is equal to (19)). This ermits an otimal olicy with a singleton suort set in the form = t n r. Secifically, we can choose the maximum for which (19) is attained to be that singleton. The same tiebreaking rule was also emloyed, for examle, by Zhao and Zheng (2000). Henceforth, t n r will secifically refer to the maximum rice attaining the minimum in (19), and we will refer to the corresonding olicies as the singleton rice. 4. Structural Proerties of Otimal Policies In this section, we consider the structural roerties of otimal olicies. We also comare the obtained olicies with those obtained for the riskneutral model of Gallego and van Ryzin (1994). Recall that the latter model is a secial case of our model with z = 0. Next, we show that the value of the righthand side of Equation (18) does not deend on r for r z. This is a reality check for the model because the revenuelevel requirement should not affect the olicy after the required level has been reached. Consider a change of variables n r = n r r, which can be interreted as the exected future increment in the objective value comared to its value in the current state (exected future revenues for r z). For values of r z, Equation (18) becomes d n r { = min t n r t t n r } n r t n 1 r+ t and the terminal condition (15) transforms, for all n and r z, to n r 0 = 0 We know that the solution exists and is uniquely determined by n r t = n r t r = J n t C z r for r z. For a fixed level of inventory n, now consider any r z and the subsystems of equations governing the collections of variables n r n 0 1 n r r r + n n max These subsystems of equations and the corresonding terminal conditions are identical for different r z. Thus, we have the following roosition that the otimal controls can be selected to be identical for all states with the same level of inventory and any revenuetodate level r z: Proosition 3. For all r > r z and all t, n, n r t = n r t. Moreover, Arg min t n r t n 1 r+ t does not deend on r, and there exist corresonding otimal controls that do not deend on r: t n r = t n r for all r>r z and all t n. This roosition formalizes the fact that a riskadjusted otimal olicy coincides with a riskneutral one after the required revenue level has been reached, a conclusion that certainly alies to the singleton rice controls as well. Let 0 t n r be an otimal singleton rice olicy in the riskneutral roblem with the same initial inventory Y T. The corresonding adjoint variables
7 332 Oerations Research 56(2), , 2008 INFORMS will be denoted by 0. As before (see Remark 4), 0 t n r denotes the largest that attains the minimum in min t 0 n r t 0 n 1 r+ t. Using standard results on the monotonicity of riskneutral olicies, e.g., decrease of rice in n at each fixed moment in time, we can conclude that after reaching the required level z, the riskadjusted olicy ossesses the same monotonic roerties. However, numerical exeriments described in 7 show that the riskadjusted otimal olicy need not be monotonic in the states where r<z(below the required level). For values of r<z, the otimal rice t n r will, in general, deend on r. The behavior of near the critical threshold r = z is of some interest. The relation between the otimal rices in the riskadjusted and riskneutral models near this threshold is given in the following roosition. The intuition behind this result is that when the current value of revenue is close to the required level z, it becomes more imortant to cross the threshold than to collect a high rice. Because a higher demand rate is generally achieved by a rice reduction, this situation creates a downward ressure on the current rice. The roosition alies to the range of r no lower than one otimal riskneutral rice below the threshold. It is intuitively clear that the downward ressure on rice may exist only within this band because, for lower values of r, it may be referable to ste over the threshold by means of higher rices. Such behavior of otimal rices is almost obvious if only one item is available for sale and is indeed observed in the numerical exeriments. Proosition 4. For all t n r such that z>r z 0 t n r, we have t n r 0 t n r. Proof. Consider, for some t n r such that z>r z 0 t n r, the otimal rices in the riskadjusted t n r and riskneutral 0 0 t n r models, resectively. Suose, on the contrary, that we have > 0. Because t n r and 0 t n r attain the minimums of the righthand sides in the corresonding differential equations for n r and 0 n r, and 0 t n r is the largest attaining its corresonding minimum, we conclude that t 0 n r 0 n 1 r+ > t 0 0 n r 0 n 1 r+ 0 t n r n 1 r+ t 0 n r n 1 r+ 0 By rewriting these inequalities, we get t t 0 0 n r > t 0 n 1 r+ t 0 0 n 1 r+ 0 (20) t t 0 n r t n 1 r+ t 0 n 1 r+ (21) 0 Because r + >r+ 0 z, we have n 1 r+ = 0 n 1 r+ and n 1 r+ 0 = 0. Therefore, the righthand sides of inequalities (20) and (21) coincide. Subtract n 1 r+ 0 ing the second inequality from the first, we get t t 0 0 n r n r >0 Note that 0 n r = r + J n t C r >r + J n t C z r = n r, and it follows that t > t 0. Due to the monotonicity of t in, wehave t t 0, a contradiction. Thus, 0 t n r 0 t n r in the original nonabbreviated notation). In the riskneutral model of Gallego and van Ryzin (1994), the rice usually increases immediately following a sale. Thus, a tyical realization of the rice rocess has uward jums at the times of sales. In the case of the riskadjusted model, however, the rice may decrease following a sale. This haens because, as the above roosition suggests, the otimal rices must be lower than the riskneutral rices when current revenues are slightly below the threshold z (within one otimal riskneutral rice below the threshold). Indeed, consider the following examle of model arameters that lead to such a rice dro. Let the set of allowed rices be and suose that the demand is time homogeneous: t 1 = 1 > 0, t 2 = 2 > 0, and t 3 = 0. Let Y T = 2 and z = 3. At the end of the lanning interval, we have n r 0 = r CI r < 3. Note that n r t will be very close to n r 0 as t aroaches zero, due to its continuity. To illustrate a dro in rice, we seek 1 and 2 such that = 1 is selected for the state n r = 1 2, and = 2 for the state n r = 2 0. That is, the otimal rice dros from 2 to 1 following a transition from 2 0 to 1 2. Att = 0, this is enforced by inequalities < > Note that the shutdown rice = 3 cannot be selected near t = 0 because n r 0 n 1 r+ 0 <0 for all n r. Because these inequalities are strict, they will also enforce the dro in rice for some range of t near 0. Substituting the exression for n r 0 in the first inequality, we get 1 2 CI 2 < CI 3 < 3 < 2 2 CI 2 < CI 4 < 3 which is equivalent to 1 1 C < 2 2 C or ( 1 > ) 1 + C Similarly, from the second inequality, we get 1 0 CI 0 < CI 1 < 3 > 2 0 CI 0 < CI 2 < 3 or 1 < 2 2 Thus, whenever demand rates satisfy ( ) < 1 + C 1 < 2 2
