Two-resource stochastic capacity planning employing a Bayesian methodology

Transcription

1 Journal of the Oerational Research Society (23) 54, r 23 Oerational Research Society Ltd. All rights reserved /3 $25. Two-resource stochastic caacity lanning emloying a Bayesian methodology E Stavrulaki 1, *, DKH Fong 2 and DKJ Lin 2 1 Bentley College, Waltham, MA, USA; and 2 Penn State University, University Park, PA, USA We examine a stochastic caacity-lanning roblem with two resources that can satisfy demand for two services. One of the resources can only satisfy demand for a secific service, whereas the other resource can rovide both services. We formulate the roblem of choosing the caacity levels of each resource to maximize exected rofits. In addition, we rovide analytic, easy-to-interret otimal solutions, as well as erform a comarative statics analysis. As alying the otimal solutions effectively requires good estimates of the unknown demand arameters, we also examine Bayesian estimates of the demand arameters derived via a class of conjugate riors. We comare the otimal exected rofits when demands for the two services follow indeendent distributions with informative and non-informative riors, and demonstrate that using good informative riors on demand can significantly imrove erformance. Journal of the Oerational Research Society (23) 54, doi:1.157/algrave.jors Keywords: alied robability; Bayesian methodology; caacity lanning; arameter estimation; single-eriod inventory models Introduction Caacity lanning is an inherently difficult roblem due to demand uncertainty, articularly in the service sector, because services often cannot be inventoried. As an examle, consider the decision of determining how many single and double rooms to build for a hotel exansion roject. Once the additional rooms of each tye have been built, the hotel has to rely on the existing rooms to satisfy the daily fluctuations in demand. (It is ractically imossible to build new rooms on a daily basis to erfectly match demand with caacity.) Recent increased customers exectations for high service levels, as well as oortunities for substitution among the available resources (eg a customer requesting a single room can stay in a double room) comlicate further the roblem of lanning for the right amount of caacity before demand materializes. Moreover, even when a manager is faced with a well-defined caacity lanning roblem, for which theoretically otimal solutions can be obtained, alying these solutions in ractice requires the estimation of unknown arameters of the demand distribution. For this reason, we examine a stochastic caacity lanning roblem from both of these ersectives: deriving otimal solutions and estimating the unknown demand arameters. We consider a roblem with two resources and two roducts where demand for one of the roducts can be satisfied by either resource. We derive the exected-rofitmaximizing caacity levels for general demand distributions and show that they corresond to modified critical fractile *Corresondence: E Stavrulaki, 23 Hamshire Road, Peabody, MA 196, USA. estavrulaki@bentley.edu ratios. These modified critical fractiles are similar to the ones derived for the classical single resource, single-eriod newsvendor model, but they also take into account the ossibility of satisfying some demand using either of the two resources. We also investigate the imact of changes to the inut and demand arameters via a comarative statics analysis, which highlights results that would have been difficult to redict otherwise. Finally, we comare the estimation of the unknown demand arameters via classical and Bayesian aroaches. A common ractice when solving the caacity lanning roblem is to substitute samle estimates for the oulation arameters in the demand distribution. It is likely, however, that rior information regarding the distribution of the demand arameters is available from various sources, such as industry trends, customer surveys, exert oinions, etc. Ideally, one could use this rior information to derive better estimates for the demand arameters than those obtained from the standard samle estimates, and to generate a distribution of maximum rofits, rather than a single exected rofit value. A owerful method for encoding rior knowledge through subjective robabilities into the decision rocess is to use the Bayesian theory. In fact, in some cases (eg the normal demand case to be discussed here), classical estimates are actually Bayesian estimates with resect to certain non-informative riors. Hence, emloying Bayesian estimates with informative riors could imrove the decision-making rocess, esecially if the noninformative riors are considered unlikely to be true. As modern comuting ower becomes inexensive, alying the Bayesian aradigm in ractice is not only feasible but also imortant. In their recent book, for examle, French and

2 E Stavrulaki et al Stochastic caacity lanning 1199 Smith 1 cited various real-world alications of Bayesian theory in diverse industries. In addition, many Bayesian alications have recently been develoed in the marketing and economics literature (see for examle, Fong and Bolton, 2 Pammer et al, 3 Fong et al, 4 and DeSabo et al 5 ). In this aer, we demonstrate how we can quantify the value of incororating rior information in a stochastic caacity lanning setting with roduct substitution. The remainder of this aer is organized as follows. First we rovide a brief literature review. Then we resent the model formulation, derive analytically otimal solutions, and erform a comarative statics analysis on the otimal solution. Next, the Bayesian rocedures that we emloy are discussed. Numerical work, with which we comare the conventional samle estimation aroach with a Bayesian aroach that assigns informative riors on the demand arameters, is then resented. The final section summarizes our findings and concludes. Related work There are two research streams related to this work: singleeriod caacity lanning roblems for multile, substitutable resources (see Sethi and Sethi 6 for a review), and singleeriod stochastic inventory models for multile, substitutable roducts (see Bassok et al 7 for a review). The works of Harrison and Van Mieghem, 8 Netessine et al 9 (stochastic caacity lanning models), Bassok et al, 7 and Parlar and Goyal 1 (single-eriod inventory models) rovide standard formulations that are closely related to ours. All of these authors, however, focus on roviding conditions for otimality and investigating the roerties of the otimal inventory olicy, but do not rovide easily interretable analytic solutions, nor erform a comarative statics analysis. In contrast, by focusing on a two-roduct, tworesource setting, we are able to rovide fairly simle equations that the otimal