Optimization and Persistence

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

ROBERT F. DELL Operations Research DepartmentNaval Postgraduate School

R, KEVIN WOOD Operations Research DepartmentNaval Postgraduate School

Most optimization-based decision support systems are used re-peatedly with only modest changes to input data from sce-nario to scenario. Unfortunately, optimization (mathematicalprogramming) has a well-deserved reputation for amplifyingsmall input changes into drastically different solutions. A pre-viously optimal solution, or a slight variation of one, may stillbe nearly optimal in a new scenario and managerially prefera-ble to a dramatically different solution that is mathematicallyoptimal. Mathematical programming models can be stated andsolved so that they exhibit varying degrees of persistence withrespect to previous values of variables, constraints, or even ex-ogenous considerations. We use case studies to highlight howmodeling with persistence has improved managerial accep-tance and describe how to incorporate persistence as anintrinsic feature of any optimization model.

T^e reasonable man /^pt imizat ion-based decision supportadapts himself to the world; % #

V^^'systems, that is, decision supportthe unreasonable one persists in trvine to adapt i u -u J U.. i J . u- If systems built around one or more mathe-the world to himself; ^

matical programming models, are pre-Therefore, all progress depends on the unrea- j • M I J •- n * j isonable man dominantly employed as follows; A model

is used to produce a plan, the plan is pub-—Man and Superman, George Bernard ShawCopyright O 1997. Institule lor C^erations Research TKOGRAMMINti—UNEAR—APPLICATIONSand the Management Sciences PROGRAMMING—INTTEGER—APPUCATIONSt)092-2102/y7/2705/UOl 5S5.00This paper was retereed,

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BROWN, DELL, WOOD

lished, revised data become available andare incorporated into the niodel, the re-vised model with many or all of the origi-nal decision variables and perhaps somenew variables is solved one or more times,and a revised plan is published. This cyclerepeats in periodic or continuous review.Confronted with revisions, managers fre-quently object, "We have committed our-selves to decisions based on prior modeladvice; don't ask us to change our plansur\less we have some compelling reason todo so." New plans that retain the featuresof prior published plans are more accept-able to decision makers than plans that re-quire drastic changes. We have developedmethods for incorporating this kind of per-sistence in modeling linear, mixed-integer,and integer linear programs.

We also use the techniques of persis-tence to incorporate managerial requestsand preferences that arise outside of thecyclic-review process. Sometimes, a man-ager has useful information about an opti-mization scenario that cannot be easilyincorporated into a model,, yet this infor-mation is critical for obtaining a usable so-lution. For instance, forecasted severeweather may affect the production of cer-tain products at a plant next week, but theplant's production-planning model doesnot encompass weather. To handle thisproblem, a manager could establish a setof weather-feasible production targets forthe affected products, make a model runthat is persistent with respect to those tar-gets, and thereby obtain a usable solution.

In our experience, lack of persistence isone of the most common sources of com-plaints about optimization. Some evidenceof our struggles with persistence can be

found in our publications on productionplanning [Avery, Brown, Rosenkranz, andWood 1992; Brown, Geoffrion, andBradley 1981; and Brown, Graves, andHonczarenko 1987], dispatching [Bausch,Brown, and Ronen 1995; and Brown, Ellis,Graves, and Ronen 1987], ship scheduling[Brown, Dell, and Fanner 1996; Brown,Goodman, and Wood 1990; and Brown,Graves, and Ronen 1987], capital budget-ing [Brown, Clemence, Teufert, and Wood1991], and global supply chain manage-ment [Amtzen, Brown, Harrison, andTrafton 1995]. Over time, we have realizedthat we need to state models like these dif-ferently to formally incorporate persis-tence. (We use persistent to mean "pertain-ing to persistence.") In addition. In theearliest design of a model we need to con-sider the cyclic-review process in whichthe model may be used, and we need todesign the models to be flexible enough tohandle unforeseeable managerial requestsor preferences. We must take into accounthow the model will be used in the realworld.

The literature, for the most part, givesshort shrift: to the likely real-world use ofan optimization model. A refreshing ex-ception is Schrage [1991, p. 129]:

Multiperiod models are usually used in a roll-ing or sliding format. In this format the modelis solved at the beginning of each period. Therecommendations of the solution for the firstperiod are implemented. As one period elapsesand better data and forecasts become availablethe model is slid forward one period.

Unfortunately, Schrage does not go on topoint out that a multiperiod model's ad-vice might need to reflect the model's ownprior prescriptions. He leaves us with noguidance about how to model and imple-

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ment persistence.The lack of advice on persistence in the

literature probably derives from thefollowing:—Many papers discuss only hypothetical,pilot, or new applications, but problemswith persistence usually emerge after amodel is used for awhile. (In many of ourapplications, we have not initially consid-ered persistence, and only over time doesthis oversight become a nuisance.)—Modelers write most papers, and theyusually focus on how to obtain an optimalsolution efficiently, rather than how thatsolution is going to be used. This focuscan bias the modeler to accept the disrup-tive consequences of an optimal solutionbecause it is, after aU, an ophmal solution.If managers wrote more papers, the focusmight change, since they normally preferusable solutions over mathematically opti-mal ones.—Everybody does it but nobody admits it.Sooner or later, most modelers deal withproblems of persistence, and typically theyresolve the problems by simply fixing cer-tain variables to their desired values. Fewmodelers are proud of doing this.We address these issues by—Using a series of case studies that dem-onstrate how persistence can mediate thedifferences in focus between managersand modelers; and—Showing how to develop models fromthe start with persistence in mind.Scheduling Coast Guard Cutters

The First United States Coast GuardDistrict has used CutS (Cutter Scheduler)to schedule cutters for three years [Brown,Dell, and Farmer 1996]. The mixed-integerlinear program within CutS assigns 16 cut-

ters to weekly patrols, maintenance, andtraining assignments over a calendar quar-ter while minimizing total time in transit.If changes occur after the Coast Guard haspromulgated a CutS schedule to the fleet,it is critical that CutS incorporates persis-tence in remodeling schedule revisions.

For instance, the summer 1994 sched-ule—developed with CutS and approvedafter slight modification by the schedulerand cutter captains—planned on the cutterSaiiibei being unavailable for three weeksbeginning in late July. This unavailabilityturned out to be delayed by three weeks.Presented with a modification of just onecutter's availability, the nonpersistent ver-sion of Cuts suggested 52 major changes,where "major change" is defined as theaddition or deletion of a week's patrol as-signment in some cutter's schedule. Thesechanges influenced 12 of the 13 weeks ofthe quarter and affected the schedules of11 of 16 cutters. This solution was mathe-matically optimal and technically impie-mentable but managerially impractical.

The need for a persistent solution in thiscase is clear: We want to retain as much ofthe already-published schedule as practi-cal, and we don't want the cost of theschedule to change significantly. In thepersistent version of CutS, we "encour-age" binary assignment variables to takeon the values they had in the solution cor-responding to the published schedule. Wedo this by converting the original vari-ables into elastic persistent variables. Each

such variable has a target value it is en-couraged to obtain and a linear penalty forany deviation from the target. The conver-sion is particularly simple for binary vari-ables because it is necessary to modify

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BROWN, DELL, WOOD

only the objective coefficients of the origi-nal variables. To keep the cost of the re-vised schedule close to the original cost,we converted the original objective func-tion into an aspiration constraint (a con-

straint on an "aspiration level," such asmentioned by Mack [1971, pp. 197-200]).Thus, the objective function of the persis-tent modei is a surrogate objective thatjust measures deviations from the originalschedule.

