A solution methodology for carpooling systems based on double auctions and cooperative coevolutionary particle swarms

Abstract

Although carpooling provides a way to achieve higher efficiency and lower costs of a transport system by grouping people and sharing cars or vehicles in use, it is still not widely accepted. Several studies indicate that cost savings and timeliness are the main incentives for acceptance of the carpooling business model. However, existing studies are deficient in supporting operations of carpooling business model from specifying the requirements of drivers and passengers, matching drivers with passengers to allocating cost savings among ridesharing participants. A pragmatic solution methodology to maximize cost savings while respecting timing constraints and allocate cost savings in carpooling systems is a critical factor for ensuring the success of carpooling business model. The problem to maximize cost savings subject to capacity constraints and timing constraints can be formulated as an integer programming problem. Although there are several works for solving carpooling problems based on metaheuristic approaches, existing studies on application of metaheuristic approaches in carpooling problems are still limited. Motivated by deficiencies of existing studies discussed above, this paper aims to (1) propose a carpooling model based on double auctions, (2) formulate the carpooling problem and develop a discrete cooperative coevolving particle swarm optimization (DCCPSO) algorithm to solve the problem (3) propose a cost savings allocation scheme to benefit ridesharing participants (4) study the influence of detour distance constraints on performance and (5) study the effectiveness of the proposed algorithm by comparing with other metaheuristic algorithms through experiments. Simulation results indicate that the proposed DCCPSO algorithm significantly outperforms several existing algorithms in solving the carpooling problem.

