Social distancing and revenue management-A post-pandemic adaptation for railways.

The SARS-CoV-2 pandemic has had a significant impact on rail operations worldwide. Adopting control measures such as a 50% occupancy rate can contribute to a safer travel environment, though at the expense of operational efficiency. This paper addresses the issues of social distancing and revenue maximization for a train operating company in a post-pandemic world. Although the two objectives appear to be highly contradictory, we believe that judicious planning can optimize both to a great extent. Existing research on social distancing on public transport has only considered the risk of virus transmission during travel. This is the first attempt to recognize the risk of virus spread in different cities along with transmission risk as part of developing a social distancing plan. We study the problem of assigning seats to passenger groups on long-distance trains while ensuring social distancing within coaches. A novel seating assignment policy is proposed that takes into account several factors that govern the spread of virus. In an effort to reduce the spread of the virus and improve revenue simultaneously, a mixed-integer programming (MIP) model is proposed to assign seats to passengers. Several families of valid inequalities and preprocessing steps are proposed to strengthen the MIP formulation, which represents a substantial contribution to the literature on group seat assignment problem. The validity of the model and the effectiveness of the valid inequalities have been evaluated using real-life data from Indian Railways. The computational results demonstrate a significant reduction in the risk of contagion and an increase in seat utilization compared to the current approach employed by operators.

Introduction

The World Health Organization declared SARS-CoV-2 outbreak a global pandemic in March 2020. This phenomenon compelled governments and institutions across the globe to restrict people’s mobility, thereby minimizing the pandemic’s severity (peakedness of the number of infections along time). However, the control measures resulted in a variety of undesired outcomes, such as an economic downturn and disruptions to the global supply chain network (SCN) [1], [2]. Researchers attempt to construct a viable SCN that has the capability to survive and persist in a changing environment with long-lasting effects [3], [4], [5], [6]. Similarly, public transport operators endeavored to adopt long-term management mechanisms that would ensure the sustainability of their operations. In this case, operational efficiency and passenger safety serve as the key determinants of sustainability. Traditionally, operational planning ascribes passenger safety to convenience and comfort [7], [8], [9]; however, the recent pandemic adds a new dimension, namely the spread of virus [10], [11], [12], [13].

Public transport sector is one of the worst-affected industries during pandemics due to travel restrictions and passenger reluctance [14]. Some inherent conditions, such as high occupant density, common contact points, extensive interaction hours and difficulty in tracking infected individuals, cause public transportation to be regarded as a crucial player in the spread of virus [15], [16]. According to studies on airborne diseases, road and rail travels play a significant role in the spread of virus to multiple cities due to extensive interaction hours during the journey [17], [18], [19]. Recent studies on SARS-CoV-2 also indicate that the potential for virus transmission from infected passengers to other occupants during travel is exorbitantly high [20], [21]. Moreover, it suggests a high likelihood of virus spread from regions of high to low intensity due to the mobility of people. This phenomenon, commonly referred to as pandemic importation, may cause rail passengers to switch to safer modes of transportation, thereby posing a greater threat to train operating companies (TOC).

The rail industry plays a crucial role in intercity transportation, and has been severely impacted by the spread of virus. Research suggests multiple strategies to prevent the spread of virus, with social distancing being one of the most widely debated [22], [23]. Here, the overarching objective is to spatially and temporally isolate people or minimize their interaction. Broadly, passenger interaction during rail travel occur at platforms and within coaches during the actual journey. The latter is of greater concern in rail operations due to the increased likelihood of contact and interaction hours, which depend directly on the passenger seating plan. The seat assignment process entails assigning exact seats to passengers from various origin-destination (OD) pairs. In general, railways employ a computerized reservation system (CRS) to regulate pricing, seat availability and tickets issuance. In the context of Indian Railways (IR), the reservation system offers two types of tickets to passengers — confirmed and waitlisted tickets, detailed in Section 3. All passengers with waitlisted tickets may not be assigned seats due to a limited number of seats; consequently passenger requests failing to obtain a seat are rejected.

The aforementioned factors motivate us to develop a seating plan for train passenger groups (passengers traveling together) that minimizes virus spread and maximizes operator’s revenue. We investigate two aspects of virus spread — virus diffusion between cities and virus transmission among passengers in coaches. To reduce the spread of virus between cities, we seat passenger groups boarding from cities with similar infection intensity together, as the risk of diffusion is exorbitantly high under the contagion of individuals from low and high infection intensity cities [24]. Further, to lower the risk of virus transmission and bolster economy, we consider different norms for minimizing interaction among passengers. This is generally attributable to regional variations in virus infection intensity [25], impact of immunization [26], [27] and the role of regional control directives [28]. The study examines SARS-CoV-2 infection intensity as a major factor in the spread of the virus, as measured by the seven-day simple moving average (SMA) of daily infections per 10 million population. In this regard, we develop a mixed-integer linear programming (MILP) model that simultaneously minimizes virus spread as well as maximizes operator’s revenue; we refer to it as the “Group Seat Assignment Problem with Social Distancing” (GSAPSD). We strongly believe that the findings of this study may assist TOC in determining effective control measures that foster a safe travel environment and enhance operational performance.

The rest of the paper is organized as follows. Section 2 presents a systematic background to social distancing, revenue management, and group seat reservation policy. Section 3 defines various risk levels depending on passenger interaction and a corresponding seating plan for each. Section 4 provides a mathematical model for the GSAPSD, and Section 5 lists several families of valid inequalities and preprocessing steps that strengthen the model. Section 6 presents detail on the cut-and-branch algorithm utilized to solve the GSAPSD and explicates enumeration techniques or separation methods for each valid inequality. In Section 7, we evaluate the seating arrangement in terms of virus spread and operator’s revenue. In addition, it demonstrates the computational performance of the proposed algorithm. Finally, the paper concludes with a discussion of future research avenues in Section 8.

Related literature

The present study is closely connected to three research themes—social distancing in public transport, revenue management and group seat reservation policy. The subsequent sections review literature pertaining to each perspective and highlight significant differences between the present study and previous research.

Social distancing in public transport

Social distancing in public transport has attracted considerable attention from the research coterie since pandemics. Kamga and Eickemeyer [29] review various control measures adopted by transport operators to facilitate social distancing. The authors contend that operators typically employ interpersonal distancing criteria, followed by modifications to operational decisions such as train length, boarding/deboarding strategies, service level, route planning, etc. Based on infrastructural settings and passenger dynamics during the journey, social distancing measures are primarily enforced at stations and on-board a vehicle. Since the present study focuses on creating social distancing among passenger groups on a train, we restrict our literature review to this subfield. However, interested reader may refer to [30], [31], [32], [33], [34], [35], [36], [37] to comprehend crowd management at rail/bus stations and airports in response to social distancing measures.