caacities must satisfy, as well as interret these equations based on the classic singleroduct newsvendor model. In addition, unlike most of the aers in the aforementioned literature, with the excetion of Hill, 11 we consider the asect of estimating the demand arameters when rior information is available. Hill emloys conjugate riors in the context of a standard newsvendor roblem and demonstrates the value of Bayesian estimation for a single unknown arameter of three different demand distributions. Similarly, we investigate the effects of arameter estimation on exected rofits in the two-resource, caacity-lanning setting. In contrast to the lack of Bayesian alications for singleeriod inventory models, the alication of Bayesian theory to estimate demand arameters of multieriod inventory models has been well studied (see Azouri, 12 for a recent literature review, and Bradford and Sugrue 13 for a twoeriod alication). Most multieriod inventory models, however, focus on the single-resource, single-roduct case, and tyically obtain otimal Bayesian olicies via stochastic dynamic rogramming formulations. In addition, most of these aers ignore the issue of lost sales, and do not consider the fixed costs associated with acquiring resources. While multieriod settings may be aroriate in certain business environments, there are cases for which singleeriod inventory or caacity lanning models, such as ours, are more aroriate; consider for examle a case for which the lanning horizon is too long to allow for reeated udates (as in our hotel exansion examle). Thus, our interest is in integrating a Bayesian estimation rocedure with a single-eriod, two-resource, caacity lanning model. In addition to single-eriod inventory models, revenue management models that focus on otimal stoing rules are somewhat related with this work (for a recent review of revenue management research see McGill and Van Ryzin 14 ). This similarity stems from the fact that identifying the otimal booking limit for discount assengers (ie the otimal stoing rule) is based on a formula qualitatively similar to the single-eriod newsvendor result. However, even though research aers by Bodily and Weatherford, 15 Brumelle et al, 16 and Ladany 17 consider two-resource revenue management roblems with some notion of substitution (ie allowing customers to ugrade) or Bayesian udating, the assumtions as well as the focus of these aers are significantly different from ours, and thus we do not review them further here; to illustrate, Brumelle et al assume that there is a ositive robability that a customer denied a discount fare will ugrade to a full fare, whereas we (and caacity lanning roblems in general) do not account for such a robability, but instead consider the shortage cost of not satisfying all demand, and allow for ugrades only if there is excess caacity. Caacity lanning model with substitution Assumtions and model structure Consider a monoolist offering two services (roducts) at fixed rices i via two resources. Resource tye 2 can only be used to satisfy demand for service 2, whereas resource tye 1 can be used to satisfy either tye of demand. That is, demand for service tye 2 can be ugraded, but demand for service tye 1 cannot be downgraded. Given this structure, the monoolist must choose the caacity level T i for each resource i, i ¼ 1, 2, to satisfy demand during a given time horizon (eg the hotel manager decides on the number of single and double rooms available in her hotel within the next 3 years). Note that this decision must be made before the monoolist observes demand. For each unit of tye i caacity the firm acquires, it incurs a fixed cost G i, even if that unit is not used to satisfy demand. The firm also incurs a variable cost v i,foreveryunitoftyei caacity it uses to satisfy demand.

3 12 Journal of the Oerational Research Society Vol. 54, No.11 After the monoolist buys caacity for each resource tye, customers arrive randomly during the fixed time horizon and lace their orders. Customers must either be served within their resective eriods of arrival or be turned away. If the firm is unable to serve a customer by the available caacity, it must comensate that customer. In other words, the firm internalizes the cost of not satisfying demand by aying a shortage cost of c i, for each unit of unmet demand. Given the above inut arameters (rice i,fixedg i and variable v i caacity costs, and shortage cost c i ) and the firm s caacity choices T 1 and T 2, the actual demand values realized during the assumed time horizon determine the actual amount of rofit (or loss) for the monoolist. However, since the monoolist will determine the actual rofit only at the end of the time horizon, she must decide on theamountofcaacitiestoacquirebasedonherexected rofits. Thus, the monoolist s objective is to determine the otimal level of caacity of each resource tye T 1 and T 2 so as to maximize exected rofits. To calculate exected rofits, the monoolist must have some knowledge of the nature of randomness for the demands. We assume that, irresective of how customers arrive during the given time horizon, the robability distribution function of the total demand for each service tye is known. Moreover, we assume that the demands D i, i ¼ 1, 2, for the two services are indeendent and continuous random variables, with robability density functions f i and cumulative distribution functions F i, resectively. We also assume the following with resect to our inut arameters: (1) all inut arameters ( i, v i, c i,andg i )are strictly ositive, (2) i v i 4, (3) v 1 4v 2,sothatitisnot otimal to substitute resource 1 for resource 2 whenever there is excess amount of resource 2 available, (4) 1 c c 2 4, so that it is more rofitable to satisfy unmet demand of tye 1 rather than of tye 2, and (5) 2 v 1 4, so that substitution of resource 2 from resource 1 when satisfying demand of tye 2 is rofitable. Model formulation We can now formally exress both the net revenue and cost associated with choosing caacities T 1 and T 2. The net revenue function, denoted by g(t 1, T 2 ), equals gðt 1 ; T 2 Þ¼ð 1 v 1 Þ minðd 1 ; T 1 Þþð 2 v 2 Þ minðd 2 ; T 2 Þ þð 2 v 1 Þ min½ðd 2 T 2 Þ þ ; ðt 1 D 1 Þ þ Š The last term of g(t 1, T 2 ) describes the net revenue generated from a customer requesting service 2 whose demand is satisfied by resource 1. Note that this substitution is made only when there is excess amount of resource 1 (ie T 1 4D 1 ) and excess demand for service 2 (ie D 2 4T 2 ). Moreover, note that g(t 1, T 2 ) is a random variable since D 1 and D 2 are also random variables. The cost function, denoted by h(t 1, T 2 ), is hðt 1 ; T 2 Þ¼c 1 ðd 1 T 1 Þ þ þ c 2 ðd 2 T 2 ðt 1 D 1 Þ þ Þ þ þ G 1 T 1 þ G 2 T 2 The first term of h(t 1, T 2 ) exresses the shortage cost of not