8 Oerations Research 56(2), , 2008 INFORMS 333 and a sale haens near t = 0 in state n r = 2 0, then as the system jums to the state 1 2, the otimal rice dros from 2 to 1. The following roosition summarizes intuitive monotonic roerties of the value function with resect to t, n, and r. Proosition 5. Let be a solution of (18). Then, for all t n r: (a) n r t n 1 r+max t ; (b) n r t is nonincreasing in t; (c) for an otimal singleton rice control t n r, t t n r > t max = 0 imlies that n r t < n 1 r+ t n r t ; (d) n r t n+1 r t ; (e) n r t n r+1 t 1. Proof. Part (a) follows from the following relaxation argument. Consider a state n r at time t and a modified roblem where one item is guaranteed to sell at rice max at the end of the lanning horizon. The otimal exected revenue of the modified roblem in the given state is certainly greater than or equal to the one in the original roblem. However, the state n 1 r + max is equivalent to the initial state of the modified roblem in terms of inventory and current level of revenue, which establishes art (a). To establish art (b), we recall that by Remark 4, the righthand side of (18) coincides with (19). From art (a), it follows that the value of (19) satisfies min t n r t n 1 r+ t t max n r t n 1 r+max t 0 Therefore, the righthand side of (18) is nonositive and d n r / 0. Part (b) follows. To establish art (c), recall that t n r denotes the largest rice attaining the minimum in min t n r t n 1 r+ t If the value of this minimum is zero, then it must be that t n r = max. The condition t t n r > t max = 0 imlies that t n r < max. Part (c) follows. The roof of (d) follows immediately from n r t = r + J n t C z r + J n+ 1 t C z = n+1 r t with the inequality in the middle obtained from a relaxation argument. Similarly, art (e) is immediately obtained from n r t = r + J n t C z r r + J n t C z r + 1 = n r+1 t 1 5. Generalization 5.1. General Terminal Value The aroach of reducing the stochastic otimization roblem (3) to a deterministic otimal control roblem can be alied to a much wider class of roblems. The existence results and otimality conditions derived for (3) can be easily obtained for an otimization roblem of the form max E F Y 0 R 0 with an arbitrary finite function F n r. The only difference in the otimality conditions is that terminal condition (15) needs to be relaced with a more general n r 0 = F n r For examle, a articularly interesting extension is an introduction of disosal costs or salvage values for unsold items into the model. For each item, let the disosal cost be D<0 (or, salvage value, D>0). Then, an immediate generalization of (3) is max E R 0 + DY 0 CP R 0 + DY 0 < z For a nonrandom D, the corresonding F n r is of the form F n r = r +Dn CI r +Dn<z.IfD is stochastic, we use the slightly more general F n r = E D r + Dn CI r + Dn < z Extended RiskNeutral Horizon Another generalization of our formulation is to consider a lanning horizon extending beyond zero, the time at which we assess the loss robability, into a later time eriod in which we only maximize exected revenue. This roblem can be written as max E R T CP R 0 < z (22) where T > 0. Note that this objective function can be rewritten as E R T R 0 + E R 0 CP R 0 <z. The value E R T R 0 is the amount of revenues earned after time 0; that is, when risk no longer affects the decision and an otimal olicy can deend on the current inventory only. Let the value of the otimal exected revenues E R T R 0 on 0 T, given the inventory n at zero, be J n T. Then, we can solve roblem (22) as the roblem of maximizing E F Y 0 R 0, where F n r = J n T + r CI r < z Variance Risk Measure We also note that the traditional meanvariance aroach to incororation of risk, while it can be analyzed by similar
9 334 Oerations Research 56(2), , 2008 INFORMS techniques, does not fit into the framework of this aer. Indeed, the meanvariance formulation would be to max E R 0 CVar R 0 This objective can be rewritten as E R 0 C E R 2 0 E R 0 2 = [ rp n r 0 C r 2 P n r 0 n r n r ( n r ) 2 ] rp n r 0 which is quadratic in P 0. Therefore, the boundary conditions for the adjoint variables will deend on P 0, and the evolution and adjoint equations governing P and will no longer searate in the sense of Remark 1. Thus, the meanvariance aroach aears to be alicable only to limitedsize dynamic ricing roblems Multiroduct Case Note also that it is easy to extend our model to the case of m erishable roducts if the minimal rice increments for all roducts have the same value, and each customer can only urchase a single item of a single roduct. Indeed, the combined revenue value will still be reresented by a single scalar r in the state descrition. The state descrition will also contain current inventory levels n i for each roduct i = 1 m. The feasible controls will be given by a discrete vector of rices 1 m 0 max m for each roduct. It can be shown that the same analytic techniques as the ones used in this aer are alicable. Of course, the comutational comlexity will increase as the number of roducts m gets larger. However, this issue is also resent in a multiroduct