To provide interpersonal distancing among passengers on board a vehicle, transport operators employ measures such as leaving alternate or middle seats vacant, operating at 50% of the capacity, etc. [38], [39]. Gioda [40] performs a uniform two-dimensional interpersonal distancing among train passengers, i.e., the distancing criteria does not contemplate passengers’ immunization status, infection intensity at different cities, interaction hours (), etc. Haque and Hamid [13] propose one-dimensional interpersonal distancing between passengers, with the gap between them determined by the level of infection at their boarding stations. Fridrisek and Janos [41] and Hörcher et al. [42] investigate the influence of travel environment (vehicle occupancy) and demand management strategies on interpersonal distancing in public transport, respectively. Hörcher et al. [42] also demonstrate that employing multiple strategies simultaneously aids in controlling the spatial and temporal occupancy. Further, in airlines sector several articles investigate the influence of seat location, interpersonal distance, passengers’ health attributes, occupants’ movements and use of protective gear (face mask) on virus transmission risk [43], [44], [45], [46], [47]. In light of the preceding discussion, we cognize the benefits of interpersonal distance in public spaces for preventing the spread of virus; however, there implementation in rail operation is meagre.

Revenue management

Revenue management (RM) refers to the conglomeration of various demand management strategies (dynamic pricing, network management, forecasting, capacity control, overbooking, cancellation policy, etc.) that a firm effectuate to improve its revenue [48], [49]. These strategies are primarily classified as pricing-based and quantity-based methods. Over the past four decades, exhaustive research has been conducted on various RM domains in different sectors, including airlines [50], [51], railways [52], [53], hotels [54], shipping [55], ambulance allocation at health centers [56], [57], etc. An overview of the classical, contemporary, and supplementary issues that arise while solving RM problems can be found in [58], [59], [60].

Since this study aims to develop an RM system that optimizes transport operator’s revenue through seat inventory control (SIC), the following paragraphs provide an overview of quantity-based methods. Initial works on quantity-based methods have been widely implemented in passenger transportation with an emphasis on resource allocation issues [61], [62]. Later studies demonstrate the application of customers’ choice behaviors as they play a crucial role in estimating the demand distribution [63], [64], [65]. In general, it aimed to forecast future demand and protect inventories that can be sold at higher prices later. Recent research emphasizes the use of stochastic demand when establishing the passengers’ choice model [62], [66].

Rail RM has been an area of interest to TOC after deregulation of the freight market. Nevertheless, the introduction of high-speed rail (HSR) and a rise in travel demand led TOC to investigate the applicability of RM in passenger services as well. Armstrong and Meissner [67] provide a summary of the various pricing and quantity-based strategies utilized by TOC for freight and passenger services. The SIC maximizes operator’s revenue by allocating seats to various OD pairs, which can be viewed as revenue per available seat mile or revenue per passenger seat mile [68]. A majority of articles examine the SIC problem with a price fixed [61], [70]; however, few studies investigate the effect of variable pricing [72], [73]. The literature also demonstrates the use of deterministic [74], [75] and probabilistic demand models [61], [70] to regulate inventory levels. Further, we observe the application of a network-level strategy, i.e., multi-train SIC model, in which demand for each OD pair is distributed across multiple trains [75], [76].

Several studies also investigate the role of SIC in demand satisfaction, with authors suggesting that reserving seats for long-distance OD pairs increases overall satisfaction [69]. For effective inventory management, Gopalakrishnan and Rangaraj [74] and Bao et al. [77] evaluate the significance of customer attributes and booking control, respectively. Further, Luo et al. [78] employ an interactive booking control system to combat demand variability. Multiple studies exploit the benefit of combining pricing and quantity-based strategies to maximize total revenue [52], [65], [72], [73]. Moreover, we observe the application of elastic capacity (flexible train composition) in revenue maximization [79]. Hu et al. [53] develop seat allocation models with dynamic adjustment based on short-term demand forecasting. Wang et al. [71] suggest a multi-stage SIC model that cogitates multiple fare classes and capacity adjustments to enhance revenue.

Other essential elements that play a crucial role in passenger seat assignment include ticket booking mechanism, joint seat regulation, customer preferences, etc. Talluri et al. [48], [80] discuss two basic control orders that regulate seat inventory: booking limits and protection levels. The former determines maximum capacity, while the latter guarantees a minimum level of service. The type of control additionally specifies the time of seat assignment, i.e., at the time of ticket purchase [66], [79], real-time or a few hours prior to departure [81]. Joint seat regulation is an operational requirement requiring passengers to occupy the same seat throughout their journey.

Group seat reservation policy

The group seat reservation policy requires members of the same family or group to be seated nearby. It has a wide range of applications, including hotel room assignments [82], amphitheater [83], public transport [84], educational institutions [85], sports arenas [86], warehouses [87], and large-scale events [88], among others. It has substantial effects on passenger satisfaction and revenue; as Yuen [89] illustrates, passenger groups increase revenue by filling seats that would otherwise be empty.

Traditional works on group reservations for passenger rail aim to maximize capacity utilization or reduce total capacity needed [84], [90]. These problems are typically modeled as knapsack or bin-packing problems. Yoon et al. [91] perform seat assignment for different types of groups categorized according to their travel patterns.

The recent pandemic has offered new insights on group reservations as they exhibit improvement in revenue without an increase in the risk of virus transmission. Fischetti et al. [83] assigns seats to groups of customers entering restaurants or amphitheaters, with the goal of minimizing transmission risk via interaction minimization. Dundar and Karakose [85] propose a two-stage algorithm for classroom seat assignment during pandemic. The initial phase aims to maximize total allocations, while the second phase maximizes the minimum interpersonal distance between students. Salari et al. [45] perform group seat assignment in airplanes under pandemic scenario and argue that an increase in passenger groups yield in a greater social distance than single passengers. In a similar vein, Gioda [40] and Moore et al. [92] suggest that relaxing social distancing norms within a family may increase occupancy without an upsurge in virus transmission risk. For the sake of convenience, we list some of the intricate characteristics of the related studies and our study in Table 1 .

Table 1

A comparison with related literature.

Ref.	Application	Travel	Objectives	Interpersonal
	area	Segment		distancing
[43]	Ar	S	SD	C
[93]	Rly	M	SD	C
[85]	Oth	NA	SD	C
[83]	Oth	NA	SD	C
[41]	Rly, Bus	S	SD	C
[40]	Rly	S	SD-GSA	C
[92]	Ar	S	SD	C
[44]	Ar	S	SD	C
[45]	Ar	S	SD-GSA	C
[46]	Ar	S	SD	C
[86]	Oth	NA	SD	C
[47]	Ar	S	SD	C
[13]	Rly	M	SD-SIC	F
This paper	Rly	M	SD-GSA-SIC	F

Application area: airlines (Ar), railways (Rly), others (Oth); Travel segment: single (S), multiple (M), not applicable (NA); Objectives: social distancing (SD), group seat assignment (GSA), seat inventory control (SIC); Interpersonal distancing: constant (C), flexible (F)

Based on the literature discussed, we affirm that the present study is essentially different from the prevailing literature in the following aspects:

To the best of our knowledge, this is the first effort to control the intercity spread of the virus and reduce the risk of virus transmission among passengers. Most studies have a singular focus, namely minimizing the risk of virus transmission or virus diffusion.