satisfying demand for service 1 from resource 1. The second term of h(t 1, T 2 ) exresses the shortage cost of not satisfying demand for service 2 given that resource 1 may also be used to satisfy some of that demand. In other words, if T 1 4D 1 then there is extra tye 1 caacity, equal to T 1 D 1,which can be used to satisfy tye 2 demand when D 2 4T 2. Thus, the shortage cost associated with tye 2 demand is incurred only if the quantity D 2 T 2 (T 1 D 1 ) þ is non-negative. The net rofit for a given demand realization (D 1, D 2 )is (T 1, T 2 ) ¼ g(t 1, T 2 ) h(t 1, T 2 ), and thus the firm s roblem is to max fe D1 ;D 2 ½ðT 1 ; T 2 ÞŠg ¼ max fe D1 ;D 2 ½gðT 1 ; T 2 Þ T 1 ;T 2 T 1 ;T 2 hðt 1 ; T 2 ÞŠg Note that the term min[(d 2 T 2 ) þ,(t 1 D 1 ) þ ]ing(t 1, T 2 ) equals D 2 T 2 if (D 2 T 2 ot 1 D 1 and D 2 T 2 4) and equals T 1 D l if (T 1 D l od 2 T 2 and T 1 D l 4). Similarly, the term (D 2 T 2 (T 1 D 1 ) þ ) þ in h(t 1, T 2 )equalsd 2 T 2 if (T 1 D l oandd 2 T 2 4) and equals D 2 (T 1 D 1 ) T 2 if (T 1 D 1 4andD 2 (T 1 D 1 ) T 2 4). Thus, we can exress the exected rofit E D1 ;D 2 ½ðT 1 ; T 2 ÞŠ function as follows: E D1 ;D 2 ½ðT 1 ; T 2 ÞŠ ð i v i Þ ¼ X2 i¼1 þð 2 v 1 Þ þ Z Ti þ T 2 x T 2 xf i ðxþdx þ T i f i ðxþdx T i ðt 1 xþf 2 ðyþdy f 1 ðxþdx T 1 þ T 2 x ðy T 2 Þf 2 ðyþdy f 1 ðxþdx ð1þ c 1 ðx T 1 Þf 1 ðxþdx T 1 c 2 ðy þ x T 1 T 2 Þf 2 ðyþdy f 1 ðxþdx T 1 þ T 2 x þ ðy T 2 Þf 2 ðyþdy f 1 ðxþdx G 1 T 1 G 2 T 2 T 2 T 1 Proosition 1 If the firm can make ositive rofits, the otimal caacity levels maximizing the exected rofit in exression (2) are found by solving the following two equations: F 1 ðt 1 Þ¼ ð2þ ð 1 v 1 þ c 1 Þ G 1 þð 2 v 1 þ c 2 ÞP½oD 1 ot 1 and D 2 4T 1 þ T 2 D 1 Š 1 v 1 þ c 1 ð3þ

4 E Stavrulaki et al Stochastic caacity lanning 121 F 2 ðt 2 Þ¼ ð 2 v 2 þ c 2 Þ G 2 ð 2 v 1 þ c 2 ÞP½oD 1 ot 1 and D 2 ot 1 þ T 2 D 1 Š ð 2 v 2 þ c 2 Þ ð 2 v 1 þ c 2 ÞP½oD 1 ot 1 Š ð4þ with resect to T 1 and T 2. Proof. See Aendix. Interreting the otimal solution By carefully looking at exressions (3) and (4), we can draw an analogy between the otimal two-recourse caacity levels and the otimal single-eriod newsvendor caacity level. Let us consider, for a moment, resource i, for i ¼ 1, 2, in isolation, as in the single-eriod newsvendor model. Then, the otimal caacity for resource i is derived by setting the critical fractile ratio ( i v i þ c i G i )/( i v i þ c i ) equal to F i (T i ), where i v i þ c i G i is the marginal gain of selling one extra unit of caacity and G i is the marginal loss of one unused unit of caacity. The otimal solutions to the tworesource, two-service caacity lanning roblem have similar interretations. To better understand these interretations, let us first exlain the meaning of two robability exressions: (1) P[oD 1 ot 1 and D 2 ot 1 þ T 2 D 1 ]istherobability of the event that there is excess tye 1 caacity, which can be used to satisfy all of excess tye 2 demand (ie no customers are left unserved), whereas (2) P[oD 1 ot 1 and D 2 4T 1 þ T 2 D 1 ] is the robability of the event that there is excess tye 1 caacity, which can all be used to satisfy excess tye 2 demand, and there is extra tye 2 demand left unserved. Keeing these meanings of the above robability exressions in mind, we now return to the two-resource roblem, and exress the marginal gain of one extra unit of tye 1 caacity that can be used to satisfy demand as Gain 1 ¼ð 1 v 1 þ c 1 Þ G 1 þð 2 v 1 þ c 2 Þ P½oD 1 ot 1 and D 2 4T 1 þ T 2 D 1 Š Observe that ( 2 v 1 þ c 2 ) is multilied by the robability of having extra tye 1 caacity (since T 1 4D 1 )andextratye2 demand, which cannot be satisfied by resource 2 alone, or even by all the excess tye 1 caacity (since D 2 4T 1 þ T 2 D 1 ). In other words, the marginal gain of one extra unit of caacity for resource 1 equals the corresonding single-resource marginal gain, lus the marginal gain associated with satisfying demand for service 2 from recourse 1, when excess tye 1 caacity is surely needed. The marginal loss of one extra unit of unsold tye 1 caacity equals Loss 1 ¼G 1 ð 2 v 1 þ c 2 Þ P½oD 1 ot 1 and D 2 4T 1 þ T 2 D 1 Š That is, the fixed cost G 1 is not considered a comlete loss if recourse 1 is used to satisfy demand for service 2. Thus, the otimal amount of caacity for resource 1 satisfies F 1 (T 1 ) ¼ Gain 1 /(Gain 1 þ Loss 1 ), which has the same intuitive structure as the critical fractile of the single-resource newsvendor model. Similarly, the marginal gain of one extra unit of tye 2 caacity that can be used to satisfy demand equals Gain 2 ¼ð 2 v 2 þ c 2 Þ G 2 ð 2 v 1 þ c 2 Þ P½oD 1 ot 1 and D 2 ot 1 þ T 2 D 1 Š In other words, the marginal gain for resource 2 consists of the corresonding single-resource marginal gain, adjusted downwards by the marginal gain associated with satisfying all excess tye 2 demand by excess tye 1 caacity (rather than by tye 2 caacity). The marginal loss of one extra unit of unsold tye 2 caacity equals Loss 2 ¼G 2 ð 2 v 1 þ c 2 Þ P½oD 1 ot 1 and D 2 4T 1 þ T 2 D 1 Š In other words, one extra unit of tye 2 caacity imlies a loss of G 2 only if the firm could not have used all of the excess tye 1 caacity to satisfy all of the excess tye 2 demand. Observe that Gain 2 þ Loss 2 ¼ ( 2 v 2 þ c 2 ) ( 2 v 1 þ c 2 ){P[oD 1 o T 1 and D 2 ot 1 þ T 2 D 1 ] þ P[oD 1 ot 1 and D 2 4T 1 þ T 2 D 1 ]} ¼ ( 2 v 2 þ c 2 ) ( 2 v 1 þ c 2 )P[oD 1 ot 1 ], which equals the denominator of exression (4). Therefore, the otimal amount of caacity for resource 2 satisfies F 2 (T 2 ) ¼ Gain 2 /(Gain 2 þ Loss 2 ), which is again similar to the single-resource newsvendor model. Comarative statics analysis In the revious subsection we enhanced our understanding of the structure of the otimal olicy based on a marginal analysis interretation of the otimal exressions, (3) and (4), as in the single-resource newsvendor model. In this subsection, as commonly done in the literature (eg Agnihothri et al 18 ), we further our insights on the behavior of the otimal solution, by erforming a comarative statics analysis with resect to the model inut and demand arameters. Unfortunately, deriving closed-form solutions from exressions (3) and (4) is not feasible even for fairly simle demand distributions, such as the uniform. Therefore, we erform our analysis numerically. We assume indeendent normal distributions for the demands of the two service tyes, because the normal distribution is so often emloyed in ractice, and is also relatively easy to handle mathematically. However (with one excetion we reort in the section Imact of changes to the demand arameters), our conclusions also hold for other symmetric two-arameter distributions (such as the Student s t-distribution). Table 1 summarizes the arameter values for the base case we considered in our numerical work.

