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[exercise.mas]

09/15/98

EXERCISES AND PROBLEMS

These exercises are carefully selected to complement the examples and case studies documented in the main bodyof the book. In many ways, they also supplement these illustrations and the answers in the Instructors' Guide canbe thought of as extensions to the main body of the book. Rather than a simple regurgitation of the basiccomputations, these exercises generally require a bit of thought and many are open-ended. We purposefully avoidexercises to the fundamental techniques covered in the appendices of the book. The reason is that we have includedthese appendices for convenience only. The referenced texts in these appendices have excellent homeworks of theirown to assist those who need more extensive review of the fundamentals. To provide an integrated view, we havedecided to place all the exercises here in this appendix, rather than at the end of each chapter. We categorize theexercises and homeworks into these topics:

Remote sensing and geographic-information systems Facility location Simultaneous location-and-routing Activity derivation, competition, and allocation Land-use models Spatial-temporal information.

We view this as a way to cut across all 17 chapters in the book, emphasizing the main streams that run through thisentire volume. For those who are more comfortable with examples (rather than concepts), this Appendix and theSolution Manual serve as a 'primer' on the subject. It also provides the opportunity for the reader to try out thesoftware that comes with this book.

For pedagogic reasons, we have broken the problems in this Appendix into two categories: the generalexercises and the advanced exercises. The latter is marked with an asterisk (*) for easy identification. The generalexercises are related to these book chapters: Introduction, Economic Methods, Descriptive Tools, Prescriptive Tools,Multi-Criteria Decision-Making, Romote Sensing and Geographic Information Systems. They are also built uponthese appendices: Optimization Schemes, Markovian Processes, Statistical Tools, and System Stability. On the otherhand, the advanced exercises are related to the following chapters: Facility Location, Spatial Separation, Location-andRouting, Location-allocation, Lowry-based models, Activity-allocation/derivation, Chaos/Catastrophe/Bifurcation/Disaggregation, Spatial Equilibrium, Spatial Econometrics, Spatial Time-Series, Spatial-Temporal Information,Retrospect and Prospects.

I. Remote sensing and Geographic-information Systems

This first group of homework problems range from the classic "Bayesian classifier" to image-processing schemessuch as "Histogram processing" on the TS-IP software, which is included in the book's software disk. Two exerciseson the "Iterative conditional mode algorithm" are included, illustrating another well-recognized classification-technique. A homework 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 Multi-criteria Decision-making procedure is explicitly incorporated as an integral part of the algorithm, showing that judgement is part andparcel to remote sensing and geographic information system.

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 the "Descriptive Tools" chapter how a decision boundary x0 can be arrivedat when there is only one attribute x such as a pixel's grey value. The concept can be extended to the case when thereare more than one attribute for classification. The equations used in this context are as follows (Gonzales and Woods1992): First the Gaussian distribution is extended to multi-dimensions by

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where µ is the mean vector and C is the covariance matrix respectively defined as

(1)

where Nj is the number of pattern vectors from class Gj (i.e. the number of pixel vectors belonging to class j), and

(2)

the summation is taken over these vectors. The multi-dimensional decision boundary now looks like

Of course, the second term is equal for all cases, and may be subsequently dropped.

(3)

Now consider a two-dimensional readings for a 3x3 set of borings monitoring a ground-water pollution-plume, with the grey values shown in italics (Wright and Chan 1994c)

y-coordinate1 2 3

x-coordinate

1 2 3 4

2 3 8 7

3 1 7 9

Can you delineate the analytic and precise boundary of the plume based on the above set of equations?

B. Iterative conditional mode algorithm. The Iterative Conditional Mode (ICM) algorithm was describedin detail in the "Remote Sensing and Geographic-information Systems" chapter under the "Contextual allocation ofpixels" section. As demonstrated in the numerical example, a β of 0 produces a non-contextual classification, whileincreasing β accentuates the contextual bias. There is a tradeoff between β and σ2, where σ2 is the variance of pixelsin a certain class. The σ should be small enough to prevent greatly overlapping regions, and at the same time β willneed to be adjusted for the noise level of the image. A 6x6 grid of grey values is given below, with high valuesrepresenting polluted ground water and low values unpolluted water. Noise is introduced into the data by virtue ofthe data-gathering procedure. For example, a value which has the same approximate grey-value as the unpollutedground-water exists in the center of pixels which are evidently polluted. The second 6x6 data-set below shows a 3x3area of apparently polluted ground-water with possible noise on one of the sides of the 3x3 area. Also a single pixel(noise) with a pollution-range grey-value exists among unpolluted pixels.

3 5 4 3 4 2 3 5 4 3 4 23 4 3 2 3 3 3 4 8 6 7 34 2 4 8 4 3 4 2 7 8 5 35 3 10 9 8 12 5 3 10 9 8 43 4 9 4 7 7 3 6 4 4 5 52 5 11 12 10 9 2 4 5 4 4 4Please perform the classification using the ICM on both of these two data-sets (Wright and Chan 1994c).

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C. Weighted iterative conditional mode algorithm. In this exercise (Wright and Chan 1994c), the weightedICM algorithm (rather than the un-weighted one used above) is to be applied to illustrate a couple of points. For thesecond data-set above, the noise pixel in the polluted area could be classified as polluted water should a low enoughβ value be applied, since three of the five neighbors of the pixel are first-order neighbors. It can be shown also thatthe noise in the unpolluted area would be easier to discern using a weighted procedure. Notice the implementationis almost identical in both the weighted and un-weighted 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 carryout the weighted ICM algorithm.

D. District-clustering model. Shown in the "Remote sensing, Geographic Information Systems" chapteris a set of non-inferior solutions for a small image entitled "Multiple subregion noninferior solutions". Examine thefile labelled S2_4a_4b and S2_4a_5b in this Figure, the first of these two code names stands for 2 subregions, thesecond and third suggest that an area of 4 pixels for subregions 1 and 2. The a and b entries specify two differentvariations on the boundary of the subregion, generating different noninferior solutions. The two noninferior imagesare drawn below sequentially, where the bolded grey values stand for one subregion and the italicized stand foranother:

11 5 6 11 5 68 12 2 8 12 25 1 1 5 1 1

Using the "Multiple-subregion model" outlined in the Sub-section under the same name,

(a) Show the "reduced constrained feasible region" model that generate 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 ground-water pollution, measurements are made at wellsplaced 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 1 shows a well located at the center ofthe 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 1. Can you combine the two sources of information to delineate thepollution pattern more accurately than you would from a single source? Specifically, perform the following:

(a) Employ the ICM algorithm with due consideration to proximity as a factor. An inverse relationship ishypothesized between distance and importance in determining the allocation of some internal pixels (i.e. pixels notat the border or fringe of the image). For internal pixels, weights are scaled against eight neighbors. Assumingunitary-distance separation between subject pixel and its first-order neighbor, and a distance of with itssecond-order neighbors. Thus the weight for first-order neighbors is 1.1716 and 0.8284 for second-order neighbors.The sum over all of its neighbors is (4)(1.1716)+(4)(0.8284)=8 and a first-order neighbors is 1.1716/0.8284=1.4142times as important as second-order neighbor in determining allocation of a pixel as specified initially.

(b) Employ multicriteria-optimization techniques as outlined in the "Multi-criteria Decision-making" chapter. Definein the decision space a binary variable which 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 ICMclassification--are captured by the "ground truth" and the choice of the β value in the ICM algorithm respectively.Here, β is a measure of forced contiguity, parametrically for a 0-1 ranged weight for combining the two sources ofinformation. The two data-sets are shown in Figure 1. Since the water is directly sampled there, one may wish totrust the data at the well 100 percent, thus at the well the weight is unitary for the well reading and zero for theremotely-sensed data. Also shown in the same Figure is the "ground truth", representing a subjective judgment bythe decision-maker.

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(c) Determine the noninferior solutions which identify the most viable image classifications. A preference structurecan be adopted whereby the smaller the deviation from the ground truth the more it represents a non-dominantsolution. Zero deviation is considered Pareto optimal. Deviation in this case is defined as the number of pixels inthe ICM-generated solution different from the ground truth. Likewise, the less the need for forced contiguity (i.e.the smaller the β value), the better.

F. Histogram processing. (Russ 1995, Gonzales & Woods 1992) The image-brightness histogram shows

Figure 1 - "Ground truth", well data, and remotely-sensed data

the number of pixels in the image having each of the 256 possible monochromatic values of stored brightness. Peaksin the histogram correspond to the more common brightness-values, which often identify particular structures thatare 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 the imagebrightness range does not necessarily cover the full 0-255 range available. Similarly, the pixels at the two tails ofthe grey values tend to contain noise, rather than the real image. Figure 2 shows an example of such a histogram.

From the standpoint of efficiently using the available grey levels on the display, some grey values are under-

Figure 2 - Several options in histogram equalization

utilized (such as those at the two tails of the given histogram). It might be better to spread out the displayed greylevels in the peak areas selectively, compressing them in the valleys (or the two tails) so that the same number ofpixels in the display shows each of the possible brightness levels. This is called histogram equalization or histogramstretch. 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 pixelshave each possible brightness value. In many cases, this spreads out the values in regions where different regionsmeet, showing details in areas with a high brightness gradient. The equalization makes it possible to see minorvariations within regions that appear nearly uniform in the original image. In this example, we show that the range60-200 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 itshistogram), the newly assigned value k is calculated as k=∑j

i=0 Ni/n', where the summation counts the number ofpixels in the image (by integrating the histogram) with brightness equal to or less than j, and n' is the total numberof pixels (or the total area of the histogram). This is graphical represented as the dashed straight line plotted fromthe left-most grey-value to the right-most value, representing the grey value range we wish to examine in detail.

In Figure 2 are shown several ways to preform histogram equalization, including controlling brightness andcontrast. Using the TS-IP image processing software provided with this book, please show on the Pentagon imagethese various options:

(a) dimmer with more contrast,(b) brighter with less contrast, and(c) both brighter and with contrast enhanced.

