Introduction
I: DC J: Depot
3.2 Model Development
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leakage through investment in fencing, and what will be the optimal investment? How profitable is to integrate the pricing, seat inventory control, and fencing investment decisions for an airline?
A related study was conducted by Zhang et al. (2010) for an unca- pacitated pricing and fencing investment decision problem of a firm.
Noticeably, airline RM modeling is significantly different from a typi- cal firm; however, both problems can resemble newsvendor problem (see Philips [2005] for details on newsvendor problem and RM rela- tionship). In airline RM, the demand arrival is assumed sequential, and therefore, the lower fare price class demand is observed prior to the respective higher fare class. In response to this, the airline exercises a nested control that reserves certain seats for passengers willing to pay a higher fare price and arrive later to purchase tickets. This control is referred in airline RM literature as nested booking control (McGill and Ryzin, 1999; Chiang et al., 2007). Furthermore, in airline RM, unlike the uncapacitated firm’s problem, there is a limited capacity rep- resented by the cabin seats. Lastly, in airline RM, the costs incurred in relation to seat inventory and related flight services are often ignored in most of the airline RM models, with no exception in this study.
Thus, the focus of this study is to revisit the problem of RM with demand leakages and fencing investments in the airline context. We first present the model for an airline RM with no fencing investment to mitigate the demand leakage, and then we extend the problem with fencing improvement decision for the airline to mitigate or augment the demand leakage through additional investment. Later, the models are analyzed, and the optimal fare pricing, seat inventory control (nested booking control), and fencing decisions are determined. Finally, a numerical experimentation study is presented to highlight the impact of some significant problem-related factors such as demand variabil- ity and leakage rate onto the airline’s RM decision. Additionally, the fencing investment decision is also studied numerically to determine the airline’s decision toward demand leakage control.
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o P timal fenCing in airline industrY
the market into two segments using differentiated prices strategy. The market segmentation is assumed imperfect; thus, the customers can- nibalize from the full fare class to discounted fare class. In order to mitigate demand leakage, a fencing investment is proposed to improve the airline’s market fences. Assuming stochastic demand, the airline performs a nested control over the single-resource capacity following Littlewood’s (1972) rule for customers’ sequential arrival. Let c denote the capacity of the airplane cabin. The airline offers seats in the cabin for two adjacent fare classes: class 1, designated for business travelers willing to pay full fare price p1, and class 2, for leisure travelers willing to pay a discounted fare price p2, where p1 ≥ p2. Like many other stud- ies from RM literature (see Choi, 1996; Chiang and Monahan, 2005;
Zhang et al., 2010), we assume a linear price-dependent demand, which, in a riskless perfect market segmentation case, is given by [αi − βipi ]+, where αi,βi > 0, ∀i = {1,2}. After the price differentiation strategy, the market segments created are assumed imperfect, and the airline observes γ proportion of passengers cannibalizing from full fare to discounted fare class. To model this behavior, we use a liner function given by γ(p1 − p2), where γ ≥ 0 represents leakage rate. If γ = 0, then the airline is considered to have a perfect fence. Thus, the determin- istic linear demand curves influenced by demand leakage would be
y p p1( 1, ,2 γ)=α1−β1 1p −γ(p1− p2) (3.1) y p p2( 1, ,2 γ)=α2−β2 2p +γ(p1− p2) (3.2) The stochastic demand Di, for fare class i, ∀i = {1,2}, is modeled from deterministic demand yi and a random factor ξi, where ξi has price- independent probability distribution fi(ξi) and cumulative probability distribution Fi(ξi), both continuous, twice differentiable, invertible, and following an increasing failure rate. Moreover, ξi is assumed in [ , ]ξ ξi i with mean μi and standard deviation σi. This study follows Mostard et al. (2005); in this study, we have assumed ξi∈ −[ 3σ, 3σ].
