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NAVAL POSTGRADUATE SCHOOLMonterey, California

/THESIS /-/

SATISFYING WAR-TIME FUEL REQUIREMENTS WITH AMINIMAL TANKER COMPLEM ENT

by

Kent A. Michaelis

September, 1997

Thesi s Advisor: RobertF. DellSecond Reader: R. KevinWood

Approved for public release; distribution is unlimited.


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September 1997 Masters Thesis

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4. TITLE AND SUBTITLE 5. FUNDING NUMBERSSATISFYING WAR-TIME UEL REQUIREMENTSWITH A MINIMAL

TANKER COMPLEMENT6. AUTHOR(S)

18. SECURITY CLASSIFI- 19. SECURITY CLASSIFI-CATION OF THIS PAGE CATION O F THIS

ABSTRACTUnckssified Unclassified

Michaelis, Kent A. I7. PERFORMING ORGANIZATION NAME@) AND ADDRESS(ES) 18. PERFORMING ORGANIZATION

20. LIMITATION OFABSTRACT

UL

Naval Postgraduate SchoolMonterey, CA 93943-5000

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solution. These results show hzit the schedule generator and the ILP can assist J4-MOB.14. SUBJECT TERMS I15. NUMBER OF

Optimization, Mixed Integer, Scheduling, Tankers, Petroleum, Fuel

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Approved for public release; distribution is unlimited

SATISFYING WAR-TIME FUEL REQUIREMENTS WITH A MINIMALTANKER COMPLEMENT

Kent A. MichaelisLieutenant Comm ander, United States Navy

B.S., University of Florida,1985M.S., Old Dom inion University,1992

Submitted in partial fulfillment of therequirements fo r the degree of

MASTER O F SCIENCE IN OPERATIONS RESEARCH

from the

NAVAL POSTGRADUATE SCHOOLSeptember, 1997

' IAuthor: &A@KentA. Michael$

Approved by:

7 epartment of Operations Research

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ABSTRACT

The United States Transportation Command (USTC) must ensure that sufficient

assets are available to transport the war-time requirements of Petroleum,Oil and

Lubrication (POL) forthe military. To be confident that sufficient assets exist to transport

POL, U STC must know the number of tankers required. The Mobility Divisionof th e

Logistics Directorate of the Joint Staff (J4-MO B)uses a simulation model, the M odel fo r

Intertheater Deployment by Air and Sea (MIDAS), to determine the required numberof

tankers. MIDAS useis problematic since many runs maybe needed, each run is

manpower-intensive, and results do not necessarily define theminimumnumberof tankers.

This thesis couples a schedule generator and an integer linear programming (ILP) model

to determine the minimum number of tankers to satisfy war-time POL requirements.

Solving a realistic scenario provided by J4-MOB (spanning75 days with 92 available

t.ankers), the ILP selects19 tankers, one-third the number initially chosen by MIDAS.

Using the ILPs recommended schedules, MIDAS confirms the ILPs solution.These

results show that the schedule generator and the ILP can assist JQMOB.

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TABLE O F CONTENTS

I. NTRODUCTION .............................................................................................................................. 1A. BACKGROUND.............................................................................................................................. 1

I . Government Relationships............................................................................................................ 22 . Rejagg ing Process....................................................................................................................... 33 Mobility Requirements Study (M RS ) 34 Current Modeling Process............................................................................................................ 45 . Problem Statement and Thesis Contribution 56. Thesis Outline............................................................................................................................... 6

II RELATED RESEARCH .................................................................................................................... 7A TANKER SCHEDULING PROBLEMS 7B . SIMULATION MODEL ................................................................................................................ 10

. .............................................................................................

.................................................................................

. ...........................................................................................

I . Model fo r Intertheater Deployment by Air and Sea..................................................................... 10

IlI . OPTIMIZATION MODEL ............................................................................................................ 13A . PROBLEM DEFINITION ............................................................................................................. 13

1 Ship Classes............................................................................................................................... 132 . Objectives................................................................................................................................... 13

B . MODELING CONSIDERATIONS ................................................................................................ 141 General Considerations.............................................................................................................. 152 . Schedule Generator Considerations............................................................................................ 163 . Integer Linear Programming Considerations.............................................................................. 17

C . MODEL FORMULATION ............................................................................................................ 181. Indices........................................................................................................................................ 182 . Data ........................................................................................................................................... 183 . Binary Variables......................................................................................................................... 204 .

Continuous Variables................................................................................................................. 206 . Description of Equations............................................................................................................ 227 Alternative Equations................................................................................................................. 228 . Penalties..................................................................................................................................... 23

lV .COMPUTATIONAL RESULTS .................................................................................................... 25A. SCENA RIO DESCRIPTION ......................................................................................................... 25B.RESULTS ....................................................................................................................................... 25

1. Methodology fo r Assigning Schedulesto Tankers (MAST ).......................................................... 262 . Modeling Groups of Tankers...................................................................................................... 263 Planning Horizon Reduction....................................................................................................... 274 . M A S T SResults .......................................................................................................................... 28

5 . MIDAS Results......................................................................................................................... 296. MASTs Iterative Process ........................................................................................................... 307 Fuel Requirements Reduced....................................................................................................... 31

V.CONCLUSIONS............................................................................................................................... 33A. CONCLUSIONS ............................................................................................................................ 33

5 . Equations ............................................... : .................................................................................. 21

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B. FUTURE RESEARC H................................................................................................................... 33LIST OF REFERENCES ...................................................................................................................... 35APPENDIX A. ACRONYMS]............................................................................................................. 37APPENDIX B. DATA DESCRIPTION] .............................................................................................. 39

I . Sea Ports ofDebarkation (SPODs)............................................................................................ 392. Sea Ports of Embarkation(SPOEs)............................................................................................. 403 Tankers....................................................................................................................................... 404 . Fuel Requirements...................................................................................................................... 415 . Distance Table........................................................................................................................... 42

INITIAL DISTRIBUTION LIST .......................................................................................................... 45

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EXECUTIVE SUMMARY

The Department of Defense (DoD), under the authority and direction of DoDDirective 5100.1, provides military forces needed to deter war and pro tect the security ofthe United States. To support the forces, DoD must maintain adequate supplies, keyamong them petroleum, oil and lubrication (POL ) products. War-time requirements forsurge (initial buildup) and sustainment (continuing requirements) may exceed the capacityof Do D POL assets. The shortfallis alleviated by the useof commercial US-flaggedvessels.

