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Recent Research for IISE Conference

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Richard Alaimo
Richard Alaimo

This document summarizes an optimization study for transporting spent nuclear fuel from 12 decommissioned nuclear plant sites in the United States to a central repository in West Texas. The study formulates two mixed integer programming models: Model 1 determines the optimal number of transportation casks and railcars to minimize total cost and time, and Model 2 provides a shipment schedule on a yearly basis that satisfies a 1500 metric ton limit and completes all shipments within the estimated time frame from Model 1. The models consider objectives, decision variables, and constraints related to sequencing shipments between sites, allocating resources, and scheduling transport over multiple years to optimize the process.

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Proceedingsof the 2016 Industrial and Systems Engineering Research Conference H. Yang, Z. Kong, and MD Sarder, eds. Optimization of Used Nuclear Fuel Shipments Abstract ID: 1345 Richard Alaimo, William Cole, Justin Ervin, Hanna Tannous, Kevin Wong, Churlzu Lim Department of Systems Engineering and Engineering Management The University of North Carolina at Charlotte Charlotte, North Carolina; 28223, United States of America Sven Bader AREVA Charlotte, North Carolina; 28262, United States of America Abstract Nuclear energy has been and still is an important source of energy for citizens all around the globe. However, once nuclear energy is deemed to have been used to its maximumpotential, it leaves a byproduct known as spent nuclear fuel that must be stored in wet and dry storage to cool off since the material is extremely warm. This process has been going on for decades and it has gotten to the point where there is too much nuclear material in storage in the United States, and it is time to move it all to a central repository facility for final disposal or recycling. In this study, we consider a shipment and transportation plan for 12 nuclear facilities in the United States that have been decommissioned and currently contain spent nuclear fuel in dry storage. This study is involved with data mining from GIS software that measure multiple railroad routes and estimating costs for transporting the spent fuel. Once this data is acquired, we formulate the problem as a mixed integer program that provides an optimal plan, which includes shipment sequencing, resource allocation, and number of resources to purchase, while minimizing the total cost of moving spent fuel. Keywords Nuclear Energy, Mathematical Optimization, Logistics, Cost Estimation, Bi-objective Optimization 1. Introduction Currently in the United States, there are twelve nuclear plant locations that are no longer processing nuclear material and have become shutdown.These plant sites hold large amounts of spent nuclear fuel (SNF) that are held in multiple types of dry storage. Before the SNF assemblies can be transported to their final destination in west Texas (Waste Control Specialists, WCS), they must sit in spent fuel pools for five years, and then dry storage areas to ensure safety of the public and environment by allowing time to cool due to heat generated from radioactive decay. Actual shipments to the permanent storage in west Texas are expected to begin in 2020 [3, 6, 7]. In this research, we analyze shipment-related attributes of currently shutdown sites in the United States (see Table 1 for information of each site) and propose a mathematical optimization model that finds a shipment and transportation plan to move all of the SNF to west Texas. In specific, we present a bi-objective optimization model that considers two objectives, total shipment costs and time to finish all shipments. The resulting plan needs to address the number of transportation casks,the number of railcars that are specialized to transport SNF, the sequence of sites for which SNF are transported,and the shipment schedule for each site. We present the optimization model in Section 2, and a preliminary result under estimated cost and time parameters in Section 3. Then, a final concluding remark is provided in Section 4.
Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader Table 1: Summary of 14 Shutdown Sites at 12 Locations 2. Mathematical Optimization Models In this section, we propose a two-step optimization method, which consists of two respective mixed integer programing (MIP) models to determine a shipment plan considering time and costs. In the first step, Model (1) determines the optimal number of transportation casks and railcars to purchase that minimizes the total cost and time of shipping SNF from each site. Subsequently in the second step, Model (2) provides a shipment schedule, on a yearly basis, that completes all sites within an estimated time frame (output from Model (1)) and satisfies the yearly inflow constraint of at most 1500 metric tons of heavy metal (MTHM) at WCS. 2.1 Model (1) Formulation To formulate the problem under general setting, consider N sites from which SNF needs to be transported to WCS. Assume that each site uses one of 𝐿 transportation cask types. Furthermore, assume that SNF fromonly up to K sites can be simultaneously processed, or equivalently, there are 𝐾 transportation crew teams. Note that this problem setting resembles a job scheduling problem on identical parallel machines (i.e., crew teams). In analogy to the job scheduling problem, let makespan denote the total estimated time to complete shipments fromall sites. Then, Model (1) determines the number of transportation casks for each type, the number of railcars, and assignment of sites to crew teams in order to optimize the cost as well as the makespan. Due to its enormous weight and safety issue, it is reasonable to limit the number of casks for each type, and let J denote this upper bound. The decision variables used throughout Model (1) formulation are as follows: π‘₯ 𝑖,οΏ½οΏ½οΏ½οΏ½,π‘˜ = binary variable representing j transportation casks that are purchased for site i and assigned to crew team k, 𝑛𝑗,π‘˜,𝑙 = binary variable representing j transportation casks of type l that are purchased and assigned to crew team k, 𝑛𝑙 = integer variable representing the number of transportation casks of type l that are purchased, 𝑑 π‘˜ = auxiliary continuous variable representing the upperbound on the makespan of crew team k, π‘₯ π‘Ÿ = integer variable representing the number of railcars to purchase, 𝐢𝑙 = continuous variable representing total shipment cost for those sites using transportation cask type 𝑙, plus the purchasing cost of cask type 𝑙, 𝐢 = continuous variable representing total cost, 𝑇 = continuous variable representing total makespan (in days). Model (1) has two objectives, minimizing the total cost and the makespan, hence constitutes a bi-objective optimization problem. To measure cost and time on a same scale, they are standardized along with a weighted parameter 𝛼 ∈ [0,1] as in (1). Objective Function (1): Minimize 𝛼 𝐢 𝐢 π‘šπ‘Žπ‘₯βˆ’ 𝐢 π‘šπ‘–π‘› + (1 βˆ’ 𝛼) 𝑇 𝑇 π‘šπ‘Žπ‘₯βˆ’π‘‡ π‘šπ‘–π‘› (1) The values of 𝐢 π‘šπ‘Žπ‘₯ and 𝐢 π‘šπ‘–π‘›were obtained by solving a single-objective optimization problem that exclusively maximizes and minimizes cost using Model (1); vice versa for 𝑇 π‘šπ‘Žπ‘₯ and 𝑇 π‘šπ‘–π‘› . The constraints in Model (1) are as follows:
Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader Constraint (2) ensures that site 𝑖 is assigned to a single crew teamand uses the number of casks between 1 and J. βˆ‘ βˆ‘ π‘₯ 𝑖,𝑗,π‘˜ = 1 for 𝑖 = 1, … , 𝑁𝐾 π‘˜ =1 𝐽 𝑗=1 (2) Constraint (3) determines the number of transportation casks for each type and the crew team that uses this type. βˆ‘ βˆ‘ 𝑛𝑗,π‘˜,𝑙 = 1 for 𝑙 = 1, … , 𝐿𝐾 π‘˜=1 𝐽 𝑗=1 (3) Constraint (4) relates the sites to transportation cask types, and ensures that sites using the same transportation cask type are assigned to the same crew team. Let 𝑁𝑙 denote the set of sites using cast type 𝑙, and 𝑠𝑙 denote the cardinality of 𝑁𝑙. βˆ‘ π‘₯ 𝑖,𝑗,π‘˜π‘–βˆˆπ‘ 𝑙 = 𝑠𝑙 𝑛𝑗,π‘˜,𝑙 for 𝑗 = 1, … , 𝐽, π‘˜ = 1,… , 𝐾, 𝑙 = 1, … , 𝐿 (4) Constraint (5) assigns the number of transportation casks of type l to purchase. 𝑛𝑙 = βˆ‘ βˆ‘ 𝑗𝑛𝑗,π‘˜,𝑙 𝐾 π‘˜ =1 for 𝑙 = 1, … , 𝐿 𝐽 𝑗=1 (5) Let 𝑝𝑖 ,𝑗 denote the total shipment time for site i when j transportation casks are available. Then, Constraint (6) calculates the upper bound on the makespan of crew team k. 𝑑 π‘˜ = βˆ‘ βˆ‘ 𝑝𝑖,𝑗 π‘₯ 𝑖,𝑗,π‘˜ for π‘˜ = 1, … , 𝐾 𝐽 𝑗=1 𝑁 𝑖=1 (6) Constraint (7) calculates the upper bound on the makespan to complete shipments from all N sites. In conjunction with the objective function that is minimized, this constraint effectively assigns the makespan to T. 𝑇 β‰₯ 𝑑 π‘˜ for π‘˜ = 1, … , 𝐾 (7) Constraint (8) computes the amount of railcars to purchase. The underlying idea is to purchase an equivalent number of railcars as the maximum sum of all combination of transportation casks that can be simultaneously shipped. Let 𝐿(𝐾) denote the collection of subsets of {1,2, … , 𝐿} having a cardinality of 𝐾. βˆ‘ π‘›π‘™π‘™βˆˆπΏ π‘˜ ≀ π‘₯ π‘Ÿ for 𝐿 π‘˜ ∈ 𝐿(𝐾) (8) Constraint (9) calculates the shipment costs fromsites using transportation cask type l, plus purchasing cost of cask type 𝑙. Let 𝑐𝑖,𝑗 denote the shipment cost of site 𝑖 when 𝑗 casks are used, and 𝑐𝑙 denote the purchasing cost of one transportation cask of type l. 