5 122 Journal of the Oerational Research Society Vol. 54, No.11 Table 1 Samle data Inut arameters Demand arameters Resource i c i G i v i m i s i Table 2 Comarative statics T 1 T 2 S 12 T 1 T 2 S 12 1 þ þ þ 2 þ þ þ c 1 þ þ c 2 þ þ v 1 þ v 2 þ þ G 1 þ G 2 þ þ m 1 þ þ m 2 þ s 1 þ þ s 2 þ þ þ Note that customer tye 1 is willing to ay a higher rice for resource tye 1 but also exects to be comensated a higher value if her demand cannot be satisfied. Moreover, note that resource tye 1 is more exensive as it has higher fixed and variable costs. In Table 2 we reort the imact on T 1 and T 2 of changes to the inut or demand arameters. We also evaluate and reort the imact on the average ercentage of tye 1 caacity used to satisfy tye 2 demand. We refer to this ercentage as the substitution rate and denote it as S 12, where S 12 E½ðT 1 D 1 Þ 1 ðd1 ot 1 ;D 2 4T 1 þt 2 D 1 Þ þðd 2 T 2 Þ1 ðd1 o T 1 ;T 2 o D 2 ot 1 þ T 2 D 1 ÞŠ= T 1 ¼ð1=T 1 Þð R T 1 ðt 1 xþf 1 ðxþð R 1 T 1 þt 2 x f 2ðyÞdyÞdxþ R T 1 f 1ðxÞ ð R T 1 þ T 2 x T 2 ðy T 2 Þf 2 ðyþdyþdxþ: The results in Table 2 rovide a number of interesting insights, which we discuss next. Imact of changes to the inut arameters. Table 2 illustrates that (with the excetion of rofits) the directions of change for the otimal decision variables T 1 and T 2 when considering small increases in rice or shortage cost are the same (ie other than the rofit column, the first two rows of Table 2 are the same). Similarly, the directions of change for T 1 and T 2 when considering small increases in the variable or fixed caacity cost are the same (see third and fourth rows of Table 2). Let us first consider increases on the fixed or variable caacity costs. As exected, when the fixed (G 1 ) or variable (v 1 ) caacity cost for resource 1 increases, the firm should otimally decrease tye 1 caacity and increase tye 2 caacity, while also decreasing the substitution rate S 12. Analogously, as v 2 and G 2 increase, both T 1 and S 12 increase while T 2 decreases. In both cases, of course, exected rofits decrease. Turning to changes in rice or shortage cost, Table 2 shows that when all other arameters remain the same, and the rice 1 of resource 1 increases (imlying that tye 1 customers are now more valuable), the firm should otimally increase its tye 1 caacity. Interestingly, when 1 increases, not only should the otimal tye 1 caacity increase but tye 2 caacity should also otimally decrease. To exlain this last, somewhat unexected result, recall that even though rice has increased, the demand distribution remains the same. Therefore, when tye 1 caacity increases, we may have excess tye 1 caacity more often, which can then be used to satisfy demand for service tye 2. To ut it differently, since the demand distribution remains constant, random demand realizations would not affect the mean demand, but would now be satisfied from higher levels of tye 1 caacity, imlying a higher frequency of realizing leftover tye 1 caacity. Therefore, as 1 increases, the firm should not only increase T 1 but at the same time decrease T 2, thereby increasing the substitution rate S 12. To give one examle of the magnitude of these otimal adjustments, we reort the imact of a 5% increase in rice or variable caacity cost v 1 from the base-case scenario we considered in Table 1: a 5% increase in 1 imlies a 4.6% increase in T 1,a 2.% decrease in T 2, a 56% increase in rofits, and a 35.3% increase in the substitution rate S 12, whereas a 5% increase in v 1 imlies a 2.8% decrease in T 1, a 4.1% increase in T 2,an 11.3% decrease in rofits, and a 34.7% decrease in the substitution rate S 12. The comarative statics results for increases in 2 are somewhat different from those for increases in 1, because resource 2 cannot be used to satisfy tye 1 demand. From Table 2 we see that as 2 increases, and service 2 becomes more valuable, both tyes 1 and 2 caacities should otimally increase to ensure better service for tye 2 customers. Interestingly, we also find that as 2 increases, T 2 increases enough to make the otimal substitution rate S 12 decrease. We can now aly arguments similar to the ones we emloyed for increases in rice to exlain what haens to the decision variables as shortage cost increases. As the shortage cost c 1 of resource tye 1 increases, it becomes more imortant to the firm to satisfy demand for service tye 1, imlying, as with increases in 1, that caacity for resource tye 1 should otimally increase, caacity for resource tye 2 should decrease (because, as before, there may be extra tye 1 caacity to use for tye 2 demand), and the substitution ercentage should increase. Similarly, as c 2 increases, both T 1 and T 2 increase while S 12 decreases. Imact of changes to the demand arameters. As we see from Table 2, changes in the mean demand of one of the resources do not affect the otimal caacity choice for the other resource, whereas changes in the standard deviation of one of the resources affect the caacity choices for both resources. Even though the first result is not necessarily counterintuitive, it is somewhat surrising. Before looking at Table 2, for examle, we may have exected that when m 2 increases, both T 1 and T 2 should increase, or that when m 1

6 increases, T 2 may decrease. However, we find that an increase in m 2 only affects T 2, but leaves both T 1 and S 12 unchanged. In contrast, as s 2 increases, the otimal values of both caacity tyes are affected; in fact, when s 2 increases, the otimal values for all three variables (T 1, T 2, and S 12 ) should otimally increase. Similarly, when the standard deviation s 1 of tye 1 demand increases (while the mean remains the same) both caacities are affected, with T 1 and S 12 increasing, and T 2 decreasing. To verify whether these results are secific to the normal distribution, we comared them with the corresonding results for t- distributed, indeendent demands. We found that, as with the normal distribution, increases in m 2 do not affect T 1 and S 12, and that changes to the standard deviation of either resource do affect the otimal caacity choices of both recourse tyes. Unlike the normal distribution