The paper presents a novel method for defining social distancing among passengers, wherein the distancing varies with infection intensity, passenger attributes (vaccination status), and interaction hours. Existing literature maintains constant interpersonal distance.

Propose multiple families of valid inequalities and preprocessing steps to strengthen the model, thereby improving computational efficiency.

The subsequent section describes some of the fundamental assumptions and settings that are used to develop the problem and mathematical model.

Problem description

The present study considers various stances of social distancing rules, group reservation policy, and SIC strategies to minimize the virus spread and maximize the operator’s revenue. Several vital assumptions made in the study while developing each viewpoint are also detailed in this section.

Distancing plan for passenger groups

We develop a novel seat assignment policy (SAP) that primarily aims to reduce the virus diffusion across cities by restricting contact between passengers from high- and low-infection areas. At the second level, the SAP aims to reduce the risk of virus transmission between passengers within coaches by minimizing static and dynamic interactions. The static interaction is reduced by a two-dimensional interpersonal distance between passengers, while the dynamic interaction, which refers to passenger interaction during boarding and alighting a train, is controlled by a seat reward system. The reward system assigns a pseudo-profit to each seat and prioritizes long-duration OD pairs passengers in the center of the coach over shorter ones. This reduces passenger dynamic interaction by restricting their mobility during boarding and alighting. In addition, the reward system ascribes preference to window seats over middle and aisle seats of the same seat column to minimize aisle interaction.

Based on the objectives and severity of the interaction, we define four levels of risk and create an appropriate seating plan for each. The risk levels outline the susceptibility of virus spread based on infection intensity at various cities/stations, vaccination status of passengers, and interaction hours. Interaction hours between passenger groups are proportional to the length of sections that coincide, where a section is the region between two consecutive stations. Some of the key assumptions used to develop the risk levels and mathematical model are as follows:

a group is considered to be vaccinated if all of its passengers have been immunized;

those boarding from the same station are likely to have similar viral load;

each coach is considered to be a bio-bubble because its air conditioning system is independent;

we only consider vaccination status when evaluating passenger attributes; other factors such as health conditions and usage of protective gears cannot be monitored.

Table 2 outlines the various risk levels and their corresponding seating plan. At the highest level, passenger groups with large variation in infection intensity of their boarding stations () are assigned seats in separate coaches to prevent the diffusion of virus. This plan is consistent with the Sharun et al. [94] recommendation, which sought to establish quarantine-free travel between countries with similar SARS-CoV-2 incidence. The next two levels provide interpersonal distancing within coaches to minimize static interaction, and consequently reduce the risk of virus transmission. The actual distance between passenger groups depend on the variation in infection intensity, interaction hours and passenger attributes, detailed in Section 7. The transmission risk is further minimized by reducing passenger dynamic interaction at the lowest level. The thresholds for variation in infection intensity specified in Table 2 are indicative. A TOC may redefine these values with the aid of virologists based on the present circumstances. Fig. 1 provides a visual representation of various risk levels and their corresponding seating arrangements.

Table 2

Risk levels and corresponding seating plan.

Risk Level	Description	Seating Plan
(1) high	high variation in infection intensity of passengers’ boarding stations, i.e., ΔI>75	coach separation between passengers; otherwise exposed to the same air-conditioning
(2) medium-high	moderate variation in infection intensity of passengers’ boarding stations, i.e., ϵ<ΔI⩽75	two-dimensional interpersonal distancing between passenger groups within coach
(3) medium-low	passenger groups boarding from stations having similar infection intensity, i.e., ΔI≈ϵ	one adjacent seat gap between two groups
(4) low	dynamic interaction minimization between passenger groups	prioritize long-duration OD pairs passengers at the center of the coaches over shorter ones; further, offer window seats precedence over aisle and middle seats

Fig. 1

Pictorial representation of seating plan for risk levels - (a) high, (b) medium-high, (c) medium-low, (d) low.

The SAP adopts a group reservation policy similar to the Moore et al. [92] model, in which the requirement for interpersonal distance between passengers in the same group is relaxed. Fig. 2 depicts the effect of interpersonal distance on capacity utilization. From Fig. 2(b) and (c), it is evident that relaxing interpersonal distance within the same group improves capacity utilisation and, consequently, increases TOC’s revenue. Further, to implement block seat assignment, we consider two blocks within a coach and employ a spiral seat mapping or numbering, as shown in Fig. 2.

Fig. 2

(a) request details. Capacity utilization with – (b) naive distancing, and (c) distancing under group reservation.

Estimation of risk score

The risk of virus diffusion between cities and transmission among coach occupants is contingent on the infection rate of the cities, passenger characteristics, interpersonal distance, and interaction hours. The risk scores computed in this paper are based on the final seat assignment within each coach. The potential for virus diffusion between cities is proportional to homogeneity of assignment within coaches. Further, the homogeneity is determined by the standard deviation and range of infection intensity of passengers’ boarding station in a coach. Consequently, a smaller value of standard deviation and range reflects a higher degree of homogenization inside the coach, indicating a lower potential for virus diffusion.

The risk of virus transmission from one group of passengers to another depends on their seating position and interaction time. We employ the Hertzberg et al. [95] model to ascertain the base risk between two passenger groups seated in a coach. The base risk represents the probability that a group will contract an infection from other groups seated within the same coach. To determine the transmission risk between two passenger groups we utilize a time factor (representing their interaction hours). It is assumed that every passenger group seated in a coach is capable of transmitting virus to every other passenger group. Thus, the sum of potential risks between each pair of groups represents the cumulative risk score, which is then transformed into a normalized score representing the average risk for any group seated inside the coach.

Seat inventory control

To maximize TOC’s revenue we employ a “seat-based control system” similar to the model proposed by Yuan and Nie [66], in which each seat is sold to multiple combinations of OD pairs (products). It works on two basic elements—bucket and pool. A bucket employs a partitioned booking limit control (PBLC) in which a portion of the available inventory is divided among various OD pairs at the beginning of the booking period. The inventory is segmented according to the estimated demand for each OD pair, capacity utilization rate, and minimum service level established by TOC. The bucket has a higher priority than the pool for ticket sales. The remainder of the initial inventory and any empty bucket seats are assigned to the ticket pool. The ticket pool employs a nested strategy in which capacity available for shorter OD pairs subsumes capacity reserved for longer OD pairs as shorter ODs are more profitable. Consequently, the booking mechanism is a hybrid of PBLC and nested approaches.

Next, we discuss the ticket booking mechanism that regulates inventory and issues tickets. The booking mechanism offers two types of tickets as discussed in Section 1. If the available inventory for the requested OD pair exceeds demand, a request is granted with confirmed ticket; otherwise, a waitlisted ticket is issued. The waitlisted requests may be confirmed or denied based on the availability of seats in the ticket pool during the final seat assignment. Fig. 3 depicts a flowchart describing how a reservation request is handled by the booking mechanism.

