A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders

Transcription

1 A Robust Optimization Approah to Dynami Priing and Inventory Control with no Bakorders Elodie Adida and Georgia Perakis July 24 revised July 25 Abstrat In this paper, we present a robust optimization formulation for dealing with demand unertainty in a dynami priing and inventory ontrol problem for a make-to-stok manufaturing system. We onsider a multi-produt apaitated, dynami setting. We introdue a demand-based fluid model where the demand is a linear funtion of the prie, the inventory ost is linear, the prodution ost is an inreasing stritly onvex funtion of the prodution rate and all oeffiients are time-dependent. A key part of the model is that no bakorders are allowed. We show that the robust formulation is of the same order of omplexity as the nominal problem and demonstrate how to adapt the nominal deterministi solution algorithm to the robust problem. Operations Researh Center, MIT, eadida@mit.edu Sloan Shool of Management, MIT, georgiap@mit.edu 1

2 1 Introdution 1.1 Motivation Fluid models provide a powerful tool for understanding the behavior of systems where the dynami aspet plays an important role. In reent years, there has been a lot of researh in an attempt to provide a deeper understanding of fluid models from a theoretial as well as an appliation point of view. In partiular, an attrative feature of these models is that they provide good sheduling, prodution and inventory poliies in a variety of settings. Furthermore, they approximate well the underlying stohastiity of problems in a deterministi way. Fluid models arise in appliations as diverse as routing, ommuniation, queueing, supply hain and transportation systems. The overall goal of this researh is to introdue and study a robust optimization model and its appliation to dynami priing and inventory ontrol. In [1] we study the problem when demand is deterministi and introdue a solution algorithm for omputing the optimal priing and prodution poliy over the time horizon for all produts. In this paper we introdue demand unertainty and use ideas from robust optimization in order to obtain the robust ounterpart of the nominal problem. Furthermore, we provide a solution algorithm for solving the robust formulation and illustrate it is of the same diffiulty as the nominal problem. To the best of our knowledge, this is the first paper introduing ideas from robust optimization in a fluid model for dynami priing. 1.2 Some Related Literature and Contributions In this setion, we will briefly review some literature on fluid models, demand unertainty and the use of robust optimization to address it. The reader should refer to [1] for additional referenes on related literature on dynami priing, inventory management and the maximum priniple for optimality. A large part of the literature has foused on the solution of linear fluid models see for example Anderson [2],[3], [4], Pullan [23], [24], Tyndall [28]. This part of the literature shows existene of an optimal solution with pieewise onstant ontrols. Pullan in partiular showed strong duality and designed a lass of algorithms. However, when fluid models are nonlinear, the dynami together with the nonlinear aspet of the problem make them harder to analyze. Nonlinear fluid models are partiularly useful for dynami priing and inventory management in supply hains. Examples of supply hain industries where fluid models of the type we disuss in this paper are relevant, inlude industries with a high volume of throughput and data on osts and demand that hange a lot. The hardware as well as the semiondutor industries are suh examples. The key ontribution of this paper is addressing demand unertainty through robust optimization. The problem of demand unertainty has motivated a signifiant amount of literature in the field of Revenue Management and Priing. A number of different approahes have been introdued to model this unertainty. Zabel [31] onsiders two models of unertain demand: a multipliative model d t = η t up t and an additive model d t = up t + η t, where d t is the demand at time t, p t is the prie at time t, up = a a b p, i.e. a downward sloping linear demand urve, and η t is assumed to be either exponentially or uniformly distributed with E[η t ] >. Young [3] and Federgruen and Hehing [18] generalize the demand model to be of the form d t = γ t pɛ t +δ t p, where γ and δ have first derivatives non positive and ɛ t is a random term with a finite mean. Gallego and van Ryzin [19], [2] as well as Bitran and Mondshein [15] assume that demand follows a Poisson proess with a deterministi intensity that depends on prie and time. Raman and Chatterjee [25] model the stohastiity of the demand by introduing an additive model where the random noise is a ontinuous time Wiener proess. For additional details and referenes, see review papers suh as [14], [17] and [29]. Robust optimization seeks an optimal solution of a problem when its data is unertain. A robust optimization formulation was first onsidered by Soyster [27] in the ase of a linear optimization problem where the data were unertain within a onvex set. He addresses unertainty by taking a worst-ase approah. Nevertheless, suh an approah dereases the performane of the solution signifiantly. Ben-Tal 2

3 and Nemirovski [9] studied robust onvex optimization. They show that some polynomial-time algorithms allow to effiiently solve exatly or approximatively some of these problems. In [1] they show that the robust ounterpart of a linear program with data within ellipsoidal unertainty sets is a oni quadrati program whih is solvable in polynomial time. Numerial examples of this approah an be found in [11]. El-Ghaoui et al [8], [16] onsider semidefinite robust optimization problems. Bertsimas and Sim [12] studied the tradeoff between robustness of a solution to a linear programming problem and the sub-optimality of the solution. Bertsimas and Thiele [13] apply robust optimization priniples to supply hain management. The ontributions of this paper are the following: 1. We introdue robust optimization ideas to a fluid model. To the best of our knowledge, this is the first paper introduing ideas of robust optimization in a fluid model setting. 2. We study a ontinuous time model for a joint priing and inventory ontrol problem that allows no bakorders and inorporates demand unertainty. We use ideas from robust optimization in order to introdue a model of demand unertainty without assuming any probability distribution on the random omponent of the demand. 3. We reformulate the robust optimization model as a deterministi model of the same order of omplexity as the nominal problem. 4. We demonstrate how to adapt the solution algorithm, introdued in [1] for the nominal problem, to the robust problem and show it is of the same order of diffiulty. 1.3 Struture of the Paper In Setion 2, we review the nominal problem and desribe a model of demand unertainty. In Setion 3, we present the robust formulation and the demand model. In Setion 4, we derive an equivalent formulation for the robust ounterpart problem. In Setion 5, we briefly explain how to adapt the algorithm desribed in [1] in order to solve the robust ounterpart. We give some numerial results in Setion 6, and we onlude in Setion 7. 2 Notations and Definitions Inputs [, T ]: time horizon; N: number of produts; Kt: shared prodution apaity rate at time t non negative; Ii : initial non negative inventory level for produt i; h i t: holding ost of one unit of produt i at time t; f i.: α i t, β i t: prodution ost funtion for produt i with respet to the prodution rate; oeffiients used for produt i at time t in the linear relationship between prie and demand: d i t = α i t β i tp i t. Outputs p i t: prie of one unit of produt i at time t ontrol variable; u i t: prodution flow rate of produt i at time t ontrol variable; I i t: inventory level number of units of produt i at time t state variable. Definitions We denote by I., p., u., α., β. the vetors with respetive omponents I i., p i., u i., α i., β i., i = 1,..., N. 3

