Synthesis Exercises and Problems

Transcription

1 Synthesis Exercises and Problems These exercises are carefully selected to complement the self-instructional modules, homework exercises, examples and case studies documented in the main body of the book. In many ways, they also supplement these varied illustrations of the concepts advance in this text; and the answers in the Solution Manual posted on my web site can be thought of as extensions to the main body of the book. Rather than a simple regurgitation of the basic computations, these synthesis exercises generally require a bit of thought, and many are open-ended case studies. To provide an integrated view, all the synthesis exercises were placed here in this section, rather than at the end of each chapter. The exercises and problems are categorized under the following topics: I. Remote Sensing and Geographic Information Systems II. Facility Location III. Simultaneous Location and Routing IV. Activity Derivation, Competition, and Allocation V. Land Use Models VI. Spatial-Temporal Information VII. Term Project We view this as a way to cut across all chapters in the book, emphasizing the main themes that run through this entire volume. For those who are more comfortable with examples (rather than concepts), the Solutions Manual on the web site will serve as a primer on the topics. (Contact the author at ychan@alum.mit.edu for information about his web site. Students and professionals should enter in the SUBJECT block: Request for sample solutions. Instructors should enter: Request for Instructor s Guide. ) The exercises also provide the opportunity to try out the software that comes with this book. I. REMOTE SENSING AND GEOGRAPHIC INFORMATION SYSTEMS This first group of problems range from the classic Bayesian classifier to image processing schemes such as histogram processing on the Training System/Image Processing (TS-IP) software, which is Image on the book s software CD. Two exercises on the Iterative Conditional Mode algorithm are included, illustrating a well-recognized classification technique. A problem is specifically introduced here to illustrate the prescriptive district clustering model advanced in this text. We then finish with a combined classification scheme in which the multicriteria Y. Chan, Location Theory and Decision Analysis, nd ed., DOI.7/ , Springer-Verlag Berlin Heidelberg 44

2 44 EXERCISES Synthesis Exercises and Problems decision-making procedure is explicitly incorporated as an integral part of the algorithm, showing that judgment is part and parcel of remote sensing and geographic information systems. A. Bayesian Classifier The Bayesian classifier is one of the ways to group pixels into different patterns thus the classifier decides that pixel j belongs to a lake while pixel i belongs to a forest. We have illustrated in the Bayesian Decision-Making section of Chapter 3 how a decision boundary x can be arrived at when there is only one attribute x such as a pixel s gray value. The concept can be extended to the case when there is more than one attribute for classification (say n attributes). The equations used in this context are as follows the (Gonzalez and Woods 99). First the Gaussian distribution is extended to multidimensions by P(x z j ) ( ) n/ C j / exp [ /(x j ) T C j (x j )] (E.) where is the mean vector and C is the covariance matrix respectively defined as j N j x C j x Gj N j x Gj xx T j j T (E.) where N j is the number of pattern vectors from class G j (i.e., the number of pixel vectors belonging to class j), and the summation is taken over these vectors. The multidimensional decision boundary now looks like d j (x) ln P(z j ) n ln ln C j [(x j )T C j (x j )] (E.3) Of course, the second term is equal for all cases, and may be subsequently dropped. Now consider a two-dimensional reading for a 3 3 set of borings monitoring a groundwater pollution plume, with the gray values shown in italics (Wright and Chan 994c), y-coordinate 3 x-coordinate Can you delineate the analytic and precise boundary of the plume based on the above set of equations?

3 Synthesis Exercises and Problems EXERCISES 443 B. Iterative Conditional Mode Algorithm The iterative conditional mode (ICM) algorithm was described in detail in Chapter 6, under the Contextual Allocation of Pixels section. As demonstrated in the numerical example, a of produces a non-contextual classification, while increasing accentuates the contextual bias. There is a tradeoff between and, where is the variance of pixels in a certain class. The should be small enough to prevent greatly overlapping regions, and at the same time will need to be adjusted for the noise level of the image. A 6 6 grid of gray values is given below, with high values representing polluted groundwater and low values representing unpolluted water. Noise is introduced into the data by virtue of the data gathering procedure. For example, a value which has the same approximate gray value as the unpolluted groundwater exists in the center of pixels that are evidently polluted. The second 6 6 data set below shows a 3 3 area of apparently polluted ground water with possible noise on one of the sides of the 3 3 area. Also a single pixel (noise) with a pollution range gray value exists among unpolluted pixels Please perform the classification using the ICM on both of these two data sets (Wright and Chan 994c). C. Weighted Iterative Conditional Mode Algorithm In this exercise (Wright and Chan 994c), the weighted ICM algorithm (rather than the unweighted one used above) is to be applied to illustrate a couple of points. For the second data set above, the noise pixel in the polluted area could be classified as polluted water should a low enough value be applied, since three of the five neighbors of the pixel are first-order neighbors. It can be shown also that the noise in the unpolluted area would be easier to discern using a weighted procedure. Notice the implementation is almost identical in both the weighted and unweighted cases. The only difference lies in the calculation of the compare value in which the summation must be broken into a first-order and a second-order summation. Now carry out the weighted ICM algorithm. D. District Clustering Model Shown in Chapter 6 under the district clustering model is a set of noninferior solutions for a small image entitled Multiple subregion noninferior solutions. Examine the file labeled S_4a_4b and S_4a_5b in Figure 6.8, the first of these two code names stands for two subregions, the second and third suggest that