The original schedule for summer 1994cost 570 transit hours, and we were able toconstrain the revision to cost no more (auser-moderated inflation of the originalschedule cost can also be used). The surro-gate cost of any assignment for the cutterwhose availability was changed (the Sani-bel) is zero. For other cutters, the surrogatecost of assignments in the revision that areidentical to those in the original schedule

is zero, exchanging patrols with nonpa-trols costs 10, and exchanging mainte-nance and training assignments costs one.For the summer 1994 revision, "PersistentCutS" prescribed only 11 major changes,five of which were for the Sanibet (Figure

1).Revisions such as this are not only more

managerially acceptable, they are typicallymuch easier to solve than the correspond-ing original schedules. Here, the revisionis about three times faster to solve thanthe original.Base Realignment and Closure ActionScheduler

BRACAS [Dell 1997] is a mixed-integerlinear program developed for the USArmy to guide it in closing and realigningmilitary installations. Realigning an instal-lation (a military base, for example) meansassigrung different units to it. BRACAS

Cutter

AdakWtangcllSanibelMonamoyJeffpfsonGrandBainbridRPPt-BonitaPt-EVaadsPt-JacksonPt-HamionPt-TimierPt-WeUsPeimbscot

Thuiidcr

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Revision with

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40 41 42 4:t 44 4f, 4f> 47 48 49 50 SI 52

Revision with persistent CutS

6 f, 6 6 AS

Sf>

s6

6

Figure 1: The First Coast Guard District revised its approved, published, summer 1994 scheduleto accommodate a three-week delay in a three-week unavailability of the cutter Sanibcl. In anoptimal solution, CutS responded with the major changes (patrol reassignments) shown at theleft on a Gantt chart with rows representing cutters, columns representing schedule weeks, andthe J indicating addition or deletion of a patrol week (a major change) in the revised schedule.Persistent CutS reduced major changes to those shown at the right with ^'s.

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maximizes the expected net present valueof savings the army accrues by schedulingexpenditures for closure and realignmentin each of six planning years. It does thiswhile satisfying a number of constraintsgoverning the way and the rate at whichthe army must spend money.

ln 1995, after a long planning processand many reviews, the Congress approvedthe army's plan to close 28 installationsand realign 13 others to save, eventually,$450 million per year. This approval in-cluded a base realignment and closure(BRAC) budget totaling about a billiondollars over six years. The budget wasbased on cost estimates the army hadmade without extensive field studies andwithout using BRACAS.

Next, the army obtained better cost esti-mates and used them to propose aninstallation-by-installation budget for eachof the six planning years. These proposedbudgets covered BRAC expenditures andanother billion dollars in separately au-thorized environmental cleanup costs. Us-ing the original (nonpersistent) BRACAS,the army discovered that, among otherthings, reallocating $100 million to an ear-lier phase of the BRAC process would in-crease savings by $233 million. In late1995, Congress approved this acceleration,and the army published the year-by-year,category-by-category BRAC andenvironmental-cleanup budgets.

In February 1996, the army received,from each target installation, revised esti-mates of annual BRAC and environmentalcosts. Compilation of the total annual costsderived from these estimates showedbudget overruns in early years (Figure 2).Clearly, the yearly budgets had to be re-

vised to be consistent with the amountsapproved by Congress.

The yearly BRAC budgets break intofour primary categories: construction, en-vironmental cleanup, operating and main-tenance, and "other." The army could real-locate BRAC funds among categorieswithin years but not between years. Un-fortunately, in these BRAC cost estimates,the target installations provided littleguidance about how they might reallocatefunds. BRACAS, with enhancements to en-courage persistence, provided a model toreallocate yearly BRAC budgets acrosscategories and installations. Prioritiesbased on estimated savings guided thereallocation, while constraints ensured thatspending stayed within yearly budgettotals.

Within the persistent BRACAS, a rangedpersistent constraint provides upper andlower limits (ranges) for each targetbudget category by installation. Theinstallations' planned costs for environ-mental cleanup covered initial studies andessential preliminary tasks that could notbe delayed. We fixed expenditures forthese categories, that is, we set upper andlower ranges in the associated persistentbudget constraints equal to establishedvalues. Construction plans are difficultto change, so if an installation had re-quested more than a million dollars,BRACAS ranged the reallocation within 10percent of plan. Operating and mainte-nance and "other" BRAC costs are some-what more flexible. We allowed yearly op-erating and maintenance requests above$2 million to range from an 80 percent de-crease to a 150 percent increase. We al-lowed requests below $2 million to be in-

September-October 1997 19

BROWN, DELL, WOOD

600

500 +

400

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00

m otherS operating and maintenance• environmental cleanup^construction

I•ra

oo

Figure 2; In late 1995, the army published an approved six-year plan for spending about $2 bil-lion to close and realign military installations (left-hand bars). Soon after, the target installa-tions submitted detailed individual schedule revisions that agree with the published plan intotal amount, but not in timing (right-hand bars). The army used persistent BRACAS to res-chedule the target installations revisions to comply with the plans Congress had approved.

creased to 35 percent of the total six-yearoperating and maintenance amount re-quested by the installation. We set therange for "other" requests above $2 mil-lion between - 90 and +150 percent andpermitted requests for lower amounts toincrease to 35 percent of total. PersistentBRACAS also constraints budget totals ineach year to Congressionally approvedlevels, exactly.

The army is following the persistentBRACAS advice.Strategic Analysis of Integrated LogisticsSystems

SAILS is an integrated decision supportsystem for building, modifying, solving,maintaining, and interpreting large-scalestrategic multicommodity logistics net-work design models [INSIGHT 19941.SAILS has been used for more than 20

years by scores of companies, includingabout half of the Fortune 50, their consul-tants, and a number of university logisticsprograms. Geoffrion and Powers [1995] re-view its remarkably long history.

The databases and models that underlieSAILS are typically large {for example,hundreds of millions of freight rates andmillions of model variables), but a modestnumber of entities are important to man-agers: These govern the go/no-go struc-tural decisions (Figure 3). SAILS has beenso successful largely because of its persis-tent features. Managers can use these fea-tures to ensure that a new network designdoes not depart very much from an olddesign and that it is based on theirguidance.

SAILS' graphical user interface cloakshuge amounts of detail and offers the user

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SUPPLIES PLANTSPLANT

WAREHOUSESFIELD

WAREHOUSESCUSTOMER

ZONES

Figure 3: A typical SAILS logistics network is best illustrated in terms of physical system enti-ties (figure adapted from Geoffrion and Powers 11995]). An optimal design can be principallyexpressed by distinguishing which of these entities are to be open and which closed. Hiddenare millions of underlying details from sole sourcing to product recipes—details that are essen-tial but not likely to be the principal focus of decision makers.

intuitive ways to influence network-designdecisions. One can fix model variables toopen or close suppliers, plants, equipment,packing lines, conversion recipes, distribu-tion centers, product bundles, sole sourc-ing, and so forth. Fixing variables is thesimplest and strongest way to insurepersistence. Customer demands for prod-ucts can be scaled, eliminated, patternedafter guide forecasts, and aggregated via ahost of georeferents. One can restrictcommodity flows to automatically followthe patterns, but not necessarily theamounts, of some prior solution. Elasticpersistetit (ranged) constraints govemthroughput limits: These are constraintsthat one can violate at some linear penaltycost per unit violation above or below thetarget ranges. (Elastic persistent (equality)constraints are a special case of the ranged

variety.)Managers can guide SAILS by fixing

certain decisions and in other ways. Forinstance, for any set of candidateopen/close decisions, one can specify theminimum and maximum number of"opens," that is, limits on the set cardinal-ity of the open/close decisions. Usingthese two features, managers can put intoeffect such statements as: "I don't knowwhere we'll relocate all of the distributioncenters, but we'll be in Columbus, Atlanta,and Denver, and we'll operate no morethan 20 locations."