4 I., p., u. will denote the optimal solution. We define: Constrained resp. unonstrained interval: interval of time where the inventory level equals zero resp. is positive; Constrained resp. unonstrained produt: produt belonging to a onstrained resp. unonstrained interval; Ative resp. inative produt: produt with a positive resp. equal to zero prodution rate. We notie that for any priing and prodution poliy and in partiular for an optimal poliy, the inventory level will lie in a sequene of intervals, where the inventory level is suessively positive and equal to zero. A onstrained interval starts at an entry time and finishes at an exit time, i.e. the time the inventory level beomes again positive. It I entry time exit time entry time τ t t unonstrained onstrained unonstrained onstrained interval interval interval interval Figure 1: Example of optimal inventory level evolution T t 3 The Nominal Problem and a Demand Unertainty Model We onsider the nominal problem P max s.t. p i tα i t β i tp i t f i u i t h i ti i t dt Ii t = u i t α i t + β i tp i t, I i t, t [, T ] i = 1,..., N p i t α it β i t u i t, p i t, t [, T ] i = 1,..., N u i t Kt, t [, T ] I i = I i, i = 1,..., N. t [, T ] i = 1,..., N We introdue an additive model of demand unertainty as follows: d i t = α i t β i tp i t + ɛ i t, 4

5 where ɛ i t is the unertain parameter. This model essentially assumes an additive demand model, i.e. that there is unertainty on the demand parameters α i., i = 1,..., N. In partiular, we model this unertainty as follows. We denote α i. the nominal funtion, α i. the realization, and we suppose that the realization belongs in an interval entered around the nominal funtion with half-length ˆα i.. In order to model the fat that the realization is unlikely to be at its worst-ase value i.e. the furthest away from the nominal value all the time, we introdue a budget of unertainty funtion Γ i. taking values in [, T ]. We assume that funtion Γ i. is an inreasing funtion of time. This funtion allows us to adjust the tradeoff between the level of onservatism sought for the robust solution and its performane. We assume it is non-dereasing in order to allow the aggregate error over time to inrease. Moreover, we assume that Γ i t 1 i, t, in order to ensure that the funtion does not grow faster than new variables are added. We suppose that and write the onstraints as follows: < ˆα i. < 1 2 α i. i α i t α i t ˆα i t t, i α i s α i s ds Γ i t t, i. ˆα i s Denoting z i t α it α i t ˆα i t the saled variation of parameter α i t, it follows that Clearly, we have z i t [ 1, 1] t, i z i s ds Γ i t t, i. 1 z i s ds t. This implies that if for a time t, Γ i t t, then onstraint 1 is unneessary as it always satisfied in that ase due to the bounds imposed on z i. by definition. As a result, in this ase at the optimal solution of the robust optimization problem, z i. will be equal to 1 on [, t] as this orresponds to the worst ase senario on that interval and it is allowed through the budget of unertainty onstraint. In partiular, the exat value of Γ i t if it is greater than or equal to t does not matter. This disussion motivates us to introdue the notion of the effetive budget of unertainty through min{t, Γ i t}. Furthermore, in order to measure the global unertainty of the problem, that is, in order to introdue a single quantity used as a metri representative of the overall budget of unertainty, we introdue the umulative effetive budget of unertainty defined by min{t, Γ it}dt. An unertainty set F is the set of vetors α = α 1.,..., α N. satisfying the onstraints above. Therefore, the budget of unertainty prevents a given produt i to have a parameter α i very different from its nominal value for an exessive part of the time horizon. However, we do not impose a budget onstraint aross produts, i.e. in terms of number of produts that may have simultaneously a demand parameter different from its nominal value. 2 5

6 4 A Robust Optimization Approah 4.1 A General Robust Optimization Model In general, a robust optimization formulation seeks a solution that is feasible for any realization of the data within the unertainty set and maximizes the realized objetive funtion. In the ase of problem P, the robust ounterpart we desribed an be written as max p,u,γ s.t. α F, γ γ p i α i β i p i f i u i h i Ĩ i dt 3 Ĩ i = u i α i + β i p i, t [, T ] i = 1,..., N Ĩ i, t [, T ] i = 1,..., N p i α i, t [, T ] i = 1,..., N β i u i, p i, t [, T ] i = 1,..., N u i K, t [, T ] Ĩ i = I i, i = 1,..., N. We observe that we may write the inventory level at time t as follows: where Ĩ i t = I i + = I i t u i s α i s z i sˆα i s + β i sp i sds I i t = I i + z i sˆα i sds u i s α i s + β i sp i sds is the nominal inventory level. The onstraints must be satisfied for all z. suh that for all i and t, where α i t = α i t + z i tˆα i t. 1 z i t 1 z i s ds Γ i t In order to reformulate this problem, we need, for eah onstraint where unertainty in involved, to determine the realization of α, that is, the worst-ase senario for example for onstraint 3, the realization that minimizes the right hand side. Then we will be guaranteed that the onstraint is satisfied for any realization. We may rewrite onstraint 3 as follows: γ p i α i β i p i f i u i h i I i + p i ˆα i z i + h i γ C + p i tˆα i tz i t + h i t z i sˆα i sds dt z i ˆα i ds dt 6

