Scheduling Coast Guard District Cutters

GERALD G. BROWN Operations Research DepartmentNaval Postgraduate SchoolMonterey, California 93943-5219

ROBERT F. DELL operations Research DepartmentNaval Postgraduate School

ROBERT A. FARMER United states Coast Guard2200 2nd Street SWWashington, DC 20593-0001

United States Coast Guard (USCG) districts schedule cutters180 feet or less in length to weekly statuses (statuses is USCGjargon for assignments) from which they primarily respond tocalls for search and rescue, law enforcement, and pollutioncontrol. The First Coast Guard District, based in Boston, hasone of the largest scheduling problems: Each of 16 cutters is as-signed weekly to one of six statuses to ensure patrol coverage,enforce equitable distribution of patrols, and honor restrictionson consecutive cutter statuses. When we state this quarterlyscheduling problem as an elastic mixed-integer linear program,we obtain face-valid schedules—superior to manually preparedschedules for all measures of effectiveness considered—withina few minutes on a personal computer. Initial acceptance of themodel was hampered by disruptive schedule revisions that re-sulted from minor changes in input. Modifications to preserverun-to-run persistence of solutions have brought success.

Semper Paratus —United States Coast less in length to weekly operational andGuard Motto . , . , , ^^_

mamtenance statuses {statuses is USCG

A ll 10 United States Coast Guard jargon for assignments) from which they

(USCG) district commands face the primarily respond to calls for search andproblem of scheduling cutters 180 feet or rescue, law enforcement, and pollution re-Copyright © 1996. Institute for Operations Research PROGRAMMING, INTEGER—APPLICATIONSand the Management Sciences GOVERNMENT—US COAST GUARD0092-3102/96/2602/0059$01.25This paper was refereed

INTERFACES 26: 2 March-April 1996 (pp. 59-72)

BROWN, DELL, FARMER

sponse. The First USCG District has one ofthe largest scheduling problems, assigningeach of 16 cutters weekly to one of six sta-tuses to ensure patrol coverage, enforceequitable distribution of patrols, and honorrestrictions on consecutive cutter statuses.The First USCG District (Figure 1) extendsfrom Maine to halfway down New Jerseyand seaward to cover the ocean areasunder US jurisdiction.

Manual scheduling of the first districtcutters for four annual quarters takes ascheduler and his assistant two to threeworking days. (For six assignments, 16 cut-ters, and 13 weeks, the number of possiblequarterly schedules is bounded by 6"" '̂̂as 10", and that's a lot.} Manual schedu-lers cannot be expected to consider all the

minutia: They often overlook details aboutindividual cutter use, do not distributemaintenance ideally, and even though pa-trolling is the primary mission of this fleet,they miss a number of required patrols.USCG issues the schedules, and they arefollowed. There is no way to know for surewhether the schedulers have overlookedbetter schedules. This motivates our searchfor improvements.

This type of cutter scheduling is differ-ent from other scheduling and timetablingproblems. Covering the patrols is para-mount, but patrols punctuated by energeticemergency responses fatigue crews andequipment, and these cutters have limitedendurance. Periods of maintenance andcrew rest are most effective when they fall

Gulf of Maine (M) Patrol

George's Bank (G) Patrol

Search and Rescue (SAR)

New York (NY) Patrol

Figure 1: The First USCG District includes coastal ocean areas off the Norlh-Eastern UnitedStates. Approximate patrol areas are shown where cutters have responsibility for search andrescue, law enforcement, and pollution response.

INTERFACES 26:2 60

u s COAST GUARD

in useful patterns, so scheduling these wellis important too. Patrol requirements andmaintenance availability change weekly, soit is impossible to design regular, cyclic ro-tations for each cutter between patrol andmaintenance statuses.

