NPS NRP Executive SummaryAnti-submarine warfare (ASW) is a core mission area for the U.S. Navy and an essential component in achieving sea control. The allocation of torpedoes in support of ASW across a U.S. Fleet is currently a manual process based on shore-based personnel's experience or historical norms. This study will analyze the impact of torpedo loadouts incorporating a complex set of constraints ranging from platform capabilities to probability of kill. A mixed-integer program will be used to compare the effectiveness of different torpedo loadouts. The goal of this work is to maximize the probability of a successful outcome in a theoretical campaign against anticipated submarine threats.N8 - Integration of Capabilities & ResourcesThis research is supported by funding from the Naval Postgraduate School, Naval Research Program (PE 0605853N/2098). https://nps.edu/nrpChief of Naval Operations (CNO)Approved for public release. Distribution is unlimited.

Hauser, Margaret M.

Kline, Jeffrey E.

Salmeron-Medrano, Javier

English

NPS NRP Executive SummaryAnti-submarine warfare (ASW) is a core mission area for the U.S. Navy and an essential component in achieving sea control. The allocation of torpedoes in support of ASW across a U.S. Fleet is currently a manual process based on shore-based personnel's experience or historical norms. This study will analyze the impact of torpedo loadouts incorporating a complex set of constraints ranging from platform capabilities to probability of kill. A mixed-integer program will be used to compare the effectiveness of different torpedo loadouts. The goal of this work is to maximize the probability of a successful outcome in a theoretical campaign against anticipated submarine threats.Approved for public release. Distribution is unlimited. This research is supported by funding from the Naval Postgraduate School, Naval Research Program (PE 0605853N/2098). https://nps.edu/nrpChief of Naval Operations (CNO)N8 - Integration of Capabilities & Resource