𝐢𝑙 = βˆ‘ βˆ‘ βˆ‘ 𝑐𝑖,𝑗 π‘₯ 𝑖,𝑗,π‘˜ +𝐾 π‘˜ =1 𝐽 𝑗=1 𝑐𝑙 π‘›π‘™π‘–βˆˆπ‘ 𝑙 for 𝑙 = 1, … , 𝐿 (9) Finally, Constraint (10) calculates the overall cost, where 𝑐 π‘Ÿ denotes the purchasing cost of single railcar. 𝐢 = βˆ‘ 𝐢𝑙 + 𝑐 π‘₯ π‘Ÿ π‘₯ π‘Ÿ 𝐿 𝑙=1 (10) 2.2 Model (2) Formulation Recall that Model (1) only determines the number of casks, the number of railcars, and the assignment of sites to crew teams by minimizing the bi-objective function (1) without considering the 1500 MTHM requirement. After the solution to Model (1) is obtained, Model (2) is formulated and solved to determine a feasible shipment schedule for each crew team by enforcing the annual 1500 MTHM requirement. The number of transportation casks that are available for site i can be determined as a result of solving Model (1), and is denoted by 𝑛𝑖 . The decision variables used in the Model (2) formulation are as follows:
Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader π‘₯ 𝑖,π‘˜,𝑀 = integer variable representing the number of canisters shipped from site i by crew team k during year w, π‘₯ 𝑖,π‘˜,𝑀 𝑏 = binary variable representing whether or not site i shipped any canisters by crew team k during year w (if π‘₯ 𝑖,π‘˜,𝑀 is greater than zero, then π‘₯ 𝑖,π‘˜,𝑀 𝑏 is equal to one. Otherwise, π‘₯ 𝑖,𝑀𝑏 π‘˜ is equal to zero), 𝑦𝑖 ,𝑀 = binary variable that assists in the sequencing of sites for all years except the last, 𝑑 π‘˜,𝑀 = continuous variable representing the time duration for crew team k to work during year w, 𝑑 𝑀 = continuous variable representing the work time duration in year w, π‘š 𝑀 = continuous variable representing the amount of SNF that is shipped during year w, 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 = integer variable representing the number of canisters that are shipped in site i’s sth shipment by crew team k during year w, 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 𝑏 = binary variable representing whether or not site i performs its sth shipment by crew team k during year w. Note that upperbound for 𝑠 and 𝑀 are calculated by 𝛾𝑖 = ⌈ number of canisters at site 𝑖 number of casks available for site 𝑖 βŒ‰, and 𝛿 is estimated by scheduling any random order of sites manually and evaluating the total number of years, respectively. This is a single-objective optimization problem which, upon finding a solution, will provide a shipment schedule to complete all N sites given the number of transportation casks and railcars along with the crew teamassignment. The idea of this optimization model is to complete entire shipments within any given year, and then continue to do so for the remaining years. The objective of Model (2) is to minimize the number of years that it takes for shipments to complete. Hence, the objective function (11) is given by Minimize βˆ‘ βˆ‘ βˆ‘ π‘₯ 𝑖,π‘˜,𝑀 𝑏 + βˆ‘ βˆ‘ 𝑦𝑖 ,𝑀 π›Ώβˆ’1 𝑀=1 𝑛 𝑖=1 𝛿 𝑀=1 𝐾 π‘˜=1 𝑁 𝑖=1 (11) The constraints that were formulated in Model (2) are explained in what follows. Constraint (12) allows the model to recognize how many canisters of SNF are to be shipped from site i. Upon finding a solution for Model (1), the time frame is then established to ship SNF from all sites over a span of 𝛿 years and maintain feasibility. Denoting the number of canisters for site i by 𝑀𝑖 , we have βˆ‘ βˆ‘ π‘₯ 𝑖,π‘˜,𝑀 𝛿 𝑀=1 𝐾 π‘˜ =1 = 𝑀𝑖 for 𝑖 = 1, … , 𝑁 (12) Constraint (13) calculates the amount of SNF (in MTHM) that is shipped during year w. Denoting the amount of MTHM per canister for site i by π‘šπ‘–, we have βˆ‘ π‘šπ‘– π‘₯ 𝑖,π‘˜,𝑀 𝑁 𝑖=1 = π‘š 𝑀 for π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (13) Constraint (14) limits the amount of SNF that can be shipped for any year. π‘š 𝑀 ≀ 1500 for 𝑀 = 1, … , 𝛿 (14) Constraint (15) allows the model to recognize whether site i shipped any canisters during year w. π‘₯ 𝑖,π‘˜,𝑀 ≀ 𝑀𝑖 π‘₯ 𝑖,π‘˜,𝑀 𝑏 for 𝑖 = 1, … , 𝑁, π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (15) Constraint (16) assists in the sequencing of sites over consecutive years.This will enforce that site i continues shipping once it begins. π‘₯ 𝑖,π‘˜,𝑀 𝑏 + 𝑦𝑖,𝑀 β‰₯ π‘₯ 𝑖,π‘˜,𝑀+1 𝑏 for 𝑖 = 1,… , 𝑁, π‘˜ = 1, … , 𝐾, and, 𝑀 = 1, … , 𝛿 βˆ’ 1 (16) For example, consider the following scenario for site 1 where it takes more than one year to complete shipping all of its canisters. Assume that site 1 begins shipping during year one. This results in decision variable π‘₯1,π‘˜,1 𝑏 = 1. With that being said, it is then possible for π‘₯1,π‘˜,2 𝑏 to equal 1 as well. However if π‘₯1,π‘˜,1 𝑏 = 0 and π‘₯1,π‘˜,2 𝑏 = 1, then it would force 𝑦1,1 to equal 1 to satisfy Constraint (16).
Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader Constraint (17) limits the sumof decision variables π‘₯ 𝑖,π‘˜,𝑀 𝑏 and 𝑦𝑖 ,𝑀 to be less than or equal to 1 for year w. π‘₯ 𝑖,π‘˜,𝑀 𝑏 + 𝑦𝑖 ,𝑀 ≀ 1 for 𝑖 = 1,… , 𝑁, π‘˜ = 1, … , 𝐾,and 𝑀 = 1, … , 𝛿 βˆ’ 1 (17) Constraint (18) allocates the amount of SNF that is shipped from site i’s sth shipment during year w. If π‘₯ 𝑖,π‘˜,𝑀 is greater than 0, then it will force 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 to be greater than 0 (for any value of 𝑠𝑖). βˆ‘ 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 = π‘₯ 𝑖,π‘˜,𝑀 for 𝑖 = 1, … , 𝑁, π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 𝛾𝑖 𝑠𝑖=1 (18) Constraint (19) limits the number of canisters that can be shipped from site i’s sth shipment. Each site is limited to shipping less than or equal to the number of transportation casks that are available. 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 ≀ 𝑛𝑙 for 𝑖 ∈ 𝑁𝑙, 𝑠 = 1, … , 𝛾𝑖 , π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (19) Constraint (20) allows the model to recognize if site i completes its sth shipment during year w. 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 ≀ 𝑀𝑖 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 𝑏 for 𝑖 = 1, … , 𝑁, 𝑠 = 1, … , 𝛾𝑖, π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (20) Constraint (21) limits site i’s sth shipment to occur only once over the span of 𝛿 years. βˆ‘ 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 𝑏 = 1𝛿 𝑀 =1 for 𝑖 = 1, … , 𝑁, 𝑠 = 1, … , 𝛾𝑖 , and π‘˜ = 1, … , 𝐾 (21) Constraint (22) calculates the time duration for crew team k to work during year w. Loading and unloading times are variable depending on the number of canisters.Shipping time is constant regardless of the number of canisters being shipped.The sum of loading and unloading times per canister can be denoted by 𝜌 (which is the same for all N sites). Shipping times for site i can be denoted by 𝑑𝑖. 𝜌 βˆ‘ π‘₯ 𝑖,π‘˜,𝑀 +𝑁 𝑖=1 βˆ‘ βˆ‘ 𝑑𝑖 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 𝑏 = 𝑑 π‘˜,𝑀 𝛾𝑖 𝑠=1 𝑁 𝑖=1 for π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (22) Constraint (23) establishes an upperbound on the number of workdays per year, which is denoted by 𝐷. 𝑑 π‘˜,𝑀 ≀ 𝐷 for π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (23) Constraint (24) enforces 𝑑 𝑀 to be an upper bound on crew team π‘˜β€™s workdays in year 𝑀. For all Ξ΄ years, the maximum allowed time is 365 days. 𝑑 π‘˜,𝑀 ≀ 𝑑 𝑀 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ π‘˜ = 1, … , 𝐾 , 𝑀 = 1, … , 𝛿 (24) 3. Empirical Result The prosed method was applied to the 12 shutdown locations with two crew teams (𝐾 = 2). There are 2 locations (Kewaunee and San Onofre) that require two different types of transportation casks, hence the total number of sites becomes 14 (see Table 1). Transportation routes from 12 locations were determined by shortest paths using Transportation Routing Analysis Geographic Information System (TRAGIS) [3, 5] (see [1] for an alternative method to select routes based on travel time, risk and population size exposed). CPLEX Solver [2] was used to solve Model (1) and Model (2). Model (1) was solved instantly, whereas Model (2) took approximately 150.95 seconds before an optimal solution was found. As a result of randomly scheduling any sequence of sites after the output fromModel (1) was obtained, Ξ΄ was found to be 8 years. Weighted parameter 𝛼 in the bi-objective function in Model (1) was set to 0.6 to obtain a solution for Model (2). Table 2 displays a summarized result from the solution to Model (1). Subsequently, Table 3 presents detailed shipment schedule for original 14 sites. In summary, the makespan is expected to be no more than 8 years while the total cost is expected to be close to $256 million. 4. Conclusion
Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader In this study, we consider a spent nuclear fuel (SNF) shipment problem. To find an optimal number of transportation casks and railcars, as well as shipment schedule, we propose a two-step method. The first step solves a bi-objective optimization to minimize both time and cost, while the second step solves a mixed integer program to determine shipment schedule. A numerical result is provided for shipping SNF from 12 shutdown nuclear power plant locations under estimated costs and shipping times per canister. Accommodating uncertainty can be an avenue for future study. Table 2: Results from the Solution to Model (1) Table 3: Results from the Solution to Model (2) Acknowledgements This study was conducted as part a Senior Design project at UNC Charlotte and sponsored by AREVA. We also acknowledge Oak Ridge National Laboratory for providing us with TRAGIS, and William Murphy from the Catawba nuclear facility for providing an informational tourof the site. References [1] Chen, Yun-Wen, Chi-Hwang Wang,and Sain-Ju Lin, A Multi-objective Geographic Information System for Route Selection of Nuclear Waste Transport,Omega, 36 (3), 363-372, 2008. [2] IBM, IBM Knowledge Center,Accessed October19th, 2015. http://www- 01.ibm.com/support/knowledgecenter/SSSA5P_12.6.2/ilog.odms.cplex.help/CPLEX/GettingStarted/topics/tut orials/InteractiveOptimizer/tutorial_synopsis.html [3] Johnson,P. E., and R. D. Michelhaugh, Transportation Routing Analysis Geographic Information System (WebTRAGIS) User’s Manual, April 2000, Accessed October14th, 2015. http://web.ornl.gov/~webworks/cpr/v823/rpt/106749.pdf. [4] Oak Ridge National Laboratory, Centralized Used Fuel Resource for Information Exchange,Accessed October 12th, 2015, https://curie.ornl.gov/. [5] Oak Ridge National Laboratory. Transportation Routing Analysis Geographic Information System, Released 2012, Gained Access October 14th, 2015. [6] Sherrell R. Greene, James S. Medford, Sharon A. Macy. β€œStorage and Transport Cask Data for Used Commercial Nuclear Fuel.” EnergX, LLC. 314 pages. Published 9 August,2013. Accessed 9 December, 2015 ATI-TR-13047. [7] Steven J. Maheras, Ralph E. Best, Steven E. Ross,Kenneth A Buxton, Jeffery L. England, Paul E. McConnell. β€œPreliminary Evaluation of Removing Used Nuclear Fuel from Shutdown Sites”. DOE (U.S. Department of Energy). 256 pages.Published 1 October, 2014. Accessed 10 December, 2015. PNNL-22676 Rev. 4.