Most ship scheduling work addressescommercial shipping problems: Ronen[1983] reviews models that essentially con-cern a fleet of ships moving goods fromone or more supply points to various de-mand points. Researchers have paid someattention to military ship scheduling, al-though the problems they considered areoften similar to those found in commercialcargo shipping, such as cargo deploymentplanning, Lally [1987], for example, as-sessed the required number of ships, andLima [1988] bounded the elapsed deploy-ment time. Some researchers have focusedon scheduling a sequence of specific tasks,such as Bechtle [1984] and Cline, King,and Meyering [1992], who looked atscheduling USCG buoy-tenders. Somehave focused on scheduling military shipsto match ship capabilities to missions ofvaried durations: Brown, Goodman, andWood [1990], Nulty and RatHff [1986], andGoodman [1985] worked on annual em-ployment periods for members of the USNavy Atlantic Fleet. Sibre [1977] scheduledUSCG high-endurance cutters in thePacific, and Darby-Dowman, Mitra, andHajian [1992] did so for the Atlantic,Farmer [1992] reviewed further contribu-tions in these and related areas.Policies for Scheduling Cutters

The First District has a fleet of 16 cutters(Figure 2):—Three are 140-foot ice-breaking tugs

(USCG designated WTGB),

—Seven are 110-foot patrol boats (WPB),and

—Six are 82-foot patrol boats (WPB),Each cutter is scheduled by week to be

either in an operational status or in amaintenance status. The operational or"bravo" statuses include—Bravo-2 (B-2), patrol status, during

which the cutter must be underway orready to get underway within two hoursof notification to sail, and

—Bravo-6, -12, -24 (B-6, B-12, B-24), lessoperational readiness, during which cut-ters must be ready to get underwaywithin six, 12, or 24 hours.The First District schedules only B-2 and

B-12 operational statuses, B-2 status weeksare assigned to cutters patrolling one offour geographical areas: Gulf of Maine,B-2(M); George's Bank, B-2(G); search-and-rescue, B-2(SAR); and New York,B-2(NY) (Figure 1), These patrol weeks fa-tigue both the crew and the cutter, so cut-ters are assigned no more than two consec-utive weeks in any B-2 status or a singleweek in B-2(SAR) status. B-2 weeks are ro-tated equitably among all available cutters.Patrols close to homeport avoid long tran-sit times to and from the patrol area. Theless stringent readiness requirement of aB-12 week allows a cutter to conduct rou-tine maintenance and training.

Not all cutter classes are eligible forevery type of patrol year round. The 140-foot WTGB tugs are reserved for ice-break-ing during the winter. The 82-foot patrolboats are not available for the SAR patrolduring severe winter weather because icingconditions can cause stability problems forthe cutters.

Cutters are scheduled to receive dedi-

March-April 1996 61

BROWN, DELL, FARMER

J ^ ^ : ^

Figure 2: The 140-foot WTGB ice-breaking lug and the 110-foot WPB patrol boat. During win-ter months, ice-breaking tugs are not available for patrols, and smaller 82-foot cutters (almostidentical in appearance to the 110-foot cutters) are not available for the SAR patrol duringsevere winter weather.

cated maintenance during "charlie" statusweeks ("C"). A cutter so assigned is notexpected to respond to anything but thegravest emergency. Maintenance periods oftwo consecutive weeks are preferred butare difficult to achieve with manual sched-uling. A cutter may be completely removedfrom the district schedule during yard ordockside maintenance periods that are

planned in advance with USCG Mainte-nance and Logistics Command, Cutters arealso scheduled to make public appearancesand patrol special events.CutS: A Cutter Scheduling Model

With the advice of the First Districtschedulers, we developed a simple decisionsupport system for scheduling cutters,called Cuts, based on an elastic mixed-

INTERFACES 26:2 62

u s COAST GUARD

integer linear program that captures muchof the detail and intent of the process (ap-pendix). The elastic formulation allowsschedules to violate, at a linear penaltycost per unit violation, requirements thatcannot be met.

Each 13-week quarterly model is pre-ceded by two weeks of fixed assignmentsfrom the end of the previous quarter. Thisis how we know whether and how toschedule consecutive weeks of patrols orpairs of maintenance weeks. The targetnumber of maintenance weeks for a cuttermay dictate scheduling a single week ofmaintenance, and if this happens we pre-fer it at the end of the quarter.

The minimum and maximum equitablenumbers of patrols for each cutter must af-ford the scheduler some flexibility. Theminimum equitable number of patrols foreach cutter is arguably the total number ofrequired patrols for all cutters divided bythe number of cutters, unless a cutter hasminimum maintenance requirements leav-ing insufficient weeks of availability; themaximum equitable number of patrols isconsidered to be the minimum plus two.

