Service Bundle Sizing and Pricing in Ride Sourcing Services
QingYang "Tom" Xiaoa , Jee Eun "Jamie" Kangb,∗ and Roger B. Chenc
a Dept. of Industrial & Systems Engineering, State University of New York at Buffalo, USA

qxiao2@buffalo.edu
b Dept. of Industrial & Systems Engineering, State University of New York at Buffalo, USA

jeeeunka@buffalo.edu
c Dept. of Civil & Environmental Engineering, University of Hawaii at Manoa, USArbchen@hawaii.edu∗ Corresponding author

Extended abstract submitted for presentation at the 11th Triennial Symposium onTransportation Analysis conference (TRISTAN XI)June 19-25, 2022, Mauritius Island

April 11, 2022
Keywords: Bundle Sizing, Pricing, Probit Model, Ride Sourcing Services

1INTRODUCTION

Ride sourcing services have emerged as a viable personalized mobility service to many travelers
and received significant attention from the research community as well as industries. Different
types of ride sourcing services are being introduced and expected to further grow in the future.
As such, in recent years, ride sourcing service operators have introduced a variety of service
bundles (e.g. Uber Ride Pass and Lyft All-Access Plan) in addition to the traditional pay-peruse (PPU) strategy. Introducing bundled services can provide several merits: offering a diversityof service types (e.g. daily/weekly/monthly passes) upon market needs can better meet travelers’
needs, potentially increase the market share, and reduce cost for travelers but at the same time,
improving efficiency of the system operations.
To our best knowledge, a number of previous research investigated these positive impacts
of introducing ride sourcing service bundles and its pricing, few have focused on formulating aservice bundle design, sizing, and pricing problems. In this paper, we formulate an optimizationbased ride sourcing service bundle sizing and pricing problem focused on commuting trips. While
we use "commuting" trips as illustration, this can simply be replaced by total miles, minutes,
or price for mobility services. Travelers’ choices of bundles are represented using a multivariate
probit (MVP) model as choices among different bundles are clearly correlated. The proposed
formulation maximizes total revenue and determines sizes and prices of the service bundles forcommuting trips (i.e., same trip characteristics). The proposed model is a nonlinear program(NLP) and KKT is applied to find optimal solution.

2METHODOLOGYLet K = {1, .., k, ..., K} be an ordered set of available service bundles based on the numberof trips it can provide (i.e. bundle sizing), Ψ = {ψ1 , ..., ψk , ..., ψK } with corresponding prices
C = {c1 , ..., ck , ..., cK }. It is noted that k = 1 with service bundle size of 1 ψ1 = 1 represents
PPU pricing at c1 .
TRISTAN XI Symposium

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QingYang "Tom" Xiao, Jee Eun "Jamie" Kang and Roger B. Chen

2.1

2

Commuting Trip Bundle Demand Specification based on MVP Model

The utility for commuter n who chooses monthly commuting bundle k consists of deterministic
part Vnk (ψk , ck ) and stochastic part (i.e. error term) εnk with mean of zero and covariance matrixΩ. Deterministic utility is defined as a linear combination of bundle size, ψ and the price, c.
Then the overall utility can be defined as:Uk (ψk , ck ) = Vk (ψk , ck ) + εk(1)= βk + βc ck + βψ ψk + εkwhere βk represents constant term for bundle k, and βc , βψ represent weights between cost andsize of a bundle.
According to MVP (Train, 2009), the probability for a commuter choosing bundle k is defined
as below.
Z Vk (ψk ,ck )−V0 (ψ0 ,c0 ) Z Vk (ψk ,ck )−VK (ψK ,cK )exp{− 12 (ηk )(Ωk )−1 ηk }···pk (ψk , ck ) =dεK −εk · · · dε0 −εk
K−11−∞−∞(2π) 2 |Ωk | 2(2)As these probabilities are not in a closed form and this integration is difficult, we rely on GHK
simulation algorithm (Cappellari & Jenkins, 2003) which enables computation of probabilities ofchoosing each choice given an observation data set. GHK simulation randomly chooses truncatedobservation ν from normal distribution. Then, the probability of choosing choice k is as follows:
1 X
pk (ψk , ck ) =|T |YΦ(βk − βk′ + βc (ck − ck′ ) + βψ (ψk − ψk′ ) −Pk ′τξ=0 lk′ ξ · νξlk′ k′τ ∈T ∀k′ ∈K, k′ ̸=k) (3)where Φ(z) stands for CDF of standard normal distribution, lk′ ξ , lk′ k′ are extracted by lower triangular matrix Lk by Cholesky decomposition of reduced variance-covariance matrix Ωk derived
from variance-covariance matrix ΩK . Equation (3) provides the likelihood of choices given priceand the size of each bundle k.