7 for all z. suh that for all i and t, where C = 1 z i t 1 z i s ds Γ i t, p i α i β i p i f i u i h i I i dt orresponds to the nominal objetive funtion and is independent of z. We seek the feasible realization of z that minimizes the right-hand side in the above inequality. We notie that we may onsider eah produt separately; in other words we minimize eah term of the sum aross produts. We thus need to find, for all i, the feasible realization of z i that solves min p i tˆα i tz i t + h i t z i z i sˆα i sds dt. Sine p i, ˆα i and h i are positive valued, it is easy to see that in the optimal solution, z i. Therefore, after transformation of variables, we may rewrite the onstraints on the new variable z i as follows z i t 1 t z i sds Γ i t t note that we abuse notations to avoid onfusion and rewrite the minimization problem as min z i max z i max z i max z i max z i max z i p i tˆα i tz i t + h i t p i tˆα i tz i tdt + p i tˆα i tz i tdt + p i tˆα i tz i tdt + p i tˆα i tz i tdt + where H i t h t i sds. Therefore, we obtain the equivalent subproblem max z i t= s= s= h i t p i t + H i tz i tˆα i tdt, s.t. s= t=s z i sˆα i sds dt z i sˆα i sds dt h i tz i sˆα i sds dt h i tz i sˆα i sdt ds H i sz i sˆα i sds p i t + H i tz i tˆα i tdt z i t 1 t z i sds Γ i t t. We notie at this point that this subproblem depends on the ontrol variable p i., and therefore, annot be diretly solved. This dependeny omes from the fat that the revenue term is non linear in both the 7

8 ontrol variable p i and the unertain parameter α i. We ould write the dual of this separated ontinuous linear program and inlude it as part of the overall robust ounterpart formulation, but its solution annot be derived independently from the rest of the problem. Consequently, the omplexity of the resulting robust formulation inreases and the new formulation is diffiult to solve. The new robust optimization problem has now a higher order of omplexity than the nominal problem. We notie that if priing was not a deision, or if demand was external i.e., not depending on priing, the general model of unertainty i.e. with realized revenues would yield a tratable formulation see for example [13] for a disretized version of the problem. This disussion motivates the model we introdue and study in the remainder of this paper. 4.2 A Simplified Robust Optimization Model The remarks in the previous setion motivate a slightly different formulation when defining the robust optimization model we onsider. For reasons of tratability, in the remainder of the paper, we onsider a robust approah that maximizes an objetive funtion onsisting of nominal revenues minus realized osts. In other words, we will take the mean value in the revenues term i.e., prie times demand of the objetive funtion. However, we still onsider demand unertainty in the ost terms of the objetive funtion i.e., we onsider realized osts and in the feasibility onstraints. That is, we onsider expeted, i.e. nominal revenues not in the probabilisti sense, but at the enter of the interval of variation. This modelling simplifiation allows us to simplify the approah and make the model tratable. As a side remark, later in the paper, we will show that a model onsidering the nominal ost as opposed to the realized ost in the objetive funtion, would lead to the same formulation. For this model, the robust ounterpart of problem P an be written as follows we omit the time argument: max p,u,γ s.t. α F, γ γ p i α i β i p i f i u i h i Ĩ i dt 4 Ĩ i = u i α i + β i p i, t [, T ] i = 1,..., N 5 Ĩ i, t [, T ] i = 1,..., N 6 p i α i, t [, T ] i = 1,..., N 7 β i u i, p i, t [, T ] i = 1,..., N u i K, t [, T ] Ĩ i = I i, i = 1,..., N. 8 We observe that onstraints 4, 6 and 7 are the ones where unertainty has an impat, equation 5, along with the initial onditions 8, simply define the state variable Ĩ. Constraints 4 and 6 link time instants by involving inventory levels. Notie that these depend on all previous ontrol deisions. In the ontrast, onstraint 7 is separable aross time. This will have an impat in the way we derive the robust ounterpart in the following sense: a onstraint that does not link together time instants needs to be satisfied at eah time for the worst realization, while for a onstraint that links time, the worst realization may not our at eah time beause of the budget onstraint involving Γ i. We will proeed, similarly to the previous setion, by determining for eah onstraint the worst ase realization of the unertain parameter α in order to reformulate the robust problem. 8

9 4.3 An Equivalent Robust Formulation We start by onsidering onstraint 7 for a given produt i and at a given time t. Clearly, the worst ase is obtained when the numerator is the smallest, i.e. z i t = 1. It may be seen that for any given time t and index i, it is possible to find a vetor of funtions z suh that z i t = 1, and α F. As a result, in the robust ounterpart, onstraint 7 is written as p i t α it ˆα i t β i t i, t. In onstraint 6, we seek for the realization of z i that minimizes Ĩit, or equivalently, that minimizes z i sˆα i sds. Clearly in the optimal solution, z i, so we an rewrite this subproblem as follows, for eah produt i and at any given time t: min z i s.t. z i sˆα i sds z i sds Γ i t z i s 1 s [, t], where the deision variable is the funtion z i over [, t]. Equivalently, max z i s.t. z i sˆα i sds z i sds Γ i t z i s 1 s [, t]. Note that t is fixed. For a given t, we want to find the realization of z i on [, t] whih makes Ĩit the smallest. This is a partiular instane of a ontinuous linear program. This lass of problems was introdued by Bellman [6], [7] to model some eonomi proesses. A dual formulation for this lass of problems was studied by Tyndall [28]. Some results by Tyndall also establish strong duality under some regularity assumptions on the data of the problem. Using these results, we obtain strong duality with the dual problem given by: or equivalently min ω i t,r i.,t max ω i t,r i.,t ω i tγ i t + r i s, tds s.t. ω i t + r i s, t ˆα i s s [, t] ω i t r i s, t s [, t] ω i tγ i t s.t. ω i t + r i s, t ˆα i s s [, t] ω i t r i s, t s [, t]. r i s, tds 9 9

