AIR FORCE INSTITUTE OF TECHNOLOGY

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1 `` PLANNING COVERAGE OF POINTS OF INTEREST VIA MULTIPLE IMAGING SURVEILLANCE ASSETS: A MULTI-MODAL APPROACH THESIS Sarah E. Jackson, Captan, USAF AFIT/GOR/ENS/03-11 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wrght-Patterson Ar Force Base, Oho APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

2 The vews expressed n ths thess are those of the author and do not reflect the offcal polcy or poston of the Unted States Ar Force, Department of Defense, or the Unted States Government.

3 AFIT/GOR/ENS/03-11 PLANNING COVERAGE OF POINTS OF INTEREST VIA MULTIPLE IMAGING SURVEILLANCE ASSETS: A MULTI-MODAL APPROACH THESIS Presented to the Faculty Department of Operatonal Scences Graduate School of Engneerng and Management Ar Force Insttute of Technology Ar Unversty Ar Educaton and Tranng Command In Partal Fulfllment of the Requrements for the Degree of Master of Scence n Operatons Research Sarah E. Jackson, B. S. Captan, USAF March 2003 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

4 AFIT/GOR/ENS/03-11 PLANNING COVERAGE OF POINTS OF INTEREST VIA MULTIPLE IMAGING SURVEILLANCE ASSETS: A MULTI-MODAL APPROACH Sarah E. Jackson, B. S. Captan, USAF Approved: /s/ Rchard F. Deckro, DBA (Charman) Professor of Operatons Research /s/ James W. Chrsss, PhD (Member) Assocate Professor of Operatons Research date date

5 Acknowledgments I would lke to frst and foremost thank my husband, and my wonderfully cute son. Also, I d lke to thank my wonderful parents who ve not only been extremely supportve n my educatonal goals, but also let ther grandson spend tme wth them allowng me tme to focus on my thess. All of my famly have been very helpful and supportve throughout my lfe. To my classmates who helped wth techncal dffcultes and just to lghten the mood many Thanks. Acknowledgements are n order for my advsor, Dr. Deckro, my reader, Dr. Chrsss, Capt Chambal for beng very helpful, and my sponsors. They have all been helpful n ths process. v

6 Table of Contents Page Acknowledgments...v Lst of Fgures...v Lst of Tables...v Abstract...x I. Introducton... 1 Background...1 Problem Statement... 3 Research Objectves... 4 Assumptons...5 Methodology... 6 Summary... 6 II. Lterature Revew...8 Overvew... 8 Project Schedulng... 9 Resource Constraned Project Schedulng Problem Generalzed Resource Constraned Project Schedulng Problem Mult-Modal Generalzed Resource Constraned Project Schedulng Problem Goal Programmng Senstvty Analyss Prevously Publshed Models Summary III. Formulaton Introducton General Approach The Dataset Key Assumptons Computatonal Effort Summary IV. Illustratve Analyss Introducton Problem Notonal Dataset v

7 Examnng the Results Flexblty of the Formulaton Summary V. Conclusons and Recommendatons Revew Recommendatons Conclusons Appendx A. Informaton Used to Develop STK Model Appendx B. Notonal Dataset Obtaned from STK Appendx C. VBA Code Appendx D. Soluton to Notonal Example Appendx E. Gantt Charts from Notonal Example Bblography v

8 Lst of Fgures Page Fgure 1. Map of POIs and startng poston of UAV usng STK Fgu Fgure 2. Sample Gantt Chart for 3-hour tme perod Fgure 3. From STK, graphcal representaton of soluton Fgure 4. From STK, changed prortes as part of senstvty analyss v

9 Lst of Tables Page Table 1. Parameters and Varables for RCPSP formulaton Table 2. Parameters and Varables for MMGRPSP formulaton Table 3. Parameters and Varables for WGP and Chebyshev GP Table 4. Parameters for LGP model Table 5. Parameter and Varable Defntons for General Formulzaton Table 6. Parameters and Varables Table 7. Notonal POIs, Lattudes, Longtudes, T-Values, Prortes, and Weghts Table 8. Allocaton for Q-west Table 9. Satellte parameters used n STK model Table 10. UAV Parameters used n STK model Table 11. Sensor parameters used n STK model Table 12. Actual values from STK model* Table 13. Modfed values used n Frontlne Premum Solver Platform Table 14. Allocaton of assets for Q-West Table 15. Allocaton of assets for Basrah Table 16. Allocaton of Assets for Mosel..78 Table 17. Allocaton of Assets for Shayka 79 Table 18. Allocaton of Assets for H-Zalah...80 Table 19. Allocaton of Assets for Hel Table 20. Allocaton of Assets for I-Corp-HQ..82 v

10 Page Table 21. Allocaton of Assets for Brdge.83 Table 22. Allocaton of Assets for DIV. 84 Table 23. Allocaton of Assets for DIV x

11 AFIT/GOR/ENS/03-11 Abstract For the Unted States to mantan nformaton superorty, t s necessary to have a means of allocatng ntellgence-gatherng assets to collect nformaton on partcular ponts of nterest. In today s geopoltcal envronment, however, the number of ponts of nterest s growng rapdly, whereas the number of avalable assets s not. To ad n mantanng nformaton superorty, ths research explores the use of a Mult-Modal Goal Programmng Resource Constraned Project Schedulng approach for allocatng magng survellance assets (land, ar, sea, and space) to a set of ponts of nterest for a gven tme perod. The multple objectves of ths formulaton are to mnmze the number of ponts of nterest not covered at any tme durng the requred perod, mnmze the devaton from the mnmum mage resoluton of each pont of nterest, and mnmze the tme between successve magng assets magng each pont of nterest. x

12 PLANNING COVERAGE OF POINTS OF INTEREST VIA MULTIPLE IMAGING SURVEILLANCE ASSETS: A MULTI-MODAL APPROACH I. Introducton Background It s sad, a pcture s worth a thousand words. In the fall of 1962, however, a pcture may have been worth over three bllon lves. In October of 1962, the U.S. was able to prove, contrary to clams made by the Sovet Unon, that offensve weapons, medum-range and ntercontnental ballstc mssles were beng placed n Cuba by the Sovet Unon. The mages taken wth a U-2 reconnassance arcraft helped to dsprove Sovet clams that the mltary buld-up n Cuba was solely defensve n nature (30). Ths knowledge, along wth other ntellgence, aded n the avodng of a potentally devastatng nuclear war Knowledge, along wth the means to successfully mplement t, s the key to preventng battles, f possble, and wnnng them f they prove unavodable. Whether by the use of force or through passve means, knowledge of enemy forces, ther composton, dsposton, ntentons, ther locatons, drecton, speed, and combat readness (38) s vtal to the sde that emerges as the vctor. In today s geopoltcal envronment, ths knowledge, also referred to as nformaton domnance, s a necessty to mantan securty and assure vctory. Informaton domnance (ID), as defned by FM 100-6, s the degree of nformaton superorty that allows the possessor to use nformaton systems and capabltes to acheve an operatonal 1