Fig. 3

A schematic representation of ticket booking mechanism.

Mathematical model

In this section, we present an integrated MILP model that minimizes the spread of virus (diffusion between cities and risk of transmission among passengers) while maximizing the operator’s revenue. Since interaction risk is determined by interaction type and corresponding interpersonal distance, the model employs parameters and decision variables that determine the exact location of passenger seats and statutory interspaces between them. Additionally, different sets are defined to address multi-level safety standards and inventory availability. Tables 3 and 4 lists various parameters and decision variables employed in model formulation. Except for the notations , and all other sets, parameters and decision variables are enumerated for each in Table 3 and 4.

Table 3

Sets and parameters used for model formulation.

Notation	Description
S	set of stations a train stops along its route, indexed by s
Q	set of classes in a train, indexed by q
R^q	set of seat rows in a coach of class q, indexed by r, r^′
C^q	set of seat columns in a coach of class q, indexed by c, c^′
G^q	set of seats in a coach of class q, |G^q|=|R^q|×|C^q|
N^q	set of passenger groups (requests) with seat demand in class q, indexed by i, j
siq	boarding station for request i∈N^q
ωiq	number of seats sought by request i∈N^q (also referred to as the request size)
hiq	travel length for request i∈N^q (defined in terms of number of stations)
piq	total ticket fare for request i∈N^q (sum of ticket fare for all passengers in a group)
tiq	time factor for request i∈N^q (based on travel hours)
Nvq	set of vaccinated requests in class q, Nvq⊆N^q
Ncq	set of requests with confirmed ticket status in class q, Ncq⊆N^q
Nsq	set of class q requests utilizing a seat at station s, i.e., Nsq={i∈N^q|siq⩽s<siq+hiq}
K^q	set of coaches of class q, indexed by k
B^q	set of request pairs (i,j) sharing station(s) along their journey, i.e., (siq⩽sjq<siq+hiq)∨(sjq⩽siq<sjq+hiq), i,j∈N^q
Bsq	set of request pairs (i,j) sharing station s along their journey, i,j∈Nsq, i≠j,∀s∈S
D^q	set of request pairs (i,j) that cannot occupy seats in the same coach due to social distancing requirements, (i,j)∈B^q, D^q⊆B^q
T^q	set of request pairs (i,j)∈B^q that can occupy seats in the same coach, i.e., T^q=B^q∖D^q
ρijq	minimum number of rows gap between requests i and j of class q, (i,j)∈T^q
δijq	minimum number of columns gap between requests i and j of class q, (i,j)∈T^q
rijq	pseudo-profit for an additional row gap between requests i and j of class q, (i,j)∈T^q
cijq	pseudo-profit for an additional column gap between requests i and j of class q, (i,j)∈T^q
m^q	minimum number of coaches of class q attached to a train
b	maximum length of a train (defined in terms of number of coaches attached)
d^q	cost of using a coach of class q
χ^qrc	seat number corresponding to row r and column c of a coach of class q
f^qrc	pseudo-profit for assigning a seat in row r and column c on a coach of class q

Table 4

Decision variables used for model formulation.

Notation	Description
ziqk	= 1 if request i∈N^q is assigned to seats in coach k∈K^q; = 0 otherwise
uiqrc	= 1 if request i∈N^q is assigned to seat corresponding to rth row and cth column on a coach of class q; = 0 otherwise
y^qk	= 1 if coach k∈K^q is utilized for seat assignment; = 0 otherwise
xiq	starting seat number for request i of class q
lijq	= 1 if requests i and j are assigned to seats in the same coach with xiq<xjq; = 0 otherwise
αiqn	lowest (n=1) and highest (n=2) row numbers of seats assigned to request i of class q
βiqn	lowest (n=1) and highest (n=2) column numbers of seats assigned to request i of class q
aijq	= 1 if row gap requirement between requests i and j of class q cannot be applied in a coach; 0 otherwise
vijq,wijq	binary variables to administer row and column gaps, respectively, between requests i and j of class q
νijq,μijq	additional row and column gaps, respectively, between requests i and j of class q

Objective function

The objective function (1) consists of six components - the first two exhibiting total revenue, the next two demonstrating pseudo-profits as part of minimizing virus transmission risk, and the last two assisting () and () in being integer-valued (discussed in greater detail later). The first component yields revenue generation through ticket sales, while the second represents the cost of using a coach. The third component aims to reduce dynamic interaction between passengers, represented as a low-risk level in Table 2. The pseudo-profit () and time factor () prioritize seating of long-length OD pair passengers in the middle of coaches over the short-length OD pairs. Additionally, it prioritizes window seats over aisle and middle seats because they are more conducive to passenger interaction. The formula (2) defines the value of , where is a class-dependent constant, , and . The next component in the objective function reduces static interaction by augmenting the actual interpersonal distance above the minimum suggested. Thus, the third and fourth components strive to reduce the transmission risk.

Operational and service constraints

The operational constraints (3) affirm that all requests with confirmed ticket status must be assigned seats in one of the coaches. Constraints (4) indicate that waitlisted requests may be assigned seats in one of the coaches if accepted. Constraints (5) limit the maximum number of coaches attached to a train. The service constraints (6) establish a minimum service level guarantee for each class specified by TOC.

Capacity constraints

The capacity constraints (7) ensure that the total number of passengers assigned to a coach at each station does not exceed coach capacity. Constraints (8) and (9) specify the upper and lower limit on the seat number for each request, respectively.

Seat positioning correlation constraints

Constraints (10)–(12) determine relative position of two requests within a coach based on their size. These constraints guarantee that if request pair () are assigned to seats in the same coach, the seat number for one of them must be higher than the other, i.e., if . Further, constraints (10) and (12) force that starting seat numbers for requests and should manifest the values of and variables, i.e., if seats for request j will start after seats for request i, and similarly for .

Group reservation policy constraints

Constraints (13) prohibit partial acceptance of requests, i.e., a request must be assigned with seats if accepted. The constraints (14) and (15) mandate the adjacency of seats assigned to request . Therefore, constraints (14), (15) and the spiral structure in seat numbering (Fig. 2) ensure a group reservation policy with block assignment.

Group seats positioning constraints

Constraints (16)–(21) assist in specifying the lowest and highest row and column numbers corresponding to seats assigned to a request. For a given request, the first two constraints (16) and (17) establish the minimum row number, while the next two constraints (18) and (19) determine the minimum column number. Similarly, constraints (20) and (21) assist in defining the highest row and column numbers for the requests, respectively. The above constraints (16)–(21) and the objective function (1) ensure that if a request is rejected, its highest and lowest row and column numbers are set to zero.

Coach incompatibility constraints

According to constraints (22), coach incompatible requests cannot be assigned to seats within the same coach. Consequently, these constraints enforce the separation requirement for high-risk level to minimize the risk of virus diffusion, discussed in Table 2.

