Distribution-free methods for multi-period, single-leg booking control

Huina Gao; Michael O. Ball; Itir Z. Karaesmen

doi:10.1057/s41272-016-0063-z

Journal of Revenue and Pricing Management

December 2016, Volume 15, Issue 6, pp 425–453 | Cite as

Distribution-free methods for multi-period, single-leg booking control

Authors
Authors and affiliations

Huina Gao
Michael O. Ball
Itir Z. KaraesmenEmail author

Research Article

First Online: 12 December 2016

Revised: 30 November 2016

Abstract

We study the multi-period, single-leg, fare class allocation problem in revenue management. Uncertainty in fare class demand is not characterized using probability distributions but by using lower and upper bounds on demand. Our multi-period model allows demand characteristics and available information to vary from period to period. Building on our previous work, static and dynamic models are developed to determine effective booking control policies based on competitive analysis of online algorithms. The underlying performance criterion used is a measure of robustness, which provides the best performance guarantee over all possible input sequences consistent with the data provided. The dynamic policies are nested by fare class in each period but the booking limits can be revised from period to period so that a fare class that is closed can be re-opened (if needed). Computational experiments compare the new policies against previously developed robust policies and also more traditional approaches. The experiments show that these new policies are effective, robust, and can provide significant gains over the existing policies.

Keywords

seat inventory control dynamic decision making competitive analysis worst-case analysis robust revenue management heuristics

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Notes

Acknowledgements

Huina Gao and Michael O. Ball acknowledge the support of the National Science Foundation (NSF) under Grants: DMI0205489 and DMI0540312.

Appendix

A: Nested booking limits

We now describe the application of standard nesting in a multi-period setting from an algorithmic point of view: The execution of the algorithm (booking control policy) in each period is driven by the period t nested booking limits $b^t=(b_1^t,\ldots ,b_m^t)$, which are specified at the beginning of the period. The algorithm processes the period t input $I^t$, where requests arrive, requesting units of only one fare class. Let C(s) and $\delta (s)$ be the fare class of and the amount of capacity requested by the $s{\mathrm{th}}$ request, respectively, for $s=1,\ldots ,||I^t||$ where $||I^t||$ denotes the total number of requests in input $I^t$ (Note that the example at the end of the previous section provides a sequence of $(C(s),\delta (s))$ pairs). When standard nesting is applied, each request in $I^t$ is processed using an effective booking limit initialized as $\hat{b}^t=b^t$. The effective booking limits are continuously updated as the algorithm executes to process the input requests in period t. Algorithm PNEST(t) below shows the evolution of the effective booking limits in period t.

PNEST(t)

Step 0: Let $n\ge b^t_1\ge \cdots \ge b^t_m$ be the period t nested booking limits, and set $\hat{b}^t=b^t$. Let $s=1$.
Step 1: After receiving request s, $\min (\hat{b}^t_{C(s)},\delta (s))$ units of it is accepted.
Step 2: Update the effective booking limit starting from class 1
$$\hat{b}^t_1\leftarrow \hat{b}^t_1 -\min (\hat{b}^t_{C(s)},\delta (s)),$$

(15)
and iteratively computing
$$\hat{b}^t_j \leftarrow \min (\hat{b}^t_{j-1},\hat{b}^t_j -\min (\hat{b}^t_{C(s)},\delta (s)) )\quad j=2,\ldots ,C(s).$$

(16)

$$\hat{b}^t_j \leftarrow \min (\hat{b}^t_{j-1},\hat{b}^t_j)\quad j=C(s)+1,\ldots ,m.$$

(17)
Step 3: If $s<||I^t||$, then $s\leftarrow s+1$ and go to Step 1. Otherwise, stop.

B: Proof of Proposition 1

Proof

Consider an input $\vec {I}=[I^1,\ldots ,I^T]$ where $I^t$ is not LBH for some t, $1\le t\le T$. There is a corresponding LBH input $I^{\prime t}$ with exactly the same profile as $I^t$. Define $\vec {I^{\prime }}=[I^1,\ldots ,I^{\prime t}, \ldots , I^T]$, which replaces $I^t$ of $\vec {I}$ with $I^{\prime t}$. The offline optimal revenues from sequences $\vec {I}$ and $\vec {I^\prime }$ are the same. Given a static nested booking limit vector $b^1$, both sequences result in the same decisions/performance prior to period t. Since a nested policy is employed, (i) the online revenues with input $I^t$ are at least as much as the online revenues with $I^{\prime t}$ at the end of period t while the remaining capacity is the same, and (ii) effective booking limits of fare classes after processing inputs $I^{\prime t}$ and $I^{t}$ are equal (Proposition 1 of LGBK’08). Therefore, the online revenues, hence the CR, is lower when input $\vec {I^{\prime }}$ is used. The proof is completed by replacing an input $I^t$ that is not LBH with the LBH input of the same profile for $t=1,\ldots ,T$. $\quad \square$

C: Proof of Proposition 2

To simplify the notation and to suppress the time index, we refer to the seller’s static decision vector as $\beta$, as opposed to $b^1$, in the remainder of this section. We first provide additional notation and state an important observation. Combining the notion of a booking limit control policy with Proposition 1, one can iteratively compute the number of requests accepted in each period and the effective booking limit at the start of each period:

Observation 1:

Consider a nested booking limit policy $\beta$ and input $\vec {I}=[I^1,..,I^T]$ where inputs arrive in LBH order in each period. Let the number of class j requests accepted in period t after processing $I^t$ be $h_j^{I^t}$. Let the effective nested booking limit vector at the beginning of period t be $\beta ^{I^t}=(\beta _1^{I^t},\ldots ,\beta _m^{I^t})$. Parameters $h_j^{I^t}$ and $\beta _j^{I^t}$ are computed iteratively as follows:

Step 0: Set $t=1$. Set $\beta ^{I^1}=\beta$, i.e., $(\beta ^{I^1}_1,\beta ^{I^1}_2,\ldots ,\beta ^{I^1}_m)=(\beta _1,\ldots ,\beta _m)$.
Step 1: Compute the number of reservations accepted in each fare class in period t, starting with class m and working backwards from class m to class 1:
$$h_m^{I^t} = \min (\beta ^{I^t}_m,I^t[m]),$$

(18)

$$h_j^{I^t} = \min (\beta ^{I^t}_j-\sum _{k=j+1}^m h_k^{I^t}, I^t[j])\quad {text{for}} \quad j=m-1,m-2,\ldots ,1.$$

(19)
Step 2: Compute the effective booking limit of each fare class at the beginning of period $t+1$, starting with class 1 and working forwards from class 1 to class m:
$$\beta ^{I^{t+1}}_1 = \beta _1^{I^t} - \sum _{k=1}^m h_k^{I^{t}},$$

(20)

$$\beta ^{I^{t+1}}_j = \min (\beta _{j-1}^{I^{t+1}},\beta _j^{I^{t}}- \sum _{k=j}^m h_k^{I^{t}})\quad {\text{for}} \quad j=2,\ldots ,m.$$

(21)
Step 3: If $t=T$, STOP. Otherwise, set $t\leftarrow t+1$ and go to Step 1.

The vector $\beta ^{{I^{T+1}}}$ provides information (in a nested fashion) on the number of unused seats at the end of the booking horizon after processing input $\vec {I}$ with policy $\beta$.

We use these relations and the corresponding notation in the proof below.

Proposition 2:

Given a static, nested booking limit policy $\beta$ and the set Q of extreme input sequences, the lowest CR is achieved by one of the inputs in set Q. That is, for any feasible input sequence $\vec {I}$ , we have

$$\frac{R(\vec {I},\beta )}{R^*(\vec {I})}\ge \min \left\{ \frac{R(\vec {I}^*,\beta )}{R^*(\vec {I}^*)}, \quad \forall \vec {I}^* \in Q\right\}.$$

Proof

Let us consider a sequence $\vec {I}=[I^1,\ldots ,I^T]$ which is period-wise LBH (without loss of generality). We compute the effective booking limits $\beta ^{I^1},\ldots ,\beta ^{I^{T+1}}$ for input $\vec {I}$ and policy $\beta$ as described in Observation 1 above. For all $t=1,\ldots ,T$, let $j^*(t)$ be the highest fare (lowest-index) fare class whose effective booking limit is zero at the end of period t, i.e., $\beta ^{I^{t+1}}_k=0$ for all $k\ge j^*(t)$ and $\beta ^{I^{t+1}}_k>0$ for all $k<j^*(t)$, $k=1,\ldots ,m$. If $\beta _k^{I^{t+1}}>0$ for all $k=1,..,m$, let $j^*(t)=m+1$. Note that if $j^*(t)=m+1$, then all requests that arrive in period t are accepted (because no effective booking limit is reached in that period). Note also $j^*(1)\ge j^*(2)\ge \cdots \ge j^*(T)$ by definition of standard nesting.

Starting with $t=T$, we show that the adversary can get a CR lower than $R(\vec {I},\beta )/ R^*(\vec {I})$ by sending one of the m extreme input sequences in period t instead of sending $I^t$. For $t=T$, the proof in LGBK’08 follows immediately: When $j^*(T)<m+1$, then the adversary cannot increase the CR by sending an input $I^{T\prime }$ such that $I^{T\prime }[k]=L^T_k$ $k< j^*(T)$ and $I^{T\prime }[k]=U^T_k$ for $k\ge j^*(T)$, $k=1,\ldots ,m$. When $j^*(T)=m+1$, all requests have been accepted throughout the booking horizon, and the CR is $R(\vec {I},\beta )/R^*(\vec {I})=1$. In this latter case, adversary can send any of the j-extreme input sequences for period T without increasing the CR.

Consider any period $t<T$.

Case 1 For all $k\ge j^*(t)$, the adversary can lower the CR by sending a LBH input in period t with $I^{t\prime }[k]=U^t_k \ge I^t[k]$, using the same argument in LGBK’08: since effective booking limits of $k\ge j^*(t)$ are all zero, additional requests in these classes are all rejected. Online revenues are unaffected by the change, while offline optimal can only increase or remain the same. Therefore CR is improved by this change.
Case 2: For all $k<j^*(t)$, the adversary can lower the CR by sending a LBH input in period t with the profile (i) $I^{t\prime }[k]=L^t_k$ if $b_k^{T+1} >0$ (in this case, a class k request in period t does not displace an equal or higher fare request in period t or any time in the remainder of the booking horizon) or (ii) $I^{t\prime }[k]=U^t_k$ if $\beta _k^{T+1} =0$ (in this case, a class k request displaces equal or higher fare requests either in period t or in the remainder of the booking horizon). The former follows from LGBK’08. In the latter, while the effective booking limit of class k is positive at the end of period t, it will become zero by the end of the planning horizon. This means that the argument for Case 1 will be valid for class k some time before the end of the booking horizon. Consequently, it is to the adversary’s advantage to send the maximum number of requests for classes that will be rejected either in this period or in the future. $\quad \square$

D: Determining the optimal booking limits in the static problem

In this section, we show that (1) can be expressed as a mixed integer programming (MIP) model to determine the optimal static booking limits. The binary variables are used in determining the correct number of fare class requests accepted in processing an input sequence, which is then used in calculating the online revenues. Binary variables are also used to ensure effective booking limits that are calculated correctly, preserving the nesting property. To simplify the notation in the problem formulation, we present the model for all extreme sequences belonging to the set Q, without referring to the property in Corollary 1.