4 444 EXERCISES Synthesis Exercises and Problems an area of four pixels for subregions and. The a and b entries specify two different variations on the boundary of the subregion, generating different non-inferior solutions. The two noninferior images are drawn below sequentially, where the bolded gray values stand for one subregion and the italicized stand for another: Using the multiple subregion model outlined in Chapter 6 in the subsection under the same name (Section X-B), (a) Show the constraint-reduced feasible region model that generated these images; (b) Verify step by step that we have generated the entire non-inferior solution set; (c) Show the equivalent weighted objective function model. E. Combined Classification Scheme In monitoring groundwater pollution, measurements are made at wells placed discretely around the study area. Interpolation (such as kriging) has been made between these readings, forming a pixel map of the pollution level throughout the study area. Figure E. shows a well located at the center of the symmetrical cluster of readings. At the same time, remotely sensed data are available for the entire area. The ground truth data are given as well in Figure E Figure E. GROUND TRUTH, WELL DATA, AND REMOTELY SENSED DATA

5 Synthesis Exercises and Problems EXERCISES 445 Can you combine the two sources of information to delineate the pollution pattern more accurately than you would from a single source? Specifically, perform the following: (a) (b) (c) Employ the ICM algorithm of Chapter 6 with due consideration to proximity as a factor. An inverse relationship is hypothesized between distance and importance in determining the allocation of some internal pixels (i.e., pixels not at the border or fringe of the image). For internal pixels, weights are scaled against eight neighbors. Assuming unitary distance separation between the subject pixel and its first-order neighbor, and a distance of.44 with its second-order neighbors. Thus the weight for first-order neighbors is.76 and.884 for second-order neighbors. The sum over all of its neighbors is (4)(.76) (4)(.884) 8 and the first-order neighbor is.76/ times as important as the second-order neighbor in determining allocation of a pixel as specified initially. Employ multicriteria-optimization techniques as outlined in Chapter 5. Define in the decision space a binary variable that labels each pixel as being polluted when the variable is unitary valued. The two criteria in the outcome space namely the value of the data and the value of contextuality in the ICM classification are captured by the ground truth and the choice of the value in the ICM algorithm respectively. Here, is a measure of forced contiguity, applied parametrically for a ranged weight for combining the two sources of information. The two data sets are shown in Figure E.. Since the water is directly sampled there, one may wish fully to trust the data at the well, thus at the well the weight is unitary for the well reading and zero for the remotely sensed data. Also shown in the same figure is the ground truth, representing a subjective judgment by the decision maker. Determine the noninferior solutions that identify the most viable image classifications. A preference structure can be adopted whereby the smaller the deviation from the ground truth the more it represents a non-dominated solution. Zero deviation is considered Pareto optimal. Deviation in this case is defined as the number of pixels in the ICM-generated solution that are different from the ground truth. Likewise, the less the need for forced contiguity (i.e., the smaller the value), the better. F. Histogram Processing Refer to the TS-IP software under the SPACE directory, located on the CD/DVD that accompany this book. The image brightness histogram shows the number of pixels in the image having each of the 56 possible monochromatic values of stored brightness (Russ 998; Gonzalez & Woods 99). Peaks in the histogram correspond to the more common brightness values, which often identify particular structures that are present. Valleys between the peaks and the two tails indicate brightness values that are less common in the image. The flat regions at the two ends of the histogram show that no pixels have those values, indicating that

6 446 EXERCISES Synthesis Exercises and Problems the image brightness range does not necessarily cover the full 55 range available. Similarly, the pixels at the two tails of the gray values tend to contain noise, rather than the real image. Figure E. shows an example of such a histogram. In order for the available gray levels to be used efficiently on the display, some will have to be removed (such as those at the two tails of the given histogram). It might be better to spread out the displayed gray levels in the peak areas selectively, compressing them in the valleys (or the two tails) so that the same number of pixels in the display shows each of the possible brightness levels. This is called histogram equalization or histogram stretch. Histogram equalization reassigns the brightness values of pixels. Individual pixels retain their brightness order (i.e., they remain brighter or darker than other pixels) but the values are shifted, so that an equal number of pixels have each possible brightness value. In many cases, this spreads out the values in regions where different regions meet, showing details in areas with a high brightness gradient. The equalization makes it possible to see minor variations within regions that appear nearly uniform in the original image. In this example, we show that the range 6 can be stretched out to occupy the entire spectrum, resulting in a dimmer image, but with better contrast. The process is quiet simple mathematically. For each brightness level j in the original image (and its histogram), the newly-assigned relative-value k is calculated as k j i N i /n, where the summation counts the number of pixels in the image (by integrating the histogram) with brightness equal to or less than j. N i is the number of pixels in the ith brightness level, and n is the total number of pixels (or the total area of the histogram). This is graphically represented as the Figure E. SEVERAL OPTIONS IN HISTOGRAM EQUALIZATION 55 Both brighter image and enhanced contrast Slope contrast of original image Brighter (less contrast) Dimmer (more contrast) 6 55