Appendices

Appendix I: Notation

Symbol	Meaning
P	the number of different passengers in the carpooling system; the set of all passengers is denoted by {1, 2, 3,....P}.
p	the index of a passenger, where p ∈{1, 2, 3,....P}.
K	the number of all pick-up and drop-off locations of passengers. Without loss of generality, it is assumed that all the pick-up and drop-off locations of passengers are distinct. In this case, there are P pick-up locations and P drop-off locations and K = 2P.
k	the index of a pick-up location or a drop-off location, where k ∈{1, 2,...,K}.
L k	the k -th location, where k ∈{1, 2,...,K}.
D	the number of drivers in the carpooling system; the set of all drivers is {1, 2, 3,....,D}.
d	the index of a driver, where d ∈{1, 2, 3,....,D}.
J d	the number of bids placed by driver d, where d ∈{1, 2,...,D}.
j	the j − th bid submitted by driver d, where d ∈{1, 2,...,D} and j ∈{1, 2,...,Jd}.
Π	a route, which consists a sequence of locations.
π d	the original route taken by driver d when driver d travels alone.
πdj(Lod, Led)	the route corresponding to the j − th bid submitted by driver d, where d ∈{1, 2,...,D} and j ∈{1, 2,...,Jd}, Lod and Led denote the origin and destination of route πdj, respectively. πdj(Lod, Led) is abbreviated as πdj whenever its is clear from the context. πdj consists of a sequence of locations $L_{1} ,L_{2} ,...,L_{\left | {M_{dj} } \right |}$.
k d j m	the m − th location visited by route πdj, $m\in M_{dj} =\{1,2,...,\left | {\pi _{dj} } \right |\}$
Πd	the set of all routes submitted by driver d, where d ∈{1, 2,...,D}, ${\Pi }_{d} =\{\pi _{dj} \left | {j\in \{1,2,...,J_{d} \}\}} \right .$.
a d	the number of available seats (capacity of the car) provided by driver d, where d ∈{1, 2, 3,....,D}.
τ d j	$\tau _{dj} = \frac {dist(\pi _{dj} (Lo_{d} ,Le_{d} ))}{dist(\pi _{d} (Lo_{d} ,Le_{d} ))}:$ the detour ratio of route πdj.
$\bar {{\tau }}_{d}$	the maximum detour ratio specified by driver d .
R d	$R_{d} = (Lo_{d} ,Le_{d} ,{\omega _{d}^{e}} ,{\omega _{d}^{l}} ,a_{d} ,\bar {{\tau }}_{d} ):$ the requirements of driver d , where $Lo_{d} ,Le_{d} ,{\omega _{d}^{e}} ,{\omega _{d}^{l}}$, ad and $\bar {{\tau }}_{d}$ denote the origin, destination, earliest departure time, latest arrival time, quantity of available seats and maximum detour ratio specified by driver d, respectively.
τ p d	estimated detour ratio $\tau _{pd} =\frac {\gamma (p,d)}{dist(\pi (Lo_{dj} ,Le_{dj} ))}$, where γ(p, d) = dist(Π(Lod, Lop)) + dist(Π(Lop, Lep)) + dist(Π(Lep, Led))
R p	$R_{p} = (Lo_{p} ,Le_{p} ,{\omega _{p}^{e}} ,{\omega _{p}^{l}} ,n_{p} ):$ the request of passenger p, where $Lo_{p} ,Le_{p} ,{\omega _{p}^{e}} ,{\omega _{p}^{l}}$ and np denote the origin, destination, earliest departure time, latest arrival time specified by passenger p and the number of passengers to be transported, respectively, and p ∈P.
$t_{p}^{dep}$	the time passenger p is picked up at Lop.
$t_{p}^{arr}$	the time passenger p is dropped of at Lep.
$t_{d}^{dep}$	the time driver d departs at Lod.
$t_{d}^{arr}$	the time driver d arrives at Led.
B d	the set of passengers $^{\prime }$ad -smallest estimated detour ratio (τpd) requests for each driver d ∈{1, 2,...,D}; $B_{d} =\{R_{p} \left | {p\in \{1,2,3,....P\},\tau _{pd} } \right .$ is ad -smallest }
c d j	the cost for transporting the bundle of passengers in the j − th bid submitted by driver d.
o d	the travel cost for taking the route πd by driver d without transporting any passengers. That is, od is the cost that driver d travels alone.
$q_{djk}^{1}$	a nonnegative integer that denotes the quantity of seats offered to pick up passengers at location k in the j − th bid submitted by driver d ; $q_{djk}^{1} =$ 0 if location k is not visited by πdj.
$q_{djk}^{2}$	a nonnegative integer that denotes the quantity of seats released due to dropping passengers at location k in the j − th bid submitted by driver d; $q_{djk}^{2} =$ 0 if location k is not visited by πdj.
$b_{dj}^{D}$	$b_{dj}^{D} = (q_{dj1}^{1} ,q_{dj2}^{1} ,...,q_{djk}^{1} ,...,q_{djP}^{1} ,q_{dj1}^{2} ,q_{dj2}^{2} ,...,q_{djk}^{2} ,...,q_{djP}^{2} ,\pi _{dj} ,c_{dj} ,a_{d} ):$ a vector generated based on Rd to represent the j − th bid submitted by driver d. The j − th bid $b_{dj}^{D}$ is actually an offer to transport passengers specified by $q_{djk}^{1}$ and $q_{djk}^{2}$ for each k ∈{1, 2, 3,....,P} at the cost cdj with capacity ad by following route πdj.
πp(Lop, Lep)	the original route from Lop to Lep that will be taken when passenger p travels alone. πp(Lop, Lep) will be abbreviated as πp whenever it is clear from the context.
n p	the number of passengers to be transported in the bid of passenger p.
$s_{pk}^{1}$	the number of seats requested in the bid of passenger p to pick up passengers at pick-up location Lk, where k ∈{1, 2, 3,....,P}.
$s_{pk}^{2}$	the number of seats to be released from in the bid of passenger p due to dropping passengers at location LP + k, where k ∈{1, 2, 3,....,P}.
f p	the price of the bid submitted by passenger p based on the original travel cost of passenger p when he/she travels alone.
${b_{p}^{P}}$	${b_{p}^{P}} = (s_{p1}^{1} ,s_{p2}^{1} ,s_{p3}^{1} ,...,s_{pP}^{1} ,s_{p1}^{2} ,s_{p2}^{2} ,s_{p3}^{2} ,...,s_{pP}^{2} ,f_{p}):$ a vector generated based on Rp to represent the bid submitted by passenger p. The bid ${b_{p}^{P}}$ is an offer to pay fp for transporting passengers.
Γdj	the set of passengers delivered in the j -th bid submitted by driver d
x d j	the variable to indicate the j − th bid placed by driver d is a winning bid (xdj = 1) or not (xdj = 0).
y p	the variable to indicate the bid placed by passenger p is a winning bid (yp = 1) or not (yp = 0).
α	cost savings allocation coefficient for carpooling information provider.
e d j k	a nonnegative integer that denotes the quantity of empty seats available after dropping and picking up passenger at location k in the j − th bid submitted by driver d.

