The performance of priority dispatching rules in a complex job shop: A study on the Upper Mississippi River

International Journal of Production Economics 216 (2019) 154–172

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International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe

The performance of priority dispatching rules in a complex job shop: A study on the Upper Mississippi River

T

Kevin D. Sweeneya,∗, Donald C. Sweeney IIb, James F. Campbellb a b

College of Business Administration, Sam Houston State University, United States College of Business Administration, University of Missouri, St. Louis, United States

ARTICLE INFO

ABSTRACT

Keywords: Complex dynamic job shop Priority dispatching rules Scheduling Real-world application Discrete event simulation

Many studies have derived and tested dynamic job shop priority dispatching rules using discrete event simulation models in the context of idealized job shop experimental designs. This paper extends research on evaluating priority dispatching rules in a completely reactive dynamic job shop by testing the performance of eight selected rules in a simulation model of a complex and real dynamic job shop: the Upper Mississippi River Inland Navigation Transportation System (UMR). The UMR incorporates many real-world complexities such as sequence dependent and seasonally varying stochastic job processing times, both capacitated and un-capacitated servers, and heterogeneous jobs with seasonally varying, interdependent stochastic arrivals that can balk (optout) at using the system in response to anticipated poor levels of service. Employing two related but different metrics, mean flow times and the total value (“utility”) of jobs processed, the results show that rules that incorporate increasingly more systemic information generally perform better as system congestion increases, particularly when balking is not allowed. However, this is not the case when customer balking is allowed, particularly for value-based priority dispatching rules. This demonstrates that the balking decision of system users has a large impact on the performance of the rules and the expected utility (value) generated by the system, particularly at high levels of system congestion.

1. Introduction There has been much literature devoted to constructing and analyzing the performance of scheduling mechanisms for dynamic job shops (Dominic et al., 2004; Ho et al., 2007; Sels et al., 2012). Dynamic job shops are characterized by the random arrival of jobs to a series of capacitated machines, or servers, for processing (Holthaus and Rajendran, 1997). Because of the random arrival of jobs, dynamic job shops typically employ reactive scheduling, where no schedule exists prior to the arrival of jobs (Ouelhadj and Petrovic, 2009). In this environment, decisions are made in real time based on priority dispatching rules where job priorities are updated dynamically and the job with the highest priority is assigned to a server when the server becomes available (Ouelhadj and Petrovic, 2009). The dynamic job shop literature has devised and tested numerous priority dispatching rules (Sels et al., 2012; Lu and Romanowski, 2013; Varga and Simon, 2014; Xanthopoulos et al., 2016). These rules may be usefully categorized along a continuum from myopic to global. Myopic rules focus on a single server, and as a result are intuitive and easy to implement, while global rules incorporate additional information from

∗

the state of jobs across the system (beyond just a single server) and are therefore more complex and often difficult to implement (Ouelhadj and Petrovic, 2009). These priority dispatching rules have been devised for, and evaluated on, a variety of performance metrics: minimization of the mean flow time of jobs, minimization of flow time variation, minimization of expected costs, due-date criteria, tardiness criteria and combination (hybrid) metrics, among others (Haupt, 1989; Kunnathur et al., 2004; Ouelhadj and Petrovic, 2009; Sels et al., 2012). However, most of these rules have only been tested in idealized simulation models of hypothetical dynamic job shops, which often impose some very restrictive assumptions such as: sequence independent job processing times, independent job arrivals and homogeneous jobs. Most real dynamic job shops are significantly more complex than the typical idealized simulated job shop. For example, they can comprise multiple servers with sequence and job dependent processing times, some servers may be capacitated while others are not, there may be systematic variation (seasonality) in both job arrival rates and processing times, jobs may be heterogeneous (have different operational requirements and values), and servers can malfunction or breakdown. Furthermore, there may even exist alternatives to using the

Corresponding author. E-mail address: [email protected] (K.D. Sweeney).

https://doi.org/10.1016/j.ijpe.2019.04.024 Received 8 August 2018; Received in revised form 19 April 2019; Accepted 22 April 2019 Available online 28 April 2019 0925-5273/ © 2019 Elsevier B.V. All rights reserved.

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job shop, which can lead to balking on the part of a specific job if a minimum threshold of expected service or value is not met. Because of the potential interactions of these real job shop possibilities, it is not clear that global scheduling rules will perform better than myopic scheduling rules for very complex real job shops, even if global scheduling rules have demonstrated superiority in simple idealized dynamic job shop models (Holthaus and Rajendran, 1997; Rajendran and Holthus, 1999; Dominic et al., 2004; Kunnathur et al., 2004; Xanthopoulos et al., 2016). This research extends previous work by testing eight priority dispatching rules in a simulation model of a complex and real job shop that features: seasonal, stochastic and interdependent job arrivals; sequence dependent job processing times at both capacitated and un-capacitated servers; random server breakdowns; and heterogeneous jobs that may balk in response to lower than desired expected service levels. This simulation model is derived from the real-world vessel traffic (jobs) that travels up and down the Upper Mississippi River (UMR) via a sequence of locks (capacitated servers) and river pools (uncapacitated servers). We derive six priority dispatching rules based on existing literature that incrementally incorporate increasingly more information regarding the state of the system when determining job priorities, and we also introduce two additional rules that prioritize jobs based on the value of each job. We evaluate the selected priority dispatching rules using a validated, verified and calibrated simulation model of a real-world transportation system, the Upper Mississippi River Inland Navigation Transportation System, as a dynamic job shop. The performance of some limited priority dispatching rules for the UMR has been examined in prior research (see Nauss, 2008; Smith et al., 2009; Smith et al., 2011), however previous models have only looked at the scheduling of a single lock at a time using simple, myopic priority rules. This research extends previous work on the UMR by comparing the efficacy of global priority rules that incorporate successively more systemic information to the simple, myopic priority rules. Furthermore, we investigate the important role that balking (opting to not use the job shop) in response to poor expected performance has on the efficacy of the priority dispatching rules by examining the priority rules under both balking and no balking conditions. To better evaluate performance, we report the results of all the priority rules using two different (but related) metrics: the total realized value produced in processing jobs and the mean flow time of jobs. In summary, the contributions of this research are four-fold:

information systems (Li et al., 2016), no research has examined the performance of these well-known priority rules in systems as complex as the UMR. Because the real-world job shop we examine includes the multiple complexities of the UMR, our model represents quite a significant extension from the idealized job shops most often considered in the literature. 2. Literature review 2.1. Dynamic job shop dispatching rules The dynamic job shop literature has proposed and evaluated many priority dispatching rules in completely reactive dynamic job shops (Ouelhadj and Petrovic, 2009). There is no single best rule for scheduling a dynamic job shop: the choice of dispatching rule should be based on which performance metrics are most important to the scheduling party (Rajendran and Holthaus, 1999; Ouelhadj and Petrovic, 2009). Common metrics that have been used to evaluate performance include mean the flow time of jobs, the variance of flow time, minimum and maximum flow time, mean tardiness, maximum tardiness and tardiness variance. Myopic dispatching rules assign priorities to jobs at a server based only on relevant criteria at that single server. Examples include FIFO (first in first out), LIFO (last in first out), LPT (longest processing time), SS (shortest set-up time) and SPT (shortest processing time). The most robust performing of the myopic processing-time based dispatching rules is SPT, which gives highest priority to the job that will take the shortest amount of time to process at the current server. While SPT has been shown to perform well under many operating conditions for multiple metrics in a dynamic job shop (Rochette and Sadowski, 1976; Dominic et al., 2004; Sels et al., 2012), it considers only the local condition of each server when assigning priority to jobs. It has been suggested that priority dispatching rules that utilize additional criteria and more system information (e.g., information at other servers) could significantly improve dynamic job shop performance compared to their myopic counterparts (Ouelhadj and Petrovic, 2009). For example, rules that consider the state of other servers in the system such as WINQ (total work in the next queue of a job) could allow for better work flow than rules that consider only the state of a single server. The reason for this is that jobs that complete more quickly than others at a local server (SPT) may warrant a lower overall priority if the remaining servers in their itinerary are congested and have significant queues. If other jobs have no remaining servers on their itinerary or will visit servers with significantly shorter queues, selecting the locally faster job (SPT) could possibly decrease system efficiency at the expense of increasing local server efficiency. Other more complex rules have been developed which take into account the status of the jobs themselves. These rules can be relatively straightforward and global, such as LWKR_FIFO (least total work remaining – first in first out, which prioritizes jobs based on the least work remaining rule first, then uses first in first out to select from jobs that have the same amount of work remaining), or they can be very complex global rules, such as those derived using genetic algorithms and simulating annealing techniques in determining priorities (Ho et al., 2007). Sels et al. (2012) identified 30 priority rules based on existing literature and evaluated their performance on flow time and tardiness metrics in different static and dynamic job shop environments using idealized settings. They found that combinations of LWKR, WINQ and local processing time-based rules such as SPT performed well in their dynamic job shop model with respect to performance on flow time related metrics. We implement and modify selected priority dispatching rules from Sels et al. (2012) to derive six priority scheduling rules, and we examine their performance in a more realistic scenario that violates many assumptions in idealized job shops. We also derive and present two new priority dispatching rules that incorporate the value of a potential job as a decision

1) We extend earlier research investigating the performance of priority dispatching rules in hypothetical and idealized dynamic job shop models to the rich environment of a very complex real-world job shop; 2) We develop two new priority rules based on the total expected value of jobs to be processed and examine the effectiveness of these and other global and myopic priority rules using both mean flow time and total realized value of jobs completed as metrics in a real dynamic job shop environment where customers can balk if they do not expect to receive sufficient value from using the job shop; 3) We document the important impact of balking on the performance of the priority dispatching rules on the metrics of mean flow time and total realized value of all jobs processed; and 4) We extend previous research on the UMR by deriving and investigating the performance of global scheduling rules relative to previously investigated myopic scheduling rules. By examining a range of priority dispatching rules in this realistic and complex dynamic job shop, we illustrate how the complexities of a more realistic system alter the effectiveness of priority dispatching rules. While there has been research that has looked at the implications of using priority scheduling rules in some real world systems, particularly in semi-conductor manufacturing (Mittler and Schoemig, 2000; Mönch et al., 2013) and scheduling real time processes in managing 155

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criterion: TOTUTIL and UTILWRK. These two priority rules are adaptations of rules that have been examined in previous literature, where TOTUTIL prioritizes jobs based on their total value contribution to the system, while UTILWRK prioritizes jobs based on the value they contribute normalized by job processing time. We evaluate the performance of all eight rules using two related metrics: (i) the mean flow time of jobs in the system, and (ii) the total value (measured in $‘s) that UMR users derive from using the system. In the context of the UMR, the total realized system value is a natural metric that measures the utility gained by UMR shippers from using the system, while mean flow time is a commonly used metric in job shop studies as well as in previous studies on priority dispatching rules on the UMR (Nauss, 2008; Smith et al., 2009).