II. Facility-location

Facility-location modelling is a key component of this book. Here we cover some less than obvious applications ofthese models. We start with "Noise-pattern recognition" of a landing aircraft, followed by an adaptation of the p-median in a "p-medoid model." This latter model is used for tracking ground-water pollution. Following the airport-location examples used extensively in the book, we further illustrate the "Nodal optimality conditions" prevalent innot only min-sum location models, but also min-max models as well. Then we turn to other min-max problems such

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as the "Vertex p-centers" model for locating such emergency facilities as fire stations. The opposite of min-maxproblems is the max-min problem, commonly found in "Obnoxious facility location," which includes "Solid wastefacilities." Another challenging facility-location model is the "Quadratic assignment problem," in which interactionbetween facilities take place. Sensor management problems bring us face-to-face with "Military intelligence facilities,"of which we focus on a nonlinear formulation and a linearized transformation of the search-and-rescue problem. Here,distress signals are detected by sensors, which pinpoint the location of the distress. We end this group of homeworkproblems by introducing the time dimension in facility location. A Markovian and non-Markovian decision-processmodels the "Repositioning" of service facility in response to changing demand patterns. Likewise, a control-theoreticmodel deals with Weber's "Location-allocation over time".

A. Pattern recognition.* (Banaszak et al. 1997) This problem describes the application of the p-medoidmethod to aid in acoustic pattern-recognition. The flight test included an array of microphones which measured theacoustic time history on the ground as the B-1 flew directly overhead. To show the sound level in decibels (DBs),the authors used the grey scales in Figure 3 to create the contour plot in Figure 4. The contour plot in Figure 4shows the sound level(in DB) by plotting 1/3 octave frequencies on the y-axis versus the time duration used tocompute the 1/3 octave frequency spectra on the x-axis. Sound level for an aircraft position and 1/3 octave frequencyband is represented as grey scales on Figure 4.

The tabulated sound-levels, from t=-0.5 seconds to t=2.8 seconds, are to be converted to a

Figure 3 - Grey scale definition (Banaszak et al. 1997)

Figure 4 - Sound levels in decibels (Banaszak et al. 1997)

mathematical-programming problem with the aid of the spreadsheet.

(a) Show that a total of 26 time segments and 17 1/3 octave frequencies could be used for a total of 442 non-emptycells. Accordingly, the actual DB levels are converted to 8 grey scales corresponding to the levels shown in Figure 3:

Grey Scale 0 1 2 3 4 5 6 7DB Level 70-74 75-79 80-84 85-89 90-94 95-99 100-104 105-109

Following the "Downed pilot" example worked out in the "Equivalence between center and median problems" sectionin the "Facility Location" chapter,

(b) Formulate this p-medoid problem as a mathematical program.(c) Solve this program to obtain the noise contour.

B. p-medoid model.* Again, refer to the two p-medoid examples discussed in the "Facility Location"chapter under the "Equivalence between center and median problems" section. Please convert the example into anetwork model. It is useful to include a graphical sketch with source, sink, arc attributes such as costs, and nodeattributes such as external flows. A network-with-side-constraint model as explained in the "Optimization technique"appendix may be necessary to operationalize the procedure (Wright and Chan 1994a).

C. Nodal optimality conditions. Consider the cities of Cincinnati and Dayton, Ohio connected by interstatehighway 75. Cincinnati has a metropolitan population of 2 million and Dayton, 1 million. A regional airport is

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proposed to serve both cities. It is to be located on I-75 such that the total person miles to travel between the twocities is to be minimized. We have shown in the book that the optimal location is Cincinnati. This is an example of"Nodal Optimality Conditions".

(a) 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?

(b) Suppose accessibility and noise exposure are of equal concerns, where should the airport be located?Accessibility is defined as the total person miles of travel while noise exposure is the total person decibel.

(c) Repeat questions (a) and (b) for the three city case where Columbus is included. Columbus has a populationof 2.1 million.

(d) Repeat the whole process for a four city case in which Indianapolis is included in addition.

D. Vertex p-centers.* In the "p-center" subsection of the "Facility Location" chapter, we solved a numericalexample for p=1, 2 and 3. We present here an efficient algorithm consisting of solving a set-covering and a greedy-heuristic procedure iteratively (Daskin 1995). The set-covering model finds the optimum number and location offacilities for a given maximum distance, as formulated in the "Facility Location" chapter under the same subsectiontitle. The maximal distance constraint is then appended: . This is a comparison between the distance from anode i to a node j, and the given maximum distance D. The modified Greedy Algorithm for solving the vertex p-center Problem (with integer distances) is stated here:

Step 1: Set an upper-bound distance, DH = (n-1) maxij{dij} where n is the number of nodes in the graph and dij isthe length of link (i,j). Also set DL=0.Step 2: Set D = (DL+DH)/2 , where is the lower integer of the argument .Step 3: Solve a set-covering problem with a coverage-distance of the argument D. Let the solution be p*(D), wherep*(D) is the optimal solution when the coverage distance is D.Step 4: If p*(D)≤p, reset DH to D; otherwise reset DL to D+1.Step 5: If DL is not equal to DH, go to step 2; otherwise, stop and use DL as the optimal value of the objectivefunction and the locations corresponding to the set-covering solution for this coverage-distance D are the optimallocations for the p-center problem.

Using this combined algorithm, solve the 1, 2 and 3 center problem For Estaboga, Alabama as given in the "p-center"subsection of the "Facility Location" chapter. Compare the results with those cited in the text.

E. Solid-waste facility. In locating a municipal-solid-waste facility, the Analytic Hierarchy Process wasoften used. Junio (1994) proposed a hierarchy of attributes as shown in Figure 5. Discuss the completeness andrelevance of such a hierarchy definition. How would you quantify this hierarchy in executing AHP?

Figure 5 - Hierarchy of a Municipal Solid Waste Problem

F. Obnoxious facility location.* Refer to the discussion of locating obnoxious facility using data-envelopment analysis (DEA) in the "Measuring spatial price" section of the "Measuring Spatial Separation" chapter.There we showed how such a max-min problem can be built into the DEA format. Can you revisit the steps,formulating the maximum average-distance facility-location problem within DEA and solve it? How is this resultdifferent from the max-min result? Why?

G. Military intelligence facilities.* In this specific problem, a number of facilities, p, can be chosen fromn possible surveillance sites. p may be any integer greater than 2 and less than or equal to n. This represents the factthat some external constraints, budget or otherwise, are at work. Another constraint being imposed is the necessityfor redundancy. This means that two sites must be monitoring a certain area, where a 'target' is located, at an

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acceptable level for the area to be considered under surveillance. It is important to note that two sites may be ableto 'see' a target, however, the combination of the two surveillance sites may still not be able to monitor the targetsufficiently. We will call the pair of sites that can monitor a target sufficiently an observing pair. Naturally, eachthreat can have different levels of importance. The goal of the problem is than to select p facilities, such that, thetotal amount of threat value, importance, is maximized. Table I below summarizes our specific problem, where theobserving pair(s) for each target is specified.

(a) Formulate the problem as a set-covering model.(b) Solve the model.

H. Quadratic-assignment problem. Refer to the quadratic-assignment problem as introduced in the

Table I - Data for military-target detection (Mehrez et al. 1997)

"Prescriptive Tools" chapter.

(a) Can you formulate the linearized version of model for the distance-separation and flow-interaction matrices asshown? (b) Now solve this linear model. (c) Is there anything peculiar about the solution to the linear model? If not, simply give the optimal assignment andthe objective function. If yes, explain the peculiarity and again give the optimal assignment and the objective functionvalue.

I. Nonlinear search-and-rescue model.* The following problem arises from a search-and-rescue (SAR)mission, where a downed pilot in enemy territory is awaiting a helicopter for evacuation (Steppe 1991). Let i be thetransmitting location of a distress signal, j be the receiving station, and k the frequency of transmission. Unlikeregular SAR in which an internationally-recognized frequency is adopted, the military application here allows fora frequency to be selected. Combined with the flexibility in the placement of sensors that detect the signal, thisbroadens the SAR problem to the generalized search-and-rescue problem (GSARP). Let d i be the largest acceptableconfidence-region-radius for transmitting-location i. Let κ be any combination of three or more receiving stationsthat acquire the signal and compute the lines of bearing (LOB), κ' be its complement, and Κ the combinations ofthree or more receiving stations. Now define Iiκ to be 1 if combination κ yields a confidence-region radius less thandi for transmission-location i, and 0 otherwise.

Specifically, bundles of (HFDFs) are to be located to a candidate set of receiving stations. Let us define theprobability that a distress signal emanating from location i propagates to station j on frequency k as Vijk. Theprobability that the distress signal is received by an HFDF is Vijkxjk, where xjk is the 0-1 binary decision-variableassigning an HFDF of frequency k to station j, with 1 representing an assignment, and 0 otherwise. The probabilitythat a target is detected is therefore

The requirement that the radius of the confidence region be less than d i is satisfied by multiplying the aboveequation by the indicator variable Iiκ. Since any three-station combinations could detect and receive a transmittedsignal, these probabilities must be summed over all combinations of three or more stations, Κ, yielding the objectivefunction: Here Fik is the probability that a distress signal emanating fromtransmitter location i is broadcast on frequency k.

All the constraints are linear. The first specifies that the number of receiving stations located does notexceed the total number of receiving stations available n, counting both new and existing stations: The

allocation of HFDFs in multiples (bundles) m j is indicated by where mj assumes the valueof unity in our example and zj is the integer decision-variable for the number of HFDF bundles in station j. The fixednumber of HFDFs, m, are to be allocated among the J receiving stations, including the reallocation of HFDFs to

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existing stations: As in other facility-location problems, service k cannot be allocated to sites j unlessthe site is open. In our case, receiving stations are opened as service sites via the allocation of HFDFs:

Integrality is specified for all variables: xj∈{0,1}, xjk∈{0,1}, zj integer.An example consisting of four stations, with possible assignment of zero, one, two or three HFDFs, is

included here in Table II and Table III. Table II contains the frequency transmission data denoted by Fik and thepropagation data denoted by V ijk in the formulation. Table III contains the confidence region indicator function datadenoted by Iiκ. Based on the data provided, please solve the nonlinear optimization problem using any nonlinearinteger-programming code.

J. Linear search-and-rescue model.* The nonlinear formulation in the last exercise is an explicit description

Table II - Transmission and Propagation Data

Table III - Confidence-region Indicator Data

of the GSARP which seeks to maximize expected geolocations. Unfortunately, the nonlinear formulation has anexplosive number of nonlinear terms for even small SAR networks. The integrality requirement also complicates thesolution, since some type of implicit-enumeration scheme must be employed. A nonlinear combinatorial formulationof this nature is clearly impractical even for small problems. A simplified formulation is required. A multi-objectivelinear integer program (MOLIP) is proposed to approximate the original formulation. The MOLIP is motivated mainlyby the physical problem itself, in that each of the criterion functions and constraints contribute toward an accuratedescription of the original problem. The GSARP has two opposing criteria when it is formulated as a MOLIP. Thefirst criterion maximizes the number of expected LOBs, while the second criterion minimizes the excess coverageof frequencies.

