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Abstract

Fuel is an essential commodity in society with an amplified role during disasters. Disasters often cause a spike in demand for fuel and impose physical and operational constraints on fuel distribution. Accordingly, public and private sector stakeholders seek disaster preparedness and response interventions to ensure an adequate supply of fuel for emergency activities. We develop models to quantify downstream operational flow capacity, i.e., the volume of fuel that can be distributed from bulk storage terminals to retail gas stations via tanker trucks. We do this for both steady state conditions and nonsteady state conditions during disasters. The operational flow capacity measures can serve as inputs to optimal relief distribution and resource allocation problems, as well as enable a cost-benefit assessment of interventions aimed at increasing system capacity. We identify the best interventions for each type of bulk storage terminal structure and determine how to prioritize bulk storage terminals for each type of intervention when resources are limited. As the downstream fuel distribution system from bulk storage terminals to retail gas stations is similar across regions, our methodology and insights are generalizable not only within the US but also in countries with comparable distribution systems. To demonstrate impact on practice, we present a case study drawn from the application of our methodology to support US federal disaster preparedness. More generally, we contribute to system level understanding of multiserver tandem cyclic queues with time-limited customers found in relief distribution systems, which is often overlooked in the disaster management literature.

Keywords

Downstream fuel distribution operational flow capacity queuing theory discrete event simulation disaster intervention evaluation

1. Introduction

Natural disasters, man-made disasters, and accidents stress supply chains that are optimized to support material flows under a reasonable range of assumptions regarding supply, demand, and operating conditions. Post-disaster surges and shifts in demand combined with unexpected constraints in supply and operating conditions inhibit distribution systems such that goods may not reach people with vital needs. Facing such threats, leaders in the public, nonprofit, and private sectors seek life-saving interventions to restore and often enhance the flow of essential goods. Identifying appropriate humanitarian action requires a scientific understanding of complex supply chain systems to assess the impact of crisis interventions on flow capacity (National Academies of Sciences, Engineering, and Medicine, 2020; Spearman and Hopp, 2021). Our research seeks such understanding by modeling a key distribution system, namely that for fuel.

To motivate the problem, we provide details on the primary components of fuel distribution and describe disaster impacts on the system. Fuel distribution systems are often decoupled into upstream and downstream systems, with inventory at bulk storage terminals. The upstream fuel distribution system begins with crude oil, which may be extracted domestically or imported from abroad and is then refined into usable fuel. After refining, fuel is transferred to bulk storage terminals via different modes of transportation, including tanker ships, barges, and pipelines. The configuration of the upstream system varies by region.

In contrast, the downstream fuel distribution system is generalizable across regions. In the downstream system, bulk storage terminals (referred to as terminals) supply wholesale fuel to retail gas stations (referred to as stations), which then sell to consumers. The terminals are usually managed by petroleum companies or retail fuel chains, whereas the majority of stations are run by individual operators. Fuel is transported by carriers that employ hazardous material (HAZMAT) certified drivers operating tanker trucks (referred to as trucks). Contractual agreements determine which drivers (term used interchangeably with trucks, since one driver per truck is common in practice) can pick up fuel from which terminals and deliver it to which stations. An overview of the crude oil and gasoline distribution in the US is shown in Figure 1 (US Energy Information Administration, 2023).

Figure 1.

Crude oil and gasoline distribution system with the downstream components shown inside the red box.

When trucks visit terminals to pick up fuel, they must first check in at one of potentially multiple gates before entering the property and obtaining fuel at one of often multiple fueling bays. After filling up with fuel, the trucks deliver the fuel to a designated station and then return to the same or sometimes a different terminal to repeat the process until the daily hours of service limit has been reached. From an operational perspective, the downstream fuel distribution system is composed of a tandem multiserver queue with two primary components: (1) the terminal gate queues, and (2) the corresponding bay queues. Additionally, the trucks, which are time-limited terminal customers, perform multiple single-stop cycles in the system. We define operational flow capacity (OFC) of the described multiserver tandem cyclic queuing system with time-limited customers for downstream fuel distribution as the volume of fuel that can be distributed from terminals to stations by trucks in one day.

During disasters, fuel demand increases since humanitarian operations such as running generators, moving emergency response crews, and hauling relief goods increase fuel consumption. Moreover, news of impending natural disasters and extreme weather often increases consumer demand for fuel due to evacuation activities and hoarding in anticipation of future shortages. For example, during Hurricane Irma in September 2017, Florida witnessed the largest evacuation in its history. Fuel terminals experienced a $45 %$ increase in wholesale demand for gasoline over the evacuation days, with some terminals experiencing up to a $91 %$ increase in wholesale demand. Retail gasoline sales during that period were estimated to be double the normal levels (DeCorla-Souza, 2018; Florida Department of Transportation, 2018). Nonweather events like leakage accidents, cyberattacks on infrastructure, and changes in trade policies can also increase fuel supply uncertainty and cause panic buying and hoarding (Colchester, 2021). The newsworthy cyberattack on the Colonial Pipeline in May 2021, for example, resulted in severe shortages at stations due to demand spikes alone (Egan and Morrow, 2021).

Unfortunately, increased fuel demand is often compounded by physical and operational capacity constraints in the upstream fuel distribution system. For example, during Hurricane Sandy in October 2012, flooding and power outages in the New York Harbor area closed down refineries and terminals (US Department of Energy, 2013). Similarly, during Hurricane Irma, port closures hindered fuel imports to Florida (Florida Department of Transportation, 2018). More recently, during Hurricane Ian in September 2022, the Central Florida Pipeline was shut down due to severe flooding in Orlando (S&P Global Commodity Insights, 2022). Likewise, a recent fire, at a natural gas liquefaction plant in Texas, in June 2022, caused the facility to be shut down completely for several months (US Energy Information Administration, 2022). It is not only physical damage of infrastructure by natural disasters or accidents that results in capacity constraints. During the cyberattack on the Colonial Pipeline, sections of the pipeline had to be shut down for security reasons (Colonial Pipeline, 2021).

Operational capabilities of terminals and transportation infrastructure can also create bottlenecks in the downstream system during disasters. Slow process times at terminals can be caused by a lack of employees (as they may be unable to work due to the disaster), drivers accessing unfamiliar facilities, power outages, and damaged equipment. For example, during Hurricane Irma some terminals had 2–4 hour long queues. Damaged roadways and evacuations can also cause traffic jams that slow down trucks. Additionally, the lack of HAZMAT or terminal-safety-certified drivers can reduce replenishment capacity for stations. Power outages and the lack of generator fuel can cause stations to shut down. Finally, the distribution system is not designed to fulfill the surges in demand that often follow disasters. During Hurricane Irma, $60 %$ of stations in major cities in Florida reported fuel shortages or power outages (DeCorla-Souza, 2018; Florida Department of Transportation, 2018). During the cyber-attack on Colonial Pipeline, up to $65 %$ of stations across several states were out of fuel due to a spike in demand above the normal level (Egan and Morrow, 2021).

As disasters constrain fuel distribution, public and private sector intervention is often required to meet the surge in fuel demand. For example, driver hours of service rules, truck weight restrictions, and fuel blend restrictions have often been waived during previous disasters. The highway patrol has been used to quickly escort trucks through evacuation traffic and checkpoints. In addition to providing operational support, government policies for managing demand spikes have also been administered. For example, during Hurricane Sandy, rationing of supplies based on consumers’ odd or even license plate numbers was put in place to discourage daily hoarding (US Department of Energy, 2013; Florida Department of Transportation, 2018; Colonial Pipeline, 2021)

Studying previous disasters, we identify that the primary limiting factors associated with retail fuel availability were the terminal and transportation operations and not the fuel inventory in terminal storage tanks. For example, in Florida, terminals have 11–16 days of demand in inventory (DeCorla-Souza, 2018; Florida Department of Transportation, 2018). Thus, we study downstream fuel distribution systems and address the following research questions as part of ongoing support for the Federal Emergency Management Agency (FEMA) preparedness efforts: (1)

How can we measure the OFC in downstream fuel distribution systems, under steady state conditions and under nonsteady state conditions during disasters? To do this we develop two models, an analytical queueing model to measure the steady state OFC as well as a discrete event simulation framework that models the nonsteady state operations for the downstream fuel distribution system under disaster conditions. Our models contribute to the theoretical understanding of multiserver tandem cyclic systems with time-limited customers, which extends more generally to relief distribution settings such as water, food, healthcare supplies, and other humanitarian supplies from staging areas where workers or fleets perform multiple cycles to meet demand following disasters. Additionally, we highlight the importance of modeling nonsteady state OFC as a more realistic constraint than storage capacity, which is commonly used in disaster relief optimization models.