Appendix II: Bid Generation Procedures

In a carpooling system based on double auction mechanism, passengers and drivers submit bids according to their preferences and constraints. This appendix focuses on the bid generation procedures for passengers and drivers.

1.1 (A) Bid Generation for Passengers

In this paper, it is assumed that each passenger has only one request. For each request, a bid will be generated according to the request. The request for the passenger p, where p ∈P, is represented by $R_{p} =(Lo_{p} ,Le_{p} ,{\omega _{p}^{e}} ,{\omega _{p}^{l}} ,n_{p} )$, where Lo_p, $Le_{p} ,{\omega _{p}^{e}} ,{\omega _{p}^{l}}$ and n_p denotes the origin, destination, earliest departure time, latest arrival time and number of passengers to be transported, respectively. For each R_p, a Generation Procedure for Passengers will be used to generate a bid ${b_{p}^{P}}$.

Let π_p(Lo_p, Le_p) denote the original route from Lo_p to Le_p that will be taken if passenger p travels alone. Generation of a bid for a passenger is a based on the route π_p(Lo_p, Le_p) from the origin Lo_p to the destination Le_p. The original distance of route π_p(Lo_p, Le_p) is denoted by dist(π_p(Lo_p, Le_p)), which can be found by applying the Google Distance Matrix API. The bid price f_p generated for the requirement R_p is based on a pricing function Pf_p(dist(π_p(Lo_p, Le_p))). Suppose the travel cost of a route is proportional to the distance of the route. Let ω denote the cost coefficient per km. The cost of taking the route π_p(Lo_p, Le_p) is f_p = Pf_p(dist(π_p(Lo_p, Le_p))) = ω × dist(π_p(Lo_p, Le_p)). The bid generation procedure generates bid price f_p for p ∈{1, 2, 3,...,P}. The bid generated for R_p is${b_{p}^{P}} =(s_{p1}^{1} ,s_{p2}^{1} ,s_{p3}^{1} ,...,s_{pP}^{1} ,s_{p1}^{2} ,s_{p2}^{2} ,s_{p3}^{2} ,...,s_{pP}^{2} ,f_{p} )$, where $s_{pk}^{1}$ denotes the number of seats requested to pick up passengers at location $L_{k} ,s_{pk}^{2}$ is a non-positive integer that denotes the number of seats to be released due to dropping passengers at location L_{P + k},,k ∈{1, 2, 3,....,K} and f_p is the bid price. Note that $s_{pk}^{1}$ is greater than zero only if L_k = Lo_p and $s_{pk}^{1}$ is zero otherwise; $s_{pk}^{2}$ is greater than zero only if L_k = Lo_p and $s_{pk}^{2}$ is zero otherwise. The Bid Generation Procedure for Passengers is as follows.