The first criterion function tends to increase the number of signals that are detected and processed by threeor more receiving stations. The expected number of transmissions from location i on frequency k received at stationj is FikVijkxjk. When summed over i and k, we turn the above expression into the total number of expected LOBs forthe SAR network. The first criterion-function incorporates an accuracy-weighting function, Wij. This declining-accuracy weighting-function captures the accuracy decline of LOBs with distance. In short, the weighting functionWij is the probability that the LOB bearing-fan-width is less than di. Correspondingly we define the first criterionas:

The second criterion-function penalizes excessive coverage of the frequencies by HFDFs. This has the effectof maximizing multiple-coverage while ensuring that coverage is evenly distributed among the receiving stations.This function requires the addition of one structural variable, yk, for each frequency to measure the extent ofexcessive coverage for that frequency. Let

The is the amount of excess coverage, where is the ceiling function or the next largest integer.

(4)

Excessive coverage can be avoided by introducing the criterion function: Without this criterion function,the HFDFs at each receiving station would be assigned to the frequencies where the probability of a transmissionand detection is the greatest, regardless of the number of HFDFs already assigned to those frequencies at other sites.

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Using the convention of the "Multi-Criteria Decision-Making" chapter and Steuer (1986), consider thefollowing SAR toy-problem with 2 criterion functions and no pairwise constraint

Number of Constraints Phase-0 # reducedProb. # # Obj. # Var. slack equality surplus switch crit cones 5001 2 23 23 2 0 0 45

Constraint matrix ---------------------------------------------------------# nonzero coeff 63

Row Col Coeff Row Col Coeff Row Col Coeff Row Col Coeff 1 3 1.0 1 4 1.0 1 5 1.0 2 6 1.0 2 7 1.0 2 8 1.0 2 9 1.0 2 10 1.0 2 11 1.0 2 12 1.0 2 13 1.0 2 14 1.0 2 15 1.0 2 16 1.0 2 17 1.0 2 18 1.0 2 19 1.0 2 20 1.0 3 3 -1.0 3 12 1.0 4 3 -1.0 4 13 1.0 5 3 -1.0 5 14 1.0 6 4 -1.0 6 15 1.0 7 4 -1.0 7 16 1.0 8 4 -1.0 8 17 1.0 9 5 -1.0 9 18 1.0 10 5 -1.0 10 19 1.0 11 5 -1.0 11 20 1.0 12 6 1.0 12 9 1.0 12 12 1.0 12 15 1.0 12 18 1.0 12 21 -1.0 13 7 1.0 13 10 1.0 13 13 1.0 13 16 1.0 13 19 1.0 13 22 -1.0 14 8 1.0 14 11 1.0 14 14 1.0 14 17 1.0 14 20 1.0 14 23 -1.0 15 3 1.0 16 4 1.0 17 5 1.0 18 6 1.0 19 7 1.0 20 8 1.0 21 9 1.0 22 10 1.0 23 11 1.0

[Right hand side]# nonzero coeff 14

Row Col Coeff Row Col Coeff Row Col Coeff Row Col Coeff 1 2.0 2 10.0 12 3.0 13 3.0 14 3.0 15 1.0 16 1.0 17 1.0 18 1.0 19 1.0 20 1.0 21 1.0 22 1.0 23 1.0

= Constraint matrix ---------------------------------------------------------# nonzero coeff 2

Row Col Coeff Row Col Coeff 1 1 1.0 2 2 1.0 [Right hand side]# nonzero coeff 2

Row Col Coeff Row Col Coeff 1 1.0 2 1.0

Constraint matrix ----------------------------------------------------------# nonzero coeff 0

[Right hand side]# nonzero coeff 0

Cost matrix for criterion functions ------------------------------------------# nonzero coeff 18

Row Col Coeff Row Col Coeff Row Col Coeff Row Col Coeff 1 6 1.7178 1 7 1.70447 1 8 1.89347 1 9 2.93833 1 10 2.9080 1 11 2.98717 1 12 .97728 1 13 1.27887 1 14 1.08597 1 15 .47198 1 16 .52436 1 17 .44037 1 18 3.12328 1 19 2.99900 1 20 3.03903 2 21 -1.0 2 22 -1.0 2 23 -1.0

Constant ---------------------------------------------------------------------# nonzero coeff 0

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(a) Now formulate the set of constraints for this MOLIP. Specifically, what are the constraints that you wish to addto or subtract from the original constraints for the singe-objective nonlinear program, if any? Considering thecomputational burden associated with multi-criteria optimization, what is the advantage of this MOLIP model overthe original NLIP model?(b) Refer to the accuracy-weighting-function data, Wij, in Table IV. Also refer to the input data for a multicriterialinear-program shown above. Due to problem-size limitations, HFDFs are dealt with as single entities rather than asbundles of entities. Another limitation in solving the test case is the linear-programming formulation and solution(rather than an integer-programming formulation and solution (rather than an integer-programming formulation andsolution). What modifications do you have to make in order to be able to solve this model in a linear-programmingcode? Now solve this linearized model using any multicriteria-linear-programming code you are familiar with.Compare the result with the nonlinear model from the exercise above.

(c) Convert the mathematical program into a network-flow formulation following the example shown in the "Facility

Table IV - Accuracy-weighting data

Location" chapter under "Another generalized version of the p-median problem" section. Specifically, sketch anetwork showing the source(s), sinks, arc attributes such as capacity, cost and gain, node attributes such as fixed orvariable external flows. What is the advantage of this network model over the MOLIP formulation, if any?

K. Repositioning.* Refer to the "Repositioning in a stochastic network" section of the "Measuring SpatialSeparation" chapter. The example on "Facility relocation" (by Mohan and Chan) was solved by an integer program.

(a) Can you reformulate and solve the problem as a linear network-flow model that guarantees integer solutions(Irish et al. 1995)?(b) In the above formulation, Markovian property is assumed. In other words, the state of the network is the soledeterminant of the location of the home nodes. When this assumption is relaxed, the operation of the service doesnot have to use the same set of p locations for the p servers every time the network is in a given state k. Thus theset of p locations to be used when the system is in state k depends not only on the current state k, but also on theentire past history of the system. Can you formulate this problem and solve it?

L. Location-allocation over time* (Tapiero 1971). In the "Discrete networks over time" section of the"Chaos" chapter, we formulated a temporal location-allocation problem due to Tapiero. Solution of his model wassketchy at best. Could you provide a step-by-step analytic solution for these two cases:

(a) when the transportation costs are linear and proportional to the Euclidean distance between origin anddestinations?(b) when costs are quadratic over distance?

III. Location-routing

The integration of facility-location and service-delivery is a key feature of this book. We use a simpletelecommunication-network maintenance-problem to lay out the integration. First, we define a region to be servedby a maintenance facility using the "Districting" technique. Then we place the facility using the "Service-facilitylocation" 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 combinedlocation-routing step. The basic building block of both steps is the quantification of spatial separation. This isillustrated in terms of "Minkowski's metric." It is through this measure that the efficacy of a location-routing schemecan be gauged, as shown in "Minkowski's metric and the generalized Leontief" problem. A number of routing modelsand algorithms are now discussed one by one, including the celebrated "Travelling salesman problem," "Space-filling

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curve," and the "Capacitated savings-insertion" heuristic. These lead toward combined location-routing models andalgorithms such as the "Spanning forest and Clarke-Wright" procedure, constructing "Nuclear power-plants and powerlines," and the "Simultaneous inventory-delivery problem." Solution techniques for these location-routing problemsare formalized in the "Benders' cut," and "Network with side constraints" illustrations. A generalized location-routingmodel is then formulated in terms of the "Transit system design" problem and solved using the RISE program in thebook-software disk. The intricacy of the solution method is highlighted in the "Integer programming vs. heuristics"problem.

A. Districting (Patterson 1995). The next three problems demonstrate a solution algorithm for improvingmaintenance depot location and service-delivery operations. Here in the first problem, we define the districts eachdepot is supposed to serve. The model is based upon enumeration and was adapted for network topology by Ahituvand Berman (1988):

where xj=1 if subnetwork j is selected to form a district and zero otherwise; aij=1 if node zone i is an element of

(5)

subnetwork j and zero otherwise; p is the number of districts or subnetworks desired, and the equity measure, 0<α<1; and fi is the fraction of demand at node i.

The algorithm consists of two different phases: Phase I determines all feasible subnetworks within the largernetwork, and Phase II determines the final subnetworks districts based upon our equity objective-function in (5).Contiguity and compactness will be bounding constraints for the first phase. One final requirement is that the psubnetworks 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 nodeand connecting contiguous nodes while enforcing the compactness requirement until the combined demandbecomes redundant. Attention must be paid to not creating separate enclaves, which are node(s) that areincapable of being separate subnetworks and cannot be connected to other subetworks without going througha previously defined subnetwork. This will prevent impossible solutions.PHASE II: The algorithm for node partitioning was developed by Garfinkel and Nemhauser (1970). Thefollowing notation is needed: X is the set of fixed variables; X is the number of fixed variables; D" isthe set of nodes in the districts, or zones of X; J is the set of districts in the current partial-solution; Nj arethe nodes in J; and · is the Cardinality of the set · general.

The computational steps are briefly outlined below:

Step 1: Initialization. Set counter L=0, and set J=X, Nj=D".Step 2: Choosing the next list . Pick the smallest number node not in Nj.Step 3: Updating set J. Add the node to J to form subnetworks.Step 4: Test for a solution. Test L = J - X. If L=0 stop, else L=L-1.Step 5: Solution is found. Pick the largest cost subnetwork in J as the current solution.Go to step2

12

Now for the network shown below, please perform the districting procedure with α=0.1 to arrive at two serviceregions.