(2)

How effective are disaster preparedness and response interventions to restore and perhaps enhance the OFC of downstream fuel distribution systems? We analyze the effects of disaster interventions, such as improving terminal process times, transportation speed, fleet size, and hours of service, on OFC. We identify which interventions are best suited for specific downstream fuel distribution systems and how to prioritize interventions in case of limited implementation resources. Our results can be used to directly influence public and private actions and enhance capacity parameters for resource allocation models that aim to optimize distribution systems. Measuring the benefits of interventions in terms of an increase in OFC provides crucial evidence when assessing implementation costs. We also present a real life case study of the application of our methodology to support preparedness efforts for FEMA.

The remainder of the paper is organized as follows, Section 2 reviews the relevant literature on disaster management and fuel distribution, with a focus on the knowledge gaps that we address. In Section 3, we describe the models used to measure the steady and nonsteady state OFC in multifacility downstream fuel distribution systems. Next, we discuss various disaster preparedness and response interventions available to stakeholders in Section 4. We analyze the effects of individual interventions on OFC for individual terminal facilities in Section 5. Then, we demonstrate the model’s applicability in a real life case study of the multifacility downstream gasoline distribution system in the state of Florida in Section 6. Finally, our research contributions and potential directions for future extensions are summarized in Section 7.

2. Literature Review

Building system-level understanding of supply chain dynamics as a foundation for effective decision support has been recommended by National Academies of Sciences, Engineering, and Medicine (2020). Gupta et al. (2016) additionally suggest finding research problems in the chaos of disaster (we identified this problem while supporting FEMA during Hurricane Irma), emphasize capacity recovery and restoration, and conclude that the underlying thread for effective disaster management is understanding integrated systems. They highlight unified models for transport, storage, and distribution, a theme highlighted again by Besiou et al. (2021). More importantly, they suggest that research must convince administrators that proposed models actually contribute to solving the problem. Our research, which played a central role in recent FEMA studies, confirms that models measuring system capacity can achieve this goal. Our work contributes to two primary areas of the literature: disaster management (Section 2.1) and fuel distribution (Section 2.2).

2.1. Disaster Management

Relief distribution and resource allocation during disasters have been extensively studied and reviewed (e.g., Anaya-Arenas et al., 2014; Baxter et al., 2020). The majority of this research develops optimization models for problems like facility and asset location, flow assignment and scheduling, vehicle routing, and fleet management (e.g., Salmerón and Apte, 2010; Doan and Shaw, 2019; Pedraza-Martinez et al., 2020). However, we must first accurately measure distribution capacity constraints, to better understand and evaluate decisions for complex and dynamic systems.

With that intent, simulation models have been applied to humanitarian logistics (Pareja Yale et al., 2020). Problems such as distribution facility layout and operations evaluation, asset and funds management, fleet mix, purchase, sales, and scheduling, and collaboration strategies evaluation for the distribution of relief goods have been modeled (e.g., Besiou et al., 2014; Stauffer et al., 2018; Gu et al., 2021). These studies either do not model facilities, or consider just one supply facility, or do not model multiple cycles of distribution vehicles. In contrast, we model multicycle operations for multiple facilities.

Some disaster research explores scheduling of multicycle fleet and workers’ movements in steady state conditions in air force operated and charity run storehouses (Mattila et al., 2008; Urrea et al., 2019). They use simulation methods, since existing queueing theory literature does not have frameworks for multiserver tandem cyclic queues with time-limited customers, only single-server tandem cyclic queues (e.g., Campbell, 1991; Boxma and Daduna, 2014) or multiserver tandem queues (e.g., Kock et al., 2008; Baumann and Sandmann, 2017). Following the call for more humanitarian study validated with simulation (Kovacs and Moshtari, 2019), we extend this body of literature and apply queueing theory for steady state approximation and discrete event simulation for nonsteady state OFC measurement of multiserver tandem cyclic systems with time-limited customers.

Furthermore, previous resource allocation problems examine retrofitting, physical expansion, and repair of distribution facilities and transportation infrastructure (e.g., Salmerón and Apte, 2010; Duque et al., 2016; Sun and Zhang, 2020). Few studies model the effects of process time or fleet size interventions (Mattila et al., 2008). Our focus on interventions that increase OFC without building infrastructure is inspired by the coronavirus disease 2019 (COVID-19) pandemic experiences where production and transportation systems increased volume quickly by using existing resources more fully and leveraging convertible capability (Ergun et al., 2022; Li et al., 2022). Utilizing dynamic interventions to improve complex adaptive systems has not yet been fully explored in disaster literature (Polater, 2021).

We contribute to the theoretical understanding of multicycle operations in multifacility disaster relief or downstream fuel distribution systems, which is often overlooked. Descriptive and quantitative research to build scientific knowledge (Spearman and Hopp, 2021) is central to our approach. By explicitly modeling the OFC (He et al., 2021) we fill the highlighted gaps in literature. Although our models are motivated by downstream fuel distribution, our framework can be applied to several other last-mile relief distribution problems for commodities such as water, food, and medical supplies as well as facility operations with multiple cycles of picking, packing, loading, and off-loading. Moreover, we analyze resource allocation in terms of interventions that improve terminal process times, transportation speed, fleet size, and hours of service for individual terminal facilities. We also present a real life case study of multifacility downstream fuel distribution system.

2.2. Fuel Distribution

Many researchers have studied fuel distribution systems using optimization methods in steady state conditions (Lima et al., 2016; Cunha de Bittencourt et al., 2021). However, there is limited research on optimal flow assignment and scheduling, infrastructure repair and resilience resource allocation, and system analysis using simulation methods for fuel distribution systems during disasters.

Existing studies focus on upstream distribution (Sabuncuoglu and Hatip, 2005; Cafaro et al., 2010), a single terminal facility (Reis et al., 2017), outlining system vulnerabilities (Costa et al., 2019), or infrastructure improvement (DeCorla-Souza, 2018). Only a few of them analyze the effects of improving terminal bay processes and fleet size on the distribution system’s performance (e.g., Reis et al., 2017; DeCorla-Souza, 2018). Additionally, none of them consider the cyclic movement of trucks in the system. We contribute to this body of literature by modeling the cyclic movement of trucks in the entire downstream distribution system and evaluating a suite of disaster interventions that improve terminal gate and bay times, transportation speed, fleet size, and hours of service.

3. Modeling OFC

In this section, we develop an analytical queueing model and a discrete event simulation model to measure the steady state (Section 3.1) and nonsteady state (Section 3.2) OFC, respectively, of multifacility downstream fuel distribution systems. We model multicycle movement of trucks, i.e., from terminals to stations and back, in a multifacility setting, i.e., multiple terminals within multiple terminal groups. We measure OFC in terms of millions of gallons of fuel per day (MMgal/day), as is relevant in practice.

Let $I$ denote the set of terminal groups. Each group, $i \in I$ , consists of a set of one or more neighboring terminals, $T_{i}$ . Each terminal, $t \in T_{i}$ , has $n_{g, t}$ gates for the check-in process and $n_{b, t}$ bays to dispense fuel. A fleet of $f_{i}$ trucks is assigned to group $i$ . Each group serves a set of stations, $s \in S_{i}$ . We assume that all the trucks assigned to a terminal group can pick up fuel at any of the terminals within that terminal group and deliver fuel to any of the stations serviced by that terminal group. We do not model distinct distributors and their contract constraints and assume that all stations are equally likely to demand fuel, since the delivery demand, schedules, and priorities of the system are proprietary.

Trucks assigned to terminal group $i$ start operating at some time $a$ and go to a terminal $t \in T_{i}$ where they either wait in the gate queue or proceed to an empty gate. At each of the multiple gates, the trucks take $r_{g}$ minutes to check in. Different terminals have different layouts and sequences of processes for entering the gates and accessing the bays, all of which are assumed to be modeled by the gate time, $r_{g}$ .