SAILS invites no-fault recommendationof a solution from past experience. Onecan dredge up this advice from detailedrecords of a past design, and perhaps sub-ject it to some qualitative editing via inter-face push buttons. Or the advice may be

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less detailed and expressed in such mana-gerial terms as "the Reno plant will likelystill stay open; try it" or "likely not." Onecan give preference to any decision, but itis not required. The interface sorts outwhat this advice means to successive orcompeting models, which do not necessar-ily share any common constraints or vari-ables but presumably have a "Renoplant."

SAILS always follows this advice first ina persistent preemptive enumeration, which is

a modification of standard branch-and-bound enumeration that incorporates andexploits persistence. Because of its elasticfeatures, SAILS guarantees completion ofat least one global design that follows allthe advice provided to the letter. Then,armed with this initial incumbent solution,the advice that suggested it, and limits onwhat digressions are permitted, SAILScontinues to make system improvementsin a customary enumeration.

Although the initial guidance alone maynot produce an incumbent solution withacceptable quality, the suggestions and thebounds that they contribute can acceleratesubsequent enumeration. In fact, good ad-vice can speed up SAILS by an order ofmagnitude or more. It's also comforting toknow that whatever SAILS finallysuggests, it has given management guid-ance primary consideration.Hamming Distance and Submarines

The sum of the absolute values of thebit-by-bit differences between two binaryvectors is called the Hamming distance be-tween them [Hamming 1986, p. 45]. Sup-pose each binary variable in a set repre-sents the decision to set up production ona machine next month or deploy a ship

next week or change customer sourcingnext year or move a berthed submarine to-morrow [Brown, Cormican, Lawphong-panich, and Widdis 1977] (Figure 4). TheHamming distance between a publishedbinary plan and a successive revision pro-vides a simple but useful gauge of the tur-bulence or lack of persistence between thetwo proposals.

We use ranged persistent constraints tolimit the Hamn^ing distance between suc-cessive solutions and call these constraintsHamming cuts. Alternately, we sometimespenalize turbulence by placing the Ham-ming distance in the objective as a Ham-ming penalty. We implement this penaltyby using an elastic persistent constraintwhose goal is to avoid any change be-tween solutions. Both Hamming cuts andHamming penalties use a simple linearfunctional: The coefficient of each binaryvariable is 1 if its prior published value is0, and it is — 1 otherwise.Kellogg Planning System

The Kellogg Planning System (KPS) isan unpublished model that relies on alarge linear program to determine the pro-duction and distribution of cereals andconvenience foods at a weekly level of de-tail. KPS models processing facilities pro-ducing base products, packaging linesconverting base products into finishedstock keeping units (skus), and shipmentsamong and inventory within processinglocations and distribution centers (DCs).The object is to meet demand at minimumcost.

KPS encompasses about a hundred baseproducts, several hundred skus, about adozen producing locations and about adozen DCs. This is a big, highly detailed

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us Naval BasePt. Loma. CA

Middle Pier

N

W-

Soufh Pier

Figure 4; A plan for submarines berthed at USNaval Base, Pt. Loma, California shows sevenSSN637 Sturgeon and five SSN688 Los Ange-les class submarines situated to receive shoreservices on a given day. Each submarineneeds different shore services day by day, andeach berth position, including a tender shipthat can act as a berth when moored asshown, has differing abilities to render suchservices. The navy minimizes berth shifts:Moving multibillion dollar submarines istime consuming and interrupts services. How-ever, new port arrivals, departures, and dailyservice schedules make some berth shifts un-avoidable. An optimization-based planner hasneeded some guidance to minimize berthshifts that only superficially improve servicebenefits. The berth-planning model expressesthe location of each submarine on any givenday as a binary decision variable and limitsundesired berth shifts between days by penal-izing the Hamming distance between existingor planned positions and suggested shifts ofthese positions.

model even though it does not explicitlydeal with raw materials. Planners solve a20-week tactical "production model" everySunday morning, with planning week onebeginning the next day. There is also astrategic what-if version of KPS, withmonthly detail, that they use to evaluatepotential major changes in production ca-pacity, inventory policy, and so forth.

KPS is persistent in several ways.

On the rolling weekly horizon, raw ma-terials and packaging materials requiredfor week one will already be arranged, soKPS cannot change week-one productionor packaging decisions. Also, "pendingstock orders" restrict week-one shippingdecisions. So, persistent variables freeze(fix) most production and distribution de-cisions in week one.

Lead times for son:ie materials exceedone week, so in some cases the system im-poses a partial freeze of the packagingschedule in weeks two and beyond. It alsouses a partial freeze of other activities toincorporate management knowledge oreven hunches about future conditions thata model cannot guess. For instance, it canreserve scarce production capabilities forkey products in critical weeks by with-holding them from products with otheralternatives.

Strategic KPS employs an 18-month ho-rizon to evaluate such issues as locatingnew production facilities or realigning ex-isting capacity. Forecasting demand 18months ahead is not easy, especially whensales promotions create spikes of highlyuncertain size. KPS responds optimally, ina mathematical sense, to estimated de-mand spikes and will even anticipatethose far in the future by adjusting deci-sions in early months. Because of this,managers initially found KPS too "ner-vous." In particular, from run to run, KPSprescribed significant changes in produc-tion across time, locations, and productswhen planners changed far-term demandestimates for only a few products.

"Nervousness" is just a lack of persis-tence caused, in this case, by KPS's "omni-science." This omniscience is mitigated, in

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BROWN, DELL, WOOD

practice, by employing a problem cascade(for example. Brown, Graves, and Ronen[1987] or the "subhorizons" suggested byChames and Cooper [1961, p. 370]). Inparticular, the strategic version of KPSuses a sliding window of five months, firstoptimizing months one through five, thenfixing month-one variables and optimizingmonths two through six, then fixingmonth-two variables, and so forth. Unlikemost cascade schemes, this one neversolves the entire (18-month) problem.

New plans that retain thefeatures of prior publishedplans are more acceptable....

Making KPS myopic has several advan-tages. Managers like the results: KPSdoesn't anticipate spikes and rearrangeproduction plans too early. Myopia alsoeliminates the need to explicitly model theshelf life of perishable inventories becauseKPS doesn't produce to inventory until itsees demand, and five months is a reason-able shelf life. The myopic model is alsomuch smaller and faster to solve.Helicopter Fleet Planning

The PHOENIX optimization modelhelped the US Army to prevail overbudget critics and modernize its agingpost-Vietnam helicopter fleet [Brown,Clemence, Teufert, and Wood 1991]. Thearmy has used PHOENIX and its progenyto plan modernization of helicopters aswell as a variety of other equipment fleets.The successes of PHOENIX, reinforced byintense scrutiny during defense budget de-bates, have also given other authors theconfidence to apply optimization to othermilitary-proctirement and equipment-

management problems. Illustrative exam-ples are presented by Dundas [1996],Faircloth [1989], and Staniec [1996].PHOENIX-Iike models are now influenc-ing planned expenditures of many billionsof dollars, and persistence is an importantfeature of most.