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Recent Research for IISE Conference

  • 1. Proceedingsof the 2016 Industrial and Systems Engineering Research Conference H. Yang, Z. Kong, and MD Sarder, eds. Optimization of Used Nuclear Fuel Shipments Abstract ID: 1345 Richard Alaimo, William Cole, Justin Ervin, Hanna Tannous, Kevin Wong, Churlzu Lim Department of Systems Engineering and Engineering Management The University of North Carolina at Charlotte Charlotte, North Carolina; 28223, United States of America Sven Bader AREVA Charlotte, North Carolina; 28262, United States of America Abstract Nuclear energy has been and still is an important source of energy for citizens all around the globe. However, once nuclear energy is deemed to have been used to its maximumpotential, it leaves a byproduct known as spent nuclear fuel that must be stored in wet and dry storage to cool off since the material is extremely warm. This process has been going on for decades and it has gotten to the point where there is too much nuclear material in storage in the United States, and it is time to move it all to a central repository facility for final disposal or recycling. In this study, we consider a shipment and transportation plan for 12 nuclear facilities in the United States that have been decommissioned and currently contain spent nuclear fuel in dry storage. This study is involved with data mining from GIS software that measure multiple railroad routes and estimating costs for transporting the spent fuel. Once this data is acquired, we formulate the problem as a mixed integer program that provides an optimal plan, which includes shipment sequencing, resource allocation, and number of resources to purchase, while minimizing the total cost of moving spent fuel. Keywords Nuclear Energy, Mathematical Optimization, Logistics, Cost Estimation, Bi-objective Optimization 1. Introduction Currently in the United States, there are twelve nuclear plant locations that are no longer processing nuclear material and have become shutdown.These plant sites hold large amounts of spent nuclear fuel (SNF) that are held in multiple types of dry storage. Before the SNF assemblies can be transported to their final destination in west Texas (Waste Control Specialists, WCS), they must sit in spent fuel pools for five years, and then dry storage areas to ensure safety of the public and environment by allowing time to cool due to heat generated from radioactive decay. Actual shipments to the permanent storage in west Texas are expected to begin in 2020 [3, 6, 7]. In this research, we analyze shipment-related attributes of currently shutdown sites in the United States (see Table 1 for information of each site) and propose a mathematical optimization model that finds a shipment and transportation plan to move all of the SNF to west Texas. In specific, we present a bi-objective optimization model that considers two objectives, total shipment costs and time to finish all shipments. The resulting plan needs to address the number of transportation casks,the number of railcars that are specialized to transport SNF, the sequence of sites for which SNF are transported,and the shipment schedule for each site. We present the optimization model in Section 2, and a preliminary result under estimated cost and time parameters in Section 3. Then, a final concluding remark is provided in Section 4.
  • 2. Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader Table 1: Summary of 14 Shutdown Sites at 12 Locations 2. Mathematical Optimization Models In this section, we propose a two-step optimization method, which consists of two respective mixed integer programing (MIP) models to determine a shipment plan considering time and costs. In the first step, Model (1) determines the optimal number of transportation casks and railcars to purchase that minimizes the total cost and time of shipping SNF from each site. Subsequently in the second step, Model (2) provides a shipment schedule, on a yearly basis, that completes all sites within an estimated time frame (output from Model (1)) and satisfies the yearly inflow constraint of at most 1500 metric tons of heavy metal (MTHM) at WCS. 2.1 Model (1) Formulation To formulate the problem under general setting, consider N sites from which SNF needs to be transported to WCS. Assume that each site uses one of 𝐿 transportation