The difficulty of establishing meaningfulmeasures of effectiveness for ship-schedul-ing problems is well documented by So-land [1982]. By adopting the same meas-ures of effectiveness as the First Districtscheduler, we sought to—Minimize the number of required patrol

weeks missed,—Minimize the transit time to patrol areas,

and—Equitably distribute the patrol weeks

among cutters.We developed costs and penalties for

Cuts to mimic the motives and rules of

thumb of a good scheduler. The cost of as-signing a cutter to a patrol is representedadequately by the hours underway at nor-mal cruising speed from homeport to thepatrol area (Table 1). Even though cuttersgenerally remain in their homeport for amaintenance week, we use a transit time oftwo hours for any cutter assigned tomaintenance: This is less than the smallesttransit time in Table 1 but greater than thezero used for the B-12 status.

We want to satisfy all requirements forweekly patrols and maintenance, regard-less of the transit time for cutters. There-fore, we derive elastic penalties for failingto satisfy those requirements from thetransit-time matrix. We set penalties forsingle-cutter patrol requirements andmaintenance penalties slightly greater thanthe largest transit time (for example, forthe data in Table 1, 40 hours). Reasoningthat violating a multi-cutter patrol require-ment by one cutter is less serious thanmissing a single-cutter patrol completely,we moderated the multi-cutter patrol pen-alty (for example, we reduced it from 40 to30 hours). The rationale for fairness penal-

The number of possiblequarterly schedules is boundedby 6^'^^' ^ lO'l

ties of 25 hours per inequitable patrol as-signment is that it is better to assign a cut-ter an extra patrol within 24 hours ofhomeport than to leave an area uncovered.

Normally, we would expect to discountpenalties over time, thus tending to pushproblems into the future so they are easierto deal with. However, we have encoun-

March-April 1996 63

BROWN, DELL, FARMER

Cutter B-2(M) B-2(G) B-2(SAR) B-2(NY)

31766

17U3445

217466

36

Table 1: Patrol transit times for each cutter are shown in hours underway at normal speedfrom homeport to each patrol area. These times are minimized by CutS as a surrogate for thetotal quarterly transit time all cutters take to reach assigned patrol areas. These times are alsoused to normalize elastic penalties; for instance, the penalty incurred by assigning less than ormore than two contiguous maintenance weeks is 40 hours, slightly longer than any singletransit time.

AdakWrangellSanibelMonomoyJefferson-lslandGrand-IsiandBainbridge-lslandPt-BonitaPt-FrancisPt-JacksonPt-HannonPt-TumerPt-WellsPenobscot-BaySturgeon-BayThunder-Bay

1736633

171814

IS4

n143636

6

96

3

3639745

1145

181818

9633639745

114•5

181818

tered so few violations using real USCGdata that this has not been necessary.Comparisons with Manual Schedules

As this research has progressed [Farmer1992], Cuts has evolved from a prototypictool evaluated by comparing its scheduleswith historical manual schedules to an ex-perimental generator of new schedules thatwere revised manually prior to publicationto a tool that produces from the model acomplete quarterly schedule that is pub-lished without change. During this evolu-tion, scores of comparisons with manualschedules have contributed more thananything else to improvements in CutSand to its credibility with the USCG.

Expert schedulers—the intended benefi-ciaries of this work—have contributed agreat deal from their experience to the de-

velopment of Cuts, But their advice wasanecdotal and usually expressed in termsof what not to do. We have had to reviseCuts frequently, sometimes on the spot,Flexibility has been key to success.

To adopt Cuts, the 10 USCG districts re-quired an extremely low-cost, site-by-site,stand-alone, single-purpose implementa-tion. Cuts had to reside on a modestly en-dowed personal computer (PC). We wereable to use a 486/66 PC with four mega-bytes of random access memory, a systemcosting about $1,400 (US).