PHOENIX-like models are all multiper-iod models that address long-term equip-ment replacement issues, including opti-mal timing of major maintenance, refit,retirement, and most important, new pro-curement. Military procurement is distin-guished by high fixed setup costs: Re-search, development, testing, andevaluation costs are large, and productioncosts are amplified by first-time applica-tion of high technology, security, and lim-ited production quantities. Alternate can-didate production and renovation rtins arearranged as campaigns, each with a startdate, duration, and level of effort. Thispartitions production methods to isolatecost-estimation, economies-of-scale, andlearning-curve effects. Military equipmentalso must be fielded in compatible units ofinter-operating equipment types. This au-gurs well for optimization.

PHOENIX uses many elastic persistentconstraints and variables to encourage de-sired spending levels, average fleet age,average "technological advantage" of thefleet, and so forth. (The persistent featureswe discuss have all been used in PHOE-NIX, but not all of these appear in the sim-plified published model.) Discount rates re-duce persistent penalties in each period,typically a year, to net present value oreven lower, reflecting the army's uncer-tainty about the future and a reasonabledesire to delay violations as long as possi-

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ble. PHOENIX'S time horizon covers manyyears, so this is important. Of course, netpresent value is not a precise conceptwhen dealing with nonmonetary units,such as technological advantage, but allelastic penalties are usually adjusted bythe same discount rate, the rate used formoney.

PHOENIX keeps track of individual hel-icopter age and not only limits it, but alsouses an elastic persistent constraint byyear on the average age of the entire heli-copter fleet: "Try to keep the average op-erational helicopter no older than Tyears." This is an example of a weightedaverage elastic persistent constraint thesystem uses to smooth or moderate fluctu-ations in plans to conform with someoverall expectation.

Some of PHOENIX'S persistent con-structs are elastic cumulant persistent con-straints with cumulant target values orranges. These constraints represent our de-sire for a sum of events to meet a sum ofsubtargets since some base event. For in-stance, consider a persistent model usingyearly retirement variables and targets forretirements. Now, the sum of yearly tar-gets from the beginning of the plaruiinghorizon to the end of each year t does im-ply a set of cumulant targets. However,such a model would allow more-or-less in-dependent deviations from its yearly tar-gets. Thus, it might miss the implied cu-mulant targets by significant amounts,especially toward the end of its time hori-zon. In contrast, PHOENIX with explicitcumulant constraints compares cumulativeretirements in each year to targets repre-senting total desired retirements from thestart of the planning horizon. So, PHOE-

NIX pays penalties in each year for the to-tal deviation since the beginning of theplanning horizon and is motivated to keepcumulative retirements on track in everyyear. (Leachman, Benson, Liu, and Raar[1996] describe another example of cumu-lant targets.)

An alternative to writing cumulant con-straints with yearly variables is to rewritethe model in terms of elastic cumulant per-sistent variables. For example, the cumulantformulation described above could be re-written to use elastic persistent variablesrepresenting cumulative retirements, eachwith a cumulant target. Any constraintneeding retirements in a single year couldbe written in terms of the difference be-tween cumulant variables in that and itspredecessor year.Persistent Partitioning

A variety of important business, engi-neering, and scientific applications employset-partitioning models. Anbil, Gelman,Patty, and Tanga [1991]; Eben-Chaime,Tovey, and Ammons [1996]; and Thuve[1981] provide illustrative examples. Thesemodels have a deceptively simple appear-ance but offer powerful modeling capabili-ties. Although set-partitioning models canbe solved with custon:iary linear integer-programming methods, they are not easyto solve reliably, and they have only be-come more fashionable as the trustworthi-ness of solution methods has improved.Predictably, real-world experience revealsissues of persistence, in particular, with re-spect to incorporating guidance from theuser.

As an illustration, suppose there areseveral hundred packages on a loadingdock, each ready to be shipped to its own

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BROWN, DELL, WOOD

destination. A fleet of identical trucks isavailable, each of which can deliver atruckload of packages to their destinationsin some order using a route that cannotexceed some maximum driving distanceor time. The problem is: How should thepackages be consolidated into a minimumnumber of feasible truckloads?

In a set-partitioning model for this, wedefine a constraint for each package tomake sure it gets delivered exactly once.We define a binary variable for each candi-date truckload, with a unit coefficient ineach constraint for a package in that truck-load. All we need do is select the minimumnumber of binary variables, that is, col-umns, so that each constraint has exactlyone unit coefficient selected (Eigure 5).

Of course, the real world is more com-plicated. Eor example, Bausch, Brown, andRonen [1995] describe a freight consolida-tion case that has a number of necessaryembellishments. Trucks are not identical,and the cost of a delivery route is a com-plicated function of which packages atruck carries and when and where it mustdeliver them. Most of these details are ex-trinsic. That is, they govem the generationof cost coefficients and the locations ofunit colunm coefficients for variables butotherwise do not appear explicitly in themodel. This is a blessing and a curse.

The number of binary variables (in thisexample, the number of subsets of pack-ages forming candidate truckloads) can beenormous. Often a key to success is a sam-pling mechanism that can generate fromthis huge population a restricted subset ofcolumns from which a good partition canbe found (for example, a good set ofroutes delivering all the packages). Barring

binary variable

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Figure 5: A set-partition puzzle: Select a set ofcolumns in this matrix so that there is exactlyone selected unit element in each row. (Mini-mizing the number of such columns is no eas-ier.) An illustrative application views eachrow as the requirement to deliver a packageand each column as an alternate deliveryroute. (We give a hint in the text and later asolution.)

exceptionally good fortune, restricted par-tition solutions will exhibit some flaws,such as ridiculously high cost or outrightinfeasibility (for example, packages not de-livered). At this point, the restricted parti-tion needs help.

One approach to improving the solutionis to retum to the column generator with"intemal," model-provided advice aboutwhat kind of columns (routes) are neededand to generate such columns. (Graves,McBride, Gershkoff, Anderson, andMahidhara [1993] use this device in sched-uling airline crews.)

Another approach accepts external guid-ance from an experienced user, who canreview a tentative solution and decidewhen it's reasonable to change the prob-lem by simply delaying a delivery, mak-ing a special delivery, hiring an outsidedelivery service, and so forth. A phonecall relaxing a bottleneck beats a clever al-gorithm every time.

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But, if an experienced user contributestime and advice to deal with a flaw andthen reoptimizes, it's not a good idea tocapriciously create new flaws. So, reoptimi-zation with human assistance makes per-sistent techniques essential.

When reoptimizing, the user should beable to suggest that the previous partialsolution be incorporated where it appearsto be sound. One way to do this is by sim-ply fixing variables. Fixing variables iso-lates the parts of the solution that lookgood, and fixing a variable to 1 can breakup a set-partitioning problem into smaller,more easily solved pieces. (For instance,you will find it easier to solve the puzzlein Figure 5 if you follow the hint that col-umn h is part of a partition.) However, weprefer to employ elastic persistent vari-ables or constraints for this problem sothat if the model does not select user-preferred routes, a penalty is inflicted.(Brown, Goodman, and Wood [1990] addelastic persistent constraints to their gener-alized set-partitioning model for annualscheduling of the US Atlantic Fleet.) Sincethe set-partitioning problem is a binary in-teger program, penalizing nonuse of pre-ferred columns is much like using aHamming penalty as part of the objective.Persistence Is Not New

Persistence is not new, but the literatureis scant. Researchers addressed the basicissue of persistence as early as the 1950sand have published recent research on is-sues of implementation.