Static interaction risk constraints

Constraints (23)–(26) are added to reduce the static interaction between passenger groups by creating a two-dimensional interpersonal distancing. The distancing plan considers an aisle to be equivalent to one row gap. The distance between two groups is measured using the Manhattan distance (number of rows or columns separating them). These constraints ensure that if request pair is assigned to seats in the same coach (), they must be separated by at least the specified number of rows () or columns (). The row and column distancing constraints are implemented in an either-or setup. These constraints implement the distancing strategy for medium-high risk level based on the desired distance between the request pairs. In addition, constraints (23)–(26) and the fourth component of the objective function (1) help to further increase the distance between the requests. Constraints (23)–(26) along with constraints (11)–(12) enforce distancing for medium-low risk level, i.e., only one adjacent seat gap, by keeping and .

Indicator forcing constraints

Constraints (27)–(30) define the upper limit for the additional rows and columns gap between every request pair in a coach. Constraints (27) and (28) affirm that additional rows and columns gap will occur between request pair if they are seated in same coach, respectively. Constraints (29) permit an additional rows gap between request pair only if the minimum prescribed row gap has occurred. Likewise, constraints (30) apply to the additional columns gap.

Non-negativity and integrality conditions

Lastly, restrictions (31)–(37) impose non-negativity and integrality conditions for each variable used in the MILP model, i.e., they define the variable domain. Constraints (16)–(21), (23)–(26) along with the fifth and sixth components of the objective function (1) ensure and variables to be integer valued automatically since any fractional solutions to and will decrease the overall gap between request pairs and the objective function value. Moreover, constraints (23)–(26) and the third component of the objective function (1) ensure and variables to be integer valued.

Extension to group seat reservation policy

The above MILP model (1), (3)–(37) performs block assignment to passenger groups via a spiral seat numbering, as shown in Fig. 2. Nevertheless, this may allow one accepted group per coach to occupy seats in multiple blocks, thereby reducing demand rejection. To strictly enforce block assignment, such that no group is assigned to seats in different blocks, the aforementioned model can be modified easily. We introduce a binary variable for each request and add constraints (38) and (39) to the model. The value for big- can be fixed at .

Model reduction and strengthening

The MILP model (1), (3)–(37) can efficiently generate optimal solution for small problem instances. However, to solve relatively large instances of real-world problems, we need valid inequalities and preprocessing to strengthen and reduce the model, respectively. This section proposes several families of valid inequalities applicable to the MILP model and fixes a number of variables via preprocessing.

Valid inequalities

Symmetry breaking (SB) inequalities

In the proposed model, if two ”identical” requests are assigned to seats in the same coach, their assignments can be swapped without a reduction in objective value. This motivates us to propose the following symmetry breaking inequalities that restrict alternate optimal solutions, thereby enhancing the computational performance. Two requests are said to be identical if they belong to the same OD pair, having similar vaccination status and size. The size of requests plays a crucial role while defining these inequalities due to interpersonal distancing requirements, as shown in Fig. 4 . Consider, for instance, eight requests from five distinct OD pairs that overlap at a section with request sizes ranging from one to six. Fig. 4(a) displays the minimum interpersonal distance requirements between passengers of the respective OD pairs, where each color represents a unique OD pair. Fig. 4(b) illustrates an optimal assignment of seats to the passengers, whereas Fig. 4(c) a reassignment by swapping seats of requests 2 and 4 having sizes six and two, respectively. It is well evident from Fig. 4(c) that initial assignments for request 3 becomes infeasible after swapping requests 2 and 4. Further, request 3 may be rejected if no feasible assignment exists or may be redirected to a location with a higher risk. This issue will not occur if both the requests have same size. Thus, we propose the following valid inequality in which the seat numbers for request are less than those of request if both requests are identical and .

Fig. 4

Implementation of symmetry breaking constraints (a) interpersonal distancing requirements (b) optimal seat assignment; (c) request rejection due to incorrect implementation.

For request pairsuch that,and, the SB inequalitiesare valid for GSAPSD.

Capacity tightening (CT) inequalities

The GSAPSD establishes two-dimensional interpersonal distance between passenger groups by specifying the number of rows () and columns () gap within a coach. However, the absence of these gaps in constraints (7) provide a weaker bound to capacity limitations. We introduce two parameters, and , that compensate for interpersonal distancing requirements and provide a tighter bound for coach capacity. Let and define the minimum number of vacant seats between request pair based on the gap in rows and columns, respectively. The parameter is defined for request at each station along its path, excluding the last, as: . Further, parameter is determined for each class and each station as: . Thus, represents the minimum seat loss at station due to request , and represents the minimum seat loss when all overlapping requests assigned to a coach at station are considered. Based on the preceding definitions, we can conclude that holds true for all .

The following proposition identifies another family of valid inequalities that provide an improved upper bound to coach capacity by accommodating interpersonal distance requirements.

Givenandas defined above, the following CT inequalitiesare valid for GSAPSD.

The proof is simple. If a request is accepted, it will consume at least seats to maintain the minimum gap with all other overlapping requests at station ; thus, its effective size is . If requests are accepted in a coach, then gaps are sufficient for allocations. However, in the above inequality, the minimum seat loss is subtracted times, once for every accepted request. Thus, the quantity is added to coach capacity to compensate for the additional loss on the left hand side of inequality (41). □

Cover (CV) and extended cover (EC) inequalities

The GSAPSD is analogous to the multiple knapsack problem for two reasons. First, the train coaches, like knapsacks, are scarce resources for which requests compete, and second, the movement of coaches from one station to another changes the knapsack problem as a result of varying packing requests. Ferreira et al. [96] discussed cover inequalities for the weighted multiple knapsack problem, which can be adapted for the GSAPSD in the following manner. Let represent a set of 3-tuples that define a knapsack. A set is a cover with respect to knapsack if and the cover is minimal if for all .

For a knapsackand coveras defined above, the following CV inequalitiesare valid for GSAPSD.

If defines a cover for the knapsack , then we can perform an extension to cover , defined as . This results to the following proposition.

For a knapsackand extended coveras defined above, the following EC inequalitiesare valid for GSAPSD.

Tightened coach incompatibility (TCI) inequalities

The coach incompatibility constraints (22) present an opportunity to improve the formulation (1)–(37). Consider three mutually coach incompatible requests , and , i.e., . Then, constraints (22) for these request pairs can be written as follows.Adding the above three constraints and dividing the ensuing inequality by 2 results inSince all the variables and coefficients in the left-hand side are integer, the above inequality can be strengthened by rounding the right-hand side down. Therefore, whenever an odd number (say , ) of mutually coach incompatible requests exists, constraints (22) can be strengthened by the following inequality.This is known as the Chvátal–Gomory procedure for constraint strengthening [97].

Additional (AD) inequalities

In addition to the aforementioned inequalities, we include several families of valid inequalities based on seat assignment to request pairs in a coach and the applicability of row or column distance between them.

