Selection of the Intensity Functions

In our computational setup we have selected different arrival intensity function empha-sizing the difference between the policies generated by the DP and the EMSR heuristics.

To evaluate the policy of the EMSR heuristic it is only important to know the expected number of type j fare class arrivals over the whole booking period. For the evaluation of the DP policy it is also important to know how these arrivals are spread over this booking period. To stress this difference we try to select intensity functions which give the same expected number of arrivals over the whole period but are differently shaped. In particular we are interested for m = 2 in the earliest and latest possible time that given an arrival this is more likely to be an arrival of an expensive fare class request. To construct these intensity functions for the different fare classes let T denote the length of the booking period and assume that the fares of the different fare classes 1, · · · , m are given by r_i. Without loss of generality these fares satisfy r₁ < r₂ < · · · < r_m and so fare class m is the most expensive while fare class 1 is the cheapest one. To select the intensity functions we first consider only two fare classes and generalize the used approach to m > 2 cases.

We start with the following normalized functions ai : R+ 7→ R.

Condition 14 The functions a_i : R+ → R, i = 1, 2 satisfy the following conditions.

• The function a₁ is a nonnegative continuously differentiable decreasing function on [0, T ] satisfyingRT

0 a₁(s)ds = 1.

• The function a₂is a nonnegative continuously differentiable increasing function on [0, T ] satisfyingRT

0 a2(s)ds = 1.

If λi : R⁺ → R denote the intensities functions for type j fare class arrivals then we set λ₁(s) := σ₁a₁(s), λ₂(s) := ασ₁a₂(s) (4.1)

with σ₁ > 0, α > 0. For these intensity functions it is clear that

E(number of arriving fare class 1 customers) = Z T

0

λ₁(s)ds = σ₁

and

E(number of arriving fare class 2 customers) = Z T

0

λ₂(s)ds = ασ₁.

Since on average cheaper fare class 1 requests arrive one should select 0 < α < 1.

To measure the capacity of the plane in comparison with the total number of expected requests the load ρ of the system is by definition

ρ := E(expected number of arriving customers)

C = σ₁(α + 1)

C

By Condition 14 and relation (4.1) the function λ₁ is decreasing and λ₂is increasing. As shown in the next lemma this monotonicity property represent the tendency of fare class 1 customers to arrive more frequently than fare class 2 customers during the beginning of the booking period while the reverse is true towards at the end of the booking period. To verify this we observe

p_i(t) = P(arrival is fare class i request | arrival at time t)

= _λ ^λi(t)

1(t)+λ2(t)

(4.2)

This shows using relation (4.1) that

p₁(t) = a₁(t)

a1(t) + αa2(t), p₂(t) = αa₂(t)

a1(t) + αa2(t) (4.3) The following result is easy to verify.

Lemma 15 If the arrival intensity function λ_i, i = 1, 2 are given by relation (4.1) and the functionsa, i = 1, 2 satisfy condition 14, then the function t 7→ p (t) is decreasing and

By the above lemma the time

t∗ = min{0 ≤ t ≤ T : p₂(t) ≥ p₁(t)} = min{0 ≤ t ≤ T : αa₂(t) − a₁(t) ≥ 0} (4.4) represents the earliest possible time that an arrival after this time is more likely to be a class 2 type arrival. To guarantee that t∗ is well defined the selected intensity func-tions a_i must satisfy Condition 14 with the additional conditions a₁(0) ≥ αa₂(0) and a₁(T ) ≤ αa₂(T ). Among the set of functions a_i satisfying all these restrictions we now would like to determine the extremal elements which minimize and maximize the value t∗. Using these extremal elements we can observe that the EMSR heuristics only take in consideration the expected number of customers whereas the policy of the DP algorithm might change due to a changing time t∗. Note it is easy to verify for the selected intensity functions that the feasible set of this optimization problem is described as follows

1. The function a1is a nonnegative continuously differentiable function on [0, T ] sat-isfying

Z T 0

a1(s)ds = 1, max0≤s≤T a⁰₁(s) ≤ 0. (4.5) 2. The function a2 is a nonnegative continuously differentiable function on [0, T ]

sat-isfying

Z T 0

a₂(s)ds = 1, min_0≤s≤T a⁰₂(s) ≥ 0 (4.6) 3. It holds that

a₁(0) − αa₂(0) ≥ 0, αa₂(T ) − a₁(T ) ≥ 0 (4.7) In the following we solve this optimization problem for special cases where a_i(t)’s are either linear or quadratic functions. We consider linear intensity functions first.