Charnes, Cooper, and Ferguson 11955]describe a linear-programming model todetermine a consistent linear formula forexecutive compensation. (Chames andCooper [1961, Chapter 10] later restate this

case.) The model selects weights foremployee-compensation factors so that theresulting formula for executive salariesmeets company-specified criteria "asclosely as possible." Examples of these cri-teria are (1) higher-ranked employeesshould be paid more than lower-rankedemployees and (2) salaries should be com-petitive on an industry-wide basis. Themodel employs both ranged and elasticpersistent constraints.

Chames and Cooper [1961, p. 215] startand end their discussion of goal program-ming with advice about persistence. Theyseek solutions such that "long-run consid-erations are not obliterated by immedi-ately attainable objectives" and conclude:

For example, constraints might be entered todemarcate regions which are "good enough,"and the objective restated to ensure that pro-grams are either (a) good enough or (b) close togood enough, etc.

Ranged persistent constraints can be usedto implement "good enough" and rangedelastic persistent constraints can be used toimplement "close to good enough."

Bowman [1963] demonstrates that onecan incorporate management's past deci-sions to produce effective present deci-sions using a linear decision rule. A refer-ee's comment quoted in that paper states"That managerial decisions might be im-proved more by making them more con-sistent from one time to another than byapproaches purporting to give 'optimal'solutions to explicit cost models . . . espe-cially for problems where intangibles (run-out costs, delay penalties) must otherwisebe estimated or assumed." Our enthusi-asm for this view is guarded. An unfortu-nate thrust of this comment. Bowman's

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work, and extensions of his work [Hurstand McNamara 1967; Jones 1967;Kunreuther 1969; and Bowman 1984] is toemphasize subjective considerations andde-emphasize or ignore objective optimi-zation altogether. Persistence can synergis-tically combine subjectivity andobjectivity.

A number of authors have suggestedconstraints to incorporate subjective con-siderations into mathematical models.Huysmans [1970] suggests adding "humanconstraints" to operations research mod-els. Trull [1966] recounts the advice of or-ganization theorists March and Simon[1958]: Once an initial decision is reached,it establishes decision rules or proceduresthat constrain future decisions. Little[1970, p. B-483] states that subjective judg-ments must be incorporated into modelsused by managers because people have away of making better decisions than theirdata seem to warrant. He concludes:

The model is meant to be a vehicle throughwhich a manager can express his views aboutthe operations under his control. Although theresults of using the model may sometimes bepersonal to the manager because of judgmentalinputs, the researcher still has the responsibili-ties of a scientist in that he should offer themanager the best information he can for mak-ing the model conform to reality in structure,parameterization, and behavior.

All of the subjective considerations wediscuss above can be implemented usingpersistent constraints.

Urban [1974] surveys over 150 articlesfrom the "Application" section of Manage-ment Science and concludes that manage-ment scientists are not building goodmodels from the decision maker's point ofview. He reminds us that a manager's

most cherished prerogative is to make de-cisions, and we must take special care toshow that the model will supplement andnot replace the manager in his or her deci-sion making. For a model in continuoususe. Urban also discusses the need to refitany new data and update any assump-tions. These requirements are the goal andguide of persistence.

Lewandowski and Wierzbicki [1989] ed-ited a series of papers that describe deci-sion support systems based on "reference-point optimizahon" and applications ofthese systems in Poland. Reference-pointoptimization incorporates managerial re-quests and preferences within the decisionsupport systems—one of the goals of per-sistence. The systems interactively formmultiple objectives based on user-suppliedreference points or aspiration constraints.Their description of aspiration constraintsis so generic that we interpret them to in-clude both persistent variables and con-straints. However, none of the edited pa-pers highlight the benefit or necessity ofpersistence, and none illustrate how per-sistence might be useful in successivemodel revisions.

Mulvey [1993] diagnoses trouble withoptimization models that rely on noisy in-put data and prescribes a technique hecalls robust optimization. Robust optimi-zation (see also Mulvey, Vanderbei, andZenios [1995]) seeks a solution that, overmany potential altemate input scenarios,is close to optimal (solution robust) and al-most feasible (model robust). Prom ourperspective of persistence, robust optimi-zation seeks a baseline solution that willpersist as best possible with a number ofaltemate forecast revisions. This is a laud-

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able goal but fraught with challenges. Weagree with Mulvey: How do you forecastfuture revisions before you solve themodel? By contrast, persistent modelinguses experience as a guide and invitesguidance for dealing with change, ratherthan depending upon precise predictions.

We must take into accounthow the model will be used.

General nonlinear-programming meth-ods and (linear and nonlinear) decomposi-tion methods solve models indirectly by it-eratively making local estimates ofdirections of improvement and takingsteps in those estimated directions. It iswise to be cautious about step length be-cause the neighborhood over which youcan trust each local approximation is usu-ally limited by ignorance of the globalproperties of the problem, and the conse-quences of prediction error can be serious.

Accordingly, nonlinear-programming al-gorithms are customarily governed bytrust regions for variable values over whichthe local approximations are assumed tohave sufficient validity (for example, seeFletcher [1981, p. 207]). Some decomposi-tion algorithms also use trust regions.There are both theoretical and heuristic ar-guments that such moderation is a virtue,and real-world computational experienceis convincing [Brown, Graves, andHonczarenko 1987; More 1983].

Even simple linear programs need trustregions because absolute linearity justdoesn't hold in the real world. Unless weapply common sense (and perhaps Tay-lor's approximation) and limit the regionover which we can expect linearity (and

perhaps Taylor's second- and higher-orderremainder terms to remain insignificant),we invite trouble. Ranged persistent vari-ables are variables bounded within de-sired ranges; the corresponding boundsconstitute a hyper-rectangular trust regionfor primal variables. We have not pre-sented a case study with this ubiquitouspersistent feature, but we always use it.

We have also suggested the use of per-sistent elastic constraints: The associatedelastic penalties constitute a trust region fordual variables. The proximal terms used insome nonlinear programming algorithms[Kiewel 1985; Mifflin 1977] and in somedecomposition algorithms [Ruszczynski1986, 1993] are similar persistent controls.

We can interpret linear regression mod-els in the light of persistence: Given a setof observations or targets, a correspondingset of independent variables, and a speci-fied metric, find a parametric function ofthe independent variables that tracks asclosely as possible with the targets. Inother words, find a parametric function ofthe independent variables that is persis-tent with respect to the desired values, thetargets. Given this interpretation, we cancompare the advantages and disadvan-tages of two standard regression models.

The least-squares (L2) model is the re-gression model that people habituallyadopt as much for its ease of computationas for the wealth of elegant statistical re-sults deriving from the usual assumptionsof homoskedastic normality of observationerrors [Draper and Smith 1966]. By con-trast, an absolute-value (LI) regression re-quires one to solve a linear program [Bar-rodale and Roberts 1973], deal withnonunique solutions, and interpret the sig-

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BROWN, DELL, WOOD

nificance of results without as much statis-tical support. An LI regression is notwithout advantages, however.

LI regression models are essentiallyelastic persistent-constraint linear pro-grams. These LPs determine model pa-rameter values that minimize the averageabsolute deviation between observed re-sponse values and forecasts of these. Thereare textbook examples [Schrage 1991, p.255]. In an LI regression, it is easy to addside constraints on estimated model pa-rameters, recourse for missing data, (elas-tic) limits on maximum estimation errorfor any observation, and any number oflinear embellishments. The resultingmodel is still a linear program, and ourexperience shows that a constrained LI re-gression is seldom much harder to solvethan its unconstrained cotmterpart. On theother hand, a standard L2 regression prob-lem is an easy-to-solve unconstrainedquadratic program, but adding linear con-straints changes it into a constrainedquadratic program that is more diffictjlt tosolve.Persistence Has Its Costs

Persistent modeling requires little addi-tional data, but you must prepare thisdata carefully. An elastic persistent vari-able needs a target value and normegativepenalties for deviations above or belowthis target. This can be generalized to al-low different per-unit penalties in differ-ent ranges, but in practice a single targetand simple linear deviation penalties usu-ally suffice and do not require additionalconstraints. The persistent-variable penal-ties can constitute a distinct objective or beweighed with other objectives.