2.2

Service Bundle Size and Pricing Model

With a demand expression that provides probabilities of choosing bundle k given its price andthe size, the service bundle sizing and pricing model is formulated as below.
Xmax Z =ck · dk (ψk , ck )Ψ,C∀k∈K
X= dK
ck · pk (ψk , ck )∀k∈KXdK X
ck=|T |
τ ∈T∀k∈KY

Φ(βk − βk′ + βc (ck − ck′ ) + βψ (ψk − ψk′ ) −∀k′ ∈K, k′ ̸=klk′ k′Pk′τ
ξ=0 lk′ ξ · νξ)

(4)
s.t.:

max
∀k‫∈ ג‬K,ψk‫> ג‬ψk{c1 , ck‫ ג‬− (ψk‫ ג‬− ψk )c1 } ≤ ck , ∀k ∈ K(5){ψk c1 , ckℶ + (ψk − ψkℶ )c1 } ≥ ck , ∀k ∈ K(6)min∀kℶ ∈K,ψkℶ <ψk+ck ∈ (0, c+k ], ψk ∈ (0, ψk ], ∀k ∈ K′ψk ̸= ψk′ , ∀k, k ∈ K, k ̸= k′

(7)
(8)

Objective is to maximize the total revenue Z across offered bundles summation of all bundle pricing ck ∈ C times corresponding quantity of subscribing respondents dk (ψk , ck ) ∈ dK incorporated
TRISTAN XI Symposium

Original abstract submittal

QingYang "Tom" Xiao, Jee Eun "Jamie" Kang and Roger B. Chen

3with bundle choice probability Equation (3) depicted in Equation (4) where bundle price ck andsize ψk are only variable vectors. The bundle price ck lower bound in inequality Constraint (5)guarantees the price difference between bundle k ‫ ג‬and k, for the case bundle size of k ‫ ג‬larger thanbundle k, would be less or equal to these two bundle passes’ difference pay by PPU because of
possible discount rate and total subscription price is no less than PPU by default. The inequality
Constraint (6) guarantees the bundle price ck upper bound offered by the charge discrepancy
between bundle k ℶ and k, with condition bundle size of k ℶ less than k, is less or equal to thedifference bundle trips pay by PPU with discount rate and bundle price is intuitively cheaper
than every time pay by PPU attached to these bundle trips. The Constraint (7) guarantees thebundle price to be positive and within a pricing range based on price per trip by empirical data+
and bundle size is bounded by travelers’ commuting needs where c+k and ψk stand for bundle k
price and size designs’ upper limit. The last Constraint (8) prevents the size of different bundle
being the same which offers diversity for bundle choices better meet travelers’ commuting needs.

2.3

Solution Method: Tocher Approximation and KKTThe proposed formulation of Commuter Bundle Sizing and Pricing problem is a nonlinear program. Here we use an approximate objective function to be in a convex form and apply KKT.
The objective function (4) with price and size constraints from (5) through (8) are still very
difficult to approach as CDF of standard normal distributions in the objective plugged withvariables ck and ψk . The Tocher approximation (Tocher, 1967, Choudhury, 2014, Yerukala &Boiroju,q 2015) further simplifies the objective maximization function shown in Equation (9) withk=2π.

XYdK X
ckmax Z =|T |Ψ,Cτ ∈T ∀k′ ∈K, k′ ̸=k∀k∈KP ′τ
βk −βk′ +βc (ck −ck′ )+βψ (ψk −ψk′ )− k
ξ=0 lk′ ξ ·νξlk ′ k ′2ke

2k1+eP ′βk −βk′ +βc (ck −ck′ )+βψ (ψk −ψk′ )− kl·ν τ
ξ=0 k′ ξ ξlk ′ k ′(9)With the Tocher approximation, the objective function in Equation (9) is concave then thisoptimization problem has global maximum. As such we can apply the KKT condition to findthe optimum bounded by constraints. The vectors of all first order conditions can be enumerated
TK ∂LKby ▽LK = [ ∂L
∂ck , ∂ψk ] = 0 with endogenous variables ck and ψk with Lagrangian function LK
integrated with the objective Z. The complementary slackness vectors functions can also bederived by equality sets respectively. Then the global optimum could be reached by KKT if theobjective function proved convexity with linear constraints for bundle size and pricing.