cask types. Furthermore, assume that SNF fromonly up to K sites can be simultaneously processed, or equivalently, there are 𝐾 transportation crew teams. Note that this problem setting resembles a job scheduling problem on identical parallel machines (i.e., crew teams). In analogy to the job scheduling problem, let makespan denote the total estimated time to complete shipments fromall sites. Then, Model (1) determines the number of transportation casks for each type, the number of railcars, and assignment of sites to crew teams in order to optimize the cost as well as the makespan. Due to its enormous weight and safety issue, it is reasonable to limit the number of casks for each type, and let J denote this upper bound. The decision variables used throughout Model (1) formulation are as follows: π‘₯ 𝑖,𝑗,π‘˜ = binary variable representing j transportation casks that are purchased for site i and assigned to crew team k, 𝑛𝑗,π‘˜,𝑙 = binary variable representing j transportation casks of type l that are purchased and assigned to crew team k, 𝑛𝑙 = integer variable representing the number of transportation casks of type l that are purchased, 𝑑 π‘˜ = auxiliary continuous variable representing the upperbound on the makespan of crew team k, π‘₯ π‘Ÿ = integer variable representing the number of railcars to purchase, 𝐢𝑙 = continuous variable representing total shipment cost for those sites using transportation cask type 𝑙, plus the purchasing cost of cask type 𝑙, 𝐢 = continuous variable representing total cost, 𝑇 = continuous variable representing total makespan (in days). Model (1) has two objectives, minimizing the total cost and the makespan, hence constitutes a bi-objective optimization problem. To measure cost and time on a same scale, they are standardized along with a weighted parameter 𝛼 ∈ [0,1] as in (1). Objective Function (1): Minimize 𝛼 𝐢 𝐢 π‘šπ‘Žπ‘₯βˆ’ 𝐢 π‘šπ‘–π‘› + (1 βˆ’ 𝛼) 𝑇 𝑇 π‘šπ‘Žπ‘₯βˆ’π‘‡ π‘šπ‘–π‘› (1) The values of 𝐢 π‘šπ‘Žπ‘₯ and 𝐢 π‘šπ‘–π‘›were obtained by solving a single-objective optimization problem that exclusively maximizes and minimizes cost using Model (1); vice versa for 𝑇 π‘šπ‘Žπ‘₯ and 𝑇 π‘šπ‘–π‘› . The constraints in Model (1) are as follows:
  • 3. Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader Constraint (2) ensures that site 𝑖 is assigned to a single crew teamand uses the number of casks between 1 and J. βˆ‘ βˆ‘ π‘₯ 𝑖,𝑗,π‘˜ = 1 for 𝑖 = 1, … , 𝑁𝐾 π‘˜ =1 𝐽 𝑗=1 (2) Constraint (3) determines the number of transportation casks for each type and the crew team that uses this type. βˆ‘ βˆ‘ 𝑛𝑗,π‘˜,𝑙 = 1 for 𝑙 = 1, … , 𝐿𝐾 π‘˜=1 𝐽 𝑗=1 (3) Constraint (4) relates the sites to transportation cask types, and ensures that sites using the same transportation cask type are assigned to the same crew team. Let 𝑁𝑙 denote the set of sites using cast type 𝑙, and 𝑠𝑙 denote the cardinality of 𝑁𝑙. βˆ‘ π‘₯ 𝑖,𝑗,π‘˜π‘–βˆˆπ‘ 𝑙 = 𝑠𝑙 𝑛𝑗,π‘˜,𝑙 for 𝑗 = 1, … , 𝐽, π‘˜ = 1,… , 𝐾, 𝑙 = 1, … , 𝐿 (4) Constraint (5) assigns the number of transportation casks of type l to purchase. 𝑛𝑙 = βˆ‘ βˆ‘ 𝑗𝑛𝑗,π‘˜,𝑙 𝐾 π‘˜ =1 for 𝑙 = 1, … , 𝐿 𝐽 𝑗=1 (5) Let 𝑝𝑖 ,𝑗 denote the total shipment time for site i when j transportation casks are available. Then, Constraint (6) calculates the upper bound on the makespan of crew team k. 𝑑 π‘˜ = βˆ‘ βˆ‘ 𝑝𝑖,𝑗 π‘₯ 𝑖,𝑗,π‘˜ for π‘˜ = 1, … , 𝐾 𝐽 𝑗=1 𝑁 𝑖=1 (6) Constraint (7) calculates the upper bound on the makespan to complete shipments from all N sites. In conjunction with the objective function that is minimized, this constraint effectively assigns the makespan to T. 𝑇 β‰₯ 𝑑 π‘˜ for π‘˜ = 1, … , 𝐾 (7) Constraint (8) computes the amount of railcars to purchase. The underlying idea is to purchase an equivalent number of railcars as the maximum sum of all combination of transportation casks that can be simultaneously shipped. Let 𝐿(𝐾) denote the collection of subsets of {1,2, … , 𝐿} having a cardinality of 𝐾. βˆ‘ π‘›π‘™π‘™βˆˆπΏ π‘˜ ≀ π‘₯ π‘Ÿ for 𝐿 π‘˜ ∈ 𝐿(𝐾) (8) Constraint (9) calculates the shipment costs fromsites using transportation cask type l, plus purchasing cost of cask type 𝑙. Let 𝑐𝑖,𝑗 denote the shipment cost of site 𝑖 when 𝑗 casks are used, and 𝑐𝑙 denote the purchasing cost of one transportation cask of type l. 𝐢𝑙 = βˆ‘ βˆ‘ βˆ‘ 𝑐𝑖,𝑗 π‘₯ 𝑖,𝑗,π‘˜ +𝐾 π‘˜ =1 𝐽 𝑗=1 𝑐𝑙 π‘›π‘™π‘–βˆˆπ‘ 𝑙 for 𝑙 = 1, … , 𝐿 (9) Finally, Constraint (10) calculates the overall cost, where 𝑐 π‘Ÿ denotes the purchasing cost of single railcar. 