The Secretary of Transportation (SecTrans)is responsible for making sufficientPO L lift capacity available, and the Maritime Administration (MARAD) provides the

oversight of this responsibility. Toensure sufficientlift capacity remains in the US-flaggedvessel inventory, MARAD must authorize any re-flagging request from the vesselsowner. Prior to approval, MARAD receivesa recommendation from Do D on whether theUS-flagged vessels re-flagging would impair POL lift capacity.

In 1991, the Mobility Requirements Study (MR S) generateda tanker study whichincluded a recommendation for tanker fleet composition. Prior to the release of the tankerstudys results, the MRS Bottom Up Review- Update (MRS BURU) study modifiedunderlying assumptions rendering the results of the MRS no longer germane. There is nopublished quantifiable numberof tankers that defines rriinimum fleetsize or capacityrequired to meet war-time commitments.

In 1997, the Mobility Division of the Logistics Directorate of the Joint Staff (54-MOB) conducteda tanker study using a simulation program, the Model for IntratheaterDeployment by Air and Sea (MIDA S). How ever, the simulation model requiresintelligent, user-influenced information prior to running the model. This informationconsists of initial starting conditions that influence the outcome (where and when toonload fuel, how much fuel to onload, etc.). Knowledgeable users determine theseconditions, without quantitative information on how these conditions may adversely affect

the final outcome.This thesis develops a schedule generator that generates all, feasible tanker

schedules. The schedulesare provided as input to an integer linear programming (ILP)model that determines a collection of schedu les that m eet the war-time fuel requirements


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with a minimum complementof tankers. For an unclassified but realistic scenarioprovided by J4-MOB (spanning75 days, sixteen onload ports, seven offload ports, twofuel types, and 92 available tankers), the schedule generator and the ILP provide

significantly better results than those initially provided by MIDAS.To satisfy the

problems fuel requirements, theILP requires only 19 tankers; MIDAS initially required60. However, utilizing the schedu les selected by the ILP, MIDASalso solves the problemusing only 19 tankers. The results are achievedin less than seven hours on an IBMRS/6000 Model 590 computer, and the results demonstrate that the schedule generatorand the ILP can assist J4-MO B.

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ACKNOWLEDGEMENTS

I wish to extend my sincerest appreciation to my advisor, Professor Rob D ell. His

depth of experience, immeasurable patience, and sense of humor were key elements in

ensurin g both a memorable experience and that a quality product resulted. ProfessorWoods critical review provided an excellent benchmark, and his review was genuinely

appreciated. I would also like to thank the following people:

0 Captain Dave C ashbaugh, USN of J4-MOB, for providing the opportunity towork along side professionalsin the hectic and changing arena of logisticsplanning and analysis.

Major Roxann O yler, USAF, of J4-MOB , for insight into the problem andupdateson how the problem could benefit from optimization.

Mr. Earl Car10 and Mr. Merlin Neff, of GRC International, Inc. for providingassistance in data organization and co llection.

Captain Stan Olsen, USA , who bore the brunt of the data collection and updateeffort required for the tanker study.

Most importantlyI

must thank my wife and boys: Denise (the Love of My Life),and Charlie, and Mark, (the greatest joys in my life) whose unwavering support,unconditional love, and inspirational enthusiasm made this possible.This thesis isdedicated to them.

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I. INTRODUCTION

This thesis provides a methodology for determining the minimum number of

petroleum, oil and lubrication (POL) tankers required for a given war scenario. (Appendix

A contains a list of applicable acronyms.) The m ethodology separates into two parts: Fora given war-time scenario, a schedule generator uses information about tankers, fuel

requirements, and portsof embarkation and debarkation, to create a set of feasible ship

schedules; an integer linear program (ILP) then determines a subset of those ship

schedules that satisfies fuel requirements with the fewest num ber of tankers.

A. BACKGROUND

The Department of Defense (DoD), under the authority and direction of DoDDirective5 100.1, provides military forces needed to de ter war and protect the security of

the United States [DoD, 1997(a)]. To support the forces, DoD ensures there exist

sufficient quantities of supplies, key among them, POL products. POL tankers (Figure1)

transport the vast quantities of this bulky, heavy p roduct.

During war time, POL requirements for surge (initial delivery and buildup) and

sustainment (long-term continuing requirements) exceed the transport capability of tankers

owned by DoD. DoD relies on the Department of Transportation (DOT ) to provide

additional POL tankers. The Shipping Act of 1916, Sections 9 and 37 (as amended

through the 1020d Congress) provides the legislative means. The DO T has relied on the

Shipping Acts resulting availability of commercial, United States flagged POL tankers

during several wars [C ongressional Budget Office, 1997, p. xi].

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Figure 1 One of over SO US-flagged POL kmkers, SS MORMACSKYis available during war-time totransport bulk POL products. SS MORMACSKY(a medium-sized tanker) has a draft of 35 feet, acruising speed of 16 h o t s , and carries 283,000 barrels of petroleum products [National Steel andShipbuilding Company (NASSCO), 1997(a)].

1. Government Relationships

The Sec retary of Transportation (SecTrans) is responsible for ensuring commercial

POL tankers are available for war-time requirements, and his conduit for executing thisresponsibility is the Maritime Administration (MARAD).MARAD approves the re-

flagging of a US-flagged vessel toa foreign flag provided it is not militarily useful, and

thereby ensures that national assets remain available. MA RAD is described as:

The maritime Administration (MARAD) is responsible for insuring that merchantshipping is available in times of war or national emergency. MARAD administersprograms to meet seal ift requirements determined by the Department of Defense (DoD)and conducts related national security activities.

The Agency mainkains inactive, Govemment-owned vessels in the National DefenseReserve Fleet (NDRF) and its Ready Reserve Force (RRF) component. The RRF wascreated to maintain a surge shipping and resupply capability available on short notice tosupport deployment of a multidivision force. [MARAD, 19971

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Figure 2 - As demonstrated during Operations Desert ShieldDesert Storm, the US Navy (USN) is a majoruser of JP5 fuel. The aircraft embarked on aircraft carriers like US S ENTERPRISE (top) and US SGEORGE WASHINGTON(second from top) use this kerosene-based jet fuel. Additionally, it is the fuelsource for main propulsion and/or electrical power generation on almost all non-nuclear powered USNavy ships, such as underway replenishment ships USS SUPPLY (second from bottom) and US SMOUNT BAKER(bottom). These underway replenishment ships receive JP5 from a shore facility forfurther trmsfer to the ships in a battle group [DoD, 1997(b)].