7 Synthesis Exercises and Problems EXERCISES 447 dashed straight line plotted from the extreme left gray value to the extreme right value, representing the gray value range we wish to examine in detail. In Figure E. are shown several ways to perform histogram equal ization, including controlling brightness and contrast. Using the TS-IP software provided with the CD, please show on the Pentagon image these various options: (a) (b) (c) dimmer with more contrast, brighter with less contrast, and both brighter and with contrast enhanced. II. FACILITY LOCATION Facility-location modeling is a key component of this book. Here we cover some less than obvious applications of these models. Following the airport location examples used extensively in the book, we further illustrate the nodal optimality conditions prevalent in not only min-sum location models, but also min-max models as well. The opposite of min-max problems is the max-min problem, commonly found in obnoxious facility location, which includes solid waste facilities. Another challenging facility location model is the quadratic assignment problem, in which interaction between facilities take place. A. Nodal Optimality Conditions Consider the cities of Cincinnati and Dayton, Ohio connected by Interstate Highway 75. Cincinnati has a metropolitan population of million and Dayton, million. A regional airport is proposed to serve both cities. It is to be located on I-75 such that the total person miles (PMT) to travel between the two cities is to be minimized. We have shown in Chapters and 4 that the optimal location is Cincinnati. This is an example of nodal optimality conditions. (a) (b) (c) (d) Per discussions in Chapter 4: if the airport is to be located on I-75 so that the total person decibels of noise pollution is to be minimized, where should the airport be built? Suppose accessibility and noise exposure are of equal concerns, where should the airport be located? Accessibility is defined as the total PMT while noise exposure is the total person decibel. Repeat questions (a) and (b) for the three-city case where Columbus is included. Columbus has a population of. million. Repeat the whole process for a four-city case in which Indianapolis is included in addition. B. Solid Waste Facility In locating a municipal solid waste facility, the analytic hierarchy process (AHP) has often been used. Junio (994) proposed a hierarchy of attributes as shown in Figure E.3. Discuss the completeness and relevance of such a hierarchy definition. How would you quantify this hierarchy in executing AHP?

8 448 EXERCISES Synthesis Exercises and Problems Figure E.3 HIERARCHY OF A MUNICIPAL SOLID WASTE PROBLEM Goal Social Environment Economics Aesthetics Property values Traffic impacts Land use Accessibility Noise and odor Ecology Health risks Air Ground water Tip fees Tax revenues Out-of-district revenues Jobs C. Quadratic Assignment Problem Refer to the quadratic assignment problem as introduced in Chapter 4. (a) (b) (c) Formulate the linearized version of model for the distance separation and flow interaction matrices as shown. Now solve this linear model. Is there anything peculiar about the solution to the linear model? If not, simply give the optimal assignment and the objective function. If yes, explain the peculiarity and again give the optimal assignment and the objective function value. III. LOCATION-ROUTING The integration of facility location and service delivery is a key feature of this book. We use a simple telecommunication network maintenance problem to lay out the integration. First, we define a region to be served by a maintenance facility using the districting technique. Then we place the facility using the service facility location model, followed by an evaluation of the entire maintenance procedure through a user performance model. To solve a real world problem, the three steps are executed repeatedly in a districting, location, and evaluation triplet. Having laid out this background, we break the problem into the service delivery step and then the combined location routing step. The basic building block of both steps is the quantification of spatial sepa ration. This is illustrated in terms of Minkowski s metric, which is also known as l p -metric as defined in Chapter 5 under the Deviational Measures subsection.

9 A. Districting Synthesis Exercises and Problems EXERCISES 449 The next three problems demonstrate a solution algorithm for improving maintenance depot location and service delivery operations (Patterson 995). Here in the first problem, we define the districts each depot is supposed to serve. The model is based upon enumeration and was adapted for network topology by Ahituv and Berman (988): Min j C j x j s. t. j x j p (E.4) j a ij x j i where x j if subnetwork j is selected to form a district and zero otherwise; a ij if node (zone) i is an element of subnetwork j and zero otherwise; p is the number of districts or subnetworks desired, and the equity measure C j i f i /p, /p (E.5) ; and f i is the fraction of demand at node i. The algorithm consists of two different phases: Phase I determines all easible subnetworks (districts) within the larger network, and Phase II determines the final subnetworks based upon our equity objective-function in Equation E.5. Contiguity and compactness will be bounding constraints for the first phase. One final requirement is that the p subnetworks must be collectively exhaustive and mutually exclusive. In other words, every node must be within one and only one subnetwork. This is accounted for in Phase II. PHASE I: Using a tree search algorithm, we find the feasible set by picking the smallest number node and connecting contiguous nodes while enforcing the compactness requirement until the combined demand becomes redundant. Care must be taken to avoid creating separate enclaves, which are node(s) that are incapable of being separate subnetworks and cannot connect to other subnetworks without going through a previously defined subnetwork. This will prevent impossible solutions. PHASE II: The algorithm for node partitioning was developed by Garfinkel and Nemhauser (97). The following notation is needed: X is the set of fixed variables; X is the number of fixed variables; D is the set of nodes in the districts, or zones of X; J is the set of districts in the current partial solution; N j are the nodes J; and is the cardinality of the set in general. The computational steps are briefly outlined below: Step : Initialization. Set counter L, and set J X, N j D. Step : Choosing the next list. Pick the smallest number node not in N j.