B. Service-facility location (Patterson 1995). The current problem is to determine where the maintenance

Figure 6 - Service network (Patterson 1995)

depot should be placed within the network. The optimal location is determined using the minimized expected-response-time for a maintenance call. The development of the stochastic-location algorithm comes from Ahituv andBerman (1988) and is discussed in the "Spatial Separation" chapter: Min (y) for all j ∈ I ,where the expectedresponse-time ( ) is the sum of the mean-queuing-delay ( ) and the mean travel-time ( ). has been definedin the "Infinite-capacity queue" subsection in the "Measuring Spatial-Separation" chapter, and y is the facility-location decision variable. It is important to note that the time to repair a facility and the regeneration time beforea next call are zero. This assumption can be made since this amount of time will be treated as a constant. When theobjective function for is minimized by taking its first derivative, it will be eliminated.

Please locate the service facility in each of the district obtained in the previous problem.

C. User-Performance Model (Patterson 1995). While the operator might have located the maintenancefacility according to their preferences, it does not mean that same sites are necessarily endorsed by the users. Themathematical representation of the model is adapted from Sanso et al. (1991). Patterson's extensions to this modelinclude: the third criterion-function for average link-delay; and the inclusion of link-availability rates to representmaintenance scheduling. While a steady-state version is given below, the model can be written for each time periodby introducing another superscript t. To introduce the entire model, the first criterion-function minimizes the numberof lost calls: Here x pq is the lost traffic between p and q. The second criterion-function callsfor computing the shortest paths: The third criterion-function minimizes the data-transmission delay (or maximizing network availability):

This criterion function can be linearized by re-writing it as , where V is the network throughput

(6)

and is the average message length. The constraints consist of those on arc capacity for all k;

those on flow conservation

and nonnegativity: xkpq, xpq≥0 for all k, p and q. Here p(k) is the availability of arc k.

(7)

Now perform the network-performance evaluation on the same network as given in the last two problems, given thetraffic demands as shown below: A = B = C = D = E = 30.

13

D. Districting, location, and evaluation (Patterson 1995). The previous three problems highlight how each

Figure 7 - Multicommodity flow (Patterson 1995)

part of the problem--districting, location, and evaluation--is solved separately. The next step is to show how eachpart is linked together and iterated to find an integrated solution.

Step 1: Optimize operator perspective(1.1) Choose number of maintenance depots (p) to be located.(1.2) Network partitioning algorithm to create p subnetworks among the I nodes.(1.3) Location algorithm to locate a single facility within the p subnetworks. The optimal solutions is

chosen according to the minimum time-to-respond objective.Step 2: Optimize user perspective

(2.1) Create network maintenance schedule for solution from Step 1.3(2.2) Run network user performance model for solution from Step 1.3 without network delay factor.(2.3) IF delay factor desired, THEN create efficient frontier by varying the network-delay factor over the

desired range.

Please revisit all the previous three problems and put the results together in one place.

E. Minkowski's metric. Consider two points y1=(14,13) and y2=(4,4) in a two-dimensional space.Employing the following general measure of "deviation" between y1 and y2, r(y;p)=[Σi yi

1-yi2 p]1/p, explore the

behavior of numerical values of r for parameter p changing from 1 to ∞:

(a) Draw a diagram of function r=f(p). What are the general properties of such a function?

(b) Perform the same analysis for p changing from 0 to 1 and also from -∞ to 0. Do these cases show anymeaningful interpretation?

(c) A more difficult but very rewarding exercise: Do these distance measures, especially for p between 1 and ∞,correspond to any particular sub-family of utility (or value) functions? Can you identify such a subclass?

(d) Perform the following graphic exercise: Define a point y2=(0,0) in a two-dimensional space. Plot all such pointsy1 whose distance from y2 is equal to a fixed number r*, that is, r=r*. Choose r*=1 and draw such loci of points y1

for p ranging from 1 to ∞. (Pay special attention to p=1,2,∞). Do the resulting "shapes" suggest any connection withutility functions?

(e) Are there some points in (d) which have the same distance from point y2 regardless of the value of p? What arethe other characteristics and possible interpretations of such points?

F. Minkowski's metric and the generalized Leontief* (Piskator and Chan 1997). The Technique for OrderPreference by Similarity to Ideal Solution (TOPSIS) is a multi-criteria distance-measuring methodology. The modelmeasures and ranks discrete alternatives based upon each alternate-criterion's Euclidean-distance from the positiveand negative ideals. The “best” alternative--or positive ideal solution--is a theoretical composite of all the bestobserved-attribute-scores for each attribute. Similarly, the “worst” alternative--or negative ideal--is a composite ofthe worst observed-scores for each attribute.

One drawback of the TOPSIS methodology is that the decision-maker must assign relative weights to eachof the evaluating criteria. This technique may be acceptable when the decision-maker has specific and justifiablerelative ranking for each input and output and the rated units are aware of these weights. With prior knowledge ofthe decision-maker’s relative ranking of outputs, units can modify their use of the various inputs and produce outputsaccordingly.

14

As introduced in the "Measuring Spatial Separation" chapter, the generalized-Leontief distance-function-model is a descriptive model which shows a unit’s efficiency in relation to an efficient frontier established by allof the units in the sample. The most efficient unit(s) lie on the boundary which the function prescribes. A unit’sefficiency is measured on a scale from 0 to 1, with 1 being the most efficient. The Leontief distance-function is amulti-criteria model--it may be used to predict the efficiency of units with multiple inputs and multiple outputs. Thismodel implicitly calculates the Marginal Rate of Transformation (MRT), or marginal value, of the output(s). TheMRTs may be derived from a ratio of the partial derivatives of the Leontief function (Grosskopf et. al., 1995) or fromthe optimal solution’s shadow prices. Unlike Data Envelopment Analysis (DEA), the generalized-Leontief model'sMRTs apply to all evaluated units and may be considered the tradeoff space for the entire industry, rather than asingle unit.

We refer to the notations of generalized Leontief-function as defined in the "Distance measure of efficiency"subsection of the "Spatial Separation" Chapter. To determine the efficient frontier we set r’=1 in a statistical fit, asdescribed in the text. Instead of a statistical fit, we calibrate this equation using a linear program as follows:

Maximize r’(y',x)=∑i∑jαijxi1/2xj

1/2+∑j∑kβ jkxjln yk'

subject to:

(i) αijxi1/2xj

1/2+β jkxjln yk'≤1 for all j,k(ii) αijxi

1/2xj1/2+β jkxjln yk'≥0 for all j,k

(iii) ∑j∑kβ jk=1(iv) αij=αji

and all αij and β jk are unrestricted in sign. Constraint (i) limits all units' maximum efficiency to 1. Similarly,constraint (ii) places a lower limit on all efficiencies. Constraint (iii) ensures output homogeneity of degree 1 andnonnegative shadow prices. Constraint (iv) guarantees symmetry.

Using the partial derivatives of the Leontief function and the formulation in Grosskopf et al. (1995), we cancalculate the Marginal Rate of Transformation (MRT) between output 1 and output 2 for each unit as: MRTmm’=r*m(y',x)/r*m’(y',x) where m is output 1 and m’ is output 2 and r*m(y',x)= r'(y',x)/ ym' and r*m’(y',x)= r'(y',x)/ ym’',where r'(.) is the Leontief function. The individual-unit MRTs indicate the slope of the production possibility frontierat the given input mix.

Now refer to the data contained in Table V. The first two columns represent outputs while the remainingthree inputs.

(a) Please perform a TOPSIS analysis assuming equal weight between the two outputs.(b) Calibrate the generalized Leontief and compute the MRTs for each unit.(c) What are the differences and commonalities between the two analyses?

G. Travelling-salesman problem.* Consider the network shown in Figure 8, where all the arcs are bi-

Table V - Minkowski metric and the generalized Leontief

directional. The optimal symmetric travelling-salesman-problem (TSP) tour has been computed as the circuit4-5-1-2-3-6-4, with a length of 18.

(a) Formulate the problem using the following model:

Figure 8 - Travelling Salesman Problem

15

min z= dijxij

s.t. xij= xij=1 (for all i and j=1,2,...,I)

and xij=0 or 1. Write out all the constraints and objective function in a tableau ready for input to a linear-programming package. Can this model be solved by network-flow computer codes? Why?

(b) What is the problem with this formulation? (Hint: Show that subtours such as 4-3-6-5-4 and 1-2-1 is a solutionto the above formulation, where subtours are defined as a tour among only a subset of the nodes.)

(c) It has been shown these additional constraints will 'fix' the problem

δi-δj+Ixij≤I-1 i=1,...,I; j=1,...,I.

δi and δj are nonnegative real numbers recording the number of 'legs' in the tour. How many of these subtourelimination constraints are necessary in the numerical example?

(d) Is there any merit in solving the problem as formulated in part (a) and disregard the constraints in part (c)?Provide your answer in quantitative terms using less than 5 lines.

H. Space-filling curve.* The medical-evacuation problem calls for local-distribution routes to deliverwounded solders from a hub to nearby hospitals. Go to the space-filling curve directory SPACEFIL, locate thePhiladelphia-hub data. Supplement the data-set PHILADEL with the following expected demands:

Hospital i Buffalo Norfolk Philadelphia Pittsburgh Syracuse Wash DC--------------------------------------------------------------------------------------------------------------------------------------Expected demand fi 83 149 384 93 13 212.

(a) Can you now use the 3-dimensional space-filling curve to configure a set of aircraft delivery-routes, given thefollowing algorithm to supplement the regular SFC-heuristic? The aircraft h capacity V(h) is 48.

Suppose each depot location j* and the associated demand clusters are fixed exogenously, as shown above. Out-and-back routes of frequency are run for each demand location where fi>V(h). For demands less than V(h),they are grouped by nodal proximity into routes of demand nodes m1,..,ml,..,mk' respectively, wherem1+..+ml+..+mk'≤I, , and with k' minimized. The leftover demands fi-V( ) aredistributed via additional frequencies scheduled for the proximal nodes m1,..,ml,..,mk" on k" additional routes, where

(fi-V( ) )≤V( ) and k" is to be minimized.

(b) Can you validate this set of routes against the solution from a vehicle-routing mathematical program?

I. Capacitated savings-insertion.* The original Clarke-Wright (CW) routing-algorithm was designed as amultiple-tour, vehicle-capacity constrained heuristic for solving single-depot vehicle-routing problems. Here, we willpresent an extension that includes range and multiple-frequency-servicing constraints. The modified CW heuristicis a savings-insertion method which goes beyond selecting optimal routes based on their relative savings versus otherroutes. On top of distance savings, it enforces vehicle capacity. The heuristic minimizes the fleet requirement andhas four steps: initialization, recording, selection, and transition (Burnes 1990, Mandl 1979). Before we start, let usdefine the remaining unsatisfied demand (in the case of the medical evacuation problem the congested remainingservice capacity) at a node i as Fi'.