case, however, we did observe a slight decrease in T 2 when m 1 increased. We, therefore, conclude that the imact of changes on the mean demand of one resource is distribution secific, and thus may or may not affect the otimal caacity choice of the other resource. In summary, our comarative statics analysis rovided us with a better understanding of how the otimal caacities should be adjusted when the roblem arameters change. One interesting finding, for examle, is that when tye 1 caacity increases (as a reaction to tye 1 demand being more rofitable, or tye 2 demand being more variable), tye 2 caacity should also be adjusted downwards. This finding is of course linked with our indeendent demands assumtion. If demands are ositively correlated, for examle, then we may see that increases in 1,ors 1,maynotnecessarily imly a decrease in T 2. Our results also suggest the imortance of alying accurate estimates for the demand distributions. Particularly, we found that changes in the mean or standard deviation of demand may imly notable differences in exected rofits; for examle, a 1% increase in the mean of tye 1(2) demand from the base level of 13(15) we considered, imlies a 5%(3%) increase in rofits. Therefore, in the next section we discuss how one could emloy an imroved estimation method for the demand arameters before calculating the otimal caacity choices. Bayesian udating for the two-resource caacity lanning model So far we focused on maximizing the exected rofit function shown in exression (1). We rovided analytic solutions for this roblem via Equations (3) and (4), which hold for any two indeendent and continuous robability distribution functions. In ractice, demands are often assumed to be normal with a given mean and variance. For examle, if we denote by z i (x) the robability density function of a normal distribution with mean m i and standard ffiffiffiffiffi deviation s i,thatis,z i ðxþ ¼1=ð 2 si Þe ð1=2þ½ðx m iþ=s i Š 2, i ¼ 1, 2, and by Z i ( ) the corresonding cumulative distribution function, then the otimal caacity levels are determined by the following relationshis: ðt 1 Þ¼ ð 1 v 1 þ c 1 Þ G 1 þð 2 v 1 þ c 2 Þ R T 1 ½1 Z 2ðT 1 þ T 2 xþšz 1 ðxþdx 1 v 1 þ c 1 ð5þ Z 2 ðt 2 Þ¼ ð 2 v 2 þ c 2 Þ G 2 ð 2 v 1 þ c 2 Þ R T 1 Z 2ðT 1 þ T 2 xþšz 1 ðxþdx ð 2 v 2 þ c 2 Þ ð 2 v 1 þ c 2 Þ½ ðt 1 Þ ðþš ð6þ In order to solve these equations we must first estimate the means and variances of the normal distributions. Obviously, the estimation of the unknown arameters is critical when maximizing the exected rofit function. Tyically, the samle mean Ẋ i and the samle variance S i 2 /(n i 1) (where S i 2 denotes the sum of squared errors and n i the samle size) are used to estimate the true demand arameters. However, this traditional estimation aroach may not be effective under certain circumstances. For examle, traditional estimation would rovide only a single value for the maximum exected rofit, whereas a Bayesian aroach would rovide a samle distribution of maximum rofits so that one can assess the effect of demand estimation on the otential maximum rofit. In addition, it is ossible that demands are not stationary and have some or all of their arameters varying with time. Are there better estimates for the arameters of the demand distributions that should be used in such cases? In articular, should managers strive to incororate rior information on the demand arameters when addressing the caacity lanning roblem? To investigate such questions, we next resent how Bayesian theory can be emloyed for two different demand scenarios. Normally distributed demands case E Stavrulaki et al Stochastic caacity lanning 123 Assume that demands are normally distributed with mean m i and standard deviation s i, that is, that D i BN(m i, s i ), i ¼ 1, 2. Instead of using the samle mean and samle variance to estimate the oulation arameters, we could emloy Bayesian estimates incororating existing rior information. A common class of riors for the mean and variance of normal distributions (see Fong and Ord 19 )is m i tð2a i ; m i ; g i b i =a i Þ and s 2 i IGða i ; b i Þ; i ¼ 1; 2; all indeendent where t denotes the Student s t-distribution, IG denotes the inverse gamma distribution (so that u i ¼ 1/s 2 i is Gammadistributed with density f ðu i Þ¼u ai 1 i b ai i exð u i b i Þ=Gða i Þ, and a i, b i, g i, and m i are constants; these constants are secified based on subjective beliefs/exert oinions regard- ð7þ

7 124 Journal of the Oerational Research Society Vol. 54, No.11 ing the demand arameters, and not necessarily based on observed data. For examle, referring back to our hotel exansion setting, if the hotel manager knows of a change in the oulation demograhics that would increase demand by a certain ercentage in the next few years, he would incororate this information into his beliefs for the rior distribution of the demand arameters, and secify a i, b i, g i, and m i accordingly (see the section Numerical results for a secific examle of how to determine these four constants). Even though we emloy conjugate riors in this aer, in ractice more realistic, non-conjugate riors can also be handled by using advanced comutational aroaches such as Markov Chain Monte Carlo methods (see Gilks et al 2 ). Note that the rior of (m i, s 2 i ) in exression (7) is equivalent to the normal-inverse gamma conjugate rior given by m i js 2 ffiffiffi i Nðm i ; s i g 2 iþ,andsi BIG(a i, b i ). Thus, the conditional density ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of m i given (s 2 i, x) is Nððg i n i X i þ m i Þ= ð1 þ g i n i Þ; s i g i =ð1 þ g i n i ÞÞ, and the marginal osterior density of s 2 i is IG(a i þ n i /2, b i þ S 2 i /2 þ ( X i m i ) 2 /2(g i þ 1/n i )), which imlies that the Bayesian estimates for m i and s 2 i are ^m i ¼ ðg in i X i þ m i Þ ð1 þ g i n i Þ ^s 2 i ¼ b i þ S 2 i =2 þð X i m i Þ 2 =ð2ðg i þ 1=n i ÞÞ a i þ n i =2 1 Note that when b i ¼, a i ¼ 1 2,andg i-n, then^m i - X i and ^s i -S 2 i /(n i 1). In other words, the traditional samle estimates of the mean and variance are indeed Bayesian estimates with resect to a non-informative rior. Thus, we are interested in comaring