There is a stream of research addressing traffic congestion on this segment of the UMR using discrete event simulation models. Sweeney (2004) presented the first discrete event simulation model designed to accurately replicate flows in the congested segment of the river by simulating the pool transit and locking operations of the various vessels that use this segment of the UMR. Several studies have examined the efficiencies to be gained (typically measured by mean system transit times for tows using the river) by applying differing myopic priority dispatch rules at the individual locks on the system, such as SPT or an equity modified version of SPT (Nauss, 2008; Smith et al., 2009, 2011). Nauss (2008) developed an optimization model to find the best method of clearing a queue at a single lock on the UMR given that a significant queue had arisen, usually as the result of a lock shutdown or repair. Smith et al. (2009) examined the performance of a single lock under a series of myopic priority rules and compared the performance of those priority rules to other potential solutions for addressing queues at individual locks, such as the use of “helper boats” to facilitate the locking process or expansion of the locks to accommodate larger vessels. Smith et al. (2011) used mixed integer programming to develop scheduling rules for vessels at a single lock by incorporating the sequence dependent processing times for each vessel. Finally, Sweeney et al. (2014) looked at myopic priority rules including a shipper's random utility function which allowed system users to “opt-out” of using the UMR in favor of alternatives (such as truck or rail transport) in reaction to expected queue and transit times. These simulations were used to evaluate myopic scheduling rules (such as FIFO or SPT at individual locks), rather than more global scheduling rules that incorporate other information such as queues at related locks or the total work content of jobs in lock queues. This research bridges that gap by investigating the performance of both myopic and global scheduling rules with respect to mean flow time and the value of jobs processed, as well as extending previous research by developing and testing priority rules based on total system value on the UMR.

2.2. The UMR as a job shop The UMR is one of the most heavily utilized shipping lanes for bulk products in the United States, moving 115.7 million tons of freight in 2016 (U.S. Army Corps of Engineers, 2016). It stretches over 1200 miles from Minneapolis, MN to the confluence of the Ohio River with the Mississippi River. The UMR is composed of a series of connected navigation “pools” with sufficient depths to permit commercial navigation by large flotillas of barges aggregated into tows. The pools are separated by dams that maintain the pool depth with vessel passage through the dams provided by lock chambers in the dams. The pools, locks and dams are numbered sequentially beginning with 1 near Minneapolis and having 25 near St. Louis; the pool upstream of a lock and dam has the same number as the lock and dam. A commercial tow operating on the UMR consists of a towboat (about 200 feet long) pushing up to 16 (unpowered) barges (each up to 195 feet long) latched together in a single unit up to 1200 feet long and 105 feet wide. The UMR also has non-commercial local traffic such as recreational vessels. Thus, a vessel in this context can refer to a commercial tow or a recreational vessel (such as privately-owned speedboat). The locks and dams are operated and maintained by the U.S. Army Corps of Engineers. Periodic congestion occurs on a heavily utilized portion of the UMR that extends from southern Iowa to mid-Missouri (approximately 102 river miles) due to the relatively small size of the five navigation locks in this segment of the system. UMR Locks 20 through 25 (there is no Lock 23) are only capable of processing 600foot-long tows in a single lock operation. The adjacent locks to the north and south are larger and are capable of processing tows that are up to 1200 feet long in a single operation. Because of economies of scale, most tows that use the river are longer than 600 feet and thus must complete a “double” lockage (processing the tow through the lock in two segmented parts) to pass through each of locks 20–25. Each double lockage can take over 2 h, compared to 45 min or less for a single lockage (for vessels less than 600 feet long), and these double lockages contribute to significant waits and congestion at these locks, particularly during the busy summer months. Over all five locks, approximately 63.7 percent of the 18,700 lockages are double lockages, while the rest are single lockages for smaller tows or recreational vessels. In viewing the UMR as a dynamic job shop, we define the following relationships: (i) a job is a vessel trip characterized by a vessel class, origination pool and destination pool; (ii) a processing operation is the movement of a job through a pool or a lock with the first operation beginning at the origination pool and the final operation the termination of the job at the destination pool; (iii) the six pools are un-capacitated servers capable of processing multiple jobs simultaneously as required; and (iv) the five locks are capacitated servers with queues as required that can only process a single job at any point in time. Fig. 1 presents a visual representation of the 11-server job shop, with the locks represented by rectangles (5 total) and the pools represented by the arrows between locks and on the left and right side of lock 20 and 25, respectively (6 total). For more information on the UMR as a complex dynamic job shop, as well as more specific information with respect to the heterogeneity of jobs and system seasonality, see Appendix A.

2.3. Customer balking in job shops Customer sensitivity to wait times has been examined in depth for service systems (Ibrahim, 2018). This literature includes investigations into the effect of delay announcements on customer behavior (Whitt, 1999; Ibrahim and Whitt, 2009), as well as modeling system users as utility maximizing decision makers (Naor, 1969; Hassin and Haviv, 1995, 2003). The main issue tackled in this research stream is that system users impose negative externalities on other system users by extending the wait times and queues faced by other users. Most of the literature on balking behaviors has examined a plethora of issues that could influence customer behavior, such as whether queue lengths are observable (Naor, 1969), unobservable (Edelson and Hilderbrand, 1975), or whether there is an optimal amount of queue information to share with users in certain circumstances (Hassin, 1986; Duenyas and Hopp, 1995; Altman and Jimenez, 2013). However, only a few papers have looked at how the information given to customers impacts the throughput of a system (Chen and Frank, 2004; Shone et al., 2013) and only one article has looked at situations where customers share information with each other on the state of the system as they make their balking decision (Hu et al., 2017). While the mechanism by which system users receive their information on the state of the UMR system is most similar to the Hu et al. (2017) scenario, none of these papers have leveraged a random choice utility model to replicate the decision making process users make as they consider using the system, and no research on customer balking has investigated the impact of customer balking on the throughput and value of a system as complex as the UMR. The next section describes the specifics of the customer balking process for shippers on the UMR. 3. Modeling customer balking behavior using a random utility model Most shippers using the UMR to move commercial products have alternatives to doing so, such as employing a different mode of 156

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Fig. 1. Dynamic job shop schematic model.

transportation to travel to the same market (such as rail or truck) or altering the final destination of their products to markets that do not involve barge transportation (Train and Wilson, 2007). The presence of these alternatives adds additional complexity to the dynamics of the system as shippers can balk or “opt-out” if they do not expect to receive sufficient value (utility) for shipping their products on the UMR. The utility each potential UMR barge shipment can expect to receive by opting-in at any point in time depends on the levels of congestion in the system, as the time and cost to complete a shipment is affected by the dynamic operating conditions of the system. To effectively handle balking, we need to model the value that system users receive from the UMR system, as this value dictates the balking behavior of potential jobs. The value a potential job expects to obtain from using the system (the opt-in utility) can be expressed as the sum of (i) a deterministic component that depends on the vessel class and expected flow time of the job from its origination pool to its destination pool and (ii) a random component associated with other unobservable factors that represents the variability in utility associated with individual jobs. In the literature, this is known as a random utility model (McFadden, 1974). The expected opt-in utility of a job is inversely related to the expected flow time of the job. More formally, the expected opt-in utility of potential job i at time t of class c is a function of the class and the expected flow time, plus a random component in, i :

OptinUtilityi = f (c, expected flow time (O, D , c, t )) +

4. Priority dispatching rules The U.S. Army Corps of Engineers collects some data on commercial vessel activities at locks, but there is no available information regarding vessel activities within the pools of the UMR system. Thus, we can only observe individual vessel arrivals or departures at the locks. Further, since vessel arrivals in the system are stochastic and dynamic with due dates and other job requirements not publicly available, we restrict our investigation of priority dispatching rules to completely reactive priority dispatching rules for processing vessels. These rules determine the next vessel to be dispatched for processing from the two-way queue of vessels (upstream and downstream) at each lock. While there are two physical queues at each lock, each lock can choose to process a vessel from either queue, effectively giving each lock one queue. Vessel priorities are dynamically updated whenever a vessel arrives at or departs from any lock in the system. It is these events at the capacitated servers (locks) that drive our completely reactive rescheduling. We first examine the implementation of six priority dispatching rules adapted from the dynamic job shop scheduling literature on the performance of this part of the UMR system. These six rules were developed to evaluate the impact on system performance of moving from relatively simple rules (locally based decision rules) to more complex system-based rules (globally based decision rules). The rules are adapted from the priority dispatching rules proposed in Holthaus and Rajendran (1997) and Sels et al. (2012) that have been shown to be robust with respect to their performance under multiple performance metrics, particularly flow time related metrics. Let.

in, i

where the expected flow time depends on the origin pool O, destination pool D, class and time. This expected opt-in utility is compared to an expected opt-out utility which is also composed of the sum of (i) a deterministic component that represents the utility the job would receive if it opted-out of using the UMR system (i.e., balked and did not enter the job shop), and (ii) a random component that represents the variability of utility associated with the opt-out option. The opt-out utility is not affected by UMR flow times. The expected opt-out utility of potential job i with random component out , i may be expressed as:

OptoutUtilityi = f (c ) +

out , i

Τ

represent the time when a dispatching decision must be made to select a job for processing from any lock queue;

tijτ

represent the expected processing time for lock operation j of job i at time τ; (in our context the expected lockage time for vessel i at lock j at time τ). This term embodies the sequence dependent and seasonally varying expected lockage times at lock j and is defined only when job i is in the queue at lock j at time τ; represent the total expected work content (the sum of expected processing times) of all vessels at time τ in the queue at the kth lock in the ordered sequence of locks required to complete the full itinerary of job i subsequent to the lock in which the vessel is currently in queue, defined for k = 1,2,3,4 as there can be at most four remaining locks in a vessels' subsequent itinerary. We define Wikτ = 0 if there is no kth lockage in the subsequent itinerary of job i. In our context, if the kth lockage in job i's remaining itinerary is lock m, then Wikτ represents the total expected minimum work content in the queue of lock m at time τ; and represent the priority value assigned to job i at time τ. Since a job can only be in a single lock queue at any time, Ziτ will have a single well-defined priority value.