For request pairthat may take seats in the same coach, the following inequalitiesare valid for GSAPSD.

Any valid inequality must be satisfied by all feasible seat assignments. Consider a solution to GSAPSD in which request pair are assigned to seats in separate coaches; then we have from constraints (10). In such a case, constraints (23)–(26) are redundant; thus forcing in constraints (45) does not eliminate any feasible seat assignment. Further, if both the requests are assigned to seats in same coach, and the variable have either a value 0 or 1 based on row or column gap, respectively. This establishes the validity of inequalities (45). □

For request pairthat may take seats in the same coach, the following inequalitiesare valid for GSAPSD.

As stated previously, a valid inequality must be satisfied by all feasible seat assignments. Consider a solution to GSAPSD, where request pair are assigned to seats in separate coaches, we have . Then, restricting the values for and to 0 does not eliminate any feasible assignment as constraints (23)–(26) are redundant. Further, if request pair are assigned to seats in the same coach (), both the variables and are free, and can take values based on the row or column gap. □

For request pairthat may take seats in the same coach, the following inequalitiesare valid for GSAPSD.

The proof of validity is similar to the proof for Propositions 6 and 7. If the request pair are assigned to seats in different coaches, whereupon and due to constraints (10) and (45), respectively. In such a case, no feasible seat assignment is eliminated, regardless of the values of and , as constraints (23)–(26) are redundant. Further, if the request pair take seats in the same coach, we have , and is free. Then, based on the actual assignments for request pair within a coach, and choose values according to the row or column gap between them. This confirms the validity of inequalities (48) and (49). □

Preprocessing

In this section we perform a number of preprocessing steps to fix some of the variables, thereby reducing the size of the model and strengthening the LP relaxation. Constraints (23)–(26) perform a two-dimensional interpersonal distancing between requests; nevertheless, the absence of request size () in these constraints provides a weak bound to the LP relaxation. Therefore, we introduce several parameters that fixes variables based on the requests size and coach configuration (Fig. 5 ).

Fig. 5

Coach configuration and seating pattern for (a) 1st class, (b) 2nd class.

Let be the set of different seating combinations for request pairs . Then, for each seating combination we have = and . Next, we define parameters and representing the maximum possible number of rows and columns gap that can be attained between request pair (), respectively. Further, and define the maximum and minimum sizes for requests and .

Consider, for instance, two requests each of size two are assigned to seats in a coach of 1st class (), as shown in Fig. 6 . Based on preceding definitions and seat assignments as shown in Fig. 6(a)–(c), we have , , and . Thus, the parameter and . Similarly, we can establish the values for and based on different possible seating combinations.

Fig. 6

Different seating arrangements for request pair () in a coach.

Property 1. In any optimal solution to the GSAPSD, variables can be fixed to 1 for all request pairs , if and .

Assume request pair having , we have . To apply row gap, the request pair must be in the same coach, this implies , see constraints (7). Also, Figs. 5 and 6 demonstrate that row gap between request pair () occur iff . Based on various seating combinations in different classes with , we can estblish that and . Since the upper bound on is , when with and , constraints (23) and (24) will be violated. This infers that must be 1 under the given condition. □

Property 2. In any optimal solution to the GSAPSD, variables can be fixed to 1 for all request pairs , if and .

Assume request pair having , we have . As discussed previously, to apply row gap between request pair () we have and . Further, depending on coach configuration (Fig. 5), request size and seating combinations (Fig. 6), two conditions exist:

Condition 1: For and , we have and .

Condition 2: For and , we have and .

Consider Condition 1, when with and , constraints (23) and (24) will be violated as cannot exceed for a feasible row gap. This ensures that must equal 1 under the given condition. Similarly, we can establish that holds true for Condition 2 as well. □

Property 3. In any optimal solution to the GSAPSD, variables can be fixed to 1 for all request pairs , if , and .

The proof is analogous to Property 2 and can be easily established. □

Property 4. In any optimal solution to the GSAPSD, variables can be fixed to 0 for all request pairs , if and .

Assume request pair having then, we have . To apply column gap between request pair (), they must be in the same coach, i.e., , see constraints (7). Also, Figs. 5 and 6 demonstrate that column gap between request pair () occur iff . Based on various seating combinations in different classes with , we can establish and . Since the upper bound on is , when with and , constraints (25) and (26) will be violated. This ensures that must be 0 under the given conditions. □

Solution approach

This section provides information regarding the algorithm used to solve the GSAPSD. The problem is solved in two phases: (i) the cutting plane phase, and (ii) branch-and-bound (B&B). We initialize the cutting plane process by solving LP relaxation of the problem defined by (1), (3)–(21) and (23)–(37). The cutting plane phase is applied only at the root node of the B&B tree, technically defined as cut-and-branch process. Let () be the solution to the current LP relaxation of GSAPSD. Then the cutting plane phase is executed as follows:

Examine for violated CV inequalities (to be discussed Section 6.1) given the current fractional solution (). If violated cover is found, extend the cover and add EC inequalities to the LP. If extended cover cannot be found, add the violated CV inequality. Solve the LP relaxation and search for more CV inequalities that are violated. If no violated CV inequality is found, go to Step 2.

Perform an exhaustive enumeration to generate SB, CT, TCI, and AD inequalities, and then append to the LP. Section 6.2 presents the detail on the generation of these inequalities.

Add constraints (22) to the LP.

Following the cutting plane phase, the problem is submitted to the B&B algorithm to obtain the integral solution. Note that constraints (22) and inequalities SB, CT and TCI are added to the LP after addition of CV and EC inequalities because they impede the separation of CV inequalities. The subsequent sections highlight details on the generation of valid inequalities as proposed in Section 5.1.

Separation of minimal CV inequalities

The separation procedure for inequalities (42) using a single knapsack is thoroughly discussed in literature [98]. The multiple knapsack problem inherits all nontrivial facet-defining inequalities associated with the single knapsack polytope. We utilize a generalized cover separation problem (GCSP) for a multidimensional knapsack problem (mKP) proposed by Bektas and Oğuz [99]. We solve the separation problem for each class and each station considering all the coaches as the mKP.

Let be the -component of the current fractional solution to the LP-relaxation of GSAPSD. The separation problem aims to identify CV inequalities that violate . The GCSP can be formulated as:

If the optimal objective value , the solution induces a violated CV inequality (42) with the cover . The above problem is a binary integer programming problem, and its solvability can be improved through variable fixing technique. Note that requests with only play active role in defining cover inequalities. Therefore, we set variables to 1 if , and 0 if .

Generation of SB, CT, TCI and additional inequalities

Due to the fact that the total number of SB, CT, TCI and AD inequalities are polynomial with , , , and respectively, a complete enumeration have been performed to generate them. This also saves the time spent in solving the separation problem. The complete set of these inequalities are added to the root node of the B&B tree, inequalities that are not violated currently may be useful during B&B process.