The MIP model

In MIP, the input parameters are $f_k$ for $k=1,\ldots ,m$, the capacity n, the period-wise profile $I^t[j]$, $t=1,..,T$, for each input $\vec {I}=[I^1,\ldots ,I^T]\in Q$, and the corresponding offline optimal revenue $R^*(\vec {I})$ for $\vec {I}\in Q$. $R^*(\vec {I})$ is easily computed a priori given the profile of the extreme sequences. The decision variable for the seller is the vector $\beta =(\beta _1,\ldots ,\beta _m)$ and the scalar z represents the CR. All other variables are auxiliary and ensure the effective booking limits follow the relations provided in Observation 1 of Appendix C.

MIP:

$$\begin{aligned} \max&\quad z \\ {\text{s.t.}}&\quad R^*(\vec {I}) z\le \sum ^{T}_{t=1}\sum ^{m}_{k=1}f_kh_k^{I^t},\quad \forall \quad t=1,\ldots ,T,\quad \forall \vec {I} \in Q \end{aligned}$$

(22)

$$h_j^{I^t}\le I^t[j], \quad j=1,\ldots ,m, \quad t=1,\ldots ,T, \quad \forall \vec {I} \in Q$$

(23)

$$\sum ^m_{k=j} h_k^{I^t}\le \beta _j^{I^t}, j=1,\ldots ,m, t=1,\ldots ,T,\quad \forall \vec {I} \in Q$$

(24)

$$I^t[j]-h_j^{I^t}-nv_j^{I^t}\le 0, j=1,\ldots ,m, t=1,\ldots ,T, \quad \forall \vec {I} \in Q$$

(25)

$$\beta _j^{I^t}-\sum ^m_{k=j} h_k^{I^t}-n(1-v_j^{I^t})\le 0, j=1,\ldots ,m, t=1,\ldots ,T, \quad \forall \vec {I} \in Q$$

(26)

$$\beta _j^{I_{t+1}}\le \beta _{j-1}^{I_{t+1}}, j=2,\ldots ,m, t=1,\ldots ,(T-1), \quad \forall \vec {I} \in Q$$

(27)

$$\beta _j^{I_{t+1}}\le \beta _j^{I^t}-\sum ^m_{k=j} h_k^{I^t}, j=2,\ldots ,m, t=1,\ldots ,(T-1), \quad \forall \vec {I} \in Q$$

(28)

$$\beta _j^{I_{t+1}}-\beta _{j-1}^{I_{t+1}}-nw_j^{I_{t+1}}\le 0,j=2,\ldots ,m, t=1,\ldots ,(T-1), \quad \forall \vec {I} \in Q$$

(29)

$$\beta _j^{I^t}- \sum ^m_{k=j} h_k^{I^t}-\beta _j^{I_{t+1}}-n(1-w_j^{I_{t+1}})\le 0, \quad j=2,\ldots ,m, t=1,\ldots ,(T-1),\quad \forall \vec {I} \in Q$$

(30)

$$\beta _1^{I_{t+1}}=\beta _1^{I^t}-\sum ^m_{k=1}h_k^{I^t}, t=1,\ldots ,(T-1), \quad \forall \vec {I} \in Q$$

(31)

$$\beta _j^{I^1}=\beta _j, \quad j=1,\ldots ,m, \quad \forall \vec {I} \in Q$$

(32)

$$\beta _j^{I^t},h_j^{I^t}\ge 0; v_j^{I^t},w_j^{I^t} \in \{0,1\}, j=1,\ldots ,m,t=1,\ldots ,T,\quad \forall \vec {I} \in Q.$$

(33)

$$0\le \beta _j\le n, \quad j=1,\ldots ,m.$$

(34)

The size of MIP is O($m^{T+1}$) as the number of constraints and binary variables are proportional to $m^{T+1}$, which is polynomial in m for fixed T. The optimal objective value in MIP is at least as much as the objective value derived in the static model of LGBK’08 when the model in LGBK’08 uses the aggregate lower and upper bounds $\sum _{t=1}^T L^t_j$ and $\sum _{t=1}^T U^t_j$, respectively, for a single-period problem. This is because there are period-wise restrictions on the input sequence. We provide an example to show the differences in the solutions of our static model and that of LGBK’08 below.

Comparison of our static model to that of LGBK’08

Here is an example to show the differences in solutions between our static model and that of LGBK’08. Both the theoretical CR and the average performance of the policies, computed using simulations, are considered: Suppose $n=10$,$f_1=5$, $f_2=1$, $T=2$, $L_1^1=0$, $L_2^1=0$, $L_1^2=2$, $L_2^2=0$, $U_1^1=8$, $U_2^1=3$, $U_1^2=6$, and $U_2^2=12$. Demand for class 2 is assumed to be uniformly distributed between $L_2^t$ and $U_2^t$ in period $t=1,2$. In two different experiments, the demand for class 1 is first uniformly distributed, and then triangular distributed, each with the range $[L_1^t,U_1^t]$ for $t=1,2$. The theoretical CR and the average performance ratio (computed as the ratio of policy revenues to offline optimal revenues, averaged across all simulation runs), as an indication of the actual CR, are computed using simulation for policies obtained by MIP and LGBK’08. The differences in booking limits, the theoretical CR, and the average performance ratios are visible in the results provided in Table E1. Although MIP guarantees a better worst-case performance, its actual average performance need not always be superior compared to LGBK’08.