After the check-in process, the trucks proceed to one of the multiple filling bays to load fuel in $r_{b}$ minutes. Different terminals have different layouts and processes within the terminal, such as positioning trucks, entering order details, filling fuel, and collecting the bill of lading. We model all these processes collectively using the bay time, $r_{b}$ . We assume that all of the filling bays at the terminal can dispense the type of fuel that is needed for delivery, and if all of the bays are busy, then the trucks wait in a common parking area with $m$ spaces. Additionally, we assume that there is sufficient fuel inventory at terminals, as discussed in Section 1.

After filling fuel at a bay, the trucks travel distance $d_{i}$ to a station $s \in S_{i}$ with speed $v_{i}$ . Fuel is then dispensed into the station tanks in $r_{s}$ minutes, and the trucks go back to a terminal $t \in T_{i}$ to perform another cycle. We assume that all trucks deliver $c$ gallons of fuel in each cycle.

Drivers can be operational for a maximum of $h$ hours in a 24-hour day. If the remaining hours of service available are less than $τ$ minutes at the terminal, then drivers end their work day. Moreover, we assume that each truck is operated by just one driver, and thus use the terms “truck” and “driver” interchangeably. We also assume that all the terminals and stations are open all 24 hours of the day in order to imitate crisis conditions as well as to model the upper bound of system capacity. Finally, to address our first research question, we measure the total volume of fuel that reaches all the stations $s \in S_{i}$ as the OFC, $γ_{i}$ , for terminal group $i$ . We focus on measuring how much total fuel can be distributed rather than on flow allocation based on specific inventory and demand. The process cycle is summarized with the flowchart in Figure 2.

Figure 2.

Flow chart of the process cycle of a truck serving terminal group $i$ .

3.1. Steady State

First, we introduce an analytical queueing model to approximate the average steady state OFC, ${\bar{γ}}_{i}^{S}$ , for terminal group $i$ using Little’s Law. Recall that Little’s Law states that the throughput of a queueing system is equal to work in progress divided by its cycle time. The described downstream fuel distribution system’s OFC will either be limited by how many trucks can be processed by the terminals or the number of trucks present in the system. Thus, we calculate the throughput from the point of view of the terminals and the fleet and avoid instability in OFC of the multiserver tandem cyclic queue by taking the minimum of the two measures.

We measure the throughput from the point of view of a single terminal $t \in T_{i}$ . At the gate queue, the work in progress is the number of gates, $n_{g, t}$ , and the cycle time is the gate time, $r_{g, t}$ . Similarly, at the bay queue, the work in progress is the number of bays, $n_{b, t}$ , and the cycle time is the bay time, $r_{b, t}$ . Thus, the independent throughput at gate and bay queues is $\frac{n_{g, t}}{r_{g, t}}$ and $\frac{n_{b, t}}{r_{b, t}}$ , respectively. Equivalently, we can interpret the gate and bay as single-server queues with cycle times equal to $\frac{r_{g, t}}{n_{g, t}}$ and $\frac{r_{b, t}}{n_{b, t}}$ , respectively. Since these queues are in tandem, they can jointly be considered as a single server with a cycle time of $\frac{r_{g, t}}{n_{g, t}} + \frac{r_{b, t}}{n_{b, t}}$ . Thus with trucks hauling $c$ gallons of fuel, the throughput $ϕ_{t}$ of terminal $t$ (in MMgal/day) is given by:

ϕ_{t} = \frac{c \times 10^{- 6} \times 60 \times 24}{\frac{r_{g, t}}{n_{g, t}} + \frac{r_{b, t}}{n_{b, t}}}

(1)

Taking the summation over all the terminals in group

i

, we get throughput

ϕ_{i}^{1}

of terminal group

i

(in MMgal/day):

ϕ_{i}^{1} = \sum_{t \in T_{i}} \frac{c \times 10^{- 6} \times 60 \times 24}{\frac{r_{g, t}}{n_{g, t}} + \frac{r_{b, t}}{n_{b, t}}}

(2)

Then, we consider the throughput from the point of view of the fleet of trucks. The cycle time for a complete cycle of a truck visiting a single terminal

t \in T_{i}

, i.e., the gate and bay processes, traveling to station

s

, unloading fuel, and traveling back to

t

will be

\frac{r_{g, t}}{n_{g, t}} + \frac{r_{b, t}}{n_{b, t}} + \frac{d_{i}}{v_{i}} + r_{s} + \frac{d_{i}}{v_{i}}

. We can consider each terminal

t \in T_{i}

as a single server with their respective cycle times. As they are parallel servers, they can collectively be considered a single server with inverse cycle time of

\sum_{t \in T_{i}} 1 / (\frac{r_{g, t}}{n_{g, t}} + \frac{r_{b, t}}{n_{b, t}} + \frac{2 d_{i}}{v_{i}} + r_{s})

. Now, if we consider the terminal group’s finite fleet,

f_{i}

, and finite hours of service,

h

, then we have a total of

f_{i} \times h

cycle hours. Thus with trucks hauling

c

gallons of fuel, the throughput

ϕ_{i}^{2}

of terminal group

i

(in MMgal/day) is given by:

ϕ_{i}^{2} = \sum_{t \in T_{i}} \frac{c \times 10^{- 6} \times 60 \times f_{i} \times h}{\frac{r_{g, t}}{n_{g, t}} + \frac{r_{b, t}}{n_{b, t}} + \frac{2 d_{i}}{v_{i}} + r_{s}}

(3)

The average steady state OFC of terminal group

i

therefore is:

{\bar{γ}}_{i}^{S} = min {ϕ_{i}^{1}, ϕ_{i}^{2}}

(4)

3.2. Nonsteady State

Next, we simulate the nonsteady state movement of trucks for $24$ hour time periods using an object-oriented program implemented in Python 3.9.6. The terminal groups, terminals gates and bays, stations, and trucks are all programmed as objects that follow the process described in Figure 2 with some additional constraints. At the end of each $24$ hour time period we output the total volume of fuel that reaches the station objects, as the nonsteady state OFC, $γ_{i}^{N S}$ , for each terminal group $i$ .

During disasters, processes can slow down and infrastructures can shut-down suddenly, and a new steady state can take a long time to set in. Using steady state OFC will therefore overestimate the system’s capability under disaster conditions. Thus, we simulate individual $24$ hour time periods, where the start of the period is the start of disaster conditions. It is nonsteady state as we simulate a fixed time horizon and a fixed number of trucks. Moreover, we assume that the trucks start in the first $6$ hours of the day, $a \in uniform [0000, 0600]$ . This is because drivers may be called to serve surge demand immediately and to accommodate $14 - 18$ hours of operations of all the drivers in the $24$ hour time period. We also assume that there are no remaining queues from the previous period.

All terminals $t \in T_{i}$ are assumed to be located at the centroid of terminal group $i$ and all trucks start and end their day at this centroid to model having a fixed home base. We also assume that the trucks go to the terminals with the shortest gate queue. This is because dispatchers usually only know information about waiting time to get inside the terminal or are called ahead of time.

In the next section, we discuss disaster preparedness and response interventions that can improve the described system’s parameters and consequently increase OFC.

4. Disaster Interventions

In this section, we describe disaster interventions that can directly improve terminal gate and bay times, transportation speed, fleet size, and hours of service, thus contributing to an increase in OFC. Table 1 summarizes these interventions by the parameters they influence, the actor responsible for implementing (private or public), the time horizon (short or long-term), and the resource requirements for implementation.

Table 1.
Summary of disaster interventions.