13 advantage n a conflct or to control the stuaton n operatons short of war, whle denyng those capabltes to the adversary. (14:Glossary-7) Wth nformaton domnance the Unted States s able to engage the adversary more precsely and wth greater lethal / non-lethal effects, because of our nformaton advantage and the enemy s correspondng nablty to move or protect tself (46). In addton, nformaton domnance allows the US to create hgh performance unts that use nformaton n a manner that allows them to accomplsh ther mssons more effectvely and quckly wth mnmum casualtes. (46) In a tme where there s no longer a sngle superpower foe, but rather a plethora of potental adversares spread across the globe, ganng and mantanng ID becomes a complex and tme consumng task. Ths task drves the need to optmze coverage of prortzed ponts of nterest usng land, ar, sea, and space assets n order to obtan knowledge that can best serve the approprate decson-makers. Mantanng contnual magng of a pont of nterest n a crtcal pre-attack perod s a complex task, nvolvng both tmng and allocaton of key space, land, ar, and sea ntellgence, survellance, and reconnassance (ISR) assets. Gven the quantty of assets and the array of possble locatons that may requre coverage, the problem of allocatng the avalable resources can become a complex and tme consumng task. The need to utlze scarce resources to mage key areas of nterest (POI) suggests the need to optmze the taskng of varous exstng assets to best serve the decson-makers. An optmzaton approach s useful n ths nstance because there are lmted resources to be dvded among several potental POIs. Optmzaton deals wth problems of mnmzng or maxmzng a functon of several varables, usually subject to 2

14 equalty and/or nequalty constrants. (30:v) Ths allows the decson maker to utlze lmted resources n a way that optmzes the resource s usefulness n achevng an objectve. Problem Statement Durng the Cold War, the Unted States prmarly faced a sngle super power opponent. Mantanng contnuous coverage of one prncpal foe and ts alles was a complex task, but was comparatvely smple compared wth current survellance requrements. In today s poltcal envronment, there s no longer one prmary foe; there are several potental allances of foes and thus an array of ponts of nterest (POI), some of whch change rapdly. Allocatng survellance assets among several potental POIs s dffcult. Mantanng contnuous or near contnuous 24-hour coverage over a multtude of potental POIs s an even more complex task. Contnuous, 24-hour coverage of a POI, therefore, may not exst when relatons between the POI and the Unted States begn to spral towards conflct. The need arses to be able to allocate survellance assets to provde the requred coverage over a new POI. There have been several attempts at optmzng coverage of a POI. The sze of the problem often makes complete enumeraton mpractcal. Branch and bound technques have been used to reduce the number of possble solutons, however the sze of the search space may stll be great due to the number of assets beng consdered (33). The most popular technque for allocatng resources s va a greedy heurstc n whch the targets are prortzed and then assgned for magng. (33:218) These approaches prove nadequate to allocate multple asset types over multple targets n a tmely manner because the analysts have to accomplsh ndependent runs for dfferent magng assets 3

15 and then combne the solutons. By not accountng for the ntegraton of varous asset types, n space, ar, land and sea, as part of the orgnal formulaton, the optmal solutons obtaned through the ndvdual asset type allocatng models tend to become sub-optmal when combned together. The approach presented here allocates an array of survellance asset types to multple target areas. Gven the need for a quck soluton to allocate avalable resources to provde contnuous 24-hour coverage of an area, the problem s ntally modeled usng mult-modal resource constraned technques to specfy all relevant optmzaton consderatons. Research Objectves The Unted States mantans a lmted number of survellance assets. The need for methods to ntegrate assets over varous domans (space, ar, land, and sea), therefore, s crtcal to effcently obtan magng survellance of areas of nterest. Addtonally, the number of potental adversares s great. Porto suggests the number of possble solutons gven N survellance assets and T targets s [( N / 2) ( T 1)! ]. (33:217) If, for example, the Unted States had fve potental survellance assets avalable to provde coverage of 20 POIs, there are possble ways to allocate the fve survellance assets to cover the 20 POIs assumng all assets were capable of observng all POIs. Should the Unted States have 10 assets avalable for allocaton and 25 POIs, the number ncreases to possbltes. The addton of tme ntervals expands the problem even further. Clearly the potental problem soluton space ncreases rapdly wth the addton of assets and POIs. To exhaustvely enumerate all possble solutons would be prohbtve. Because the tme 4

16 nterval of nterest n ths research s the 72 hours precedng the ntaton of actons, ths research focuses on those assets that are avalable to the theater n queston. The approach however can be expanded to multple theaters. A lst of the ntellgence assets n a partcular theater s gven n the Theater Intellgence Archtecture Plan. However, for use n ths thess, an unclassfed generc database has been used. It contans a fcttous number of space, ar, land, and sea-based magery survellance assets along wth capabltes based on unclassfed sources. These assets were modeled usng a program whch has already been ndependently verfed and valdated for use n aerospace applcatons, Analytcal Graphc, Inc s, Satellte Toolkt (STK). Assumptons To ad n the completon of ths research, some prncpal assumptons were made. The frst assumpton made was that all ponts of nterests are terrestral. No ponts of nterest such as launched mssles, arcraft n flght, launched space vehcles, or other vehcles or objects already n flght are taken nto consderaton n ths model. Secondly, ths model does not take nto consderaton the retaskng of satelltes. Ths could however be ncorporated nto the approach by some manpulaton to the data and the addton of some constrants. Ths ssue s addressed n Chapter 5. The thrd major assumpton s that tmes are all ntegers. For example, the duraton of an access of an asset to a pont of nterest s an nteger value, and the start tmes of the accesses are also nteger n nature. Ths was done to smplfy some of the calculatons accomplshed n the model. Chapter 5 suggests ways of adjustng ths assumpton for more realstc representaton of access tmes and duratons. 5

17 Fnally, s t assumed that there exst some proposed routes for UAVs. These routes may not be all the possble routes, however they are assumed to represent the frst choce routes for the UAVs. Senstvty analyss can then be used to provde some constrants for routes that can then be generated wth the approprate route generaton programs. Methodology To allocate survellance assets to provde near-contnuous to contnuous coverage of a pont(s) of nterest, a modelng approach was developed based on Project Schedulng Program (PSP) prncples. More specfcally, a Mult-Modal Resource Constraned Project Schedulng Program (MMRCPSP) approach was extended to ths survellance allocaton problem. The model at mnmzes the uncovered tme gaps between assets coverng the pont of nterest. In order to take nto consderaton the prorty of the varous areas of nterest, a Goal Programmng (GP) approach was also ncorporated nto the model. Ths was done to ensure that the pont of nterest wth the hghest prorty are covered pror to assgnng assets to lower prorty targets. Summary Ths chapter has gven an overvew of the background of the problem, the purpose of ths research, the scope of the research accomplshed, the fundamental assumptons made durng the research, and a bref overvew of the approach and methodology used to provde a model to solve the allocaton of magery survellance assets. Chapter 2 provdes a lterature revew n order to provde the reader wth a background of the approaches used to develop the model. Chapter 3 presents the methodology of developng the model. Chapter 4 llustrates the approach by analyzng a notonal 6