10 45 EXERCISES Synthesis Exercises and Problems Figure E.4 SERVICE NETWORK 4 78 Rate:. 4 Rate:. 5 5 Rate:.35 Legend Rate: Maintenance call arrival rate 3 Rate: Rate:. SOURCE: Patterson (995). Reprinted with permission. Step 3: Step 4: Step 5: Updating set J. Add the node to form subnetworks. Testing for a solution. Test L J X. If L stop, else L L. Finding a solution. Pick the largest cost subnetwork in J as the current solution. Go to Step. Now for the network shown below in Figure E.4, please perform the districting procedure with. to arrive at two service regions. B. Minkowski s Metric Consider two points y (4, 3) and y (4, 4) in a two-dimensional space. Employing the following general measure of deviation between y and y, r(y; p) [ i y i y i p ] /p, explore the behavior of numerical values of r for parameter p changing from to : (a) (b) (c) (d) Draw a diagram of function r f(p). What are the general properties of such a function? Perform the same analysis for p changing from to and also from to. Do these cases show any meaningful interpretation? Perform a more difficult but very rewarding exercise: Do these distance measures, especially for p between and, correspond to any particular subfamily of utility (or value) functions? Can you identify such a subclass? Perform the following graphic exercise: Define a point y (, ) in a two-dimensional space. Plot all such points y whose distance from y is equal to a fixed number r*, that is, r r*. Choose r* and draw such loci of points y for p ranging from to. (Pay

11 Synthesis Exercises and Problems EXERCISES 45 (e) special attention to p,, ). Do the resulting shapes suggest any connection with utility functions? Are there some points in (d) which have the same distance from point y regardless of the value of p? What are the other characteristics and possible interpretations of such points? IV. ACTIVITY DERIVATION, ALLOCATION AND COMPETITION The transition from facility location to land use models can be marked by activity derivation, allocation, and competition. Thus economic activities such as population and employment are generated at an activity center. Residential neighborhoods then compete to provide housing for these people, resulting in a distribution of residents among these neighborhoods. Here in this group of exercises, we solve a matrix multicriteria game, in which there is more than one payoff among the competitors. The gravity model is a traditional way to analyze competition among geographic areas. Using the gravity versus transportation models exercise, one can see that the gravity model is an extension of the all or nothing assignment of activities. Assignment from one single supply exclusively to one single demand is performed by the Hitchcock- Koopman transportation model. This is complemented by the calibration of a doubly constrained model. A. Multicriteria Game Consider the following game decision maker (DM) maximizes his minimum gain while decision maker (DM) minimizes her maximum loss. Gain of DM is exactly equal to the loss of DM (i.e., a zero-sum game). Instead of the single metric used in the conventional payoff matrix, there is more than one criterion in measuring payoffs. These multiple payoffs are therefore expressed in terms of a vector (rather than a scalar). An example appears below in Table E., where the cells contain the two payoffs for each pair of decisions reached between DM and DM: Thus if both DMs decide to play their second option, DM wins 3 units in the first criterion and in the second. DM loses the same number. The symbols p i and q j denote the probability DM and DM will play the ith and jth strategy Table E. MULTICRITERIA GAME DM q q q 3 p (3, ) (, ) (4, ) (3, 4) (3, ) (, 3) (, 5) (, ) (3, ) DM p p 3

12 45 EXERCISES Synthesis Exercises and Problems respectively. When p and q assume fractional values, the game is a called a mixed strategy game. A pure strategy is when p s and q s are or in value. Let each vector payoff a ij (a ij, a ij ) T be replaced by a convex combination of both components: wa ij ( w), a ij, where w is a ranged weight. For example, a w3 ( w) w, and so on. It can be shown that, similar to a conventional zero-sum two-person game an LP can be set up to solve this problem, where the primal and dual solutions correspond to the strategy taken by the two decision makers. If nonnegative variables p and q are defined such that p pz and q qz, the equivalent LP is: Max q q q 3 s.t. (w ) q (4 w) q (5 4w) q 3 (w ) q (w ) q q 3 (3w ) q (3 w) q (w ) q 3 (a) Solve this LP by varying the weights w from to. (b) Is there an equilibrium defined here as a pair of decisions with which both sides are happy? At this equilibrium, z is a nonegative number representing the gain to DM and the loss to DM. B. Gravity versus Transportation Model Refer to the doubly constrained gravity model as discussed in the subsection bearing the same title in Chapter 3. When the value of becomes in the propensity function F(C ij ), the function becomes a special function of travel cost, F(C ij ) C ij C ij, and the doubly constrained gravity model can be written as V ij (k i l j V i V j ) F (C ij ) z ij C ij or z ij C ij V ij z ij z ij ij C ij V ij n' V ij V j j,,..., n (E.6) i n' j V ij V i i,,..., n z in Equation (E.8) is interpreted as the total travel cost now. By minimizing total travel cost (for instance, veh-min), we have the classical transportation model. Notice this model reflects the system optimum rather than user optimum as obtained by conventional gravity-model calibration. Now answer the following questions: (a) (b) (c) What value would assume in the propensity function to have maximum accessibility? What value would assume to have minimum accessibility? For a prescriptive model, what is the resulting trip distribution for case (a)?