Calhoun, Institutional Archive of the Naval Postgraduate School

Calhoun: The NPS Institutional ArchiveDSpace RepositoryFaculty and Researchers Faculty and Researchers' Publications2022Theater Torpedo Inventory OptimizationSalmeron-Medrano, Javier; Hauser, Margaret M.; Kline,Jeffrey E.Monterey, California: Naval Postgraduate Schoolhttps://hdl.handle.net/10945/71914This publication is a work of the U.S. Government as defined in Title 17, UnitedStates Code, Section 101. Copyright protection is not available for this work in theUnited States.Downloaded from NPS Archive: CalhounNPS NRP Executive Summary Theater Torpedo Inventory Optimization Period of Performance: 10/24/2021 – 12/31/2022 Report Date: 12/30/2022 | Project Number: NPS-22-N260-A Naval Postgraduate School, Operations Research (OR) This research is supported by funding from the  Naval Postgraduate School, Naval Research Program (PE 0605853N/2098).  Approved for public release. Distribution is unlimited.     MONTEREY, CALIFORNIA  THEATER TORPEDO INVENTORY OPTIMIZATION  EXECUTIVE SUMMARY   Principal Investigator (PI):  Dr. Javier Salmeron and Dr. Moshe Kress, Operations Research (OR)  Additional Researcher(s):  No additional researchers participated in this research project.  Student Participation:   LT Violeta Lopez, USN, OR  Prepared for:  Topic Sponsor Lead Organization:  N8 - Integration of Capabilities & Resources Topic Sponsor Organization(s): Integration of Capabilities & Resources Topic Sponsor Name(s): LCDR Jeffrey Kee Topic Sponsor Contact Information: 571-256-9566, Jeffrey.kee@navy.mil.NPS NRP Executive Summary Theater Torpedo Inventory Optimization Period of Performance: 10/24/2021 – 12/31/2022 Report Date: 12/30/2022 | Project Number: NPS-22-N260-A Naval Postgraduate School, Operations Research (OR)    Page 2 of 4   Project Summary Torpedo loadout decisions for anti-submarine warfare (ASW) depend on threat scenarios, capacities, capabilities of ASW platforms, and constraints on inventory and/or budget. These decisions are crucial as U.S. adversaries continue to grow their fleets and expand their global operations. Such torpedo loadout decisions are typically made in advance of the detection of an adversary submarine or even before an ASW unit is deployed. Currently, loadout plans for the Mk-54 lightweight torpedoes are made manually, and the goal of the project was to develop a decision aid, using formal mathematical optimization, for allocating ASW torpedoes in an uncertain environment.  We developed the Torpedo Allocation Stochastic Optimization Model (TASOM), which has two versions. TASOM is a mixed-integer, stochastic optimization model that seeks to minimize expected failure to meet operational goals measured by probability of kill. The optimization is subject to budget and inventory constraints, as well as other operational constraints.     Keywords: anti-submarine warfare, ASW, torpedoes, optimal loadout, stochastic optimization  Background  The Mk-54 torpedo type is employed by the MH-60R Seahawk helicopter, P-8 maritime patrol aircraft, and surface ships when conducting ASW operations. The “mod 0” and “mod 1” variants of the Mk-54 are currently in use. The delivery of a “mod 2” variant is expected in fiscal year 2026. Of particular interest is the procurement and allocation of mod 2 torpedoes, as they are anticipated to have significantly improved performance compared to previous variants, but at a substantially higher cost.  This trend of high-effectiveness, high-cost torpedoes underscores the need for efficient allocation of these weapons among the various platforms: helicopters, patrol aircraft and surface ships.  Given a stockpile of torpedoes, a set of weapon-delivery platforms, each with its own capacities and capabilities, and a set of possible scenarios, each of which may realize, with a certain probability, the problem is how best to allocate the stockpile of torpedoes among the platforms. The problem addressed in this research belongs to a large family of weapon-allocation (Page, 1991; Avital, 2004; Brown and Kline, 2021), and target-assignment (Manne, 1958; Ahuja et al., 2007) problems. Our model advances the existing literature on this topic by developing analytic (vs. simulation) models, which consider multiple shooters against multiple targets in an uncertain environment.   Findings and Conclusions  Two optimization models were developed: TASOM-1 and TASOM-2. Their measures of effectiveness are combinations of torpedoes’ cost and expected cost of not meeting operational goals of adversary’s targets killed. The key difference between both models relates to the assessment of fire engagements that do not reach a desired probability of kill threshold. TASOM-1 minimizes the expected number of missed submarines when the consideration is binary: kill probability threshold fully met or not (in which case the target is considered safe). TASOM-2 minimizes the expected shortfall from (i.e., deviation below) the probability of kill threshold. In this case, partial credit is given to engagements that do not fully achieve the desired threshold. Both models also consider constraints on budget for: purchasing torpedoes; inventory of torpedoes already in stock; limited NPS NRP Executive Summary Theater Torpedo Inventory Optimization Period of Performance: 10/24/2021 – 12/31/2022 Report Date: 12/30/2022 | Project Number: NPS-22-N260-A Naval Postgraduate School, Operations Research (OR)    Page 3 of 4   magazine capacity for P-8 squadrons; ships with embarked helicopter detachments; and a limited salvo size for aircraft units.   