A ranged persistent constraint requires

a target range governing its value. Rangedelastic persistent constraints also need per-unit linear penalties for violations aboveand below the target range. Such con-straints can also be generalized to allowdifferent per-unit penalties in differentranges, but we find that linear penaltieswith a single range usually suffice.

The challenge is to express persistentfeatures, especially penalties, to create aninsightful, unified model.Conclusions

Optimization models respond unpre-dictably to seemingly inconsequentialchanges in input. This is especially trou-blesome when an optimized plan has beenadopted and inevitable small changes indata necessitate a revision. The model userdesires a revised plan that is "not too dif-ferent" from the old, but reoptimizationmay prescribe massive revisions com-pletely out of proportion to the changes ininputs. At best, this is annoying. At worst,good models lose credibility with theirusers.

Our prescription for this problem is per-sistence horn the root persist:

1. to continue steadily or firmly in some state,purpose, course of action, or the like, esp. inspite of opposition, remonstrance, etc.: to per-sist in a belief; to persist in one's folly. [Web-ster's 1989]

By making a model persistent, a new solu-tion may be obtained that is not too differ-ent from the previous solution yet isnearly optimal with respect to standardcriteria. The persistent techniques that weuse are summarized below.—A subset of a model's variables may befixed or frozen to their preferred values.{Preferred means "previous" or "desired"

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depending on how persistence is inter-preted in a given model.) A hyper-rectangular trust region for the modelvariables, centered around a preferred so-lution, implements another simple form ofpersistence.—Some variables can be converted intoelastic persistent variables that incur linearpenalties for deviating from their pre-ferred values. When binary variables allincur the same penalty for changing fromtheir preferred values, the total penaltymeasures the Hamming distance betweenthe preferred and new solutions.—Ranged persistent constraints ensurethat certain aggregate quantities do notdeviate too far from preferred values.These quantities may or may not be partof an originating nonpersistent model.Ranged persistent constraints with binaryvariables can limit the Hamming distanceof a new solution from a preferred solu-tion; we call such constraints "Hammingcuts."

—Ranged persistent constraints can be toostrict. To allow a persistent model moreflexibility, we employ elastic persistentconstraints that may or may not beranged. Such constraints encourage aggre-gate quantities to achieve preferred valuesor ranges of values, but allow penalizeddeviations to occur, too. A discount factoris often applied to penalties on such con-straints (or variables) when they are in-dexed by time or proximity.—Sometimes we convert an original objec-tive function into an aspiration constraintin a persistent model. This is just a persis-tent constraint, elastic or ranged, that en-courages or forces a new solution not todeviate too far in cost from a preferred so-

lution's cost.^Por the purpose of introducing subjec-tive judgments into a model, we some-times limit the number of binary variablesthat can be set to one or zero using set car-dinality constraints.—Rather than penalizing or restricting de-viations of variables or constraints fromprevious values, it may be desirable to pe-nalize or restrict deviations that accumu-late over time. Por instance, we may notwant to retire exactly 10 aircraft in eachyear t in a model, but we might like to seethat about lOt aircraft are retired on aver-age year-by-year through year t in themodel. In such a case, the persistent con-structs described above may be applied tocumulant variables or cumulant con-straints. When elastic penalties are usedwith a sequence of cumulant variables orconstraints, they imply penalties on aweighted average of their noncumulantcounterparts. Weighted average con-straints can also directly govern linearcombinations of arbitrary performancemeasures.

There are also two rather different tech-niques that we use to achieve persistentbehavior in certain models:—By solving a time-indexed model with aproblem cascade, that is, by sequentiallysolvuig such a model over a subset of themodel's time periods (say, 1 through t, 2through t+h...,T-t-\-1 through T),we induce a f-period myopia in the modelthat reduces "nervousness" of solutions tominor changes in data.—A persistent preemptive enumerationcan be used to solve mixed-integer pro-grams by branch and bound: The solverinvestigates preferred solutions before

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BROWN, DELL, WOOD

widening its focus and considering less(subjectively) desirable solutions.

Textbooks, even those with case studies,don't offer much advice about persistencein optimization models. Thus, we havepresented a collection of case studies thatmotivates the need for persistence in suchmodels and shows how to incorporate it.These case studies reflect a long, slowawakening on our part that persistenceshould be a model enhancement, ratherthan a crippling oversight. A persistent so-lution is a more usable solution and, de-spite the need to solve a larger model, of-ten takes less time to compute than itsnonpersistent counterpart.

Persistence is usually added to an opti-mization model too late, after it misbe-haves, and almost all optimization modelseventually do misbehave. After the fact,repairing the model and regaining thefaith of sponsors can be complicated. It isbetter to plan for persistence from the out-set. To this end, we have provided newvocabulary and new mathematical nota-tion that should help simplify describingand exploring the issues of persistence inoptimization models. This vocabulary isdefined as used in the body of the paperand is summarized above; the mathemati-cal notation is defined as part of a helicop-ter fleet-modemization model described inthe appendix.

(In Figure 5, columns c, e, h, and / forma partition.)Acknowledgments

Some of this work has been supportedby the Air Force Office of Scientific Re-search and the Office of Naval Research.We thank Professor Rick Rosenthal (NavalPostgraduate School) for his careful re-

view. We keep examples of Mary Haight'seditorial markups handy to show our the-sis students that we, too, are humbled bysuch.APPENDIX: A PERSISTENTFORMULATION, PHOENIX REDUX

A simplified PHOFNIX-like prototype(Figure 6) demonstrates persistence in aninitial model formulation. This exampledoes not incorporate all the persistent fea-tures we have identified, but it is a goodexample of how persistence can be ex-pressed concisely to yield an easily under-stood formulation. Our persistent formula-tion extends the NPS (Naval PostgraduateSchool) standard format, a format we en-force on ourselves, our clients, and ourstudents. (Similar formulation formats ap-pear in the literature.) NPS format followsa define-before-use subdivision of modelentities, including indices (dimensions),data (and units), variables (and units), andthe model's objective and constraints. Typ-ically, we follow this subdivision with ver-bal descriptions of the objective functionand constraints.

We denote elastic persistent variables bya "•" but otherwise do not distinguishthem to simplify the model presentation—the persistent data requirements are de-ferred to a later section of the formulationas shown in Figure 7. For instance, eachcontinuous inventory variable is intro-duced as (a v for c < t, and amplified laterto include an associated target value X l,,-and penalties per aircraft under (PXi ), orover this target (PX, ,.). For this model,only the production and inventory vari-ables are persistent. Their targets mightrepresent values obtained from a previousrun using slightly different data.

Persistent variables (see Figure 8 for apictorial representation) can be directly ac-commodated by specialized solution meth-ods (Brown and Olson [1996] and Fourer[1985, 1988, 1992]). Lacking these tools,conventional methods can be used with

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Indices:

p - production line,a = aircraft type,t = planning year,c - cohort year (year of manufacture).Data:(Note: All costs include an appropriate discount fac-tor based on t.)