13 Synthesis Exercises and Problems EXERCISES 453 (d) (e) For a prescriptive model, what is the trip distribution for case (b)? Interpret the result of (c) and (d). C. Calibration of a Doubly Constrained Model Given the following data on interzonal trips V ij and the associated costs C ij, please calibrate a doubly-constrained gravity model: [V ij ] zone and [C ij ] zone zone 3 5 zone Suppose F(C ij ) c ij, carry out the calculations as far as you can, following the procedure described in the doubly constrained model subsection of Chapter 3. Give the final four equations for the four unknowns, and solve the equations. V. LAND USE MODELS Analysis of land use models is a center piece of this book. Here, the economic-base and activity distribution exercise shows how the activity derivation, distribution, and competition concepts can be used to simulate the housing requirements of a college town over time. This set of calculations is then formalized in the iterative Lowry model calculation, which is encoded on the software CD. A. Economic-Base and Activity Allocation In a study of a college town, State College, Pennsylvania, Chan and Rasmussen (979) forecasted housing requirements. Using the basic concepts of the Lowry model, they derived the subareal housing requirement of the town using the university enrollment as the basic activity. Their algorithm follows a two-part procedure: Part I. Housing Demand Factor. Define the zoning types of all residentially zoned developable land.. Establish the number of students, blue-collar employees, and whitecollar employees from tract i working at employment center c labeled here as E S ic, E B ic, E W ic respectively. 3. Determine the separation d between each tract centroid and employment center. 4. Obtain the percentage of student, blue-collar, and white-collar commuters traveling a distance of d miles to the related employment center labeled f S i (d), and f B i (d), and f W i (d) respectively. 5. Determine the percentage of students, blue-collar workers, and white-collar workers in residential type t labeled p S t, p B t and p W t, respectively 6. Compute the housing demand factor: V d it k c d E k ic f k i (d) p k t.

14 454 EXERCISES Synthesis Exercises and Problems Part II. Allocation of Housing Demand. Determine the excess housing supply in tact i, N i. The excess is equally distributed among the number of zoning types t Max : N it N i /t Max.. Determine the maximum holding capacity for developable dwelling units: N c it (developable average) (average dwelling units per acre). 3. Allocate the total housing demand N to each tract i: N it NV d it / i t V d it. The housing demand for housing type t in tract i can either be accommodated by the excess housing supply N it or new construction. Housing demand exceeding the holding capacity of a tract would have to be located elsewhere. The additional developable capacity of a tract for housing type t is N c it N c it N it N it. 4. Additional iterations are necessary as long as one or more N c it is negative (i.e., there is spill over from a tract), and excess capacity still exists in the region to accommodate the excess. Otherwise, the algorithm terminates. Chan and Rasmussen then compared their forecast with the ones by the Centre Region Planning Commission (CRPC). The housing projection performed by the CRPC is computed by a two-step procedure: () The future population for the region is computed; and () the number of dwelling units is derived from that figure. The derivation process is generally founded on a extrapolation forecasting techniques. The CRPC population forecast takes into consideration a cohort survival model and a straight-line proportional model. (These techniques are discussed in the Econometrics Modeling section of Chapter.) The following assumptions are made among both studies: (a) No substantial in or out migration would take place, which implies the student enrollment at Penn State University would stabilize at 3,5 by 985. (b) Existing trends, including birthrates/death rates and other coefficients and ratios, will remain constant over time for each township of the Centre Region. Since the study, the dwelling units that were actually observed became available. These figures are tabulated beside the Chan and Rasmussen and CRPC forecasts in Table E.. Can you perform a before-and-after analysis as to the accuracy of the forecasts by the Chan-Rasmussen model vis-a-vis the CRPC study? Table E. COMPARISON OF FORECASTS AND OBSERVED HOUSING UNITS 985 Figures College Ferguson Halfmoon Harris Patton State College Single family Chan & Rasmussen CRPC Observed Multiple family Chan & Rasmussen CRPC Observed