18 example and demonstrates how the results can be used by the decson makers. Chapter 5 gves the conclusons arrved from the research and recommendatons for future studes. 7

19 II. Lterature Revew Overvew In 2000 the Rand Corporaton conducted a study to address the ablty of the Ar Force to effectvely attack tme crtcal targets (TCTs) and the extent to whch ts successes and falures n ths area can be attrbuted to dynamc command and control and battle management capabltes (or ther lack) (19:) t was noted that a major shortfall to optmal allocaton of scarce ISR (Intellgence, Survellance, and Reconnassance) resources and to cross-cued and / or smultaneous collectons s the lack of agreed-upon (by DoD [Department of Defense] and the ntellgence communty) CONOPS [Concept of Operatons], TTP[Tactcs, Technques, and Procedures], and automated tools for ntegrated taskng and battle management of (1) sensors from multple ntellgence dscplnes (mult-int), (2) sensors from multple platform domans (cross-doman), and (3) the assocated PEDS [Processng, Explotaton, and Dssemnatons Systems] to support mltary montorng, assessment, plannng, and executon processes and tmelnes. (19: 28) Mult-INT refers to sgnals, measurement and sgnature, human, and magery ntellgence dscplnes (19:28). Cross-doman refers to the ntegraton of land, sea, ar, and space (19:28). Ths research focuses on magery ntellgence, whch s obtaned from electrooptcal, radar, nfrared, and photography systems (28:16). Ths research does, however, take nto consderaton cross-doman allocaton of magery survellance assets. The man focus of ths research s to develop a mathematcal formulaton that ntegrates land, ar, sea, and space magng survellance assets to provde near-contnuous to contnuous coverage of POIs. The ntegraton of Mult-Modal Resource Constraned Project Schedulng and Goal Programmng accomplshed ths. 8

20 In ths secton, a descrpton of the Project Schedulng Problem (PSP) s gven to provde an ntroducton to approaches used. Ths ntroducton leads nto the necessary background of Resource Constraned Project Schedulng Problem (RCPSP), Generalzed RCPSP (GRCPSP), and Mult-Modal RCPSP (MMRCPSP). Next, a descrpton of Multple-Crtera Decson Makng (MCDM) s gven and lends tself to the dscusson of Goal Programmng (GP). Concludng the chapter s an ntal look at the result of ntegratng the approaches for the purpose of ths research, a Mult-Modal Goal Programmng Resource Constraned Project Schedulng Problem (MMGPRCPSP) formulaton. Project Schedulng There are four prmary objectves of Project Schedulng: 1) mnmze the completon tme of a project, 2) determne the capactes of the renewable resources so that a deadlne s met and resource costs are mnmzed, 3) maxmze the net present value of a project, and 4) mnmze both the estmated rework tmes and costs (13:65). To better understand these objectves, and n turn the essence of the PSP, some defntons are requred. A project s a set of actvtes (tasks) that are performed under a set of requrements (constrants) n order to complete a process satsfyng a partcular objectve, typcally wthn some tme horzon. For example, constructon of a satellte could be a project. In terms of ths thess, the project s gatherng magng of a POI or of a set of POIs near contnuously or contnuously. The objectve s to mnmze the number of POIs not covered at any tme durng the observaton perod, mnmze the devaton from the requred mage resoluton of each POI, and mnmze tme gaps between successve 9

21 magng survellance assets. Ths objectve, as wll be seen, s accomplshed usng Goal Programmng rather than through a standard PSP formulaton. An actvty (task) s one of possbly several events, wth a gven duraton, needed to accomplsh the project. For the satellte constructon project agan, examples of some of the actvtes necessary for the project nclude the desgn phase, assembly of the parts, and testng. For the purpose of ths research, the actvty(s) s the collecton of magng from the POI(s). A resource s somethng that s used durng the actvty. There are three types of resources: renewable, non-renewable, and doubly constraned. A renewable resource s somethng that can be re-used durng the project. In the satellte example, one renewable resource s the people assemblng the parts. The people work durng a certan shft and after they have fnshed a shft return at the start of ther next scheduled shft. The land, ar, sea, and space magng survellance assets are the renewable resources n ths research. A non-renewable resource s a resource whch can be used only once durng an actvty. For nstance, n assemblng the satellte, the ndvdual parts are non-renewable. After a part s attached to one satellte, t wll not return to be used on another satellte. For the purpose of ths research, a nonrenewable resource would be a one-way msson for an magng asset. Doubly constraned resources are resources that are constraned by both actvty and the project. The most common example of ths s money. There could be an allotted amount of money to be used for each the desgn phase, assembly phase, and the test phase, and then a specfc budget for the entre project. Each amount of monetary 10

22 allotment cannot be exceeded. For the purpose of ths research, doubly constraned resources (.e. a satellte that was tasked to dfferent ponts of nterest each orbt) are not used; however, could easly be ncorporated at a later tme f deemed necessary. Fundamentally, the Project Schedulng Problem s to optmally schedule multple actvtes, requrng varous types of resources, whch completes a project whle achevng the objectve defned by the decson-maker. In attemptng to accomplsh ths, there typcally are constrants on the resources. Ths type of PSP s called the Resource- Constraned PSP (RCPSP). RCPSP s used n ths research and therefore s descrbed n more detal n the followng secton. Resource Constraned Project Schedulng Problem In the majorty of projects undertaken, there s some lmt or constrant on the resources avalable. Due to the addton of such constrants, whch are not taken nto consderaton n the basc Program Evaluaton Revew Technque (PERT) or Crtcal Path Method (CPM) approaches to PSP, an extenson on the basc PSP needs to be ncluded to compensate for the lmted resources. When the constrants on the resources are consdered, the basc PSP may not provde an optmal soluton; therefore, the use of the Resource Constraned Project Schedulng Problem (RCPSP) formulaton s necessary. The RCPSP formulaton gven s adapted from Prtsker, Watters, and Wolfe (1969). The objectve functon of ths formulaton, expresson [2.1], (one of several gven n Prtsker, Watters, and Wolfe, 1969) s to mnmze the total project throughput tme. Mnmzng the total throughput tme s equvalent to maxmzng the number of tme perods remanng after the project s completed (34:96). 11