1.2 Bid Generation Procedure for Passengers

1.3 (B) Bid Generation for Drivers

Generation of bids for drivers is a complex task. Two sub-problems are involved to generate bids for a driver. The first sub-problem is to select passengers and the second one is to determine the sequence to pick up and drop the selected passengers for ridesharing. The computational complexity to generate all possible bids for a driver grow explosively with respect to the number of passengers as well as the number of seats of the car. To reduce the computational complexity in generation of bids for a driver, a simple rule is applied to assign requests of passengers according to the origin and destination of the driver and the pick-up/drop-off locations of the passengers’ requests. As drivers usually will not share rides with passengers that are far away, an index called ‘detour ratio’ is defined for drivers to select potential passengers. The proposed procedure calculates an estimated detour ratio $\tau _{pd} =\frac {\gamma (p,d)}{dist(\pi _{d} (Lo_{d} ,Le_{d} ))}$ of each passenger’s request, where γ(p, d) = dist(Π(Lo_d, Lo_p)) + dist(Π(Lo_p, Le_p)) + dist(Π(Le_p, Le_d)), which is the sum of the distance from the origin of driver d to the pick-up location of passenger p, the distance from the pick-up location of passenger p to the drop-off location of passenger p and the distance from the drop-off location of passenger p to the destination of the driver d. Let $\bar {{\tau }}_{d}$ denote the maximum detour ratio specified by driver d . Only passengers’ requests that satisfy the detour constraint will be selected as the potential passengers for a driver. That is, driver d will consider to meet the requirements of ${b_{p}^{P}}$ only if $\tau _{pd} \le \bar {{\tau }}_{d}$.

For each driver d, the procedure generates bids based on the requirements of driver $d,R_{d} =(Lo_{d} ,Le_{d} ,{\omega _{d}^{e}} ,{\omega _{d}^{l}} ,a_{d} )$ and the requirements of passengers’ request $R_{p} =(Lo_{p} ,Le_{p} ,{\omega _{p}^{e}} ,{\omega _{p}^{l}} ,n_{p} )$. The bid generation algorithm first calculates γ(p, d) and τ_pd to find the set B_d of requests with a_d smallest γ(p, h, d) for driver d. The algorithm then constructs a set of feasible routes Π_d to pick up $q_{djk}^{1}$ and drop off $q_{djk}^{2}$ passengers associated with B_d subject to the maximum detour ratio $\bar {{\tau }}_{d}$ specified for driver d and the time constraints described by ${\omega _{p}^{e}} \forall p\in \{1,2,3,....P\}$ and ${\omega _{p}^{l}} \forall p\in \{1,2,3,....P\}$ of individual passengers. Finally, our algorithm calculates the cost c_dj for each driver’s route π_dj ∈Π_d as follows. Suppose driver d ∈ D travels by following the route π_dj to pick up and drop off passengers. The distance of taking the route π_dj(Lo_d, Le_d) is denoted by dist(π_dj(Lo_d, Le_d)), which can be found by applying the Google Distance Matrix API. The bid price placed by driver d ∈ D is based on a pricing function c_dj = Df_dj(dist(π_dj(Lo_d, Le_d))). Let c_dj = Df_dj(dist(π_dj(Lo_d, Le_d))) = ω × dist(π_dj(Lo_d, Le_d)), where ω is the cost coefficient per km. The bid generated for driver d ∈ D is$b_{dj}^{D} =(q_{dj1}^{1} ,q_{dj2}^{1} ,...,q_{djk}^{1} ,...,q_{djP}^{1} ,q_{dj1}^{2} ,q_{dj2}^{2} ,...,q_{djk}^{2} ,...,q_{djP}^{2} ,$π_dj, c_dj, a_d), where $q_{djk}^{1}$ is greater than zero only if $q_{djk}^{1}$ passengers are picked up at L_k in route π_dj and $q_{djk}^{2}$ is greater than zero only if $q_{djk}^{2}$ passengers are dropped at L_k in route π_dj. The algorithm to generate bids for drivers is as follows.

1.4 Bid Generation Algorithm for Drivers