Affected Parameter Actor Time Horizon Intervention Resource Requirements

Terminal process Private Short Employee management Labor

Time ( $r_{g}, r_{b}$ ) Long Technology and equipment improvement Equipment purchase

Terminal facility ( $n_{g}, n_{b}$ ) Private Long Additional gate and bay construction Facility expansion

Transportation Public Short Traffic escorts Labor

Speed ( $v$ )

Fleet size ( $f$ ) Private Short Additional drivers Labor

Public Short Military fleet Labor

Hours of service ( $h$ ) Public Short Requirements waiver Policy change

Affected Parameter	Actor	Time Horizon	Intervention	Resource Requirements
Terminal process	Private	Short	Employee management	Labor
Time ( $r_{g}, r_{b}$ )		Long	Technology and equipment improvement	Equipment purchase
Terminal facility ( $n_{g}, n_{b}$ )	Private	Long	Additional gate and bay construction	Facility expansion
Transportation	Public	Short	Traffic escorts	Labor
Speed ( $v$ )
Fleet size ( $f$ )	Private	Short	Additional drivers	Labor
	Public	Short	Military fleet	Labor
Hours of service ( $h$ )	Public	Short	Requirements waiver	Policy change

First, we consider terminal process time interventions that directly reduce gate time, $r_{g, t}$ , and bay time, $r_{b, t}$ , for a particular terminal, $t$ (DeCorla-Souza, 2018; Florida Department of Transportation, 2018). In the short-term, terminal employees can be positioned to check in truck drivers while they wait in gate queues, thus reducing $r_{g}$ when they reach the gate. Alternately, having employees manage traffic and queues inside the terminal while helping drivers with paperwork and efficient fuel loading can reduce $r_{b}$ . During disasters, when more unfamiliar drivers access the terminals, clear instructions and support are known to help drivers navigate the facility faster.

Terminal operators can incrementally invest in equipment like larger pipes or extra loading arms at the bays to decrease $r_{b}$ . Technological improvements for order management systems at the gates and bays can reduce or eliminate manual data entry, thus reducing errors and time for drivers, ultimately reducing $r_{g}$ and $r_{b}$ . To incentivize action by private sector operators, state or federal agencies can mandate minimum service level requirements for disasters, based on the terminal structure, or provide financial or fiscal incentives to fund improvements at targeted terminals. Facility expansion to add gates or bays may require more significant regulations or incentives if terminal operators do anticipate sufficient return on investment. However, creative alternatives like providing additional check-in system points to open the same physical entry gate effectively increases the number of gates, $n_{g, t}$ , at a much lower cost.

To directly improve the transportation speed, $v$ , public sector actors can employ law-enforcement escorts to move trucks safely and quickly through evacuation traffic and damaged roads (DeCorla-Souza, 2018; Florida Department of Transportation, 2018). To increase fleet size, $f$ , National Guard or active duty military personnel can drive idle trucks during disasters (Colchester, 2021). Public sector agencies can also provide subsidies for driver retention and additional employment by private sector carriers (DeCorla-Souza, 2018). Policymakers can also waive Federal Motor Carrier Safety Administration (2022) hours of service rules that limit truck drivers’ daily operation time ( $14$ hours) and driving time ( $11$ hours), thus increasing $h$ and enabling all trucks to conduct more trips in a day (US Department of Energy, 2013; Florida Department of Transportation, 2018).

In Section EC.1 of the e-companion, we discuss these interventions in further detail, including the implementation considerations. Such information provides a foundation to further develop implementation cost scenarios as part of a rich cost-benefit analysis for public and private sector intervention. We also mention some additional policy, infrastructure, and operational interventions that do not directly affect model parameters, but remove barriers to improvement.

In the following sections we quantify the benefits of the discussed interventions, in terms of increase in OFC, for individual terminal facilities and a real life case study of multifacility downstream distribution system.

5. Numerical Analysis

In this section, we quantify the increase in OFC by improving terminal gate and bay times, transportation speed, fleet size, and hours of service for individual terminal facilities. We consider different levels of improvement by changing corresponding parameter values and identify which type of intervention is most beneficial for which terminal structure. Although it is difficult to collect precise proprietary data, we recognize the call for research using field data, with all its “noise, dirt, and missing elements” by Besiou and Van Wassenhove (2020). Similar to Biller et al. (2019) and Costa et al. (2019) we rely on public sources, industry insights, visits to terminals in the US, and validation from industry practitioners to identify parameter values.

5.1. Individual Interventions

We model a single terminal group that has a single terminal with different number of gates, $n_{g}$ , number of bays, $n_{b}$ , and station distances, $d$ . Specifically, we consider $n_{g} = {1, 2}$ , $n_{b} = {1, 2, 3, \dots, 15}$ , and $d \in uniform [0, μ] \forall μ \in {25, 50, 100, 150, 200, 250, 300} miles$ . Most terminals in the US fall in this range of structural parameter values.

In the nonsteady state model we simulate a set of stations to be served by the terminal group, $S$ , and sample the station distances from $d$ . The size of $S$ is assumed to be a large number, $| S | = 275$ , such that it does not constrain the OFC measurement. We also assume that there are $m = 3$ parking spaces inside the terminal.

We fix the start time of truck objects in the simulation by sampling $a \in uniform [0000, 0600] hours$ as discussed earlier and set the stopping criteria to be $τ = 30 minutes$ , so that trucks do not enter the terminal when they can not complete the consequent trip to a station. We assume truck capacity to be fixed at $c = 9000 gallons$ (DeCorla-Souza, 2018; Florida Department of Transportation, 2018) and assume that every delivery constitutes a full truckload (Li et al., 2017). Additionally, we assume that the time taken to empty fuel at stations is $r_{s} = triangular (50, 70, 60) minutes$ (DeCorla-Souza, 2018).

Next, we set the baseline values for the intervention parameters. The time to check in at terminal gates is assumed to be $r_{g} = triangular (6.3, 7.7, 7) minutes$ and the time to fill fuel at terminal bays is assumed to be $r_{b} = triangular (31.5, 38.5, 35) minutes$ . Transportation speed is modeled as $v = triangular (40.5, 49.5, 45) miles/h$ . We choose to model processes as triangular distribution in the simulation model following Mattila et al. (2008), and Biller et al. (2019). Then, we assume a fleet size of $f = 50$ trucks. Finally, truck driver hours of service rules (Federal Motor Carrier Safety Administration, 2022) mandate $h = 14$ hours of service.

We study the relationship between the average nonsteady state OFC, ${\bar{γ}}^{N S}$ , computed across $50$ simulation runs, and individual intervention’s parameter values. The intervention parameters, i.e., $r_{g}, r_{b}, v, f$ and $h$ , are changed one at a time, while the remaining parameters are fixed at baseline values as summarized in Table 2. Each $24$ hour day replication, without a burn-in period, takes up to 1 second to simulate, depending on the parameter values.

Table 2.
Baseline parameter values for individual terminals.

Parameter Description Values

$n_{g}$ Number of check-in gates ${1, 2}$

$n_{b}$ Number of fuel fueling bays ${1, 2, \dots, 15}$

$d$ Station distance in miles Uniform $[0, μ],$

$\forall μ \in {25, 50, 100, 150, 200, 300}$

$S$ Set of stations $| S | = 275$

$m$ Number of parking spaces $3$

$a$ Start time for trucks in hour of day Uniform $[0000, 0600]$

$τ$ Stopping criteria for trucks in mins $30$

$c$ Capacity of trucks in gallons $9000$

$h$ Hours of service of drivers $14$

$r_{s}$ Time taken at stations in mins Triangular $(50, 70, 60)$

$r_{g}$ Time taken at terminal gates in mins Triangular $(6.3, 7.7, 7)$

$r_{b}$ Time taken at terminal bays in mins Triangular $(31.5, 38.5, 35)$

$v$ Speed of trucks in miles/h Triangular $(40.5, 49.5, 45)$

$f$ Fleet size $50$

Parameter	Description	Values
$n_{g}$	Number of check-in gates	${1, 2}$
$n_{b}$	Number of fuel fueling bays	${1, 2, \dots, 15}$
$d$	Station distance in miles	Uniform $[0, μ],$
		$\forall μ \in {25, 50, 100, 150, 200, 300}$
$S$	Set of stations	$\| S \| = 275$
$m$	Number of parking spaces	$3$
$a$	Start time for trucks in hour of day	Uniform $[0000, 0600]$
$τ$	Stopping criteria for trucks in mins	$30$
$c$	Capacity of trucks in gallons	$9000$
$h$	Hours of service of drivers	$14$
$r_{s}$	Time taken at stations in mins	Triangular $(50, 70, 60)$
$r_{g}$	Time taken at terminal gates in mins	Triangular $(6.3, 7.7, 7)$
$r_{b}$	Time taken at terminal bays in mins	Triangular $(31.5, 38.5, 35)$
$v$	Speed of trucks in miles/h	Triangular $(40.5, 49.5, 45)$
$f$	Fleet size	$50$

In addition, we use Welch’s 1-tail $t$ -test at a significance level of $0.05$ to determine the parameter value, called the improvement threshold, above or below which there is no statistically significant increase in nonsteady state OFC, $γ^{N S}$ . Moreover, we calculate the steady state OFC, ${\bar{γ}}^{S}$ , using Equation 4 and expected values of the parameters described, and compare it to $γ^{N S}$ . Again, using 1-tail $t$ -test at a significance level of $0.05$ we determine whether ${\bar{γ}}^{S}$ is statistically significantly greater than $γ^{N S}$ . We plot the results for commonly present terminal structures, $(n_{g}, n_{b}, d)$ , in the US, i.e., a) $(1, 4, 100)$ and b) $(2, 10, 150)$ .