23 Table 1. Parameters and Varables for RCPSP formulaton Parameters: project number, = 1,2,,I; I = number of projects j job number, j = 1,2,,N ; N = number of jobs n project tme perod, t = 1,2,,max G ; G s the absolute due date t g e d j l j u j k r jk R kt a desred due date of project earlest possble perod by whch project could be completed number of perods requred to perform job j of project earlest possble perod n whch job j could be completed latest possble perod n whch job j could be completed resource or faclty number, k = 1,2,,K; K = number of dfferent resource types amount of type k resources requred on job j of project amount of type k resource avalable n perod t k s the amount of resource k avalable for the project Varables: x 1 f job j of project s completed n perod t; 0 otherwse jt x t 1 n perod t f all jobs of project have been completed by perod t, 0 otherwse nmt tx 0 for t max{ a + d 1;max ( a + d + d 1)} < n n j Pn j j n where P n s the set contanng other jobs of project that must precede job n, 1 otherwse t > mn j F { G d m j where F m s the set contanng other jobs of project that must follow job m, 1 otherwse x 0 for } subject to u j t = l j x u t m I = 1 G max x [2.1] t = e t xjt = 1 for = 1,..., I; j = 1,..., N [2.2] (1 / N ) N t 1 j = 1 q = l j x n mt n t = l t = l I m u jq for = 1,.. I; t = e, e + 1,..., G [2.3] t * x + d t * x [2.4] N = 1 j = 1 t + d j q = t 1 r jk x jq n R x, x, xnt, x {0,1} jt t mt nt kt for k = 1, h, K; t = mn a, h, G j [2.5] 12

24 Ths quantty of remanng tme perods s represented by the summaton of x t as t vares from the earlest possble start of a project, e, to G, project s absolute due date. Ths objectve s then subject to precedence and resource constrants. The frst constrant, expresson [2.2], dctates each job havng only one completon tme for each project. In other words, the jobs, whch are beng used n the set of projects, are only to be accomplshed once for each project. Expresson [2.3] prevents project to be completed untl all of the jobs are completed for that project. Ths s to ensure that the project s completed and no jobs are skpped that are requred to be done. The precedence constrant, gven by expresson [2.4], s to ensure that a job s not started pror to the completon of another job(s), f there exst such a requrement. For nstance, n the constructon of a buldng, t s requred to have the walls n place pror to the roof beng added. Fnally, expresson [2.5] s a resource-constraned set, therefore, there must exst a constrant on the avalablty of resources n each tme perod. Suppose a job s beng processed n perod t f the job s completed n perod q. (34:98) Then expresson [2.3] lmts the consumpton of resource k by all jobs for tme t to the amount of k avalable. The model forces actvtes to be shfted n the schedule f the resource lmts are reached. Generalzed Resource Constraned Project Schedulng Problem The Generalzed Resource Constraned Project Schedulng Problem (GRCPSP) allows for more flexblty n the precedence requrements of the actvtes. Precedence requrements allow the user to defne one of two tme-lag requrements between two actvtes, mnmal or maxmum tme-lag. The mnmal tme-lag denotes the mnmum 13

25 tme allowed between two actvtes. The maxmal tme-lag denotes the maxmum tme allowed between two actvtes. There are three mnmal / maxmal tme-lag classfcatons used n GRCPSP; 1) a fnsh-start relaton dctates a mnmal / maxmal tme must elapse between the fnsh f actvty and the start of actvty j, 2) a fnsh-fnsh relaton defnes the requrement for an amount of tme that must occur between the completon of actvty and the completon of actvty j to allow for cope wth the output of actvty, 3) a start-fnsh relaton represents the requrement for a tme-lag between the start of actvty and the completon of actvty j. The mnmal tme-lag relatons can be used n combnaton and the maxmal tme-lag relatons can be used n combnaton f needed to accurately defne the relatons between actvtes and j (13). Mult-Modal Generalzed Resource Constraned Project Schedulng Problem In some nstances, there may be more than one way to accomplsh an actvty. Such nstances can be represented by a Mult-Modal Generalzed Resource Constraned Project Schedulng Problem (MMGRCPSP) formulaton. The RCPSP s actually a generalzed formulaton of a MMGRCPSP n that t assumes the exstence of only one way to accomplsh an actvty. The formulaton gven s adapted from Sprecher (41:8) Expresson [2.6], the objectve functon, represents the desre to mnmze the makespan, the duraton of the project. Ths objectve s subject to a set of constrants. Expresson [2.7] constrans the problem by allowng only one mode assgnment to each actvty, and only one completon tme of that mode. To ensure the usage (consumpton) of renewable (non-renewable) resources do not exceed the per-perod avalablty of each resource type, Expressons [2.8] and [2.9] are ncluded n the formulaton. 14

26 Parameters: J M j Table 2. Parameters and Varables for MMGRPSP formulaton number of jobs set of modes n whch job j can be performed d jm duraton of job j beng performed n mode m R ( N, D) set of renewable (non-renewable, doubly constraned) resources T upper bound on the projects makespan υ δ K r 0 ( K r 0) number of unts of non-renewable (double constraned) resource r, r n R (r n D) ρ δ K rt 0 ( K rt 0) number of unts of renewable (double constraned) resource r, r n R (r n D) avalable n perod t, t=1,, T EF j ( LF j ) the earlest (latest) fnsh tme of job j based on the modes wth smallest duraton ρ δ k jmr 0 ( k jmr 0) number of unts of renewable (doubly constraned) resource r, r n R (r n D), used (consumed) by job j beng performed n mode m at the perod the job s n process υ k jmr number of unts of non renewable resource r, r n N, consumed by job j beng performed n mode m Varables: x 1 f job j s performed by mode m and completed n perod t; 0 otherwse jmt subject to M j LF m = 1 t = EF j M J m= 1 LF J mn tx [2.6] t = EF x jmt = 1 j = 1,2,, j J Jmt J [2.7] J M j = 1 m = 1 j k ρ jmr t + d jm 1 q = t x jmq K ρ rt r R; t = 1,2,, T [2.8] J M j LF j ν kjmr j = 1 m = 1 t = EF x jmt {0,1} j x jmt K ν r r N [2.9] 15

27 Goal Programmng In the aforementoned formulatons, each constrant was a hard constrant. In other words, each constrant must be satsfed as equal or less than the rght hand sde. However, what f the rght-hand-sde values were desred, but the decson maker would allow some devaton from these attanment levels? Goal programmng allows the decson maker some flexblty n defnng achevement levels or target values of varous parameters n the problem. Wthn the approprate constrants, a devatonal varable s ntroduced to model whether or not the target value s obtaned exactly, falls short or s exceeded. Allowng ths controlled relaxaton of the constrants opens up the possblty of a feasble soluton where an nfeasble soluton prevously exsted when the constrants had to be met wth strct regard to the nequalty or equalty specfcatons. The overall purpose of GP s to mnmze the devatons between the achevement of the goals and ther aspratonal levels. (36:3) There are three basc forms of GP: 1) Archmedean GP, 2) Chebyshev GP, and 3) non-archmedean GP (21:12). The Archmedean form s used to mnmze the sum or weghted sum of all devatons from the goals (21:12). Ths s also known as Weghted GP (WGP). The WGP formulaton gven here s adapted from Romero,

28 Table 3. Parameters and Varables for WGP and Chebyshev GP Parameters: s the set of constrants F α β (x) f b d s the weghtng factor for the negatve devaton s the weghtng factor for the postve devaton s the th constrant functon s the target value of the th constrant s the maxmum devaton Varables: s the negatve devaton from goal n p s the postve devaton from goal subject to mn k = 1 ( α * n + β p ) [2.10] f ( x) + n p = b [2.11] x {F} [2.12] Ideally, n ths formulaton n and p wll equal zero. However, f ths s not possble, then the objectve of the formulaton s to mnmze the postve and negatve devatons, n proportonal relaton to the values of α and β. Chebyshev GP, or mnmax GP, s used to mnmze the maxmum of the unwanted goal devatons. (21:13) The followng formulaton s adapted from Romero subject to mn d [2.13] α n + β p d [2.14] f ( x) + n p = b [2.15] x {F} [2.16] 17