16

Step 1. Initialization: Construct an out-and-back route for each demand, recording the time required D, the capacityremaining on each vehicle h, V'(h), and the unsatisfied demand at each demand-node Fi'. If Fi'>0 or if Fi'=V'(h)=0,the route is optimal, provided the vehicle-range U(h) is not exceeded. Otherwise, reconstruct all routes until Fi'=0.

Step 2. Recording: Enumerate the number-of-nodes in a route and record the routes with the maximum number ofnodes Mmax.

Step 3. Selection: Combine these routes with routes having fewer than Mmax nodes. Calculate route-time savingsδij=D0i+D0j-Dij, where D0i is the time or distance from the depot 0 to demand i, D0j is the time from the depot todemand j, and Dij is the separation between demands i and j. Pick the route with the largest savings among allfeasible routes. Also record vehicle-capacity-remaining V'(h), and the remaining unsatisfied demand Fi'. A feasibleroute is defined as one which visits nodes with Fi'>0 and which does not violate the range-constraint U(h). If thereare no remaining routes with less than the maximum-number-of-nodes Mmax, stop.

Step 4. Transition: If a feasible combined-route exists at step 2 or 3, prohibit the demand nodes in the route frombeing used in any other route and returns to step 2. If no combined feasible-route is found in step 3, subtract 1 fromthe maximum node-number Mmax←Mmax-1 and return to step 2. If Mmax=0, select the remaining out-and-back routesas optimal and stop.

Once the modified CW heuristic has found a solution, a split-delivery heuristic will search for opportunitiesto split deliveries to single-demand points between two or more routes:

Step 1. If the excess demand at a node Fi' is less than the remaining vehicle-capacity of two or more routescombined, other than the one visiting the particular demand-node, m, then the demand-node becomes a split-deliverycandidate: V1'(h)+V2'(h)+..V'm-1(h), V'm+1(h)..+Vk'(h)≥Fi' where V k'(h) is the remaining capacity on route k, and Fi'(k+1)is the remaining demand at demand-node i on route k+1.

Step 2. Pick the savings on route k: , where x i and x j are twoconsecutive-demands on route x for x=1,...,k; and b and a are demands immediately before and after the candidatedemand-node q on route k. Naturally, all routes need to remain feasible.

It is interesting to note that the computational complexity of this CW procedure remains at O(I3) in spite of theserefinements.

Now use this modified CW procedure to solve the Philadelphia-hub medical evacuation problem for Day 1 of theWar as shown below. Can you compare the result with the expected demand obtained from the SFC? You canperform this manually or you may wish to modify the CW code in the SPANFRST directory on the computer diskthat comes with this book.

Philadelphia Syracuse Buffalo Pittsburgh Washington Norfolk---------------------------------------------------------------------------------------------------------------

- 5 68 73 183 123

J. Spanning forest and Clarke-Wright.* Refer to the Baker-Chan case-study of Defense Courier Servicein the "Simultaneous location-routing" chapter. Can you reproduce the validation result as shown in the "ValidationRun 1" Table? Notice the data and heuristics are encoded in the computer disk in the directory SPANFRST andunder the subdirectory VALID. Simply follow the directions in the README files to execute region-1 data. Noticethat while the Spanning Forest and Clarke-Wright codes are stand-alone programs, you need to have access to amixed-integer-programming code to execute the mathematical program that produces line 1 of the Table.

17

K. Nuclear power-plants and power lines.* In the Pacific Gas and Electric case study in the "Location-routing" chapter, we formulated the problem of power-coverage redundancy. In this formulation, the demand nodesat north and south bays of San Francisco have their electricity supplied by two different power plants, in case oneof them should fail.

(a) Can you re-run this model using a mathematical-programming package familiar to you?(b) Does your result support the conclusions spelled out in the chapter?

L. Simultaneous inventory-delivery problem.* Using any computer package you may be familiar with,perform six non-duplicating trial runs for the "Single-depot/multi-tour inventory-replenishment problem". In otherwords, add six more lines to the "Seven sample solutions for the routing/allocation model" Table in the "Location-Routing" chapter. Does a regular LP run yield integer solution?

M. Bender's Cut. Refer to the generalized Benders' solution to the numerical problem under the "Singlefacility/multi-tour/allocation" section of the "Location-Routing" chapter.

(a) Can you carry out one more iteration of the solution algorithm and show that the next cut actually carry us toy0

2=y13=y2

1=1 (with the remaining y values at zero), and the corresponding upper bound is 3990?

(b) We mentioned in the text that there is a simpler way to solve the same problem using regular Benders'partitioning technique (instead of the generalized Benders' procedure above). Can you carry out this procedure? Startout with the same y vector as used in the generalized Benders, show that the first cut takes us to y1

3=y21=y2

2=1 andan upper bound of 4717. With additional cuts, verify the optimal solution is as shown in the book.

N. Network with side constraints.* Refer to the suggestion in the "Multiple-facility/multiple-tour/allocationmodel" section of the "Location-Routing" chapter. Solution using modern software such as CPLEX detected anetwork with three nodes and seven arcs among 22 constraints, with the remaining constraints forming a mixedinteger program. The symmetric, compact formulation actually obscures most of the network structure. While CPLEXcan automatically detect any underlying network structure, it is felt that the network structure within the model canbe exploited more fully in the solution algorithm. Can you describe step-by-step how you would

(a) reformulate the problem as an asymmetric model with directed arcs, following the suggestions in the "Depotlocation in commodity distribution" section of the same chapter?

(b) Solve this model following the "network with side constraints" approach described in the "Optimization" book-appendix, in which the network part of the model is explicitly identified?

(c) Is there any advantage following the approach described in (b) above?

O. Transit system design.* Refer to the RISE directory in the accompanying computer disk. The purposeof this exercise is to examine a method to generate alternative transit plans, whereby the route structure andfrequencies of service are to be configured. The study area consists of seven downtown zones in York, Pennsylvania,labelled as zones 1 through 7 on the "York, highway network" figure in the "Activity Allocation and Derivation"chapter. For the purpose of this assignment, zone centroids can be placed at the closest street intersection. ADowntown People Mover (DPM) system is proposed, with additional alignment between zones 2 and 7 and zones1 and 6 beyond the existing streets.

The following data files are to be used: RISE1.DAT, RISE2.DAT, and RISE3.DAT. RISE1.DAT containsa test data-set intended to generate a star-shape route network. RISE2.DAT contains a test data-set intended togenerate a two-loop route network. RISE3.DAT contains a test data-set intended to generate an open-loop routenetwork. Each file includes the demands, the distance separation between zones, the seat capacity of the vehicle fleet,the travel-time formula for DPM, and the operating cost. It is proposed that a flat fare of 35 cents is charged, asshown in the "yield" formula of the data-set.

18

(a) Please input the specifications for the desired DPM route-structure;

(b) Configure the routes and their frequencies of service (in which each zone centroid is considered as a possibletransit station);

(c) Plot these routes on the York map provided;

(d) What is the rationale upon which you input the specifications in (a)? What is the coverage of the system (i.e.which origin-destination pairs are not served, if any)? Discuss the viability of the route system, including financialconsiderations and convenience to the public.

P. Integer programming vs. heuristics* (Memis et al. 1997). Refer to the "Comparison between integerprogramming vs. heuristic solution" Table in the "Location-Routing" chapter. Please formulate the integer programto the problem tackled in Experiment 1. Verify the results against that from the RISE-heuristic procedure. Contraryto conventional wisdom, why does the heuristic solution yield a higher profit than the RISE heuristic?

Refer to the network in Figure 9, please write out the mathematical-programming formulation for the dualobjectives of (a) Maximizing demand coverage in the routing from starting node 1 to terminal node 4, with differentnodal demand for each state k, and (b) Minimizing the travel time on this route.

Can you guess at the solution of this mathematical program by inspection?

IV. Activity derivation, allocation and competition

Figure 9 - Stochastic Facility-Location and Routing

The transition from facility-location to land-use models can be marked by activity derivation, allocation andcompetition. 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 residentsamong these neighborhoods. Here in this group of exercises, "Braess' Paradox" serves as the introduction tooligopolistic competition among participants for the limited resources in a network. Then we solve a matrix"Multicriteria game," in which there is more than one 'payoff' among the competitors. The idea is now carried overto "Gaming in obnoxious facility location," in which the city government and the public debate over anything fromsolid waste to airport siting based on different perspectives. The gravity model is a traditional way to analyzecompetition among geographic areas. Using the "Gravity vs. transportation models" exercise, one can see that thegravity model is an extension of the "all or nothing" assignment of activities from one single supply exclusively toone single demand as performed by the Hitchcock transportation model. The transition from 'discrete' transportationto 'continuous' gravity model is also illustrated by the "Extension to TOPAZ" homework for the doubly-constrainedgravity-model. Once its role is properly identified, we show the intricacies of "Gravity model calibration" througha numerical example. This is complemented by the "Calibration of a doubly-constrained model." When thegeneration, distribution and competition concepts are put together, we apply the integrated package toward a casestudy entitled "Visitation to Washington State Parks."

19

A. Braess' paradox.* One of the simplest illustration of gaming is the Braess' Paradox. In the transportationnetwork shown in Figure 10, let us say 6 units wish to move from origin 1 to the destination 2. There are two routesavailable for the movement, one on the left consisting of nodes 1-3-2 and the other on the right consisting of 1-4-2.

(a) If each unit optimizes its own travel-time based on the current system congestion, formulate and solve the

Figure 10 - Illustrating Braess' Paradox

problem as a mathematical program similar to the "Braess' paradox with a perfectly inelastic demand" section in the"Equilibrium and Disequilibrium" chapter. Do it again for a central authority who optimizes the travel time for theentire system.

(b) Now a new link is added from node 3 to node 4, thus making available a third path for travel 1-3-4-2. Solve theproblem again for both user and system optimization as performed in (a) above.

(c) Re-formulate parts (a) and (b) in terms of variational inequality. What insights can you gain from this re-formulation?