15 Synthesis Exercises and Problems EXERCISES 455 B. Forecasting Airbase Housing Requirements Now that you are familiar with the Chan-Rasmussen housing model, can you use the same model to forecast housing requirements for an Air Force base? Similar to the college town model, this new model is based on the hypothesis that the foundation of the local economy is an Air Force base (Bahm et al. 989). Whiteman Air Force Base (AFB) near Knob Noster, Missouri is chosen for the study. Whiteman was picked because the base is a major source of employment for the region and was expected to grow at the time of the study in 989. The source of the increase in military- and civilian-employment is the new B- bomber wing. Three types of economic activities are envisioned to increase: military and their dependents, civilian Department of Defense (DOD) employees, and civilian non-dod employees. There are 5 housing tracts or zones in the region. There are four employment centers: Warrensburg, Sedalia, Knob Noster, and Whiteman AFB. Commuting distance is measured in one-mile (.6-km) increments, with the longest commuting distance being 46 miles (73.6 km). There are five residential types: single family, double family, multiple family, dormitories and non-residential. Additional developable capacities, excess housing, and resident profiles are documented in Table E.3. The information is listed by each tract/zone i. By resident profile we mean the percentage of military, DOD civilian, and non-dod civilians in each type of housing whether it be single family, double family, multiple family, or dormitory. Commuting distances from each of the 5 tracts/zones to the four employment centers are shown in Table E.4. The trip distribution, or the percentage of workers traveling distance d to an employment center, is shown in Table E.5. Included in the table are the increases in military, DOD civilian and non-dod civilian jobs in each of the four employment centers. Now forecast the housing requirements at the study area based on these assumptions: (a) insignificant projected increase in employment from manufacturing in Warrensburg and Sedalia, (b) insignificant projected increases in employment or student enrollment at Missouri State University, and (c) only a small amount of associated cross-commuting from Whiteman to other points in the study region. All these make Whiteman AFB the major employer in the projected future, attracting the local population to the base. VI. SPATIAL-TEMPORAL INFORMATION The unifying theme throughout this book is really how one analyzes spatialtemporal information in general. In this last block of problems, we let the data guide us in the analysis. The first problem eloquently shows the difference between spatial and univariate forecasts, particularly regarding their respective accuracies. Subsequently we worry about the calibration of a spatial forecasting model, an area so demanding that much more research is still needed.

16 456 EXERCISES Synthesis Exercises and Problems Table E.3 ADDITIONAL DEVELOPABLE CAPACITIES, EXCESS HOUSING* AND RESIDENT PROFILE Tract/zone Single family Double family Multiple family Dormitory () () (9) (9) (6) 5 () 5 3 (9) 3 5 (7) 4 5 (4) 5 3 Resident profile Single family Double family Multiple family Dormitory % Military % Civilian/DOD % Civilian/non-DOD *Excess housing numbers are in parentheses.

17 Synthesis Exercises and Problems EXERCISES 457 Table E.4 COMMUTING DISTANCES a TO EMPLOYMENT CENTERS Tract/zone Warrensburg Sedalia Knob Noster Whiteman AFB a In integral miles (or multiples of.6 km). A. Cohort Survival Method The cohort survival method is an econometric technique introduced in Chapter, in the Interregional Growth and Distribution subsection. Please review the discussions in the text and answer these questions (Jha 97): (a) Suppose these statistics are gathered for York County, Pennsylvania during the period. The number of births is, and the number of deaths is 5. The average population for the period is,. There were,4 people migrating to York and,95 migrating out. Define the following terms for a certain forecast time period: crude birthrate, crude death rate, and net migration.

18 458 EXERCISES Synthesis Exercises and Problems Civilian/non-DOD Employment distribution Warrensburg Sedalia Knob Noster Whiteman AFB Civilian/DOD Civilian/non-DOD Trip distribution Military Military Civilian/DOD Table E.5 TRIP DISTRIBUTION AND JOB PROFILES AT THE EMPLOYMENT CENTERS

19 Synthesis Exercises and Problems EXERCISES 459 (b) Check the population for females in 945 in York County, Pennsylvania by the cohort survival method. Use the population statistics shown in the following table. Note that these numbers are in hundreds, i.e., means. VII. TERM PROJECT Age Total The entries in this table represent the number of people in each age group. The surviving ratio of the 4 year group is 98% and the percentage of female children is 49%. The fertility rate of the 5 9 year group is 43%; the rate for 4 groups and 5 9 groups is 56%. As a capping stone, the purpose of this term project is to (a) Show how Multi-criteria Decision Analysis can be used in spatial information technology such as image processing. (b) Demonstrate the practical use of a Bayesian classification model coded in MATLAB and provided in the book CD/DVD under the PATTERN directory. (c) Demonstrate via real-life example that is downloaded from the NOAA GOES satellite dish using the GVAR image-acquisition software. Alternatively, a real-life image can be obtained from the default sample image called TESTG, or other sources, including the collection of satellite images on the book CD under the folder IMAGEFILES. Step. Organize the class into individual teams as follows: Name Team Name Name 3 Name 4 Team Name 5 Name 6 Name 7 Team 3 Name 8 Name 9 Name Team 4 Name Name Name 3 Team 5 Name 4 Name 5 Etc. Professional responsibility dictates that each team member participates.