We implemented CutS in GAMS[Brooke, Kendrick, and Meeraus 1992] andoptimized it with a variety of solvers. Wereport results achieved with the mixed-integer linear programming solver XA[Sunset Software Technology 1993]. This

INTERFACES 26:2 64

u s COAST GUARD

Quarter

SpringSummerFallWinter

Total CutterWeeks

208208208208

AvailableCutter Weeks

162

155150133

MaintenanceWeeks

Manual

6g637064

Cuts

&«627064

Missed PatrolWeeks

Manual

1468

Cuts

0003

Table 2; "Total Cutter Weeks" shows the product of 16 cutters working 13 weeks, "AvailableCutter Weeks" deducts weeks when cutters are unavailable for work. The comparison of man-ual with Cuts schedules (at five percent integrality tolerance) shows the number of mainte-nance (charlie) weeks assigned and the patrol weeks missed for each system. CutS missed farfewer patrols.

software and a reasonable complement ofutilities cost about $4,000 (US) per site.

Early in this effort, the First District pro-vided manually prepared, publishedschedules for four consecutive quarters be-ginning with spring 1991. For each cutterand each week, these schedules show pa-trol weeks (but not specific patrol areas),maintenance weeks, and weeks unavail-able (Table 2).

Cuts scheduled each quarter using thesame data the manual schedulers hadused, except that we altered maintenancerequirements (with permission) to evennumbers to make it easy to schedule pairsof weeks. The resulting problems, aftereliminating unnecessary variables and con-straints, have about 850 equations, 650 bi-nary variables, 550 continuous variables,and 7,100 nonzero elements in the con-straint matrix.

Cuts produces quarterly schedules withreasonable responsiveness (Table 3). How-ever, things can slow down if CutS mustseek too small an integrality gap—the dif-ference between the cost of the scheduleproduced and a lower bound on that cost.

expressed as a percentage of schedule cost.The First District schedulers find CutSschedules face valid, even when CutS onlyguarantees them to be within 10 percent ofoptimal. We can generate an annual sched-

Minutes to Guarantee integralityTolerance (%)

Quarter

SpringSummerFallWinter

10%

1.11.13.23.6

5%

1.11.13.26.1

1%

1.11.13.2

37.7

0%

1.11.1

57.1253.9

Table 3: It takes only a few minutes to com-plete a quarterly schedule with CutS using a486/66 PC, GAMS 2.25, and the XA solver. In-tegrality tolerance is the largest admissibledifference between the cost of a schedule andthe lower bound on this cost, expressed as apercentage of schedule cost. The actual inte-grality gap achieved may be better than thistolerance. It takes manual schedulers morethan two days to complete an equivalent setof four quarterly schedules. Even CutS sched-ules from the coarsest, 10-percent toleranceare face valid; this Is fortunate because re-sponse time is generally much quicker forthese than for other tolerances.

March-April 1996 65

BROWN, DELL, FARMER

ule within five percent of optimal by run-ning the quarterly versions of CutS consec-utively, including the required inputs foreach model, in under two hours. This is avast improvement over the two to threedays for an equivalent manual exercise.

The fall and winter versions of CutSconsistently take longer: This is probablybecause winter restrictions on the 140-footice-breaking tug, WTGB, and 82-foot WPBclasses of cutters leave us with far feweravailable cutter weeks and a more con-strained, harder scheduling problem(Table 2).

Our primary mission is to assign re-quired patrols to minimize the number ofmissed patrols. For the sample year (Table2), manual schedules missed 19 patrol

weeks, but CutS (at five percent integralitytolerance) missed only three. This is anenormous improvement.

Patrols should be equitably assignedamong cutters. Equity is subjective andcrews are hkely to judge it by glancing ataggregate overall assignments (Table 4),

A quick comparison reveals the distribu-tion of patrols an:\ong cutters, but not de-tails like cutter-by-cutter unavailability. Wecan approximate the visual equity testquantitatively by comparing the standarddeviations of patrols assigned. For in-stance, among WPB cutters available year-round, the sample standard deviation fortotal annual manual assignments is 2.55patrol weeks, while for CutS this is 2,29.During the spring, the manually scheduled

Cutter

AdakWrangellSanibelMonomoyJefferson-IslandGrand-IslandBainbridge-IslandPt-BonitaPt-FrancisPt-]acksonPt-HannonPt-TurnerPt-WellsPenobscot-BaySturgeon-BayThunder-Bay

Spring

4(4)4(4)6(6)3(4)5(4)5(5)5(4)2(4)5(4)2(4)5(4)2(4)6(4)4(4)4(4)2(2)