To generate TCI inequalities we utilize an undirected graph where each vertex represents a request and an edge is present if vertices and correspond to two coach incompatible requests and , i.e., . Each odd-length cycle on this graph generates a TCI inequality (44) where denotes the length of the cycle. Fig. 7 shows an example of graph where vertices represent requests and black edges define incompatibility between two requests. Here, each odd length cycle is represented by a different color.

Fig. 7

Incompatibility graph to generate TCI inequalities.

Note that if is bipartite, TCI inequalities cannot be obtained as such a graph does not contain a cycle of odd length. Given a non-bipartite graph , we employ a depth-first search to identify all the cycles of odd-length.

Computational study

This section aims to demonstrate the validity of the proposed model through computational experiments on real-world examples. We also exhibit the computational performance of the cut-and-branch procedure to solve GSAPSD using different problem instances.

Data description

We conduct computational tests on the New Delhi (NDLS) - Howrah (HWH) rail corridor of Indian Railways (IR). The NDLS - HWH rail corridor stretches approximately 1454 km and is one of IR’s networks with the highest traffic density. The rail corridor accommodates numerous train movements with distinct stoppages. The computational tests were conducted using data from a single train, but they are easily adaptable to any train with different stoppage schedules and coach configurations. Fig. 8 depicts the NDLS - HWH rail corridor as well as the infection intensity in cities along the line. Further, Table 5 provides infection intensity based on the number of cases reported in various cities between 21 April and 02 May 2021 (accessed from covid19.theavtar.com on 10 April 2022).

Fig. 8

Infection intensity at various cities along NDLS - HWH rail corridor.

Table 5

Infection statistics at stopping stations of train under study.

Stations	7-day SMA
NDLS (1)	117.3
CNB (2)	37.7
MGS (3)	25.4
GAYA (4)	26.4
DHN (5)	19.0
ASN (6)	7.8
DGR (7)	7.0
HWH (8)	27.3

The details concerning train movement utilized in the study include passenger demand, ticket pricing, stoppage plan, train length, etc. Passenger demand data was obtained from the Centre for Railway Information System (CRIS) of IR, and ticket prices for various OD pairs are based on the minimum baseline fare declared in [100]. The ticket pricing policy utilizes a telescopic (non-linear) fare structure in which the ticket price per kilometer for short OD pairs is greater than that of long OD pairs. Regarding service type and stoppage plan, the train encompasses two seating classes and eight stops along its route. The maximum length of the train is eight coaches. Table 6 draws attention to additional characteristics pertaining to coach configuration and service levels utilized in the experiments. Further, the cost of using a coach () is equivalent to the revenue generated at 30% occupancy.

Table 6

Coach configuration settings for computational tests.

Seating	Seats per	Rows per	Columns per	Limit on coaches
classes (Q)	coach (|G^q|)	coach (|R^q|)	coach (|C^q|)	[m^q,|K^q|]
1st	24	4	6	[1, 6]
2nd	48	6	8	[2, 8]

The parameter settings for medium-high risk level is detailed in Table 7 . These values represent the interpersonal distance between request pairs where both groups are considered to be vaccinated. Moreover, any change in the vaccination status of passenger groups causes a proportional change in the interpersonal distance between request pairs. Consequently, the row and column gap considered in the experiments have the property: and , where subscripts and represent the vaccinated and non-vaccinated passenger groups, respectively. This distancing strategy is consistent with Eyre et al. [27] and Subbaraman [101] studies indicating that vaccinated passengers pose a lower risk of virus transmission than non-vaccinated passengers. Since vaccination data for passenger groups are unavailable, we utilize a Bernoulli distribution to generate the vaccine status of passenger groups for our experiments. However, a TOC may collect this information from passengers during ticket purchase or prior to the final seat assignment. Next, the pseudo-profits per unit additional row and column gaps (, ) that depend on the interaction hours between passenger groups are defined in Table 8 . Finally, the constant term in the pseudo-profits associated with seats () (see formula (2)) is fixed to 1 for both the classes.

Table 7

Distancing plan for medium-high risk level.

ΔI	Interaction	Row gap		Column gap
	hours	ρ^1	ρ^2	δ^1	δ^2
[25,75]	ΔT>4	3	4	3	4
	2⩽ΔT<4	3	4	2	3
	ΔT<2	2	3	2	3
<25	ΔT⩾4	2	3	1	2
	2⩽ΔT<4	1	2	1	2
	ΔT<2	1	1	1	1

Table 8

Pseudo-profit per additional row and column gap.

Interaction hours	r	c
ΔT>10	36	60
7.5<ΔT⩽10	24	48
5.0<ΔT⩽7.5	12	24
2.5<ΔT⩽5.0	6	12
ΔT⩽2.5	3	6

Results and discussion

In this section we analyze the results obtained from the model on three fronts - (i) virus diffusion and transmission risk, (ii) revenue, and (iii) the effectiveness of the cut-and-branch method in solving real-life GSAPSD. We conduct a comparative analysis of the risk of virus diffusion and transmission between the proposed SAP and the TOC seating plan. Presently, to reduce passenger interaction, TOC leave the middle seats empty on each block of a coach. To conduct a comprehensive analysis, we generate additional test instances by varying the demand from 0.5 to 1.2 times the base demand by increments of 0.10-unit. These instances are labeled d0.5, d0.6, d0.7, d0.8, d0.9, d1.1 and d1.2, with d1.0 serving as the base case. The algorithm has been implemented using Visual C++, and the callable library of CPLEX 12.10.0 is used as the MILP solver. Experiments are performed on a desktop computer with a 3.00 GHz Intel Core i7 processor and 32 GB of RAM running Windows 10 Pro.

First, we investigate the quantum of virus diffusion in each coach under base demand. As discussed in Section 3.2, the the risk of virus diffusion depends on the standard deviation and range of infection intensity of passengers’ boarding stations within coaches. We compute both metrics in each coach and at each section of the train journey, where the likelihood of virus diffusion decreases as their value decreases. Fig. 9 compares the risk of diffusion for both seating policies. It is readily apparent that the GSAPSD significantly reduces the variation in infection intensity across each section and coach, thereby reducing the risk of virus diffusion between cities. The results exhibit zero variation (standard deviation and range) in certain coaches (1st-C1, 2nd-C2 and 2nd-C3) due to end-to-end demand occupancy. Thus, the proposed SAP provides greater control over the spread of viruses between cities. In addition, the sectional analysis may assist TOC in identifying distress zones along the route and instituting preventive measures.

Fig. 9

(a) standard deviation and (b) range of infection intensity for the GSAPSD; (c) standard deviation and (d) range of infection intensity based on TOC’s approach.

Next, to validate the robustness of the proposed model with regard to the risk of diffusion, we conduct sensitivity analysis on additional test instances for both seating policies. We present a comparison between the worst (coach with the highest risk at a section) and best (coach with the lowest risk at a section) outcomes at different sections for GSAPSD and TOC, respectively. Fig. 10 illustrates a comparison of standard deviation for 2nd class coaches at different sections along the route. Since Section (1-2) consists solely of demand from the origin city, the variation is null under both methodologies. It is well evident from Fig. 10 that the proposed model offers better control of the diffusion risk on all fronts (varying demand and at all sections). However, we did not observe any significant differences in diffusion risk between the two methodologies at lower demand (0.5 times the base demand). This may be due to low demand variation and consequently low infection intensity variation among coaches.

