# A central problem in network revenue management

Save this PDF as:

Size: px
Start display at page:

Download "A central problem in network revenue management"

## Transcription

1 A Randomized Linear Programming Method for Computing Network Bid Prices KALYAN TALLURI Universitat Pompeu Fabra, Barcelona, Spain GARRETT VAN RYZIN Columbia University, New York, New York We analyze a randomized version of the deterministic linear programming (DLP) method for computing network bid prices. The method consists of simulating a sequence of realizations of itinerary demand and solving deterministic linear programs to allocate capacity to itineraries for each realization. The dual prices from this sequence are then averaged to form a bid price approximation. This randomized linear programming (RLP) method is only slightly more complicated to implement than the DLP method. We show that the RLP method can be viewed as a procedure for estimating the gradient of the expected perfect information (PI) network revenue. That is, the expected revenue obtained with full information on future demand realizations. The expected PI revenue can, in turn, be viewed as an approximation to the optimal value function. We establish conditions under which the RLP procedure provides an unbiased estimator of the gradient of the expected PI revenue. Computational tests are performed to evaluate the revenue performance of the RLP method compared to the DLP. A central problem in network revenue management is determining optimal decision rules for sequentially accepting or denying itinerary requests. Bid-price controls are one such class of decision rules. In a bid price control, threshold values (called bid prices) are set for each leg of a network, and an itinerary (path on the network) is accepted only if its fare exceeds the sum of the bid prices along the path. SIMPSON (1989) and WILLIAMSON (1992) first studied this method and proposed approximations to generate bid prices based on dual prices of various mathematical programming formulations of the problem. 1. INTRODUCTION AND OVERVIEW IN TALLURI AND VAN RYZIN (1996), we analyzed the structure of an optimal policy for a dynamic stochastic model of network revenue management. The same model is used in this paper. Time is discrete. We consider an m-leg network, and let x (x 1,..., x m ) denote the (integer) vector of remaining leg capacities and k denote the number of time periods 207 remaining. As a convention, we use superscripts to index components of a vector and use subscripts to denote time or elements of a sequence. We use x T to denote the transpose of a vector x. As in Talluri and van Ryzin (1996), we model demand as a sequence of requests for itineraries (paths) over time. An itinerary consists of a collection of legs together with an associated fare. To model multiple fare classes, we define several itineraries (one for each fare class), each having an identical set of legs but different fares. As a simplification, we assume itineraries that are rejected are lost to the network. In particular, we do not model diversion among itineraries. Define A [a ij ] where a ij 1 if itinerary j uses leg i and a ij 0 otherwise; the jth column of A, A j, is the incidence vector for itinerary j. Abusing this notation somewhat, we shall also use i A j to indicate the legs i that are used by itinerary j. We assume A j has at least one nonzero component, i.e., all itineraries use at least one leg. Each itinerary j contributes revenue r j. Define r (r 1,...,r n ). In this paper, we will not need to specify the Transportation Science / 99 / \$01.25 Vol. 33, No. 2, May Institute for Operations Research and the Management Sciences

2 208 / K. TALLURI AND G. VAN RYZIN arrival process in detail; the only demand information we need is the distribution of total demand to come at each point in time. However, as a canonical example of an arrival process, one can consider the case where only one itinerary arrives in a period and the probability that itinerary j arrives in period k is p j n j k, where j 1 p k 1 for all k. Let Y j denote the total number of requests for itinerary j in the remaining time k, and define Y (Y 1,..., Y n ). To avoid excessive notation, we do not explicitly index Y with the remaining time k; however, Y will always represent the total demand to come over the remaining k units of time. Let J k (x) denote the value function (optimal expected revenue give x and k). This function is defined formally for a general network model with Markovian arrivals in Talluri and van Ryzin (1996). However, for our purposes, an informal definition of J k (x) will suffice. In Talluri and van Ryzin (1996), we showed that it is optimal to accept a reservation for itinerary j with a revenue r j in period k if and only if r j J k 1 x J k 1 x A j. (1) This condition is intuitive: r j J k 1 (x A j ) is the expected revenue if we accept the reservation, and J k 1 (x) is the expected revenue if we do not; therefore, if the former is greater than the latter, it makes sense to accept the reservation. A bid price control, as defined in Talluri and van Ryzin (1996), specifies an m-vector of values (bid prices), k (x), for each remaining capacity x and remaining time k, such that an itinerary j is accepted if and only if r j i k x. i A j That is, itinerary j is accepted if and only if its revenue exceeds the sum of the bid prices on the legs it uses. This type of control is appealing on both intuitive and practical grounds. Intuitively, bid prices represent the marginal value of capacity (also called displacement cost) on each leg; if the current request exceeds the sum of the expected marginal values of the legs it requires, then it makes intuitive sense to accept it, because the current revenue is larger than our estimate of the next best use for the same capacity. Bid price controls are also appealing from an implementation standpoint, because they require only a small number of control values (one bid price for each leg), and the decisions to accept or reject arriving requests are trivial to determine given these values. However, in general, the optimality condition, Eq. 1, does not lead to a bid price control. (See Talluri and van Ryzin (1996) for counter examples.) At the same time, it is easy to see how condition 1 motivates a bid price structure. Indeed, if we imagine x is a real vector and J k 1 (x) is a differentiable function of x, then a first-order approximation to Eq. 1 yields r j J k 1 x J k 1 x A j T x J k 1 x A j x i J k 1 x. (2) i A j If this first-order approximation is good, then using the gradient x J k 1 (x) as a vector of bid prices should produce close-to-optimal decisions. Although this reasoning is very informal, it lies at the heart of the concept of bid price controls. Moreover, it was proved in Talluri and van Ryzin (1996) that a bid price policy is, in fact, asymptotically optimal as the demand volume and leg capacities tend to infinity, which provides theoretical support for the use of bid price controls. 1.1 Approximation Methods To implement a bid price control strategy, one needs a method for generating bid price values. Computing J k ( ) is not feasible in practice because of the enormous size of the state space. (The state space is of size O(C m ), where C is the maximum leg capacity and m is the number of legs. Typical numbers for a major airline network are C 300 and m 5000 or more.) Therefore, the only realistic option in practice is to use approximation methods. Such methods can generally be interpreted as consisting of two steps: 1) approximate the value function J k (x); and 2) use the associated gradient (or subgradient) information from the approximation as a vector of bid prices. That is, form an approximation J A k (x) to the value function and use J A k (x) (or a subgradient if J A k (x) does not exist) for the vector of bid prices. Typically, these approximations are recomputed periodically throughout the booking horizon in response to changes in the remaining capacity or demand forecast. Perhaps the simplest example of such an approximation is the deterministic linear programming (DLP) method, which was initially investigated by Williamson (1992). The DLP approximation to the value function is obtained by finding a tentative itinerary allocation, y, that solves J k LP x max r T y (3) Ay x (4) 0 y EY. (5)