B. Multi-criteria game. Consider the following game (Zelany 1982). Decision maker 1 (DM1) maximizeshis minimum gain while decision maker 2 (DM2) minimizes her maximum loss. Gain of DM1 is exactly equal tothe loss of DM2 (i.e. a zero-sum game). Instead of the single metric used in the conventional payoff matrix, thereare 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, where the cells contain the two payoffs for each pair of decisionsreached between DM1 and DM2:

Thus if both DMs decide to play their second option, DM1 wins 3 units in the first criterion and 2 in the second.

Table VI - Multi-criteria game

DM2 loses the same amounts. p'i and q'j denote the probability DM1 and DM2 will play the ith and jth strategyrespectively. When p' and q' assume fractional values, the game is a called a mixed strategy game. A pure strategyis when p's and q's are 1 or 0 in value.

Let each vector payoff be replaced by a convex combination of both components:, where w is a 0-1 ranged weight. For example, a11= , and so on. It can be shown

that, similar to a conventional zero-sum two-person game, a linear program (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 nonnegativevariables p and q are defined such that p'=pz' and q'=qz', the equivalent LP is:

Max q1+q2+q3

s.t. .

(a) Solve this LP by varying the weights w from 0 to 1.

(b) Is there an 'equilibrium', defined here as a pair of decisions which both sides are happy with? At thisequilibrium, z' is a nonnegative number representing the gain to DM1 and the loss to DM2?

20

C. Gaming in obnoxious facility location* (Clough et al. 1997). Refer to the obnoxious facility locationproblem discussed in the subsection under the same name in the "Facility Location" chapter. In terms of theobnoxious facility problem, Player 1 in a zero-sum game such as the one introduced in the "Multi-criteria game"above can be thought of as the people of Estaboga, Alabama. These people are trying to maximize the shortestdistance between the undesirable facilities and their communities. The reward in this case can be thought of as thedistance (in miles or km) that a given facility is from their homes after its location is determined. In the 'game',Player 1 (the local residents) improves his/her objective by placing the obnoxious facility farther from the affectedcommunities. Player 1, therefore, chooses the strategy which maximizes the distance between the facility and anyof the communities. Meanwhile, Player 2, which can viewed as the owner of the obnoxious facility, is trying to placethe structure as close to the communities it serves as possible to reduce transportation costs. Hence, Player 2 willexercise his/her best strategy to thwart Player 1’s objective. In other words, Player 2 is trying to minimize themaximum distance the facility is located from the community.

In the payoff matrix, the reward for Player 2 is actually the amount s/he concedes to Player 1. For thisproblem, the payoff matrix is simply the distance matrix of the primal problem formulation. This matrix isreproduced and modified for game theory in Table VII. Strategies in this problem for Player 1 are interpreted aslocations for the obnoxious facility, while strategies for Player 2 indicate community locations. One minor deviationfrom standard game theory is required in this problem--Player 2 is not allowed to choose the same location for thecommunity as Player 1 chooses for the facility. Normally in game theory, there are no such restrictions; however,the game theory interpretation of this problem breaks down without this stipulation.

Now Player 1 will pick a strategy that provides the best minimum payoff. Since Player 2 is trying to keep

Table VII - Payoff matrix for Estaboga, Alabama

Player 1’s reward as small as possible, he will naturally choose the strategy that yields the smallest payoff for thestrategy chosen by Player 1. To illustrate, first consider the single-facility-location 'game'. If Player 1 chooses strategy(location) 3 for the obnoxious facility, Player 2 will naturally choose strategy (location) 4 for the community (thisis the minimum-distance strategy for Player 2). The resulting payoff for Player 1 is 3 miles (4.8 km). The two facilitylocation game is only slightly more complex. In this game, Player 1 and 2 must choose two strategies each. Thecombined payoff of both strategies is awarded to Player 1. Player 2 receives a slight advantage in this game becausePlayer 1 receives the minimum payoff attained by combining the two sets of strategies. For example, referring toTable VII, if Player 1 chooses locations 1 and 2, Player 2 will consequently choose options 3 and 6.

(a) What is the payoff? Which strategy selected by Player 2 matches with Player 1’s choices?(b) Formulate the dual problem mechanically setting aside integrality requirements.(c) Solve the dual linear program.(d) To the extent possible can you interpret this dual LP?(e) Is there computational advantage to examine the dual LP formulation?

D. Gravity vs. transportation model. Refer to the doubly constrained gravity-model as discussed in the Sub-section bearing the same title in the "Descriptive Tools" chapter. When the value of α becomes 1 in the propensityfunction F(Cij), the function becomes a special function of travel cost, F(Cij)=Cij

-α=Cij-1, and the doubly-constrained

gravity model can be written as

21

z in Equation (9) is interpreted as the total travel-cost now. By minimizing total travel-cost (say in veh-min), we have

(8)

(9)

(10)

the classical Hitchcock-Koopman transportation model. Notice this model reflects the system-optimum rather thanuser-optimum as obtained by conventional gravity-model calibration. Now answer the following questions:

(a) What value would α assume in the propensity function to have maximum accessibility?(b) What value would α assume to have minimum accessibility?(c) For a prescriptive model, what is the resulting trip-distribution for case (a)?(d) For a prescriptive model, what is the trip distribution for case (b)?(e) Interpret the result of (c) and (d).

E. Extension to TOPAZ.* Review the TOPAZ model as described in the "Optimal placement of activities"Sub-section in the "Location-Allocation" chapter. Suppose we write Vij=kiljViVjexp(-bτij) for a doubly-constrainedgravity model (instead of a singly-constrained model). Show that the TOPAZ objective function now looks like

and the constraints become

(11)

F. Gravity-model calibration. In calibrating a logit model (Kanafani 1983) to calculate the probability of

(12)

(13)

making a trip for purpose m, pm= exp(vm)/Σk exp(vk), a survey is taken of travellers from city 1 to city 2. Of the 10respondents, it is known that 4 are recreational travellers and 6 are workers commuting to their places ofemployment. Average intercity travel-times (t) for work trips (A) and nonwork trips (B) are available from secondarysources as:

(a) Compute the one-way travel-time for both work and nonwork trips, counting the intercity times and intracitytimes for both ends of the trip, where intracity times at city i are defined as τii and intercity times τij. From the datagiven, is this an aggregate or disaggregate model?

22

(b) Write out the 'propensity' functions exp(vm) for both work and nonwork trips. How many coefficients must becalibrated in these functions in the logit equations assuming the valuation of time is different for work trips andnonwork trips?

(c) Estimate these coefficients by the maximum-likelihood technique.

(d) Can you solve the equation(s) for the unknowns? Give the reason(s) for your answer.

G. Calibration of a doubly-constrained model. Given the following data on inter-zonal trips Vij and the

associated costs Cij, please calibrate a doubly-constrained gravity-model: [Vij] = and [Cij] =

Suppose F(Cij)= , carry out the calculations as far as you can, following the procedure described in the "Doubly-constrained model" Sub-section of the "Descriptive Tools" chapter. Give the final 4 equations for the 4 unknowns,and solve the equations.

H. Visitation to Washington State parks.* The patronage of the New York Metropolitan State Parks wasanalyzed in depth in the "Generation, Competition and Distribution" chapter under the "A case study of state parks"Section. Can you revisit the steps for the Washington State Parks? Refer to the STATEPRK directory in theaccompanying computer disk, invoke the WASPRKS.BAT file and execute to final completion. Compare and contrastthe park-visitation pattern with that of New York. Now test the policy sensitivities such as parking-fee increases, parkimprovements and park closures, and again compare between the two state-park systems.

V. Land-use models

Land-use models is a center piece of this book, and the above section adequately prepares the reader for the currentblock of homework problems. Here, the "Economic base and activity distribution" exercise shows how the activityderivation, distribution and competition concepts can be used to simulate the housing requirements of a college townover time. This set of calculations is then formalized in the "Iterative Lowry-model calculation," which is encodedin the book-software disk. Through its analytic, matrix form, the behavior of the model is further investigated viathe homework "Bifurcation in the Garin-Lowry model." Again, this disaggregate formulation is operationalized inthe book-software disk. Using this software, we carry out a case study in "Bifurcation in York," Pennsylvania,verifying the results reported in the main body of text. The Garin-Lowry modelling tradition is further extended inthe "Kansai Airport regional input-output model," where the spatial input-output model is used for yet another real-world case study.

A. Economic base and activity allocation. In a study of a college town, State College-Pennsylvania, Chanand Rasmussen (1979) forecasted housing requirements. Using the basic concepts of the Lowry model, they derivedthe subareal housing requirement of the town using the university enrollment as the basic activity. Their algorithmfollows a two-part procedure:

I. Housing Demand Factor

1. Define the zoning types of all residentially-zoned developable-land.2. Establish the number of students, blue-collar employees, and white-collar employees from tract i working atemployment center c--labelled here as ES

ic, EBic, and EW

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 travelling a distance of d miles to therelated employment center--labelled fi

S(d), fiB(d), and fiW(d) respectively.

5. Determine the percentage of students, blue-collar workers, and white-collar workers in residential type t--labelledpS

t., pBt. and pW

t. respectively.6. Compute the Housing Demand Factor: Vd

it=∑k∑c∑d Ekicfi

k(d)pkt.

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II. Allocation of Housing Demand

1. Determine the excess housing supply in tact i, ∆Ni. The excess is equally distributed among the number of zoningtypes tmax: ∆Nit=∆Ni/tmax.2. Determine the maximum holding capacity for developable dwelling units: Nit

c=(developable average)(averagedwelling units per acre).3. Allocate the total housing demand N to each tract i: N it=NVd

it/∑i∑tVdit. The housing demand for housing type t

in tract i can either be accommodated by the excess housing-supply ∆Nit or new construction. Housing demandexceeding the holding capacity of a tract would have to be located elsewhere. The additional developable capacityof a tract for housing type t is ∆Nit

c=Nitc-Nit-∆Nit.

4. Additional iterations are necessary as long as one or more ∆Nitc 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 Center Region Planning Commission(CRPC). The housing projection performed by the CRPC is computed by a two-step procedure: (1) The futurepopulation for the region is computed; and (2) the number of dwelling units is derived therefrom. The derivationprocess is generally founded on an extrapolation forecasting techniques. The CRPC population forecast takes intoconsideration a Cohort-Survival Model and a straight-line proportional model1. The following assumptions are madeamong both studies: (a) No substantial in or out migration would take place, which implies the student enrollmentat Penn Sate University would stabilize at 31,500 by 1985. (b) Existing trends, including birth/death rates and othercoefficients and ratios, will remain constant over time for each township of the Centre Region.