Wikτ

where f(c) represents the (assumed constant per class) deterministic component of the opt-out utility of job class c. The job opts-in to use the system when the expected opt-in utility is greater than the expected opt-out utility, and balks when the reverse is true. Representing the random components of the opt-in and opt-out utilities with standard Extreme Value Type I distributions yields a standard logit model to determine the probability of each potential job balking or not that depends on the expected flow time of the job. Appendix B provides details on the specifics of the random utility model used in this simulation, including the data used to derive it.

Ziτ

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Rule 1 (FIFO): This priority rule is a simple application of first in first out. If a job is completed and the queue at a server (lock) is empty, then the first arriving vessel is selected for immediate processing. If a job is completed and there are jobs in queue, then the job with the longest wait time in that queue is selected for processing. This scheduling mechanism relies only on local information to select a job for processing at each lock and serves as our benchmark. (A slightly modified form of first in first out is currently used on the UMR under normal operating conditions, where the modification gives some priority to recreational vehicles over commercial tows.) Rule 2 (SPT): This priority rule simply selects the job for processing at each lock from its queue that has the shortest expected, sequence dependent processing time at that lock at time τ. Mathematically, the priority value is defined as Zi = tij and the job with the lowest Ziτ is selected from the queue for processing at time τ. Ties are broken by selecting the job with the earliest arrival time at the lock. This scheduling mechanism relies only on local lock information to select a job for processing at each lock. Rule 3 (SPTWIN1Q): This priority rule essentially incorporates SPT at each lock augmented with additional information about the expected amount of work remaining in the queue at the next lock of each job's remaining itinerary. The priority value at lock j for job i is defined as Zi = tij + Wi1 and the job with the lowest Ziτ value is selected for processing at lock j at time τ. Ties are broken by selecting the job with the earliest arrival time at the lock. This rule considers the state of the next lock in the itinerary of each job, in addition to the local lockage time for the job when prioritizing vessels for service at each lock. With this rule, the job at each lock requiring the smallest amount of server time at the current lock plus the expected service time for all the jobs in queue at the next lock of that job (looking only one lock ahead) is given priority for processing. This scheduling rule does not look ahead to future arrivals at the queues after time τ. Rule 4 (SPTWIN2Q): This priority rule takes into account the work in other lock queues within two locks of the local lock (one more lock upstream or downstream than SPTWIN1Q). In effect, it considers the state of the system up to two locks away if a job's remaining itinerary includes those locks. The priority value at lock j is defined as Zi = tij + Wi1 + Wi2 and the job with the lowest Ziτ value is selected for processing at lock j at time τ. Ties are broken by selecting the job with the earliest arrival time at the lock. With this rule, the job at each lock requiring the smallest amount of server time at the current lock plus the expected server time for all the jobs in queue at the next two locks of job (or one lock if that lock is the last in its itinerary) is given priority for processing. Rule 5 (SPTWIN3Q): This priority rule considers the state of the system up to three locks away if the job's itinerary includes those locks (one more lock than SPTWIN2Q). The priority rule is defined as 3 Zi = tij + k = 1 Wik and the job with the lowest Ziτ value is selected for processing at lock j at time τ. Ties are broken by selecting the job with the earliest arrival time at the lock. With this rule, the job at each lock requiring the smallest amount of server time at the current lock plus the expected server time for all the jobs in queue at up to the next three locks of that job (or two locks/one lock if that lock is the last in their itinerary) is given priority for processing. Rule 6 (SPTWIN4Q): This priority rule takes into account the total work in queue at all locks to be visited by the job as well as the processing time of the job at the local lock. This scheduling rule utilizes the most system information in order to select a job for processing. The 4 priority rule is defined as Zi = tij + k = 1 Wik and the job with the lowest Ziτ value is selected for processing at time τ. Ties are broken by selecting the job with the earliest arrival time at the lock. With this rule, the job at each lock requiring the smallest amount of server time at the current lock plus the expected server time for all the jobs in queue at the next four locks (or three locks/two locks/one lock if that lock is the last in their itinerary) is given priority for processing.

The motivation behind the selection of these scheduling rules is to explore how including incremental information about the state of the capacitated servers farther away (in time) in priority dispatching rules impacts system performance. While FIFO and SPT both utilize only local lock information to dynamically prioritize jobs, the other priority rules incorporate increasingly more systemic information to augment the job priority decision made at each lock. We also include in our investigation two new priority dispatching rules that prioritize jobs in lock queues based on the total utility each job expected to receive when it opted in to use the system. We term these dispatching rules TOTUTIL and UTILWRK. Rule 7 (TOTUTIL): This prioritization rule gives priority to selecting jobs from the lock queues that expect to receive the most utility (value) from using the system. The priority rule is defined as Ziτ = the expected utility of job i when it first opted-in to use the system, and the job with the greatest value is selected. Ties are broken by selecting the job with the earliest arrival time at the lock. This rule is designed to expedite the movement of jobs through the system that have the greatest expected value for using the system and therefore a significant direct impact on the total value received by system users. Rule 8 (UTILWRK): This prioritization rule gives priority to selecting jobs from the lock queues that expect to receive the most utility (value) from using the system per unit of total expected processing time required to process the job at the capacitated servers. The priority rule is defined as Ziτ = the expected utility of job i when it first opted-in to use the system divided by the total expected processing time required at all capacitated servers on the vessel's itinerary. The job with the greatest value is then selected for processing. Ties are broken by selecting the job with the earliest arrival time at the lock. This rule is designed to expedite the movement of jobs through the system that have the greatest expected value for using the system per unit of capacitated server time required to process the job and incorporates aspects of both the value of jobs and their processing time requirements. 5. The UMR simulation model In this research, we leverage a simulation model of the UMR that embeds a random utility model within a discrete event simulation model of this segment of the UMR system (Sweeney et al., 2014). The utility model explicitly incorporates the incentives of potential users to opt-in to using the river and also provides a natural metric for evaluating the performance of the system: the total utility (value) received by users that opt-in to use the system. The simulation model is constructed using Alion Science and Technology's Micro Saint Sharp® Version 3.6 discrete event simulation software (Alion Science and Technology, 2011). Each execution of the discrete event simulation model replicates one calendar year of the operation of this segment of the UMR and tracks the movements of jobs (vessels) that opt-in to use the system as they proceed from their origination pool (first server) to their destination pool (last server). It incorporates the systematic and stochastic variation in potential job arrivals, pool travel times and sequence dependent lockage times throughout the entire year with one of the eight priority dispatch rules governing the disposition of vessels (jobs) from lock queues. The model also calculates the “realized” utility of each job that optsin to use the system as the sum of (i) a flow time related component of utility using the simulated flow time of that job and (ii) the randomly drawn component of utility when the job was created. More formally, the realized utility of job i may be expressed as:

RealizedUtilityi = f (c, simulatedflow timei ) +

in, i

Essentially, the simulated flow time of a job through the system is recorded at the end of the job's transit through the system and is then used in the value function to compute the value that the job actually received from using the system. The simulated flow time of the most 158

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combination of priority dispatching rule and exogenous arrival rate of potential jobs, we perform 200 annual replications and analyze the average total realized utility for UMR users across the exogenous arrival rates and priority dispatching rules. We also examine and report mean flow times for jobs to provide comparisons with metrics in the existing literature. We then examine the expected performance of the system using the realized utility and flow time metrics employing a robust ANOVA based technique.

Table 1 Mean UMR total realized utility in dollars ($2007 millions) by priority rule. Exogenous Job Arrival Rate

Priority Rule

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL UTILWRK

1.0

1.1

1.2

1.3

1.4

1.5

Total Utility

Total Utility

Total Utility

Total Utility

Total Utility

Total Utility

Mean

Mean

Mean

Mean

Mean

Mean

2826 2825 2834 2832 2841 2839 2830 2825

3064 3067 3083 3085 3082 3087 3089 3069

3228 3249 3268 3270 3269 3273 3310 3278

3306 3349 3367 3371 3373 3374 3444 3364

3338 3402 3425 3435 3435 3436 3461 3336

3359 3451 3473 3479 3482 3482 3405 3260

7. Results and analysis Table 1 presents the mean total realized utilities (measured in millions of year 2007 dollars) of all tows that opt-in for system use (i.e. do not balk) for each of the eight dynamic priority dispatching rules and six exogenous arrival rates of potential jobs. Clearly evident in Table 1 is the increase, at a decreasing rate, of expected total realized utilities for increased exogenous arrival rates across the priority rules with mean annual total realized utilities for UMR users’ ranging from approximately $2.8 billion at baseline arrival rates up to nearly $3.5 billion with exogenous arrival rates increased 50 percent. We note, however, the decrease in mean annual total realized utilities for UMR users associated with the TOTUTIL rule as exogenous arrival rates increase from 40 to 50 percent and the UTILWK rule with exogenous arrival rates above 30 percent. These decreases are the direct result of the greater levels of balking behavior that result from the very large increases in flow times that occur under these two priority scheduling rules due to the very high levels of congestion; this phenomenon is discussed in more detail in section 7.2. Also apparent in Table 1 is the increase in realized utilities associated with the more global non-utilitybased priority rules for each exogenous arrival rate of potential shipments, though the relationship between the more global priority rules is less regular than that between the more myopic priority rules. Table 2 presents the mean flow times for the eight priority rules for the six different levels of exogenous job arrival rates. We note the relatively poor performance of UTILWRK and TOTUTIL across the six exogenous arrival rates and the relatively good performance of SPT in minimizing mean flow times, especially under the circumstances of relatively high capacitated server loads. These mean flow time results are consistent with existing literature. Table 3 presents the average proportion of potential users (tows) that opt-in for UMR shipment from the 200 annual replications of each combination of priority dispatching rule and level of exogenous mean arrival rate. Clearly evident in Table 3 is the decrease in the proportion of potential users that opt-in to use the UMR as exogenous arrival rates increase. This decreasing relationship holds across all priority rules. Interesting to note is that at baseline levels of exogenous job arrivals the percentage of potential users that opt-in to use the UMR system is fairly consistent for all dispatching rules at near 67 percent. However, as exogenous arrival rates increase and congestion on the UMR builds, the

recently completed job of the same class, origin, and destination pool is then employed as the expected flow time of the opt-in utility for the next potential job with the same vessel class, origination pool and destination pool. Sweeney et al. (2014) describe in detail the validation, verification, and calibration of the integrated simulation model. The eight proposed priority scheduling rules are implemented in the simulation model by appropriately altering the manner in which vessels are dispatched from lock queues. Each priority rule is verified that it performs as intended through detailed inspection of the selection of vessels for processing from lock queues. 6. Experimental design In previous research in less complex job shop environments, the effectiveness of priority dispatching rules has been shown to be sensitive to the metrics employed for system performance, as well as the loads faced by the capacitated servers (Holthaus and Rajendran, 1997; Ouelhadj and Petrovic, 2009; Sels et al., 2012). This research investigates the performance of these priority dispatching rules using both the standard mean flow time metric and the total realized value performance metric for UMR users (measured in $). We explore the performance of these rules at different loads on the servers by increasing the “exogenous” mean arrival rates of potential jobs (shipments utilizing tows) by 10, 20, 30, 40 and 50 percent relative to baseline arrival rates. This is an exogenous job arrival rate because some tows can balk due to long expected transit times from congestion. Thus, a 50% increase in the exogenous arrival rate may mean less than a 50% increase in use of the job shop, as the increased traffic level can create enough congestion to cause increased balking. For each Table 2 Mean UMR flow times (hours) by priority rule. Exogenous Job Arrival Rate