3 COMPUTING BID PRICES / 209 Note this approximation corresponds to assuming that demand to come, Y, is always equal to its mean, EY. The primal itinerary allocation, y, is not used directly. Rather, let ( 1,..., m ) denote the dual prices associated with constraint (4). Provided the linear program is not degenerate at the optimal solution, J k LP (x) exists and is equal to. Therefore, the vector of dual prices is used for the bid prices. Although this method is quite simple and intuitive, it has several obvious weaknesses. First, it only uses the mean demand and ignores all other distributional information on Y. This is clearly unrealistic, because it is known that, for single leg problems, the expected marginal value of seat capacity depends on the entire distribution of future demand BELOBABA (1989), BRUMELLE and MCGILL (1993), LITTLEWOOD (1972), ROBINSON (1991), and WOLLMER (1992). As a result, the DLP can produce poor approximations to the true marginal expected value. For example, one can easily show that, if the expected demand on a leg is strictly less than the capacity, then the deterministic linear program will return a zero bid price for the leg. Yet, with highly variable demand, the expected marginal value can be much higher than zero higher than the fares of some of the lower fare classes perhaps even though the mean demand is less than capacity. Such discrepancies can lead to poor accept/deny decisions. However, the linear programming method is appealing precisely because it is so simple and computationally efficient. It is desirable, therefore, to have a method that is nearly as simple and efficient as the deterministic linear programming method but which also captures more distributional information about future demand. 1.2 A Randomized Linear Programming Method One idea for incorporating more stochastic information into the linear programming method is to replace the expected value of Y in constraint (5) by the random vector Y itself. Then, the expected value of the resulting optimal solution can be used as an approximation to the value function. That is, define v k x, Y max r T y (6) Ay x (7) 0 y Y. (8) The optimal value v k (x, Y) is a random variable. Let (x, Y) denote an optimal vector of dual prices for the set of constraints (7), and note that (x, Y) is also a random vector. Next, consider the approximation to the value function, J k PI x Ev k x, Y. (9) We call this the perfect information (PI) approximation, because it corresponds to a case in which future allocations (and revenues) are based on perfect knowledge of Y; however, at time k, Y is not yet realized. Assuming the gradient exists, we then use x Ev k (x, Y) as our vector of bid prices. However, this method is viable only if we can efficiently compute x Ev k (x, Y). One appealing approach is to consider interchanging differentiation and expectation. Assuming such an interchange is justified (sufficient conditions are given in Lemma 1 below), we have x Ev k x, Y E x v k x, Y. (10) This interchange, in turn, suggests a procedure for estimating x Ev k (x, Y). Simply simulate N independent samples of the demand vector, Y 1,...,Y N, and solve Eqs. 6 8 for each sample. Then estimate the gradient using N 1 x, Y N i. (11) i 1 That is, simply average the dual prices from the PI allocation over a series of randomly generated demands. We call this idea the randomized linear programming (RLP) method. The RLP method has several appealing features. First, it is a simple modification to the DLP method, so it can be easily incorporated into production revenue management systems that use DLP. Second, it has the flexibility to model very general demand distributions, because one only needs the ability to simulate demand to apply the method (e.g., one could allow for different coefficients of variation and/or correlations among the components of Y). Finally, it incorporates distributional information on future demand. To see why, consider the simple case where m 1 (a single leg) and A [1] (one itinerary). In this case, r, x, Y, and are all scalars, and it is easy to see that (x, Y) r if Y x and (x, Y) 0ifY x, so if Y has a continuous distribution, then d dx Ev k x, Y E x, Y rp Y x, which is Littlewood s (1972) expression for the expected marginal value of capacity. That is, in this simple case, the RLP method produces the correct expected marginal value. In contrast, the DLP