29 postve devatons from the goal (k)the non-archmedean form, also known as lexcographc goal programmng, allows for prortzed goals. Ths concept of pre-emptve prortes lets the decson maker specfy that one prorty s preferred over another prorty such that the goals should be fulflled n a specfc order, (.e. hgher prorty goals are satsfed frst and t s only then that lower prortes are consdered. (36:4)) The LGP gven s adapted from Ignzo Table 4. Parameters for LGP model A Parameters Tu an ordered vector such that the kth, uk term s of prorty k coeffcent matrx b rght hand sde value/goal Tv s the vector [x n p] where x s the vector of varables, n and p are the vectors of negatve and Tc the row vector of weghts assocated wth devatons at rank k subject to T (1)T (k)t lex mn u = { c v,, c v} [2.17] Av = b [2.18] v 9 By utlzng one of the aforementoned GP models, t s possble to allow some controllable devatons from the desred rght-hand-sde values. Ths controllable devaton allows the decson maker to have some flexblty n the mathematcal representaton of the problem, whch n turn gves some nsght to alternate solutons. Senstvty Analyss Senstvty analyss of solutons to optmzaton problems gves the decson maker nsght nto the robustness of a soluton. It tests the effect of model assumptons and asssts n measurng senstvty to the precson of the data. It also allows the 18

30 nvestgaton of some lmted varatons n the operatonal condtons wthout the need for re-solvng the model. Perhaps at the tme of the formulaton, crcumstances were less stressed, (.e. plentful resources, no deadlnes). Now, however, after the formulaton s complete, crcumstances have changed, resources have been depleted unexpectedly, tme s now crtcal. The decson maker s not gong to wat for a new formulaton to be developed and solved. Utlzng senstvty analyss on the orgnal formulaton and soluton could provde adequate alternatve solutons for the new crcumstances. Addtonally, senstvty analyss may be used to provde solutons to what-f scenaros; what f there were more resources or more tme. By lookng ahead to such what f scenaros, the decson maker mght opt to make mnor changes n certan values n order to gan hgher benefts from the objectve functon. In GP there are seven dscrete changes that allow senstvty analyss to be performed: 1) change n the weghtng factor at prorty level, 2) change n the weghtng factor of the devaton varable, 3) change n the orgnal rght-hand-sde goal, 4) change n the coeffcents, 5) addton of a new goal, 6) addton of a new decson varable, and 7) reorder the orgnal prorty levels. (20:453; 39:62) Utlzng such changes, beng proactve n provdng a lst of suggested changes wth correspondng benefts, allows the decson maker to have potental contngency plans f unforeseen crcumstances arse. Ths s an extremely mportant edge f the decsons made, based on the results, mean lfe or death. Prevously Publshed Models There have been two models, whch have been developed recently, n an attempt to ncorporate multple asset types, Teledyne Brown Engneerng (TBE) developed the 19

31 Sensor-Platform Allocaton Analyss Tool (SPAAT) and ALPHATECH Inc. developed the Mult Asset Synchronzer (MAS). SPAAT utlzes mxed nteger programmng technques to determne the optmum mx of sensors, platforms, and ground statons (35:37) to be used n varous scenaros. SPAAT offers selecton of smple objectves that can drve the model. These objectve functons nclude: mnmzng cost, maxmzng area coverage, mnmzng coverage, and feasblty goals (whch s used as a dagnostc tool (35:39)). MAS modfes a networkng technque known as the Vehcle Routng Problem wth Tme Wndows (VRPTW). Ths program, however, s used to allocate only arborne survellance assets. Addtonally, t prmary goal s to resolve tradeoffs n platform route plannng, sensor resource allocaton, and collecton schedulng to construct hghly effcent ISR [magng, survellance, and reconnassance] plans. (27:44) Summary The purpose of ths chapter was to revew the basc background n fundamental elements that were used n the development of the mathematcal program used n ths thess. Chapter 3 descrbes the methodology used to acheve the mathematcal formulaton developed n ths research to allocate magng assets to cover varous POIs. 20

32 III. Formulaton Introducton The concept of determnng an optmal mx of magng assets to provde requred coverage of a POI seems dauntng. There s a fnte number of magng survellance assets that are allocated to a theater. A theater tself could contan upwards of 50 countres. Each country alone can have many ponts of nterests, be they mltary bases, mltary support factores, fghter squadrons, or even troops movng from one place to another. The vast array of possbltes complcates the process of allocatng lmted magng survellance assets. Addtonally, determnng whch objectve functon would best capture the decson-makers needs and preferences s dffcult. Should cost be mnmzed? Should area of coverage be maxmzed? Should tme spent observng targets be maxmzed? Should assets used be mnmzed? Should the number of targets be maxmzed? These are just a few of an array of possble objectve functons. Ths chapter develops a mathematcal formulaton for allocatng magng assets over a fxed tme horzon. Frst, a general approach for a Mult-Modal Goal Programmng Resource Constraned Project Schedulng Problem (MMGPRCPSP) s gven. Followng the general formulaton development, the specfc formulaton that was used n ths research s gven. The data needed for ths formulaton, the key assumptons made n ths program, and computatonal effort follows the formulatons. 21

33 The general problem beng looked at s to observe a set of POIs wth a fnte set of assets for a fxed tme. Addtonally, there s a desred resoluton for each POI, whch s to be met by each asset magng that POI. General Approach To optmze the allocaton of resources to accomplsh requred tasks, the concept of Mult-Modal Resource Constraned Project Schedulng Goal Programmng s used. The MMRCPSP allows for the schedulng of assets (constraned resources) n such a way that dfferent assets (multple modes) can be used to accomplsh a task. The goal programmng (GP) aspect of the formulaton ncorporates the varous pre-emptve prortes assgned to dfferent tasks. The overall ntent of the formulaton s to allocate pre-assgned assets to accomplsh a set of tasks durng an establshed tme wndow. Gven such a settng, three man goals are addressed n ths thess: 1) mnmze the number of POIs not maged at any tme durng the requred perod, d, 2) mnmze the devatons from resoluton requrements of the POIs, d2, and 3) mnmze the tme gap between assets magng a POI, d3. The followng sectons dscuss the constrants used to determne the value of these devatonal varables. Mnmze POIs not Imaged Durng Tme Horzon As part of the objectve functon, t s necessary to mnmze the number of POIs that are not maged durng the observaton horzon. To assst n ths goal, there must exst a means of countng the POIs not maged. Expresson [3.1] accomplshes ths requrement. d s a goal programmng devaton varable that wll equal one f no asset s 22