variation of the model of Gallego and van Ryzin (1994). There is no inherent difference in the treatment of a revenuelevel requirement in 1 and mroduct situations Moving Revenue Target Generalization Our analysis of roblem (3) under the discrete rice assumtion leads to a HamiltonJacobiBellman (HJB) equation of the form familiar from the stochastic control literature. However, it is not obvious how to handle the roblem in the form (1) (2) directly by standard stochastic control techniques, and, in this section, we describe a ractically relevant generalization of model (1) (2) that does not have an immediate solution through the standard method but can be aroached with the techniques of this aer. Consider a comany that has a moving revenue target z t. The target could reresent cash flow requirements that result from the need to cover ongoing oerational costs. Due to the stochastic nature of sales, it is imossible to guarantee that the target is met with certainty; however, management may have some robability t with which it wants to exceed the revenue target z t at time t. This naturally leads to a roblem of the form max E R 0 (23) subject to P R t >z t t for almost all t T 0 (24) We will assume that z t and t are continuous functions of t. We use R t >z t in this formulation instead of R t z t for technical reasons to ensure uer semicontinuity of the function t P R t >z t, a requirement for known otimal control results. This formulation can be cast as an otimal control roblem with a state constraint. Indeed, using robability vector P in the same way as before, objective (23) can be exressed as E R 0 = rp n r 0 (25) n r while constraint (24) can be rewritten as a state constraint P R t >z t = P n r t I r >z t t n r for almost all t T 0 (26) Therefore, we obtain an otimal control roblem with the objective to maximize the functional (25) of the robability vector P, whose evolution is described by system (11) with the initial conditions (7) (8), and whose value is restricted by the state constraint (26). We will refer to this roblem as PG. This formulation satisfies the conditions of Theorem of Clarke (1990,. 211). This includes uer semicontinuity of the function g t P = t P n r I r >z t n r secifying the state constraint as g t P t 0 because in our roblem, z t is continuous, P is guaranteed to be nonnegative, and, therefore, each term of the form P n r I r > z t is uer semicontinuous. Whereas Theorem of Clarke (1990) is very general and alies to the case of nonsmooth roblem data (even including nonsmooth dynamics of the system), our roblem is much simler because all functions, excet for one used in the state constraint, are smooth. Thus, the main difference in otimality conditions for this formulation as comared to Proosition 2 will be due to the state constraint: Proosition 6. For an otimal state trajectory P and controls in roblem PG, there exist a measurable function from T 0 to dimensional real sace, a nonnegative measure on T 0, a constant 0 1,
10 Oerations Research 56(2), , 2008 INFORMS 335 and an adjoint trajectory such that: (i) satisfies the adjoint system of differential equations d n r d n r = t t n r [ n r t + n r s ds n 1 r+ t T t ] n 1 r+ s ds n r (27) T t = 0 n r \ (28) (ii) t n r delivers the maximum in the auxiliary roblem max P n r t [ t n 1 r+ t + n r s ds T t ] n r t n 1 r+ s ds t n r T t (29) s.t. t n r = 1 (30) for all n r and for almost all t T 0 ; (iii) for almost all t T 0 with resect to measure, we have n r t = I r >z t for r z t and n r t 1 0 for r = z t, and measure is suorted on the set { t T 0 n r } P n r t I r >z t = t (iv) the following transversality conditions hold: n r 0 + n r s ds = r n r (31) T 0 (v),, and are not simultaneously equal to zero. As one can see, the main comlexity for the analysis of these otimality conditions lies in the resence of measure, which lays the role of a Lagrange multilier for state constraint (26). 6. Otimal Policy Aroximation Techniques The otimal dynamic ricing olicy for the riskadjusted roblem may be hard to comute because of the size of the state sace and the large number of stes required for time discretization. Such a olicy may also be hard to aly in ractice if frequent rice changes are not ossible. Thus, a decision maker may want to have an assortment of methods that rovide aroximate solutions of a simler structure. A articularly simle tye of olicy is the fixedrice olicy in which the rice remains constant during the entire lanning horizon. It is also very easy to find an otimal olicy in this restricted class. Indeed, under fixedrice, the total number of customer arrivals is a Poisson random variable N with the rate = 0 T t (32) The exected value of the number of sales N can be exressed as Y T E N = E min Y T N = np N =n +Y T P N >Y T Y T n=1 = n n e + Y n=1 n! T P N >Y T Y T 1 n e = + Y n=0 n! T P N >Y T = P N <Y T + Y T P N >Y T and the hybrid objective value as E R 0 CP R 0 <z = P N <Y T + Y T P N >Y T CP min Y T N < z/ For each given value of, it is easy to evaluate using numerical integration techniques. The objective function is also easy to evaluate because it is exressed via the cumulative distribution function (and its comlement) of the Poisson random variable with a given rate. We will denote the otimal fixedrice olicy selected according to this method as FP. While the comutation of the FP olicy is easy, it is hard to get analytic results directly. On the other hand, it is ossible to aroximate the total demand random variable N using the normal distribution: N + X, where X is N 0 1. In the following, we consider the uncaacitated case. The analysis can be generalized by considering rices >z/y T (otherwise, reaching the revenue target is imossible). Using x and x to denote the standard normal cumulative distribution and density functions, resectively, and = to denote the exected revenues in the uncaacitated case, we can aroximate the hybrid objective by C f, where f = ( ) z 1 Note that is tyically assumed to be decreasing where it is ositive, whereas is assumed to be unimodal and to satisfy the regularity condition lim = 0. To simlify the analysis, we additionally assume that has the unique maximizer, and is strictly increasing (decreasing) for < ( > ). Using firstorder otimality conditions, we immediately get the following roosition