Summer

4(4)1(4)5(4)6(6)4(4)3(4)6(4)7(4)2(4)3(3)4<4)6(4)H0i2(4)1(4)4(4)

Fall

4(4)5(6)5(6)3(3)5(6)5(6)5(5)5(4)3(6)6(4)4(5)4(6)5(4)0(0)0(0)0(0)

Winter

4(3)6(6)5(5)5(6)4(6)2(3)7(6)0(1)6(6)4(5)6(6)4(4)4(5)0(0)0(0)0(0)

Annual

16(15)16 (20)21 (21)17(19)18 (20)15 (18)23(19)14(13)16 (20)15 (16)19(19)16 (18)18(17)6(8)5(8)6(6)

Table 4: Quarterly patrol weeks were assigned manually and (in parentheses) by CutS andtotaled for the year. Cutters assigned three or fewer patrols probably had limited availability;the last three cutters are 140-foot WTGB ice-breaking tugs, and they are not available at all forfall or winter patrols. People often judge equity at a glance, and the standard deviation reflectsthis cursory judgment. Among 80-foot and 110-foot WPB cutters available year round, the sam-ple standard deviation for total annual manual assignments is 2.55 patrol weeks, while forCuts this is 2.29.

INTERFACES 26:2 66

u s COAST GUARD

standard deviation among all cutters is1.41 patrol weeks but only 0.77 weekswith CutS; in the summer, CutS reduces1.83 weeks to 0.57.

These equity statistics are simplistic andsuperficial. A meaningful comparison re-quires a complete analysis of the detailsand reasoning underlying the schedule.Nonetheless, people judge schedules at aglance, ignoring for the most part cutter-by-cutter details, and standard deviationsreflect this cursory judgment.

Clearly it's preferable for cutters to pa-trol close to homeport: This is the motivefor minimizing equivalent transit hours inCuts. The USCG manual schedule forwinter 1992 was assessed by CutS to re-quire 458 equivalent transit hours, whilethe Cuts optimization needed only 310.These values included penalties for missingpatrols. During one four week periodwhen neither schedule had penalties, CutSreduced transit hours from 105 to 83.How to Deal with Changes

Published schedules must sometimes berevised. CutS uses optimization, and opti-mization has a well-earned reputation foramplifying small input data changes intowholesale revisions of prior solutions. Thisis a serious problem because major revisionof an already published schedule can ne-cessitate more planning, negotiation, andmessage traffic than the original did. Wehave had to deal with this.

Consider just one change. The scheduleCuts suggested for summer 1994—aschedule the scheduler and cutter captainsapproved after slight modification—^hadconsidered the cutter Sanibel to be un-available for three weeks beginning in lateJuly (weeks 42-44 on our fiscal calendar).

This unavailability was delayed for threeweeks (to weeks 45-47).

Presented with this slight change, CutSsuggested a breathtaking number of sched-ule revisions. For our purposes, a revisionthat inserts or deletes a B-2 patrol week isa major change. CutS prescribed 52 such

We have had to revise CutSfrequently, sometimes on thespot.

major changes—changes in all but oneweek of the entire quarter—affecting theschedules of 11 of the 16 cutters. This revi-sion is mathematically optimal and techni-cally implementable but nonintuitive andmanagerially impractical.

We modified the optimization model inCuts to respond to small changes whileretaining as much of an already-publishedschedule as practical—we call this desir-able property "persistence" [Brown, Dell,and Wood 1995].

In our experience with CutS, a minorchange in input does not alter the objectivefunction value substantially, even when itcauses lots of schedule revisions. Ourproblem isn't lack of good solutions butrather too many of them. Accordingly, forminor changes we can replace our objec-tive function with an equivalent aspirationconstraint to insure that revisions cost nomore than the original schedule; we thenmodify CutS to revise with a surrogate ob-jective, defined below, which preservespersistence.