Table E1

Booking limits, theoretical CR, and average performance ratio

Model	Booking limit	Theoretical	Average ratio	Average ratio
Model	$b_2$	CR	$D_1$ is uniform	$D_1$ is triangular
LGBK’08	3.28	0.73	0.878	0.9788
MIP	3.68	0.76	0.84	0.99

E: Proof of Proposition 4

Proof

Through induction, we will show that for a m fare, T-period problem, the lowest competitive ratio is achieved by one of the $m+1$ extreme input sequences. We use the notation $I \prec \prec I^\prime$ to denote that given booking limit vector $b^T$ of the seller and two inputs I and $I^\prime$; I results in a CR that is no-higher than $I^\prime$ (i.e., $I^\prime$ is dominated by I). We focus only on LBH inputs.

Step 1: We will prove that in the last period and for the lowest fare, adversary will send in the amount of $L_T^m+\Delta ^m$, $\Delta ^m \in \{0,U_T^m-L_T^m\}$. Consider three LBH input sequences, $S_1, S_2, S_3$ with the following profiles: $S_1^T[m]=U^T_m$, $S_2^T[m]=L_T^m+\Theta ^m$ for $0<\Theta ^m<U_T^m-L_T^m$, $S_3^T[m]=L_T^m$, and $S_1^T[j]=S_2^T[j]=S_3^T[j]$ for all $j=1,..,m-1$. $S_2$ is dominated by either $S_1$ or $S_3$:
- Case 1.1: If $0\le b^T_m \le L_T^m+\Theta ^m$, $S_1\prec \prec S_2$ because the online revenues are equal with inputs $S_1$ and $S_2$ and the offline optimal revenue $R^*(S_1)$ is the highest with $S_1$.
- Case 1.2: Consider $b_T^m > L_T^m+\Theta ^m$. (i) If processing $S_2$ with $b^T$ and accepting $L_T^m+\Theta ^m$ requests in class m does not displace any higher fare requests, then $S_3$ also does not displace any higher fare requests when processed by $b^T$. In this situation, online revenues satisfy $R(S_2,b^T)\ge R(S_3,b^T)$ and offline revenues satisfy $R^*(S_2)\ge R^*(S_3)$. Notice that $R^*(S_2)-R^*(S_3)\le f_m*\Theta ^m$ and $R(S_2,b^T)= f_m*\Theta ^m+R(S_3,b^T)$. This implies that $\frac{R^T+R(S_2, b^T)}{R^*(S_2)}$ $\ge$ $\frac{R^T+R(S_3,b^T)}{R^*(S_3)}$. Therefore, $S_3\prec \prec S_2$. (ii) If processing $S_2$ with $b^T$ and accepting $L_T^m+\Theta ^m$ requests in class m displaces higher fare requests, then $S_1$ also leads to displacement of higher fare requests. In this situation, $R^T+R(S_1,b^T)\le R^T+R(S_2,b^T)$ because $S_1$ accepts a higher number of class m requests, and $R^*(S_1) \ge R^*(S_2)$, which leads to $\frac{R^T+R(S_1,b^T)}{R^*(S_1)}$ $\le$ $\frac{R^T+R(S_2, b^T)}{R^*(S_2)}$. Thus,$S_1\prec \prec S_2$.
- Therefore, a LBH sequence $I^T$ with profile $L_m^T<I^T[m]<U_m^T$ is dominated.
Step 2: Through backward induction on fare classes, we will show that a LBH sequence $I^T$ with profile $L_j^T<I^T[j]<U_j^T$ is dominated. The statement is true for class m based on STEP 1. Suppose it is true for classes $m,m-1,\ldots ,k$. We will show this is true for class $k-1$. Consider six LBH input sequences $S'_1,\ldots ,S'_6$ with profiles $S'_1[k]=L_k^T$, $S'_1[k-1]=L_{k-1}^T$, $S'_2[k]=L_k^T$, $S'_2[k-1]=L_{k-1}^T+\theta$ for $0<\theta < U^T_{k-1}-L^T_{k-1}$, $S'_3[k]=L_k^T$, $S'_3[k-1]=U_{k-1}^T$, $S'_4[k]=U_k^T$, $S'_4[k-1]=L_{k-1}^T$, $S'_5[k]=U_k^T$, $S'_5[k-1]=L_{k-1}^T+\theta$ for $0<\theta < U^T_{k-1}-L^T_{k-1}$, and $S'_6[k]=U_k^T$, $S'_6[k-1]=U_{k-1}^T$.
- Case 2.1: Consider $S'_1,S'_2,S'_3$ where $L_k^T$ requests of class k arrive. (i) If all of the $S'_2[k-1]$ requests are accepted by $b^T$ and none of the requests in classes $k-2$ to 1 are displaced, then all of the $S'_1[k-1]$ requests are also accepted by $b^T$ without displacing higher fare requests. This is similar to Case 1.2(i) above. Therefore, $S'_1 \prec \prec S'_2$. (ii) If all of the $S'_2[k-1]$ requests are accepted by $b^T$, leading to displacement of requests in any of classes $k-2$ to 1, then at least $S'_2[k-1]$ requests of class $k-1$ will be accepted when $S'_3$ is processed by $b^T$, leading to at least as many displacements of higher fare. This is similar to Case 1.2(ii) above. Thus, $S'_3\prec \prec S'_2$.
- Case 2.2: Consider $S'_4,S'_5,S'_6$ where $U_k^T$ requests of class k arrive. Using the same argument as in Case 2.1 above, one can show that either $S'_4 \prec \prec S'_5$ or $S'_6 \prec \prec S'_5$. Therefore, an input $I^T$ with $L_j^T< I^T[j] < U_j^T$ for any j is dominated. Next, we show there are only $m+1$ of the non-dominated sequences.
Step 3. We know any extreme LBH input $I^T$ has profile $I^T[m]=L_m^T$ or $I^T[m]=U_m^T$. Consider LBH sequences $S''_1$, $S''_2$, and $S''_3$ with profiles $S''_1[m]=U_m^T$, $S''_1[m-1]=U_{m-1}^T$, $S''_2[m]=L_m^T$, $S''_2[m-1]=U_{m-1}^T$ , and $S''_3[m]=L_m^T$, $S''_3[m-1]=L_{m-1}^T$.
- Case 3.1: If processing $S''_2$ with $b^T$ displaces any request in classes $m-1$ to 1, then $S''_1$ leads to at least the same number of displaced requests in these classes. Then, using the logic in Case 1.2(ii) above, we have $S''_1 \prec \prec S''_2$.
- Case 3.2: If processing $S''_2$ with $b^T$ does not result in displacement of any request in classes $m-1$ to 1, $S''_3 \prec \prec S''_2$ by Case 1.2(i) above.
- Combining Cases 3.1 and 3.2, we see that $S''_2$ is dominated. Therefore, a non-dominating sequence $I^T$ has a profile such that if $I^T[m]=L_m^T$, then $I^T[m-1]=L^T_{m-1}$. Note that, this observation can be extended to any two consecutive class j and j-1.
Therefore, the total number of non-dominated input streams in period T is only $m+1$ and $Q^{*T}$ is the set of all non-dominated sequences for period T. $\quad \square$