First, we explore terminal gate time interventions by modeling $r_{g} \in triangular (0.9 {\bar{r}}_{g}, 1.1 {\bar{r}}_{g}, {\bar{r}}_{g}) \forall {\bar{r}}_{g} \in {1, 2, 3, \dots, 10, 15, 20} minutes$ . Values of ${\bar{r}}_{g}$ greater than baseline correspond to disasters when processes are slowed down, and values smaller than baseline correspond to different levels of interventions that speed up the check-in process. We model the bounds of the triangular distribution based on conversations with practitioners. We assume that during disasters and under interventions the mean gate time, ${\bar{r}}_{g}$ , changes but the upper and lower bounds are always $\pm 10 %$ of the mean.

The results for common terminal structures are shown in Figure 3, and additional terminal structures in Section EC.2 of the e-companion. We also plot the improvement threshold of ${\bar{r}}_{g}$ and corresponding ${\bar{γ}}^{N S}$ for all terminal structures, in Section EC.2 of the e-companion. We note that the $99 %$ confidence interval of ${\bar{γ}}^{N S}$ is within ${\bar{γ}}^{N S} \pm 0.01$ MMgal/day for all the parameter combinations, which is less than $2$ truckloads of fuel. Additionally, we find that ${\bar{γ}}^{S}$ is always statistically significantly greater than $γ^{N S}$ .

Figure 3.

Operational flow capacity (OFC) vs terminal gate time, ${\bar{r}}_{g}$ , for terminals with different number of gates, $n_{g}$ , number of bays, $n_{b}$ , and station distances, $d$ . Improvement thresholds indicate terminal gate time values below which there is no statistically significant improvement in nonsteady state OFC.

We expect OFC to increase with decrease in ${\bar{r}}_{g}$ . For the terminals shown in Figure 3 we see that decreasing ${\bar{r}}_{g}$ below baseline does not increase $γ^{N S}$ statistically significantly. However, for terminals with either a single gate and a large number of bays or for terminals with $2$ gates, a large number of bays, and short station distances, the increase in $γ^{N S}$ is statistically significant with decrease in ${\bar{r}}_{g}$ . Additionally, the improvement threshold gets lower and $γ^{N S}$ gets higher with increase in number of bays and decrease in station distance. Moreover, for cases like terminal structure $(1, 4, 100)$ , ${\bar{γ}}^{S}$ could even estimate improvement threshold to be lower than that from ${\bar{γ}}^{N S}$ , as seen in Figure 3(a).

Similarly, we examine the effects of bay time interventions and simulate $r_{b} \in triangular (0.9 \bar{r_{b}}, 1.1 \bar{r_{b}}, \bar{r_{b}}) \forall \bar{r_{b}} \in {15, 17.5, 20, \dots, 45} minutes$ . Values of $\bar{r_{b}}$ greater than baseline correspond to disasters when processes are slowed down, and values smaller than baseline correspond to different levels of interventions that speed up the fuel filling process. Again, we assume that during disasters and under interventions the mean bay time, ${\bar{r}}_{b}$ , changes but the upper and lower bounds are always $\pm 10 %$ of the mean.

The results for common terminal structures are shown in Figure 4, and additional terminal structures in Section EC.2 of the e-companion. We also plot the improvement threshold of ${\bar{r}}_{b}$ and corresponding ${\bar{γ}}^{N S}$ for all terminal structures, in Section EC.2 of the e-companion. The $99 %$ confidence interval of ${\bar{γ}}^{N S}$ is within ${\bar{γ}}^{N S} \pm 0.01$ MMgal/day for all cases, which is less than $2$ truckloads of fuel. Moreover, ${\bar{γ}}^{S}$ is always statistically significantly greater than $γ^{N S}$ .

Figure 4.

Operational flow capacity (OFC) vs terminal bay time, ${\bar{r}}_{b}$ , for terminals with different number of gates, $n_{g}$ , number of bays, $n_{b}$ , and station distances, $d$ . Improvement thresholds indicate terminal bay time values below which there is no statistically significant improvement in nonsteady state OFC.

Similar to ${\bar{r}}_{g}$ , we expect OFC to increase with decrease in ${\bar{r}}_{b}$ . For the terminals shown in Figure 4 we see that decreasing ${\bar{r}}_{b}$ below baseline increases $γ^{N S}$ statistically significantly for terminal structure $(1, 4, 100)$ but not for $(2, 10, 150)$ . For terminals with a small number of bays or terminals with $2$ gates and short station distances, the increase in $γ^{N S}$ is statistically significant with decrease in ${\bar{r}}_{b}$ . The improvement threshold also gets lower with decrease in number of bays and decrease in station distance. Moreover, for cases like terminal structure $(1, 4, 100)$ , ${\bar{γ}}^{S}$ could even estimate improvement threshold to be lower than that from ${\bar{γ}}^{N S}$ , as seen in Figure 4(a).

Next, we consider transportation speed interventions. We simulate $v \in triangular (0.9 \bar{v}, 1.1 \bar{v}, \bar{v}) \forall \bar{v} \in {25, 35, 45, 55, 65, 75} miles/h$ . Values of $\bar{v}$ lower than baseline correspond to disasters when evacuation traffic and transportation infrastructure damage slow down trucks, and values larger than baseline represent transportation speed under interventions. Once again, we assume that during disasters and under interventions the mean transportation speed, $\bar{v}$ , changes but the upper and lower bounds are always $\pm 10 %$ of the mean.

The results for common terminal structures are shown in Figure 5, and additional terminal structures in Section EC.2 of the e-companion. We also plot the improvement threshold of $\bar{v}$ and corresponding ${\bar{γ}}^{N S}$ for all terminal structures, in Section EC.2 of the e-companion. For all the cases, the $99 %$ confidence interval of ${\bar{γ}}^{N S}$ is within ${\bar{γ}}^{N S} \pm 0.02$ MMgal/day, which is less than $3$ truckloads. Again, ${\bar{γ}}^{S}$ is statistically significantly greater than $γ^{N S}$ for all parameter combinations.

Figure 5.

Operational flow capacity (OFC) vs transportation speed, $\bar{v}$ , for terminals with different number of gates, $n_{g}$ , number of bays, $n_{b}$ , and station distances, $d$ . Improvement thresholds indicate transportation speed values above which there is no statistically significant improvement in nonsteady state OFC.

As $\bar{v}$ increases, OFC is expected to increase, as is observed for the terminals shown in Figure 5. The increase in $γ^{N S}$ is statistically significant for terminals with long station distances. Moreover, the improvement threshold gets higher with $2$ gates and increases in station distance. For example, in Figure 5 we observe that improvement threshold for terminal structure $(1, 4, 100)$ is $65$ miles/h, whereas for $(2, 10, 150)$ it is $75$ miles/h.

We then study the effect of fleet size interventions on OFC and simulate $f \in {10, 25, 50, 75, \dots, 200} trucks$ . Values of $f$ lower than baseline indicate disruptive events when drivers become survivors of the disaster and fleet size is compromised, while values larger than baseline model fleet size under interventions.