10 We notie that in this ase the primal and thus the dual subproblem takes as inputs only the known parameters Γ i. and ˆα i.. Let s now rewrite onstraint 4 omitting the time argument to ease the reading: γ p i α i β i p i f i u i h i I i + h i γ C + h i t z i sˆα i sds dt z i ˆα i ds dt for all z. suh that for all i and t, 1 z i t 1 z i s ds Γ i t, where C = p i α i β i p i f i u i h i I i dt. We seek the feasible realization of z that minimizes the right-hand side in the above inequality. It follows that the onstraint will be satisfied for any feasible realization. As previously, we separate this problem aross produts. We thus need to find, for all i, the feasible realization of z i that solves min h i t z i z i sˆα i sds dt. It is easy to see that in the optimal solution, z i. Therefore, after a variable transformation, we may rewrite the onstraints on the new variable z i as follows z i t 1 t z i sds Γ i t t and rewrite the minimization problem as min z i max z i h i t H i tz i tˆα i tdt, z i sˆα i sds dt where H i t h t i sds, by using the same tehnique as in the previous setion. Therefore, we obtain the equivalent subproblem max z i s.t. H i tz i tˆα i tdt z i t 1 t z i sds Γ i t t. 1

11 This is an instane of a separated ontinuous linear program. This lass of problems was introdued by Anderson [2] to study a job shop sheduling problem. Pullan [24] studied duality properties for these problems. Using these results, we an write the dual as: max Π i,q i s.t. Γ i tdπ i t q i tdt 1 Π i t + q i t H i tˆα i t t q i t t Π i T = Π i non-dereasing. Pullan showed that if the funtions H i.ˆα i. and Γ i. are pieewise analyti, for the optimal solutions z i lying in the spae of measurable bounded funtions, strong duality holds. In what follows, we will assume that these assumptions hold. This model is interesting and useful in our analysis sine both the primal and the dual problems are independent of the prie i.e. independent of the ontrol variable of the initial problem. This remark will allow us to greatly simplify the robust ounterpart problem. In the robust problem, strong duality allows us to replae the minimization problems primal subproblems by their respetive dual maximization subproblems for eah onstraint. Moreover, at the optimum, the maximum will be realized, therefore, we an simply replae the maximization subproblems by their objetive funtions and integrate the onstraints on the dual variables into the onstraints of the robust ounterpart. Therefore, after moving the first onstraint bak to the objetive funtion, we obtain the following. Theorem 1. The robust optimization formulation for problem P is: max p,u,π,q,ω,r s.t. p i tα i t β i tp i t f i u i t h i ti i t q i t dt Γ i tdπ i t i i Ii t = u i t α i t + β i tp i t i t [, T ] I i t ω i tγ i t + r i s, tds i t [, T ] Π i t + q i t H i tˆα i t i t [, T ] ω i t + r i s, t ˆα i s i s [, t] t [, T ] ω i t i t [, T ] r i s, t i s [, t] t [, T ] q i t i t [, T ] Π i T = i Π i non-dereasing i p i t α it ˆα i t i t [, T ] β i t p i t, u i t i t [, T ] u i t Kt t [, T ] I i = I i H i t = t i h i sds i t [, T ] 11 11

12 4.4 Simplifiation of the Robust Formulation We notie that ˆα i, h i. and Γ i., i = 1..., N are data, therefore, eah of the two dual subproblems 9, 1 takes only known parameters as inputs. Let s suppose we have at our disposal an effiient algorithm for solving ontinuous linear programs and therefore, we are able to solve these dual subproblems. Let s denote ω, r the optimal solutions of problem 9. We observe that when we plug the optimal value of problem 1 into the robust formulation 11, it appears as a onstant in the objetive funtion. Moreover, the onstraints of the dual subproblem 1 are no longer neessary sine they are independent of the remaining state and ontrol variables. Remark: For this reason, an approah maximizing the nominal objetive funtion, i.e. onsidering both revenues and osts as nominal, but preserving the demand unertainty impat on the inventory level in the no-bakorder onstraint, would lead to the same robust optimization formulation. As a result, we obtain the following robust optimization formulation: max p,u s.t. [ i ] p i tα i t β i tp i t f i u i t h i ti i t dt 12 Ii t = u i t α i t + β i tp i t i t [, T ] 13 I i t J i t i t [, T ] 14 p i t α it ˆα i t i t [, T ] 15 β i t p i t, u i t i t [, T ] u i t Kt t [, T ] I i = I i i J i t = ω i tγ i t + r i s, tds i t [, T ]. In this formulation, the unertainty of demand has an effet only on the no bakorder onstraint and the upper limit on pries. This makes sense intuitively sine when the inventory osts are unertain, given no additional information, it seems sensible for the system to optimize using as objetive the expeted ost, i.e. the osts obtained using the nominal demand. The unertainty of demand translates into protetion levels for the pries and the inventory levels see 14, 15 that are stronger than in the nominal ase. That is, protetion levels ensure that the inventory remains above level J i >, and pries below the limit αit ˆαit β i t < αit β i t. As a result, even with some variation in the demand - within the introdued unertainty onstraints - the inventory level will remain positive, and pries will remain below their upper bound. These protetion levels depend on the budget of unertainty Γ and on the length ˆα of the interval of variation for the demand parameter α. They are determined through the solution of the dual subproblem 9. The following theorem follows after a hange of variable Īi = I i J i. Theorem 2. Assuming that ri is pieewise differentiable with respet to its seond argument, and that and Γ i are pieewise differentiable, we obtain the following equivalent robust formulation for problem ω i 12