4 210 / K. TALLURI AND G. VAN RYZIN method produces a marginal value of zero if EY x and a marginal value of r if EY x. This example suggests why the RLP method may produce a somewhat better approximation of the marginal value of capacity than the DLP method. It turns out that the RLP method, in fact, received some brief attention over a decade ago in a study of network control methods at American Airlines conducted by SMITH and PENN (1988). However, the idea is not widely known, and we are not aware of any extensive study of its theoretical properties or practical performance. 1 In the remainder of the paper, we provide such a study. In particular, in Section 2, we give sufficient conditions under which both Eq. 10 is justified and Eq. 11 is an unbiased (and consistent) estimator of the gradient of the expected PI revenue. These conditions also guarantee the existence of x Ev k (x, Y). In Section 3, we examine the situation where x Ev k (x, Y) fails to exist. In Section 4, we then perform a series of computational tests to assess the revenue performance of the RLP method relative to the DLP method. Our conclusions are presented in Section PROPERTIES OF THE RLP ESTIMATOR WE NEXT ESTABLISH conditions under which the RLP estimator gives an unbiased estimate of the gradient of Ev k (x, Y). This is important because, in general, E (x, Y) is only a subgradient of Ev k (x, Y). If E (x, Y) is not a gradient, then the logic derived from Eq. 2 for using E (x, Y) as a vector of bid prices fails because the directional derivative (if it exists), D x; d lim h30 1 h Ev k x hd, Y Ev k x, Y, (12) is no longer given by E T (x, Y)d. The following two conditions, however, are sufficient to both justify Eq. 10 and to guarantee that Eq. 11 is an unbiased estimator of the gradient: CONDITION 1. If x j S j A j, then {A j : j S} must have rank m. That is, x does not lie in any subspace defined by a subset of fewer than m columns of A. CONDITION 2. P(Y j y) is continuous in y, for all j 1,..., n. 1 Smith and Penn (1988) concluded that the RLP method was too time consuming relative to the improvement it provided, and they focused their testing on the DLP method. (Examples of failures of these conditions are provided in Section 2.2.) Our main result is summarized in Theorem 1 below. The proof of the theorem requires a sequence of lemmas, which we present next. 2.1 Proof of Unbiasedness First, we require a lemma by GLASSERMAN (1994), which provides sufficient conditions for interchanging expectation and differentiation. LEMMA 1. (Glasserman, 1994). Let X( ) be a random function satisfying the Lipschitz condition: There exists a K 0 such that X h X Kh, for all h 0. Suppose that X( ) is differentiable at (a.s.). Then (d/d )EX( ) exists and d d EX E d X. d The first condition of the lemma is easy to establish for the RLP method. Indeed, it is easy to see that if r max{r j : j 1,..., n} and e i is the ith unit vector, then, from Eqs. 6 8 and the fact that elements of A are0or1, v k (x he i, Y) v k (x, Y) hr for all x and i. That is, increasing x i by h allows n at most an increase of h in j 1 y j, which contributes at most r h to the objective function value. Therefore, we have LEMMA 2. The function v k (x, Y) defined in Eqs. 6 8 satisfies the Lipschitz condition in Lemma 1. From Lemma 1, it then follows that if x v k (x, Y) exits with probability one, Eq. 10 is justified. Note that x v k (x, Y) exists if and only if the dual price (Y) is unique. Therefore, we must establish conditions under which (Y) is unique with probability one. To do so, consider the Lagrangian dual of the linear program, Eqs. 6 8, L x, max 0 y Y n y j r j T A j T x j 1 n Y j r j T A j T x, (13) j 1 where (x) max{0, x}. Note that L(x, ) is convex in. Let * denote a solution to the problem min L x,. (14) 0

5 COMPUTING BID PRICES / 211 By strong duality, v k (x, Y) L(x, * ) and * is always a subgradient of v k (x, Y). If * is unique, then it is also a gradient, and x v k (x, Y) exists. [See Chapter 6 of Bazaraa, Sherali, and Shetty (1993).] Therefore, we must show that * is unique with probability one. To this end, let d 0 denote a feasible direction from *, i.e., there exists an 0 such that Define the sets * hd 0, h 0,. (15) E 0 j: r j * T A j 0, (16) E 1 j: r j T * A j 0, r j T * A j hd T A j 0, (17) E 2 j: r j T A j 0, r j T * A j hd T A j 0, (18) and note from Eq. 13 that L x, * hd L x, * hd T x Y j A j E 0 E 1 j. (19) The solution * is unique if the right hand side above is strictly positive for all feasible directions d. We need the following lemma. LEMMA 3. Suppose that C1 holds, d T A j E 0 E 1, and d T x 0. Then d 0. Proof. By Eq. 19 and the optimality of *,itmust be true that the system, T A j 0, j E 0 E 1 T x 0, is unsolvable, else is a strictly improving direction. Thus, by Farka s lemma, the system j A j x (20) j E 0 E 1 is solvable. But, by condition C1, {A j : j E 0 E 1 } has rank m, sod T A j E 0 E 1 implies d 0. e The almost sure existence of the gradient is established in the next lemma. LEMMA 4. Suppose C1 and C2 hold. Then x v k (x, Y) exists with probability one. Proof. As mentioned above, it suffices to show that * is a unique optimal solution to Eq. 14 with probability one. To do so, we must show that, with probability one, there are no directions d 0 for which L(x, hd) L(x, ). The proof is by contradiction. Indeed, if d 0 and L(x, hd) L(x, ), then, from Eq. 19, we T must have d x Y j A j E 0 E 1 j 0. By Condition 2, it is clear that P(x j E0 E 1 Y j A j ) 0. Hence, the only way the above equality can hold with positive probability is if d T x 0 and d T A j 0, j E 0 E 1. But then, Lemma 3 implies d 0, a contradiction. e THEOREM 1. Suppose conditions C1 and C2 hold, then x Ev k (x, Y) exists and x Ev k x, Y E x v k x, Y E x, Y. Proof. This follows directly from Lemmas 1, 2, and 4. e Theorem 1 shows that, under C1 and C2, the RLP estimator, Eq. 11, is indeed an unbiased estimator of the gradient x Ev k (x, Y); by the strong law of large numbers, the estimator is also consistent. 2.2 Examples of Nondifferentiability To gain some insight into Conditions 1 and 2, it is useful to consider simple examples in which the linear program, Eqs. 6 8 is degenerate with positive probability, and thus x Ev k (x, Y) does not exist. First, consider the following example which violates Condition 1. r 100, 1, 1 A x 2, 2 Note x 2A 1, so Condition 1 is violated as claimed. Assume Condition 2 holds, however, and that P(Y 1 2) 1 2. Then, for realizations with Y 1 2, it is clear that the optimal solution is y (2, 0, 0), a degenerate optimal solution. Therefore, the linear program is degenerate with probability at least 1 2. Next, suppose r 100, 1000, 1 A , x 2, 3 so Condition 1 is satisfied, but suppose P(Y 2 1) 1 2, so Condition 2 is violated. Also, assume P(Y 2 2) 1 2. Then, when Y 2 1 and Y 2 2, the linear program has optimal solution x (1, 1, 0), which is