11 336 Oerations Research 56(2), , 2008 INFORMS (the roof is in the online aendix): Proosition 7. Suose that is defined and differentiable for all +. Then, the rice that maximizes the normal aroximation C f to the hybrid objective satisfies the equation ( ( ) r z C f 2 1 zr ) = 0 (33) r 2 Moreover, if z, then. Ifz<, there exists 1 < < 2 such that 1 = 2 = z, and either > 2 or 1 < <. An imrovement uon olicy FP is ossible if we artition the entire lanning horizon into k equal intervals and use a fixed rice on each of them. We define a myoic olicy with k rices (denoted as Mk) as the olicy that uses k rices selected in the beginning of each interval as the otimal fixed rices from that oint until the end of the lanning horizon. We call this olicy myoic because it ignores the ossibility of future rice changes. Such a olicy maintains the advantages of the FP olicy: ease of comutation and imlementation. It will be subotimal comared to the otimal state feedback olicy with k fixed rices selected at the beginning of each interval (we will denote such a olicy as Ok). Because the olicy Ok has to be comuted using dynamic rogramming, it will be considerably harder to comute if the number of different rices max is large (resulting in a very large state sace of the size Y T max ). On the other hand, the olicy Ok will lead to significantly simler rice aths than a fully dynamic ricing olicy, and although lacking the transarency of the FP or Mk olicies, may still be reasonably easy to imlement in ractice. Exerimental comarisons of the aroximate olicies mentioned above to the true otimum is resented in the next section, along with other numerical exeriments. 7. Numerical Exeriments In this section, we resent a series of numerical exeriments with our model. In these exeriments, we utilize a discretetime aroximation of Equation (18), as described in the online aendix. The online aendix also shows how we can organize calculations so that the otimal solution for a range of required level of revenues z from zero u to a secified maximum value can be obtained simultaneously. We consider this feature to be very imortant in ractice because it saves comutation time, lets a user of the model observe the effect of different values for z, and facilitates the selection of an aroriate value. Section 7.1 gives an examle of the otimal olicy in the case of a stationary demand rocess. We will see that a managerially significant imrovement in the robability of reaching a required level of revenue can be achieved at some moderate exense in terms of the exected value of revenue. However, we will also see that the effects of the riskadjusted objective comared to the riskneutral case decrease as the initial inventory increases. The effects, however, may also deend on the structure of demand. Thus, 7.2 describes an exeriment in which we comare the following tyes of demand: one stationary and two nonstationary ones with generally increasing/decreasing willingness of customers to ay. This comarison is done based on the families of efficient frontiers for different values of z. We see that better results can be achieved when willingness to ay is increasing in time. Sections 7.1 and 7.2 also rovide comarisons of the aroximate olicies with the otimal one. Finally, 7.3 resents a largerscale examle with 250 items and 25,000 time stes and shows that a managerially significant imrovement in the robability of reaching the required level of revenue can still be attained The Behavior of the Model in the Case of a Stationary Demand Process We first study the effects of introducing a risk term into a numerical examle resented in 3.4 of Gallego and van Ryzin (1994). In that examle, the demand function is stationary of the form t = ex a (34) The value of is selected so that ex 1 T =10. The initial inventory ranges from 1 to 20. The above choices corresond to those of the Gallego and van Ryzin (1994) examle. We have also secified a = 0 1, max = 100, C = 100, and z = 200. The value of max = 100 signifies that 101 discrete rice levels were used: = The discretization ste size was t = 10 3 T, resulting in K = time intervals. The comutational rocedure was imlemented in C++ and comiled with g++ version on a 3.4 GHz P4 workstation with 1 Gb of RAM running Linux. The comutation of the otimal olicy for this examle took 2.54 seconds. The otimal exected revenues for each Y T 1 20 and z can be obtained as 1 Y T r r, where r = 200 z (the quantity 1 n r denotes the exected revenues comonent of the adjoint variable n r under an otimal ricing olicy; see the online aendix for comutational details). Note that all of this sensitivity information is obtained in a single run. These values are shown in Table 1. Naturally, the revenues tend to dro with a decrease in Y T. In addition, we see that for many values of Y T, the revenues decrease, then increase slightly, dro again, and finally, increase as z varies. That is, the otimal E R 0 is higher when z slightly exceeds E R 0, with smaller exected revenues for zvalues immediately above and below. Such behavior of the otimal exected revenues can be exlained by the fact that one does not need to sacrifice as much in exected revenues to achieve a given increase in P R 0 z when z is near the center