13 P : max p,u [ i ] p i tα i t β i tp i t f i u i t h i tīit dt s.t. Ī i t = u i t α i t + β i tp i t D i t i t [, T ] Ī i t i t [, T ] p i t α it ˆα i t i t [, T ] β i t p i t, u i t i t [, T ] u i t Kt t [, T ] Ī i = Ī i i where Ī i = I i ω i Γ i and D i t = ω i tγ i t + ωi t Γ i t + ri ri t, t + s, tds. t This problem is very similar to the original problem, in terms of type and number of onstraints and variables. However, it differs in the fat that we annot introdue a parameter ᾱ i t suh that we have: I i t = u i t ᾱ i t + β i tp i t be the evolution of inventory levels ᾱit β i t be the upper limit of the prie p it the revenue term of the objetive funtion be of the form p i tᾱ i t β i tp i t. Therefore, a straightforward appliation of the algorithm we propose in [1] is not possible. In what follows we propose a modified algorithm solving the nominal problem that will also solve this new robust optimization reformulation. The disussion that follows also allows us to illustrate that solving the robust optimization model is of the same diffiulty as the nominal one. 4.5 Preliminary Result Proposition 1. The funtion J i. is non dereasing, i.e. the funtion D i. is non negative for all i. Proof. We onsider the dual problems P t and P t+dt respetively at times t and t + dt: max ω it,r i.,t ω i tγ i t r i s, tds s.t. ω i t + r i s, t ˆα i s s [, t] ω i t r i s, t s [, t], max ω i t+dt,r i.,t+dt s.t. ω i t + dtγ i t + dt +dt r i s, t + dtds ω i t + dt + r i s, t + dt ˆα i s s [, t + dt] ω i t + dt r i s, t + dt s [, t + dt]. 13

14 We denote by ωi t, r i., t, ω i t + dt, r i., t + dt the respetive optimal solutions. It is lear that ωi t + dt, r i., t + dt is feasible for P t, therefore, we have or equivalently ω i tγ i t As a result, we observe that ω i t + dtγ i t + r i s, tds ω i t + dtγ i t J i t + dt = ω i t + dtγ i t + dt + r i s, t + dtds ω i tγ i t + +dt r i s, t + dtds = ω i t + dtγ i t + ω i t + dt Γ i tdt + ω i tγ i t + = J i t + ω i t + dt Γ i tdt + r i s, tds + ω i t + dt Γ i tdt + +dt t r i s, t + dtds r i s, t + dtds + r i s, t + dtds. +dt t r i s, tds. +dt t r i s, t + dtds r i s, t + dtds Sine Γ i t by assumption, and for feasibility of P t+dt, ωi t+dt, r i s, t+dt s [t, t+dt], we obtain that J i t + dt J i t. Next we show that the robust optimization formulation an be solved using a method whih is of the same diffiulty as the method in [1] for solving the original nominal problem. 5 Solution of the Robust Formulation Consider the following problem: [ max p,u s.t. i ] p i tα i t β i tp i t f i u i t h i ti i t dt Ii t = u i t α i t + β i tp i t D i t i t [, T ] I i t i t [, T ] p i t α it ˆα i t i t [, T ] β i t p i t, u i t i t [, T ] u i t Kt t [, T ] I i = Ī i i where ˆα i t 1 2 α it, D i t i, t are assumed to be known. Assumption 1. For all produts i, α i t, ˆα i t, β i t, h i t, D i t as well as Kt are positive, ontinuous funtions of time. Furthermore, ˆα i t 1 2 α it i, t. 14

15 Assumption 2. For produt i, i = 1,..., N, funtion f i is stritly onvex, non-negative and inreasing. Assumption 3. f i < α it β it, i = 1,..., N t [, T ] and thus f i < α it+2d i t β it, i = 1,..., N t [, T ]. Assumption 4. The following inequality holds at all times t ˆα i t + D i t Kt. i s.t. > α i t 2 ˆα i t f β i t i ˆαit+Dit This last assumption ensure that the prodution apaity level if suffiiently large to guarantee that the minimum inventory level onstraints an be satisfied, i.e. that there exists a feasible solution to this problem. In order to solve this problem, we will use ideas from ontinuous dynami programming and the Maximum Priniple. More speifially, we first write the Lagrangian and the neessary onditions for optimality involving the deision variables, Lagrange multipliers, and adjoint variables. Then we derive a method to solve simultaneously all the neessary onditions, i.e. determine the multipliers, adjoint variables, and deision variables that solve the system of onditions. We do so by first writing the optimal poliy as a funtion of the multipliers and adjoint variables. Then we assume that some harateristis of the system are observable and determine the optimal solution under these assumptions. Finally, we relax this assumption. Appendix A provides the details of this algorithm. For the sake of brevity, we do not give the details of the proof that this algorithm finds the optimal solution, but the derivation is very similar to [1]. One differene is that onstrained produts never idle in the robust problem, while in the nominal problem they either a idle or b are produed in order to satisfy the demand. We do reognize two possible states however, one a in whih they are produed in order to keep the inventory level at zero and allow for demand unertainty the rate depends on D i and ˆα i only, and a seond one b where they are produed in order to satisfy the demand with unertainty. This disussion leads to onlude that the new algorithm for solving the robust optimization formulation is of the same order of omplexity as the algorithm for solving the nominal problem. It should be noted that onsidering a ontinuous time system makes the problem solution signifiantly more omplex than in a disrete time setting, in whih the problem would be a quadrati program under linear onstraints. For suh a model there exists readily available software for determining a solution. A ontinuous time approah has the advantage of not introduing any approximation to the real setting: it provides the exat solution of the system. When taking a disrete time approah, one has to deide what a reasonable time step should be, and to allow pries hanges and prodution only at those time, while in reality a supplier may have more flexibility. In order to avoid being too restritive, the time step needs be very small, and if the time horizon is large the size of the problem may beome exeedingly large. Moreover, we believe that a similar approah an be applied to problems in areas other than dynami priing and inventory ontrol, where the evolution of the system evolves dynamially and justify a ontinuous time approah. We believe that the tehniques presented here may be helpful to those areas as well. 6 Numerial Results 6.1 Choie of Parameters In this setion, we onsider a numerial example for two produts on a time horizon [, 1] that is similar to the example we onsidered in [1]. In this paper we also introdue demand unertainty. Our goal is to understand the relationship between the optimal objetive value and the budget of unertainty Γ i.. As a result, we will onsider only one demand senario and demand unertainty model, and a apaity level that is onstant at 75% of the maximum of the umulative prodution rate ahieved in the nominal 15