34 magng a POI at any tme durng the plannng perod,.e. x mt = 0. d wll be mnmzed n the objectve functon. There wll be one such constrant for each POI. d + x 1 POI [3.1] m M t ST ma mt Parameters: P P Table 5. Parameter and Varable Defntons for General Formulzaton an ordered vector such that the kth term s of prorty k pre-emptve weght for prorty k POIs k (k) w the vector of weghts assocated wth devatons at rank k w R j weght of devaton j for POI the set of possble routes for a partcular asset NumSmulRt number of smultaneous routes an asset n R can travel M POI s R set of all assets set of all POIs ST ma set of access tmes for mode m magng POI Satelltes set of modes representng known satelltes wth known access tmes MQ mnmum requred mage resoluton for POI IQ mt mage qualty of mode m of POI at tme t Varables: d the s the vector of the devaton varables d devaton varable j assocated wth POI j Routes R mt 1 f route c s chosen for UAV, 0 otherwse DX 1 f mode m s magng s magng POI at tme t, 0 otherwse DEV mt t 1 f no assets are magng POI at tme t F devaton from mnmum requred mage qualty of mode m on POI at tme t Mnmze Devatons from Desred Resoluton of each POI It s also desrable to mnmze the devaton from the mnmum mage qualty desred by each asset magng a partcular POI. We must frst determne the devaton from a partcular requrement for each asset at every tme the asset s magng a POI. 23

35 Towards ths, the expressons [3.2] and [3.3] are used. F mt s the goal programmng devatonal varable. The number of constrants [3.2] n the formulzaton wll be equal to the sum of all the duratons of each access tme of each asset to each POI. x IQ ' F x MQ t ST[ m,, a], t' = t,..., t + Duraton [3.2] mt mt mt mt F mt m M t ST [ m,, a] Mnmze the Tme Gap Between Successve Imagng Assets = d 2 POI [3.3] The fnal purpose of the objectve functon s to mnmze the amount of tme between successve assets magng a partcular POI. Constrants [3.4] and [3.5] work toward accomplshng ths goal. Expresson [3.4] ntroduces the devatonal varable DEV t, whch s equal to one f there are no assets magng task at tme t, to determne the tmes when no asset s magng POI at tme t. Expresson [3.5] then sums these devatons for each POI. Ths sum, selectng asset assgnments that mnmzes d3, s mnmzed n the objectve functon, thus d3, the total tme between assets magng all POIs n the set. There wll one constrant [3.4] for each tme nterval n tme for each POI. ma DX mt + DEVt 1 POI, t tme [3.4] m j DEV j = d3 POI [3.5] Addtonal Constrants It s necessary to have a constrant that lmts assets, whch are only able to accomplsh a set number of smultaneous routes. Equaton [3.6] dctates that the sum of 24

36 the routes a partcular asset travels must be less than or equal to the allowable number of smultaneous routes for that asset. Gven a partcular route, t s not possble for the resource to start a task at a tme that s not n the route selected. Ths lmtaton s gven by equaton [3.7]. In the formulaton, there wll be a constrant [3.6] for each asset wth more than one possble route, and a constrant [3.7] for each asset wth multple route possbltes, for each possble start tme of that asset to mage POI at tme t. c R Routes NumSmulRt [3.6] c s R x ct = Routes c R, POI, t tme [3.7] c If there exsts a set of assets that have unchangeable access tmes for each POI, then t s necessary to set x mt equal one for those assets at the respectve access tmes. Ths would be the acqure tme of a satellte that s not to be re-establshed, for example. Mathematcally, ths s done va equaton [3.8]: x mt = 1 t ST[ m,, a], m Satelltes, AOI, a access [3.8] In the overall formulzaton, there wll be a constrant [3.3] for every asset wth set start tmes, for every start tme of that asset to mage POI at tme t. Objectve Functon The goals n the objectve functon are numercally weghted, w j, wthn a prorty class, and lexcographcally weghted by prorty class. As ths s a pre-emptve goal program, lexcographc weghts of the tasks to be accomplshed control the prortzaton. The tasks assocated wth the hghest prorty are requred to be accomplshed frst; only after the frst prorty goals are attaned does the solver consder accomplshng the tasks n the second hghest prorty. These pre-emptve weghts are gven as P k, where 25

37 k=1,2,, total number of prorty classes and P P >> P 1 >> 2 >> m k. Mathematcally, these goals and weghts can be represented by the objectve functon gven as expresson [3.9]. Table 5 s a lst of parameter and varable defntons used n ths general formulzaton. (1) T ( k ) T lexmn { Pw d,, P w } [3.9] 1 k d Specfc Formulzaton The followng s a complete mathematcal formulaton used n ths thess. Table 5 gves the parameters and varables that are used n the specfc formulaton of ths thess. In ths case, the project s to mage a set of POIs for a gven observaton horzon. The resources are the dfferent magng survellance assets and the tasks are the magng of the varous POIs. The objectves, n the example shown, are to mnmze the number of POIs not maged at all durng the observaton horzon, mnmze the devaton from the mnmum magng resoluton, and mnmze the amount of tme between each successve magng asset for a partcular POI. Of course, other objectves can be modeled wth ths approach. The arborne magng assets are only allowed to fly one route at a tme. Ths requrement s adapted from expressons [3.2] and [3.3]. The specal operatons forces (SOF) team, also utlzes expressons [3.2] and [3.3], but the SOF team s allowed to mage two POIs smultaneously, the equvalent of havng two smultaneous routes. 26

38 Table 6. Parameters and Varables Parameters: pre-emptve weght for prorty k POIs P k W weght of havng POI maged at any tme durng observaton perod W 2 weght of havng POI maged wthn mnmum magng requrements W 3 weght of havng POI maged wth mnmal tme gaps R the set of possble routes for UAV S the set of 12-member SOF teams ST ma set of access tmes for mode m magng POI Satelltes set of modes representng known satelltes wth known access tmes MQ mt mnmum requred mage resoluton for POI IQ mage qualty of mode m of POI at tme t Varables: d Ftotal TotGAP RTUAV mt 1 f POI s not maged at any tme durng observng horzon, 0 otherwse total devaton form mnmum mage resoluton for POI total gap tme devaton from allowable tme gap for POI 1 f route c s chosen for UAV, 0 otherwse DX 1 f mode m s magng s magng POI at tme t, 0 otherwse DEV mt t 1 f no assets are magng POI at tme t F devaton from mnmum requred mage qualty of mode m on POI at tme t 1 3 POIsn P1 POIsn P3 lexmn{ P W d + W2 Ftotal + W3 TotGAP, l, P W d + W2 Ftotal + W3 TotGAP} [3.10] subject to RTUAV = 1 [3.11] c R x ct m t= 1 x mt c = RTUAV c R, POI, t tme [3.12] 432 x mt c 2 m S, POI [3.13] = 1 t ST[ m,, a], m Satelltes, POI, a access [3.14] d + x 1 POI [3.15] mt m M t STma mt IQmt' Fmt xmt MQ t ST[ m,, a], t' = t,..., t Duratonma [3.16] x + F mt m M t ST [ m,, a] = Ftotal POI [3.17] 27