F: Proof of Proposition 5

Preliminaries

In this section, we present elementary observations and results that are used in the proof of Proposition 5. First of all, we make use of duality in proving the optimality of the suggested solution for LPM. The dual formulation of LPM, called DLPM, is given below.

$$DLPM:\,\min \, n^T\nu +\sum _{j=1}^{m}U_j \varpi _j +\sum _{j=1}^m \left( R^T+\sum _{i=1}^{j-1}f_iL_i^T\right) \lambda _j + \left( R^T+\sum _{i=1}^{m+1}f_iL_i^T\right) \lambda _{m+1}$$

(35)

$${\mathrm{s.t.}}\quad \sum _{j=1}^{m+1} R^*_j \lambda _j \ge 1,$$

(36)

$$\quad \quad \quad \varpi _j + \nu - f_j*\sum _{i=1}^j \lambda _i \ge 0,\quad j=1,..,m,$$

(37)

$$\quad \quad \qquad \nu \ge 0, \quad \lambda _j,\varpi _j \ge 0, j = 1,..,m.$$

(38)

Second, the solution to LPM is closely related to the solution of the following linear program, which disregards the $(m+1)^{st}$ extreme sequence:

$$\begin{aligned} GBM: \max \quad z \quad \quad \quad \quad \quad \qquad & \\ {\text{s.t.}} \quad \quad \quad \quad \quad \quad \qquad & \\ R^*(I^1\cdots I^{T-1}I^{*T,k}) z\le & R^T+\sum ^{k-1}_{i=1} f_i L^T_i+\sum ^m_{i=k} f_i x^T_i ,\quad k=1,\ldots ,m, \end{aligned}$$

(39)

$$\sum _{j=1}^m x_j^T\le n^T$$

(40)

$$0 \le x^T_j \le U^T_j, \quad j = 1,\ldots ,m$$

(41)

When $R^T=0$, this is equivalent to the formulation in LGBK’08 that determines the static nested booking limits in a single-period problem. The closed-form solution to GBM is given by

$$z^{*GBM}= \theta _{u}$$

(42)

$$x^{*GBM}_k= \left\{ \begin{array}{ll} g_k z^* + L^T_k & \quad 1\le k< u\\ (R^*_k z^* - \sum _{i=1}^{k-1}f_iL_i^T-R^T )/f_k & \quad k=u \\ 0 &\quad k > u \end{array} \right.$$

(43)

where the parameters are defined as

$$g_m=R^*_m, \quad g_k = (R^*_k - R^*_{k+1})/f_k \ge 0, \quad k=1,\ldots ,m-1$$

$$u = \max \left\{ j:\left( R^T+\sum _{i=1}^{j-1}f_iL_i^T\right) \sum ^{j-1}_{i=1}g_i < N_j^T R^*_j\right\},$$

(44)

$$\theta _u=\frac{ (R^T+\sum _{i=1}^{u-1}f_iL_i^T)/f_u + N^T_u}{R^*_u/f_u + \sum ^{u-1}_{i=1} g_i}.$$

We omit the proof of optimality of the solution given in (42) and (43) for GBM here; that result follows from the proof in LGBK’08 with the addition of the scalar $R^T>0$ to the online policy revenues for each extreme input. Notice that the solution of GBM only partially characterizes the optimal solution of LPM.

Finally, we use one additional parameter in the solution of LPM:

$$\pi =\frac{R^T+\sum _{i=1}^{m-1} f_iL_i^T+f_m*\min (L^T_m,N^T_m)}{R^*_{m+1}}.$$

Note that $\pi$ is a constant and it represents the highest competitive ratio possible for any booking limit policy when $(m+1)^{st}$ extreme sequence is processed. This is what is represented in constraint (6) of LPM. We have the following observation on parameter $\pi$.

Lemma 1:

If $u=m$ and $\theta _m \ge \pi$ , then $\pi =\frac{R^T+\sum _{i=1}^{m} f_iL_i^T}{R^*_{m+1}}$.