6 212 / K. TALLURI AND G. VAN RYZIN degenerate. Therefore, the linear program is degenerate with probability at least SOME COMMENTS ON THE NONDIFFERENTIABLE CASE LET R n DENOTE the set of demand vectors Y for which the linear program, Eqs. 6 8, is degenerate. Let c R n, i.e., c is the complement of. The nondifferentiable case occurs when P(Y ) 0. In this section, we discuss some issues related to this case. 3.1 Convergence to a Subgradient Estimator If the linear program, Eqs. 6 8, is degenerate for some Y, then the dual solution (x, Y) will depend on the solution algorithm (e.g., interior point methods may return a different dual solution than simplex-based methods). Thus, we consider (x, Y) to be defined as the dual solution returned by a given linear programming algorithm. Note that, for any realization of demand Y, (x, Y) is a subgradient of v k (x, Y), i.e., v k z, Y v k x, Y T x, Y z x for all z. Because all terms above are uniformly bounded in Y, taking expectations on both sides implies J k PI z J k PI x E T x, Y z y, and hence E (x, Y) is a subgradient of J PI k (x). However, the existence of lim N3 (1/N) N i 1 (x, Y i ) will depend on how Eqs. 6 8 is solved at each step i. For example, it is possible that the linear programming algorithm is initialized with a previous solution (warm started), and hence (x, Y i ) becomes a function of the history of the sequence of demand vectors {Y 1,..., Y i 1 }. [Our experience in numerical testing is that such warm starts significantly speed up the computation of (x, Y i ).] Under such conditions, it is possible that the estimator, Eq. 11 does not converge. However, the following proposition is clearly true. PROPOSITION 1. Let A denote the linear programming algorithm used to solve Eqs If algorithm A is initialized with the same data on each run (e.g., it is not warm started with data from prior solutions), then lim N3 (1/N) N i 1 (x, Y i ) always exists and is a subgradient of Ev k (x, Y). 3.2 Directional Derivatives As mentioned, when E (x, Y) is only a subgradient, then the logic derived from Eq. 2 for using E (x, Y) as a vector of bid prices fails because the directional derivative (if it exists) is no longer given by E T (x, Y)d. We next show that the directional derivatives D(x; d) (see Eq. 12) always exist. The proof leads to a natural algorithmic modification to the estimator, Eq. 11, to make it an unbiased estimator of directional derivatives. Define u(x, Y; d) as the optimal value of the linear program Then we have u x, Y; d min d T (22) T x v k x, Y T A r 0. PROPOSITION 2. The directional derivative, D(x; d), exists and D x; d Eu x, Y; d. Proof. Let v k (x, Y) denote the set of subgradients of v k (x, Y) and note that v k (x, Y) { : T x v k (x, Y), T A r, 0}. Observe that v k (x, Y) is uniformly bounded for all Y, provided there are no zero columns of A. It is well known (Bazaraa, Sherali, and Shetty (1993), page 218.) that the directional derivative, D(x, Y; d), satisfies D x, Y; d lim h30 1 h v k x hd, Y v k x, Y inf d T : v k x, Y. Note the right hand side above is simply u(x, Y; d). Because v k (x, Y) is uniformly bounded for all Y, it follows that the D(x, Y; d) exists for all Y and is also uniformly bounded. Applying Lemma 1 establishes the result. e Proposition 2 shows that, by solving the auxiliary linear program, Eq. 22 for each sample, we can obtain the unbiased estimator of the directional derivative, N 1 u x, Y N i ; d. i 1 The disadvantage of this approach, of course, is that this estimator is for a single direction d. To make effective use of such information in network revenue management, we need to evaluate the derivative for every itinerary j (i.e., every direction A j ) by solving a separate sequence of linear programs. For a large airline network, the number of itineraries, n, can be