12 Oerations Research 56(2), , 2008 INFORMS 337 Table 1. The otimal exected revenues for several levels of initial inventory Y T and desired levels of revenue z: stationary demand examle of Gallego and van Ryzin (1994). z Y T of the distribution of R 0 (which roughly corresonds to E R 0 ). As z deviates from the center of the distribution, a more rominent decrease in E R 0 results from otimizing the riskadjusted objective (3). As the deviation of z from the center becomes more extreme, it is difficult Table 2. to achieve a significant increase in P R 0 z. Therefore, the term CP R 0 <z will not significantly affect the olicy, and, as a consequence, the revenues. In Table 2, we also resent the difference in the exected revenues under the riskneutral exected revenue maximization versus the The ercentage difference in the otimal exected revenues in the riskneutral and the riskadjusted models relative to the riskneutral case for each level of initial inventory Y T and desired level of revenue z. z Y T
13 338 Oerations Research 56(2), , 2008 INFORMS Table 3. The difference in the otimal P R 0 z in the riskneutral and the riskadjusted models for each level of initial inventory Y T and desired level of revenue z. z Y T maximization of the riskadjusted objective (3). The difference becomes smaller for larger values of Y T because the uncertainty in revenues relative to the exected value and, consequently, the effects of the risk term on olicies, decrease. Table 3 shows the effect of the riskadjusted objective in terms of the increase in P R 0 z versus the riskneutral case. As one can see, a significant increase in this robability can be achieved for various values of z (esecially for the smaller Y T s). Unfortunately, the achieved increase in P R 0 z becomes smaller with an increase in the initial inventory. Table 4 rovides a summary of these comarisons for the value of initial inventory Y T = 10. One can see that the ractically significant imrovements (above 5%) in P R 0 z are obtained in the wide range of z from 60 to 160. Smaller imrovements can also be imortant if they corresond to the lowrange ercentiles of the distribution of R 0, such as z = For examle, the robability of failing to reach z = 50 has decreased from 4.79% to 1.99% (corresonding to an increase in P R 0 z from to ). We also examine the features of the otimal rice T n r at time T (beginning of the lanning horizon). Of course, in the original roblem formulation, the initial value of the revenue is zero, but due to our simultaneous sensitivity analysis, we have the otimal rice available for different initial levels of revenue as well. The rice exhibits strikingly different behavior from the rice otimal under the riskneutral maximization. The rice rofile is shown in Figure 1. As suggested by Proosition 4, the otimal rice is below the one for the riskneutral case for values of initial revenue r immediately below z. However, we see that the rices for even smaller revenues r can become significantly higher and subsequently dro. This behavior is intuitively reasonable because even a small ossibility of reaching the critical revenue threshold z can encourage a rice increase. However, as the chance of reaching Table 4. The summary of the otimal E R 0 and P R 0 z for different values of z, and corresonding changes E R 0 and P R 0 z comared to the riskneutral case (C = 0) for Y T = 10. z E R 0 E R 0 P R 0 z P R 0 z
14 Oerations Research 56(2), , 2008 INFORMS 339 Figure 1. The initial otimal rice T n r for each n r n z becomes smaller, maximizing exected revenue becomes more imortant, which results in a lower rice. Another imortant observation is that even under the exonential demand model emloyed in this exeriment, T n r is not necessarily decreasing in n for a fixed r < z. One ossible exlanation is that having more items on hand increases the chance to reach the level z, thus driving the rice higher under some scenarios. Finally, in Table 5, we resent the erformance of the aroximate olicies discussed in 6 alied to this examle. Along with the fixed rice, we consider myoic and otimal olicies with 2, 4, 10, and 20 rices (denoted as FP, M2, M4, M10, M20, O2, O4, O10, and O20, resectively). The entries in the table reresent the maximum ga between the otimal value of the hybrid objective and its aroximation measured as the ercentage of the otimal riskneutral revenues (which give an uer bound for the otimal hybrid objective function) over all values of z = Generally, the erformance of every aroximate method imroves with the initial inventory Y T. This is natural because uncertainty in revenues decreases with Y T. The otimal fixedrice olicy, FP, results in a much larger ga (at least 9%) than Mk or Ok olicies (below 4% and and 1% for k = 20 and larger values of Y T ). Increasing the r number of rices in the myoic olicy usually imroves the ga. For some values, it actually increases the ga because the myoic olicy does not take into account likely future rice changes. The erformance of Ok olicies imroves monotonically and is much better than that of Mk olicies with the same number of rices The Behavior of the Model in the Case of Nonstationary Distribution of the Reservation Price Our second exeriment is related to the observation that imrovements in P R 0 z tended to get smaller with an increase in Y T under the exonential stationary demand structure. This, however, is strongly related to the roerty of stationarity of the reservation rice distribution. To recover a reservation rice distribution from the rate of the sales rocess, let t = t 0 be the (finite) maximum of t with resect to at each t T 0. Then, the quantity t / t can be interreted as the robability that a customer arriving according to a Poisson rocess with rate t buys a roduct riced at (see, for examle, Gallego and van Ryzin 1994 and Zhao and Zheng 2000). This interretation is consistent with the view that a customer comares a directly unobservable quantity, his or her (T,n,r)