16 β h γ I αt ˆαt produt t t 2.1t+.2 produt t t 2.1t+.2 Table 1: Data hosen as input in the numerial implementation senario 1 senario 2 senario 3 senario 4 senario 5 senario 6 Γ i t, i = 1, t 1+.5t 1+.2t.5+.8t.5+.5t.5+.2t min{t, Γ it}dt Table 2: Multiple senarios of budget of unertainty ase under not apaity onstraint: Kt = This guarantees that the apaity onstraint is tight for most of the time horizon. We onsider the same prodution ost struture where the ost f i. is a quadrati funtion of the prodution rate. ˆα i. whih represents the half-length of the allowed range for parameter α i., must satisfy < ˆα i. < 1 2 α i.. For ease of omputations in this example, we onsider input parameters ˆα i. that are linear funtions of the time nevertheless, the linearity assumption is not neessary: ˆα i t = a i t + b i, where a i, b i. Indeed, it is reasonable to onsider that in pratie, the auray of a foreast for the demand is non inreasing on the time horizon, i.e. that the length of the interval of feasible outomes is non dereasing, hene a i. We hoose a i, b i, i = 1, 2 suh that, the unertainty on α i is rather small initially, when the foreast should be rather aurate, and is about 4% of its nominal value at the end of the time horizon. The input data are summarized in the following table. Notie that f i u i = γi 2 u2 i. The hoie of Γ i. satisfies Γ i t 1. In this example, we first onsider these input parameters Γ i. to be linear funtions of the time: Γ i t = g i t + i, where g i, i, g i < 1. Indeed, it an be seen that as soon as the graph of Γ i. is above the 45 line with the horizontal axis, its atual value does not matter and z i t takes the value 1, meaning that the realized α i lies at the extreme of its allowed range worst ase senario. To avoid this from happening on muh of the time horizon, we hoose g i < 1. In order to study the effet of the budget unertainty on the optimal objetive value i.e. performane, we will onsider multiple senario in whih only the parameter Γ i. varies, and in whih it varies in the following way: on the one hand we let the value at time hange but keep a onstant slope, and on the other hand we keep a onstant slope, and hange the value at time. In all senarios we assign the same budget of unertainty to both produts. We will ompute the umulative effetive budget of unertainty min{t, Γ it}dt as a measure of the global unertainty in eah senario. 6.2 Closed-Form Solution for the Sub-Problem 9 In order to implement our algorithm, we need to ompute the derivative of the inventory safety level D i. and the modified initial inventory level Ī i. The reader an refer to Appendix B for a derivations 16

17 6 nominal value range limit Funtion alpha: nominal value and range of unertainty alpha i t Figure 2: Demand unertainty of the following results. and Ī i = I i D i t = { at + b if t < a1 ggt + + gat + b if < t T. It is also easy to verify that in all senarios we have at all times whih implies that Assumption 5 holds. 6.3 Results ˆα i t + D i t < Kt, 2 Using these inputs, we run the algorithm and obtain the prodution rates, pries, and inventory levels under the optimal poliy for the robust formulation we desribed in 12. Reall that in this formulation, the inventory levels are onstrained to remain above a safety level J i t. 17

18 Figure 3: Choie of budget unertainty funtion Γ.. 15 Produt 1 inventory level for nominal demand Produt 2 inventory level for nominal demand senario 7 senario 6 senario 3 senario 5 senario 2 senario 4 senario 1 Figure 4: Optimal inventory levels over time results 18

19 3.5 Produt 1 prodution rate Produt 2 prodution rate senario 7 senario 6 senario 3 senario 5 senario 2 senario 4 senario 1 Figure 5: Optimal prodution rates over time results 5 Produt 1 prie Produt 2 prie senario 7 senario 6 senario 3 senario 5 senario 2 senario 4 senario 1 Figure 6: Optimal pries over time results 19

20 Similar to the nominal problem, the system starts by building up inventory in antiipation of the demand peak that ours in the middle of the time horizon. Then the inventory levels are kept at the minimum level - J i t in the robust ase - for the remaining time. Notie that pries have the same shape as the demand urves. We notie that in all senarios, the apaity onstraint is tight exept at the very end of the time horizon. Sine produt 2 is heaper to produe and to hold, its prodution rate is maintained at a lower value than for produt 1 as the produts are unonstrained. In that stage, the budget of unertainty has no signifiant influene on the prodution rates. This makes sense beause we saw in formulating the problem that unertainty played a role in the no bakorder onstraint and the upper bound on pries. As a result, while the inventory level is positive, the unertainty does not matter. In the onstrained phase, the safety level J i t for the inventory inreases as the budget of unertainty inreases, whih is onsistent with our interpretation that the more demand unertainty we impose to the system, the larger should be the minimum value of the inventory level guaranteeing no bakorder for any realization of the demand. We observe that in that phase, as the umulative budget of unertainty inreases, the prodution rate for produt 2 inreases more and more beause maintaining the level of inventory for that produt at J 2 t requires to produe more and more, given that prodution rate for produt 2 starts from a lower value that prodution rate for produt 1. Finally, we notie that pries slightly inrease as the umulative budget of unertainty inreases, refleting the inreasing diffiulty of satisfying all onstraints as the unertainty inreases. These numerial results suggest that the umulative effetive budget of unertainty might be a relevant metri for measuring the global unertainty. This is further onfirmed by the following omputation on the objetive value. We ompute in eah senario the optimal objetive value and the umulative effetive budget of unertainty see Table 2. Cumulative effetive Budget of unertainty budget of unertainty Objetive value Γ i t, i = 1, 2 min{t, Γ it}dt senario t senario t senario t senario t senario t senario t senario Table 3: Objetive value results Table 2 and Figure 7 seem to suggest that the umulative effetive budget of unertainty may be a reasonable way of measuring the unertainty in the problem. Notie that as the unertainty inreases, the optimal objetive value dereases. This illustrates the trade-off between optimality high optimal objetive value and robustness high level of unertainty. 7 Conlusion In this paper, we have proposed and studied a robust optimization approah for inorporating demand unertainty in a fluid model of dynami priing and inventory ontrol. Using ideas from robust optimization, we reformulate the problem as a deterministi problem of a similar form as the original nominal formulation. Our approah does not make assumptions on the probabilisti distribution of the demand 2