𝐢 = βˆ‘ 𝐢𝑙 + 𝑐 π‘₯ π‘Ÿ π‘₯ π‘Ÿ 𝐿 𝑙=1 (10) 2.2 Model (2) Formulation Recall that Model (1) only determines the number of casks, the number of railcars, and the assignment of sites to crew teams by minimizing the bi-objective function (1) without considering the 1500 MTHM requirement. After the solution to Model (1) is obtained, Model (2) is formulated and solved to determine a feasible shipment schedule for each crew team by enforcing the annual 1500 MTHM requirement. The number of transportation casks that are available for site i can be determined as a result of solving Model (1), and is denoted by 𝑛𝑖 . The decision variables used in the Model (2) formulation are as follows:
  • 4. Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader π‘₯ 𝑖,π‘˜,𝑀 = integer variable representing the number of canisters shipped from site i by crew team k during year w, π‘₯ 𝑖,π‘˜,𝑀 𝑏 = binary variable representing whether or not site i shipped any canisters by crew team k during year w (if π‘₯ 𝑖,π‘˜,𝑀 is greater than zero, then π‘₯ 𝑖,π‘˜,𝑀 𝑏 is equal to one. Otherwise, π‘₯ 𝑖,𝑀𝑏 π‘˜ is equal to zero), 𝑦𝑖 ,𝑀 = binary variable that assists in the sequencing of sites for all years except the last, 𝑑 π‘˜,𝑀 = continuous variable representing the time duration for crew team k to work during year w, 𝑑 𝑀 = continuous variable representing the work time duration in year w, π‘š 𝑀 = continuous variable representing the amount of SNF that is shipped during year w, 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 = integer variable representing the number of canisters that are shipped in site i’s sth shipment by crew team k during year w, 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 𝑏 = binary variable representing whether or not site i performs its sth shipment by crew team k during year w. Note that upperbound for 𝑠 and 𝑀 are calculated by 𝛾𝑖 = ⌈ number of canisters at site 𝑖 number of casks available for site 𝑖 βŒ‰, and 𝛿 is estimated by scheduling any random order of sites manually and evaluating the total number of years, respectively. This is a single-objective optimization problem which, upon finding a solution, will provide a shipment schedule to complete all N sites given the number of transportation casks and railcars along with the crew teamassignment. The idea of this optimization model is to complete entire shipments within any given year, and then continue to do so for the remaining years. The objective of Model (2) is to minimize the number of years that it takes for shipments to complete. Hence, the objective function (11) is given by Minimize βˆ‘ βˆ‘ βˆ‘ π‘₯ 𝑖,π‘˜,𝑀 𝑏 + βˆ‘ βˆ‘ 𝑦𝑖 ,𝑀 π›Ώβˆ’1 𝑀=1 𝑛 𝑖=1 𝛿 𝑀=1 𝐾 π‘˜=1 𝑁 𝑖=1 (11) The constraints that were formulated in Model (2) are explained in what follows. Constraint (12) allows the model to recognize how many canisters of SNF are to be shipped from site i. Upon finding a solution for Model (1), the time frame is then established to ship SNF from all sites over a span of 𝛿 years and maintain feasibility. Denoting the number of canisters for site i by 𝑀𝑖 , we have βˆ‘ βˆ‘ π‘₯ 𝑖,π‘˜,𝑀 𝛿 𝑀=1 𝐾 π‘˜ =1 = 𝑀𝑖 for 𝑖 = 1, … , 𝑁 (12) Constraint (13) calculates the amount of SNF (in MTHM) that is shipped during year w. Denoting the amount of MTHM per canister for site i by π‘šπ‘–, we have βˆ‘ π‘šπ‘– π‘₯ 𝑖,π‘˜,𝑀 𝑁 𝑖=1 = π‘š 𝑀 for π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (13) Constraint (14) limits the amount of SNF that can be shipped for any year. π‘š 𝑀 ≀ 1500 for 𝑀 = 1, … , 𝛿 (14) Constraint (15) allows the model to recognize whether site i shipped any canisters during year w. π‘₯ 𝑖,π‘˜,𝑀 ≀ 𝑀𝑖 π‘₯ 𝑖,π‘˜,𝑀 𝑏 for 𝑖 = 1, … , 𝑁, π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (15) Constraint (16) assists in the sequencing of sites over consecutive years.This will enforce that site i continues shipping once it begins. π‘₯ 𝑖,π‘˜,𝑀 𝑏 + 𝑦𝑖,𝑀 β‰₯ π‘₯ 𝑖,π‘˜,𝑀+1 𝑏 for 𝑖 = 1,… , 𝑁, π‘˜ = 1, … , 𝐾, and, 𝑀 = 1, … , 𝛿 βˆ’ 1 (16) For example, consider the following scenario for site 1 where it takes more than one year to complete shipping all of its canisters. Assume that site 1 begins shipping during year one. This results in decision variable π‘₯1,π‘˜,1 𝑏 = 1. With that being said, it is then possible for π‘₯1,π‘˜,2 𝑏 to equal 1 as well. However if π‘₯1,π‘˜,1 𝑏 = 0 and π‘₯1,π‘˜,2 𝑏 = 1, then it would force 𝑦1,1 to equal 1 to satisfy Constraint (16).