B, = budget available in year t,FR, = total aircraft required in year (,FA/ = desired average age of all aircraft in year (,

RC,,,^- = cumulativenumber of aircraft of type (I, cohortc to be retired by year (,

tt,, = annual survival fraction of aircraft a./„ = lag in years between the year when air-craft n is

paid for and the year it joins the fleet,PC,, = cumulative aircraft to be produced in a pro-

duction campaign on line p,

CP^i = the unit cost of producing aircraft a in year (,CR,,,,. = the cost of retiring aircraft a, cohort c, in year

t,CO,a^. = annual operating and maintenance cost for an

aircraft a of age t - c - !„ years,CFpf, = the fixed cost paid in year f when line p is

started in year f,/ = relative weight for standard costs versus linear

persistent elastic penaltiesVariables:

^ui- = for c < /, inventory of operational aircraft oftype a of cohort year c in year / (X,,,, is the numberof aircraft a produced in year t).

R,nt = number of aircraft a in cohort c that are retiredat the beginning of year t,

Y,., = 1 if production line p is opened at the beginningof year t, 0 otherwise.

Mode! (U and v indicate persistent features):minimize

+ (1 - -I) (linear persistent elastic penalties)

subject to

- FR, (1)

B,

PC,Y,,,Y,,,

V( (2)

(3)

V;' (4)

^ 0, > 0

V^ a, c

V/, a. c

Vf, a. c

(5)

(6)

Additional data for persistence follows in Figure 7.

Explanations for the objective function and con-straints folhiw in Figure 7.

Figure 6: An example of PHOENIX, the model used to modernize the army's helicopter fleet,with modifications for persistence. Conventional notation is used with persistent features dis-tinguished by " V " or "U". Elastic persistent variables (witb a " v " over the variable) bave atarget value, and finear penalties for deviations from this in either direction. Ranged persistentconstraints (symbolically, =) may vary in value over a limited range. Elastic persistent (ranged)constraints (symbolically, ^) signify constraint ranges that can be violated at a linear penaltyper unit violation.

the addition of some auxiliary variables.For instance, a continuous persistent vari-able X may be expressed as the sum, X ' +X^ ~ X-, with X*' the fixed target, X^ apositive deviation costing PX"X^, and X"a negative deviation costing PX"X . Therepresentation is even simpler for binaryvariables.

Ranged persistent constraints in Figure6 are denoted =, signifying that a range

will be provided. For example, each in-stance of constraint (1) superficially stipu-lates that fKf aircraft must be procured;however, the notation " = FR/' alerts usthat a persistent constraint range isimplied, and in the following persistencesection of the formulation, this is ampli-fied to be a number no less than FR, andno greater than FR,. Ranged constraintsare supported by virtually all commercial

September-October 1997 33

BROWN, DELL, WOOD

Additional data for persistent mode! features

Data for elastic persistent variables:(Note: All penalties include an appropriate discountfactor based on t.)X^^, PX^, PX,;,, target and penalties per unit above and

below target for X,,,,-, (X ',,. is production target whenc = t and is an inventory target when c < (,)

Data for ranged persistent constraints:

FR,. FR, allowed range for FR,.Data for elastic persistent constraints:

FA I, FA,, PFA,\ PFA;~ range and penalties per unitabove and below range for FA,,

B,, B,, PB,*, PB," range and penalties per unit aboveand below range for B,,

PCp, PCp, PPC^, PPCp range and penalties per unitabove and below range for PC^,,

RC,^^. RC,a,., PRC,+., PRC;~^, range and penalties per unit

Explanation of objective function and constraints

Objective: Minimize a weighted sum of operating andretirement costs plus linear persistent elastic penal-ties.

Eqn. (1): Total aircraft inventory must fall within agiven range in each year,

Eqn. (2): Average fleet age should fall within a desiredrange in each year.

Eqn, (3): Expenditures in each year should fall withinacceptable budget ranges,

Eqn. (4); Cumulative production on each opened lineshould fall within efficient ranges.

Eqn, (5): Production and attrited inventory mustbalance between years.Eqn, (6): Cumulative retirements should fall withindesired ranges each year.

above and below range for RC,,^^..

Figure 7: Additional data underlying the persistent PHOENIX reformulation, and an explana-tion of the objective function and constraints.

Optimizers and can be also be formulatedas a standard equality constraint with abounded slack variable.

The elastic persistent (ranged) con-straints (^) indicate that a range and pen-

alties for violating the range will be pro-vided (Figure 8 pictorially represents this).Specialized linear programming algo-rithms handle elastic constraints directly[Brown and Olson 1996], but any such

Pen , s:

slope

XV

elastic persistent variable: X

implementation: X = X -X +X

B

elastic persistent constraint: V =

implementation:B_-S~ <V <.^

persistent penalty: PXX

implementation: = PX

elastic penalty: Pen

implementation: Pen ~ D~S +Figure 8: In this pictorial representation of elastic persistent variables (left), and elastic persis-tent constraints (right), the variable V could represent a single variable but likely represents amore complex constraint value, such as 2 ( •=( R,P, ' " equation (6) of Figure 6.

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constraint can be represented for standardsolvers as an equality constraint u'ith pe-nalized artificial variables and a boundedslack variable.ReferencesAnbil, R.; Geiman, E.; Patty, B.; and Tanga, R.

1991, "Recent advances in crew-pairing opti-mization at American Airlines," Interfaces,Vol. 21, No. 1 Ganuary-February), pp. 62-74.

Amtzen, B. C; Brown, G. G.; Harrison, T. P.;and Trafton, L. L. 1995, "Global supply chainmanagement at Digital Equipment Corpora-tion," interfaces, Vol. 25, No. 1 Qanuary-Feb-ruary), pp. 69-93.

Avery, W.; Brown, G. G.; Rosenkranz, I. A.;and Wood, R. K. 1992, "Optimization of pur-chase, storage and transmission contracts fornatural gas utilities," Operations Research, Vol.40, No. 3 (May-]une), pp. 446-462.

Barrodale, I. and Roberts, F. D. K. 1973, "Animproved algorithm for discrete /j linear ap-proximation," SIAM Journal on NumericalAnalysis, Vol. 10, No. 5 (October), pp. 839-848.

Bausch, D, O,; Brown, G. G.; and Ronen, D.1995, "Consolidating and dispatching truckshipments of Mobil heavy petroleum prod-ucts," Interfaces, Vol. 25, No. 2 (March-April),pp. 1-17.

Bov^man, E, H. 1963, "Consistency and opti-mality in decision making," Managemetit Sci-ence, Vol. 9, No. 2 Ganuary), pp. 310-321.

Bowman, E. H, 1984, "Content analysis of an-nual reports for corporate strategy and risk,"Interfaces, Vol. 14, No. 1 Ganuary-February),pp. 61-71.

Brown, G. G.; Clemence, R. D.; Teufert, W. R.;and Wood, R. K. 1991, "An optimizationmodel for modernizing the Army's helicop-ter fleet," Interfaces. Vol. 21, No. 4 Guly-Au-gust), pp. 39-52.

Brown, G, G.; Cormican, K. ].;Lawphongpanich, S.; and Widdis, D, B. 1997,"Optimizing submarine berthing with persis-tence incentive," Naval Research Logistics, Vol.44, No, 4 dune).

Brown, G. G.; Dell, R. F.; and Farmer, R. A.1996, "Scheduling Coast Guard district cut-ters," Interfaces, Vol. 26, No. 2 (March-April),pp. 59-72.

Brown, G. G.; Ellis, C. I.; Graves, G. W.; andRonen, D. 1987, "Real-time, wide area dis-

patch of Mobil tank trucks," Interfaces, Vol.17, No. 1 Oanuary-February), pp! 107-126.