In this case, the original schedule costs570 transit hours, and we constrain the re-vision to cost no more. The surrogate cost

March-April 1996

BROWN, DELL, FARMER

Weeks

Cutter 40 41 42 43 44 45 46 47 48 49 50 51 52

Revision with CutS

A A AA A

AA

AA

AA

AAA

AA

A

A

AAA

AAA

AA

AA

AdakWrangell ASanibel A AMonomoyJefferson-Is! and A A AGrand-IslandBainbridge-IblandPt-BonitaPt-Francis APt-Jackson APt-Hannon A APt-Turner A APt-WellsPenobscot-BaySturgeon-BayThunder-Bay

Revision with Persistent CutS

AdakWrangellSanibel 5 5 5 6 5Monomoy §Jefferson-lsland 5Grand-Island 5Bainbridge-lslandPt-BonitaPt-FrancisPt-JacksonPt-Hannon §Pt-Turner ^Pt-Wells 5Penobscot-BaySturgeon-BayThunder-Bay

Table 5: Persistence between schedule revisions can be enhanced. A revision of the summer1994 First District schedule delays a three-week unavailability for the Sanibel from weeks42-44 to weeks 45-47. A A represents a resulting major change to the original schedule (that is,B-2 patrol inserted, or deleted): These schedule revisions are counterintuitive and dispropor-tionate to the slight input modification. After CutS is enhanced for persistence, a 6 shows re-maining major changes. The original schedule cost 570 transit hours with a three percent inte-grality gap: The revision is constrained to cost no more than 570 transit hours. The modified,persistent revision is much easier to promulgate and no more expensive to execute.

INTERFACES 26:2 68

u s COAST GUARD

of any Sanibel assignment is zero. Forother cutters, the surrogate cost of assign-ments in the revision that persist with theoriginal schedule is zero, exchanging B-2patrols with B-12 or C status costs 10, andexchanging B-12 and C status costs 1. Forthe summer 1994 revision, "persistent"Cuts prescribed only 11 major changes(Table 5). Such revised models are typi-cally much easier to solve than the corre-sponding original schedules: Here, the re-vision was about three times faster.

We have also been asked to fix individ-ual assignments. Fixing assignment vari-ables is easy. However, tinkering with an

The goal is better schedules,not replacing schedulers.

entire existing schedule by fixing individ-ual variables is a tedious way to fine tuneand too often unleashes unpleasant sur-prises.

In the real world, changes are inevitable.Decision support systems must accommo-date changes, or their utility is limited, ifnot crippled. Optimization models, in par-ticular, are much improved by incorporat-ing explicit features for dealing withchanges.Conclusions and Caveats

Cuts devises quarterly schedules that aresuperior to manually developed schedules:Cuts misses fewer required patrols, assignspatrols closer to homeport, and more equi-tably distributes patrols among cutters.Cuts has recommended face-valid sched-ules that have been implemented withoutany changes by a scheduler. However, thegoal is better schedules, not replacing

schedulers. The real benefit may be help-ing schedulers evaluate feasible alterna-tives quickly. CutS offers far better alterna-tives.

Will the Coast Guard continue to useCuts? Although Cuts offers impressive im-provements, USCG districts have sched-uled their cutters for a long time withoutbothering with computers, let alone math-ematical models. Thus far, we have pro-vided on-call support to the First District.Considering how quickly schedulers turnover, it may not be feasible for us to dothis if all 10 districts adopt CutS. CutS iseasy to use but admittedly suffers cosmeti-caliy for lack of a visually appealinggraphical user interface: Our circular di-lemma is that it makes sense to develop aslick interface only if the Coast Guardreally wants to use it widely, but lack ofsuch an interface limits demand.

Optimization has a nasty habit of pre-scribing tumultuous revisions based onsmall scenario-to-scenario input changes.We believe this explains much of the resis-tance to adoption of optimization-baseddecision support systems: Our modified,persistent version of CutS is an example ofmodeling for increased managerialacceptance.AcknowledgmentsWe thank the following USCG First Dis-trict schedulers, in order of their tenure:Lieutenants Robert Patton, Mike Sabellico,John Sharon, and Bill Stinehour. Farmerdiscovered this topic while sponsored bythe USCG Research and DevelopmentCenter. Dell has been supported, in part,by the Naval Postgraduate School researchinitiation program. Professor Kevin Wood(Naval Postgraduate School) has been a