exected rofits when using arameter estimates with informative riors (^m i, ^s i ), or non-informative riors ( X i, S 2 i /(n i 1)). To summarize, assuming that the true demand distribution is normal (we refer to this case as the normally distributed demands (NDD) case), we want to comare the following two estimation scenarios when maximizing exected rofit: Scenario NDD-NIP: Substitute simle samle estimates (ie Bayesian estimates with a non-informative rior (NIP)) for the mean and variance of the normal demand distributions in the exected rofit ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exression (2). In other words, substitute X i and S i = ðn i 1Þ for m i and s i, resectively, in Equations (5) and (6). Scenario NDD-IP: Substitute the Bayes estimates of the informative riors (IP) for the mean and variance (see exression (7)) in the exected rofit exression (2). In other words, substitute ^m i and ^s i and for m i and s i, resectively, in Equations (5) and (6). We resent numerical results for these two demand scenarios in the section Numerical results. t-distributed demands case In the revious subsection we assumed that demands follow a stationary normal distribution. In certain cases, however, it ð8þ ð9þ is ossible that the demand distributions are non-stationary. We consider a secial non-stationary demands case here, in which the true demands are t-distributed, D i Bt(2a i, m i,2b i ), i ¼ 1, 2; this case occurs when the means of normally distributed demands remain constant, but the variances are assumed to vary with time and follow an inverse gamma distribution. In other words, we assume that D ij s ij B N(m i, s ij )ands 2 ij BIG(a i, b i ), where the subscrit j ¼ 1,y, n i denotes the current time eriod. Given this class of t-distributed true demands, and using the same class of conjugate riors that we emloyed in the revious subsection, the redictive distribution of future demand conditional on existing data X for service tye i (for any time eriod j4n i )isd i XBt(2a i þ n i, ~m i, ~s 2 i ), where the location arameter ~m i, and the scale arameter ~s 2 i are given as follows: ~m i ¼ ðg in i X i þ m i Þ ð1 þ g i n i Þ ð1þ ~s 2 i ¼ ½ð1 þ g in i Þð2b i þ S 2 i Þþn ið X i m i Þ 2 Š½1 þ g i þ g i n i Š ð2a i þ n i Þð1 þ g i n i Þ 2 ð11þ Werefertothiscaseasthet-distributed demands (TDD) case, and so we want to comare the following two estimation scenarios when maximizing exected rofit: Scenario TDD-NIP: Use a normal distribution to aroximate the distribution of each demand and substitute samle estimates for the mean and variance of the normal demand distributions in exression (2). In other words, substitute X i and S i = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn i 1Þ for m i and s i, resectively, in Equations (5) and (6). Scenario TDD-IP: Use the redictive distributions t(2a i þ n i, ~m i, ~s 2 i ), t ¼ 1, 2. In other words, use this t- distribution with location and scale estimates ~m i and ~s i in Equations (3) and (4). For both the NDD and the TDD cases, we next comare the use of informative and non-informative riors by (i) simulating random samles from the true distributions, (ii) deriving otimal caacity decisions based on these samles, and (iii) comaring exected rofits for each scenario. Numerical results Our main motivation in numerically comaring the various demand scenarios described in the revious section is to verify and quantify the value of using informative riors on the demand arameters for our caacity lanning roblem. Our urose is not to erform a large simulation study, but rather to illustrate that good (not erfect) rior information can be valuable and can imrove erformance when making caacity decisions under uncertainty. We also give a secific examle of how the arameters of the common class of riors introduced in the revious section can be calculated.

8 E Stavrulaki et al Stochastic caacity lanning 125 Method descrition A detailed seudoalgorithm for the method we use to comare the estimation scenarios is shown in the aendix. We should emhasize that the Bayesian aroach we described in the revious section is based on the joint osterior distribution of m i and s i for both the NDD and the TDD cases. Thus, one can generate samles from the osterior distribution of {m i, s i }, and calculate the corresonding otimal values for the caacities {T 1, T 2 }foreach samle; therefore, one can generate a samle distribution for {T 1, T 2 } and a corresonding samle distribution for maximum exected rofits. This aroach can rovide the decision maker with much more information on rofits than when using classical estimation, where only one value of maximum exected rofit can be generated. In our numerical results, however, for simlicity, we only consider the osterior mean and variance and evaluate the corresonding exected rofit (see the algorithm described in the aendix). To give a secific numerical examle, let us first consider the normally distributed demands case, and let us assume that the true mean demand for service 1 is m 1 ¼ 13 and the true variance is s 2 1 ¼ 22 2 ¼ 484. Let us consider a decision maker who believes that the average demand is E[m 1 ] ¼ m 1 ¼ 13, give or take 2 (ie Stdev[m 1 ] ¼ 2), and that the mean variance is E[s 2 1 ] ¼ s 2 1 ¼ 484, give or take 225 (ie Stdev[s 2 1 ] ¼ 225). In other words, the decision maker has good subjective information (because her beliefs regarding the average demand E[m 1 ] and average variance E[s 2 1 ]are accurate), but not erfect information (because she also assigns rior demand distributions around the values of E[m 1 ]ande[s 2 1 ]). Given these rior beliefs, the values for the arameters of the conjugate riors a 1, b 1, g 1,andm 1 can be derived as follows. We have assumed that s 2 1 BIG(a 1, b 1 ); thus, the mean E[s 2 1 ] ¼ b 1 /(a 1 1) and the standard deviation Stdev½s 2 1 Š¼ðb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=ða 1 1ÞÞ ða 1 2Þ. Solving these two equations for a 1 and b 1 yields a 1 ¼ 2 þ (E[s 2 1 ]/Stdev[s 2 1 ]) 2 and b 1 ¼ (a 1 1)E[s 2 1 ]. Substituting E[s 2 1 ] ¼ 484 and Stdev[s 2 1 ] ¼ 225 in these exressions we get a 1 ¼ 6.63 and b 1 ¼ To calculate