Priority Rule

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL UTILWRK

1.0

1.1

1.2

1.3

1.4

1.5

Flow Time

Flow Time

Flow Time

Flow Time

Flow Time

Flow Time

Mean

Mean

Mean

Mean

Mean

Mean

23.85 22.86 22.66 22.86 22.96 23.05 25.01 23.79

31.16 28.75 28.95 29.27 29.25 29.71 33.01 29.98

47.97 42.30 44.40 45.92 45.88 46.40 59.09 47.50

81.58 67.42 73.40 75.09 76.16 75.27 144.13 95.72

120.97 96.44 105.12 106.98 109.65 108.91 326.37 198.26

156.49 121.54 134.00 139.77 142.83 146.34 526.72 350.97

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dispatching rules with respect to the mean total system value and mean flow time metrics we analyze the experimental results as independent one-way ANOVA models, with one model for each metric associated with each of the exogenous job arrival rates. We analyze the six exogenous arrival rates independently to increase the likelihood of not violating important assumptions underlying ANOVA analysis such as the approximate normality and similar variances of sub-population distributions of the target variable (Howell, 2010 pages 320–321). All statistical tests are conducted using SPSS 25.0 software (IBM Corp., 2017). We first check that the realized UMR utilities and flow times are approximately normally distributed. We verified that this is the case for our data using the Shapiro-Wilk's test (Shapiro and Wilk, 1965) finding that at the p < 0.05 level we accept the approximate normal distributions of the sub-populations of the utilities and flow times. We next employ the Levene test statistic (Levene, 1960) to test the homogeneity of the variance of realized UMR utilities and job flow times across the eight scheduling priority rules within each of the six exogenous job arrival rate models. As the Levine test statistic is significant (p < 0.05) for some exogenous arrival rate models, we employ an ANOVA analysis for the models that uses an appropriate Welch statistic that is robust to differences in variances across groups (Howell, 2010, page 335 and Field, 2013). Estimating the six ANOVA models we find there is a statistically significant difference between the mean total realized utilities of UMR users and mean job flow times across the eight priority rule groups as determined by each model's Welch statistic with all models exhibiting a p < 0.001. Therefore, we reject the null hypothesis that there is no difference in the performance of the priority rules with respect to the mean total realized utility and flow times for each level of exogenous arrival rate. An ANOVA does not indicate which priority scheduling rules perform significantly different from the others, but only indicates that at least one significant difference exists amongst all the rules. To investigate which priority rules are significantly different from the others, we employ a post hoc test for each of the six different exogenous job arrival rates to ascertain the significance of the differences in mean realized utilities and flow times amongst the priority rules. We then seek to identify patterns in the differences in the performance of the priority dispatching rules over the full set of exogenous arrival rates. We first present the results for the total utility metric. We employ the Ryan-Einot-Gabriel-Welch Q (REGWQ) method to identify homogeneous groups of priority rules with respect to the mean realized total utility metric for each level of exogenous job arrival rates (Howell, 2010 page 393). The REGWQ method employs a step-down procedure of sequential significance tests to group the eight priority scheduling rules based on total realized utility such that, within each group, the means of total realized utility between the priority scheduling rules are not significantly different from each other. However, the means of total realized utility between the different groups of rules are significantly different from those in other groups. Table 4 shows the homogeneous subsets generated by the REGWQ procedure for each of the six different levels of exogenous arrival rates for the total utility metric using the familywise error rate of α = 0.05. The worst performing group of rules forms group 1, and subsequent groups of rules are statistically significantly better (they generate higher mean utility values) than rules of the lower numbered groups. We note that REGWQ procedure permits the subsets of homogeneous rules to have non-empty intersections between successive groups of rules. At baseline levels of exogenous job arrival rates, 1.0, we find that the priority scheduling rules fall within one of three overlapping groups. SPTWIN1Q, SPTWIN2Q, and TOTUTIL are in all three groups, which indicates that these three priority dispatching rules are not statistically different from any other priority dispatching rules. Furthermore, the first group contains both FIFO and SPT, the second

Table 3 Proportion of potential jobs that opt-in by exogenous job arrival rate. Exogenous Job Arrival Rate Priority Rule

FIFO SPT SPTWN1Q SPTWN2Q SPTWN3Q SPTWN4Q TOTUTIL UTILWRK

1.0

1.1

1.2

1.3

1.4

1.5

Mean

Mean

Mean

Mean

Mean

Mean

.667 .670 .670 .670 .671 .670 .665 .665

.661 .664 .667 .666 .667 .667 .658 .658

.647 .655 .657 .657 .657 .658 .642 .644

.625 .639 .641 .641 .641 .642 .607 .612

.600 .619 .622 .623 .624 .624 .581 .573

.578 .601 .605 .606 .607 .607 .564 .541

percentage of potential users that choose to utilize the UMR differentiates on the basis of dispatching rules, with a low of 54 percent choosing to opt-in to use the UMR under UTILWRK to approximately 61 percent of potential users opting to use the UMR under SPTWIN3Q and SPTWIN4Q with exogenous job arrival rates increased by 50 percent. We note that while a smaller percentage of jobs use the UMR when exogenous demand rates are higher, the total number of jobs processed and total system utility generally increase. This differentiation highlights the importance of incorporating users’ utility-based opt-in responses to the dynamic operating conditions in the simulation model. We observe that across the priority rules and exogenous arrival rates that average annual lock utilization rates range from approximately 63.4 percent of the time over the year (at baseline arrival rates under FIFO) to nearly 79.4 percent (with a 50 percent increase of exogenous arrival rates with TOTUTIL). We note that even with an annual 63.4 percent utilization rate, the extreme seasonality present in the utilization of the locks means that there are periods of time, particularly during the busy summer months, when vessel arrivals at the locks can outpace the capacity of the locks to process those arrivals. With a 50 percent increase in exogenous arrival rates, individual lock queues for the TOTUTIL rule can exceed 100 vessels waiting for processing in some months. As a final model validation check, we examined whether the actual utilities received by system users aligned with the expected opt-in utilities of system users with the different priority rules over the exogenous arrival rates. We did this validation because the mechanism that is used to update the expected opt-in utility function (the realized flow time of the last identical vessel type to complete the same itinerary as the newly generated vessel) might systematicaly differ from the realized flow time of the last vessel as the system increasingly congests over time. To do this, we compared the expected opt-in utility and the realized utility of all users that opted-in to use UMR system to look for systemic differences across the priority rules and levels of exogenous demand. We found very close agreement between the expected opt-in and realized utilities. The realized utility for system users that opted-in was statistically significantly slightly less than the expected utility for those same users at the higher exogenous arrival rates, but even though it was statistically significant, the difference between realized and expected utility never exceeded −0.3 percent. This very small difference indicates that while the levels of exogenous demand are related to differences in realized and expected utility for system users, that difference is very small compared to the actual realized utilities of the system users. 7.1. ANOVA and post-hoc means analysis While the previous section details how the priority rules perform relative to each other in absolute terms, it does not provide any insight into whether these differences in performance are statistically significant. In order to compare the performance of the different priority 160

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Table 4 Homogeneous groups of priority rules – total realized value by exogenous job arrival rate. Performance Group

Group 1

Group 2

Group 3

Group 4

Exogenous Job Arrival Rate 1.0

1.1

1.2

1.3

1.4

1.5

FIFO SPT SPTWIN1Q SPTWIN2Q TOTUTIL UTILWRK FIFO SPTWIN1Q SPTWIN2Q SPTWIN4Q TOTUTIL SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL

FIFO SPT UTILWRK

FIFO

FIFO

FIFO UTILWRK

UTILWRK

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL

SPT

SPT

SPT

FIFO

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK TOTUTIL

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

TOTUTIL

TOTUTIL

SPT

Group 5

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK TOTUTIL

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

value of incorporating at least some systemic information when developing priority dispatching rules for this relatively busy, complex, real world system. Interestingly, increasing the amount of system wide information in the global rules doesn't appear to significantly improve system performance with respect to total mean value produced as the global rules form a single or overlapping homogeneous groups for each level of exogenous demand. Also, interesting to note is the performance of the two utility based dispatching rules, TOTUTIL and UTILWRK. In particular, TOTUTIL performs dramatically better than other rules at medium levels of exogenous job arrivals but becomes one of the worst performing scheduling rules with respect to total mean utility when facing very high levels of exogenous job arrivals. The reason for this is evident when examining the mean flow times results reported in Table 2: the vessels with the most utility (double tows) also have the longest processing times at the capacitated servers. Prioritizing the slower, higher utility vessels at the expense of greatly increasing flow times begins to dramatically congest the system at very high levels of exogenous job arrivals, causing increased balking on the part of potential other users evidenced in Table 3. For example, at 1.4 and 1.5 levels of exogenous job arrival rates, the mean flow times for a vessel on the system is almost 3 times higher under TOTUTIL than with the next worst priority dispatching rule. These very long wait times discourage other vessels from using the system, as can been seen from the opt-in percentages from Table 3. In other words, when the system is heavily congested the direct value gains made by prioritizing the slower but higher utility vessels are, in effect, cancelled out by the more frequent opt-out decision of potential system users, particularly the lower valued small tows. Table 5 shows the homogeneous subsets generated by applying the REGWQ procedure across the different exogenous job arrival rates using mean job flow time as the performance metric where smaller mean flow times indicate better performance. Interesting to note here is the very poor performance of TOTUTIL, UTILWRK and FIFO across all levels of exogenous job arrival rates, with TOTUTIL forming the worst performing group at every level of exogenous job arrivals and FIFO and UTILWRK members of the next two worst performing group for every level of job arrival rate. The myopic SPT is always a member of the best performing group, with the global