The results for common terminal structures are shown in Figure 6, and additional terminal structures in Section EC.2 of the e-companion. We also plot the improvement threshold of $f$ and corresponding ${\bar{γ}}^{N S}$ for all terminal structures, in Section EC.2 of the e-companion. For all the cases, the $99 %$ confidence interval of ${\bar{γ}}^{N S}$ is within ${\bar{γ}}^{N S} \pm 0.01$ MMgal/day, which is less than $2$ truckloads. Furthermore, ${\bar{γ}}^{S}$ is statistically significantly greater than $γ^{N S}$ for all terminal structures and fleet size values, except in two cases.

Figure 6.

Operational flow capacity (OFC) vs fleet size, $f$ , for terminals with different number of gates, $n_{g}$ , number of bays, $n_{b}$ , and station distances, $d$ . Improvement thresholds indicate fleet size values above which there is no statistically significant improvement in nonsteady state OFC.

We expect $γ^{N S}$ to increases with $f$ , as seen in Figure 6. We observe that increasing $f$ above baseline increases $γ^{N S}$ statistically significantly for terminals with a large number of bays. The improvement threshold also gets higher when the terminal has $2$ gates and a larger number of bays. Moreover, for cases like terminals in Figure 6, ${\bar{γ}}^{S}$ might estimate improvement threshold to be lower than that from ${\bar{γ}}^{N S}$ .

Finally, we evaluate the relationship between OFC and $h$ by simulating $h \in {14, 15, 16, 17, 18} hours$ . We simulate maximum $h = 18$ hours to accommodate the last truck start time at $0600$ hour of the day assumed in our nonsteady state model. The results for common terminal structures are shown in Figure 7, and additional terminal structures in Section EC.2 of the e-companion. We also plot the improvement threshold of $h$ and corresponding ${\bar{γ}}^{N S}$ for all terminal structures, in Section EC.2 of the e-companion.

Figure 7.

Operational flow capacity (OFC) vs hours of service, $h$ , for terminals with different number of gates, $n_{g}$ , number of bays, $n_{b}$ , and station distances, $d$ . Improvement thresholds indicate hours of service values above which there is no statistically significant improvement in nonsteady state OFC.

For all combinations of structural parameters, the $99 %$ confidence interval of ${\bar{γ}}^{N S}$ is within ${\bar{γ}}^{N S} \pm 0.01$ MMgal/day, which is less than $2$ truckloads. Additionally, ${\bar{γ}}^{S}$ is statistically significantly greater than $γ^{N S}$ for all terminal structures and hours of service values, except in four cases. Finally, we observe that for all terminal structures, increasing $h$ statistically significantly increases $γ^{N S}$ till $h = 18$ .

We make an important note that the analytical approximation of steady state OFC overestimates the capabilities of the terminals during nonsteady state of disasters and different levels of all interventions. Thus, we propose using nonsteady state OFC, that considers multicycle operations, as a tighter constraint as opposed to steady state OFC for optimal fuel or relief distribution problems. Moreover, if there are limited resources to implement a particular intervention, then policymakers can use sensitivity analysis results, in Section EC.2 of the e-companion, to prioritize terminals which will have the largest increase in OFC. We note that these results also quantify benefits for long-term infrastructure investment, like adding an extra check-in gate or a fueling bay, since we include a wide range of $n_{g}$ and $n_{b}$ values.

Apart from sensitivity analysis on intervention parameters to measure effects on OFC, we also conduct sensitivity analysis on the rest of the fixed parameters in Section EC.2 of the e-companion. We observe that changing number of parking spaces, $m$ , and truck stopping criteria, $τ$ , does not affect OFC significantly. Decreasing truck start window, $k$ , allows trucks at almost all terminal structures to do more trips and increase OFC. Similarly, decreasing station time, $r_{s}$ , increases OFC for almost all terminal structures. Thus, the general insights discussed above stay consistent.

5.2. Prioritizing Interventions

Various interventions might independently (and significantly) increase OFC for different terminal structures. If stakeholders have limited resources, then they will likely prioritize the intervention that increases OFC by the largest magnitude for terminals present under their jurisdiction. Thus, we summarize for each terminal structure which intervention leads to the highest OFC at the improvement threshold in Figure 8. The shape of the marker indicates the intervention that increases OFC the most; the text next to the marker indicates the improvement threshold value of the corresponding intervention parameter; and the shade in gray-scale indicates the magnitude of OFC at the improvement threshold. We ignore hours of service in this analysis since it statistically significantly increases OFC for all terminal structures for all values, and it does not have direct costs attached to it that would be considered in implementation budgets.

Figure 8.

Recommended interventions to prioritize for each terminal with different number of gates, $n_{g}$ , number of bays, $n_{b}$ , and station distances, $d$ . Texts indicate the improvement threshold values of the corresponding intervention parameter.

In general, terminal gate time interventions should be targeted at terminals with a single gate, a large number of bays, and short station distances. Terminal bay time interventions should be targeted at terminals with a small number of bays. Transportation speed improvement increases OFC the most when terminals have long station distances. Increasing fleet size is more beneficial when terminals have multiple gates, a larger number of bays, or serve longer station distances.

Implementing multiple interventions can increase OFC even further. Thus, we study multiple interventions and apply the generalized insights discussed above to the real-life case study of the multifacility downstream gasoline distribution in Florida in the following section.

6. Florida Case Study

In this section we examine the real-life case study of the multifacility downstream fuel distribution system in the state of Florida. Florida experiences tropical storms and hurricanes with high frequency (National Oceanic and Atmospheric Administration, 2023b). In the last two decades, six major hurricanes (category 3-5 on Saffir-Simpson Hurricane Wind Scale) have made landfall in Florida (National Oceanic and Atmospheric Administration, 2023a). Moreover, most of the terminals in Florida are located near the coast, making the downstream gasoline distribution system highly vulnerable to operational and infrastructure capacity constraints in addition to demand surges during hurricanes. Thus, it is especially important to be prepared for extreme disaster scenarios for regions like Florida.

We use data from Internal Revenue Service (2022), Oil Price Information Service (OPIS), and IHS Markit company, as part of our ongoing collaboration with FEMA to support preparedness efforts. The downstream gasoline distribution network of Florida consists of $37$ terminals and $7, 416$ stations, which are grouped geographically into $6$ terminal groups, as shown in Figure 9.

Figure 9.

Map of gasoline terminals and stations by terminal group in and around Florida.

We note that OPIS assign stations to terminal groups based on pricing and do not consider OFC. The station distances, $d$ , are modeled by the empirical distribution of station to terminal group distances measured using Open Source Routing Machine (2017). We refer to the longest station to terminal group distance as the delivery radius. DeCorla-Souza (2018) report the number of bays, $n_{b, t}$ , for most terminals, $t$ . For the rest, we identify $n_{b, t}$ from Google Maps satellite view. We assume that terminals with less than $10$ bays have $1$ gate, and terminals with $10$ or more bays have $2$ gates based on conversations with practitioners:

n_{g, t} = {\begin{cases} 1 & \forall t : n_{b, t} < 10, \\ 2 & \forall t : n_{b, t} \geq 10 \end{cases}

(5)

National Tank Truck Carriers (2019) report that

27, 248

trucks carry gasoline in the US. Moreover, Florida accounts for

6.4 %

of the nation’s taxed gasoline gallons (Federal Highway Administration, 2021). We assume that the national fleet is divided proportional to gasoline consumption among all states, thus estimate the total fleet in Florida as

F = 1744

. Similarly, we assume that

F

is distributed among terminal groups,

i \in I

, proportional to the number of stations that they serve,

| S_{i} |

, since we assume that all stations have equal demand. Thus, baseline fleet size,

f_{i}

, for each terminal group

i

is estimated as:

f_{i} = \frac{F | S_{i} |}{\sum_{i \in I} | S_{i} |} \forall i \in I

(6)

We calculate the baseline steady state OFC,

{\bar{γ}}^{S}

, for each terminal group using Equation 4 and operational parameter values from Table 2. Additionally, from DeCorla-Souza (2018) we know the storage capacity of each terminal group. The terminal group information of Florida is summarized in Table 3.

Table 3.

Details of gasoline terminal groups in and around Florida.