Proof

We show this by proving that when $u=m$ and $\theta _m \ge \pi$, then it is impossible to have $N_m^T<L_m^T$. Suppose $N_m^T<L_m^T$, $u=m$, and $\theta _m \ge \pi$. Then, by definition,

$$\theta _m=\frac{R^T+\sum _{i=1}^{m-1}f_iL_i^T + N^T_m*f_m}{R^*_m + \sum ^{m-1}_{i=1}g_i*f_m}=\frac{R^T+\sum _{i=1}^{m-1}f_iL_i^T + N^T_m*f_m}{R^*_{m+1} + \sum ^{m}_{i=1}g_i*f_m},$$

$$\pi =\frac{R^T+\sum _{i=1}^{m-1} f_iL_i^T+N^T_m*f_m}{R^*_{m+1}},$$

and $\theta _m < \pi$, which is contradictory to the condition that $\theta _m \ge \pi$. Thus, in this case, $N_m^T \ge L_m^T$ and the resulting $\pi$ is $\frac{R^T+\sum _{i=1}^{m} f_iL_i^T}{R^*_{m+1}}$. $\quad \square$

In addition to the solution to GBM, consider the alternative

$$z^{*ALT}= \pi$$

(45)

$$x^{*ALT}_k= \left\{ \begin{array}{ll} g_k z^{*ALT} + L^T_k &\quad 1<k\le m\\ \min (U_T^1,n^T- \sum _{i=2}^{m}(L_i^T+g_i\pi )) & \quad k=1 \\ \end{array} \right.$$

(46)

which if feasible for LPM under certain conditions:

Lemma 2:

When $u=m$ and $\theta _m\ge \pi$ , $x^{*ALT}$ of equation (46) and $z^{*ALT}=\pi$ are feasible for LPM.

Proof

To prove this, we first have to prove that $x_1^{*ALT}\ge L^1_T+g_1\pi$: When $u=m$, we have $N_m^T>0$, that is, $n^T>\sum _{i=1}^{m-1}L^i_T$. Let’s define $R_m^{+}=\sum ^{m-1}_{i=1}f_iL_T^i$. In this case, we know that $\theta _m \ge \pi$, so based on Lemma 1,

$$\theta _m=\frac{R^T+R_m^{+}+(n^T-\sum ^{m-1}_{i=1}L_i^T)*f_m}{R^*_m+\sum ^{m-1}_{i=1}g_i*f_m} \ge \frac{R^T+\sum ^{m}_{i=1}f_iL_i^T}{R^*_{m+1}}=\pi.$$

Then, we get the following series of relations by algebraic manipulations:

$$\begin{aligned} \left( n^T-\sum ^{m-1}_{i=1}L_i^T\right) f_mR^*_{m+1}\ge & (R^T+R_m^+)(R^*_m-R^*_{m+1})+R^*_mf_mL_m^T+f_m\sum ^{m-1}_{i=1}g_i(R^T+R_m^+)+\sum ^{m-1}_{i=1}g_if_mL_m^T \\ \left( n^T-\sum ^{m-1}_{i=1}L_i^T\right) R^*_{m+1}\ge & (R^T+R_m^+)\sum ^{m}_{i=1}g_i+R^*_mL_m^T+\sum ^{m-1}_{i=1}g_if_mL_m^T\\ f_mg_m+R^*_{m+1}= & R^*_m \\ \left( n^T-\sum ^{m-1}_{i=1}L_i^T\right)\ge & \frac{(R^T+R_m^+)\sum ^{m}_{i=1}g_i}{R^*_{m+1}}+\frac{\sum ^{m}_{i=1}g_if_mL_m^T}{R^*_{m+1}}+L_m^T\\ n^T\ge & \sum ^{m}_{i=1}L_i^T+\sum ^{m}_{i=1}g_i*\pi . \end{aligned}$$

Then

$$x_1^{*ALT}=n^T-\sum ^{m}_{i=2}L_i^T+\sum ^{m}_{i=2}g_i*\pi \ge L_1^T + g_1\pi.$$

(47)

Next we prove the feasibility of $x^{*ALT}$. Here are the observations:

Constraint (8) is satisfied: We have $g_i\le U_T^i - L_T^i$ for $i=1,\ldots ,m$; this follows from the proof in LGBK’08. By construction, $x^{*ALT}_1\le U_T^1$. We also have $z^*=\pi \le 1$ by design. This implies $x^{*ALT}_k=\pi * g_i + L_T^i \le U_T^i$ for $1<i\le m$.
Constraint (5) is satisfied: We have $u=m$,$\pi \le \theta _m= (R^T+\sum _{i=1}^{m}f_iL_i^T)/R^*_{m+1}$. So for any k, $2\le k\le m$, we have
$$R_k^*\pi \le R^T+\sum _{i=1}^{m}f_iL_i^T+R_k^*\pi -R_{m+1}^*\pi.$$
Further algebra yields
$$R_k^*\pi \le R^T+\sum _{i=1}^{m}f_iL_i^T+\sum _{i=k}^m(R_i^*-R_{i+1}^*)\pi$$

$$R_k^*\pi \le R^T+\sum _{i=1}^{k-1}f_iL_i^T+\sum _{i=k}^mf_i(L_i^T+g_i\pi)$$

$$R_k^*\pi \le R^T+\sum _{i=1}^{k-1}f_iL_i^T+\sum _{i=k}^mf_ix_i^{*ALT}.$$
For $k=1$, we have $L_T^1+g_1\pi \le x_1^{*ALT}\le U_T^1$ which follows from Equation (47) above. This yields $R_1^*\pi \le R^T+\sum _{i=1}^mf_ix^{*ALT}_i$.
Constraint (6) is satisfied: This follows from the definition of $\pi$:
$$R_{m+1}^*\pi \le R^T+\sum _{i=1}^{m-1}f_iL_i^T+f_m \,\min (L_m^T,N_m^T).$$
The remaining constraint $\sum _{i=1}^{m}x_i^{*ALT}\le n^T$ is satisfied.

$\quad \square$

Proof of Proposition 5

Proof

We first prove the feasibility of the solution suggested, then prove its optimality.