4. Current Modeling Process

The Jo int S taff is the lead organization currently coordinating efforts to determine

the requisite number of POL tankers. The Joint Staf fs Logistics Directorate, MobilityDivision (J4-MOB), conducted numerous simulationsusing the Model for IntertheaterDeployment byAir and Sea (MIDAS)to determine the minimum number of tankersrequired forwar. MIDAS is a deterministic (no random aspects) model that attempts toanswer the following question, Can the given fuel requirementsbe met with the given setof sea-lift assets? Beeker, et al.[19961 describe MIDAS:

MIDAS is a strategic deployment-scheduling model developed for analysis ofairlift, sealift, prepositioning mobility progrcms of the Dep


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When simulating a large scheduling problem, objectivesare developed andweighted with respect to relative importance. The two objectives that receive the heaviestweights in MID ASare the efficient use of ships and aircraft and the arrival of the units assoon as possible [Beeker, e t al., 1996, p.2-31. As an example, a problem solvedin this

thesis, when initially run on MIDAS, delivers all fuel for the entire 75-day windowin 30days, utilizing60 tankers, each tanker making a single trip. MIDAS feasibly satisfies thefuel requirements. However, any attempt to determine a minimal tanker complementusing MIDAS would require numerous runs with numerous changes to the initial startingconditions. Post-run analysis would be manually interpreted , and intelligent changes madeto guide MIDA S to select the specific ships, por ts to onload fuel, and times to onload fuel,that result in fewer tan kers being selected to satisfy fuel requirements.

Enumerating all possible schedules within MIDAS is not practical for many

scenarios,so MID AS uses heuristic decision rules to route ships. Beeker,et al. [ 1996, p.2-41 specifically addressa limitation of this type of model: Heuristic methods maybe lessrigorous than optimization techniques and do not guarantee obtaining an optimalsolution. Th e genesis for this thesis derives from the inability of the Joint Staff toimmediately confirm that the feasible schedule provided by MIDASis optimal. Theprimary impetusis to create a model that answers How many POL tankersare requiredby the DoD to figh t and sustain a given war scenario?

5. Problem Statement and Thesis Contribution

To confidently determine theminimal number of tankers required to satisfy war-time fuel requirements, numerous runsare required for each of a set of representative warscenarios. A war scenario,in this thesis, is defined as a set of tankers available for use byDoD, a set of ports for onloading and offloading fuel, and a given set of daily fuelrequirements. Each run varies some aspect, or aspects, of the initial conditions thatinclude: fuel requirem ents (when, where, and type of fuel), tanker availability (when andwhere they are), t he number of available tankers, tanker characteristics (size, speed, draft,fuel capacity, etc.), and port characteristics (draft, production and storage capacities,

number of available berths, etc.). Post-run analysis provides a number of tankers, or rangeon the number of tankers, that would satisfy the fuel requirements across the variedscenarios. If 25 tankers satisfy the requirements across all scenarios, then it reasonable toconclude that 25 tankers would suffice.

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MIDAS is accepted by J4-MOB, OS D, and USTC as a valid too l to eva luate war-time planning, and J b M O B is committed to validating any tanker study results withMIDA S. T o determine a minimum number of tankers required, even approximately, for asingle scenario could require a very large number of runsin MIDA S. A reductionin the

numberof runs fo r a given scenario would expedite the overall time required to determinethe minimum number of tankers across various scenarios. This thesis do es notdemonstrate results across many different scenarios. Rather, it shows results for a realisticscenario provided by J4-MOB, and thereby demonstrates the usefulnessof themethodology.

This thesis provides a methodology to help J4-MOB reduce the number of M IDASruns required to find a minimum number of tankers for any given scenario.Themethodology is broken down into two steps. A schedule generator (referred to asSkedGen) uses known information about tankers, fuel requirements, and ports ofembarkation and debarkation, to create output files consisting of feasible schedules forindividual tankers. An integer h e a r program (referred to as ILP) then determinesasubset of these schedules that satisfies fuel requirements with the fewest (approximately)number of tankers. When referring to the collection of Skedgen and ILP, the term M AST(Methodology for Assigning Schedules to Tankers) is used.

6. Thesis Outline

Chapter I1 includes a review of tanker scheduling problems relevant tothis thesis.Chapter III includes modeling considerations, assumptions, the mathematical formulationof the schedule generator andthe optimization model. Chapter IV describes theapplication of the model to the J4-MOB Tanker Study problem, the origin of theunclassified data, and results fromthe computations. Chapter V encompassesrecommendations for further research and conclusions drawn froma specific instance ofthis model. Appendix A contains a list of acronyms, and Appendix B contains the d ata forthe scenario solvedin this thesis.

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McKay and Hartley[ 19741 minimize operating and purchasing costs associatedwith the transportation of bulk petroleum products by the Defense Fuel Supply Center(DFSC) and the Military Sealift Comm and (MSC). Their formulation allows for multipledeliveries of multiple products at multiple locations, and partial pick-up and delivery of

products (providedit is cost-beneficial). They use an approximate solution technique tosolve a specific integer linear problem fora typical DFSC task. The approximationtechniqueis: Solve the linear programming (LP) relaxation of the problem; look a t the sizeof the fuel loads carried; round up or down any variables thatare close to one or zero;then re-solve the LP relaxation with these set values. Optimality can notbe assured underthis technique. The dimensions of the problem they solve include 700-900 integer

variables,2,500continuous variables, and1,000 constraints.McKay and Hartleys model differs somewhat from MAST. MA ST allows for

multiple deliveries of multiple products at multiple locations, but does not allow for partialdeliveries of multiple products (The generated schedulesf i U the tanker as fullas possible,with one type of fuel.A tanker may pick upa different fuel type on another trip, butthemodel only allows one SPOE, one SPOD, and one type offuel per trip.). Otherdifferences are thatthe McKay and Hartley problemis substantially smaller thanthe one inthis thesis, and their objective of minimizing costis not necessarily equivalent tominimizing tankers.

Ronen [ 19831 provides a com prehensive review of m odels and problemsassociated with scheduling cargo ships. He proposesa classification scheme for cargo

scheduling problems and addresses works in the literature (prior to 1983) in these classes.He draws out the differences between cargo-ship scheduling, and other types oftransportation scheduling (e.g., bus scheduling, train routing). H e groups the cargo-scheduling problems in three different categories of operations: liners, tramps, andindustrial. MA ST wouldbe classified under Ronens industrial category. His review ofworks in the industrial category includes the Dantzig-Fulkerson and McKay-Hartleymodels discussed above. In this category, Ronen also reviews Floods[ 19541 model thatminimizes empty transit by a cargo ship and thereby minimizes the num ber of tankers used,Briskins [ 19661 model that allows for multiple discharge ports, Bellmore, Bennington and

Lubores [ 19713 model that allows for amix of tanker types, and partially loaded tankers[Ronen, 1983, pp. 119-1261. Th e last three articles each address different aspects of theproblem in this thesis and provide background for the more pertinent articles reviewedbelow.