  • 5. Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader Constraint (17) limits the sumof decision variables π‘₯ 𝑖,π‘˜,𝑀 𝑏 and 𝑦𝑖 ,𝑀 to be less than or equal to 1 for year w. π‘₯ 𝑖,π‘˜,𝑀 𝑏 + 𝑦𝑖 ,𝑀 ≀ 1 for 𝑖 = 1,… , 𝑁, π‘˜ = 1, … , 𝐾,and 𝑀 = 1, … , 𝛿 βˆ’ 1 (17) Constraint (18) allocates the amount of SNF that is shipped from site i’s sth shipment during year w. If π‘₯ 𝑖,π‘˜,𝑀 is greater than 0, then it will force 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 to be greater than 0 (for any value of 𝑠𝑖). βˆ‘ 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 = π‘₯ 𝑖,π‘˜,𝑀 for 𝑖 = 1, … , 𝑁, π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 𝛾𝑖 𝑠𝑖=1 (18) Constraint (19) limits the number of canisters that can be shipped from site i’s sth shipment. Each site is limited to shipping less than or equal to the number of transportation casks that are available. 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 ≀ 𝑛𝑙 for 𝑖 ∈ 𝑁𝑙, 𝑠 = 1, … , 𝛾𝑖 , π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (19) Constraint (20) allows the model to recognize if site i completes its sth shipment during year w. 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 ≀ 𝑀𝑖 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 𝑏 for 𝑖 = 1, … , 𝑁, 𝑠 = 1, … , 𝛾𝑖, π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (20) Constraint (21) limits site i’s sth shipment to occur only once over the span of 𝛿 years. βˆ‘ 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 𝑏 = 1𝛿 𝑀 =1 for 𝑖 = 1, … , 𝑁, 𝑠 = 1, … , 𝛾𝑖 , and π‘˜ = 1, … , 𝐾 (21) Constraint (22) calculates the time duration for crew team k to work during year w. Loading and unloading times are variable depending on the number of canisters.Shipping time is constant regardless of the number of canisters being shipped.The sum of loading and unloading times per canister can be denoted by 𝜌 (which is the same for all N sites). Shipping times for site i can be denoted by 𝑑𝑖. 𝜌 βˆ‘ π‘₯ 𝑖,π‘˜,𝑀 +𝑁 𝑖=1 βˆ‘ βˆ‘ 𝑑𝑖 𝑧𝑖,𝑠𝑖,π‘˜,𝑀 𝑏 = 𝑑 π‘˜,𝑀 𝛾𝑖 𝑠=1 𝑁 𝑖=1 for π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (22) Constraint (23) establishes an upperbound on the number of workdays per year, which is denoted by 𝐷. 𝑑 π‘˜,𝑀 ≀ 𝐷 for π‘˜ = 1, … , 𝐾, and 𝑀 = 1, … , 𝛿 (23) Constraint (24) enforces 𝑑 𝑀 to be an upper bound on crew team π‘˜β€™s workdays in year 𝑀. For all Ξ΄ years, the maximum allowed time is 365 days. 𝑑 π‘˜,𝑀 ≀ 𝑑 𝑀 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ π‘˜ = 1, … , 𝐾 , 𝑀 = 1, … , 𝛿 (24) 3. Empirical Result The prosed method was applied to the 12 shutdown locations with two crew teams (𝐾 = 2). There are 2 locations (Kewaunee and San Onofre) that require two different types of transportation casks, hence the total number of sites becomes 14 (see Table 1). Transportation routes from 12 locations were determined by shortest paths using Transportation Routing Analysis Geographic Information System (TRAGIS) [3, 5] (see [1] for an alternative method to select routes based on travel time, risk and population size exposed). CPLEX Solver [2] was used to solve Model (1) and Model (2). Model (1) was solved instantly, whereas Model (2) took approximately 150.95 seconds before an optimal solution was found. As a result of randomly scheduling any sequence of sites after the output fromModel (1) was obtained, Ξ΄ was found to be 8 years. Weighted parameter 𝛼 in the bi-objective function in Model (1) was set to 0.6 to obtain a solution for Model (2). Table 2 displays a summarized result from the solution to Model (1). Subsequently, Table 3 presents detailed shipment schedule for original 14 sites. In summary, the makespan is expected to be no more than 8 years while the total cost is expected to be close to $256 million. 4. Conclusion
  • 6. Richard, Cole, Ervin, Tannous,Wong, Lim, and Bader In this study, we consider a spent nuclear fuel (SNF) shipment problem. To find an optimal number of transportation casks and railcars, as well as shipment schedule, we propose a two-step method. The first step solves a bi-objective optimization to minimize both time and cost, while the second step solves a mixed integer program to determine shipment schedule. A numerical result is provided for shipping SNF from 12 shutdown nuclear power plant locations under estimated costs and shipping times per canister. Accommodating uncertainty can be an avenue for future study. Table 2: Results from the Solution to Model (1) Table 3: Results from the Solution to Model (2) Acknowledgements This study was conducted as part a Senior Design project at UNC Charlotte and sponsored by AREVA. We also acknowledge Oak Ridge National Laboratory for providing us with TRAGIS, and William Murphy from the Catawba nuclear facility for providing an informational tourof the site. References [1] Chen, Yun-Wen, Chi-Hwang Wang,and Sain-Ju Lin, A Multi-objective Geographic Information System for Route Selection of Nuclear Waste Transport,Omega, 36 (3), 363-372, 2008. [2] IBM, IBM Knowledge Center,Accessed October19th, 2015. http://www- 01.ibm.com/support/knowledgecenter/SSSA5P_12.6.2/ilog.odms.cplex.help/CPLEX/GettingStarted/topics/tut orials/InteractiveOptimizer/tutorial_synopsis.html [3] Johnson,P. E., and R. D. Michelhaugh, Transportation Routing Analysis Geographic Information System (WebTRAGIS) User’s Manual, April 2000, Accessed October14th, 2015. http://web.ornl.gov/~webworks/cpr/v823/rpt/106749.pdf. [4] Oak Ridge National Laboratory, Centralized Used Fuel Resource for Information Exchange,Accessed October 12th, 2015, https://curie.ornl.gov/. [5] Oak Ridge National Laboratory. Transportation Routing Analysis Geographic Information System, Released 2012, Gained Access October 14th, 2015. [6] Sherrell R. Greene, James S. Medford, Sharon A. Macy. β€œStorage and Transport Cask Data for Used Commercial Nuclear Fuel.” EnergX, LLC. 314 pages. Published 9 August,2013. Accessed 9 December, 2015 ATI-TR-13047. [7] Steven J. Maheras, Ralph E. Best, Steven E. Ross,Kenneth A Buxton, Jeffery L. England, Paul E. McConnell. β€œPreliminary Evaluation of Removing Used Nuclear Fuel from Shutdown Sites”. DOE (U.S. Department of Energy). 256 pages.Published 1 October, 2014. Accessed 10 December, 2015. PNNL-22676 Rev. 4.