21 22 Objetive value as a funtion of the effetive umulative budget of unertainty 21 2 Objetive value Effetive umulative budget of unertainty Figure 7: Trade-off between robustness and performane but rather assumes an additive demand model whose unertain parameter lies in a given interval, and a budget of unertainty. Furthermore, we were able to adapt the algorithm for solving the deterministi problem and show that the adapted algorithm is no more omplex than the original one. We implemented this algorithm on a numerial example that illustrates the trade-off between robustness and performane. 8 Aknowledgements Preparation of this paper was supported, in part, by the PECASE Award DMI from the National Siene Foundation and the Singapore MIT Alliane Program. We would like to thank the editors and the referees for their insightful omments that have helped us improve the ontent and exposition of this work. Referenes [1] E. Adida and G. Perakis. A Nonlinear Fluid Model of Dynami Priing and Inventory Control with no Bakorders, submitted for publiation, Operations Researh Center, MIT, 24. [2] E. J. Anderson. A New Continuous Model for Jobshop Sheduling. International Journal of Systems Siene, 12: , [3] E. J. Anderson and P. Nash. Linear Programming in Infinite Dimensional Spaes, Wiley-intersiene, Chihester, [4] E. J. Anderson and A. B. Philpott. A Continuous-Time Network Simplex Algorithm. Networks, 19: , [5] K. J. Arrow and M. Kurz. Publi Investment, the Rate of Return, and Optimal Fisal Poliy. Baltimore: The Johns Hopkins University Press,

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23 [27] A. L. Soyster. Convex Programming with Set-Inlusive Constraints and Appliations to Inexat Linear Programming. Operations Researh, 21: , [28] W. F. Tyndall. A Duality Theorem for a Class of Continuous Linear Programming Problems. SIAM J. Appl. Math., 13: , [29] C. A. Yano and S. M. Gilbert. Coordinated Priing and Prodution/Prourement Deisions: A Review. A. Chakravarty, J. Eliashberg, Eds. Managing Business Interfaes: Marketing, Engineering and Manufaturing Perspetives. Kluwer Aademi Publishers, Boston, MA. [3] L. Young. Prie, inventory and the struture of unertain demand. New Zealand Operations Researh, 62: , [31] E. Zabel. Monopoly and unertainty. The Review of Eonomi Studies, 37:25-219, 197. A Appendix: Algorithm Input: T N time horizon; number of produts produts are indexed by i; and for all i: α i t, β i t suh that d i t = α i t β i tp i t; f i. prodution ost funtion for produt i, takes the prodution rate as argument; stritly onvex, inreasing, non-negative funtion suh that f i α it + 2D i t, β i t h i t holding ost per unit of produt i; ψ i t φ i t, η ψ i t = φ i t = suh that f 1 i ψ i t α it + β it 2 2 ψ it D i t =. suh that φ i t, η = F 1 t,i η where F t,ix = X f i αi t β i tx { 2 ψt if f i ˆα it + D i t α it 2ˆα i t β it f i ˆα it + D i t otherwise. { φ i t, ηt ηt + f i ˆα it + D i t I i initial level of inventory for produt i. if ηt αit 2ˆαit β i t otherwise. f i ˆα it + D i t + D i t ; It is important to notie that the definition of φ i t involves ηt. To make this learer we will use also the notation φ i t φ i t, ηt. Output: u i t, p i t, i = 1,..., N, t [, T ]. 23

24 Notations: Ī i ηt ρ i t ritial value of the initial inventory above whih it is optimal not to produe produt i on the entire time horizon; Lagrange multiplier for the apaity onstraint; Lagrange multiplier for the onstraint on the non negativity of I i t; q i t adjoint variable for the dynami onstraint for produt i; τ first time the apaity onstraint beomes tight γ i t binary variable equal to 1 when produt i has a positive initial inventory level lower than Ī i and is on the first unonstrained interval at time t, equal to otherwise; δ i t binary variable equal to 1 when produt i is on a onstrained interval at time t, equal to otherwise; t i first entry time on a onstrained interval for produt i; t 2 i subsequent entry time on a onstrained interval for produt i; t 1 i exit time from a onstrained interval for produt i; I set of ative onstrained produts; I set of ative unonstrained produts; Remarks: We observe that ψ i t = φ i t, ; We may derive dφ i dt t, ηt = φ i t t, ηt + ηt φ i t, ηt η Algorithm for 1 produt Initialization 1. Let 2. Let 3. Let Ht = hs ds. t { αt Dt if Ht < αt βt zt = 1 2 αt + βtht D t if Ht αt βt Ī = zs ds.. If I Ī, go to 4. If < I < Ī, go to 5. If I =, go to 9. 24

25 4. Large initial inventory level Let qt = Ht t [, T ]; ρt = t [, T ]; δt = t [, T ]; γt = 1 t [, T ]. Go to Small initial inventory level First unonstrained interval Define qt = q + hs ds αt Dt 1 2 αt + βtqt Dt yt = f 1 qt if qt < αt βt if αt βt qt min{f ; αt 2ˆαt βt } αt 2 + βt 2 qt Dt if f qt αt 2ˆαt βt if αt 2ˆαt βt qt f ˆαt Dt f 1 qt ˆαt Dt if qt > max{ αt 2ˆαt βt ; f }. Solve for q and t smallest feasible solution the following nonlinear system: ψt = qt ψt ht yt dt = I 6. Let qt as given above, γt = 1, δt = and ρt = on [, t ], Go to Next intervals Define qt = ψt 1 + hs ds t 1 αt Dt 1 2 αt + βtqt Dt ȳt = f 1 qt if qt < αt βt if αt βt qt min{f ; αt 2ˆαt βt } αt 2 + βt 2 qt Dt if f qt αt 2ˆαt βt if αt 2ˆαt βt qt f ˆαt Dt qt ˆαt Dt f 1 if qt > max{ αt 2ˆαt βt ; f }. Attempt to solve for t 1 t and t 2 > t 1 smallest feasible solutions the following nonlinear system: ψt 2 = qt 2 ψt 2 ht 2 2 t 1 ȳt dt = If there is no solution, let t 1 = t 2 = T and go to 1. If there is a solution, go to 8 25