3CASE STUDY

We develop a case study that determines a monthly commuter travel bundles. One thousand
travelers are generated with their intended commuting days (between 1 to 20). Travelers areassumed to choose bundle k, based on their intended commuting days. Given two generated
scenarios of bundle sets (i.e., two sets of sizes and prices), travelers’ choose one bundle k basedon log-normal probabilities given price and size.
The generated sample is used for the MVP model estimation for constants βk , coefficientsβc and βψ by maximum likelihood estimation (MLE) and the result is presented in Table 1.
The estimated parameters indicate all constants positive and respondents are sensitive to bundle
price changing and discouraged by increment of price, also demonstrating at close bundle pricelevel subscribers are prone to attach to bundles with more passes. Further the corresponding
scaled covariance matrix ΩK can be derived where all diagonals parameters are scaled to be one.
With these parameters, results of the bundle sizing and pricing model is presented in Table
2. The computation requires limited computing time. Bundle K is fixed with five bundle choices
TRISTAN XI Symposium

Original abstract submittal

QingYang "Tom" Xiao, Jee Eun "Jamie" Kang and Roger B. ChenMVP model

Estimate
Std. Error
p-value

4β0

β1

β2

β3

β4

βcβψ

0.63
0.17
0.00611.31
0.34
0.00811.98
0.60
0.01311.72
0.44
0.06931.25
0.25
0.0230-0.0300.006
0.00000.32
0.07
0.0000

Table 1 – Bundle Choice MVP Model Estimationincluding PPU by empirical ride sourcing services bundle design data having K = {0, 1, 2, 3, 4}
where each bundle’s size and price are solved optimal. The result shows the bundle size optimum
to be reasonable as the bundle size diversities and spreads appropriately to better satisfy a varietyof travelers’ commuting trips. Solved by bundle subscription price, the corresponding optimal
trip per price for each bundle is also rational as travelers are usually more sensitive to price pertrip and all solved bundle price per trip offers sound discount making the travel fair per trip
cheaper than standard price (i.e. PPU price) meanwhile maximizing stakeholder’s revenue.
Optimal bundle size

PPU

4.18-day bundle

9.87-day bundle

17.04-day bundle

Unlimited(20-day bundle)

Optimal bundle
price

$12

$49.41$114.69$196.13$227.8

Price per trip

$12

$11.82$11.62$11.51$11.39

Discount

-1.5%

3.2%

4.1%

5.1%

Table 2 – Optimum Bundle Size and Price by KKT

4Conclusion

In this paper, we formulate a commuting trip service bundle sizing and pricing problem basedon travelers’ choices of bundles based on MVP model. MVP model accounts for traveler choicesthat are correlated among different bundle choices. The proposed formulation is able to generatereasonable outcomes for bundle sizing and prices and the computation is efficient. It is noted thatwhile we have illustrated this as "commuting trips", these can simply be extended to total miles,minutes, or prices of mobility service bundles it covers. The next potential step is characterizing
trips (e.g., Origin-Destination, time, etc.). Currently in this approach, all trips are represented
as days (commuting trips) or can be extended as miles, minutes, or prices. However, the ridesourcing operators are highly constrained by trip requirements due to spatial-temporal mismatchof fleet and demand as well as vehicle relocations. Then, this bundle sizing and pricing can befurther integrated with fleet management and relocation operations.

References
Cappellari, Lorenzo, & Jenkins, Stephen P. 2003. Multivariate probit regression using simulated maximum
likelihood. The STATA journal, 3(3), 278–294. https://doi.org/10.1177/1536867X0300300305.
Choudhury, Amit. 2014. A simple approximation to the area under standard normal curve. Mathematicsand Statistics, 2(3), 147–149.
Tocher, Keith Douglas. 1967. The art of simulation. English Universities Press.
Train, Kenneth E. 2009. Discrete choice methods with simulation. Cambridge university press. https://doi.org/10.1017/CBO9780511805271.
Yerukala, Ramu, & Boiroju, Naveen Kumar. 2015. Approximations to standard normal distributionfunction. International Journal of Scientific & Engineering Research, 6(4), 515–518.

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