(a)

Proof of feasibility: Case 1: If the optimal solution of GBM has $u<m$, then constraint (5) for $k=m$ is not binding because only the first u out of those m constraints in the GBM are binding. This follows from the proof in LBGK’07. Thus, in this case, given $x^{*GBM}$, the corresponding $z^*=\theta _u$ satisfies constraint (5) when $k=m$, and also satisfies constraint (6) of LPM because constraint (6) yields the same online revenue but lower offline optimal revenue given the $(m+1)^{st}$ extreme sequence. So $z^*=\theta _u$ and $x^{*GBM}$ constitute a feasible solution to LPM.

Case 2: If the optimal solution of GBM has $u=m$ and $\theta _m<\pi$, then constraint (6) in LPM is not binding for the solution $x^{*GBM}$ and $z^*=\theta _m$. Therefore the solution $x^{*GBM}$ and $z^*=\theta _m$ of GBM is again feasible for LPM.

Case 3: If the optimal solution of GBM has $u=m$ and $\theta _m\ge \pi$, then Lemma 2 proves $x^{*ALT}$ and $z^{*ALT}$ are feasible for LPM.
(b)

Proof of optimality: Cases 1 and 2: Consider the following solution for DLPM:
$$\varpi ^*_j=0,\quad j=1,\ldots ,m, \quad \lambda ^*_{j}=0\quad j=u+1,\ldots ,m+1$$

(48)

$$\nu ^* = 1/\left( R_u^*/f_u + \sum _{i=1}^{u-1}g_i\right)$$

(49)

$$\lambda _j^*=\nu ^* (1/f_j - 1/f_{j-1})\quad j=1,\ldots ,u.$$

(50)
It can be showed that this solution is feasible for DLPM and its objective function value is equal to $\theta _u$. Therefore, the solution $x^{*GBM}$ and $z^{*GBM}$ is the optimal solution for LPM.

Case 3: Consider the following feasible solution to DLPM:
$$\begin{aligned} \lambda _j= & \left\{ \begin{array}{ll} 0 & \quad j \le m\\ 1/R^*_{m+1} & \quad j = m+1 \end{array} \right. \\ \nu= & 0, \\ \varpi _j= & 0 \quad \forall j=1,\ldots ,m. \end{aligned}$$
The objective function value of DLPM is equal to $\pi$ for this trivial solution. Therefore, the solution $x^{*ALT}$ and $z^{*ALT}$ is the optimal solution to LPM. $\quad \square$

G: Proof of Proposition 6

Proof

We consider the same cases introduced in the proof of Proposition 5.

Cases 1 and 2: If $u<m$ or $u=m$ and $\theta _m<\pi$in the solution of GBM, then constraint (6) is not binding in LPM. In these cases, GBM solution provides the optimal booking limit and its CR is correctly calculated in GBM as shown in LGBK’08.
Case 3: If solution to GBM has $u=m$ and $\theta _m\ge \pi$, then $x^{*ALT}$ is the optimal booking limit policy. In this case, we know $u=m$, $z^*=\pi =\frac{R^T+\sum _{i=1}^{m}f_iL_i^T}{R^*_{m+1}}$ is the optimal objective value of LPM, and constraint (6) is binding in LPM. Given the $k{\mathrm{th}}$ extreme input sequence $I^{*T,k}$ and a nested booking limit policy $b^{*T}$ obtained from $x^{*ALT}$ of Equation (46), the online policy revenue satisfies $R^T+R(I^{*T,k}, b^{*T})= R^T+\sum _{i=1}^{k-1}f_i \min (x_i^{*ALT},I^{*T,k}[i])$ by construction of $x^{*ALT}$. Note that $L_i^T\le x_i^{*ALT}\le U_i^T$ and $R^T+R(I^{*T,k}, b^{*T})= R^T+\sum _{i=1}^{k-1}f_i L_i^T +\sum _{i=k}^{m}f_i x_i^{*ALT}$ for extreme sequence k. Therefore, the constraints of LPM correctly calculate the online revenues of policy $b^{*T}$ for inputs $k=1,..,m+1$. The minimum CR is achieved by input $m+1$ in this case, and the corresponding CR is equal to $\pi$. $\quad \square$

H: Analysis of decisions in periods t < T

Proposition 7:

For periods $t<T$ , the minimum CR is achieved when input sequences in each period are LBH.

Proof

Proof is similar to that of Proposition 1 and is omitted. $\quad \square$

Unfortunately, further sequence reduction is not possible for a general, dynamic problem because the adversary’s non-dominated choices in period $t<T$ depend on $(R^t, n^t)$ and cannot be characterized by a predefined set of extreme sequences. This is in contrast to our finding for period T where we reduce the number of extreme sequences to m; see Proposition 4. The reduction for period T is possible because the adversary always chooses the number of requests to be equal to either the lower or the upper bound of demand in each fare class to minimize the CR. This is not true in period $t<T$ as we prove below using an example.

Proposition 8:

Given that the seller restricts her choices to a nested booking limit policy $b^t$ in period $t<T$ , there are problem instances where the optimal (CR minimizing) adversary policy must choose an LBH input $I^t$ in period t with the profile $L^t_k<I^t[k]<U^t_k$.

Proof

We provide an example to prove this. Consider $T=2$, $m=2$, $n=14$, $f_1=2$, and $f_2=1$. The demand information is $U^1_1=4$,$L^1_1=2$,$U^1_2=L^1_2=5$, $U^2_1=3$,$L^2_1=1$,$U^2_2=5$, and $L^2_2=2$. By Proposition 4, there are three extreme input sequences in the second period, denoted as $I^{*2,1},I^{*2,2},I^{*2,3}$, respectively. Consider an LBH input $I^1$ with the profile $I^1[2]=L^1_2=5$ and $I^1[1]=L^1_1=2$ for $t=1$. We call this Input-1. Consider two other alternative LBH inputs for $t=1$, with profiles $I^1[2]=U^1_2=5$, $I^1[1]=U^1_1=4$ , and $I^1[2]=5$,$I^1[1]=3$, called Input-2 and Input-3, respectively. Note that Input-1 and Input-2 represent all the inputs for $t=1$ where the demand for each fare class takes either the lowest or the highest possible value. (If one were to define the set of extreme sequences for $t=1$, these two inputs would constitute its superset.)