7 COMPUTING BID PRICES / 213 on the order of 100,000, making such an approach somewhat impractical. 4. SIMULATION EXPERIMENTS OUR ANALYSIS THUS FAR provides a formal basis for interpreting the estimator, Eq. 11, and the RLP method in general. However, what ultimately matters in practice is how well the RLP method performs on real-world networks. In this section, we describe a series of simulation experiments comparing the revenue performance of the DLP and RLP methods. We first describe the simulation methodology and the test networks used in our tests. We then present and discuss the numerical results. Our conclusion is that the RLP method provides a small but significant improvement over DLP and, thus, warrants further consideration. 4.1 Simulation Methodology To evaluate the performance of the RLP method, we conducted a series of simulation tests using both real-world and randomly generated networks. In our simulations, each request is for one seat (no group bookings) and we did not simulate cancellations or no-shows (no overbooking effects). Each day consists of 1000 minutes (roughly 18 hours) of booking time. Booking requests are generated over a booking horizon of approximately 20 days. Booking requests are randomly generated for each instance using a two-step process. First, the final demand for an itinerary is generated based on a given distribution. We used both a Poisson distribution and a truncated normal distribution with a fixed coefficient of variation in our tests. Each itinerary has an associated booking curve, that specifies the fraction of total demand observed as a function of time. In the second step, this booking curve is used as a probability distribution to determine the arrival time of a request. That is, for each one of the generated requests, we generate a uniform random number between 0 and 1, and the arrival time of the request is then determined from the inverse of the booking curve function. Note that this process produces a (nonhomogeneous) Poisson process when the final distribution is Poisson; for non-poisson cases, it produces demand that statistically conforms to both the final demand distribution and the booking curve. The resulting stream of itinerary requests is then sorted based on their arrival times and processed sequentially by a simulated reservation system that uses a bid-price control rule to make each accept/ deny decision. Bid prices are constant for each simulated day, but are recalculated at the start of each simulated day by rerunning the bid price optimization (either DLP or RLP) using updated forecasts and capacities. This procedure mimics the overnight processing performed by airlines in practice. Although actual arrivals were generated according to these discrete distributions, within the RLP we used continuous, truncated-normal distributions. That is, each component of Y in Eq. 8 within our RLP computations was modeled as a truncatednormal random variable with the appropriate mean and standard deviation. We did this because continuous distributions better match the assumptions of the linear programming model (e.g., continuous allocation of continuous capacity) and they also tend to produce less degeneracy. To make the test more realistic, we also simulated a simple forecasting process. That is, the optimization did not use actual demand means and variances; rather, estimates of the means and variances were determined based on observations of past simulated demand. In this way, we tried to mimic the effect of forecast error on the control policy. Forecasts were obtained as follows. Twenty files of arrival histories were generated according to the given distributions, corresponding to twenty past departures of each itinerary. These twenty histories were used to estimate means and variances for the first simulation run. Then, a new demand process was simulated and added to the history. These twenty-one history files were then used to estimate means and variances for the second simulation run, and so on. This forecasting methodology basically assumes stationarity and independence of demand. Given that there is no seasonality or trend in the sequence of historical files being generated, this is an accurate forecasting method. In practice, forecasting methods would need to take into account nonstationarities in the demand process. This process was used to generate 20 simulated departure days of each network under both the RLP and DLP policy. The same sample paths of data were used for each method (coupled simulations) to reduce the variance in our performance comparisons. It appeared that 20 runs were sufficient to get a reasonably accurate estimate of the revenue differences, because increasing the number of runs showed little difference in the relative performance of the methods. For our RLP implementation, 30 samples of the dual prices (Y) were averaged on each call to the optimization module. Based on some preliminary testing, it did not seem that the performance of the RLP method improved much with larger samples, though this behavior could very well be problem dependent.

8 214 / K. TALLURI AND G. VAN RYZIN Fig. 1. Network 1 with Poisson demand: percent improvement over DLP. 4.2 Test Networks We used three different test networks for our simulations. The first one (Network 1) is based on data from a hub-bank (a set of arriving flights connecting to a set of departing flights) of a major international airline with 102 flight legs, 1066 itineraries, 7 fare classes, and 1 compartment. The forecast of the demand means for each itinerary (called an origin destination fare class, or ODF) were given along with capacities for each flight leg. Booking curves based on historical observations were provided for each itinerary. From these data, we generated two different demand scenarios: one with Poisson demand for each ODF and another in which demand for each ODF has a rounded and truncated normal distribution (truncated by 0 on the left) with a coefficient of variation (CV) of The parameters of the normal distribution are determined so that the mean and the standard deviation of its truncated version match the desired mean and standard deviation. The second dataset (Network 2) comes from a large U.S. airline s domestic hub-bank with 62 legs, 517 itineraries, 11 fare classes, and 1 compartment. The fare structure for this network was quite different from Network 1, exhibiting significantly lower fare dispersion. Again, both Poisson and truncated normal demand were simulated for this network. Finally, we generated a random airline network with a hub-and-spoke topology (Network 3), consisting of 20 legs, 120 itineraries, and 8 fare classes. The demand means for each ODF were randomly generated, uniformly spread between 0.0 and 3.0. The fares for each ODF were also randomly generated from a uniform distribution in specific ranges for each fare class. The fares varied between \$30 and \$1,100. The booking curves enforced a strict lowbefore-high fare pattern for each O D pair. Only the Poisson demand case was simulated for Network 3. For each network, various load factors were simulated by scaling the mean demands by a constant. We generated load factors approximately between 60 and 85%. For each run, an upper bound on the maximum achievable revenue was also calculated by solving the linear programming relaxation of the so-called perfect-hindsight integer program. (That is, the linear program that allocates capacity based on perfect information on the total realized demand.) This bound is likely to be loose for two reasons. First, it assumes perfect information on future demand, which is clearly overly optimistic. Second, for a huband-spoke network, there may be a gap between the optimal value of the integer program and its linearprogramming relaxation. Nevertheless, the perfecthindsight upper bound does give some measure of the absolute optimality gap. 4.3 Results The results of our simulation experiments are shown in Figs These graphs show both the RLP and perfect-hindsight upper bound revenue, expressed as a percentage difference from the DLP revenue. The numbers in parentheses are the revenue upper bounds expressed as a percentage difference from the DLP revenue. The RLP generates a slight improvement in revenue over the DLP policy for the two airline networks (Networks 1 and 2). The revenue improvement Fig. 2. Network 1 with demand from a truncated normal distribution (CV 1.414): percent improvement over DLP.