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Brown, Graves and Ronen [1987] solve a major oil companys crude oil tankerrouting and scheduling problem. Th e minimization of the following costsare included:

daily cost of ow ned vessels, the cost of expending fuel during transit (speed dependen t),port and canal dues, spot charter costs, and the cost of owningan idle ship. The test

problems solved using their m odel include an80 day planning horizon,50 cargoes (up to25 of which maybe spot chartered),a fleet size of 24 ships, and three loading and ninedischarging ports. They solve problems with up to 7 ,349 schedules on anIBM 3033 in

under five seconds.Their model and the one developed in this thesis are very similar. The major

difference between their model and MAST is the minimization of cost, and the greatfidelity with which it is modeled. Additionally, based on the cos t of crude o il beingtransported, they may change the destination of the ship as the price changes; this isbeyond the modeling scope needed in this thesis.

Fisher and Rosenwein [1989] solve a more generic ship scheduling problem byminimizing costs of cargoes carried. Their model, though designed for any bulk cargo ,successfully solves anMSC tanker scheduling problem. Their costs include theoperating cost of the ship in the available fleet, and the cost ofa spot charter. Theygenerate a menu of all possible schedules, resulting in the formulation ofa set-packingproblem. They solve the set-packing problem using the dua l of a Lagrang ian relaxation ofthe problem. They solve anMSC scheduling problem of delivering 28 cargoes with 17tankers using less than800 schedules, using cost data provided byMSC. The solver was

written in PASCAL, and run ona V A X8600 in less than five minutes for the test problemconsidered.

Differences between the Fisher and Rosenwein model and MAST include: theproblem size(30 day planning period, less than800 schedules, 17 ship tanker fleet); theminimization of costs versus tankers; the consideration of spot charters for transportrather than relying on a given fleet of tankers; and the treatment of cargoes as fixedquantities of fuel (given two 300,000 barrel cargoes and only one550,000 barrelcapacity ship, the Fisher-Rosenwein model would deliver one cargo of 300,000 barrels. Incontrast, MAST would transport the maximum ship capacity,550,000 barrels.). Thegreatest similaritiesare the inclusion of port and tanker characteristics, and that theyaremodeled in great detail (depth, storage capacity, load/unload times, ship availabilitywindows, etc.)in both the Fisher-Rosenwein model and MAST.

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Pagonis strategic sealift optimization model determines the best set of schedulesfor cargo vessels, minimizing penalties associated with: port loading, ship berthing, andcargo not transported [Pagonis, 19951. The structure for his model was the mostinsightful in developing MAST. He uses data similar to the type used during war

planning, and considers single- and dual-front war scenarios, using implicit and explicitdelays for delivery of units. The most demanding scenario from a computationalstandpoint was a dual-front war scenario. The 2,027 schedules take just under twentyminutes to generate, and yield a solution guaranteed to be within 6.33 percent of theoptimal solution in 2 hours and 15 minutes on an IBM RS/6000 Model 590. [Pagonis,

The major differences between Pagonis model and MAST are that unit-type cargo

modeled in Pagonis (tanks, vehicles, etc.) requires a single delivery, is only available at

one SPOE,and is to be delivered to only one SPOD. In contrast, POL tankers modeled inMAST may make multiple trips from a single SPOE to the same SPOD, with the samecargo. Or, POL tankers may make multiple trips from various SPOEs to various SPODs,with various cargoes.

1995, pp. 34-43].

B. SIMULATION MODEL

1. Model for Intertheater Deployment by Air and Sea

The Model for Intertheater Deployment by Air and Sea (MIDAS) program is thetool used by J4-MOB to conduct simulations to determine the minimumnumber of tankersrequired for war.

The main objectives in MIDAS are the earliest possible delivery of forces, thearrival of forces in the order required (e.g., FT Benning troops must arrive prior to thetroops from FT Hood), the on-time arrival of supplies for sustainment, efficient use ofships and aircraft, and maintaining the integrity of the military units [Beeker, et al., 1996,p. 2-31. MIDAS uses a heuristic, a greedy search algorithm, to maximize the utilization

of each ship [Beeker, et al., 1996, p. 2-41.

A limitation of this type of model is that Heuristic methods may be less rigorousthan optimization techniques and do not guarantee obtaining an optimal solution.[Beeker, et al., 1996, p. 2-41. This limitation can be problematic: When MIDAS providesa satisfactory solution to a scenario, MIDAS has served its purpose. However, when

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MIDAS provides an unsatisfactory answer, it couldbe caused by the scenario or by theheuristics. Th e inability to guarantee that the schedule derived is optimal is the primaryimpetus for the developmentof MAST.

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1. General Considerations

The current usage of PO L by DoD during peacetime operations is 98 percent bulkfuel and 2 percent other [Quiroga and Strength, 1996, p.11. It is expected that these

usage percentages will continue to hold, and that the modeling of these major fuel types(JP5 andJP8 ) is sufficient (Figure 4).

A single speed is assumed for each tanker, its most efficient speed. This speed isused in the inter-port time computations as follows:A tanker that transits from Pearl

Harbor, Hawaii to Diego Garcia travels 9,775 nautical miles(nms). Divide this distanceby the ships assumed 16-knot speed and the result is 610.9375hours, or 25.46 days. Forpurposes of this thesis, this is rounded up to 26 days.

To determineonload/offload times, the amount of fuel to be transferred is divided by 200 mbbls, and thetime is rounded up to the nearest whole day. The model accounts for the fuel productionlimits and storage capacities at the S PO Es andSPODs. The fuel requirements are the netrequirements, for any given day, at the SPOD s. There is a desirable buffer or fuelreserve of 15 days, specific to an SPOD, it is pre-designated, and is includedin these netfuel requirements.

A pumping day is 200 mbbls (thousand barrels) per day.

Figure 4 - The Landing Craft-Air Cushion (LCAC) offloads cannored and conventional personnel carrierswhich both utilize JPS fuel. The LCAC, a hovercraft, CNI carry a 60 ton MlAl Abrmsnautical miles, at speeds of up to 60 knots, on a cushion of air. This capability of transporting troops pastthe beach and inland to more secure areas requires a large expenditure of fuel [DoD,1997(c)].

up to 60

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2. Schedule Generator Considerations

To generate schedules for the individual tankers, SkedG en allows, time perm itting,for each ship to make up to five deliveries of fuel. The time window for creating

schedules is set at75 days (this canbe changed).A unique combination of SP OE, pickupdate, amount of fuel, fuel type, SPOD, and drop off datemake up each delivery.SkedGens sm allest unit for measuring time is a day.