15 340 Oerations Research 56(2), , 2008 INFORMS Table 5. The maximum ga between the aroximate and otimal hybrid objective values as a ercentage of the otimal riskneutral revenues for different values of Y T : the stationary demand examle of Gallego and van Ryzin (1994). Aroximate olicy (%) Y T FP M2 M4 M10 M20 O2 O4 O10 O reservation rice, to, and makes a urchase if the reservation rice is higher than. Accordingly, for each t, a robability distribution function of the form F t = 1 t / t can be thought of as a distribution of reservation rice. Gallego and van Ryzin (1994) considered a case in which F t is stationary, i.e., indeendent of t, while Zhao and Zheng (2000) studied a general, nonstationary case. In their analysis of the stationary case, Gallego and van Ryzin (1994) showed that the fixedrice heuristic is asymtotically otimal. This suggests that it may be harder to achieve significant imrovements in P R 0 z by a dynamic (versus fixed) ricing olicy for larger initial inventories. As well, the fixedricing olicy is itself not useful for imroving P R 0 z when the inventory is large. Thus, there is little hoe of imroving P R 0 z when the reservation rice distribution is stationary and inventory is large. In contrast, Zhao and Zheng (2000) resent an examle in which the reservation rice distribution is not stationary, and, accordingly, the dynamic ricing strategy is extremely relevant even for large inventories. Thus, we will comare the stationary model form used reviously: 0 t = ex a 0 with two nonstationary models: 1 t = ex a 1 + bt 2 t = ex a 2 bt where b>0. The arameters were chosen so that a 0 = 0 2, a 1 = 0 3, a 2 = 0 1, and b = 0 2/ T. The function 0 t is a standard exonential demand function. The second function, 1 t, corresonds to a situation when the value of each item deteriorates with time and customers are willing to ay less and less for it; and the third one, 2 t, exhibits the oosite behavior. We used Y T = 50, z = 300, max = 100, and = ex 1 Y T / T. The calculations were done for 5,000 time stes for each of the values of C from the set (total comutation time was 384 seconds for each model). This resulted in an aroximation to the efficient frontier in the sace of E R 0, P R 0 z. Again, note that the simultaneous sensitivity analysis feature of our aroach rovided us with the otimal E R 0, and P R 0 z for all values of z in the range from 0 to 300. Therefore, the efficient frontiers were simultaneously comuted for 0 z 300. Because it is imractical to resent these curves for each value of z, we only lot them for values z in Figure 2. The efficient frontiers of the most favorable shae corresond to the increasing value demand structure because they show a ossibility of significant imrovement in P R 0 z without sacrificing much of the exected revenue. One can see this in the length of the shoulders in the to ortion of the curve (for each resective value of z in all three figures) as well as the sharness of the curve s dro from the shoulders. In contrast, the case of decaying value demand structure aears to be the most unfavorable because the maximum ossible imrovement in robability is smaller and requires
16 Oerations Research 56(2), , 2008 INFORMS 341 Figure 2. The family of efficient frontiers for Y T = 50: stationary 0 t, decaying value 1 t, and increasing value 2 t demand structures. Exectation Exectation Exectation Probability Probability z = 300 Stationary demand structure z = 250 Decaying value demand structure Increasing value demand structure z = 200 z = 300 z = 250 z = 200 z = 300 z = 250 z = Probability a significant sacrifice in exected revenue. Thus, one can see that the resence of nonhomogeneity of demand and its tye may significantly change the effects from introduction of a risk term, even for larger inventories. Finally, we examine the erformance of the aroximate methods as a function of cost C of failure to reach the desired level of revenues and the tye of nonhomogeneity of demand structure. Table 6 shows the maximum ga (over z) for each demand structure, each level of C, and the aroximations FP (fixed rice), M10 (myoic with 10 rices), and O10 (otimal with 10 rices). Generally, the difficulty of the roblem increases with C for each of these aroximations. The FP olicy erforms much better for the stationary demand structure relative to decreasing or increasing value structures. M10 does not imrove over FP significantly when nonstationarity is resent, whereas there is a noticeable imrovement for the stationary demand structure. O10 significantly imroves uon both FP and M10 for either demand structure. This shows that successful otimization of the hybrid objective generally requires some form of feedback, but the feedback has to be otimal when the reservation rice distribution is nonstationary. A smaller ga for the decaying value structure in the case of the O10 olicy is exlained by limited sace for imrovement in robability to reach the desired level z comared to other demand structures The Behavior of the Model for Large Values of Initial Inventory It is interesting to examine the behavior of the model for much larger values of initial inventory. In this exeriment, we studied the erformance of our model on the examle with Y T = 250. We used a linear demand function, with the reservation rice distribution changing in time in a manner similar to 2 t : 3 t = 1 0 a 2 bt + where = ex 1 Y T / T, a 2 = 0 1, b = 0 2/ T (the same as in the revious exeriment). These arameters result in a shutdown rice of at most 10 because a 2 bt for all t T 0. We note that although we also erformed a similar large inventory exeriment with the exonential demand function, we saw little imrovement in P R 0 z. Other model arameters were z = and C = Again, we note that the value z = is simly the uer range of z values considered in the exeriment, and the results for all smaller z are obtained simultaneously due to the setu of the numerical rocedure. The discretetime aroximation utilized K = time stes (total comutation time was 2,265 seconds). We first show, in Figure 3, the robability mass function of R 0 before the risk term is incororated into the model. The mean, which is equal to , and the median of 852 are indicated by dashed and solid lines, resectively. Note a skew of the distribution to the left. This left skewness means that one has a somewhat higher robability of getting low revenues even though most of the realizations of the revenue distribution belong to the area with higher revenue values. Figure 4 suerimoses the robability mass functions P R 0 = r of the total revenue in the riskneutral (solid line) and riskadjusted with z = 800 (dashed line) roblems. These distributions are comuted by solving the initial value roblem for state system (11) under the riskneutral and riskadjusted otimal controls, resectively. The robability that the total revenue is below z = 800 (that is, the area under the curve to the left of z = 800) reduces significantly under the riskadjusted olicy. It is achieved by shifting the bulk of the total revenue distribution to the left. The mean of the total revenues decreases from to , and the standard deviation from to Figure 5 shows an (absolute) imrovement in P R 0 z