Terminal	Number	Delivery	Fleet Size	Steady State Baseline	Storage
Group	of Bays	Radius (miles)	(trucks)	OFC (MMgal/day)	Capacity (MMgal)
Panhandle	${2, 2, 2, 3}$	$84.60$	$127$	$2.28$	$20.6$
Bainbridge	${4, 5}$	$97.55$	$60$	$1.75$	$25.9$
Jacksonville	${4, 5, 6, 6, 8}$	$92.71$	$226$	$4.91$	$143.8$
Orlando	${3, 10, 13}$	$112.81$	$438$	$4.64$	$81.7$
Tampa	${3, 3, 4, 4, 4, 4,$	$128.97$	$443$	$10.69$	$202.1$
	$5, 6, 7, 8, 10}$
Miami	${2, 3, 3, 3, 3, 4,$	$92.10$	$450$	$9.80$	$165.5$
	$4, 4, 4, 5, 8, 8}$

We then measure the average nonsteady state OFC, ${\bar{γ}}^{N S}$ , over $50$ simulation runs for each terminal group for various scenarios. We start by studying the “Non-Steady State Baseline Scenario” wherein the operational parameter values are as described in Table 2 and the terminal structure parameters are as described in Table 3. Then, we investigate how different levels of disasters decrease the ${\bar{γ}}^{N S}$ and what interventions can restore it back to baseline (Ibarra et al., 2021). The results are summarized in Table 4.

Table 4.

Nonsteady state OFC (MMgal/day) for terminal groups in and around Florida under various scenarios.

	Scenario
						$50 %$ Impact +
Terminal	Nonsteady State	$20 %$	$20 %$ Impact +	$20 %$ Impact +	$50 %$	Interventions +
Group	Baseline	Impact	Interventions	HoS Extension	Impact	HoS Extension
Panhandle	$1.52$	$1.21$	$1.44$	$1.67$	$0.73$	$1.60$
Bainbridge	$1.25$	$0.87$	$1.14$	$1.21$	$0.39$	$1.28$
Jacksonville	$4.06$	$3.22$	$3.81$	$4.44$	$1.72$	$4.25$
Orlando	$3.86$	$3.08$	$3.51$	$4.27$	$2.16$	$3.91$
Tampa	$8.51$	$6.59$	$7.62$	$9.20$	$3.29$	$8.78$
Miami	$7.64$	$6.07$	$6.87$	$8.38$	$3.47$	$7.93$

	Scenario
				Interdiction +
Terminal	Nonsteady State		Interdiction +	All Interventions +
Group	Baseline	Interdiction	All Interventions	HoS Extension
Orlando	$3.86$	$3.96$	$8.35$	$11.04$
Tampa	$8.51$	$0$	$0$	$0$

OFC = operational flow capacity; HoS = hours of service.

Similar to the results in Section 5, we find that for terminal groups in Florida, baseline ${\bar{γ}}^{S}$ (Table 3) is statistically significantly larger than baseline ${\bar{γ}}^{N S}$ (Table 4). Using ${\bar{γ}}^{S}$ overestimates the system’s capacity by $20 %$ – $50 %$ . The largest difference in magnitudes of ${\bar{γ}}^{S}$ and ${\bar{γ}}^{N S}$ is for terminal groups with a larger number of terminals as the individual differences add up. We also note that the storage capacity (Table 3) of these terminal groups is orders of magnitude larger than the baseline ${\bar{γ}}^{N S}$ (Table 4). Thus, we further argue for the usage of nonsteady state OFC as a tighter constraint of the system’s capabilities instead of steady state OFC or storage capacity.

Next, we consider a “ $20 %$ Impact Scenario” where terminal gate and bay times are $20 %$ higher than baseline, and transportation speed and fleet size are $20 %$ lower than baseline. During disasters, terminal employees and truck drivers focus on surviving, leading to slower process times and a smaller fleet. Additionally, damaged terminal and transportation infrastructure inhibit process times and transportation speed. We set gate time to $r_{g} =$ triangular $(7.56, 9.24, 8.4)$ minutes and bay time to $r_{b} =$ triangular $(37.8, 46.2, 42)$ minutes. We also lower the transportation speed to $v =$ triangular $(32.4, 39.6, 36)$ miles/h and the fleet size of each group $i$ is set to be $⌈ 0.8 f_{i} ⌉$ trucks. We observe that Bainbridge suffers the largest decrease of $31 %$ in ${\bar{γ}}^{N S}$ compared to baseline, followed by Tampa that experiences $23 %$ reduction from baseline ${\bar{γ}}^{N S}$ .

Then, we look at the “ $20 %$ Impact + Interventions Scenario.” We execute various interventions (except hours of service intervention), to the terminal groups under $20 %$ disaster impact, and discuss ones that result in a ${\bar{γ}}^{N S}$ close to the baseline ${\bar{γ}}^{N S}$ . Terminals with larger number of gates and bays have larger OFC, thus we focus on the largest terminal (terminal with largest number of bays) of each terminal group. We select the type of intervention to be implemented based on the insights of Section 5. Additionally, we determine the level of change in corresponding intervention parameters from public sources, industry insights, informal visits to terminals in the US, and validation from industry practitioners, as discussed in the previous section.

In Panhandle, the largest terminal has $3$ bays, thus for that terminal we consider bay time intervention such that $r_{b} =$ triangular $(15.75, 19.25, 17.5)$ minutes (DeCorla-Souza, 2018). The resultant ${\bar{γ}}^{N S}$ is $95 %$ of baseline. Similarly, in Orlando we consider gate time intervention for the largest terminal since it has $13$ bays. We set $r_{g} =$ triangular $(3.15, 3.85, 3.5)$ minutes and the resultant ${\bar{γ}}^{N S}$ is $91 %$ of baseline. In contrast, for Jacksonville and Miami, we need to improve both the gate time and bay time of the largest terminal of the group such that resultant ${\bar{γ}}^{N S}$ are $94 %$ and $90 %$ of baseline, respectively. For Bainbridge, we apply bay time intervention and increase transportation speed to $v =$ triangular $(54, 66, 60)$ miles/h (Davenport, 2007) in order get to $91 %$ of baseline ${\bar{γ}}^{N S}$ . Finally, for Tampa we need to apply both gate and bay time intervention, as well as provide $25$ extra trucks to its compromised fleet since its largest terminal has $10$ bays and its delivery radius is the longest in Florida. The resultant ${\bar{γ}}^{N S}$ is $90 %$ of baseline.

However, when there is $20 %$ disaster impact, if we just extend the hours of service (HoS), and implement no other interventions, such that trucks are operational for $h = 18$ hours instead of $14$ , then we find in the “ $20 %$ Impact + HoS Extension Scenario” that baseline ${\bar{γ}}^{N S}$ is exceeded for most terminal groups. Only Bainbridge’s resultant ${\bar{γ}}^{N S}$ is $97 %$ of baseline. Different terminal groups may need different levels of interventions depending on the existing terminal structures and fleet size, thus case specific modeling is important for decision making.

Similarly, we also model a “ $50 %$ Impact Scenario.” We set gate time to $r_{g} =$ triangular $(9.45, 11.55, 10.5)$ minutes and bay time to $r_{b} =$ triangular $(47.25, 57.75, 52.5)$ minutes. We lower the transportation speed to $v =$ triangular $(20.25, 24.75, 22.5)$ miles/h and the fleet size of each group $i$ is set to be $⌈ 0.5 f_{i} ⌉$ trucks. We observe that again, Bainbridge and Tampa have the largest decrease of $69 %$ and $61 %$ from baseline ${\bar{γ}}^{N S}$ , respectively.

Then, we model the “ $50 %$ Impact + Interventions + HoS Extension Scenario.” We identify what levels of interventions can exceed baseline ${\bar{γ}}^{N S}$ given $50 %$ disaster impact. We consider that all terminal groups have extended HoS such that $h = 18$ hours, since it increases $γ^{N S}$ significantly.