The following events associated with SPOEs (SPODs) that take time areaggregated: pullingin and out of port; setting up (breaking down) pumping stations; andtime to onload (of fload ) fuel. This sum is then rounded up to the nearest day, and addedto the inter-port transit time to determine when the tanker is ready to onload (offload) fuelat the next SPOE (SPOD).

The first day a tanker is available for onloading fuel, and the closest portto thattanker for onloading fuel are both determined prior to running SkedGen and provided asdata (the initial SPOE, and the day the tanker arrives at theSPOE). The calculationsinclude the time to com plete the current delivery of fuel, the transit from initial position attime 0 to original destination, the offload time, the transit to initialSPOE (applicable toUS-flagged, MSC and EUSC ships with fuel onboard) and any time required to activateships (RRF only). A tankers initial availability (both the place and time) are initialcond itions that specifically define a given run.

SkedGen creates schedules that include the following information: when a shiponloads a specified amount of a certain fuel, at anSPOE,for delivery to anSPOD, on aspecific day. Figure5 shows an example of a specific schedule for theSS COASTRANGE.

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SS COAST RANGE, Schedule 4S03

Time (in days)0 Enroute to Pearl Harbor3 Arrive at Pearl Harbor34 Complete Onload4 Enroute to Diego Garcia26 Anive at Diego Garcia2627 Complete Offload

Onload 220 mbbls, of JPS

Offload 220 mbbls of JP S

27 Enroute Pearl Harbor49 Arrive at Pearl Harbor49SO Complete OnloadSO EnrouteJeddah72 Arrive at Jeddah7273 Complete Offload

Onload 320 mbbls of JP8

Offload 320 mbbls of JP8

Figure 5 - The schedule generator (SkedGen) computes the transit time between ports based on distancetables and tanker speeds. SkedGen does not: send a tanker to a draft prohibiting port, overfill tankers,onload more fuel than available, offload more fuel than carried onboard, nor offload more fuel thm heSPOD can store. A specific ship, SS COASTRANGE, transits to Pearl Harbor to onload 220 mbbls of

JPS for delivery to Diego Garcia


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C. MODEL FORMULATION

1. Indices

d

e

f

i

S

t

2. Data

Pen i,s

DevPen 6f,t

UnDelPen 6f,t

ShipSkedi

Rqmt d,f,t

FuelProd e,f

StoragePOE e,f

StoragePOD d,f

POD, (d= Cairo, Diego Garcia,. . , Thailand);POE, (e = A1 Jubail, Am uay Bay,. . . UK);fuel type (f=JPS, JP8);

ship, (i= ALATNA, ALMA,. . . , VEGA);

schedules,(s = 1,2, . . , ); andtime (in days), (t= 1,2, . . . T).

Total penalty associated with tankeri, using schedules;

Penalty for fuel requirements satisfied from th e buffer atSPOD d, fuel type f, on dayt;

Penalty for unsatisfied fuel requirements at SPOD d, fueltype f, on day t;

Number of schedules for tankeri. It is 1 if i corresponds to

a single tanker, and ifi corresponds toa group, then it isequal to the number of tankersin the group;

Daily fuel requirement at SPO D d , of fueltype f, at time t,in mbbls;

Daily fuel production capacity, at S PO E e, of fuel type f, inmbbls;

Fuel storage capacity at SPOE e , of fuel type f,inmbbls;

Fuel storage capacityat SPOD d , of fueltype f, inmbbls;

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5. Equations

Minimize:

(1)

Subject To:

c i . s I hipSkediS

V i (2)

MBblsDevd,, I ufferd. V d, f, t (4)

FuelAvailPOE,j,,= FuelAvailPOE,jvt.l+ FuelPOE,j,( -

cMBblsIni,s,e,/,t X i . s

i s

'J e , f, t > O (5 )

FUelPOE,j,t I uelProd,d V e , f, t ( 6 )

FuelAvailPOE f,' I toragePOE J V e , f, t (7)

i ,st '


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FuelPOE et,t V e, f, t (10)

MBblsDevdf,r 2 V d, f, t (1 1)UnDlvrdd4.t 2 V d, f, t (12)

FuelAvailPOE 2 V e, f, t (13)

6. Description of Equations

The objective function (1) minimizes the sum of all penalties, which are described

Constraint (2) ensures that each tanker is assigned at most one schedule;ShipSkedi has value 1 when i corresponds to a single tanker or it is the number of tankersin the tanker group. Constraint (3) ensures fuel requirements are met on time, but

contains elastic variables for delivery shortages. There are two types of delivery

shortages, designated MBblsDev and UnDlvrd. MBblsDev is the amount of fuelreserves required to satisfy fuel type f requirements at SPOD d, on day t. UnDlvrd isunsatisfied fuel type f requirements at SPOD d, on day t. Constraint (4) ensures fuelreserve use does not exceed the fuel buffer. Constraint (5) is the daily inventory flow-balance constraints associated with the SPOEs. Constraint (6)precludes the production of

more oil than is feasible. Constraint (7) precludes the model, on a daily basis, fromonloading more fuel from an SPOE than can be stored there. Constraint (8) precludes themodel from delivering more fuel to an SPOD than can be stored. Sufficient SPOD storagewas found for the given scenario, thus constraint (8) was removed from the ILP to reducethe size of the model.

in section 8 below.

7. Alternative Equations

Alternative equations can be used when formulating this problem. Although not

used in the solution of the scenario in Chapter IV, they may prove more effective on adifferent scenario.

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for the MS C tankers, D OT forth e RRF tankers), and available for war.As a result of thispreference, no penalties a re assigned for the useof RR Fand MSC tankers.

The ship class with the next highest usage preference is the EUSC class.Thepenalty for a ship in the EUSCfleet is set equal tothe tankers capacity.

To encourage the use of US-flagged tankers last, the penalty fora tanker in thisclass is set to the individual tankers capacity plus that of the maximum capacity of theEUS C flag fleet tankers(676).

The penalties per mbbl of fuel in the MBblsDev category (fuel requirementsatisfied by the fuel reserves) and in the UnDlvrd category (unsatisfied fuel requirements)are 1 and 2 respectively. If the fuel shor tage for a given day canbe satisfied by usingsome of the fuel reserves, then the forces can still operate (the reserveis depleted to somedegree). But, if the amount of fuel shortage exceeds the fuel reserve capacity, theoperational commander does not have sufficient fuel available to conduct operations (thereserve is empty). Thus thereis distinction, and larger penalty, on the amount ofundelivered fuelin excess of the reserve level.MBblsDev and UnDlvrdare each measuredin mbbl-day, or thousands of barrels per day of requirements thatare not met.