group contains FIFO, and the third group contains SPTWIN3Q. At 1.1 times baseline exogenous job arrivals, we find that the priority scheduling rules fall into two distinct groups, with FIFO, SPT and UTILWRK composing the first group while the second group is composed of the four more complex priority scheduling rules and TOTUTIL. At 1.2 and 1.3 times baseline exogenous job arrivals, we find that the priority scheduling rules fall into five and four distinct groups respectively, with the first two groups composed of a single rule (FIFO in the first group and SPT in the second group), the more global dispatching rules falling into intermediate groups, and the best performing group containing only TOTUTIL. For 1.4 times exogenous levels of job arrivals, we find that the priority scheduling rules fall into 4 distinct groups, with UTILWRK joining FIFO in the worst performing group and UTILWRK, SPT alone in the second group, the more global dispatching rules forming a third group, and TOTUTIL forming its own best performing group. Interestingly, at the 1.5 level of exogenous job arrivals, UTILWRK, FIFO, TOUTIL and SPT respectively form the four worst performing groups individually and the more global dispatching rules form a distinct best performing group. We note that TOTUTIL performance relative to the other rules falls dramatically at this level of job arrivals. We note that as the exogenous arrival rate of potential jobs increases so does systemic congestion, and the differences in the performance of the more globally based priority dispatching rules with respect to the total realized utility metric become more apparent. At lower levels of exogenous job arrivals, the differences in performance between the priority scheduling rules is small, as demonstrated by the stratification of the priority scheduling rules into only three overlapping groups for baseline demand levels. As congestion increases, the more complex global priority scheduling rules begin to more clearly separate themselves from the less complex rules. Furthermore, the priority dispatching rules which incorporate more information on the state of other capacitated servers begin to significantly outperform the priority scheduling rules that consider only one capacitated server (FIFO and SPT). In our complex dynamic job shop, we observe that more complex (global) priority dispatching rules outperform less complex (myopic) priority dispatching rules with respect to total realized utility as utilization of the capacitated servers increase, indicating the additional 161

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Table 5 Homogeneous groups of rules – mean flow time by exogenous job arrival rate. Performance Group

Exogenous Job Arrival Rate 1.0

1.1

1.2

1.3

1.4

1.5

Group 1 Group 2

TOTUTIL FIFO UTILWRK

TOTUTIL FIFO

TOTUTIL UTILWRK

TOTUTIL UTILWRK

TOTUTIL UTILWRK

Group 3

SPT SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q

SPTWIN2Q SPTWIN4Q UTILWRK

TOTUTIL FIFO SPTWIN4Q UTILWRK SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT

FIFO

FIFO

FIFO SPTWIN4Q

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT

SPTWIN2Q SPTWIN3Q SPTWIN4Q

Group 4

Group 5

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q

Group 6

SPT

rule is a robust performing rule and performs well in minimizing flow times in our busy, complex real system.

Table 6 Mean realized total UMR utility in dollars ($2007 millions) by priority rule (without job balking).

7.2. The importance of balking

Exogenous Job Arrival Rate

Priority Rule

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL UTILWRK

SPTWIN1Q SPTWIN2Q SPTWIN3Q

1.0

1.1

1.2

1.3

1.4

1.5

Total Utility

Total Utility

Total Utility

Total Utility

Total Utility

Total Utility

Mean

Mean

Mean

Mean

Mean

Mean

2442 2438 2446 2440 2448 2442 2444 2442

2668 2673 2677 2679 2678 2673 2683 2678

2849 2862 2871 2878 2878 2878 2903 2894

2842 2909 2937 2945 2946 2947 3059 3029

2655 2852 2913 2938 2950 2963 3115 3022

2464 2736 2873 2919 2954 2970 3048 2884

To investigate the effect of balking on the results of the performance of the priority dispatching rules we run the simulation model with the utility of the opt-out alternative degraded to such a low level that no potential jobs actually balk. To provide a fair comparison, we also decrease the baseline exogenous job-arrival rates such that when no can jobs opt-out the baseline performance of the system still approximates the observed baseline performance of the system. See Sweeney et al. (2014, page 162) for details of how the exogenous arrival rates are adjusted to preserve the observed baseline performance. We then increase these adjusted arrival rates by 10, 20, 30, 40 and 50 percent relative to the baseline arrival rates to investigate the performance of the system without balking under each of the priority dispatching rules as the utilization of the capacitated servers is increased. Without balking we note that the system will congest much more rapidly as all potential jobs must enter the system. Tables 6 and 7 resent the mean total realized utilities and flow times of tows compiled from 200 replications for each of the eight dynamic priority dispatching rules and six exogenous arrival rates of potential jobs without balking. The importance of including the possibility of job balking in the simulation model is clearly evident in the striking differences between the results presented in Tables 6 and 7 compared to Tables 1 and 2 When potential jobs have an alternative and can balk in response to

dispatching rules forming various middle groups between FIFO and SPT as exogenous job arrival rates increase. At levels of exogenous job arrivals greater than or equal to 1.2 times the baseline, we find that SPT significantly outperforms all other rules in reducing mean flow times. Furthermore, we notice that global rules that consider fewer servers tend to perform slightly better on the mean flow time metric than rules that consider more servers, particularly as exogenous job arrivals rates increase. We conclude that the myopic SPT Table 7 Mean UMR flow times (hours) by priority rule (without job balking). Exogenous Job Arrival Rate

Priority Rule

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL UTILWRK

1.0

1.1

1.2

1.3

1.4

1.5

Flow Time

Flow Time

Flow Time

Flow Time

Flow Time

Flow Time

Mean

Mean

Mean

Mean

Mean

Mean

21.34 20.70 20.48 20.58 20.74 20.69 21.99 21.06

26.71 24.92 24.72 25.27 25.46 25.54 27.86 25.39

48.14 42.11 42.03 42.92 43.07 43.67 45.56 41.08

147.73 116.30 113.07 114.91 116.65 118.79 149.28 116.33

349.91 240.52 229.86 231.89 233.73 236.32 419.27 319.75

568.74 396.23 360.05 363.35 357.50 360.42 726.08 596.69

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Table 8 Homogeneous groups of rules – total mean realized value by exogenous job arrival rate (without job balking). Performance Group

Group 1

Group 2

Group 3

Exogenous Job Arrival Rate 1.0

1.1

1.2

1.3

1.4

1.5

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK TOTUTIL

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK

FIFO

FIFO

FIFO

FIFO

SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK TOTUTIL

SPT SPTWIN1Q

SPT

SPT

SPT

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK TOTUTIL

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK

SPTWIN1Q

SPTWIN1Q UTILWRK

SPTWIN2Q SPTWIN3Q SPTWIN3Q SPTWIN4Q UTILWRK TOTUTIL

SPTWIN2Q

Group 4 Group 5

TOTUTIL

Group 6 Group 7

SPTWIN3Q SPTWIN4Q TOTUTIL

incorporate system wide information. Only at very high levels of exogenous demand (1.5 times baseline levels) do the global rules begin to significantly outperform the myopic rules. Especially interesting is the comparison of the no balking results to the balking results. Comparing Tables 4 and 8 with the total value metric shows that at lower levels of exogenous demand (1.0–1.4 times baseline levels), the performance hierarchy of the scheduling priority rules are largely consistent across the balking and no balking cases with only small variations in the total number of significantly different performing groups. However, at large levels of exogenous demand, the performance of the utility-based scheduling rules drops off dramatically in the balking case compared to the no balking case. This demonstrates the importance of balking to the performance of the system in terms of generating value: there is an inherent tradeoff when balking is allowed between prioritizing higher utility jobs (which take longer to process) and encouraging more jobs to use the system by reducing mean flow times (as expected flow times are part of the value function). In other words, when demand is high, the higher expected flow times of the utility-based priority rules which occur by prioritizing slower moving and higher value traffic encourage more balking and thus decrease total system utility (through fewer jobs processed) as an unintended consequence. When jobs cannot balk, users are forced onto the system and scheduling by utility maximizes the total value of the system. This is supported by the mean flow time results across the balking and no balking cases (Tables 5 and 9), which have largely consistent performance hierarchies across exogenous demand levels, regardless of whether balking is permitted.

expected poor levels of service, then even at significantly elevated exogenous job arrival rates, system congestion (as measured by the mean flow times of jobs) is significantly dampened as lower utility jobs opt-out of using the congested system. Thus, including the possibility that jobs may balk suggests a much less congested system that still produces significant value for its users. Potential jobs that do not benefit sufficiently from using the congested system opt-out, thereby reducing realized levels of system congestion for remaining users. Table 8 shows the homogeneous subsets generated by the REGWQ procedure for each of the six different levels of exogenous arrival rates for the total realized utility metric without balking. At baseline levels of exogenous arrival rates (1.0), there is no significant difference between the scheduling rules in terms of mean total realized system value. However, as the exogenous arrival rate and system congestion increases, the performance of the scheduling rules becomes significantly different. The utility based scheduling rules of TOTUTIL and UTILWRK are statistically significantly better than all processing time based scheduling rules for exogenous levels of demand from 1.2 to 1.4 times baseline levels, with TOTUTIL performing statistically significantly better than UTILWRK. For the processing time based scheduling rules, those that incorporate more system wide information outperform those that incorporate less system wide information. In contrast to the case when balking is allowed, the priority rules that incorporate more system wide information statistically outperform those that incorporate less, especially as exogenous demand increases. Also, interesting to note is that the performance of UTILWRK as a scheduling rule decreases dramatically when system congestion is high at exogenous demand levels of 1.5 times baseline demand. Table 9 shows the homogeneous subsets generated by the REGWQ procedure for each of the six different levels of exogenous arrival rates for the mean flow time metric. Unlike the value metric, at baseline levels of exogenous arrival rates (1.0) there is significant differences between the scheduling rules in terms of mean flow time, with the utility-based metrics and FIFO performing significantly worse than processing time based scheduling rules. At lower levels of exogenous demand (1.0–1.4 times baseline levels), the performance of the myopic SPT is not significantly different from the other top performing scheduling rules, including those that