the value of g 1 as a function of a 1 and b 1,we note that if a random variable is t-distributed with k degrees of freedom, location arameter m, and scale arameter ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2, t(k, m, s 2 ), then its standard deviation is ½k=ðk 2ÞŠs. Thus, since we assume that m 1 Bt(2a 1, m 1, g 1 b 1 /a 1 ), see exression (7), we have that Stdev½m 1 Š¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a 1 =ð2a 1 2Þ g 1 b 1 =a 1 which imlies that g 1 ¼ (Stdev[m 1 ]) 2 (a 1 1)/b 1 ). Therefore, substituting the values of a 1 and b 1 in the exression for g 1 yields g 1 ¼.83. Finally, the location arameter m 1 is set equal to E[m 1 ]som 1 ¼ 13. Summarizing, given the rior beliefs regarding the average mean and average variance, the demand arameters for service tye 1, (a 1, b 1, g 1, m 1 ) ¼ (6.63, 2724,.83, 13). Given these arameter values, let us assume that a random samle of size n 1 ¼ 5 is drawn from the normal distribution N(m 1, s 1 ) ¼ N(13, 22), which roduced the samle estimates ð X 1 ; S 1 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 1 1ÞÞ¼ð125; 15Þ. Then from Equations (8) and (9), the Bayesian estimates are (^m 1, ^s 1 ) ¼ (125.97, 19.8). Similarly, if m 2 ¼ 15, s 2 2 ¼ 25 2 ¼ 625, and subjective information suggests that E[m 2 ] ¼ 15, Stdev[m 2 ] ¼ 3, E[s 2 2 ] ¼ 625, and Stdev[s 2 2 ] ¼ 225, the arameters (a 2, b 2, g 2, m 2 ) ¼ (9.72, 5447, 1.44, 15), and if a random samle of size n 2 ¼ 5 from the normal distribution N(m 2, s 2 ) ¼ N(15, 25) roduces the estimates ð X 2 ; S 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 2 1ÞÞ¼ð145; 35Þ, then ( ^m 2, ^s 2 ) ¼ (145.61, 26.55). Based on these estimates, we comare the two scenarios for the NDD case as summarized in Table 3. Note that we assumed a small samle size to reflect cases for which there is only a small amount of available demand data, erhas due to changing business conditions, or due to the services offered being relatively new. Moreover, for both service tyes, the coefficient of variation imlied by the chosen arameter values is low so that the robability of negative demand is very small. Let us now consider an examle for the TDD case. For comarison, suose that the same subjective beliefs resented above for the NDD case aly here as well, leading to the same arameter values of (a 1, b 1, g 1, m 1 ) ¼ (6.63, 2724,.83, 13) and (a 2, b 2, g 2, m 2 ) ¼ (9.72, 5447, 1.44, 15). Moreover, suose that random samles of sizes n 1 ¼ n 2 ¼ 5 were drawn from the true t-distributions t(2a i, m i, 2b i,), i ¼ 1, 2, and roduced the same samle estimates ð X 1 ; S 1 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 1 1Þ; X 2 ; S 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 2 1ÞÞ¼ð125; 15; 145; 35Þ. Then from Equations (1) and (11), the Bayes estimates for the demand arameters are (~m 1, ~s 1 ) ¼ (125.97, 2.13) for service 1 and (~m 2, ~s 2 ) ¼ (145.61, 27.58) for service 2. Based on these estimates, we comare the two scenarios for the TDD case as summarized in Table 4. These numerical examles illustrate that neither of the estimation scenarios (NIP or IP) has erfect information on the demand rocesses. Both rely on samle data, but in addition, the informative rior estimates incororate subjective beliefs regarding the rior distribution of the mean and variance. Based on these beliefs, the arameters of the conjugate riors are estimated. Thus, it is imortant to emhasize that our algorithm for comaring the two estimation scenarios is not designed to favor one versus the other. At the same time, however, we chose the subjective beliefs regarding the mean and variance of the unknown Table 3 Scenario Demand scenarios for the NDD case Estimated distribution NDD-NIP Service 1: N(125, 15) Service 2: N(145, 35) NDD-IP Service 1: N(125.97, 19.8) Service 2: N(145.61, 26.55)

9 126 Journal of the Oerational Research Society Vol. 54, No.11 Table 4 Scenario arameters to be reasonably close to the true values because we want to know how valuable good information can be. If the decision makers have no secific information regarding their stochastic rocesses (somewhat unlikely in most business settings), then they might simly aly the classical estimates. Results Demand scenarios for the TDD case Estimated distribution TDD-NIP Service 1: N(125, 15) Service 2: N(145, 35) TDD-IP Service 1: t(18.25, , ) Service 2: t(24.43, , ) We emloy here the same inut arameter values we reorted in the section Comarative statics analysis (see Table 1). For each demand case (normally distributed and stationary, NDD or t-distributed and non-stationary, TDD) and each estimation scenario (use of informative rior, IP, versus use of non-informative rior, NIP) we erform 3 simulations runs. Table 5 summarizes the results for the base case described in Table 1. It reorts the average, maximum, and minimum ercentage differences when comared with the true exected rofit for each scenario. For examle, the Average of the NDD-NIP scenario is calculated as the ratio of the difference between the true exected rofit and the average exected rofit (based on 3 samles of the NDD- NIP case), over the true exected rofit. Similarly, the max of the NDD-IP scenario is calculated as the ratio of the difference between the true exected rofit and the maximum exected rofit (based on 3 samles of the NDD-NIP case), over the true exected rofit. We have also run simulations with larger samle sizes (eg n i ¼ 1 or 15, instead of 5), larger coefficient of variations (eg 25 or 2% instead of about 16%), and various values for arameters such as variable and fixed costs. The results remain qualitatively the same in that the use of informative riors reduces the difference with the true exected rofit significantly. Thus, our numerical results suggest that it is desirable to use good rior information, if available. Summary and conclusions We investigate both the modeling and the demand-estimation asects of a caacity lanning roblem, in which two available resources rovide two services. Demand for one of the services can be satisfied by both tyes of caacity, whereas demand for the other service can only be rovided by a