Following Petruzzi and Dada (1999), an additive approach is assumed for Di, ∀i = {1,2}, such that
D yi
(
i,ξi)
= yi +ξi, ∀ =i { , }1 2 (3.3) A list of notations relevant to the model is presented in Table 3.1.5 8 sYed asif r a z a and mihael a turiaC
3.2.1 No Fencing Investment
As specified earlier, Di ∀i = {1,2} is assumed with sequential arrivals so that the airline observes discounted fare class demand prior to the full fare class demand. Thus, the airline’s revenue from offering two fare classes to its passengers while performing a nested control of the inventory capacity would be given as:
ˆ min min , , min ,
π = p1
{
x1+{
x D2 2}
D1}
+ p2{
x D2 2}
(3.4)The revenue function from this equation can be simplified as in the following equation (see Appendix):
ˆ
(
π ξ ξ
ξ ξ
ξ
ξ ξ
= + +
(
−)
−
( )
( )
−
+
∫
p x p x p p F d
p F d
x y
x F
1 1 2 2 1 2 2 2 2
1 1 1 1
2
2 2
1 2
1 22
2 2 2
) 1 ξ
x y
− y
∫ −
∫
(3.5)Table 3.1 Model Parameters and Notations PARAMETERS
c Inventory capacity
αi Maximum perceived demand in fare class i, ∀i = {1,2}
βi Price sensitivity of deterministic demand in fare class i, ∀i = {1,2}
yi = yi(p1,p2,γ) Deterministic demand in fare class i, ∀i = {1,2}
ξi∈[ξ ξi, i] Stochastic demand factor for fare class i, ∀i = {1,2}
fi(ξi) Probability distribution function of stochastic factor ξi , ∀i = {1,2}
Fi(ξi) Cumulative probability distribution of stochastic factor ξi , ∀i = {1,2}
Di = Di(p1,p2,γ,ξi) Price-dependent stochastic demand in fare class i, ∀i = {1,2}
≠ˆ Revenue without fencing investment E(π) Revenue with fencing investment
G(γ) Cost of fencing
G0 Initial cost of fencing
* Optimal of a decision control parameter DECISION VARIABLES
pi Price in fare class i, ∀i = {1,2}
xi Capacity allocation for fare class i, ∀i = {1,2}
γ Demand leakage factor, 0≤ ≤γ γ
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o P timal fenCing in airline industrY
The first two terms in Equation 3.5 represent the deterministic risk- less profit; the third term is the expression of revenue gain from nested capacity control mainly due to price differential (p1 > p2), where expected demand x y F2 2 d 2
2
2 2
( )ξ ξ
ξ
∫
− is protected from discounted fare class 2 and reserved for full fare class 1. The last term represents the loss in revenue due to an observed demand for full fare class, which is lower than the actual capacity allocated x1 x y F2 2 d 22
2 2
+
∫
ξ − ( )ξ ξ . Theairline problem in this case is formulated as follows:
P: p p x x , , ,
Max
1 2 1 2≠ˆ (3.6)
subject to : x1+x2 ≤c (3.7) 3.2.2 With Fencing Investment
Given that the price differentiation strategy results in imperfect fences and hence, in demand leakage, the airline’s problem extends to dimin- ishing the customers’ shifting from full fare class to discounted fare class. Without loss of generality, we presume that the airline decides to increase fencing levels through an investment of specific costs.
Suppose that for reaching γ leakage, the airline must bear a cost, G(γ), assumed nonnegative, continuous and monotonically decreasing in γ. Thus, the revenue function from Equation 3.5 is adjusted by the fenc- ing cost G(γ), and the airline problem is formulated now as a con- straint nonlinear optimization problem, P′:
ʹ = −
P G
p p x x
: ( )
,Max, , ,
1 2 1 2γπ πˆ γ (3.8)
subject to : x1+x2 ≤c (3.9) The optimal expected revenue when fencing investment decisions are taken would be π*( *, *, *, *, *)p p x x1 2 1 2 γ , and the airline’s problem is to determine the optimal integrated decisions on fare prices p1* andp2*, seat inventory control x1* and x2*,and investment G(γ*) for demand leakage γ*. It is important to notice here that the optimality of reve- nue, ≠ˆ , from P would be an upper bound on the optimal total expected revenue, π, from P′, when the airline decides on fencing investment.
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