7.3. Supplemental experiments We provide robustness checks of the performance of the priority rules in the context of the operations of the UMR by conducting two supplemental experiments: the smoothing of seasonal job arrival rates that reduce peak season arrivals and increase off peak season arrivals and the decrease of transit time variability of jobs as they transit the pools between locks. The first experimental scenario, for example, could mimic a pricing scheme which incentivizes shippers to use the river in a less seasonal fashion, while the second experiment could 163

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Table 9 Homogeneous groups of rules – mean flow time by exogenous job arrival rate (without job balking). Performance Group

Exogenous Job Arrival Rate 1.0

1.1

1.2

1.3

1.4

1.5

Group 1

TOTUTIL

TOTUTIL

TOTUTIL

FIFO

FIFO

FIFO

UTILWRK

Group 3

UTILWRK

SPT SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK

TOTUTIL FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK

TOTUTIL

Group 2

TOTUTIL FIFO SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL

UTILWRK

FIFO

Group 4

SPT SPTWIN2Q SPTWIN3Q SPTWIN4Q

SPT SPTWIN1Q SPTWIN2Q

SPT

Group 5

SPT SPTWIN1Q SPTWIN2Q SPTWIN4Q

SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

from the performance of the myopic priority rules. Once exogenous demand increases to 1.5 times baseline levels, the priority rules separate themselves into a grouping that looks very much like the 1.2 times baseline exogenous demand case with seasonality in the job arrivals. The reduction in peak demands dramatically decreases system congestion and flow times by increasing lock utilizations during seasonal periods of low lock utilization. Spreading job arrivals over the entire calendar year has the effect of muting the performance of all the rules until the system begins to congest at the higher arrival rates of jobs. Table 11 shows the homogeneous subsets generated by the REGWQ procedure for 200 annual simulations of each of the six different levels of exogenous arrival rates and eight priority rules for the mean flow time metric with steady random job arrival rates. At baseline levels of exogenous demand, the global priority rules and SPT outperform the utility-based scheduling rules (and FIFO) by a significant margin, a pattern that continues as exogenous demand levels increase to 1.5 times baseline levels. While TOTUTIL is always the worst performer of the priority rules, SPT is always a member of the highest performing group and separates itself from the global rules when exogenous demand reaches 1.4 times baseline demand; this behavior also closely mirrors the performance of the priority rules that looks like the 1.2 times baseline demand case with seasonal arrivals. Overall, this experiment shows that the seasonality of arrivals serves to greatly increase congestion on the UMR during peak arrival periods and influence the efficacy of the priority scheduling rules: the performance groupings of the priority rules with non-seasonal arrivals at exogenous demand 1.5 times baseline levels closely mimic the performance groupings of exogenous demand at 1.2 times baseline levels with seasonal arrivals. Finally, we note that across the priority rules and exogenous arrival rates that average annual lock utilization rates range from approximately 62.1 percent of the time over the year (at baseline arrival rates under SPT) to nearly 90.1 percent (with a 50 percent increase of exogenous arrival rates with TOTUTIL). Compared to the average annual lock utilizations rates with seasonal arrivals we find similar lock utilizations with relatively low potential job arrival rates and higher average annual lock utilizations with high potential job arrivals.

mimic the imposition of a strict speed limit or other policy which forces vessels to travel at similar speeds as they traverse the pools between locks. We choose these two scenarios as they permit us to investigate the impacts of seasonality in the arrival of jobs and variability in the transport times between capacitated servers on the performance of the priority rules and both scenarios represent policies which could theoretically be implemented in the real UMR system. The removal of seasonality in job arrival rates redistributes demand across all 12 months of the year. In short, this robustness check removes the large spikes in demand in the spring and summer months and replaces them with a flat stochastic demand distributed across all months. Due to the reduction in periodic large demand spikes, we expect to see a more consistent congestion throughout the year, with fewer periods of very long flow times, even with exogenous demand rates increased to 1.5 times their base levels. Tables C1 and C2 in Appendix C present the mean total realized utilities and flow times of tows compiled from 200 replications for each of the eight dynamic priority dispatching rules and six exogenous arrival rates of potential jobs with seasonality removed. As expected, we do observe a less congested system with greatly reduced flowtimes across all the priority rules and exogenous job arrival rates. We also note that for the lower exogenous job arrival rates that the mean total realized utilities are lower than those seen in Table 1 despite the reduced flow times. This is because the population of tows in the early and later months are skewed towards small tows (which earn lower utility); in other words, small tows are more likely to use the system in the spring and fall months. As the exogenous job arrival rates increase the system eventually demonstrates greater realized utilities than those in Table 1 due to the reduced flow times. Table 10 shows the homogeneous subsets generated by the REGWQ procedure for 200 annual simulations of each of the six different levels of exogenous arrival rates and eight priority rules for the total utility metric with steady stochastic job arrival rates. At baseline levels of exogenous arrival rates (1.0), there are two overlapping performance groups. These two overlapping performance groups stay relatively stable for exogenous demand levels of 1.1 and 1.2 times baseline arrival rates. As exogenous job arrival rates further increase, the performance of the global priority rules (as well as the TOTUTIL priority rule) begin to separate themselves 164

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Table 10 Homogeneous groups of priority rules – total realized utility by exogenous job arrival rate (seasonal arrivals removed). Performance Group

Group 1

Group 2

Group 3

Exogenous Job Arrival Rate 1.0

1.1

1.2

1.3

1.4

1.5

FIFO SPT SPTWIN2Q SPTWIN4Q UTILWRK TOTUTIL FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK

FIFO SPT SPTWIN1Q SPTWIN4Q UTILWRK TOTUTIL FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK

FIFO SPT SPTWIN1Q SPTWIN2Q UTILWRK TOTUTIL FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL

FIFO SPT SPTWIN2Q SPTWIN3Q UTILWRK TOTUTIL FIFO SPT SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL

FIFO SPT UTILWRK

FIFO

SPT SPTWIN1Q TOTUTIL

SPT UTILWRK

SPTWIN1Q SPTWIN2Q SPTWIN4Q TOTUTIL

SPTWIN1Q UTILWRK

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL

Group 4

SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

Group 5

Table 11 Homogeneous groups of priority rules – mean flow time by exogenous job arrival rate (seasonal arrivals removed). Performance Group

Exogenous Job Arrival Rate 1.0

1.1

1.2

1.3

1.4

1.5

Group 1 Group 2 Group 3

TOTUTIL UTILWRK FIFO

TOTUTIL UTILWRK FIFO

TOTUTIL UTILWRK FIFO

TOTUTIL UTILWRK FIFO

Group 4

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

SPTWIN3Q SPTWIN4Q

SPTWIN2Q SPTWIN3Q SPTWIN4Q

TOTUTIL UTILWRK FIFO SPTWIN3Q SPTWIN4Q SPTWIN1Q SPTWIN2Q

Group 5

SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q

SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

TOTUTIL UTILWRK FIFO SPTWIN3Q SPTWIN4Q SPTWIN1Q SPTWIN2Q

SPT SPTWIN2Q SPTWIN3Q

SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q

SPT

SPT

Group 6

SPT SPTWIN1Q SPTWIN2Q

The second additional experiment involves reducing variability in vessel pool transit times, effectively making seasonal travel times between locks the same for all vessels of similar types. Reducing this source of systemic variability might boost the performance of the global scheduling rules relative to the case where the variability in transit times is included by decreasing the variability in tow arrivals at subsequent locks. Tables C3 and C4 in Appendix C present the mean total realized utilities and flow times of tows compiled from 200 replications for each of the eight dynamic priority dispatching rules and six exogenous arrival rates of potential jobs with reduced variability in vessel pool transit times. We note that both realized utilities and flow times are very similar to those presented in Tables 1 and 2 Table 12 shows the homogeneous subsets generated by the REGWQ procedure for 200 annual simulations of each of the six different levels of exogenous arrival rates and eight priority rules for the total utility metric with reduced variability in pool processing times. Interestingly, there is no statistically significant difference in performance between any priority rules at 1.0 times baseline exogenous

demand. However, as exogenous demand increases, the priority rules separate themselves in a manner very similar to the case where actual transit time variability is included in the model (including the sudden dip in performance of the TOTUTIL priority scheduling rule at 1.5 times baseline exogenous demand). Table 13 shows the homogeneous subsets generated by the REGWQ procedure for 200 simulations of each of the six different levels of exogenous arrival rates and eight priority rules for the mean flow time metric with reduced variability in pool processing times. The results for mean flow time with reduced pool transit variability are also very similar to the results for the case where actual pool transit variability is included in the model: for all levels of exogenous demand, TOTUTIL, FIFO, and UTILWRK form the worst performing group of priority rules. As exogenous demand increases, the global priority rules separate themselves from all of the myopic priority rules except SPT, which ultimately performs better than all other priority rules, especially when exogenous demand (and therefore congestion) is high. It is interesting to note the general 165

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Table 12 Homogeneous groups of priority rules – total realized utility by exogenous job arrival rate (reduced pool transit time variability). Performance Group

Group 1

Group 2

Exogenous Job Arrival Rate 1.0

1.1

1.2

1.3

1.4

1.5

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK TOTUTIL

FIFO

FIFO

FIFO

UTILWRK

UTILWRK

SPT UTILWRK SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL

SPT

SPT

FIFO

FIFO

SPTWIN1Q SPTWIN4Q UTILWRK

SPTWIN1Q SPTWIN2Q SPTWIN3Q UTILWRK

SPT

TOTUTIL

SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK TOTUTIL

SPTWIN2Q SPTWIN3Q SPTWIN4Q

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL

SPT

Group 3

Group 4

Group 5

TOTUTIL

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q

Table 13 Homogeneous groups of priority rules – mean flow time by exogenous job arrival rate (reduced pool transit time variability). Performance Group

Group 1 Group 2 Group 3

Group 4

Group 5

Exogenous Job Arrival Rate 1.0

1.1

1.2

1.3

1.4

1.5

TOTUTIL FIFO UTILWRK SPTWIN2Q SPTWIN3Q SPTWIN4Q

TOTUTIL FIFO

TOTUTIL FIFO

TOTUTIL UTILWRK

TOTUTIL UTILWRK

TOTUTIL UTILWRK

SPTWIN2Q SPTWIN3Q SPTWIN4Q UTILWRK SPT SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q