9 COMPUTING BID PRICES / 215 Fig. 3. Network 2 with Poisson demand: percent improvement over DLP. Fig. 5. Random Network 3 with Poisson demand: percent improvement over ranges from 0.04 to 0.32%. Although small in absolute terms, these improvements are significant, especially at the higher load factors. The performance of the RLP method on the randomly generated network, however, is somewhat erratic and does not uniformly dominate the DLP method. However, it seems to be at least comparable to DLP. One factor that could explain such behavior on this network is the difference in the arrival process of bookings. The real-world datasets from Networks 1 and 2 have booking curves that are closer to being uniform over time. In the random network (Network 3), bookings were generated using a strict low-before-high fare booking pattern. It is plausible that the expected perfect information revenue is a better approximation of the value function in the former case. That is, assuming perfect future allocations is a particularly optimistic assumption in the low-before-high case. Finally, we note that, on these three networks, we almost never encountered degeneracy of the linear programs of the RLP method. We attribute this to the continuous distributions used in our RLP computations combined with the irregular network capacities and itineraries on these networks. Again, however, such behavior is likely to be highly problem dependent. 5. CONCLUSIONS THE RLP METHOD is a simple and appealing alternative to DLP for large-scale network revenue management applications. Overall, our numerical tests indicate that RLP provides a small but significant improvement over DLP, though this conclusion is preliminary and warrants further investigation. It would be interesting to see if there are other variations of this approach that would improve revenue performance further. Also, there may be some interesting work in using variance reduction techniques to improve the efficiency of the estimator, Eq. 11. More generally, simulation-based optimization techniques may prove to be a fruitful approximation approach for future research in this area. REFERENCES Fig. 4. Network 2 with demand from a truncated normal distribution (CV 1.414): percent improvement over DLP. M. S. BAZARAA, H. D. SHERALI AND C. M. SHETTY, Nonlinear Programming: Theory and Algorithms, John Wiley, New York, 1993.

10 216 / K. TALLURI AND G. VAN RYZIN P. P. BELOBABA, Application of a Probabilistic Decision Model to Airline Seat Inventory Control, Opns. Res. 37, (1989). S. L. BRUMELLE AND J. I. MCGILL, Airline Seat Allocation with Multiple Nested Fare Classes, Opns. Res. 41, (1993). P. GLASSERMAN, Perturbation Analysis of Production Networks, in Stochastic Modeling and Analysis of Manufacturing Systems, Yao, D. (ed.), Springer-Verlag, Berlin, K. LITTLEWOOD, Forecasting and Control of Passengers, 12th AGIFORS Symposium Proceedings, (1972). L. W. ROBINSON, Optimal and Approximate Control Policies for Airline Booking with Sequential Nonmonotonic Fare Classes, Opns. Res. 43, (1991). R. W. SIMPSON, Using Network Flow Techniques to Find Shadow Prices for Market and Seat Inventory Control, MIT Flight Transportation Laboratory Memorandum M89-1, Cambridge, Massachusetts, B. C. SMITH AND C. W. PENN, Analysis of Alternative Origin Destination Control Strategies, in AGIFORS Symposium proc. 28, Airline Group of the International Federation of Operational Research Societies, K. T. TALLURI AND G. J. VAN RYZIN, An Analysis of Bid- Price Controls for Network Revenue Management, Management Sci. 44, (1996). E. L. WILLIAMSON, Airline Network Seat Control. Ph.D thesis, MIT, Cambridge, Massachusetts, R. D. WOLLMER, An Airline Seat Management Model for a Single Leg Route when Lower Fare Classes Book First, Opns. Res. 40, (1992). (Received: May 1997; revisions received: February 1998, January 1999; accepted: January 1999)

### Role of Stochastic Optimization in Revenue Management. Huseyin Topaloglu School of Operations Research and Information Engineering Cornell University

Role of Stochastic Optimization in Revenue Management Huseyin Topaloglu School of Operations Research and Information Engineering Cornell University Revenue Management Revenue management involves making

### Airline Revenue Management: An Overview of OR Techniques 1982-2001

Airline Revenue Management: An Overview of OR Techniques 1982-2001 Kevin Pak * Nanda Piersma January, 2002 Econometric Institute Report EI 2002-03 Abstract With the increasing interest in decision support

### A Randomized Linear Programming Method for Network Revenue Management with Product-Specific No-Shows

A Randomized Linear Programming Method for Network Revenue Management with Product-Specific No-Shows Sumit Kunnumkal Indian School of Business, Gachibowli, Hyderabad, 500032, India sumit kunnumkal@isb.edu

### A Lagrangian relaxation approach for network inventory control of stochastic revenue management with perishable commodities

Journal of the Operational Research Society (2006), 1--9 2006 Operational Research Society Ltd. All rights reserved. 0160-5682/06 \$30.00 www.palgrave-journals.com/jors A Lagrangian relaxation approach

### A Statistical Modeling Approach to Airline Revenue. Management

A Statistical Modeling Approach to Airline Revenue Management Sheela Siddappa 1, Dirk Günther 2, Jay M. Rosenberger 1, Victoria C. P. Chen 1, 1 Department of Industrial and Manufacturing Systems Engineering

### Models and Techniques for Hotel Revenue. Management using a Rolling Horizon

Models and Techniques for Hotel Revenue Management using a Rolling Horizon Paul Goldman * Richard Freling Kevin Pak Nanda Piersma December, 2001 Econometric Institute Report EI 2001-46 Abstract This paper

### Randomization Approaches for Network Revenue Management with Customer Choice Behavior

Randomization Approaches for Network Revenue Management with Customer Choice Behavior Sumit Kunnumkal Indian School of Business, Gachibowli, Hyderabad, 500032, India sumit kunnumkal@isb.edu March 9, 2011

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### THEORY OF SIMPLEX METHOD

Chapter THEORY OF SIMPLEX METHOD Mathematical Programming Problems A mathematical programming problem is an optimization problem of finding the values of the unknown variables x, x,, x n that maximize