Brown, G. G.; Geoffrion, A. M.; and Bradley, G.H. 1981, "Production and sales planning withlimited shared tooling at the key operation,"Management Science, Vol. 27, No. 3 (March),pp. 247-259.

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

Brown, G. G.; Graves, G. W; and Honczarenko,M. D, 1987, "Design and operation of amulti-commodity production/distributionsystem using primal goal decomposition,"Management Science, Vol. 33, No. 11 (Novem-ber), pp. 1469-1480.

Brown, G. G.; Graves, G. W.; and Ronen, D.1987, "Scheduling ocean transportation ofcrude oil," Management Science, Vol. 33, No. 3(March), pp. 335-346.

Brown, G. G. and Olson, M. P. 1996, "A sim-plex algorithm to maximize persistence andface validity among related model solutions,"paper presented at INFORMS National Meet-ing, Washington, DC (May 5-8).

Chames, A. and Cooper, W. W. 1961, Manage-ment Models and Industrial Applications of Lin-ear Programming, John Wiley and Sons, NewYork.

Chames, A.; Cooper, W. W.; and Ferguson, R.O. 1955, "Optimal estimation of executivecompensation by linear programming," Man-agement Science, Vol. 1, No. 2 (January), pp.138-151.

Deli, R. F. 1997, "Optimizing base realignmentand closure implementation," working pa-per. Naval Postgraduate School, Monterey,California.

Draper, N. R. and Smith, H. 1966, Applied Re-gression Analysis, John Wiley and Sons, NewYork.

Dundas, J. R. 1996, "Optimal fleet modemiza-tion modei for the US Navy's combat logis-tics support helicopter," masters thesis. Na-val Postgraduate School, Monterey,California.

Eben-Chaime, M.; Tovey, C. A,; and Ammons,J. S. 1996, "Circuit partitioning via set parti-tioning and column generation," OperationsResearch, Vol. 44, No. 1 Ganuary-February),pp. 65-76.

September-October 1997 35

BROWN, DELL, WOOD

Faircloth, J. A. 1989, "An integer programmingapproach to long range shipbuilding sched-uling," masters thesis. Naval PostgraduateScbool, Monterey, California.

Fletcher, R. 1981, Practical Methods of Optimiza-tion, Volume 2: Coiistriuned Optimization, JohnWiley and Sons, New York.

Fourer, R. 1985, "A simplex algorithm forpiecewise-linear programming I: Derivationand proof/' Matliematicnl Programming, Vol.33, No. 2 (November), pp. 204-233.

Fourer, R. 1988, "A simplex algorithm forpiecewise-linear programming II: Finiteness,feasibility and degeneracy," hAathematical Pro-gramming, Vol. 41, No. 3 (September), pp.281-315.

Fourer, R. 1992, "A simplex algorithm forpiecevk ise-linear programming III: Computa-tional analysis and applications," Mathemati-cal Programming, Vol. 53, No. 2 (June), pp.213-235.

Geoffrion, A. M. and Powers, R. F. 1995,"Twenty years of strategic distribution sys-tem design: An evolutionary perspective,"Interfaces, Vol, 25, No. 5 (September-Octo-ber), pp. 105-127.

Graves, G. W.; McBride, R. D.; Gershkoff, I.;Anderson, D.; and Mahidhara, D. 1993,"Flight crew scheduling," Miiiuigement Sci-ence, Vol. 39, No. 6 gune), pp. 736-745.

Hamming, R. W. 1986, Coding and InformationTheory, second edition, Prentice-Hall, Engle-wood Cliffs, New Jersey.

Hurst, E. G. and McNamara, A. B. 1967, "Heu-ristic scheduling in a woolen mill," Manage-ment Science, Vol. 14, No. 4 (December), pp.B182-B203.

Huysmans, J. H. B. M. 1970, The Implementationof Operations Research, Wiley Interscience,New York.

INSIGHT 1994, SAILS Concepts: A Handbook forSAILS Users, Insight, Inc, Alexandria,Virginia.

Jones, C. H. 1967, "Parametric production plan-ning," Management Science, Vol. 13, No. 11(July), pp. 843-866.

Kiewel, K. C. 1985, "Methods of descent fornondifferentiable optimization," Lecture Notesin Mathematics No. 1133, Springer-Verlag,Berlin.

Kunreuther, H. 1969, "Extensions of Bowman'stheory on managerial decision-making,"

Management Science, Vol. 15, No. 8 (April),pp. B415-B439.

Leachman, R. C; Benson, R. F.; Liu, C; andRaar, D. J. 1996, "IMPReSS: An automatedproduction-planning and deli very-quotationsystem at Harris Corporation—Semiconduc-tor Sector," Interfaces, Vol. 26, No. 1 (Janu-ary-February), pp. 6-37.

Lewandowski, A. and Wierzbicki, A. P. 1989,Aspiration Based Decision Support Systems,Springer-Verlag, New York.

Little, J. D. C. 1970, "Models and managers:The concept of a decision calculus," Manage-ment Science, Vol. 16, No. 8 (April), pp. B466-B485.

Mack, R. P. 1971, Planning on Uncertainty,Wiley-Interscience, New York.

March, J. G. and Simon, H. A. 1958, Organiza-tions, John Wiley and Sons, New York.

Mifflin, R. 1977, "An algorithm for constrainedoptimization with semismooth functions,"Mathematics of Operations Research, Vol. 2, No.2 (May), pp. 191-207.

More, J. J. 1983, "Recent developments in algo-rithms and software for trust region meth-ods," in Mathematical Programming—The Stateof the Art, Bonn 1982, eds. A. Bachem, M.Grotschel and B. Korte, Springer-Verlag, Ber-lin, pp. 258-287.

Mtilvey, J. M. 1993, "Robust optimization onPC supercomputers," SIAM News, Vol. 26,No. 7 (November), pp. 1,12-14.

Mulvey, J. M.; Vanderbei, R. J.; and Zenios, S.A. 1995, "Robust optimization of large-scalesystems," Operations Research, Vol. 43, No. 2(March-April), pp. 264-281.

Ruszczyriski, A. 1986, "A regularized decom-position method for minimizing a sum ofpolyhedral functions," Mathematical Prognim-ming. Vol. 35, No. 3, (July), pp. 309-333.

Ruszczyriski, A. 1993, "Regularized decomposi-tion of stocbastic programs: Algorithmictechniques and numerical results," workingpaper 93-21, (April), International Institutefor Applied Systems Analysis, Laxenburg,Austria.

Scbrage, L. 1991, LINDO: An Optimization Mod-eling System, fourth edition, Boyd and Fraser,San Francisco, California.

Staniec, C. J. 1996, "Optimizing anti-armorweapons procurement under constrained De-partment of Defense budgets," paper pre-

INTERFACES 27:5 36

PERSISTENCE

sented at INFORMS National Meeting,Washington, DC, May 5-8.

Thuve, H. 1981, "Frequency planning as a set-partitioning problem," European journal ofOperntioiml Research, Vol. 6, No. 1 (January),pp. 29-37.

Trull, S. G. 1966, "Some factors involved in de-termining total decision success," Manage-ment Science, Vol. 12, No. 6 (February), pp.B270-B280.

Urban, G. L. 1974, "Building models for deci-sion makers," Interfaces, Vol. 4, No. 3 (May),p p . i - n .

Webster's Encyclopedic Unabridged Dictionary ofthe English Uinguage 1989, Random House,New York.

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