In Table 1, we fix the seller’s and adversary’s decisions in period $t=1$, and compute the optimal decisions in the second period. When we start with $b_2^1=0$ and Input-1 in $t=1$, the resulting CR is 0.6154 after computing optimal decisions in period $t=2$. When $b_2^1=0$ is used to process Input-2 and Input-3 in $t=1$, the CR is 0.7059 and 0.6666, respectively, after second period optimal decisions are computed for the seller and the adversary. Therefore, if seller chooses $b_2^1=0$ in $t=1$, the adversary would choose Input-1 over Input-2 and Input-3 to achieve the lowest CR. However, if the seller chooses $b_2^1=5$ in $t=1$, the CRs are 0.9474, 0.9474, and 0.9444, for Input-1, Input-2, and Input-3, respectively. In this case, Input-3 is clearly a better choice for the adversary: It yields a lower CR compared to Input-2 and Input-3. This shows that given a nested booking limit policy in period $t=1$, the adversary has non-dominated inputs where the number of requests in each fare class is not equal to the minimum or the maximum feasible amount (Table H1).

Table H1

An example to characterize the solution to the two-period dynamic problem

Input-1 in $t=1$: $I^1[1]=L^1_1=2$ $I^1[2]=U^1_2=L^1_2=5$				Input-2 in $t=1$: $I^1[1]=U^1_1=4$ $I^1[2]=U^1_2=L^1_2=5$				Input-3 in $t=1$: $L^1_1<I^1[1]=3<U^1_1$ $I^1[2]=U^1_2=L^1_2=5$
$b_2^1$	Optimal in $t=2$		CR	$b_2^1$	Optimal in $t=2$		CR	$b_2^1$	Optimal in $t=2$		CR
$b_2^1$	$b_2^{2}$	Input	CR	$b_2^1$	$b_2^{2}$	Input	CR	$b_2^1$	$b_2^{2}$	Input	CR
0	12	$I^{*2,3}$	0.6154	0	10	$I^{*2,3}$	0.7059	0	11	$I^{*2,3}$	0.6666
1	11	$I^{*2,3}$	0.6923	1	9	$I^{*2,3}$	0.7649	1	10	$I^{*2,3}$	0.7333
2	10	$I^{*2,3}$	0.7692	2	8	$I^{*2,3}$	0.8235	2	9	$I^{*2,3}$	0.8000
3	9	$I^{*2,3}$	0.8461	3	7	$I^{*2,3}$	0.8825	3	8	$I^{*2,3}$	0.8666
4	8	$I^{*2,3}$	0.9231	4	4	$I^{*2,3}$	0.9412	4	7	$I^{*2,3}$	0.9333
≥5	7	$I^{*2,1}$	0.9474	≥5	3	$I^{*2,2}$	0.9474	≥5	4	$I^{*2,2}$	0.9444

$\quad \square$

This property is expected in dynamic decision making. Thus, non-dominated sequences appear difficult to characterize for $t<T$.

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Authors and Affiliations

Huina Gao
- 1
Michael O. Ball
- 2
Itir Z. Karaesmen
- 3
Email author

1.MITRE Corporation, CAASDMcLeanUSA
2.Robert H Smith School of Business and Institute for Systems ResearchUniversity of MarylandCollege ParkUSA
3.Kogod School of BusinessAmerican UniversityWashingtonUSA

Distribution-free methods for multi-period, single-leg booking control

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Model	Booking limit	Theoretical	Average ratio	Average ratio
Model	\(b_2\)	CR	\(D_1\) is uniform	\(D_1\) is triangular
LGBK’08	3.28	0.73	0.878	0.9788
MIP	3.68	0.76	0.84	0.99

Input-1 in \(t=1\): \(I^1[1]=L^1_1=2\) \(I^1[2]=U^1_2=L^1_2=5\)				Input-2 in \(t=1\): \(I^1[1]=U^1_1=4\) \(I^1[2]=U^1_2=L^1_2=5\)				Input-3 in \(t=1\): \(L^1_1<I^1[1]=3<U^1_1\) \(I^1[2]=U^1_2=L^1_2=5\)
\(b_2^1\)	Optimal in \(t=2\)		CR	\(b_2^1\)	Optimal in \(t=2\)		CR	\(b_2^1\)	Optimal in \(t=2\)		CR
\(b_2^1\)	\(b_2^{2}\)	Input	CR	\(b_2^1\)	\(b_2^{2}\)	Input	CR	\(b_2^1\)	\(b_2^{2}\)	Input	CR
0	12	\(I^{*2,3}\)	0.6154	0	10	\(I^{*2,3}\)	0.7059	0	11	\(I^{*2,3}\)	0.6666
1	11	\(I^{*2,3}\)	0.6923	1	9	\(I^{*2,3}\)	0.7649	1	10	\(I^{*2,3}\)	0.7333
2	10	\(I^{*2,3}\)	0.7692	2	8	\(I^{*2,3}\)	0.8235	2	9	\(I^{*2,3}\)	0.8000
3	9	\(I^{*2,3}\)	0.8461	3	7	\(I^{*2,3}\)	0.8825	3	8	\(I^{*2,3}\)	0.8666
4	8	\(I^{*2,3}\)	0.9231	4	4	\(I^{*2,3}\)	0.9412	4	7	\(I^{*2,3}\)	0.9333
≥5	7	\(I^{*2,1}\)	0.9474	≥5	3	\(I^{*2,2}\)	0.9474	≥5	4	\(I^{*2,2}\)	0.9444