26 8. Let q + ρt = ψt t [t, t 1 ] γt = t [t, t 1 ] δt = 1 t [t, t 1 ] ρt = t [t 1, t 2 ] qt as desribed in step 7 t [t 1, t 2 ] γt = t [t 1, t 2 ] δt = t [t 1, t 2 ] Do t t 2 ; go to Zero initial inventory level Let t = ; go to Final step Let if qt + ρt < αt βt 1 pt = 2 qt + ρt + αt βt if αt αt 2ˆαt βt qt + ρt βt αt ˆαt βt if qt + ρt > αt 2ˆαt βt t [, T ] ut = { if qt + ρt f f 1 qt + ρt if qt + ρt > f. t [, T ] Algorithm for multiple produts: Initialization 1. Do the algorithm for 1 produt above, for eah of the N produts. Output δ i, γ i, u i. Remove all produts for whih I i > Īi; update N. 2. Let { τ = min min { t : u i t = Kt } } ; T 3. If τ = T, stop. Otherwise, let S 1 {i : γ i τ = 1, δ i τ = } S 2 {i : γ i τ =, δ i τ = } S 3 {i : δ i τ = 1}. Parameters τ, q i, t i, i S 1 and the urrent t 1 i, t2 i, i S 2 S3 need to be updated simultaneously with the omputation of ηt for t τ. 4. We determine ηt along with τ, qi, t i, i S 1 and t 1 i, t2 i, i S 2 S3 where t i, t2 i solutions suh that all of the following holds: are the smallest i S 1, we have 26

27 y i t = q i t = q i + h is ds, ρ i t = t [, t i ], q i t i = φ i t i, ηt i, d φ i dt t i h it i and i y it dt = I i where α i t D i t 1 2 α i t + β i tq i t D i t q i t ηt f 1 i f 1 i if qt < α it β it if α it β i t q it min{f i + ηt; α it 2ˆα i t β i t } αit 2 + βit 2 q it D i t if f i + ηt q it αit 2ˆαit β i t q i t f i + ηt ˆα i t D i t q i t ηt ˆα i t D i t if α it 2ˆα i t β it i S 2, we have q i t = ψt 1 i + h t 1 i sds, ρ i t = on [t 1 i, t2 i ], i q i t 2 i = φ i t 2 i, ηt2 i, d φ i dt t2 i h it 2 i and 2 i y t 1 i t dt = i i S 3, we have q i t + ρ i t = ψ i t on [t i, τ], q i t + ρ i t = φ i t, ηt on [τ, t 1 i ], q i t = φ i t 1 i, ηt1 i + h t 1 i sds, i ρ i t = on [t 1 i, t2 i ] q i t 2 i = φ i t 2 i, ηt2 i, d φ i dt t2 i h it 2 i and 2 i y t 1 i t dt = i ηt = on [, τ], if q i t > max{ α it 2ˆα i t β i t ; f i + ηt}. N u it reahes Kt for the first time at time τ, where u i t = max{, f 1 i q i t + ρ i t} η is omputed by the following algorithm: a Reorder the indies from 1 to N by inreasing value of {q i τ + ρ i τ f i, i : δ iτ = } { α it 2ˆα i t β it f i ˆα it + D i t, i : δ i τ = 1}. Define q τ + ρ τ f = αt 2ˆαt β t f ˆα t + D t. b Let I = {i : δ i τ = 1, α iτ 2ˆα i τ β i τ Ī = {i : δ i τ = 1, α iτ 2ˆα i τ β i τ I = {i : δ i τ =, q i τ f i > } i 1 = min I if I = i 1 = min I if I = k = min{i 1, i 1} f iˆα i τ + D i τ } f iˆα i τ + D i τ < } 27

28 Find ηt suh that 1 2 α it β i tφ i t, ηt + D i t + ˆα i t + D i t I Ī + f 1 i qi t ηt = Kt 16 I If ητ [max{; q k 1 θ + ρ k 1 τ f k 1 }; q kτ + ρ k τ f k ] [max{; α k 1τ 2ˆα k 1 τ β k 1 τ f k 1 ˆα k 1τ+D k 1 τ}; α kτ 2ˆα k τ β k f k ˆα kτ+d k τ] ητ [max{; q k 1 τ + ρ k 1 τ f k 1 }; q kτ + ρ k τ f k ], go to 4d. Otherwise, if k = i 1, do I I \ {i 1 }; Ī Ī {i 1}; k = min{i 1, i 1} ; go to 4. Otherwise, if k = i 1, do I I \ {i 1}; k = min{i 1, i 1}; go to 4. Note that the sets S 1, S 2, S 3 need to be updated eah time we onsider a new stage. d If at a time no greater than t i, i S 1, than t 2 i, i S 2, and than t 1 i, i S 3, we observe that respetively ηt takes negative values, or that for some produt i unonstrained, q i t + ρ i t ηt hanges sign, or that for some produt i onstrained, α it 2ˆα i t β it f i ˆα it + D i t ηt hanges sign, we set respetively η to zero or we update the sets of ative unonstrained produts, or we update the set of onstrained produts in eah lass, and the value of η. In the first ase, we hek whether the apaity onstraint beomes tight again. In the last 2 ases, we iterate this step. e Compute the derivative of ηt using equation Final step Let if q i t + ρ i t < α it β it 1 p i t = 2 q i t + ρ i t + α it β it if α it 2ˆα i t β it q i t + ρ i t α it β it αt ˆα i t β i t if q i t + ρ i t > α it 2ˆα i t β i t t [; T ] { if qi t + ρ i t ηt f i u i t = qi t + ρ i t ηt if q i t + ρ i t ηt > f i t [; T ]. f 1 i B Appendix: Numerial results: losed form solution for D i. In this setion, we use the notation and data introdued in Setion 6 to derive a losed form solution for the derivative of the inventory safety level D i.. We showed in Setion 4.4 that D i t is obtained by solving for all fixed t: min ω i t,r i.,t ω i tγ i t + r i s, tds s.t. ω i t + r i s, t ˆα i s s [, t] ω i t r i s, t s [, t]. 28