39 DX mt + DEVt 1 POI, t tme [3.18] m j DEV j = TotGAP POI [3.19] Ftotal, TotGAP, F d, x mt, RTUAV, DX c mt 0 mt, DEV t {0,1} Throughout ths formulaton, goal programmng s used wth both numerc and pre-emptve weghtng purposes. The pre-emptve prortzaton s to ensure the frst prorty set of tasks s accomplshed before the second prorty tasks are consdered. The second prorty set of tasks s then satsfed pror to consderng the thrd prorty set of tasks, and so forth. Ths s helpful f the hgher prorty tasks change. Pre-emptve weghts are based on expert opnon, target values or other approprate consderaton. The numerc weghts, or dfferental weghts, gve precedence wthn the set of tasks at the same prorty. They allow dscrmnaton wthn a prorty, just as they would n a regular lnear program. Ths weghtng can be done relatve to wthn each prorty class, or over all tasks. A smple example of the use of these weghts s as follows; one scenaro mght requre that the mnmzaton of the tme gap s of hghest prorty, then mnmzng devatons from certan requrements, then tasks not accomplshed at all durng the observaton perod and n each case, the tasks need to be accomplshed n a partcular order. The numercal weghts, however have to be determned usng an acceptable scale. Ths formulaton allows a great deal of flexblty to the analyss. It can accommodate a wde range of objectves and requrements, allowng the fne tunng of the model and the analyss. 28

40 The Dataset Data needed to run ths model ncludes: 1) desred objectves, 2) a lst of POIs that are prortzed and weghted, 3) total length of observaton tme, 4) a lst of all assets (.e. satellte wth IR magng, UAV wth SAR capabltes) avalable to theater, f there are multple assets of the same type, they are lsted ndvdually, f choosng from potental routes, each route s lsted as a separate entry, 5) access tmes of each asset to the POIs throughout the duraton of the observaton perod, 6) the duraton of the access each asset to each POI, 7) mnmum requred resoluton of each POI, and 8) qualty of mages avalable from asset based on common scale A lst of POIs, the length of the observaton horzon, and the mnmal resoluton requrements for each POI are nputs from the persons nterested n gatherng the magery. Ths could be a Theater Commander, a CINC, natonal leadershp, or other approprate authorty. Prortzaton of the POIs s based on approprate classes and scales developed by the nterested party. Scales should be developed usng approprate decson analyss and measurement theory (see Burke, Krkwood, and others), or another tested means of assgnng unbased weghts to dfferent POIs based on characterstcs of the POI n queston and the commander s ntent. In the notonal dataset used, the process of prortzng and weghtng the dfferent POIs s assumed to have been accomplshed based on one of the aforementoned methods. The lst of assets and the number of each asset type n each theater are gven by the Theater Intellgence Archtecture Plan (TIAP). The duraton of access of each asset to each POI s known to the operators of the assets. The qualty of the mages avalable from each asset s based on a predetermned common scale. The acquston tme of each 29

41 asset can be calculated gven a startng locaton. The number of assets of one type that can be used n one tme perod s based on the Theater Commander s doctrne. Fnally, the objectves used n the analyss wll be based on the Commander s operatonal requrements. Key Assumptons As mentoned n Chapter 1, there were four man assumptons made n ths research: 1) terrestral ponts of nterest, 2) the retaskng of satelltes s not allowed, 3) all tmes are ntegers, and 4) there exsts predetermned routes for the UAV(s) to chose. However these are not the only assumptons made durng the process of ths research. The weghtng of POIs s assumed to have been accomplshed usng an acceptable analytcal method. Addtonally, ths model does not account for any potental breakdowns of equpment, or the retaskng of satelltes. If an asset s n the database as avalable to the theater n queston, t s assumed to be fully operatonal. If there exst multple areas of nterest n dfferent theaters, then those POIs are consdered as separate problems and can be handled ndependently. Ths s because the model consders only the assets avalable n one partcular theater and not across theaters. Wth ths basc model, however, a decomposton approach could be used to coordnate theaters. Computatonal Effort The MMGRCPSP s known to be NP-hard. (see Schrmer) NP s a class of decson problems. Decson problems are problems that ask the queston s there a feasble soluton? These decson problems are analogous to the optmzaton problem of fndng a feasble soluton. The decson problem s not 30

42 computatonally harder than the correspondng optmzaton problem. (13:22). However, the two problems are computatonally smlar; f the decson problem s computatonally hard, then so s the optmzaton problem. An NP-class of decson problems s a set of problems for whch no polynomal tme algorthms are known but for whch the yes answer can be verfed n polynomal tme. (13:23) The term polynomal tme comes from complexty theory, a means of classfyng computatonal problems as ether easy or hard based on the runtme of the approach. Polynomal tme s bascally runtme that s a polynomal functon (n the amount of steps requred of the algorthm) dependent on tme that bounds the tme an algorthm works to solve a problem. If ths bound s not present, then the functon s consdered an exponental-tme algorthm. NP-complete problems are the hardest problem n NP (13:23). If an optmzaton problem s NP-hard, then the decson problem s NP-complete. Due to ths hardness, computng a soluton usng ths approach could prove tme consumng and/or dffcult to fnd a feasble soluton. In such cases, usng a heurstc to fnd a startng soluton can speed up the process. If the problem s large enough, t may requre a soluton va a heurstc approach. The concept of developng a usable heurstc for the formulaton presented n ths chapter s dscussed n Chapter 5. Summary Ths chapter provded the methodology used to develop a mathematcal formulaton to optmally allocate magng assets to cover partcular POIs durng an 31

43 observaton perod, subject to POI, magng and tme gap constrants. Chapter 4 wll apply ths methodology to a notonal example and analyze the solutons provded. 32

44 IV. Illustratve Analyss Introducton Ths chapter demonstrates the methodology presented n Chapter 3 usng a small notonal dataset. Frst, an overvew of the problem s gven. The overvew s followed by a descrpton of how the notonal dataset was created. Fnally, the results and analyss based on the proposed methodology, s presented. Problem As tensons slowly begn to ncrease, Command desres to obtan magng survellance on ten ponts of nterest for a 72-hour perod. The 72-hour perod s to be dvded nto 10-mnute ntervals. There are, however, only sx magng assets avalable for use n coverng the ten POIs: four satelltes, one UAV, and one 12-member specal operatons forces (SOF) team. The SOF team can be broken nto two teams of sx members each, brngng the potental number of magng assets to seven. Due to other commtments, the four satelltes are unable to be retasked at ths tme; the access tmes from each satellte to each target and the duraton of these accesses are, therefore, known and fxed. The use of a UAV route generator, such as one descrbed n Grmm (1992) or Knney (2000), has been employed to provde two potental routes for the UAV. The sx-member specal operatons teams are currently able to montor one POI each, and under current condtons, the teams have no restrcton as to whch POIs they can montor. The ten POIs have been weghted usng Decson Analyss, Measurement Theory or some other approprate method (see Krkwood, 1997; Burke, 1999), whch takes nto 33