March-April 1996 69

BROWN, DELU FARMER

tough editor but generously donated histime to sharpen our exposition.APPENDIX

The Cuts formulation follows.Indicesi ^ cutter (for example, Adak, Wrangell,

. . .);k = statuses (for example, B-2(M), B-2(G),

B-2(SAR), B-2(NY), C) (the B-12 status isnot explicitly contained in this index set);and

( = week (for example, 1,2, • • •, T = 15).DataCOSTiii ^ "cost" of assigning cutter / to sta-

tus k (units for these costs and penaltiesare discussed below);

REQi, - recommended number of cuttersfor patrol status k;

RPENi,, ^ penalty for scheduhng less thanthe recommended number of cutters{REQk) for patrol status k in week ((costper cutter);

FAIR,, FAlRj ^ minimum, maximum equi-table number of patrols for cutter i;

FPEN ^ penalty for under- or over-utilization of an equitable number of pa-trols (cost per patrol);

CHARLIEj = minimum number of mainte-nance weeks required for cutter i;

CPEN, ^ penalty incurred in week f due toassignment of less than, or more thantwo contiguous maintenance weeks (costper instance);

F/X,i,, F/X,j, ^ lower, upper bound on as-signment of cutter / to status k duringweek (: This can be used to restrict as-signments or to fix partial schedules suchas the known statuses for two weekspreceding each quarter. In the absenceof a restriction, FIX = 0 and FIX = I;

ASPIRE = the maximum desired total"cost" of a quarterly schedule;

CHANGEii,, = given a prior incumbentschedule with assignment of cutter / tosome status in week t, CHANGE,^, is thepenalty for changing the old incumbentstatus to a new candidate status k. In the

absence of comparison with a priorschedule, CHANGEiu = 0 V iX t; and

CHANGE ^ the maximum desired total"change" of a quarterly schedule.

Decision Variablesassigtin,, ^ 1 if cutter i is assigned status k

in week t, and 0 otherwise;= elastic variable measuring the

number of cutters less than the recom-mended number for patrol statusk in week (;

n = elastic variable representing as-signment of cutter i to a noncontiguousmaintenance week f;

d3c,, = elastic variable representing assign-ment of cutter i to a third contiguousmaintenance week t;

unfair, = elastic variable representing thenumber of inequitable patrols under- orover-assigned to cutter i;

cost = total "cost" of a quarterly schedule:This cost may be minimized, and isbounded by ASPIRE; and

change = total changes between a prior in-cumbent schedule and a new candidateschedule: This cost may be minimized,and is bounded by CHANGE.

FormulationSubject to

2 assign^,, - REQ,, - dreq,,, V/r ¥^ C, t. (1)

assign,,, < ] V), t.

igriia > CHARllEj Vi,

(2)

(3)

igniC2 ^ 0 + dcoun V(,

assignee, - assigtiia-^ -

< 0 -H dcon,, Vi, \ <t (4)

< 0 + dcon,T Vi.

INTERFACES 26:2 70

u s COAST GUARD

assignee, + assign^o-i +

< 2 + dZc,, VI, t > 1.

FAlRj - unfair, Vi,

(5)

(6)ignikt ^ FAIR, + unfair, Vi.

< 2 VI, f > 2.

Vi, f > 1.

(7)

(8)

it ^ 0 Vi, t;

^O VI, f; unfairi>0 Vi.

(9)

2 Z

K Z Zk t

K Z Z ,, -̂ d3c,,) (10)

unfair, - cosf.

cost < ASPIRE,

Z Z ikt = change,

(11)change < CHANGE.

Minimize

cosf

or minimize

change. (12)

The constraints and objectives are de-scribed below:(1) The recommended number of cuttersmust be assigned to patrol k or the short-fall assessed.(2) Each cutter may be assigned to at mostone status each week. (Any cutter not as-signed a status is assumed to be in statusB-12.)(3) A minimum number of maintenanceweeks must be assigned to each cutter dur-ing the quarter.(4-5) Maintenance should be assigned intwo consecutive weeks: (4) assesses anelastic violation if a cutter i is assignednon-contiguous maintenance in week t,even if this is the last week of the quarter,while (5) assesses a violation for a thirdcontiguous maintenance week (.(6) An equitable number of patrols shouldbe assigned to each cutter, or the magni-tude of inequity assessed.(7) At most two consecutive patrols maybe assigned to each cutter.(8) A cutter should not be assigned to con-secutive weeks of search-and-rescuepatrols.(9) Assignment variables are binary; elasticvariables are non-negative.(10) The full cost of a schedule includesassignment costs and elastic penalty costs.This cost is bounded by ASPIRE.(11) This penalty function assesses howmuch a revised candidate schedule differsfrom a prior incumbent schedule. Thispenalty is bounded by CHANGE.(12) The objective may be to minimizetotal cost or to minimize changes from aprior incumbent schedule.