### SIMULATING CANCELLATIONS AND OVERBOOKING IN YIELD MANAGEMENT

CHAPTER 8 SIMULATING CANCELLATIONS AND OVERBOOKING IN YIELD MANAGEMENT In YM, one of the major problems in maximizing revenue is the number of cancellations. In industries implementing YM this is very

### Re-Solving Stochastic Programming Models for Airline Revenue Management

Re-Solving Stochastic Programming Models for Airline Revenue Management Lijian Chen Department of Industrial, Welding and Systems Engineering The Ohio State University Columbus, OH 43210 chen.855@osu.edu

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### Principles of demand management Airline yield management Determining the booking limits. » A simple problem» Stochastic gradients for general problems

Demand Management Principles of demand management Airline yield management Determining the booking limits» A simple problem» Stochastic gradients for general problems Principles of demand management Issues:»

### Revenue Management for Transportation Problems

Revenue Management for Transportation Problems Francesca Guerriero Giovanna Miglionico Filomena Olivito Department of Electronic Informatics and Systems, University of Calabria Via P. Bucci, 87036 Rende

### MATHEMATICAL BACKGROUND

Chapter 1 MATHEMATICAL BACKGROUND This chapter discusses the mathematics that is necessary for the development of the theory of linear programming. We are particularly interested in the solutions of a

### MIT ICAT. Airline Revenue Management: Flight Leg and Network Optimization. 1.201 Transportation Systems Analysis: Demand & Economics

M I T I n t e r n a t i o n a l C e n t e r f o r A i r T r a n s p o r t a t i o n Airline Revenue Management: Flight Leg and Network Optimization 1.201 Transportation Systems Analysis: Demand & Economics

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### On the Interaction and Competition among Internet Service Providers

On the Interaction and Competition among Internet Service Providers Sam C.M. Lee John C.S. Lui + Abstract The current Internet architecture comprises of different privately owned Internet service providers

### REVENUE MANAGEMENT: MODELS AND METHODS

Proceedings of the 2008 Winter Simulation Conference S. J. Mason, R. R. Hill, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. REVENUE MANAGEMENT: MODELS AND METHODS Kalyan T. Talluri ICREA and Universitat

### Airline Reservation Systems. Gerard Kindervater KLM AMS/RX

Airline Reservation Systems Gerard Kindervater KLM AMS/RX 2 Objective of the presentation To illustrate - what happens when passengers make a reservation - how airlines decide what fare to charge to passengers

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### Routing in Line Planning for Public Transport

Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany MARC E. PFETSCH RALF BORNDÖRFER Routing in Line Planning for Public Transport Supported by the DFG Research

### Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets

Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that

### Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

### S = {1, 2,..., n}. P (1, 1) P (1, 2)... P (1, n) P (2, 1) P (2, 2)... P (2, n) P = . P (n, 1) P (n, 2)... P (n, n)

last revised: 26 January 2009 1 Markov Chains A Markov chain process is a simple type of stochastic process with many social science applications. We ll start with an abstract description before moving

### Stochastic Inventory Control

Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Cargo Capacity Management with Allotments and Spot Market Demand

Submitted to Operations Research manuscript OPRE-2008-08-420.R3 Cargo Capacity Management with Allotments and Spot Market Demand Yuri Levin and Mikhail Nediak School of Business, Queen s University, Kingston,

### On spare parts optimization

On spare parts optimization Krister Svanberg Optimization and Systems Theory, KTH, Stockholm, Sweden. krille@math.kth.se This manuscript deals with some mathematical optimization models for multi-level

### Optimal Booking Limits in the Presence of Strategic Consumer Behavior

Cornell University School of Hotel Administration The Scholarly Commons Articles and Chapters School of Hotel Administration Collection 2006 Optimal Booking Limits in the Presence of Strategic Consumer

### LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Revenue Management with Correlated Demand Forecasting

Revenue Management with Correlated Demand Forecasting Catalina Stefanescu Victor DeMiguel Kristin Fridgeirsdottir Stefanos Zenios 1 Introduction Many airlines are struggling to survive in today's economy.

### An Improved Dynamic Programming Decomposition Approach for Network Revenue Management

An Improved Dynamic Programming Decomposition Approach for Network Revenue Management Dan Zhang Leeds School of Business University of Colorado at Boulder May 21, 2012 Outline Background Network revenue

### By choosing to view this document, you agree to all provisions of the copyright laws protecting it.

This material is posted here with permission of the IEEE Such permission of the IEEE does not in any way imply IEEE endorsement of any of Helsinki University of Technology's products or services Internal

### 1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

CS787: Advanced Algorithms Topic: Online Learning Presenters: David He, Chris Hopman 17.3.1 Follow the Perturbed Leader 17.3.1.1 Prediction Problem Recall the prediction problem that we discussed in class.

### A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem

A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,

### Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach

MASTER S THESIS Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach PAULINE ALDENVIK MIRJAM SCHIERSCHER Department of Mathematical

### Minimize subject to. x S R

Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such

### Demand Forecasting in a Railway Revenue Management System

Powered by TCPDF (www.tcpdf.org) Demand Forecasting in a Railway Revenue Management System Economics Master's thesis Valtteri Helve 2015 Department of Economics Aalto University School of Business Aalto

### Min-cost flow problems and network simplex algorithm

Min-cost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.