45 consderaton the decson makers values concernng each POI, ncludng the prorty of the POI. Addtonally, due to operatonal requrements, the decson maker has provded mnmum mage resoluton requrements for each POI. The requrement s to allocate the avalable magng assets to cover as many of the POIs as possble for as much tme as possble durng the 72-hour observaton horzon. If complete coverage s not possble, precedence wll be gven to the POIs wth greater prortzed weghts over the lesser-weghted POIs. Included n allottng the assets, a choce as to whch route the UAV should use s determned based on the potental routes gven. Notonal Dataset The ten POIs used n the notonal dataset were taken from Fuller (1997). Fuller has two unclassfed scenaros; an Iraq scenaro and a Mddle East scenaro. The POIs used n ths notonal dataset were extracted from the Iraq scenaro. Included n the POI data were the lattudes, longtudes, and notonal target or T-values of the ndvdual POIs, whch are gven n Table 7. For more nformaton on the POIs, lattudes, longtudes, and notonal values, refer to Fuller (1997). Fgure 1 s a map of the POIs and the startng locaton for the UAV. Command has prortzed the POIs based on ther T-values. If the POI has a T- value of eght or hgher, t s consdered a prorty 1 POI; a value of four to seven corresponds to a prorty of 2, and a target value of three or less was classfed as a prorty 3 target. Addtonally, the notonal weghts for each POI are n the form of ntegers between 1 and

46 Fgure 1. Map of POIs and startng poston of UAV usng STK 35

47 Table 7. Notonal POIs, Lattudes, Longtudes, T-Values, Prortes, and Weghts Name POI # Lattude Longtude T-Value Prortes Weghts Q-West Basrah Mosel Shayka H-Zalah Hel I-Corp-HQ Brdge DIV DIV The mnmum resoluton requred s dependent on operatonal requrements assocated wth the POIs. A resoluton of sx-meters allows the decson maker or mage analyzer to recognze large buldngs. Dependng on the ntended use of the magery, dfferent mnmal requrements are assgned to each POI. For use n ths notonal dataset a prorty 1 POI s assumed to requre a mnmal resoluton of one meter, and no POI requres a resoluton of less than sx meters. These resoluton requrements were arbtrarly chosen based on the capabltes of a certan resoluton. To determne the varous access tmes of the assets to the POIs, data was collected usng STK. One of the sponsors of ths thess provded orbtal nformaton for the notonal set of satelltes, as well as notonal sensor data (ncludng focal length and pxel ptch) for the satelltes and UAV. COL Wllam Klmack,USA, of the Unted States Mltary Academy, provded nformaton concernng the specal operatons team (See Appendx A for detals). Ths nformaton was programmed nto STK and a three-day smulaton was run. The smulaton provded access tmes, the duraton of each access, and the ground samplng dstance (GSD) of each asset to each POI throughout the three-day perod. The 36

48 GSD corresponds to the resoluton obtanable from the sensors to the POI. Ths data s provded n Appendx B. In an actual operatonal settng ths data would be obtaned from varous databases and analyses. The data from STK was used as nput to a Mcrosoft Excel Worksheet. A code was wrtten n Vsual Basc Applcatons to generate the coeffcent and varable matrces for the optmzaton problem (see Appendx C). Those matrces were transferred to a worksheet for nput to Frontlne s Premum Solver Platform to solve the problem. More specfcally, the Large-Scale LP Solver of the Frontlne Premum Solver Platform was used. Usng a Dell INSPIRON 8100 laptop wth a Pentum 3 processor, t took under two mnutes for the solver to fnd an optmal soluton. In ths notonal example, there was no dstncton between mnmzng the number of POIs not maged durng the tme horzon and mnmzng the devaton from the desred mage resoluton nor mnmzng the tme between successve magng assets for a partcular POI. Each devaton for a gven POI was weghted equally. Addtonally, the POIs n the pre-emptve prorty 1 set are weghted more heavly than the POIs n preemptve prorty 2 and 3 sets. Expresson [4.1] gves the objectve functon used n ths partcular formulaton. mn P (98d 1 98TotGAP 51FTotal 40FTotal d TotGAP + 52TotGAP + 34FTotal + 70d d TotGAP + 51TotGAP ) + P (50d + 11FTotal + 98FTotal TotGAP TotGAP + 87FTotal 7 ) + P (52d + 40d d + 40TotGAP + 70FTotal d + 11d FTotal + 34TotGAP + 66FTotal + 50FTotal [4.1] TotGAP Examnng the Results Runnng the Large-Scale LP Solver produced a soluton for allocaton of the seven assets. The soluton assgned the UAV to route 2, the soft team was dvded nto 10 ) 37

49 two teams; one coverng Q-West, and the other coverng Basrah. By revewng the soluton, the analyst s capable of determnng when each POI s beng maged, for how long and at what resoluton. Table 8 gves the allocaton of assets for Q-west, the remanng POIs are gven n Appendx D. Addtonally, the nformaton provded by the soluton shows the analyst when a POI s not beng maged. Ths could prove useful to show when coverage s lmted, and the decson-maker can plan accordngly. For example, Fgure 2 s a Gantt chart for a 3- hour perod, utlzng data provded n the soluton for the notonal example (see Appendx E for complete Gantt chart from notonal example). Target 1 Target 2 Target 3 Target 4 Target 5 Target 6 Target 7 Target 8 Target 9 Target 10 Tme Fgure 2. Sample Gantt chart for a 3-hour perod (n 10 mnute ntervals) Ths type of chart could be used by decson makers to determne how much coverage of a partcular POI s avalable f a msson s planned for a partcular tme. Furthermore, f tme allows, the analyst could gve ths nformaton to the UAV route planners, so that another route could be planned or perhaps the decson-maker can deploy another soft team to provde addtonal coverage. 38

50 Table 8. Allocaton for Q-west POI Asset coverng Start tme Duraton Image Resoluton Intervals not covered Total ntervals not covered Q-West Quckbrd Quckbrd Quckbrd Quckbrd to 9 65 ntervals UAV route to to to SOF team Fgure 3 graphcally demonstrates another way the data from the soluton can be used. It shows the soluton for the notonal example color-coded to show how many ntervals each POI s not beng maged. Fgure 3 s used to graphcally demonstrate the soluton when the prortes are changed, dscussed n the next secton. These types of fgures could be used to easly compare and contrast dfferent alternatve solutons, whch could arse from goal programmng senstvty analyss descrbed n the next secton. 39

51 40 Blue ndcates less than 100 uncovered ntervals Whte ndcates between 100 and 300 uncovered ntervals Red ndcates between 300 and 400 uncovered ntervals Green ndcates over 400 uncovered ntervals Fgure 3. From STK, graphcal representaton of soluton

52 41 Blue ndcates less than 100 uncovered ntervals Whte ndcates between 100 and 300 uncovered ntervals Red ndcates between 300 and 400 uncovered ntervals Green ndcates over 400 uncovered ntervals Fgure 4. From STK, changed prortes as part of senstvty analyss