ReferencesBechtle, J. M. 1984, "Routing a USCG buoy-

tender to service aids to navigation: A case ofthe traveling salesman problem," MS tbesis.Naval Postgraduate School, Monterey, Cali-fornia.

Brooke, A.; Kendrick, D.; and Meeraus, A.1992, GAMS: A User's Guide. Release 2.25.The Scientific Press, South San Francisco,

March-April 1996 71

BROWN, DELL, FARMER

California.Brown, G. G.; Dell, R. F.; and Wood, R. K.

1995, "Optimization and persistence," work-ing paper.

Brown, G. G.; Goodman, C. E.; and Wood, R.K. 1990, "Annual scheduling of Atlantic fleetnaval combatants," OperaHons Research, Vol.38, No. 2, pp. 249-259.

Cline, A. K.; King, D. H.; and Meyering, J. M.1992, "Routing and scheduling coast guardbuoy tenders," Interfaces, Vol. 22, No. 3, pp.56-72.

Darby-Dowman, K.; Mitra, G.; and Hajian, M.1992, "Solution strategies for zero-one pro-gramming with a class of vehicle problem,"presented at the EURO/TIMS Joint Interna-tional Conference, Helsinki, Finland.

Farmer, R. A. 1992, "Scheduling coast guarddistrict cutters," MS thesis. Naval Postgrad-uate School, Monterey, California.

Goodman, C. E. 1985, "Annual scheduling ofAtlantic fleet naval combatants," MS thesis.Naval Postgraduate School, Monterey, Cali-fomia.

Laity, M. J. 1987, "Strategic allocation of sealift:A GAMS based integer programming ap-proach," MS thesis. Naval PostgraduateSchool, Monterey, California.

Lima, N. R. 1988, 'A column generation tech-nique for a crises deployment planning prob-lem," MS thesis. Naval Postgraduate School,Monterey, California.

Nulty, W. G. and Ratliff, H. D. 1986, "Interac-tive optimization methodology for fleetscheduling, PDRC 86-11," School of Industrialand Systems Engineering, Georgia Institute ofTechnology, Atlanta, Georgia.

Ronen, D. 1983, "Cargo ships routing andscheduling: Survey of models and problems,"European lournal of Operational Research, Vol.12, pp. 119-126.

Sibre, C. E. 1977, "A quadratic assignment/lin-ear programming approach to ship schedul-ing for the US Coast Guard," MS thesis. Na-val Postgraduate School, Monterey, Califor-nia.

Soland, R. M. 1982, Proceedings of a Symposiumon Gargo Ship Routing and Scheduling, Serial-TM-69250. Institute for Management Scienceand Engineering, School of Engineering andApplied Science, George Washington Univer-sity, Washington, DC.

Sunset Software Technology 1993, XA: Profes-sional linear Programming System GAMSVersion 2.2. San Marino, California.

S. N. White, Captain, US Coast Guard,

Chief, Search and Rescue Branch, First

Coast Guard District, 408 Atlantic Avenue,

Boston, Massachusetts 02210-3350, v\̂ rites,

"Dear Dr. Dell:

"I would like to thank you for the time

and effort you spent to develop a patrol

boat schedule for the First Coast Guard

District during July, 1995. You produced a

very workable and realistic schedule de-

spite having to factor in numerous individ-

ual cutter requests and various scheduling

constraints, The schedule which you pro-

duced met First Coast Guard District

scheduling needs in all aspects. Although

last minute operational needs required the

original schedule to be n:iodified slightly,

the methodology behind your mathemati-

cal model has proved its effectiveness.

"Your assistance in this matter was

greatly appreciated and I look forward to

the opportunity to work with you again in

the near future."

INTERFACES 26:2 72