UTILWRK SPTWIN2Q SPTWIN3Q

FIFO

FIFO

FIFO

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT

SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q SPT

SPT SPTWIN1Q SPTWIN2Q SPTWIN4Q

results almost completely mirror the results of the case where transit time variability is included in the model. To summarize, we find that the performance of the priority scheduling rules with reduced variability in pool travel times is similar to the performance with the actual variability for both the mean annual value metric and mean flow time metrics. Therefore, we conclude that the performance of the priority of the rules are robust with respect to the variability of pool travel times.

heterogeneous requirements and values of processing individual jobs. In our simulation of a very complex real dynamic job shop, the UMR Inland Navigation System, we evaluate the performance of eight priority dispatching rules in an environment with five capacitated servers, connected by six un-capacitated servers where the arrival rates of heterogeneous jobs are stochastic, seasonal and dependent on meeting differentiated service requirements. Additionally, the processing time for a job at each capacitated server is stochastic, sequence dependent and seasonal. We find that as loads on the capacitated servers increase through increased job arrival rates, the more complex (global) priority scheduling rules significantly outperform the less complex (myopic) priority dispatching rules based on the metric of total utility realized for users of the UMR system. This finding supports and extends earlier results from experiments with less complex job shops that global priority scheduling rules can frequently outperform their myopic counterparts (Holthaus and Rajendran, 1997; Sels et al., 2012). In contrast, we note that with respect to the metric of the mean flow times of jobs, the myopic rule of SPT is always among the best

8. Discussion and conclusions While there have been many evaluations of the performance of myopic and global priority scheduling rules in idealized job shops, these job shop models typically have employed some restrictive analytical assumptions. In real world job shops, there may be many simultaneous complexities such as: customers balking in response to unmet service requirements, sequence dependent job processing times at capacitated servers, seasonal and stochastic job processing times and arrival rates, random server breakdowns, and 166

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performing rules and actually performs significantly better than any other rule as expected loads on the capacitated servers increase. This finding also supports and extends earlier results from experiments with less complex job shops that SPT is a good priority scheduling mechanism for minimizing flow times in dynamic job shops (Holthaus and Rajendran, 1997; Sels et al., 2012). However, we also note that while SPT performs at least as well as all other priority dispatching rules in terms of minimizing mean system flow times, to maximize total realized utility for system users SPT is clearly outperformed by the more global priority dispatching rules at high levels of exogenous demand, and also outperformed by the utility based priority dispatching rules at low and intermediate levels of exogenous job arrival rates. In addition, we also find that including balking behavior in a complex dynamic job shop has dramatic consequences for the relative performance of priority dispatching rules. When levels of exogenous job arrivals are low or intermediate and balking is possible, a priority rule incorporating the expected utility of individual jobs, TOTUTIL, maximizes the total system utility received by users. However, when exogenous job arrival rates become very high and balking is possible, the TOTUTIL rule performs significantly worse than non-utility-based priority rules, such as SPTWIN4Q, due to the utility-based rule creating very high flow times that then decrease the actual utility received by all those that do not balk. In contrast, when balking is not possible (so all potential jobs are forced to enter the job shop), the TOTUTIL rule maximizes the total utility received by users for all evaluated exogenous job arrival rates. However, with respect to the flow times of jobs metric, the utility-based priority rules perform very poorly and the more global job processing timebased priority rules eventually outperform all other rules including SPT at high exogenous job arrival rates. Furthermore, for all levels of exogenous demand with or without balking, a priority dispatching rule that integrates information regarding both the expected utility of individual jobs and the processing time requirements of jobs, UTILWRK, does not ever provide significantly better performance than rules based entirely on either job processing time requirements or rules based on the expected utility of individual jobs. These results help extend previous research on system throughput and performance in the presence of queue sensitive

customers (Chen and Frank. 2004; Shone et al., 2013; Ibrahim, 2018). Finally, we also support findings in the literature that there is no single best rule for scheduling a dynamic job shop. The choice of rule should be based on which performance metric (or metrics) is most important to the scheduling party (Rajendran and Holthaus, 1999; Ouelhadj and Petrovic, 2009) and the particular operating characteristics of the job shop. 9. Limitations and future research Because the only data available for the dynamic simulation of the UMR system is information regarding vessels arriving or departing from a lock, we are limited to investigating completely reactive scheduling mechanisms such as priority dispatching rules where schedules cannot be constructed ahead of time. Furthermore, these reactive scheduling mechanisms can only be employed at the occurrence of a very limited set of events: a vessel arriving or departing a lock. One avenue for future research of the UMR system is to investigate implementing different priority rules at different times at different locks in the UMR system, in the spirit of Romero-Silva et al. (2018). With additional information, researchers in the future could explore the performance of predictive reactive or robust proactive scheduling mechanisms, which would partially or completely build schedules in advance based on relevant performance metrics and modify schedules as necessary, in the context of this complex real-world job shop. Additionally, researchers could explore the performance of priority scheduling mechanisms in real dynamic job shops that are more complex than the typical experimental settings in the literature, but not as uniquely complex as the UMR Inland Navigation System. This line of research would facilitate the identification of the factors that directly impact the performance of the priority scheduling mechanisms. Furthermore, it would be interesting to examine how other utility-based priority dispatching rules, particularly in the context of user opt-in decisions, might impact different performance metrics, and whether other factors such as the ratio of utility or processing times between heterogeneous classes of jobs impact the efficacy of any utility-based priority dispatching rules.

Appendix Appendix A. The UMR as a Complex Dynamic Job Shop This Appendix provides details on the study section of the UMR and its modeling as a dynamic job with (i) interdependent, heterogenous, seasonal and stochastic job arrivals that may opt out of entering the job shop if not expecting to receive a required level of service; (ii) seasonal, sequence dependent and stochastic job processing times at five capacitated servers; and (iii) seasonal and stochastic job processing times at six uncapacitated servers linking the five capacitated servers. Together, these complexities make this job shop (i.e. congested segment of the UMR) quite different from dynamic job shops investigated in the literature and provide the opportunity to evaluate the performance of different priority dispatching rules in a rich environment. Sweeney et al. (2014) first noted the dynamic job shop aspects of processing tow traffic through the congested portion of the UMR system, and in the present paper we exploit this to examine the performance of selected dynamic job shop priority schemes in this complex operating environment. Jobs in the form of vessel trips enter the system stochastically at one of six pools with each job characterized by a vessel type (type 1 = recreation or local vessel, type 2 = single lockage commercial tow, and type 3 = double lockage commercial tow), its origination pool and its destination pool. Once entering the UMR in its origination pool, each job is processed through a specific sequence of servers (alternating pools and locks) based on the specified origination and destination pool. The pools are un-capacitated servers as many jobs may be processed through a pool simultaneously, and the locks are capacitated servers as only a single job can be processed at any lock at any point in time (moving either upstream or downstream). The U.S. Army Corps of Engineers records detailed information regarding all vessel lockages in the UMR system, but does not explicitly track the activities of vessels within the pools. The most recent detailed lockage data that has been made available is from calendar year 2000, all the following data analyses are based on these data. Using this detailed lockage data, Sweeney et al. (2014) derived the implied individual trip itineraries (jobs) for commercial tows and other vessels on the UMR system. Table A1 presents summary information regarding the origin and destination pools for the 8893 jobs that utilized at least one lock in this segment of the UMR. This shows the 30 possible vessel itineraries and indicates that approximately 59.1 percent of the vessel trips (5256/8893) are processed by only a single lock, while 18.4 percent of all vessel trips (1640/8893) are processed by all 5 locks.

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Table A1

Counts of Baseline Vessels by Origination and Termination Pools Terminal Pool

Origin Pool

Total

20 21 22 24 25 26

Total

20

21

22

24

25

26

0 677 123 167 19 754 1740

680 0 491 59 5 194 1429

114 552 0 400 5 130 1201

88 49 422 0 404 342 1305

20 5 3 459 0 566 1053

886 178 133 263 605 0 2065

1788 1461 1172 1348 1038 1986 8893

Table A2 presents the arrival of these jobs characterized by vessel type and month. The strong seasonality present in the different monthly arrival rates of these jobs and for different vessel types is clearly evident, with very low arrival rates in the winter months that increase dramatically in the early spring, peak in the summer and early fall, and then decline again as winter arrives. This seasonality is the result of the unfavorable navigation conditions in the UMR in winter due to extreme cold and river ice and is represented in the model by intensifying the stochastic job arrivals modeled with exponential interarrival times to their maximum monthly rates and then thinning the maximum arrival rates to appropriate monthly rates. Table A2

Baseline Arrivals by Month and Vessel Type.

Month

Total

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Double Tows

Single Tows

Local Vessels

Total

4 34 367 397 439 403 421 389 329 410 358 139 3690

4 20 109 124 119 104 117 128 164 138 166 87 1280

7 26 152 181 327 431 683 740 602 395 265 114 3923

15 80 628 702 885 938 1221 1257 1095 943 789 340 8893

Processing times of jobs at each of the capacitated servers are represented by random draws from lognormal distributions differentiated by the direction of travel, the vessel type (a double tow, a single tow, or a local vessel), the month of the year, and whether the lockage is a fly, turnback, or exchange lockage. Those lognormal distributions were determined by regression models that estimated the mean and the standard deviation of a lognormal distribution that characterizes the specific lockage of that vessel. Table A3 presents summary statistics for the operation of each of the five locks in this segment of the UMR in minutes by vessel type and shows the wide variation in lockage times across the three vessel classes at the individual locks. Over all five locks, approximately 63.7 percent of the 18,700 lockages required to process these 8.893 jobs are double tow lockages requiring an average of 117.3 min to complete, 15.3 percent are single tow lockages requiring an average of 42.8 min to complete, and 21.0 percent are recreation and local traffic lockages requiring an average of 17.1 min to complete. Table A3

Selected Lockage Statistics Lock

Vessel Type

Mean (minutes)

N

Std. Deviation

20

Double Local Single Total Double Local Single Total Double Local Single Total Double Local Single

110.56 18.07 41.69 81.40 114.79 15.81 40.15 82.38 127.73 19.09 52.69 97.00 116.81 16.39 41.24

2266 677 585 3528 2340 749 615 3704 2360 633 524 3517 2468 795 537

44.06 33.47 33.11 56.75 28.80 9.68 24.70 50.00 39.06 13.58 57.85 59.78 35.32 8.36 22.40