20 46 EXERCISES Synthesis Exercises and Problems Step. Please review the following documents:. Read Chapter 3, Section VII.E, in our textbook on Bayesian classifier, and try to understand these concepts. (a) Use an observed distribution to estimate the underlying distribution of a set of data. (b) Review a bimodal distribution of gray-value intensities as a precursor to the Forest vs. Lake classification example. (c) Relate the methodology to the Forest-Lake example in the following reading assignment.. Read Chapter 6, Section VIII, in our textbook on Pattern Recognition, and try to understand these concepts: (d) Spectral Classification vs. Spatial/Contextual classification (e) Spectral Classification example (f) Spatial Classification example 3. Read Chapter 6, Section IX, in our textbook on District Clustering Model, and try to understand these concepts: (g) The Benabdallah-and-Wright model (h) Apply the model to analyze the Washington DC Mall image. Step 3.. Now answer the trailing questions briefly and to the point. The only exception is Question 4, in which annotated output of the computer runs are required, both in hard copy and software copy.. Write a technical paper to systematically document the theories, methods, as well as the results based on a satellite imagery provided as part of the software. In other words, embed answers to the questions in Part as a complete technical essay. The paper should be typewritten and submitted in both hard and soft copies. Part I of the Project REFER TO FIGURE 6.4, ENTITLED CONTEXTUAL VS. NON-CONTEXTUAL IMAGE CLASSIFICATION. Question : The above Figure illustrates the difference between spectral vs. contextual classification of an image. Can you explain these two terms in your own language? REFER TO FIGURE 6.3, ENTITLED PDF IN BAYESIAN CLASSIFIER. Question : We used a Bayesian classifier to perform a combined spectral and contextual classification. Explain in your own terms how this is applied toward the example on Lake vs. Forest pixels as shown by the previous and following illustrations.

21 Synthesis Exercises and Problems EXERCISES 46 Figure E.4 k = IMAGE CLUSTERING REFER TO THE TWO-CLASS 4-PIXEL IMAGE EXAMPLE WORKED OUT IN CHAPTER 6 SECTION VIII.B, ENTITLED CONTEXTUAL ALLOCATION OF PIXELS, and also FIGURE 6.33, ENTITLED SPOT SUB-IMAGE GRAY VALUES Question 3. Aside from Bayesian decision theory, multi-criteria optimization has been employed to perform image classification, as illustrated in Figure 6.33 on SPOT image of the Washington DC Mall. Can you explain the results as shown in the three frames for Channels,, and 3? Part II of the Project REFER TO FIGURE E.4, ENTITLED K = IMAGE CLUSTERING. Question 4. Under the directory PATTERN, a MATLAB program has been provided on the CD/DVD to perform the k- medoid classification based on multi-criteria optimization and Bayesian decision theory, as illustrated in the above Figure E.4 example. Based on this demonstration of a k-medoid Algorithm, please run the k-medoid software, IMGKMED, for the GOES satellite image TESTG as provided on the CD/DVD. Assisted by the trailing instructions, please (a) (b) (c) Execute the algorithm for k = 3 until it converges. Explain how you know it converges. In the MATLAB IMGKMED algorithm, the placement of your seed medoids is random. Correspondingly, you would expect the number of iterations to reach convergence different from one run to another, even for the same initial image. Perform three runs corresponding to the following weight sets and discuss the differences between your results. Discuss the possible application of the k-medoid classification technique in terms of preventing natural hazards such as storms.

22 46 EXERCISES Synthesis Exercises and Problems In addition to the annotated outputs, please make sure you answer each of the five questions in your technical paper. Weight set w w Operating Instructions for the MATLAB program IMGKMED Open the MATLAB software. Go to the directory/folder in which you have placed the IMGKMED folder. At the command line, enter IMGKMED. Press the Load image button. Select the TESTG.BMP file. Set k = 3. Set the weights for proximity or for Channel (monochromatic grayscale) by moving the lever of the weight scale. Determine the number of iterations required to reach convergence. Execute by pressing the Cluster button. Notice that since the provided image(s) is black and white, channels and 3 are nonfunctional. It is suggested that you leave the setting for Proximity at the half way point when you start out. For simplicity, please leave the channels, and 3 settings to the middle point (5 5), and only change w and w (the proximity setting). When you run the IMGKMED software, make sure you re-load the original test image every time. In other words, anytime you change your input parameters, such as the number of iterations, you need to re-load the test image. Otherwise, erratic behavior will result from the IMGKMED software. Question 5. Pick a downloaded, image-processed satellite image other than the testg.bmp file. Can you speculate how that image is actually constructed? Please simply pick the IVHRR image in the IMAGEFILES folder on the CD/DVD and analyze it. (By the way, the disk that comes with the book has a lot more images. The disk also contains an image-processing software called TS-IP, complete with a User s Manual.) REFERENCES Ahutuv, N.; Berman, O. (988). Operations management of distributed service networks A practical quantitative approach. New York: Plenum Press. Bahm, P.; Ross, M.; Chan, Y. (989). Forecasting housing requirements for an Air Force installation. Working Paper. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio.