The TASOMs are stochastic. Instead of assuming perfect information regarding a specific threat scenario, which includes number of targets (adversary submarines) and identity of ASW platforms that will engage them, our stochastic models assume probabilistic information about the likelihood of several possible threat scenarios. The solution that TASOM recommends is a torpedo loadout plan that is not tailored to a deterministic threat scenario; instead, it reaches a compromise among all potential scenarios that are considered in the analysis. Thus, the solution that minimizes the total cost of (a) torpedo purchases; and (b) either expected number of missed submarines (for TASOM-1) or expected shortfall from the desired probability of kill threshold (for TASOM-2), applies to a wide range of possible threat scenarios and not just a single one. The assessed cost for not meeting a probability of kill threshold for a submarine is a planner’s input.   To show the value of the stochastic programming approach over a typical deterministic planning, we present a notional case designed to represent an ASW operation where four destroyers with embarked MH-60R detachments and two P-8 squadrons are patrolling an area for adversary submarines. We assume the adversary fleet comprises 20 submarines of different classes, out of which 5 to 10 submarines are deployed. The desired probability of kill threshold is 90% for all submarine threats. One hundred threat scenarios are randomly generated where a subset of the submarine fleet deploys and appears to the patrolling ASW units. Here, we define a scenario as a configuration of deployed adversary submarines, available ASW units, and subsets of ASW units that can engage a certain submarine.   From running simulations, we observe that TASOM-1 loadout performs marginally better than the average loadout solution that can be obtained using a deterministic approach. The TASOM-2 value over manual or deterministic planning is more apparent.  Our models are combined with an accessible user interface, which facilitates generating scenarios and accessing pertinent results. TASOM uses mathematical optimization, explicitly deals with uncertain scenarios, and facilitates sensitivity analysis. Together, this research provides planners with a decision aid tool that can be used to guide torpedo allocation and budget decisions under uncertainty.   Recommendations for Further Research We recommend further development of Torpedo Allocation Stochastic Optimization Model-2 (TASOM-2) over TASOM-1. The soft constraint used for the desired kill probability threshold is more realistic than the hard, binary, constraint in TASOM-1. Additionally, TASOM-2 solves significantly faster than TASOM-1 and has demonstrated greater performance improvements when compared to the deterministic model for the average loadout and “all-targets” loadout.   Recommended future work includes exploring the effects of penalizing the probability of kill shortfall differently.  For example, an exponential penalty of degree n would incentivize spreading the probability of kill shortfall among different targets (n >1) or concentrate them on fewer targets (n <1). This would involve pre-calculating the penalty on the shortfall of a combination in an engagement.  NPS NRP Executive Summary Theater Torpedo Inventory Optimization Period of Performance: 10/24/2021 – 12/31/2022 Report Date: 12/30/2022 | Project Number: NPS-22-N260-A Naval Postgraduate School, Operations Research (OR)    Page 4 of 4   Finally, only one test case was considered in this thesis, but additional cases should be tested to verify model results. Additional cases can be expanded to consider more realistic anti-submarine warfare (ASW) operations. In our case, all ASW units and all deployed adversary submarines were confined to one area. A more realistic case may involve ASW units patrolling their own separate areas, as in a barrier defense plan. Adversary submarines appearing in one of many areas can easily be controlled when creating scenarios. Additionally, friendly submarines, with their own, separate torpedo inventory, operating in their own, separate area, can be incorporated in a case study to represent a more comprehensive ASW operation. Multiple cases with multiple areas can be run in a series to consider a campaign-level setting.  References Ahuja, R. K., Kumar, A., Jha, K. C., & Orlin, J. B. (2007).  Exact and heuristic algorithms for the weapon target assignment problem. Operations Research, 55(6), 1136–1146.  Avital, I. (2004). Two-period, stochastic supply-chain models with recourse for naval surface warfare. [Master’s thesis, Naval Postgraduate School]. NPS Archive: Calhoun. http://hdl.handle.net/10945/1707 Brown G. G., & Kline, J. E. (2021). Optimizing missile loads for U.S. Navy combatants. Military Operations Research, 26(2). https://www.jstor.org/stable/27028854 Manne, A. S. (1958). A target-assignment problem. Operations Research, 6(3), 346–351.  Page, J. M. (1991). The optimal force mix and allocation of fires for the future field artillery. [Master’s thesis, Naval Postgraduate School]. NPS Archive: Calhoun. http://hdl.handle.net/10945/26671 Acronyms ASW   anti-submarine warfare TASOM  Torpedo Allocation Stochastic Optimization Model 

Theater Torpedo Inventory Optimization

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Theater Torpedo Inventory Optimization

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