Since the study, the dwelling units that are actually observed became available. These figures are tabulatedside-by-side the Chan-Rasmussen and CRPC forecasts in Table VIII. Can you perform a "before-and-after" analysisas to the accuracy of the forecasts by the Chan-Rasmussen model vis-a-vis the CRPC study?

B. Forecasting air-base housing-requirements (Bahm et al. 1989). Now that you are familiar with the Chan-

Table VIII - Comparison of forecasts and observed housing units

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 localeconomy is an Air force base (AFB). Whiteman 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 is expected to grow atthe time of the study in 1989. The source of the increase in military- and civilian-employment is the new B-2 bomberwing.

Three types of economic activities are envisioned to increase: military and their dependents, civilianDepartment-of-Defense (DOD) employees, and civilian non-DOD employees. There are 25 housing tracts or zonesin the region. There are four employment-centers: Warrensburg, Sedalia, Knob Noster, and Whiteman AFB.Commuting distance is measured in one-mile (1.6 km) increments, with the longest commuting distance being 46miles (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 IX. Theinformation is listed by each tract/zone i. By resident profile is meant the percentage of military, DOD-civilian, andnon-DOD civilians in each type of housing--whether it be single family, double family, multiple family or dormitory.Commuting distances from each of the 25 tracts/zones to the four employment-centers are shown in Table X. Thetrip-distribution, or the percentage of workers travelling distance d to an employment center, is shown in Table XI.Also shown in the Table are the increases in military, DOD-civilian and non-DOD-civilian jobs in each of the fouremployment-centers.

1These techniques are discussed in the "Econometric modelling" Section of the "Economic Methods" Chapter.

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Now forecast the housing requirements at the study area based on these assumptions: (a) insignificant

Table IX - Additional developable-capacities & excess housing

Table X - Commuting distances to employment centers

Table XI - Trip distribution and job profiles at the employment centers

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 ofassociated cross-commuting from Whiteman to other points in the study region. All these make Whiteman AFB themajor employer in the projected future, attracting the local population to the base.

C. Iterative Lowry-model calculations.* Refer to the Lowry-model example worked out in the "A numericalexample" Section of the "Activity Allocation and Derivation" Chapter. Following the first cycle as given in the book,can you complete the entire six iterations required for final convergence?

D. Traditional Lowry software. One of the most common functions of a land-use model is forecasting. Inthis assignment, you are to forecast the land-use of York, Pennsylvania for a forecast year corresponding to atransportation-system improvement. A set of data (named DATA2.DAT) is located in the directory LOWRY of theaccompanying computer disk. The data file represents the implementation of an ubiquitous Personal Rapid Transit(PRT) system in York. The system operates at a constant speed of 25 miles per hour (40 km/h) on existing streets,replacing all roadway traffic (particularly the automobile) and becoming the city's only means of transportation. Sucha precipitous policy (henceforth referred to as the "PRT alternative") will undoubtedly redirect land developmentsin York. Compared with the 'null' or "do-nothing" alternative (in which the existing auto-oriented system prevailsas shown in DATA1.DAT,) the PRT-alternative will probably improve the accessibility at certain congested partsof the city but prolong the travel time at other less congested places. The land-use pattern is expected to changedramatically accordingly to the theory of gravitational interaction.

You are commissioned by the City to evaluate the impact of the proposed PRT-alternative. Specifically, youare asked to:

(a) make a forecast corresponding to the PRT-alternative;(b) compare the forecast with the 'null' or "do-nothing" alternative; and(c) explain the development changes from the base year in terms of the difference in transportation policies.

Aside from a graphic plot to show the effects of a policy change for the forecast year, you are to document yourcomparison and explanation ((b) and (c) above) in a writeup of no more that five typed-pages (double spaced). Thefollowing outline for your submission is suggested: (i) difference in input, (ii) difference in output, (iii) accessibility,(iv) development opportunity, (v) overall interpretation, and (vi) visual display. For the last part (part (vi)), the datasetYORK.DAT, containing both the base-year and the null- and PRT-alternative forecast-year data--may be useful.

E. The Projective Land Use Model (PLUM).* Review the Lowry and PLUM discussions in the "Activityallocation and Derivation" and "Chaos, Catastrophe, Bifurcation and Disaggregation in Locational Models" chaptersin the text.

(a) What are some of the major advantages of PLUM over previously discussed Lowry-type models? Whatsuggestions for improvement could one make based on comparison with other Lowry-type models?

25

(b) Allocation probability-functions, population per housing-unit, land-absorption coefficients and factors influencingregional-growth are all dependent on past behavior of the region in the PLUM-model system-of-equations. Explainthe implications of this technique as far as model output is concerned. Are there any mechanisms within PLUM thatwould tend to counter-balance negative implications of the dependency on past-performance for future-allocation ofactivities? If so, explain how these mechanisms operate and evaluate their effectiveness in dealing with the problempreviously cited in the book.

(c) One obvious difference between other Lowry-type models and PLUM is the latter's use of zone-specific functionfor allocation of employees and for land-use allocation. Explain why this method would be empirically moreacceptable than average regional-values for corresponding functions in other models. What data requirements arenecessary for the use of zone-specific parameters?

(d) The probability of locating a non-basic worker or a resident in a particular zone is a function of the probabilitymatrix as defined the "Spatial allocation" subsection of the "Activity allocation and Derivation" chapter. This matrixis decided also by two other functions in PLUM. Identify these functions and describe the variables involved in each.

(e) What factors may influence over allocation in the PLUM method of basic and non-basic employee-allocation?

F. Bifurcation in the Garin-Lowry model.* In the "Chaos, Catastrophe and Bifurcation" chapter is a setof sample calculations to show the convergence of a Lowry-Garin model. While we have discussed Groups 1 and2 results of these experiments in the Sub-section "Different zonal multipliers", we have not touched on the Group-3results.

(a) Can you, first of all, reproduce these Group-3 results?(b) Explain the implications of these Group-3 tabulations in the context of bifurcation.

G. Bifurcation in York. Refer to the directory on the disk called YICHAN. This directory contains a state-of-the-art code on the disaggregate, constrained Garin-Lowry Model.

(a) As a first step, re-run the above 'Bifurcation' exercise using the data-sets for the 4-zone example to familiarizewith the workings of these computer programs. Validate against the results shown in the "Chaos, Catastrophe,Bifurcation and Disaggregation" Chapter under "A numerical example" Sub-section.

(b) Then perform the same with the 42-zone data-set of York, Pennsylvania. Validate against the bifurcationbehavior documented in the "Spatial-Temporal Information" Chapter under the "Case study 3 - development in York,PA" Sub-section. What are the major differences between the disaggregate results from current runs vis-a-vis thosefrom the previous, aggregate runs sought by the "Traditional Lowry Software" exercise above?

H. Kansai Airport regional input-output model.* To supplement the discussion in the "Spatial EconometricModel" chapter, we will document the Kansai model (Suzuki et al. 1989, Pak et al. 1988) formally. The followingrepresents the regional aspatial input-output equation-set

where ρ=[ρpq] is the matrix of input-output coefficients, R is the vector of gross output, O is the final or 'export'

(14)

demand- vector. These notations have been previously defined and an introduction to input-output models was madein the "Spatial Equilibrium and Disequilibrium" chapter under the "Growth of regional economic activities." InEquation (14) the 12 economic sectors in Kansai define the dimensions of these quantities--ρ is 12x12, R is 12x1,and so is O. The gross output R is obtained from Equation (14) as , where I is the identity matrix. Akey component of the input-output model is the multiplier (or value-added) effect of trade among industrial sectors.If we denote the commodity-value-added vector as R" and the labor-force-value-added vector as Y', then

26

and where C' is a diagonal matrix converting the gross-output vector to value-added vector, and inturn converts R" to Y'. The vector of employment levels, denoted by E, can then be obtained by

where W" is the diagonal matrix consisting of per-capita value-added productivity (wage rate).Now based on this model, we wish to estimate the economic impact of the new airport operation.

(a) How would you calibrate the model?(b) Thus far, the model is aspatial, how would you distribute the economic activities among each subarea of thestudy region?

VI. Spatial temporal information

The unifying theme throughout this book is really how one analyzes "Spatial temporal information" in general. Inthis last block of homeworks, we let data "speak for themselves" and hopefully have them guide us in the analysis.The first homework eloquently shows the difference between "Spatial vs. univariate forecasts," particularly regardingtheir respective accuracies. Where there are precipitous external influences beyond the underlying driving force thatmay disrupt a traditional forecast, the effects of such influences are captured in the "Univariate forecast withintervention" exercise. The inadequacy of such a non-spatial approach is then discussed and redressed in the "Spatialforecast with intervention" exercise. Subsequently we worry about the "Calibration of a spatial forecasting model",an area so demanding that much more research is still needed. We end this appendix with "Voronoi diagrams andmulti-facility location model." This exercise brings out the glamour of a latest technique (Voronoi diagrams) as wellas the practicality of a traditional technique (multi-facility location model). It compares the advantages anddisadvantages of identifying subway-station locations using either one of these two approaches.

A. Cohort-survival method (Jha 1972). The Cohort-survival method is an econometric technique introducedin the "Economic Methods of Analysis" chapter under the "Interregional growth and distribution" subsection. Pleasereview the discussions in the text and answer these questions:

(a) Suppose these statistics are gathered for York County, Pennsylvania during the 1940-1945 period. The numberof births is 2,000 and the number of deaths is 500. Average population for the period is 210,000. There were 1,400people migrating to York and 1,295 migrating out. Define the following terms for a certain forecast time-period:crude birth-rate, crude death-rate, and net migration.

(b) Check the population for female in 1945 by the cohort-survival method for York county, Pennsylvaniacharacterized by these population statistics:

Age 0-4 5-9 10-14 15-19 20-24 25-29 Total-------------------------------------------------------------------------------------------------19402 10 14 15 18 22 24 10319452 7 10 12 14 16 21 80

The entries in this Table represent the number of people in each age group. The surviving ratio of the 0-4 year groupis 0.98 and the percentage of female children is 0.49. The fertility rate of 15-19 year group is 43, the rate for 20-24groups and 25-29 groups is 56.

2These numbers are in hundreds, i.e. 10 means 1000.