Fig. 10

Comparison of diffusion risk between TOC’s approach and GSAPSD.

A similar analysis is conducted to validate the effectiveness of the proposed approach in terms of the risk of virus transmission. The transmission risk is based on the static interaction among passengers, as discussed in Section 3.2. Here, we compare the weighted average risk for all coaches at each section. The risk is computed for each coach and weighted by coach occupancy for aggregation. Fig. 11 illustrates the transmission risk for both seating policies in different sections, revealing that the transmission risk under GSAPSD is lower than that under TOC’s approach. This reduction is relatively greater in the initial and final sections than in the middle sections, where the interaction hours between requests are considerably higher. In addition, the proposed SAP reduces transmission risk significantly for higher demand, which may be advantageous when normalcy returns after the pandemic. Results demonstrate the proposed model’s effectiveness in restricting virus spread and outperform the seat assignment policy adopted in practice by TOC. Specifically, our model reveals a greater capacity to control virus diffusion between cities, which is useful for reducing the global spread of virus.

Fig. 11

Comparison of transmission risk between TOC’s approach and GSAPSD.

Further, we compare the GSAPSD and TOC approaches in terms of total revenue generated from ticket sales and seat kilometers gained. The seat kilometer gain is the sum of the product of additional demand accepted times the distance travelled by each demand. Fig. 12 (a) and (b) exhibit a comparison of total revenue and additional seat kilometer gain, respectively, for both the seating policies under varying demand. It is evident from Fig. 12(a) and (b) that the proposed method offers a higher capacity utilization and consequently increases revenue.

Fig. 12

A comparison on revenue and seat kilometer gain for GSAPSD and TOC seating policies.

Finally, we evaluate the computational efficiency of the proposed solution approach (discussed in Section 6) on the previously described test instances. The basic modeling parameters specified in Table 6, Table 7, Table 8 remain constant across all instances. For a comparative evaluation, the instances are solved using three distinct solution strategies outlined in Table 9 . Strategy S1 employs the standard CPLEX branch-and-cut (B&C) algorithm with default settings, whereas strategy S2 uses the proposed valid inequalities and preprocessing steps. Thus, strategy S1 serves as a benchmark for comparing our inequalities to standard CPLEX cuts. Strategy S3 combines CPLEX cuts and our inequalities, where our inequalities are being added only at the root node of the B&B tree, whereas CPLEX cuts being added throughout the B&C process. The objective of strategy S3 is to determine whether our inequalities and CPLEX cuts have any synergy.

Table 9

Different solution strategies.

Name	Cuts Added
	Cplex Cuts	Our Cuts & preprocessing
S1	✓
S2		✓
S3	✓	✓

We consider percentage optimality gap at the conclusion of the B&B algorithm as a performance indicator to evaluate different solution strategies. The time limit to solve each instance is set at 3600 seconds. Table 10 exhibits the performance metric for each strategy on different problem instances. Comparing S2 to S1, it is evident that the proposed valid inequalities have significantly reduced the optimality gap, approximately 57%. Further, strategy S3 performs better in large problem instances, resulting in a 61% reduction in optimality gap compared to S1. Therefore, some synergy is obtained between our inequalities and CPLEX cuts with increase in problem size. The second column represents the revenue obtained from the ticket sales. As the demand increases, the revenue improves; however, the marginal gain diminishes since the capacity remains constant. This makes the problems difficult to solve; also evidenced by the higher optimality gap for instances d1.1 and d1.2 (strategy S3). It is observed that an optimality gap of 5–6% is reasonably acceptable for this problem since the lower bound is already at the optimal solution in majority of the instances.

Table 10

Performance of different solution strategies.

Problem	Revenue		B&B Gap %
Name			S1	S2	S3
d0.5	5,42,785		13.94	0.02	0.95
d0.6	6,05,450		12.27	2.16	3.15
d0.7	7,15,845		16.53	5.51	6.04
d0.8	7,83,280		14.07	6.12	5.43
d0.9	8,55,395		14.47	7.72	6.15
d1.0	8,88,415		15.32	8.52	6.42
d1.1	9,17,650		15.87	10.27	8.39
d1.2	9,25,785		16.54	11.19	9.56
Average			14.87	6.43	5.76

Conclusions and future work

In this paper we investigate the significance of social distancing and revenue maximization for public transport in post-pandemic world. Our objective is to provide TOC with an optimization tool that facilitates an efficient trade-off between the risk of virus spread and capacity utilization. The proposed model offers an intelligent method for determining the seating arrangement of passengers and fostering a safe travel environment. Thus, it may assist TOC in developing strategies that bring confidence to commuters and promote sustainable operations.

Articles published hitherto focus on controlling virus transmission to passengers where the factors that govern virus spread have been neglected in the development of a distancing strategy. The present study first considers a multifaceted approach to virus spread, i.e., diffusion to different cities and transmission among passengers. Further, the seating plan contemplates a variety of factors (infection intensity in different cities, passenger attributes, interaction hours, etc.) and enables a suitable and customizable distancing plan among passengers. We expect that the inclusion of surrounding characteristics, such as ventilation and air circulation parameters, will bring exciting and challenging research while establishing safety standards. Since SARS-CoV-2 is an airborne disease, the introduction of these parameters will generate new insights into the subject matter. Moreover, dynamic control of the distancing plan based on the occupancy level of coaches can be incorporated into the optimization model.

A basic limitation of the model is that virus outbreaks depend solely on the variation in infection intensity between cities. Further, the effectiveness of the seating plan against the risk of diffusion is determined by the standard deviation and range of infection intensity. If we have mathematical functions that characterize SARS-CoV-2 diffusion based on seat location, we can circumvent this setting. In addition, the transmission risk is measured based on the passengers’ seat location, ignoring mobility within coaches and other potential contact points. Therefore, we emphasize that virologists and epidemiologists may propose improved metrics for quantifying the risk of virus spread and developing a more effective seating plan.

In the context of the modeling approach, the proposed MILP model for the GSAPSD draws parallels to the multidimensional knapsack problem, leading to the derivation of a number of valid inequalities to strengthen the formulation. The computational results utilizing real-world problem instances advocate the same. It may be worthwhile to investigate the possibility of developing heuristics based on these valid inequalities to solve large-size instances in a reasonable amount of CPU time.

Author Statement

The authors would like to express their gratitude to the editors and reviewers for their helpful, insightful, and constructive comments on this article, particularly during the period when SARS-CoV-2 was prevalent. We also acknowledge the Centre for Railway Information Systems, Indian Railways, for providing the valuable inputs necessary to carry out the research. However, the views expressed here are of the authors alone.