secialized resource (ie downgrades are not allowed). We rovide easy-to-interret, analytic solutions that resemble the familiar critical fractile ratio of the single-resource newsvendor model, but also account for the ossibility of substituting one of the resources with the other. We also enhance our understanding of how changes to demand or inut arameters imact the otimal caacity choices and rofits by erforming a Comarative statics analysis. We then turn our attention to understanding the imact of estimating the unknown arameters of the demand distributions. The traditional estimation aroach is to use samle mean and samle variance, resectively, to aroximate the unknown oulation mean and variance of a demand distribution. It is often the case, however, that useful rior information is available that could imrove the caacity lanning rocess significantly. Thus, we describe how rior information can be used to rovide the otimal caacity levels and a distribution of maximum rofits based on a Bayesian methodology, and we quantify the value of using such rior information. In our numerical exeriments, we consider the case of indeendent NDDs and the case of indeendent TDDs, with the latter case corresonding to non-stationary demands. For each case, we comare exected rofits under scenarios that aly Bayesian estimates or standard samle estimates. Our simulations show that the Bayesian estimates yield exected rofits that are consistently closer to the true exected rofit and thus demonstrate that informative riors can imrove exected rofits significantly. One future direction related to this work would be to consider the correlated demands case. In this case, it would be interesting to identify the circumstances under which the benefits of rior information are more ronounced. Acknowledgments The authors would like to thank the editor and two anonymous referees for their helful comments. Table 5 Percentage difference between true rofits and exected rofits NDD-NIP NDD-IP TDD-NIP TDD-IP Average 1.22%.9% 1.97%.21% Max 4.31%.36% 5.73%.76% Min.1%.%.27%.2% Aendix Proof of Proosition 1. It is easy to show that the objective function in exression (2) is concave in {T 1, T 2 }(seefor examle Bassok et al 7 and Parlar and Goyal 1 ). Thus, if the firm makes ositive rofits, the first-order conditions are both necessary and sufficient. Taking the artial derivative

10 E Stavrulaki et al Stochastic caacity lanning 127 of the rofit exression (2) with resect to T 1 yields q½exressionð2þš ¼ð 1 v 1 Þð1 F 1 ðt 1 ÞÞ c 1 ðf 1 ðt 1 Þ 1Þ qt 1 G 1 þð 2 v 1 Þ ð1 F 2 ðt 1 þ T 2 xþþ f 1 ðxþdx þ c 2 ð1 F 2 ðt 1 þ T 2 xþþf 1 ðxþdx Thus, setting the artial with resect to T 1 equal to zero we get that ð 1 v 1 þ c 1 G 1 Þ ð 1 v 1 þ c 1 ÞF 1 ðt 1 Þ þð 2 v 1 þ c 2 Þ ð1 F 2 ðt 1 þ T 2 xþþ f 1 ðxþdx ¼ which imlies Equation (3). Similarly, taking the artial derivative of the rofit exression (2) with resect to T 2 yields q½exressionð2þš ¼ð 2 v 2 Þð1 F 2 ðt 2 ÞÞ qt 2 G 2 ð 2 v 1 Þ ½F 2 ðt 1 þ T 2 xþ F 2 ðt 2 ÞŠf 1 ðxþdx þ c 2 ½1 F 2 ðt 1 þ T 2 xþš f 1 ðxþdx þ ½1 F 2 ðt 2 ÞŠ f 1 ðxþdx T 1 Observing that the last R term within the braces of the 1 revious equation, T 1 ½1 F 2 ðt 2 ÞŠf 1 ðxþdx, equals ð1 F 2 ðt 2 ÞÞ R T 1 ð1 F 2ðT 2 ÞÞf 1 ðxþdx, we can simlify the first derivative with resect to T 2 as q½exressionð2þš ¼ð 2 v 2 þ c 2 Þð1 F 2 ðt 2 ÞÞ qt 2 G 2 ð 2 v 1 þ c 2 Þ ½F 2 ðt 1 þ T 2 xþ F 2 ðt 2 ÞŠ f 1 ðxþdx and setting the artial with resect to T 2 equal to zero we get that ð 2 v 2 þ c 2 Þð1 F 2 ðt 2 ÞÞ G 2 þð 2 v 1 þ c 2 ÞF 2 ðt 2 Þ ð 2 v 1 þ c 2 Þ which imlies Equation (4). f 1 ðxþdx F 2 ðt 1 þ T 2 xþf 1 ðxþdx ¼ & Algorithm for calculating our numerical results (1) Choose a demand scenario R to be either the NDD or the TDD (ie RA{NDD, TDD}). Choose values for the true m i and s i, i ¼ 1, 2. Denote the true demand distribution for service i by f R i (.), i ¼ 1, 2 (in other words, f R i (.) has mean m i and variance s 2 i ). For every demand scenario R, reeat stes (2) and (3). (2) Choose an estimation scenario S(R) for a given true demand distribution R, wheres(r)a{nip, IP}. For every estimation scenario S(R), reeat stes (3a) (3f) several times. (3a) Generate a random samle of size n i following f R i (.), i ¼ 1, 2. (3b) Derive the estimated demand distributions f E(S(R)) i (.), i ¼ 1, 2, for aroach S(R), denoted for simlicity by f E i (.) (in other words, the arameters of f E i (.) are ð^m i ; ^s i Þ or ð~m i ; ~s i Þ deending on S(R)). (3c) Derive otimal caacity levels, denoted by ^T 1 and ^T 2, that maximize exected rofit for scenario S(R) given the estimated demand distributions f E i (.) ð ^T 1 ; ^T 2 ; f E 1 ð:þ; f E 2 ð:þþ ¼ max T 1 ;T 2 n o E f E 1 ð:þ;f2 Eð:Þ ½ðT 1 ; T 2 ÞŠ : (3d) Evaluate the true exected rofit ð ^T 1 ; ^T 2 ; f R 1 ð:þ; f R 2 ð:þþ given caacity levels ^T 1 and ^T 2 and the true demand distributions f i R (.), i ¼ 1, 2. (3e) Derive the true otimal caacity levels T 1 R and T 2 R that maximize exected rofit for scenario S(R) given the true demand distributions f i R (.): ðt1 R ; T 2 R ; f 1 R ð:þ; f 2 R ð:þþ n o ¼ max E f R T 1 ;T 2 1 ð:þ; f R 2 ð:þ ½ðT 1 ; T 2 ÞŠ : (3f) Evaluate the difference between the true exected rofit for aroximately otimal caacities, ð ^T 1 ; ^T 2 ; f R 1 ð:þ; f R 2 ð:þþ, and the true exected rofit, (T 1 R, T 2 R, f 1 R (.), f 2 R (.)), given the otimal caacities. When iteration limit is reached, go to ste (2). References 1 French S and Smith JQ (eds) (1997). The Practice of Bayesian Analysis, J Wiley and Sons: New York. 2 Fong DKH and Bolton GE (1997). Analyzing ultimatum bargaining: a bayesian aroach to the comarison of two otency curves under shae constraints. J Bus Econ Stat 15: Pammer SE, Fong DKH and Arnold SF (2). Forecasting the enetration of a new roduct: a Bayesian aroach. J Bus Econ Stat 18: Fong DKH, Pammer SE, Arnold SF and Bolton GE (22). Reanalyzing ultimatum bargaining: comaring nondecreasing curves without shae constraints. J Bus Econ Stat 2: DeSabo WS, Kim Y and Fong DKH (1999). A bayesian multidimensional scaling rocedure for the satial analysis of revealed choice data. JEconom89:

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