17 342 Oerations Research 56(2), , 2008 INFORMS Table 6. The maximum ga between the aroximate and otimal hybrid objective values as a ercentage of the otimal riskneutral revenues for different values of C: stationary 0 t, decaying value 1 t, and increasing value 2 t demand structures. Aroximate olicy FP (%) M10 (%) O10 (%) C 0 t 1 t 2 t 0 t 1 t 2 t 0 t 1 t 2 t , relative to the riskneutral model for various values of z and a relative decrease in revenue with resect to the base value of (comuted with C = 0). The imrovement in P R 0 z can be significant, reaching close to 9% for intermediate values of z = We also note that, for examle, a comany with a target level of revenue z = 817 facing a 20% loss robability could reduce this to 12% with relatively little (2.3%) loss in exected revenues. In general, the overall cost of imrovement in robability in terms of exected revenue is not very high. On the other hand, as seen from the grah, the decrease in revenue is always less than 3%. 8. Conclusions and Directions for Future Research This aer resents a new dynamic ricing model that ermits control of both exected revenues and the risk that total revenues will fall below a desired minimum. This model would ermit decision makers in a number of emerging alication areas for revenue management to seek Figure 3. Distribution function The robability mass function P R 0 = r of the total revenue in the riskneutral roblem for linear increasing value demand structure 3 t and Y T = ,000 1,050 R Note. The mean and the median are indicated by dashed and solid lines, resectively. ricing olicies that balance maximization of exected revenues against risk of oor erformance. We show that this model falls within a general class of continuoustime otimal control roblems, demonstrate the existence of otimal solutions under reasonable assumtions, rovide otimality conditions, and exlore some of the structural roerties of solutions. We also resent an efficient comutational rocedure that allows evaluation of this model on roblems of realistic size. Our aroach also rovides useful and numerically tractable analytical tools that can aid in decision making. The future research related to this model may include the following toics: (1) embedding this model into a gametheoretic framework that models the effects of risk in a cometitive setting; (2) studying the effects of rice guarantees (Levin et al. 2007) on the behavior of our model; (3) extending the model to include such factors as multile roduct tyes, returns, exchanges, and cancellations; and (4) incororating lossrobability risk into general inventory management systems including variable and stochastic costs, and stochastic interest rates. Figure 4. Distribution function The robability mass functions P R 0 = r of the total revenue in the riskneutral (solid line) and riskadjusted with z = 800 (dashed line) models for linear increasing value demand structure 3 t and Y T = R
18 Oerations Research 56(2), , 2008 INFORMS 343 Figure 5. Imrovement in P(R(0) z) Decrease in E[R(0)], % The imrovement in P R 0 z, and the decrease in E R 0 for linear increasing value demand structure 3 t and Y T = ,000 1,050 z ,000 1,050 z 9. Electronic Comanion An electronic comanion to this aer is available as art of the online version that can be found at htt://or.journal. informs.org/. References Agrawal, V., S. Seshadri Imact of uncertainty and risk aversion on rice and order quantity in the newsvendor roblem. Manufacturing Service Oer. Management 2(4) Belobaba, P Airline yield management: An overview of seat inventory control. Transortation Sci Bitran, G., R. Caldentey An overview of ricing models for revenue management. Manufacturing Service Oer. Management 5(3) Chan, L., Z. Shen, D. SimchiLevi, J. Swann Coordination of ricing and inventory decisions: A survey and classification. D. Simchi Levi, D. Wu, Z. Shen, eds. Handbook of Quantitative Suly Chain Analysis: Modeling in the EBusiness Era. Kluwer Academic Publishers, Boston, MA, Chen, F., A. Federgruen Meanvariance analysis of basic inventory models. Working aer, Columbia University, New York. Chen, X., M. Sim, D. SimchiLevi, P. Sun Risk aversion in inventory management. Oer. Res. Forthcoming. Clarke, F Otimization and Nonsmooth Analysis. Classics in Alied Mathematics, Vol. 5, 2nd ed. Society for Industrial and Alied Mathematics, Philadelhia, PA. Eeckhou, L., C. Gollier, H. Schlesinger The riskaverse (and rudent) newsboy. Management Sci. 41(5) Elmaghraby, W., P. Keskinocak Dynamic ricing in the resence of inventory considerations: Research overview, current ractices and future directions. Management Sci. 49(10) Feng, Y., B. Xiao Maximizing revenues of erishable assets with a risk factor. Oer. Res. 47(2) Feng, Y., B. Xiao. 2000a. A continuous time yield management model with multile rices and reversible rice changes. Management Sci. 46(5) Feng, Y., B. Xiao. 2000b. Otimal olicies of yield management with multile redetermined rices. Oer. Res. 48(2) Feng, Y., B. Xiao A combined risk and revenue analysis for managing erishable roducts. Working aer, Chinese University of Hong Kong, Hong Kong, China. Gallego, G., G. van Ryzin Otimal dynamic ricing of inventories with stochastic demand over finite horizons. Management Sci. 40(8) Gīhman, I., A. Skorohod Controlled Stochastic Processes. Sringer Verlag, New York. Gnedenko, B The Theory of Probability. Chelsea Publishing Co., New York. Kahneman, D., A. Tversky Prosect theory: An analysis of decision under risk. Econometrica 47(2) Lee, E., L. Markus Foundations of Otimal Control Theory. John Wiley & Sons Inc., New York. Levin, Y., J. McGill, M. Nediak Price guarantees in revenue management and dynamic ricing. Oer. Res. 55(1) Lim, A., G. Shanthikumar Relative entroy, exonential utility, and robust dynamic ricing. Oer. Res. 55(2) Martínezde Albéniz, V., D. SimchiLevi Meanvariance tradeoffs in suly contracts. Naval Res. Logist. 