23 Synthesis Exercises and Problems EXERCISES 463 Banaszak, D.; Cordeiro, J.; Chan, Y. (997). Using the K-medoid and covering approaches in pattern-recognition problems. Working Paper. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Burnes, M. D. (99). Application of vehicle routing heuristics to an aeromedical airlift problem. Master s Thesis. (AFIT/GST/ENS/9M-3). Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Chan, Y. (5) Location transport and land-use: Modeling spatial-temperal information. Berlin and New York: Springer. Chan, Y.; Rasmussen, W. (979). Forecasting housing requirements in a college town. Journal of the Urban Planning and Development Division (American Society of Civil Engineers) 5:9 3. Clough, J.; Millhouse, P.; Chan, Y. (997). The obnoxious facility location problem. Working Paper. Department of Operational Sciences. Air Force Institute of Technology. Wright- Patterson Air AFB, Ohio. Daskin, M. (995). Network and discrete location: Models, algorithms, and applications. New York: Wiley. Francis, R.; McGinnis, L.; White, J. (999). Facility layout and location: An analytical approach, 3rd ed. Englewood Cliffs, New Jersey: Prentice-Hall. Garfinkel, R. S.; Nemhauser, G. L. (97). Optimal political redistricting by implicit enumeration techniques. Management Science, 6: Gonzalez, R. C.; Woods, R. E. (99). Digital image processing. Reading, Mass.: Addison-Wesley. Grosskopf, S.; Magaritis, D.; Valdmanis, V. (995). Estimating output substitutability of hospital services: a distance function approach. European Journal of Operational Research 8: Irish, T.; May, T.; Chan, Y. (995). A stochastic facility relocation problem. Working Paper. Department of Operational Sciences. Air Force Institute of Technology. Wright- Patterson Air AFB, Ohio. Jha, K. (97). Demographic models. Working Paper. Department of Civil Engineering. Pennsylvania State University. University Park, Pennsylvania. Junio, D. F. (994). Development of an analytic hierarchy process for siting of municipal solid waste facilities. Master s Thesis. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Kanafani, A. (983). Transportation demand analysis. New York: McGraw-Hill. Mandl, C. (979). Applied network optimization. New York: Academic Press. Memis, T.; Eravsar, M.; Chan, Y. (997). Integer programming solution to Route Improvement Synthesis and Evaluation (RISE). Working Paper. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Mitchell, E. J. (969). Some econometrics of the Huk rebellion. American Political Science Review 63:59 7. Patterson, T. S. (995). Dynamic maintenance scheduling for a stochastic telecommunications network: Determination of performance factors. Master s Thesis. Department of Operational Sciences. Graduate School of Engineering. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio.

24 464 EXERCISES Synthesis Exercises and Problems Pfeifer, P. E.; Bodily, S. E. (99). A test of space-time ARMA modelling and forecasting of hotel data. Journal of Forecasting 9:55 7. Piskator, G. M.; Chan, Y. (997). Estimating a production function and efficient frontier for the United States Army recruiting battalions. Working Paper. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Russ, J. C. (998). Image processing handbook, 3rd ed. Boca Raton: CRC Press. Steppe, J. M. (99). Locating direction finders in a generalized search and rescue network. Master s Thesis. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Steuer, R. E. (986). Multiple criteria optimization: Theory, computation, and application. New York: Wiley. Tapiero, C. S. (97). Transportation-location-allocation problems over time Journal of Regional Science : Wright, S. A. (995). Spatial time-series: Pollution pattern recognition under irregular intervention. Master s Thesis. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Wright, S. A.; Chan, Y. (994a). A network with side constraints for the k-medoid method for optimal plant location applied to image classification. Working Paper. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Wright, S. A.; Chan, Y. (994b). Multicriteria decision-making applied to the ICM contextual image classification technique. Working Paper. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Wright, S. A.; Chan, Y. (994c). Pure and polluted groundwater classification on a pixel map. Working Paper. Department of Operational Sciences. Air Force Institute of Technology. Wright-Patterson Air AFB, Ohio. Zelany, M. (98). Multiple criteria decision making. New York: McGraw-Hill.

25 Appendix Control, Dynamics, and System Stability While the main body of the text concentrates on Location Theory and Decision Analysis, there are some computational aspects of model solution that the readers may wish to review. Four appendices are provide here for that purpose. The first appendix follows our self-instructional module on Empirical Modeling. In this appendix, we review the basic theories that govern the evolution of complex systems, wherein systems transition from one state to another over time. Systems may evolve on their own or external influence may be brought to bear upon their development. In both cases, there can be smooth transitions as well as precipitous happenings. We discuss the conditions under which a system may change between these two types of evolution namely from smooth to precipitous changes and vice versa. Most importantly, we wish to effect these changes where we can, so as to direct the development toward a desired goal. Stochastic, nonlinear system is a powerful tool for location theory and decision analysis. We have seen an example in Chapter 4 under the topic of Optimal Control of Spatial Interaction. Other examples can be found under Economic Base Theory, Facility Expansion, and Competitive Location and Games. These are scattered throughout this book and the accompanying CD/DVD. For the curious, Chan (5) applies the methodology in depth while discussing the Garin-Lowry Model and Spatial Equilibrium. I. CONTROL THEORY The concept of control theory was introduced in Appendix 3, where an example of inventory control was worked out in the context of vehicle dispatching. In the example, trucks deliver a stock of cargo X(t) at the loading dock over the afternoon between hours t and t. The cargo is to be airlifted to a destination. We wish to construct a schedule to minimize operating cost and schedule delay. The problem was solved by discrete dynamic programming, wherein the optimal dispatch schedule as indicated by the control variable U(t) is determined. Here we will generalize and ormalize the results in a more systematic way using control theory (Silberberg 99). The general form of a control theory problem is expressed as a maximization problem instead of minimization: Max t f(x(t), U(t), t) dt U(t) t (A.) subject to the state equation 465