27

B. Spatial vs. univariate forecasts.* Pfeifer and Bodily (1990) applied STARMA to demand-related datafrom eight hotels from a single hotel chain in a large US city. The time-sequence plots of pickup percentages areplotted in Figure 11. These hotels are separated by the following symmetrical distances:

Suppose spatial weights are simply inversely proportional to distances. Model the time-series data in terms

Table XII - Spatial vs. univariate forecasts

Figure 11 - Time sequence plots of pickup percentages (Pfeifer and Bodily 1990)

of both univariate analysis and using spatial weights. Compare the results. Use the first 300 data points for modelbuilding, and the hold-out sample of 70 points to compare the one-step-ahead forecasting performance of the twoapproaches.

C. Univariate forecast with intervention.* Auto-Regressive Moving Average (ARMA) and Spatial-TemporalARMA (STARMA) empirical analyses are used to model the level of arsenic contamination found in ground water(Wright 1995). The contamination levels at wells 1, 2 and 3 are recorded in Figure 12. Pumping is used to containthe contamination, constituting an intervention. The predicted drawdown (in feet) at each of the seven monitoringwells through each of the pumping scenarios over time (S1 through S6) are shown below. The first five rows (W1through W5) represent strict monitoring wells which are located generally downstream from the two most upstreampumping wells. The last two rows represent the drawdown at these two wells which are pumping wells themselves.

Figure 12 - Contamination time-series for wells 1, 2 and 3 (Wright 1995)

Next, the drawdowns are scaled against the nominal case when all pumps are operating at 300 gallons (1.20m3) per minute. Nominal pumping occurs in the first 26 time periods. Intervention occurs at time-period 27. The finalintervention input-time-series for each monitoring well is shown below with the value of each pumping-scenario foras many time-periods as that pumping-scenario occurred:

28

The intervention of various levels of pumping is modelled with a transfer function to create an intervention-transfer-function input-series to give physical meaning to the impulse-response-weights found.

Three wells were monitored. Please show that among the most parsimonious univariate ARMA/Transferfunctions obtained for well 1 is an AR(2) series:

Zt=5.50+0.1776Zt-1+0.2344Zt-2+522.875Xt+599.798Xt-1+392.735Xt-2+At

where Zt is the original time-series, and Xt is the intervention time-series due to pumping, and At is the residualseries. Show that wells 2 and 3 exhibit an AR(2) time-series as well:

Zt=5.53+0.1104Zt-1+0.3313Zt-2+180.974Xt+283.278Xt-1+214.2925Xt-2+At

Zt=7.69+0.1444Zt-1+0.2703Zt-2+105.094Xt+87.414Xt-1-93.260Xt-2+At

D. Spatial forecast with intervention.* In question VI.C directly above, instead of three univariate-series,it is more satisfactory to model the ground-water contamination problem as a spatial-temporal series. Spatial weightsemployed in the STARMA modelling are created using analytic methods. These spatial weights are used to modelthe relationship between neighbors--i.e. determining which neighbors of the site will be the lth-order neighbors is

the next task. An adjacency matrix is constructed between all the seven wells: This matrix describes

the set of neighbors reachable in one step, where a unitary entry suggests that well i potentially exerts influence overwell j. The set of neighbors reachable in two steps is defined by squaring the adjacency-matrix. The sum of theoriginal adjacency-matrix and the squared matrix define the matrix of neighbors reachable in one or two steps.Neighbors reachable in one or two steps would show a "2" in the matrix, but a "1" is displayed for clarity below:

The column with the most entries, column 2, represents the well with the largest number of influential

neighbors. The site, well 2, is therefore selected as the site of interest.Well 2's first-order neighbors are selected as those which exhibit the smallest lateral distance from their

flow-lines. The following matrix shows the lateral distance from the predicted flow-line of each influential well tothis one, including two-step neighbors. Zero entries imply that no relationship exist between the two sites:

The closest neighbors to the site of interest, well 2, are wells 1 and 3, which are classified as

first-order neighbors. All other wells constitute second-order neighbors. The weights of the member of each first andsecond-order neighbors are created by inverting the lateral distances so that heaviest weights are assigned the closerneighbors. The weights assigned to the elements of a particular lth order neighborhood are then scaled to sum tounity. In this way, the stream of data from a particular lth-order neighborhood becomes a weighted sum of its

29

members and therefore remains indicative of the values of its members. The resulting spatial weights are:

Following the procedure outlined in the "Spatial Time-Series" Chapter under the "Extended numericalexample" section, show how these spatial weights are used in the spatial-temporal modelling of well 2. Find the mostparsimonious univariate spatial time-series. Can you interpret the results of this model?

E. Calibration of a spatial forecasting model.* As cited in Cliff and Ord (1981), Mitchell (1969) studiedthe pattern of insurgent control during the Huk rebellion in the Philippines, linking control to a variety of culturaland economic factors. Doreian and Hummon (1976) pointed out that control of any given area either by thegovernment or by the insurgents has immediate relevance for the control in adjacent areas. In other words, one wouldexpect the insurgency to spread, or to contract, through adjoining areas. Thus Doreian and Hummon proposed aregression model with a spatial component to analyze this phenomenon. The regressor variables are:

P the proportion of the population speaking the Pampangan dialect,F farmers as a percentage of the population,O owners as a percentage of farmers,S the percentage of cultivated land given over to sugar cane, M the presence of mountainous terrain (dummy),X the presence of swamps (dummy).

In the regression equation, P is used multiplicatively with the other variables, so that the exogenous variables arePF, PO, PS, PM, and PX.

Here are three sets of estimates for a (linear) model linking insurgent (Huk) control to the cultural,demographic, ethics, and physical exogenous-variables for each municipality i. Each regression coefficient has anaccompanying standard error given in brackets. The coefficient-of-multiple-determination is also shown for eachmodel.

Nonspatial ordinary-least-squares (OLS) model (R2=0.73):

Y= 1.147 +3.794F -1.912PO +0.461PS +38.38M +17.17PX(2.94) (0.939) (0.438) (0.161) (7.02) (7.94)

Simultaneous maximum-likelihood spatial-model (R2=0.80):

Y= -1.316 +0.571(w(l))T(W(l)y)~Y +1.942PF -0.889PO +0.118PS +28.75PM +11.41PX(2.39) (0.008) (0.762) (0.355) (0.132) (5.69) (6.44)

Conditional-OLS spatial model (R2=0.80):

Y= -1.382 +0.586(w(l))T(W(l)y)~Y +1.892PF -0.862PO +0.108PS +28.49PM +11.26PX(2.62) (0.138) (0.928) (0.453) (0.164) (6.51) (7.01)

In the above, the non-spatial model is the standard linear-model, fitted by OLS without any spatial autoregressive-component. In the second model, the simultaneous spatial-scheme has been fitted by maximum likelihood. In the thirdmodel, conditional OSL has been used to estimate the parameters of a spatial model. In both models 2 and 3, datahave been 'filtered' ahead by a spatial 'mask' defined by the weight vector w(l) for each municipality i.

Compare and contrast the results of these three models.

30

F. Voronoi diagrams and multi-facility location model.* In the Voronoi-assignment model, it is assumedthat candidate sites for a facility are identified ahead of time. This is a restrictive assumption, in that these sites arenot finite and discrete. They have to be determined from a continuous distribution of demands over the entire studyarea. Ideally, location and allocation are simultaneous, interdependent decisions. One possibility of creating a discreteset of customer locations is to minimize the average distance between the n generator points and their respectivevertices. The motivation is the vertices represent discrete customer locations for each Voronoi region and providea reasonable approximation to the continuous customer distribution within the region. The corresponding objective-function would look like: min where xj is the jth vertex of the ith Voronoi region. Since thenumber of vertices is an unknown, however, the summation limits are now variables, which greatly complicates themodel.

Even if the number of vertices per region is known, there is still the task of writing constraints to locatethe vertices. A possible solution is to use the Voronoi property that the boundary of a Voronoi region is theperpendicular bisector to the line linking two neighboring generators. With this property, it is possible to writeconstraints relating the perpendicular bisector slopes to the slopes of the lines joining neighboring generators. Butthis procedure introduces quadratic terms into the problem. Regardless of the quadratic terms, it is still necessaryto further contain the problem so that the vertices actually form boundaries of adjoining Voronoi regions as opposedto regions that do not encompass the entire population of customers. This requirement is difficult to meet, though,since there is no clear rule as to what vertices form which boundaries.

Another property of Voronoi diagrams is that a circle centered at a vertex will pass through exactly threegenerators as long as no generators fall inside the circle and the Voronoi diagram is not degenerate. The drawbackto this approach, though, is there is no rule as to which vertices will fall on the circle. To use this property, then,a 0-1 integer programming approach is needed to allow more than three generators to be candidates for falling ona particular circle. In other words, the problem become a mixed integer problem.

As suggested in the "Voronoi diagrams" Sub-section of the "Spatial Temporal Information" chapter, the cityof Dayton is interested in building a light rail system to service the greater Dayton area. Six stations are to be locatedto serve the population centers of Trotwood, Englewood, Vandalia, Huber Heights, Fairborn, Beavercreek, Kettering,Centerville, West-Carrolton/Miamisburg, Moraine/Oakwood, and downtown Dayton, each with population weightof 2, 3, 2, 3, 9, 8, 3, 7, 10, 2, 7, respectively. Given the grid system of streets, it is proposed that the stations belocated so as to minimize the rectilinear distance from the population centers to the stations, taking into account thenumber of people per center. It is desired that the farthest rectilinear distance for a population center to the closeststation be no greater than four miles. We wish to locate the stations using the most straightforward optimization-scheme, and to determine the shortest path between stations using Voronoi diagrams. In this homework problem, weonly examine the first phase of this location routing problem, namely locating the stations.

Since a vectilinear metric is used, we are no longer dealing with ordinary Voronoi diagrams. Francis et al.(1992) proposed a multi-facility location method with rectilinear distances. We adopt this model for locating the sixstations among the 11 population centers:

Here dwji is westerly distance between station i and population center j, and de

ji is easterly distance between station

(15)

i and population center j. dsji is the southerly distance between station i and population center j, and dn

ji is thenortherly distance between station i and population center j. It can be seen that for an optimal solution to be reached,at least one of the W-E variables, and one of the S-N variables must be zero. This will allow a geometrically soundsolution to exist.

31

Portion the Dayton population-centers into six regions centered around six stations. Approximately onestation would serve two centers. The centers are then paired by geographic proximity. Now determine the station thatwill service the center, and the center location.