4.1.1.1 Linear Intensity Functions Let the functions a₁(s) and a₂(s) given by

a₁(s) = a₁₁− a₁₂s a₂(s) = a₂₁+ a₂₂s (4.8) and by relation (4.1) the arrival intensities are;

λ1(s) = σ(a11− a12s), λ2(s) = σα(a21+ a22s).

For the linear function case, optimization of t∗ can be transformed into simple fractional programming and the solution can be found analytically. In the Appendix C.1 we provide extensive analysis of the solution. The analytical solution reveals that the minimum t∗ =

T

1+α can be reached by the following two linear functions

λ1(s) = σ(2T⁻¹− 2sT⁻²) and λ1(s) = 0 + σα(2sT⁻²).

Also the following functions intersect at the maximum point t∗ = T ; λ₁(s) = σ((2 − α)T⁻¹− 2(1 − α)T⁻²) and λ₂(s) = σT⁻¹ In Figure 4.1, we depict the extreme cases.

4.1.1.2 Quadratic Intensity Functions

Let the functions ai : [0, T ] → R+, i = 1, 2 satisfy the parametric representation

ai(s) = ai1+ ai2s + ai3s² (4.9) We clarify the feasibility conditions of the functions in relations (4.5), (4.6) and (4.7). For construction of feasible region and solution procedure we refer to Appendix C.2. Feasi-ble region of this setting denoted by polytope Pq. Introduce for a = (a1(0), ...., a⁰⁰(0))

(a) minimum t_∗ (b) maximum t_∗

Figure 4.1: Example of fare class arrival probabilities in extreme linear cases

belonging to P_qthe function h : P × R → R given by h(a,t) := a₁(t) − αa₂(t)

Observe for every t the function a 7→ h(a,t) is a linear function. Also we know by Lemma 15 that p₁(t) ≥ p₂(t) if and only if h(a,t) ≥ 0. Now introduce the function t(a) : R⁶ 7→ R+

t(a) = h(t∗) = a₁₁+ a₁₂t∗+ a₁₃t²_∗− α(a₂₁+ a₂₂t∗+ a₂₃t²_∗) = 0.

We can easily verify the following lemma;

Lemma 16 The function t : P → [0, T ] is continuous and quasiconcave.

The objective function is

min{t(a); a ∈ P_q} (4.10)

Unfortunately, this problem does not have an analytical solution. However we know that the minimum t∗ is attained at a vertex of P_qbecause the function in 4.10 is quasiconcave on the polytope Pq. Hence, enumeration of all vertices will lead us to the optimum point.

4.1.1.3 Comparing Linear and Quadratic Arrival Intensity Functions

Clearly type of intensity function affects on the intersection point t∗. Also as we mention above section, the overall booking demand has an influence on the intersection point.

Then, t∗ changes according to value of α which is the function of expected number of customers arrivals. Figure 4.2 shows that for quadratic arrival intensities the intersection points t^∗ are lower compared to those for linear arrival intensity functions.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t*

α Quadratic functions Linear Functions

Figure 4.2: Values of t^∗ according to different α

4.1.1.4 Generalization to More Than 2 Fare Classes

To apply the above procedure to more than 2 fare classes let there be m fare classes and introduce the numbers 1 := α₁ > α₂ > ... > α_m > 0 and σ₁ > 0. Also consider two functions a_i, i = 1, 2 satisfying Condition 14 and select the intensity functions λ_i : R+ →

for i ≤ q and

λ_i(t) = α_iσ₁a₂(t) (4.12)

for i ≥ q + 1. This means that the cheaper fare classes 1, ·, q have decreasing arrival intensity function and the more expensive fare classes p + 1, ..., m have increasing arrival intensity functions.

If we select the arrival intensities as in(4.11) and (4.12), both quadratic and linear cases boil down to the two fare classes. Intersection point of p₁(t) and p_m(t) directly implies that the other fare class functions have already intersected with each other. Hence, we only pay attention to select an appropriate α_i, 1 ≤ i ≤ m.