### 10.1 Integer Programming and LP relaxation

CS787: Advanced Algorithms Lecture 10: LP Relaxation and Rounding In this lecture we will design approximation algorithms using linear programming. The key insight behind this approach is that the closely

### Airline Schedule Development

Airline Schedule Development 16.75J/1.234J Airline Management Dr. Peter Belobaba February 22, 2006 Airline Schedule Development 1. Schedule Development Process Airline supply terminology Sequential approach

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### CHAPTER 17. Linear Programming: Simplex Method

CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2

### 6 Scalar, Stochastic, Discrete Dynamic Systems

47 6 Scalar, Stochastic, Discrete Dynamic Systems Consider modeling a population of sand-hill cranes in year n by the first-order, deterministic recurrence equation y(n + 1) = Ry(n) where R = 1 + r = 1

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

### Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities

Combinatorics, Probability and Computing (2005) 00, 000 000. c 2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom Uniform Multicommodity Flow through the Complete

### Single-Period Balancing of Pay Per-Click and Pay-Per-View Online Display Advertisements

Single-Period Balancing of Pay Per-Click and Pay-Per-View Online Display Advertisements Changhyun Kwon Department of Industrial and Systems Engineering University at Buffalo, the State University of New

### Date: April 12, 2001. Contents

2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### Optimizing Call Center Staffing using Simulation and Analytic Center Cutting Plane Methods

Submitted to Management Science manuscript MS-00998-2004.R1 Optimizing Call Center Staffing using Simulation and Analytic Center Cutting Plane Methods Júlíus Atlason, Marina A. Epelman Department of Industrial

### TOWARDS AN EFFICIENT DECISION POLICY FOR CLOUD SERVICE PROVIDERS

Association for Information Systems AIS Electronic Library (AISeL) ICIS 2010 Proceedings International Conference on Information Systems (ICIS) 1-1-2010 TOWARDS AN EFFICIENT DECISION POLICY FOR CLOUD SERVICE

### By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

### 4: SINGLE-PERIOD MARKET MODELS

4: SINGLE-PERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market

### Absolute Value Programming

Computational Optimization and Aplications,, 1 11 (2006) c 2006 Springer Verlag, Boston. Manufactured in The Netherlands. Absolute Value Programming O. L. MANGASARIAN olvi@cs.wisc.edu Computer Sciences

### Lecture 6: Approximation via LP Rounding

Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### LINEAR PROGRAMMING WITH ONLINE LEARNING

LINEAR PROGRAMMING WITH ONLINE LEARNING TATSIANA LEVINA, YURI LEVIN, JEFF MCGILL, AND MIKHAIL NEDIAK SCHOOL OF BUSINESS, QUEEN S UNIVERSITY, 143 UNION ST., KINGSTON, ON, K7L 3N6, CANADA E-MAIL:{TLEVIN,YLEVIN,JMCGILL,MNEDIAK}@BUSINESS.QUEENSU.CA

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### Multiple Linear Regression in Data Mining

Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

### Teaching Revenue Management in an Engineering Department

Teaching Revenue Management in an Engineering Department Abstract: Revenue management is one of the newly emerging topics in the area of systems engineering, operations research, industrial engineering,

### Evaluating the Lead Time Demand Distribution for (r, Q) Policies Under Intermittent Demand

Proceedings of the 2009 Industrial Engineering Research Conference Evaluating the Lead Time Demand Distribution for (r, Q) Policies Under Intermittent Demand Yasin Unlu, Manuel D. Rossetti Department of

### A Production Planning Problem

A Production Planning Problem Suppose a production manager is responsible for scheduling the monthly production levels of a certain product for a planning horizon of twelve months. For planning purposes,

### Duality in Linear Programming

Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow

### 1 Portfolio mean and variance

Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a one-period investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Moral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania

Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically

### 1 Polyhedra and Linear Programming

CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

### Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

### Revenue Management in the airline industry: problems and solutions

Revenue Management in the airline industry: problems and solutions Guillaume Lanquepin-Chesnais supervisors: Asmund Olstad and Kjetil K. Haugen Høgskolen i Molde 23 th November 2012 G.LC (himolde) RM in

### O&D Control: What Have We Learned?

O&D Control: What Have We Learned? Dr. Peter P. Belobaba MIT International Center for Air Transportation Presentation to the IATA Revenue Management & Pricing Conference Toronto, October 2002 1 O-D Control:

### Offline sorting buffers on Line

Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

### Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA

A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce

### Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Load Balancing and Switch Scheduling

EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### Appointment Scheduling under Patient Preference and No-Show Behavior

Appointment Scheduling under Patient Preference and No-Show Behavior Jacob Feldman School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853 jbf232@cornell.edu Nan

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### Section Notes 4. Duality, Sensitivity, Dual Simplex, Complementary Slackness. Applied Math 121. Week of February 28, 2011

Section Notes 4 Duality, Sensitivity, Dual Simplex, Complementary Slackness Applied Math 121 Week of February 28, 2011 Goals for the week understand the relationship between primal and dual programs. know

### Revenue management based hospital appointment scheduling

ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 11 (2015 No. 3, pp. 199-207 Revenue management based hospital appointment scheduling Xiaoyang Zhou, Canhui Zhao International

### Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

### Department of Industrial Engineering

Department of Industrial Engineering Master of Engineering Program in Industrial Engineering (International Program) M.Eng. (Industrial Engineering) Plan A Option 2: Total credits required: minimum 39

### Random graphs with a given degree sequence

Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

### On generalized gradients and optimization

On generalized gradients and optimization Erik J. Balder 1 Introduction There exists a calculus for general nondifferentiable functions that englobes a large part of the familiar subdifferential calculus

### Chapter 7. Sealed-bid Auctions

Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### Completion Time Scheduling and the WSRPT Algorithm

Completion Time Scheduling and the WSRPT Algorithm Bo Xiong, Christine Chung Department of Computer Science, Connecticut College, New London, CT {bxiong,cchung}@conncoll.edu Abstract. We consider the online

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all