This penalty structure ensures that asmall amount of undelivered fuel at a SPOD,or SPOD s, does not force the utilization of a previously idle tanker. How ever, a totalshortfall of undelivered fuel-days that, for instance, exceeds the capacity of an EUSCtanker that can satisfy the shortfall, forces theILP to select the tanker. For example,ifthere is an undelivered am ount of fuel at a po rt for two consecutive days of60 mbbls, and

the ports Buffer is 25 mbbls,th e computed penalty using constraint(3) is 190 mbbl-days.A penalty of 50 mbbl-days, associated with MBblsDev, isthe product of 25 (for thedeviation,in mbbls), two (number of days), and one (amount of penalty per mbbl). And apenalty of 140 mbbl-days, associated with UnDlvrd, is the product of35 (for th eundelivered amount,in mbbls), two (number of days), and two (amount of penalty permbbl). Th e sum of these two penalties is 190 mbbl-days. The reforethe ILP would selectany tanker schedule that deliversat least 60 mbbls prior tothe first day of shortage andhas a penalty less than 190 mbbls (allRRF and MSC tankers have zero penalty, and anyEU SC tanker with penalty less than 190 would be appropriate).

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IV. COMPUTATIONAL RESULTS

A. SCENAR IO DESCRIPTION

The scenario solvedin this thesis is provided by J4-MO B. It spans a 75-dayplanning period, andis defined by the set of SPOEs that produce fuel, the set of SPODs

that require fuel, the tanker assets available, and the fuel requirements (per fuel type and

day) at the SPOD s. The set of seven SPODs that require fuel, with characteristics, are

listed in Appendix B, Table3. The set of 16 SPOEs that provide fuel, with their

characteristics, are listedin Appendix B, Table 4. The available tanker fleet consists of 92

tankers, each categorizedin one of four ship classes,RRF, MSC, EUSC, or US-flagged

(AppendixB, Table 5 contains a subset of these tankers, and their characteristics). Asubset of the 13,518 mbbls totalfuel requirement, listed by SPOD, fuel type and day is

contained in AppendixB, Table 6. The distances used in the thesisare contained in

AppendixB, Table 7 with the SPOEs down the first column and the SPODs across the

top.

B. RESULTS

J4-MOB provided the unclassified data used in this thesisin Microsoft ExcelSpreadsheets. After rearranging the da ta format, it was saved as space delimited files(*.prn extension) and comma delimited files (*.csv). The space delimited format served asthe input for the model generator whichis written using the General Algebraic ModelingSystem (GAM S) [Brooke , et al., 19921. The comma delimited files serve as input forSkedG en. SkedG en was writtenin PASC AL, and writes output filesin ASCII text format(*.txt). SkedG enruns on a Dell OptiPlex GXPro Personal Com puter with a Pentium Pro200 megaHertz processor. The ILP is solved on an IBM R S/6000 Model 5 90 w orkstationusing GAMS to generate the model and either OSL [Wilson, et al., 19921 or CPL EX

[CPLE X Optimization, Inc., 19941 to solve it.

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cumulative sizeof the 30 output files was 172 megabytes. Th e largest numberof

schedulesfor a tanker group was 56,263. An attempt to solve the correspondingILP

took over 30,000 central processor unit (cpu) seconds(8 hours and 20 minutes of cpu

time) and returned an integer solution guaranteed tobe within 17 percent of the optimal.

Generating this problem required 1.9 gigabytes of RAM, and thefull generatiordsolution

process required roughly 16 hours. Th e results were promising, but the determination of a

subset of schedules that would enable theILP to solve the problem more qu ickly required

more work.

3. Planning Horizon Reduction

The next attempt to reduce the numberof schedules generated involved reducing

the planning horizon from 90 to 75 days. The war may last longer than 75 days, but the

early part of the conflict, when meeting the surge phase requirements, requires the greatest

number of tankers. The fuel requirements in the sustainment phase begin to approach a

steady-state condition. Thus, it is reasoned that the minimum number of tankers

required for surge will suffice during the sustainment phase, and 75 days is sufficient to

model the surge phase.

SkedGen, limited to a 75-day scheduling window, created 26,900 feasible

schedulesin 4 minutes, 20 seconds. The cumulative size of the30 output files was 14

megabytes. Th e largest number of schedules for any tanker grou p was 5,910. Th e linear

programming(LP) relaxation of this set of schedules was solved in 469 cpu seconds,

requiring only 249 megabytesof RAM, ut an integer solution was not obtained in

100,000 cpu seconds (a limitof 100,000 cpu seconds was set).

This baseline iteration (Run 1, Table 1) includesthe schedule reduction due to

tanker grouping and reducing the window to 75 days. Th eLP relaxation of the 26,900

feasible schedules resultsin an objective function value of 9,476. It uses 15.38 tankers(2.01 RRF, 7.33 MSC, 4.47 EUSC, and 1.57 US-flag).AU fuel requirementsare met, but

not on time, with 438 m bbl-days of fuel being delivered late. (Mbbl-days is not the

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the grouping process), could reveal tankers in this group that deliverJP8 to Gu am by day

30. If such a schedu le is found , it could be re-introduced into theILP.

Run Schedule

Generation

(seconds)

1 260

2 260

3 260

4 260

5 260

Number Optimization Number of Planning Number Number of US Undelivered

Schedules (cpu secs) Avail Tanker Horizon of Flag Tankers Fuel

Groups (days) Tankers (mbbl-days)

26,900 >100,000 26 75 NIA NIA NIA

2,875 739 26 75 16 7 1,916

8,467 6,050 26 75 17 8 829

17.23 1 18,949 26 75 19 8 608

17,231 25,186 26 75 15 5 284

Table 1 -- Final Results. The time for the schedule generator run on a personal computer represents real-world time, rather th"~ cpu seconds. The optimization time w


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MORSKY

FALCONMONSPRAYJHAMhER

Table 2 -- The tankers selected by the ILP,


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integer solution was returned with a better objective function value, 18,637, and the total

number of tankers selected increased to 17 (an additional US-flagged tanker). The

amount of fuel not delivered on time was829 mbbl-days, (6.1 percent), and theILP failed

to deliver only 434 mbbls.

The final run resulted from re-introducing more schedules, those schedules that

delivered fuel to the ports with shortages, on the days the shortages occurred.This

resulted in the number of feasible schedules increasing to 17,231, as described in

paragraph5 above.