21

22

24

168

Lock Utilization

54.6%

58.1%

64.9%

(continued on next page)

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Table A3 (continued) Lock 25

Overall

Vessel Type

Mean (minutes)

N

Std. Deviation

Lock Utilization

Total Double Local Single Total Total

85.12 116.35 16.54 39.52 79.60 84.88

3800 2485 1069 597 4151 18,700

53.00 36.52 9.28 29.32 54.85 55.23

61.5%

62.9% 60.4%

Table A3 also indicates that the locks were utilized approximately 60.4 percent of the available time during the year processing vessels. Not shown in Table A3, but evident in the detailed data, is the large differences in monthly utilization of the locks that varies from a low of 0.94 percent in January at Lock 20 to nearly 81.93 percent in July at Lock 24. These locks process bi-directional (both upstream and downstream) traffic through a single lock chamber and each lockage involves either raising (for upstream travel) or lowering (for downstream travel) the vessel to match the next pool in the trip. The bi-directional traffic creates sequence dependent job processing times, as processing two sequential vessels moving in the same direction requires the lock operator to readjust the water level in the lock chamber back to the starting level of the first vessel in order to accommodate the subsequent vessel for processing. Processing two consecutive vessels in the same direction is known as a turnback lockage. If the next vessel to be processed is moving in the opposite direction of the previous vessel, then the lock operator does not need to readjust the water level inside the lock chamber before processing the subsequent vessel, but the approaching vessel must wait for the exiting vessel to completely clear the lock area. Processing two consecutive vessels in opposite directions is known as an exchange lockage. Table A4 shows selected statistics for turnback and exchange lockages in minutes by direction of vessel travel at each lock and documents the sequence dependent processing times. Further still, lockage times for all vessels vary systematically in seasonal patterns during the year, with longer lockage times observed at the beginning and end of the year and shorter lockage times observed during the middle of the year. Table A4

Summaries of Lockage Times by Lockage Type and Direction of Travel Lock

Lockage Type

Direction

Mean (minutes)

N

Std. Deviation

20

Exchange

Downbound Upbound Downbound Upbound Downbound Upbound Downbound Upbound Downbound Upbound Downbound Upbound Downbound Upbound Downbound Upbound Downbound Upbound Downbound Upbound

86.71 85.42 88.89 80.88 83.46 83.48 88.07 80.93 100.52 98.61 104.00 93.27 108.30 93.85 114.11 92.17 96.91 86.81 99.81 87.55

671 642 578 552 696 721 631 548 762 772 672 648 646 682 640 571 804 770 665 679

77.644 57.489 47.823 37.690 51.074 48.529 47.163 39.402 63.652 57.328 67.937 42.559 49.956 47.707 53.252 31.908 48.420 59.654 43.467 40.512

Turnback 21

Exchange Turnback

22

Exchange Turnback

24

Exchange Turnback

25

Exchange Turnback

The river pools that connect successive locks (see Fig. 1) are a source of significant travel time for vessel jobs processed through multiple locks with travel times varying by pool, season (due to differing river flow conditions), vessel type and direction of travel. Commercial tows are the only vessels that are uniquely identified and tracked lock-by-lock in the Corps’ data. Recreation vessels and other local lock traffic are not consistently identified at locks and consequently it is not possible to accurately track their activities in the system. Therefore, we model these local vessels as entering the system in a specific pool in a specific direction to use a specific lock and then exiting the system immediately in the adjacent pool. We employ the lock-by-lock data from the Corps to estimate the elapsed time for the 9807 commercial tows that were observed in lockages at successive locks in their itinerary to estimate pool processing times as lognormal random variables. Table A5 presents selected summary statistics for the time between successive lockages for commercial tows by lock and previous lock. Table A5 indicates that commercial tows spend an average of approximately 4.71 h in the system between successive lockages with a wide variance amongst pools. We also observe systematic and seasonal variations in these mean pool processing times, but do not report them in Table A5. We note the pools are large enough (approximately 20 miles in length) with sufficient carrying capacity that vessels do not face queues or congestion while moving in a pool between the locks. Thus, the pools between the locks are modeled as un-capacitated servers that can process multiple vessels simultaneously.

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Table A5

Selected Summary Statistics Baseline Pool Transit Times Lock

Previous Lock

Mean (hours)

N

Std. Deviation

20 21

21 20 22 21 24 22 25 24

4.21 4.25 4.62 3.15 5.02 4.51 6.81 4.72

1063 1108 1198 1226 1107 1225 1420 1460

6.86 47.55 2.68 4.17 2.24 18.34 2.86 13.77

22 24 25

Appendix B. The Utility (Value) Model The simulation model implements this random utility model using data and a logit model compiled from the most recent survey of grain shippers that utilize the UMR (Train and Wilson, 2007). Because grain shipments comprise the vast majority of commercial freight moved on this segment of the UMR, we use the random per ton utilities estimated from this survey to estimate the opt-in utility of potential UMR jobs. Jobs utilizing commercial tows that opt-in to use this segment of the UMR expect to receive an average deterministic utility equal to $34.56 per ton of product (at 2006 price levels) under baseline operating conditions, deterministic utility equal to $26.66 per ton when opting-out and a random unobservable utility component averaging $6.52 per ton with either option. This $7.90 per ton net difference in expected utility reflects the attractiveness of the UMR system for capturing potential freight shipments that can move in tows. The Corps data indicate that double tows hauled an average load of 15,298 tons and single tows hauled an average load of 2078 tons yielding a net expected utility per vessel of $120,854.20 and $16,416.20, respectively. With these net expected levels of utility and the random variation of the non-flow time related utility components Sweeney et al. (2014) estimate that approximately 66.7 percent of potential jobs that could utilize commercial tows will opt in to use the UMR. The expected utility for potential transportation jobs using double and single tows decreases with increased levels of congestion, as the increased flow times make the UMR a relatively less attractive shipping alternative. This increases the likelihood that individual jobs will opt out to utilize a non-UMR shipping alternative. Sweeney et al. (2014) estimate that each additional hour of expected system time in a tow's itinerary on the UMR decreases the expected utility by approximately $0.035 per ton (Sweeney et al., 2014). There is no available published information regarding balking values for local vessel traffic, which is mostly privately-owned recreation vessels that only utilize a single lock. Consequently, it is likely the utilities of these jobs are very small relative to jobs utilizing tows. Therefore, we assume that none of these vessels balk (all opt-in to use the local lock), but the opt-in utilities of these jobs are negligible compared to the utilities of the commercial tows. Appendix C. Detailed Model Results from the Supplemental Experiments This appendix provides the mean annual total realized utility for UMR users and flow times for each combination of priority dispatching rule and exogenous arrival rate of potential jobs compiled from 200 annual replications of the UMR model for the supplemental experiments. Tables C1 and C2, respectively, present the mean total realized utility and flow time results of the model runs associated with seasonal arrivals being removed from the model. Table C1

Mean Realized UMR Users' Total Utility in Dollars ($2007 Millions) by Priority Rule (Seasonal Arrivals Removed) Exogenous Job Arrival Rate

Priority Rule

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL UTILWRK

1.0

1.1

1.2

1.3

1.4

1.5

Total Utility

Total Utility

Total Utility

Total Utility

Total Utility

Total Utility

Mean

Mean

Mean

Mean

Mean

Mean

2687 2686 2695 2692 2695 2693 2682 2684

2953 2950 2951 2956 2960 2951 2941 2948

3208 3206 3210 3212 3214 3217 3204 3199

3450 3460 3468 3462 3461 3463 3450 3449

3674 3683 3695 3696 3703 3700 3691 3676

3861 3874 3887 3892 3892 3893 3907 3880

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Table C2

Mean UMR Flow Times (Hours) by Priority Rule (Seasonal Arrivals Removed) Exogenous Job Arrival Rate

Priority Rule

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL UTILWRK

1.0

1.1

1.2

1.3

1.4

1.5

Flow Time

Flow Time

Flow Time

Flow Time

Flow Time

Flow Time

Mean

Mean

Mean

Mean

Mean

Mean

17.59 17.17 17.24 17.26 17.27 17.29 18.06 17.77

18.79 18.24 18.27 18.28 18.34 18.33 19.49 19.14

20.42 19.71 19.69 19.76 19.89 20.02 21.70 20.92

23.14 22.10 22.14 22.29 22.30 22.50 25.27 23.97

27.94 25.91 26.66 26.68 27.44 27.47 32.16 28.98

37.99 33.64 35.71 36.09 37.40 37.70 48.24 40.12

Tables C3 and C4, respectively, present the mean total realized utility and flow time results of the model runs associated with the assumption of reduced transit time variability of vessels in the lock pools. Table C3

Mean Realized UMR Total Utility in Dollars ($2007 Millions) by Priority Rule (Reduced Pool Transit Time Variability) Exogenous Job Arrival Rate

Priority Rule

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL UTILWRK

1.0

1.1

1.2

1.3

1.4

1.5

Flow Time

Flow Time

Flow Time

Flow Time

Flow Time

Flow Time

Mean

Mean

Mean

Mean

Mean

Mean

23.40 22.43 22.39 22.49 22.80 22.72 24.91 23.48

31.05 28.61 28.52 29.04 29.23 29.40 33.14 29.85

49.33 42.53 44.61 45.14 45.24 44.73 58.30 46.91

81.69 67.62 73.11 73.83 74.60 75.48 144.22 94.29

119.10 94.87 104.61 106.02 107.69 109.71 326.15 197.98

157.86 121.88 132.62 140.10 139.95 142.52 545.77 350.38

Table C4

Mean UMR Flow Times (Hours) by Priority Rule (Reduced Pool Transit Time Variability) Exogenous Job Arrival Rate

Priority Rule

FIFO SPT SPTWIN1Q SPTWIN2Q SPTWIN3Q SPTWIN4Q TOTUTIL UTILWRK

1.0

1.1

1.2

1.3

1.4

1.5

Total Utility

Total Utility

Total Utility

Total Utility

Total Utility

Total Utility

Mean

Mean

Mean

Mean

Mean

Mean

2827 2825 2836 2838 2836 2836 2834 2827

3059 3073 3087 3088 3095 3086 3087 3074

3232 3254 3266 3276 3275 3275 3313 3273

3310 3342 3366 3375 3373 3378 3439 3368

3342 3410 3428 3439 3429 3434 3453 3330

3356 3451 3478 3479 3485 3483 3393 3254

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