In the case of Bainbridge, we exceed baseline ${\bar{γ}}^{N S}$ by restoring transportation speed to $v =$ triangular $(40.5, 49.5, 45)$ miles/h and adding $25$ more trucks to the compromised fleet. With Orlando, we only need to apply gate and bay time interventions to the largest terminal such that $r_{g} =$ triangular $(3.15, 3.85, 3.5)$ minutes and $r_{b} =$ triangular $(15.75, 19.25, 17.5)$ minutes. Panhandle requires gate and bay time interventions for the largest terminal as well as restoration of transportation speed. Whereas Jacksonville requires $20$ more trucks in addition to gate and bay time interventions for the largest terminal and restoration of transportation speed. Miami and Tampa require more effort than the other terminal groups. They require gate and bay time interventions for the two and three largest terminals, respectively, restoration of transportation speed, and additional $30$ and $60$ more trucks in their compromised fleets, respectively. As expected, more intervention effort is required for larger disaster impact.

Next, we consider the “Interdiction Scenario” where we assume that under catastrophic conditions all of Tampa’s terminals are shut down, possibly due to power outages or irreparable damages by hurricanes. For example, the Port of Tampa Bay was shut down for five days during Hurricane Irma in September 2017 (Kolpakov et al., 2022) and for four days during Hurricane Ian in September 2022 (NOAA Office of Response and Restoration, 2023). From the OPIS data, we know that $99 %$ of Tampa’s stations have Orlando as their secondary terminal group. We assume that when Tampa is shut, Orlando has to serve Tampa’s stations in addition to its own. Thus, we only analyze terminal group Orlando in the “Interdiction Scenario” and consequent scenarios. All of the parameters for Orlando are the same as baseline, except the station distances are sampled from the new distribution considering Tampa’s stations as well. In the “Interdiction Scenario” Orlando’s ${\bar{γ}}^{N S}$ is only $32 %$ of the combined baseline ${\bar{γ}}^{N S}$ of Orlando and Tampa. Thus, many interventions may be required to make up for Tampa’s lost capacity.

We model the “Interdiction + All Interventions Scenario” where we apply gate and bay time interventions to all of Orlando’s terminals such that $r_{g} =$ triangular $(3.15, 3.85, 3.5)$ minutes and $r_{b} =$ triangular $(15.75, 19.25, 17.5)$ minutes, we increase the transportation speed to $v =$ triangular $(54, 66, 60)$ miles/h, and assume that half of Tampa’s original fleet is operational and access Orlando terminals thus making Orlando’s total fleet $f = 660$ trucks. We observe that despite improving the whole system, Orlando’s resultant ${\bar{γ}}^{N S}$ is only $67 %$ of the combined baseline ${\bar{γ}}^{N S}$ of Orlando and Tampa.

In addition, if we also extend HoS such that $h = 18$ in the “Interdiction + All Interventions Scenario + HoS Extension,” Orlando’s resultant ${\bar{γ}}^{N S}$ is still just $89 %$ of the combined baseline ${\bar{γ}}^{N S}$ of Orlando and Tampa.

In the above scenarios, we assume that the intervention terminal gate and bay times are achievable, all trucks move with increased speed, all trucks utilize the extended operational hours, and additional trucks are available to be added to the fleet. Thus, these results show an upper bound of OFC that can be achieved. We additionally study the case of the multifacility downstream gasoline distribution in Utah, in Section EC.3 of the e-companion, to demonstrate the applicability of our methodology across geographies.

The case study employs insights from the numerical analysis to identify the best interventions for each terminal group. We observe that singular interventions are sufficient to achieve an adequate increase in OFC under lower disaster impact scenarios. However, in the case of more severe disasters, multiple interventions become essential. We find that restoration of transportation speed is a crucial intervention across all scenarios. More obviously, terminal groups with a larger number of terminals and/or terminals with a larger number of bays require more interventions. It is also noteworthy that in catastrophic scenarios such as the interdiction of a terminal group, implementing all interventions for the entire secondary terminal group may still fall short in fully restoring lost capacity, as the secondary group might not be sizable enough. In such instances, sourcing from an additional terminal group may be more beneficial than enhancing the existing backup terminal group. Thus, demand allocation to multiple terminal groups and optimal intervention allocation should additionally be modeled to prepare for catastrophic disasters.

7. Conclusion

Natural and man-made disasters often cause a spike in demand for fuel as well as impose physical and operational constraints on fuel distribution. The literature on fuel distribution lacks models to rigorously assess the flow to retail gas stations following disaster. Thus, we develop models to measure the downstream fuel OFC, i.e., the volume that can be distributed from bulk storage terminals to retail gas stations by tanker trucks in one day, for steady and nonsteady states. Results contribute to a system level understanding of fuel distribution that continually faces bottlenecks during disaster response. Moreover, this work contributes more generally to scientific understanding of multiserver tandem cyclic queues with time-limited customers found in relief distribution systems, which is often overlooked in disaster management literature. Although we focus on fuel, the approach can be applied to evaluate OFC for distribution of water, food, healthcare supplies, and other humanitarian supplies from staging areas where fleet drivers must perform multiple cycles to meet demand. We additionally make the case of utilizing OFC as a tighter constraint for resource allocation problems in distribution systems, especially during the nonsteady state of disasters.

Policymakers seek evidence regarding the impact of specific preparedness and response interventions when developing plans to improve fuel availability following disaster. In response, we assess the benefits of various operational improvement interventions, as opposed to infrastructure resilience that is common is existing literature. We identify the best interventions for each type of bulk storage terminal structure. We also prioritize bulk storage terminals for each type of intervention when resources are limited. To further demonstrate practical impact, we present a case study drawn from application of our methodology to support US federal disaster preparedness. As the downstream fuel distribution system from bulk storage terminals to retail gas stations is similar across regions, our methodology and insights are generalizable to improve fuel management during disasters not only within the US but also internationally in countries with comparable distribution systems. In addition, we discuss resource requirements and implementation details for each intervention.

We note a few limitations in our research. First, we do not explicitly consider financial costs of interventions in our analysis. Obtaining proprietary data from the private sector or empirical data to precisely estimate financial and political costs for the public sector has not been possible. To avoid providing potentially inaccurate and misleading analyses, we do not assign explicit values to these costs. However, we hope that our extensive exploration and quantification of the benefits of interventions, in terms of increase in OFC, is the foundation to encourage cost-benefit analysis. Second, inability to obtain data also leads us to make approximations based on conversations with industry practitioners for baseline values and distributions of system parameters. Third, we assume all retail gas stations to be equal as we first focus on measuring the upper limit of the OFC. Lastly, we conduct scenario planning in our case study but do not consider the probability of the scenarios.

This study suggests further research opportunities. Since the private sector owns and operates most fuel infrastructure, intervention requires engagement between private and public sector to build trust and a basic understanding of each other’s context. Public-private partnership to facilitate data sharing, parameter validation, articulation of intervention costs (financial and political), and mutual cost-benefit analysis would strengthen evidence to justify early action. An important extension is to leverage realistic nonsteady state OFC in further optimizing interventions across bulk storage terminals based on disaster conditions and demand surge probability. Routing decisions for tanker trucks based on disaster scenario and consequent retail gas station demand should be considered as well. Finally, application of the modeling approach to food, water, and other essential goods would not only improve humanitarian outcomes but also further scientific understanding of distribution systems during disasters.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478241231876 - Supplemental material for Modeling Operational Flow Capacity and Evaluating Disaster Interventions for Fuel Distribution

Supplemental material, sj-pdf-1-pao-10.1177_10591478241231876 for Modeling Operational Flow Capacity and Evaluating Disaster Interventions for Fuel Distribution by Shraddha Rana, Timothy Russell, Justin J Boutilier and Jarrod Goentzel in Production and Operations Management

Footnotes

Acknowledgment

The authors would like to thank Connor Makowski at the MIT Computational Analytics, Visualization & Education (CAVE) Lab for his efforts in developing the discrete event simulation tool used in our analysis, and Lauren Finegan at the MIT Humanitarian Supply Chain Lab for her work in compiling the list of interventions discussed in this paper.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received the following financial support for the research, authorship and/or publication of this article: This work was supported by the US Department of Homeland Security, Federal Emergency Management Agency [grant number 70FB8018D0000037].

ORCID iDs

Shraddha Rana

Timothy Russell

Supplemental Material

Supplemental material for this article is available online ().

How to cite this article

Rana S, Russell T, Boutilier JJ and Goentzel J (2024) Modeling OFC and Evaluating Disaster Interventions for Fuel Distribution. Production and Operations Management 33(3): 682–700.

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