7. Fuel Requirements Reduced

To demonstrate the abilityof MAST to accommodate similar scenarios, thescenario providedby J4-MOB was modified (Run5, Table 1). The reduction of fuelrequirements by ten percent resultsin a reduction of the required tankers from 19 (eightUS-flagged) to 15 (five US-flagged). The total fuel requirements droppedfrom 13,518mbbls to 12,172 mbbls. The amount of undelivered fuel dropped to 77 mbbls, and 99.39percent of the fuel requirements were satisfied during the 75 day window. Therelationship between the fuel requirements and the number of required tankers is notprecisely linear. If so, the expected number of tankers wouldbe 17, versus the 15 thatwere selected. However, MASTS results are intuitive, the reduction in fuel requirements

resultedin a reduction in the number of required tankers.

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from optimality) could be compared to determine the point of diminishing returns withrespect to time and the reductionin the number of tankers.

The conversion of the schedule generator into a more robust object-orientedprogram, coupled with a Graphical User Interface (GUI) wouldbe beneficial to a user

with little or no knowledge of mathem atical programming. The GUI shouldlink theschedule generator and the ILP to allow the user to launch thetw o programs with asingle command. Th e GU I could initiate the schedule generator, the ILP , and output theresults in an environment that would allow th e user to easily conduct post-run analysis.

J4-MOB currently conducts POL tanker runs and container ship runsindependently. Ideally, these shouldbe run simultaneously, to help eliminate portoverloading that might occur with separately optimized scheduling problems. Thesimultaneous runningof the programs could provide insight into optimal ship schedulingproblem, versus just the segmented POL tanker and container ships scheduling problems.

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LIST OF REFERENCES

Appelget, J. A., Explicit-Constraint Br,mching for Solving Mixed-Integer Progr,uns, Ph.D. Dissertationin Operations Research, Naval Postgraduate School, Monterey, CA, 1997.

. Beeker, E., Bohli, P., Kershaw, S., Keyfauver, C., Marcotte, J., Sims,S. Model for IntertheaterDeployment by Air and Sea (M IDA S)- Users Manual Version2.3, GRC International Inc., April 1996.

Bellmore, M., Bennington, G., Lubore, S., A Multi-Vehicle Tulker Scheduling Problem,Transportation Science ,vol.5 , no. 1, pp. 36-47, 1971.

Briskin, L. E., Selecting Delivery Dates in the Tanker Scheduling Problem, Management Science,vol.12, no. 6, pp. B224-B233,1966.

Brooke, A., Kendrick, D., Meeraus, A., GAMS,A Users Guide, Release2.25, The Scientific Press, SanFrancisco, CA, 1992.

Brown, G. G., Graves, G. W., Ronen, D., Scheduling Ocean Transportation of Crude Oil, ManagementScience,vol. 33, no. 3, pp. 335-346, 1987.

CPLEX Optimization, Inc., Using theCPLEXCallable Library,CPLEX Optimization, Inc., InclineVillage, NV, 1994.

Congressional Budget Office, MovingU S .Forces: Options o r Strategic Mob ility,U.S. GovernmentPrinting Office, Washington, DC, February, 1997.

Dantzig, G.B., Fulkerson, D.R., Minimizing the Number of Tulkers to Meet a Fixed Schedule, NavalResearch Logistics Quarterly,vol. 1, no. 1, pp. 217-222, 1954.

Department of Defense, The Department of Defense Org,dzatioml Structure, http://www.dtic.dla.mil/

defenselink/pubs/ofg/os-dod.html,3 April 1997(a).

Department f Defense, Enterprise and George Washington Battle Group Turnover,http://www.dtic.dla.mil/defenselink/photos/Jull996/9607 2-N-7340V-00 1 html, 2 1 July 1997(b).

Department of Defense, Muines Offload Equipment from LCAC,http://www.dtic.dla.mil/defenselink/photos/Aug 1996/96061S-N-3 149V-002.hbnl,21 July 1997(c).

Fisher, M. L., M. B. Rosenwein, An Interactive Optimization System for Bulk-Cargo Ship Scheduling,Naval Research Logistics,vol. 36, no. 1, pp. 27-42, 1989.

Flood, M. F., Application of Transpomtion Theory to Scheduling a Military Tanker Fleet, OperationsResearch,vol. 2, no. 1, pp. 150-162, 1954.

Kross,W., Mobility Requirements Study, Bottom Up Review-Updzzte, Volume 111, Office of theChairman of the Joint Chiefs of Staff, p. ES-1,15 March, 1995.

Kross, W., Mobility Supplement to the Joint Strategic Capabilities Plrm for FY 1996 (JSCP FY 96),Chairman of the Joint Chiefs of Staff nstruction (CJCSI) 3110.1 B, Enclosure C, 1996.

35


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Lloyds of London, Lloyds Ports ofthe World,Lloyds of London Press, Ltd., Colchester, Essex, UnitedKingdom, 1996.

Macke, R.E., Mobility Requirements Study, Volume I, Office of the Chairman of the Joint Chiefs ofStaff, p. ES-1,23 J a n ~ r ~ y992.

McKay, M.D., Hartley, H. O., Computerized Scheduling of Seagoing Tankers, Naval ResearchLogistics Quarterly,vol. 21 , no. 2, pp. 25.5-264, 1974.

Maritime Administration, MARAD 9.5ANNUAL REPORT, Chapter 1:National Security,http://mxad.dot.gov/CHPTOCl HTM, 01 April 1997.

National Steel and Shipbuilding Compmy (NASSCO), Coronado Class T,mkers, SS MORMACSKY,http://www.n,?ssco.com/design_p6b.Jpg,18 July 1997(a).

National Steel and Shipbuilding Compmy, Cxlsbad Class Tmkers, S S COAST RANGE,http://www.nassco.com/design-pSb.jpg, 8 July 1997(b).

Pagonis, G .W., Optimizing Strutegic Seal$ ,Masters Thesis in Operations Research, Naval PostgraduateSchool, September 199.5.

Quiroga, J.E., Strength, J.T., Determining an Optimal Bulk-Cargo Schedule to Satisfy GlobalU S .Military Fuel Requirements,Masters Thesis in operations Research, Naval Postgraduate School,September, 1996.

Ronen, D., Cargo Ships Routing and Scheduling:Survey of Models and Problems, European JournalofOperational Research,vol. 12, no. 2, pp. 119-126, 1983.

Wilson, D. G., Rudin, B. D., Introduction to the Optimization Subroutine Library, IBM SystemsJournal, vol. 31 , no. 1,1992.