53(7) McGill, J., G. van Ryzin Revenue management: Research overview and rosects. Transortation Sci. 33(2) Mohler, R Bilinear Control Processes with Alications to Engineering, Ecology, and Medicine. Mathematics in Science and Engineering, Vol Academic Press, New York/London, UK. Phillis, R Pricing and Revenue Otimization. Stanford University Press, Stanford, CA. Phillis, R Personal communication. Nomis Solutions, CA. Secomandi, N., K. Abbott, T. Atan, E. A. Boyd From revenue management concets to software systems. Interfaces 32(2) Simon, H Rational choice and the structure of the environment. Psych. Rev. 63(2) Talluri, K., G. van Ryzin The Theory and Practice of Revenue Management. Kluwer, Boston, MA. Vinter, R Otimal Control. Systems & Control: Foundations & Alications. Birkhäuser, Boston, MA. Weatherford, L., S. Bodily A taxonomy and research overview of erishableasset revenue management: Yield management, overbooking, and ricing. Oer. Res Yano, C., S. Gilbert Coordinated ricing and roduction/ rocurement decisions: A review. A. Chakravarty, J. Eliashberg, eds. Managing Business Interfaces: Marketing, Engineering Issues and Manufacturing Persectives. Kluwer Academic Publishers, Norwell, MA, Zhao, W., Y. Zheng Otimal dynamic ricing for erishable assets with nonhomogeneous demand. Management Sci. 46(3)
19 OPERATIONS RESEARCH doi /ore ec. ec1 ec4 ecomanion ONLY AVAILABLE IN ELECTRONIC FORM informs 2007 INFORMS Electronic Comanion Risk in Revenue Management and Dynamic Pricing by Yuri Levin, Jeff McGill, and Mikhail Nediak, Oerations Research 2007, doi /ore Online Aendix: Mathematical Proofs To establish notation, we first briefly resent a standard statement of the wellknown Pontryagin maximum rincile for a minimization roblem Pontryagin Maximum Princile For a generic otimal control roblem on the fixed interval T 0 min g x 0 dx s.t. = f t x t u t (EC2) x T = x 0 u t U t (EC1) (EC3) (EC4) one introduces adjoint variables y for equations (EC2), and a Hamiltonian function H t x y u = y T f t x u The maximum rincile claims that for an otimal x u there exists y and a constant 0 1 not both identically equal to zero such that y satisfies the adjoint system of differential equations dy = H t x t y t u t x with the boundary condition y 0 = g x x 0 u t delivers the maximum of H t x t y t u with resect to u U t almost everywhere on T Proof of Proosition 2 The above roosition is formulated for a minimization roblem, so one needs aroriate adjustments to be able to use the maximum rincile. We simly select g P = rp n r + C P n r n r n r r<z To aly the maximum rincile to our roblem, we introduce the adjoint variables n r t for each of the evolution equations in (11). The Hamiltonian function has the form H t P = [ n r t n r P n r + t n+1 r P n+1 r ] n r n+1 r Note that H is linear in. The term corresonding to n r in H is of the form t P n r n 1 r+ n r n r Moreover, we can rewrite the Hamiltonian as H t P = P n r t n 1 r+ n r n r n r ec1
20 ec2 Levin, McGill, and Nediak: Risk in Revenue Management and Dynamic Pricing. ec1 ec4; sul. to Oer. Res., doi /ore , 2007 INFORMS and observe that the roblem of maximizing H with resect to for a given and P searates into comletely indeendent linear subroblems of the form (16) (17) for each n r. The resulting adjoint system has the form (13) (14) and the terminal condition is n r 0 = r CI r < z n r We observe that = 0 would imly = 0, which would contradict the requirement of the maximum rincile that both and are not identically zero. We can select, without loss of generality, = 1 and get (15) Proof of Lemma 1 We first assert that the following function is constant along the trajectories of P satisfying the system formed by (11) and (13) (14), and arbitrary boundary conditions (that is, not only (7) (8) and (15)): G P = Indeed, due to bilinearity of H in P we have n r P n r n r H t P = H n r = n r n r n r By construction of H and the definition of the adjoint system, we have P n r H P n r dp n r = H n r and d n r = H P n r Therefore, dg P t t = n r [ dp n r ] d n r n r + P n r = [ H n r n r P n r n r = H t P H t P = 0 H P n r ] We conclude that G P t t is constant as a function of t. Consider now G P t t evaluated for t satisfying terminal conditions (15), and P t satisfying initial conditions P n r t = 0 P n r t = 1 n n r r corresonding to the system being in state n r at time t with robability one. Then n r t = G P t t = G P T T = n r r CI r < z P n r T = E R T CI R T < z Y t = n R t = r 1.4. Proof of Proosition 7 The otimality condition (33) can be rewritten as ( r 1 + C f z ) = C f r 2 2 ( ) z 1 Note that if z then the righthandside of this equation is nonositive for all. The lefthandside is nonositive for. Thus, the otimal solution that solves the equation must satisfy. On the other hand, if z< then, by continuity of and its strict increasing (decreasing) behaviour for < (> ), there exist 1 and 2 such z = 1 = 2 and 1 < < 2. Note that the righthandside is ositive for 1 << 2 and negative for > 2 and < 1. The lefthandside is ositive for < and negative for >. Thus, we either have > 2 or 1 < <.
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