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APPENDIX A. [ACRONYMS]

bbls: barrels, a measurement of liquid products

BUR: B ottom Up Review

BURU: B ottom U p Review- Update

cpu: Central Processing Unit

CJCS: C hairman, Joint Chiefs of Staff

DCNO: Deputy Chief of Naval Operations

DFM: Diesel Fuel Marine

DFSC: Defense Fuel Supply Center

DoD: Department of Defense

D O TDepartment of Transportation

EUSC: Effective US Controlled

GAMS: General Algebraic Modeling System

L A : Intratheater Lift Analysis

ILP: Integer Linear Programming

54: Joint Staff's Director for Logistics

J4-MOB: Joint Staff Director for Logistics, Mobility Division

JF5: A kerosene based je t fuel used for US Navy aircraftJP8: A kerosene based jet fuel, used for non-US Navy aircraft due to greater volatility

JS: Join t Staff

JSCP: Joint Strategic Capabilities Plan

LCAC: Landing Craft-Air Cushion

LP: Linear Program

MARAD: Maritime Administration

MAST: Methodology for Assigning Schedules to Tankersmbbls: 1000's of bbls

MIDA S: Model for Intertheater Deployment by Air and Sea

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MRS: M obility Requirements Study

MRS BURU (MRS Bottom Up Review, Update)

MSC: M ilitary Sealift Command

N4: DC NO for Logistics

NASSCO: NationalS tee1 and Shipbuilding Company

nm: Nautical Mile

NDRF: National Defense Reserve Fleet

OOTW:Operations Other ThanWar

OSD:Office of the Secretary of Defense

OSD(PA&E): Office of the Secretary of Defense Program Analysis and Evaluation

POL: Petroleum, Oil and Lubrication

ROS: Reduced Operating Status

RRF: Ready Reserve Force

SecTrans: Secretary of Transportation

SkedGen:The Schedule Generator

SPOD: Sea Portof Debarkation

SPOE: Sea Portof Embarkation

US: United States

USA: United States ArmyUSAF: United StatesAir Force

USMC: United States Marine Corps

USN: United S tates Navy

USTC: U nited States Transportation Comm and

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APPENDIX B. [DATA DESCRIPTION]

For ease of reading and consistencyin this appendix, text inbold print represents

pieces of data and variables in MAST.

1. Sea Ports of Debarkation (SPODs)

The data for the sevenSPODs is summarizedin Table 3. The ports are listed

down the first column, with their characteristics across the top two rows. Th eName,

Draft, and Number Berths columns are straightforward (the draft informationis drawn

from LloydsPorts of the World [Lloyds, 19961). ThePump Time is conservatively

estimated as the number of eight-hour days required to empty a200,000 barrel tanker.

Buffer and InitPODInv data is contained in the JPSBuf, JP8Buf, JPSInit and

JP8Init columns. Fuel Type is either 8 for JP8 only, or b for both (noport requires

only JP5). Storage capacitiesof JP5 and JP8 (JPSSto and JP8Sto) are in the last two

columns, yieldingStoragePOD data. Th e data derives from MRS BURU andDFSC

(updated November, 1996).

SPOD Data File

Table 3 -- SPOD Data. The port Name, Draft, Pum p Time, and Number of Berths, are as describedabove. JPSBuf, and JP8Buf contain the data for the Buffer panmeter (fuel reserves). InitPODInvgets its data from JPSInit and JP8Init (initial inventories). Fuels is as described above, and JPSSto

and JP8Sto are the values used by StoragePOD (storage capacities).

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2. Sea Ports of Embarkation (SPOEs)

Name

The data for the sixteen SPOEs is summarized in Table4, similar in design toTable 3, with one exception. FuelProd data is containedin columns seven and eight,

JP5Prod and JP8Prod, and is measuredin mbbls of daily production. The data derivesfrom M RSBURU and DFSC (updated November,1996).

Draft Pump Num ber Fuel JPSProd JPIProd JP5Sto JPSSto

Time Berths Types (mbbls) (mbbls) (mbbls) (mbbls)

SPOE Data File

AL JUBAIL 6 5 1 4 8 0 88 0 1,372

Table 4 -- SPOE Data. The port Name and Draft, Pump Time and Number of Berths are describedabove. The Fuel Types column indicates whether the port accommodates JP8 (8) or both (b) fueltypes. The last two pLiTs of columns contain the amount of JPS and P 8 roduction ( Fuelprod) andstorage (StoragePOD).

3. Tankers

The data for the tankersis summarizedin Table 5. A subset of the entire list ofships is provided (only10 of the 92 are shown). MSC (MSC controlled tankers) andMARAD (all other tankers) provided the data on tanker characteristics, usage, and

availability, viaUSTC.

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5. Distance Table

The distance data (Table 7) derives from an algorithm for computing the distancesbetween ports within the MIDAS model. The distance table assumes both canals (Suez

and Panama) are open and transit remains unimpeded.

Distance D ata File

the SPODson the first row. Distancesare in nautical miles.

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M

Table 8 -- The 92 tankers are aggregated to reduce the size of the problem, yet still maintain 8 closesemblance of the feasible set of schedules. For the twelve projected tankers, Projected Tanker (PT) Sevenrepresents the schedules of the group. The number of schedules, is reduced from over 798,000 to 289,661.

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INITIAL DISTRIBUTION LIST

1.

2.

3.

4.

5.

6.

7.

8.

Defense Technical Information Center... . . . . 28725 John J. Kingman Rd.,ST E 944Ft. Belvoir, Virginia 22060-62 18

DudleyKnoxLibrary ................................................................................... 2Naval Postgraduate School41 1 Dyer Rd.Monterey, California 93943-5 101

Defense Logistics Studies Information Exchange........... ... . ... .......... ,....... 1Army Logistics Management CollegeFort Lee, Virginia 23801-6043

5-4 Mobility Division...................................................................................... 1Logistics D irectorate, 5-4Attn: CAPT Cashbaugh4000 Joint S taff, PentagonWashington, D.C., 20318-4000

HQ USTRANSCOM TCJ5............................................................................ 1Attn: USTC J5-AS, Mr. Rob Saylor508 Scott DriveScottAFB, Illinois 62221

Deputy Chief of Naval O perations (Logistics).................................... ............ 1Attn: CDR Millie Simons, N412C2000 Navy PentagonWash ington, D.C., 20350-2000

Professor Dell code O w e........................................................................... 2Naval Postgraduate SchoolMonterey, California 93943-5000

Professor Wood code O W d........................................................................ 1Naval Postgraduate SchoolMonterey, California 93943-5000

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9. Professor Schrady, CodeOWSO .................................................................... 1Naval Postgraduate SchoolMonterey, California 93943-5000

10.LCDR Kent Michaelis....................................................................................

2227 S Charlene D rPanama City, Florida 32404

11. Gnazio Brangaccio......................................................................................... 113504 W Robin LaneSun City West, Arizona 85375

97 michaelis master

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